数学prep1DS

macaudavid

Practice Test #1 Data Sufficiency
(218 Questions)

1.        86-!-item-!-187;#058&000002
A garden store purchased a number of shovels and a number of rakes.  If the cost of each shovel was $14 and the cost of each rake was $9, what was the total cost of the shovels and rakes purchased by the store?

(1)  The ratio of the number of shovels to the number of rakes purchased by the store was 2 to 3.

(2)  The total number of shovels and rakes purchased by the store was 50.
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【答案】C
【思路】
設Qs為shovels的數量
設Qr為rakes的數量
題目問: 14Qs + 9 Qr =?
(1) Qs=2a, Qr=3a ,只知比例,不知各為多少,無法求得數量,不充分
(2) Qs + Qr = 50,只知總和,不知各為多少,也無法求數total cost,不充分
(1)+(2)
將(1)代入(2)可得a=10 , 即Qs=20,Qr=30,所以total cost已經可以算出, 充分!
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2.        142-!-item-!-187;#058&000140
Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts.  Of these students, dislike lima beans; and of those who dislike lima beans,   also dislike brussels sprouts.  How many of the students like brussels sprouts but dislike lima beans?

(1)  120 students eat in the cafeteria.

(2)  40 of the students like lima beans.
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【答案】D
【思路】
        beans        ~ beans       
sprouts        a        b       
~ sprouts        c        d       
                 t

註: ~代表dislike
   設t為total students
   設like beans & like sprouts人數 為a
   設dislike beans & like sprouts人數 為b
   設like beans & dislike sprouts人數 為c
   設dislike beans & dislike sprouts人數 為d
已知d=  
題目問: b=?
(1) t= 120, 得知b+d=80且d=50, 所以得出b=30, 充分
(2) a+c = 40 =   得出t=120 同(1)也可以算出b, 充分
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3.        196-!-item-!-187;#058&000141
How many different prime numbers are factors of the positive integer n ?

(1) Four different prime numbers are factors of 2n.

(2) Four different prime numbers are factors of n2.
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【答案】B
【思路】
(1) 如果2n=2.3.5.7的話
n的prime factor為3.5.7=>三個
如果2n= .3.5.7的話
n的prime factor為2.3.5.7=>四個 ,不充分
(2)  ,a,b,c,d不管是幾次方,都可以確定質數只有四個,充分
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4.        250-!-item-!-187;#058&000160
Is the product of a certain pair of integers even?

(1)  The sum of the integers is odd.

(2)  One of the integers is even and the other is odd.
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【答案】D
【思路】
設2個整數為x,y
題目問 xy = 2n?
(1) x + y = 2n+1  ,即至少有一個為奇數;一個為偶數, 則xy相乘,一定為偶數,充分
(2) 有一個為奇數;一個為偶數, 則xy相乘,一定為偶數,充分
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5.        304-!-item-!-187;#058&000163
If k is an integer and 2 < k < 8, what is the value of k ?

(1) k is a factor of 30.

(2) k is a factor of 12.
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【答案】E
【思路】
(1) 30因數有 2,3,5 ; 所以k=3 or 5 or 6,無法確定,不充分
(2) 12因數有 ,3 ; 所以k=3 or 4 or 6,無法確定,不充分

(1)+(2) 同樣有 3 or 6所以無法確定, 不充分
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6.        363-!-item-!-187;#058&000183

On the number line above, is the product of w, x, y, and z negative?

(1) z is positive.

(2) The product of w and x is positive.
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【答案】E
【思路】
題目問: wxyz = 負?
(1) z=負,不知 w,x,y是否為正負,所以不充分
(2) wz=正,表示wz可能為+ +或 - -,不知x,z所以一樣不充分

(1)+(2) 合起來不知y是否為正負,所以不充分
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7.        509-!-item-!-187;#058&000355
If y and z are integers, is y(z + 1) odd?

(1) y is odd

(2) z is even.
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【答案】C
【思路】
(1) y等於奇數,不知z,所以無法判斷,不充分
(2) z等於偶數,不知y,所以無法判斷,不充分
(1)+(2) y等於奇數, z等於偶數,可確定知道答案,充分
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8.        563-!-item-!-187;#058&000364
If n and m are positive integers, what is the remainder when 3^(4n + 2 + m) is divided by 10 ?

(1) n = 2

(2) m = 1
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【答案】B
【思路】
(1) n = 2 , 不知 m = ? m 的變化會影響到餘數的結果，餘數會不一樣，帶幾個不同數字進去，然後除10看看......不充份
(2) m=1
  的餘數都是7
因為
3的1次方個位數等於3
3的2次方個位數等於9
3的3次方個位數等於7
3的4次方個位數等於1
3的5次方個位數等於3 (又回到3了,所以4次一個循環)
所以4n是4的倍數算出來餘數都是1
(4n+2) 的話餘數變9 (例如n=1時 , 4*1+2=6, 餘數是9, n=2時, 4*2+2=10, 餘數還是9)
(4n+2+1) 的話餘數都變7, 充分
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9.        617-!-item-!-187;#058&000483
If x is a positive integer, is x < 16 ?

(1) x is less than the average (arithmetic mean) of the first ten positive integers.

(2) x is the square of an integer.
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【答案】A
【思路】
(1) x <  =5.5 ,確定小於16,充分
(2) x =   可能小於16,也可能大於16,不充分
(1)+(2) n/a
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10.        719-!-item-!-187;#058&000515
Of the 800 students at a certain college, 250 students live on campus and are more than 20 years old.  How many of the

800 students live on campus and are 20 years old or less?

(1)  640 students at the college are more than 20 years old.

(2)  60 students at the college are 20 years old or less and live off campus.
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【答案】C
【思路】
        campus        ~ campus       
>20        250        b       
20
c        d       
                        Total:800
設live off campus & old>20人數 為b
設live campus & old 20人數 為c
設live off campus & old 20人數 為d

題目問c=?

(1) 250 + b= 640 可知 b,但無法知道c,不充分
(2) d=60 c無從得知,不充分
(1)+(2)
由(1)得知 250 + b= 640,所以可以算出 c+d= 160
由(2)得知d = 60 代回c+d= 160,所以C可以算出答案,充分!
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11.        959-!-item-!-187;#058&000583
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.

(2) p is the sum of the squares of two positive integers.
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【答案】D
【思路】
(1) P=8n+5 =4(2n+1) + 1  得到餘數為1,充分
(2) p= ,餘1,充分
Ps: P總和要為奇數, 兩數相加至少有一個數為奇數
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12.        1013-!-item-!-187;#058&000609
Of the 25 cars sold at a certain dealership yesterday, some had automatic transmission and some had antilock brakes.  How many of the cars had automatic transmission but not antilock brakes?

(1)  All of the cars that had antilock brakes also had automatic transmission.

(2)  2 of the cars had neither automatic transmission nor antilock brakes.
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【答案】E
【思路】
        transmission        ~ transmission       
Brake        a        b       
~Brake        c        d       
                        Total:25
設有transmission & 有 Brake車子數量 為a
設沒有transmission & 有 Brake車子數量 為b
設有transmission & 沒有 Brake車子數量 為c
設沒有transmission & 沒有 Brake車子數量 為d
題目問 c=?
(1) 有brake汽車中也有transmission,表示 b=0, 但不知c為多少,不充分
(2) d=2, 也不知c為多少,不充分
(1)+(2)
可知b=0,d=2,但沒有其他算式可以算出c是多少,所以不充分!
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13.        1067-!-item-!-187;#058&000618
Is x to the right of -5 on the number line?

(1) x is to the right of -7 on the number line.

(2) x is between -4 and -3 on the number line.
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【答案】B
【思路】題目問: x在數線上位在-5的右邊嗎?
(1) x位在-7的右邊,表示可能在-5的左邊or右邊,不充分
(2) x在-4和-3的中間,確定在-5右邊,充分
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14.        1121-!-item-!-187;#058&000619
Is y between -2 and 1 on the number line?

(1) y is to the right of -1 on the number line.

(2) y is to the left of 2 on the number line.
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【答案】E
【思路】題目問: 數線上,y在-2和1的中間嗎?
(1) y位在-1的右邊,所以可能在-2和1的中間,也有可能在1的右邊,所以不充分
(2) y位在2的左邊,所以可能在-2和1的中間,也有可能-2在的左邊,所以不充分
(1)+(2) 取交集,一樣不充分
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15.        1175-!-item-!-187;#058&000621
Is r to the right of -6 on the number line?

(1) r is between -4 and -1 on the number line.

(2) r is between -3 and 1 on the number line.
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【答案】D
【思路】題目問: r在數線上位在-6的右邊嗎?
(1) r位在-4和-1的中邊,表示確定在-6的右邊,充分
(2) r在-3和1的中間,確定在-6右邊,充分
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16.        1275-!-item-!-187;#058&000656
Lines n and p lie in the xy-plane.  Is the slope of line n less than the slope of line p ?

(1)  Lines n and p intersect at the point (5,1).

(2)  The y-intercept of line n is greater than the y-intercept of line p.
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【答案】C
【思路】
(1)兩線同過一個點, 根本比較不出來, 所以不充份...
(2) 兩線的與Y所截取的值N線比P線大, 也比較不出來, 因為不知兩線的各另外一點, 所以不充份
(1)+(2)合併即能算出, 因為條件一已經給了兩個線段的一點, 再加上條件二給了兩個線段分別在Y座標上點的相對位置, 畫個圖即能算出來, 線段較陡的斜率一定比線段較平的斜率大, 所以選選項(C)
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17.        1329-!-item-!-187;#058&000662
Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives.  Of the six countries, if Country A sent the second greatest number of representatives, did Country A send at least 10 representatives?

(1)  One of the six countries sent 41 representatives to the congress.

(2)  Country A sent fewer than 12 representatives to the congress.
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【答案】E
【思路】
(1)當有一個地區是41, 表示剩下的5個區域=75-41=34
故41的那個區域就會是greatest
當A=9時,這34可以分配給剩下的區域 EX: (9,8,7,6,4) 符合題目要求 但A小於10
當A=11時,則可分配成(11,8,7,6,2) 符合題目要求 但A至少等於10
所以(1)不充分

(2)可由上面的舉例發現 A小於12不成立 因為等於9和等於11都可以
所以(2)不對

(1)+(2) 沒有資料可以顯示其餘國家的人數比例,所以資料不充分
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18.        1383-!-item-!-187;#058&000675
What is the hundredths digit of the decimal z ?

(1)  The tenths digit of 100z is 2.

(2)  The units digit of 1,000z is 2.
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【答案】E
【思路】
(1) 100z=xx.2xx  除以100  z= xx.xx2 ,可知千分之一位數是2,還是不知百分之一位,不充分
(2) 1000z=2.xxx 除以1000  z= x.xx2, 可知千分之一位數是2,還是不知百分之一位,不充分
(1)+(2) 還是不知百分之一位,不充分
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19.        1437-!-item-!-187;#058&000677
Is z equal to the median of the three positive integers x, y, and z ?

(1) x < y + z

(2) y = z
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【答案】B
【思路】代數字推法
(1) 3<4+5 4為median,但4不知是y或z,所以無法判斷,不充分
(2) 344或 445 不管x大於y,z或小於y,z都可確認z為median ,充分
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20.        1491-!-item-!-187;#058&000755
A certain one-day seminar consisted of a morning session and an afternoon session.  If each of the 128 people attending the seminar attended at least one of the two sessions, how many of the people attended the morning session only?

(1)  of the people attended both sessions.

(2)  of the people attended the afternoon session.
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【答案】B
【思路】
        morning        ~ morning       
afternoon        a        b       
~ afternoon        c        d       
                        Total:128
設參加morning & 參加 Afternoon人數 為a
設沒有參加morning & 參加 Afternoon人數 為b
設參加morning & 沒有參加 Afternoon人數 為c
設沒有參加morning & 沒有參加 Afternoon人數 為d
題目已告知至少都有參加一門課所以d = 0, 問c = ?
(1)  ,即a=96 ,沒有a+c或c+d的合來推出c,所以不充分
(2)  ,即a+b=112,又Total=128可推出c+d = 16 ,因為d=0,所以c可得答案,充分
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21.        1545-!-item-!-187;#058&000766
If a certain charity collected a total of 360 books, videos, and board games, how many videos did the charity collect?

(1) The number of books that the charity collected was 40 percent of the total number of books, videos, and board games that the charity collected.

(2) The number of books that charity collected was 66  percent of the total number of videos and board games that charity collected.
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【答案】E
【思路】設book=b, videos=v, board games=b
b+v+g = 360,問v=?
(1) b=360 x 40% ,還是不知v,所以不充分
(2) b=  (v+g) , 還是不知v,所以不充分
(1)+(2) 由(1)代入(2)可知v+g的合,但還是不知道v等於多少,所以不充分
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22.        1599-!-item-!-187;#058&000810
Of the 800 sweaters at a certain store, 150 are red. How many of the red sweaters at the store are made of pure wool?

(1)  320 of the sweaters at the store are neither red nor made of pure wool.

(2)        100 of the red sweaters at the store are not made of pure wool.
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【答案】B
【思路】
        Red        ~ Red       
wool        a        b       
~ wool        c        d       
                        Total:800
設顏色Red & 是 wool 數量 為a
設顏色不是Red & 是 wool 數量 為b
設顏色Red & 不是 wool 數量 為c
設顏色不是Red & 不是 wool 數量 為d
已知a+c=150, a+b+c+d = 800 ,求a=?
(1) d=320 ,不知a,不充分
(2) c=100,因為a+c=150,可求出a, 充分
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23.        1653-!-item-!-187;#058&000843
At least 100 students at a certain high school study Japanese. If 4 percent of the students at the school who study French also study Japanese, do more students at the school study French than Japanese?

(1)  16 students at the school study both French and Japanese.

(2)        10 percent of the students at the school who study Japanese also study French.
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【答案】B
【思路】
        French        ~ French       
JP        a        b       
~ JP        c        d       
                        Total:
設study French & study JP 人數 為a
設沒有study French & study JP 人數 為b
設study French & 沒有study JP 人數 為c
設沒有study French & 沒有study JP 人數 為d
題目已知  , ,求a+c > a+b ?
(1) a=16, 不充分
(2)  ,已知 ,可算出 ,可得a+c>a+b,充分
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24.        1952-!-item-!-187;#058&002488
At a certain restaurant, if each hamburger costs the same amount, what is the cost, excluding sales tax, of 1 hamburger?

(1)  The total cost, including a 6 percent sales tax, is $4.77 for 3 hamburgers.

(2)  The total cost, including a 6 percent sales tax, is less than $6.50 for 4 hamburgers.
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【答案】A
【思路】設hamburger成本=h, 問 h=?
(1) 3h(0.06) = 4.77,可得h,充分
(2) 4h(0.06) < 6.50 ,不知h多少,不充分
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25.        2055-!-item-!-187;#058&002563
What is the value of n ?

(1)  n is between 0 and 1.

(2)    is   more than n.
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【答案】B
【思路】
(1) n=小數or分數,太多答案,不充分
(2)  =n +  ,可得n,充分
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26.        2109-!-item-!-187;#058&002567
In a certain conference room each row of chairs has the same number of chairs, and the number of rows is 1 less than the number of chairs in a row.  How many chairs are in a row?

(1)  There is a total of 72 chairs.

(2)        After 1 chair is removed from the last row, there is a total of 17 chairs in the last 2 rows.
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【答案】D
【思路】
在說，有一個房間裡面放了椅子，
每一排有K張椅子，然後有(k-1)排，問你說，總共有多少椅子呢？ 也就是要求k(k-1)
(1) 總共有72張椅子。可列示為k(k-1)=72 只有唯一正數解。
(2) 當一張椅子移開以後，最後兩排的椅子數變成17, 所以也可以知道，原來的時候，一排有(17+1)/2=9張椅子喔。充分
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27.        2260-!-item-!-187;#058&002626
What is the price for a certain meal listed on a menu?

(1)  The total paid for the meal, sales tax, and gratuity is $10.84.

(2)        The sales tax on food is 6 percent.
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【答案】E
【思路】設meal price=m
(1) 設tax=t, gratuity=g, m+t+g=10.84 ,無法得知m,不充分
(2) t=0.6m, 無法得知m,不充分
(1)+(2) 兩個式子還是不知m等於多少,不充分
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28.        2365-!-item-!-187;#058&002728
Which of Company X and Company Y earned the greater gross profit last year?

(1) Last year the expenses of Company X were   of the expenses of Company Y.

(2) Last year the revenues of Company X were $6 million less than the revenues of Company Y.
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【答案】E
【思路】
(1) 設X費用=Xe, Y費用=Ye,  ,不知profit多少,不充分
(2) 設X收入=Xr, Y收入=Yr, Xr = Yr-6 , 不知profit多少,不充分
(1)+(2)四個未知數,只有兩個方程式,無法知道profit多少
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29.        2517-!-item-!-187;#058&002771
What is the value of 6x - 10 ?

(1)  3x - 5 = 16

(2)        12x - 10 = 74
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【答案】D
【思路】
(1) x可算出答案,充分
(2) x可算出答案,充分
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30.        2663-!-item-!-187;#058&002855
Is w greater than 1 ?

(1) w + 2 > 0

(2) w2 > 1
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【答案】E
【思路】題目問w>1嗎?
(1) w>-2, 可能大於1也可能小於1,不充分
(2) w>1 or w<-1,一樣不充分
(1)+(2) 取交集, 得到-2<w<-1 或 w>1 沒有唯一解,不充分
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31.        2764-!-item-!-187;#058&002889
At a refreshment stand, each can of soda sells for the same price and each sandwich sells for the same price.  What is the total price for 2 sandwiches and 3 cans of soda at the stand?

(1)  At the stand the total price for 1 sandwich and 1 can of soda is $3.

(2)        At the stand the total price for 3 sandwiches and 2 cans of soda is $8.
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【答案】C
【思路】設sandwiches =X, soda=Y 問 2X+3Y=?  
(1)X+Y=3   -------------------insufficient
(2)3X+2Y=8 -------------------insufficient
(1)(2)解聯立得X Y ------------sufficient
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32.        2818-!-item-!-187;#058&002974
Al, Pablo, and Marsha shared the driving on a 1,500-mile trip.  Which of the three drove the greatest distance on the trip?

(1)  Al drove 1 hour longer than Pablo but at an average rate of 5 miles per hour slower than Pablo.

(2)        Marsha drove 9 hours and averaged 50 miles per hour.
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【答案】E
【思路】Ad+Pd+Md=1500,問 誰的距離最長   
(1) Ah=Ph+1, As=Ps-5 ---------------------insufficient
(2)Md=9 50=450     --------------------insufficient
(1)(2) ------------------------------------------insufficient
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33.        2873-!-item-!-187;#058&002998
If x = , is y equal to 1 ?

(1)  y2 ( x + ) = 1

(2)  y ( 2x – 1 ) = 2x - y
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【答案】B
【思路】直接代入
(1) y2 =1,  y=     ---------------------insufficient
(2)1-y=0,  y=1       -------------------- sufficient
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34.        2927-!-item-!-187;#058&003024
During an experiment, some water was removed from each of 6 water tanks.  If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

(1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.

(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.
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【答案】A
【思路】 = 標準差
(1)  =10, 每項移走30% =7--------sufficient
(2) 只知 mean,不知各tank移走多少水  ----------------------------------------insufficient
P.S 項數標準差,若同時增加或減少某數,例如加5,則標準差不變
    項數標準差,若同時增加或減少某比例,例如5%,則標準差會等比增加或減少
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35.        3076-!-item-!-187;#058&003114
If p and n are positive integers and p > n, what is the remainder when p2 - n2 is divided by 15 ?

(1)  The remainder when p + n is divided by 5 is 1.

(2)        The remainder when p - n is divided by 3 is 1.
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【答案】E
【思路】 =(p+n)(p-n)
(1) p+n=5a+1 ----------------------------------------------------------------------insufficient
(2) p-n=3b+1 -----------------------------------------------------------------------insufficient
(1)(2)(p+n)(p-n)=(5a+1)(3b+1)=15ab+5a+3b+1 ---------------------------insufficient
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36.        3130-!-item-!-187;#058&003116
If x is positive, is x > 3 ?

(1)  (x - 1)2 > 4

(2)        (x - 2)2 > 9
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【答案】D
【思路】
(1)X-1> , X-1>2, X>3  -------------------sufficient
(2)X-2> , X-2>3, X>5  -------------------sufficient
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37.        3184-!-item-!-187;#058&003224
When 1,000 children were inoculated with a certain vaccine, some developed inflammation at the site of the inoculation and some developed fever.  How many of the children developed inflammation but not fever?

(1)  880 children developed neither inflammation nor fever.

(2)        20 children developed fever.
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【答案】C
【思路】設 A—only inflammation, B—only fever, C—both, D—neither
           A+B+C+D=1000 ,問 A=?
(1) D=880   ----------------------------------------------------------------------insufficient
(2) B+C=20 -----------------------------------------------------------------------insufficient
(1)(2) A=100 ----------------------------------------------------------------------sufficient
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38.        3481-!-item-!-187;#058&003324
If r and s are positive integers, is   an integer?

(1)  Every factor of s is also a factor of r.

(2)  Every prime factor of s is also a prime factor of r.
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【答案】A
【思路】數字代入法
(1)   -----------------------------------------------------------------sufficient
(2)   -----------------------------------------------------------------insufficient
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39.        3582-!-item-!-187;#058&003470
For the students in class A, the range of their heights is r centimeters and the greatest height is g centimeters.  For the students in class B, the range of their heights is s centimeters and the greatest height is h centimeters.  Is the least height of the students in class A greater than the least height of the students in class B ?

(1) r < s

(2) g > h
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【答案】C
【思路】range= greatest height – least height 問 g-r>s-h
(1)r<s --------------------------------------------------------------------------------insufficient
(2) g>h ------------------------------------------------------------------------------insufficient
(1)(2) yes ----------------------------------------------------------------------------sufficient
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40.        3636-!-item-!-187;#058&003488
Beth's bank charges a service fee on a regular checking account for each month in which the balance on the account falls below $100 at any time during the month.  Did the bank charge a service fee on Beth's regular checking account last month?

(1)  During last month, a total of $1,000 was withdrawn from Beth's regular checking account.

(2)        At the beginning of last month, Beth's regular checking account balance was $500.
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【答案】E
【思路】
(1) We do not know the beginning account, so we cannot know this answer---------insufficient
(2) We do not know how much Beth withdraw or deposit -------------------------------insufficient
(1)(2)We do know how much Beth deposit-------------------------------------------------insufficient
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41.        3690-!-item-!-187;#058&003497
Machines X and Y work at their respective constant rates.  How many more hours does it take machine Y, working alone, to fill a production order of a certain size than it takes machine X, working alone?

(1) Machines X and Y, working together, fill a production order of this size in two-thirds the time that machine X, working alone, does.

(2) Machine Y, working alone, fills a production order of this size in twice the time that machine X, working alone, does.
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【答案】E
【思路】設work=1, machineX=Xhour, machineY=Yhour, 問y-x=?
(1)    --------------------------------------------------------------insufficient
(2)   -----------------------------------------------------------------------insufficient
(1)(2)------------------------------------------------------------------------------------insufficient
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42.        3744-!-item-!-187;#058&003565
John and Mary own shares of stock in a certain company.  Does John own more shares of the company's stock than Mary?

(1) Mary owns more than 500 shares of the company's stock.

(2) The number of shares of the company's stock that John owns is 400 less than twice the number of shares of the company's stock that Mary owns.
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【答案】C
【思路】問J>M
(1)M>500     ----------------------------------------------------------------------insufficient
(2) J=2M-400  --------------------------------------------------------------------- insufficient
(1)(2)用數字代入 J>M--------------------------------------------------------------- sufficient
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43.        3798-!-item-!-187;#058&003572
Each person in a certain group supports only one of the two candidates R and T. Of the people in the group, 45 percent support Candidate R and the rest support Candidate T.  How many people in the group are in favor of a flat tax?

(1)  Of the people in the group who support Candidate R, 58 percent are in favor of a flat tax.

(2)        Of the people in the group who support Candidate T, 22 are in favor of a flat tax.
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【答案】E
【思路】 問 Rtax+Ttax=? 設 total population=X人
(1)     ----------------------------------------------insufficient
(2)  -------------------------------------------------- insufficient
(1)(2)we do not know X,-----------------------------------------------------------------------insufficient
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44.        3852-!-item-!-187;#058&003578
In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis?

(1) a + b = -1

(2) The graph intersects the y-axis at (0, -6).
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【答案】C
【思路】intersect the x-axis, 其方程式的y=0
       y=(x+a)(x+b) y= +x(a+b)+ab
(1) y=  -x+ab ----------------------------------------------------------------------insufficient
(2) -6=ab      -----------------------------------------------------------------------insufficient
(1)(2)
x^2-x-6=0 x=3 or-2 -----------------------------------------------------------------sufficient
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45.        3906-!-item-!-187;#058&003579
How many of the 42 people in a group are employed students?

(1)  29 of the 42 people are employed.

(2)        24 of the 42 people are students.
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【答案】E
【思路】A—only employed, B—only student, C- both, D—neither,A+B+C+D=42,問 C=?
(1) A+C=29 ----------------------------------------------------------------------insufficient
(2) B+C=24 -----------------------------------------------------------------------insufficient
(1)(2) because we do not know the number of people who are neither employed nor students----------------------------------------------------------------------------insufficient
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46.        3960-!-item-!-187;#058&003582
How many hours did it take Helen to drive from her house to her parents' house?

(1)  Helen's average speed on this trip was 72 kilometers per hour.

(2)  If Helen's average speed on this trip had been 8 kilometers per hour greater, it would have taken her 1 hour less.
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【答案】C
【思路】 ,問hour
(1)      ------------------------------------------------------------------------insufficient
(2)          -------------------------------------------------------insufficient
(1)(2) S=72代入得D  --------------------------------------------------------------sufficient
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47.        4068-!-item-!-187;#058&003673
Is m ≠ n ?

(1) m + n < 0

(2) mn < 0
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【答案】B
【思路】
(1)-2+(-2)<0 ; -2+(-1)<0 --------------------------- -------------------------------insufficient
(2) MN<0則MN必為 一正一負 --------------------------------------------------------sufficient
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48.        4122-!-item-!-187;#058&003746
Last Friday each of the pets at a certain veterinary clinic was given either 1 treat or 2 treats.  What was the total number of treats given to pets at the clinic last Friday?

(1) The total number of pets at the clinic last Friday was 90.

(2)  of the pets at the clinic last Friday were given 2 treats each.
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【答案】C
【思路】設1 Treat-X Pets, 2Treat-Y Pets 問X+2Y=?
(1)X+Y=90     ---------------------------------------------------------------------insufficient
(2)    ---------------------------------------------- insufficient
(1)(2)解聯立得XY              -------------------------------------------------- sufficient
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49.        4176-!-item-!-187;#058&003769
Does x + y = 5 ?

(1) 4x + y = 17

(2) x + 4y = 8
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【答案】C
【思路】
(1) 4x + y = 17  ---------------------------------------------------------------------insufficient
(2) x + 4y = 8------------------------------------------------------------------------- insufficient
(1)(2)解聯立得XY              -------------------------------------------------- sufficient
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50.        4230-!-item-!-187;#058&003784
A construction company was paid a total of $500,000 for a construction project.  The company's only costs for the project were for labor and materials.  Was the company's profit for the project greater than $150,000 ?

(1)  The company's total cost was three times its cost for materials.

(2)        The company's profit was greater than its cost for labor.
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【答案】C
【思路】profit= total – (labor+ material)  問profit>150000
(1) (l+m)=3m    -------------------------------------------------------------------insufficient
(2)  p>l   ---------------------------------------------------------------------------- insufficient
(1)(2)p=500,000-(l+m), p=500000- (l+ l), P=500000-1.5l
數字代入法P=15(千), l=23(千) ; P=25(千),l=16(千)  -------------------------- sufficient
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51.        4284-!-item-!-187;#058&003822
What is the total value of Company H's stock?

(1) Investor P owns  of the shares of Company H’s total stock.

(2) The total value of Investor Q’s shares of Company H’s stock is $16,000.
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【答案】E
【思路】問h=?
(1) p=  h ----------------------------------------------------------------------------insufficient
(2) Q=16000 ------------------------------------------------------------------------- insufficient
(1)(2)             ------------------------------------------------------------------insufficient
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52.        4338-!-item-!-187;#058&003827
If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30 ?

(1)  None of the four numbers is greater than 60.

(2)        Two of the four numbers are 9 and 10, respectively.
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【答案】C
【思路】 , 問幾個數超過30?
(1)  ;     --------------------------------insufficient
(2)   ;     -------------------------------- insufficient
(1)(2)             ---------------------------------------------- sufficient
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53.        4392-!-item-!-187;#058&003851
If wx = y, what is the value of xy ?

(1) wx2 = 16

(2) y = 4
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【答案】A
【思路】
(1) W X X=16, Y X=16-------------------------------------------------------------sufficient
(2) WX=4    ------------------------------------------------------------------------- insufficient
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54.        4538-!-item-!-187;#058&004024
Is x - y + 1 greater than x + y - 1 ?

(1) x > 0

(2) y < 0
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【答案】B
【思路】x - y + 1> x + y – 1, 1>y ?
(1) x > 0--------------------------------------------------------------------------------insufficient
(2) y < 0----------------------------------------------------------------------------------sufficient
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55.        4687-!-item-!-187;#058&004202
Malik's recipe for 4 servings of a certain dish requires 1  cups of pasta.  According to this recipe, what is the number of cups of pasta that Malik will use the next time he prepares this dish?

(1)  The next time he prepares this dish, Malik will make half as many servings as he did the last time he prepared the dish.

(2)  Malik used 6 cups of pasta the last time he prepared this dish.
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【答案】C
【思路】4 serving/cup, how many cup next time?
(1) how many serving last time?  -------------------------------------------------insufficient
(2) 6 4=24 serving last time------------------------------------------------------ insufficient
(1)(2) this time serving=24    =12   =3cup------------------------------ sufficient
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56.        4741-!-item-!-187;#058&004206
During a sale, a clothing store sold each shirt at a price of $15 and each sweater at a price of $25.  Did the store sell more sweaters than shirts during the sale?

(1)  The average (arithmetic mean) of the prices of all of the shirts and sweaters that the store sold during the sale was $21.

(2)  The total of the prices of all of the shirts and sweaters that the store sold during the sale was $420.
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【答案】A
【思路】設shirt = x, sweater = y 問y>x?
(1)  ,15x+25y=21x+21y,4y=7x,y>x-----------------------------------------sufficient
(2) 15x+25y=420    ------------------------------------------------------------------------- insufficient
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57.        4795-!-item-!-187;#058&004223
During a sale, for each shirt that Mark purchased at the regular price, he also purchased a shirt at half the regular price.  How many shirts did Mark purchase during the sale?

(1)  The regular price of each of the shirts that Mark purchased during the sale was $21.50.

(2)  The total of the prices for all the shirts that Mark purchased during the sale was $129.00.
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【答案】C
【思路】S=?
(1) P=21.5       ---------------------------------------------------------------------insufficient
(2) S P+ P=129-------------------------------------------------------------------- insufficient
(1)(2)解聯立得X              -------------------------------------------------- sufficient
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58.        4895-!-item-!-187;#058&004290
What is the remainder when the positive integer x is divided by 3 ?

(1)  When x is divided by 6, the remainder is 2.

(2)  When x is divided by 15, the remainder is 2.
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【答案】D
【思路】X=3A+R,問R=?
(1) X=6B+2, X=3(2B)+2, R=2  ------------------------------------------------------sufficient
(2) X=15C+2,X=3(5C)+2, R=2--------------------------------------------------------sufficient
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59.        4950-!-item-!-187;#058&004329
Ann deposited money into two new accounts, A and B.  Account A earns 5 percent simple annual interest and account B earns 8 percent simple annual interest.  If there were no other transactions in the two accounts, then the amount of interest that account B earned in the first year was how many dollars greater than the amount of interest that account A earned in the first year?

(1)  Ann deposited $200 more in account B than in account A.

(2)        The total amount of interest that the two accounts earned in the first year was $120.
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【答案】C
【思路】interest B- interest A=?
(1) 設account A=X dollars, account B=X-200, X  0.05-(X-200)  0.08---------insufficient
(2) 設account A=X dollars, account B=Y, X  0.05+Y  0.08=120-------------- insufficient
(1)(2)  X  0.05+(X-200)  0.08=120   解聯立得 X             --------------sufficient
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60.        5054-!-item-!-187;#058&004422
What was the percent increase in the population of City K from 1980 to 1990 ?

(1)  In 1970 the population of City K was 160,000

(2)  In 1980 the population of City K was 20 percent greater than it was in 1970, and in 1990 the population of City K was 30 percent greater than it was in 1970.
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【答案】B
【思路】設1970=A,1980=B,1990=C,問 =?
(1) A=160000---------------------------------------------------------------------------insufficient
(2)  
5B=6A,10C=13A, B= ,C= 代入--------------------------------------------- sufficient
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61.        5157-!-item-!-187;#058&004472
Of the 60 animals on a certain farm,   are either pigs or cows.  How many of the animals are cows?

(1)  The farm has more than twice as many cows as it has pigs.

(2)  The farm has more than 12 pigs.
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【答案】C
【思路】由系統資訊可得知一方程式: x+y =40 (x為pig, y為cows)
(1) 由此可得知 y> 2x，加上原本資訊 x+y =40 可得知 (x,y) =(11,29)或(12,28),或(13,27) =>無法得知為哪一組合
← 選項(1)不成立
(2) 由可可得知 x >12，加上原本資訊 x+y =40 可得知 (x,y) =(13,27)…(39,1) =>無法得知為哪一組合← 選項(2)不成立
(1) + (2) 由此可得知 y> 2x , x >12 , 且x+y =40 =>(x,y)的組合中只有(13,27)符合所有需求 ← 選項(1)+(2)成立
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62.        5212-!-item-!-187;#058&004501
In isosceles  RST what is the measure of  R

(1)  The measure of  T is 100°.

(2)        The measure of  S is 40°.
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【答案】 A
【思路】
(1) 等腰三角形中有一角為100°，表示另外兩角必為40°、40°。(因總合為180°，100°不可能為等邊的角度) ← 選項(1)成立
(2) 等腰三角形中有一角為40°，此角可能為對等的角度(40°、40°、100°)，也有可能是頂角(40°、70°、70°) ← 選項(2) 不成立
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63.        5266-!-item-!-187;#058&004544
The cost of delivery for an order of desk chairs was $10.00 for the first chair, and $1.00 for each additional chair in the order.  If an office manager placed an order for n desk chairs, is n > 24 ?

(1)  The delivery cost for the order totaled more than $30.00.

(2)  The average (arithmetic mean) delivery cost per chair of the n chairs was $1.36.
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【答案】 B
【思路】 根據題目可知計算運費的方程式為 : Delivery Cost = 10 + (n-1) 1

(1) 套入方程式 Delivery Cost = 10 + (n-1) 1 > 30 => n> 21 (n 可能為22,23,24)← 選項(1) 不成立
(2) 套入方程式，  = 1.36 ，可得出n 的值 ← 選項(2) 不成立
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64.        5320-!-item-!-187;#058&004563
Are at least 10 percent of the people in Country X who are 65 years old or older employed?

(1) In Country X, 11.3 percent of the population is 65 years old or older.

(2) In Country X, of the population 65 years old or older, 20 percent of the men and 10 percent of the women are employed.
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【答案】B
【思路】假設65歲(含)以上男生數為x,女生數為y
(1) 由此可得知 x+y=11.3，但未知employed資訊。← 選項(1) 不成立
(2) 由此可得知題目所確認(65 years old or older & employed)的資訊為: 確認 是否>0.1
  =   = 0.2-0.1 >0.1?
=> 2>1+  由此式可得知 2一定大於1+ 因 一定小於1(在x,y皆為正整數的情形下)
所以由此式可推得符合條件1.65 years old or older & 2. Employed 的比率一定大於10% ← 選項(2) 成立
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65.        5374-!-item-!-187;#058&004565
If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10 ?

(1) On the number line, z is closer to 10 than it is to x.

(2) z = 5x
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【答案】A
【思路】確定此方程式是否成立 z >   ，目前已知 1. x < 10 2.X > 0

(1) 得知 10-Z < Z-X => 2Z > 10 +X =>   z >   ← 選項(1)成立
(2) 得知 z = 5x 帶入原方程式 為 5x>   => 9x > 10 。代入兩個條件x < 10 、X > 0 皆無法得出 9x -10 是否大於0或小於0 ← 選項(2)不成立
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66.        5428-!-item-!-187;#058&004588
In the finite sequence of positive integers K1, K2, K3, . . . , K9, each term after the second is the sum of the two terms immediately preceding it.  If K5 = 18, what is the value of K9 ?

(1) K4 = 11

(2) K6 = 29
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【答案】D
【思路】如果將此有規則的數列用x與y表示:設K1=x,K2=y,K3後面的都可以用x和y的方程式表示。已知K5=2x+3y=18
(1) 此選項提供另一方程式 K4=x+2y=11，所以可聯立解出 x 和 y ← 選項(1)成立
(2) 此選項提估另一方程式 K6=3x+5y=29，所以可聯立解出 x 和 y ← 選項(2)成立
※ 如果沒有時間，其實只要想著從第三個開始就有規則，變數只有二個(即沒辦法套規則的K1和K2)，所以只要有兩個數值即可聯立解二聯方程式。
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67.        5482-!-item-!-187;#058&004634
Is x + y negative?
       
(1) x is negative.

(2) y is positive.
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【答案】E
【思路】兩數相加為負數的組合: 負數+負數 或 負數絕對值大於正數
(1) 不確定y ← 選項(1)不成立
(2) 不確定x ← 選項(2)不成立
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68.        5536-!-item-!-187;#058&004638
What is the tens digit of the positive integer r ?

(1) The tens digit of   is 3.

(2) The hundreds digit of 10r is 6.
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【答案】B
【思路】
(1) 由此選項可得知 =30+x => r=300+10x ← 因不知道x，選項(1)不成立
(2) 由此選項可得知 10r = 600+x+y => r=60+x+ ←知道十位數為6，選項(2)成立
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69.        5825-!-item-!-187;#058&004737

In triangle ABC above, what is the length of side BC ?

(1) Line segment AD has length 6.

(2) x = 36
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【答案】B
【思路】用三角形角度合為180°來推算 ABD是否等於 DAB。發現: 1.  CBD=180°-4x° 2.  ABD=180°- x°-2 x°-(180°-4x)°=x°= DAB，因而得知 ADB為等腰三角形。
(1) 因已得知 ADB和 BDC皆為等腰三角形，所以邊AD = 邊BD= 邊BC。即知邊AD=6即可推邊BC。← 選項(1)成立
(2) 不知道任何邊長資訊，所以無法得知邊BC。← 選項(2)不成立
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70.        6067-!-item-!-187;#058&004849
Is the integer x divisible by 6 ?

(1) x + 3 is divisible by 3.

(2) x + 3 is an odd number.
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【答案】C
【思路】整數x可被6整除表示符合兩個條件: 1.x可被3整除 且 2.x可被2整除
(1) 此選項表示x可被3整除( 為整數表示 為整數=>x為3的倍數)，但x+3如果為單數(ex:x=6)即不能被2整除。←選項(1)單獨不成立

(2) 此選項表示x可被2整除 (因x+3為偶數)，但不表示可被3整除(ex:x=7)。←選項(2)單獨不成立

(1)+(2) : 表示可被2也可被3整除 =>因此可被6整除 ←選項(1)+(2)成立
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71.        6125-!-item-!-187;#058&004856


In the figure above, if x and y are each less than 90 and PS∥QR  is the length of segment PQ less than the length of segment SR ?

(1) x > y       

(2) x + y > 90
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【答案】A
【思路】已知PS和QR平行 => 角度越大的在平行的兩條線之間會越短
(1) 得知 x > y，所以 SR > PQ  ←選項(1)成立
(2) 得知 x + y > 90，但無法得知哪一個角度較大 ←選項(2)不成立
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72.        6179-!-item-!-187;#058&004900
A bookstore that sells used books sells each of its paperback books for a certain price and each of its hardcover books for a certain price.  If Joe, Maria, and Paul bought books in this store, how much did Maria pay for 1 paperback book and 1 hardcover book?

(1)   Joe bought 2 paperback books and 3 hardcover books for $12.50.

(2)        Paul bought 4 paperback books and 6 hardcover books for $25.00.
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【答案】E
【思路】假設paperback book 的單價為x, hardcover book的單價為y=
(1) 得知 2x+3y=12.5，但無其他資訊可幫助求得x,y ←選項(1)不成立
(2) 得知 4x+6y=25，但無其他資訊可幫助求得x,y ←選項(2)不成立
(1) + (2) 兩個方程式聯集但無法求出x,y。(因為兩個式子一模一樣) ←選項(1)+(2)不成立
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73.        6279-!-item-!-187;#058&004943
What is the average (arithmetic mean) of eleven consecutive integers?

(1)  The average of the first nine integers is 7.

(2)  The average of the last nine integers is 9.
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【答案】D
【思路】求連續11個整數的平均值。套用平均值的公式可得知:   *設第一個值為k
(1) 知道前九個數值的平均數套用到公式中 可得知k的值，可帶回原方程式求題目11個數的平均值。
←選項(1)成立
(2) 知道後九個數值的平均數套用到公式中可得知k的值，帶回後其他求11個數的平均值。 ←選項(2)成立
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74.        6333-!-item-!-187;#058&004944
If x and y are integers, what is the value of x + y ?

(1)   690 < x < y < 696

(2)        692 < x < y < 695
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【答案】B
【思路】已知x 和 y 皆為整數
(1) x和y的可能組合落在691、692、693、694，所以無法確定x+y為多少←選項(1)不成立
(2)在此條件下，x一定為693，y一定為694 =>所以可得知x+y ←選項(2)成立
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75.        6387-!-item-!-187;#058&004958
Is x less than 20 ?

(1) The sum of x and y is less than 20.

(2) y is less than 20.
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【答案】E
【思路】
(1)得知 x+y<20，但不能得知x、y是否為負數。(如有一方為負數另一方仍有可能會 20) ←選項(1)不成立
(2)得知 y<20，但不夠充足來判定x是否小於20。←選項(2)不成立
(1)+(2) 得知 x+y<20 且y<20，但仍無法確定x是否為負數 ←選項(1)+(2)不成立
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76.        6441-!-item-!-187;#058&005031
Is the integer k divisible by 4 ?

(1)   8k is divisible by 16.

(2)   9k is divisible by 12.
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【答案】B
【思路】
(1)  為整數，表示k可被2整除。但不表示一定可被4整除。←選項(1)不成立
(2)  為整數，表示3k可被4整除 =>即k可被4整除。←選項(2)成立
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77.        6741-!-item-!-187;#058&005163
If n is an integer between 10 and 99, is n < 80 ?

(1)  The sum of the two digits of n is a prime number.

(2)  Each of the two digits of n is a prime number.
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【答案】B
【思路】10 <n < 99，求 n <80?

(1)兩個數相加為值數，有可能為: 94(相加為13) 已大於80。←選項(1)不成立

(2)兩位數皆為質數，表示選擇在一位數的質數 {2,3,5,7} 所得出最大的兩位數為75，小於80。←選項(2)成立
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78.        6841-!-item-!-187;#058&005248
For each customer, a bakery charges p dollars for the first loaf of bread bought by the customer and charges q dollars for each additional loaf bought by the customer.  What is the value of p ?

(1)  A customer who buys 2 loaves is charged 10 percent less per loaf than a customer who buys a single loaf.

(2)  A customer who buys 6 loaves of bread is charged 10 dollars.
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【答案】C
【思路】由題目可得知價格公式為: p+q (n-1) *n為總購買數量

(1)由此可另外得知   = (1-0.1)p，但無其他資訊可求出p、q ←選項(1)不成立

(2) 由此可另外得知 p+5q=6，但無其他資訊可求出p、q。←選項(2)不成立

(1) + (2) 可聯立解兩方程式，因此可求出p ←選項(1))+(2)成立
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79.        6946-!-item-!-187;#058&005390
If the terms of a sequence are t1, t2, t3, . . . , tn, what is the value of n ?

(1)  The sum of the n terms is 3,124.

(2)  The average (arithmetic mean) of the n terms is 4.
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【答案】C
【思路】

(1)由此可得知 t1+…+ tn = 3124，但無其他資訊求得n ←選項(1)不成立
(2)由此可得知  =4，但無其他資訊求得n。←選項(2)不成立
(1) + (2) 可聯立解兩方程式，因此可求出p ←選項(1))+(2)成立
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80.        7243-!-item-!-187;#058&005480
If x is a negative number, what is the value of x ?

(1) x2 = 1

(2) x2 + 3x + 2 = 0
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【答案】A
【思路】已知x為負數
(1) x2 = 1，且x為負數 => x= -1 ←選項(1)成立
(2) x2 + 3x + 2 = 0 => (x+2)(x+1)=0 => x=-2 或 -1 ←選項(2)不成立
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81.        7301-!-item-!-187;#058&005482

In the figure above, what is the ratio  

(1)   The perimeter of rectangle KLMN is 30 meters.

(2)        The three small rectangles have the same dimensions.
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【答案】B
【思路】*設其他點以利說明
(1) 已知長方形KLMN的周長為30，但無法得知其他資訊←選項(1)不成立
(2) 已知三個小的面積一樣,且有共用邊(ex:AD為LDAC和MDAB的共用邊)可推出AC=AB。三個長方形的面積相等，表示LDAC=CKNB (設共用邊BC為2k)=> 長*k=2K*寬 =>可得知長為2k，寬為k。KN=2k，MN=3k即可求出 . ←選項(2)成立
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82.        7356-!-item-!-187;#058&005489
Is  > ?

(1) x2 > y2

(2) x > y
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【答案】A
【思路】
(1) x2 > y2   平方已將正負數的影響拿掉(等同於絕對值) ←選項(1)成立
(2) x > y 的情況下，y有可能為負數。如此一來y的絕對值不一定小於x。 ←選項(2)不成立
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83.        7463-!-item-!-187;#058&005565
If u, v, and w are integers, is u > 0 ?

(1) u = v2 + 1

(2) u = w4 + 1
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【答案】D
【思路】已知u,v,w皆為整數
(1) 在此情況下，v的平方一定大於0且1也大於0，所以兩的大於0的數相加一定大於0 ←選項(1)成立
(2) 在此情況下，w的四次方一定大於0且1也大於0，所以兩的大於0的數相加一定大於0 ←選項(2)成立
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84.        7517-!-item-!-187;#058&005572
If p is a prime number greater than 2, what is the value of p ?

(1)   There are a total of 100 prime numbers between 1 and p + 1.

(2)   There are a total of p prime numbers between 1 and 3,912.
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【答案】D
【思路】
(1) 從1開始署到第101(2不是)的質數即是p  ←選項(1)成立
(2) 數1~3912有幾個質數，即可得知p ←選項(2)成立
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85.        7713-!-item-!-187;#058&005641
Is k positive?

(1) k is between -2 and 3 on the number line.

(2) k is between 1 and 2 on the number line.
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【答案】B
【思路】
(1) k仍有可能為-1 =>所以無法得知是否為正數  ←選項(1)不成立
(2)  =>  => 已可知道k為正數。 ←選項(2)成立
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86.        7767-!-item-!-187;#058&005660
Each of the students in a certain class received a single grade of P, F, or I.  What percent of the students in the class were females?

(1)  Of those who received a P, 40 percent were females.

(2)  Of those who received either an F or I, 80 percent were males.
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【答案】E
【思路】
(1) 只知道P的女生比率為40%，但無P:F:I資訊  ←選項(1)不成立
(2) 只知道F或I的男生比為80%，但無P:F:I資訊。 ←選項(2)不成立
(2) + (2) 還是無法得知 P:F:I資訊  ←選項(1)+(2) 不成立
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87.        7824-!-item-!-187;#058&005664

In the figure above, segments PQ and PR are each parallel to one of the rectangular coordinate axes.  What is the sum of the coordinates of point P ?

(1)   The x-coordinate of point Q is -1.

(2)        The y-coordinate of point R is 1.
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【答案】C
【思路】已知PO和PR分別和y軸、x軸平行，所以可得知P點的x座標和Q點一樣、y座標和R點一致。
(1) 知道Q點的x座標 => 即知道P的x座標，但無法得知y座標 ←選項(1)不成立
(2) 知道R點的y座標 => 即知道P的y座標，但無法得知x座標 ←選項(2)不成立
(1)+(2)  知道P點的x、y座標，即可得知 x+y 的合 ←選項(1)加(2)成立
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88.        8066-!-item-!-187;#058&005745
What is the greatest common divisor of positive integers m and n ?

(1) m is a prime number.

(2) 2n = 7m
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【答案】C
【思路】求正整數m、n的最大公約數
(1) 知道m是質數，但不知到n ←選項(1)不成立
(2) 知道2n=7m，即n= 。因n為整數，雖知道m一定為2的倍數但不知確切數 ←選項(2)不成立
(1) + (2) 知到1.m是質數且為2的倍數，m一定為2  2.知到n和m的關係 ←選項(1)+(2)成立
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89.        8212-!-item-!-187;#058&005846
What is the retail price of a certain calculator?

(1)  The retail price of the calculator is $2.00 more than the wholesale price.

(2)  The retail price of the calculator is 50 percent more than the $4.00 wholesale price.
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【答案】B
【思路】設retailer price 為x ，wholesale price為y
(1) x=y+2，但無其他資訊求得x ←選項(1)不成立
(2) x=1.5 4，所以可得知x ←選項(2)成立
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90.        8266-!-item-!-187;#058&005947
On Jane's credit card account, the average daily balance for a 30-day billing cycle is the average (arithmetic mean) of the daily balances at the end of each of the 30 days.  At the beginning of a certain 30-day billing cycle, Jane's credit card account had a balance of $600.  Jane made a payment of $300 on the account during the billing cycle.  If no other amounts were added to or subtracted from the account during the billing cycle, what was the average daily balance on Jane's account for the billing cycle?

(1)  Jane's payment was credited on the 21st day of the billing cycle.

(2)  The average daily balance through the 25th day of the billing cycle was $540.
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【答案】D
【思路】平均值= (每天的餘額平均，昨天的期末於今天的期初餘額)，求平均值。已知:1.d1=600，其中dn=-300。且其中無任何金額變動。

(1) 300元是在第21天中付出，所以可得知1~20天每日餘額是600，21~30每日餘額是300即可求總平均值 ←選項(1)成立
(2) =540  且300是在第25天以前付掉(因期初為600)，第26~30天每日餘額為540，即可求總平均值， ←選項(2)成立
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91.        8417-!-item-!-187;#058&005991
What is the remainder when the positive integer n is divided by 6 ?

(1) n is a multiple of 5.

(2) n is a multiple of 12.
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【答案】B       
【思路】若甲是乙的倍數，乙又是丙的倍數，則甲一定是丙的倍數。

(1) n是5的倍數，5卻不是6的倍數。
(2) n是12的倍數，12是6的倍數，故n是6的倍數。
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92.        8471-!-item-!-187;#058&005997
Is the integer r divisible by 3 ?

(1) r is the product of 4 consecutive positive integers.

(2) r < 25
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【答案】A
【思路】相當於在問：r是不是3的倍數？

(1) r=四個連續正整數的乘積：r一定是3的倍數，因為每三個連續的數中間就有一個會是3的倍數。
(2) 條件很明顯不充分。
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93.        8525-!-item-!-187;#058&006006
If the integer n is greater than 1, is n equal to 2 ?

(1) n has exactly two positive factors.

(2) The difference of any two distinct positive factors of n is odd.
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【答案】B
【思路】若兩數相減為奇數，則兩數必為一奇數，一偶數。
兩奇數或兩偶數相減都會是偶數。

(1) n有兩個正因數  只能得知n是個質數。
(2) n = 2 時，兩因數相減的結果有 2 – 1 = 1，2 – (-1) = 3，(-2) – 1 = -3，(-2) - (-1) = -1 四種，全是奇數。
        若n > 2，為偶數：n - 2 會是偶數。
        若n > 2，為奇數：n - 1 會是偶數。
   由以上推理可知2是符合條件要求的唯一數字，故條件充分。
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94.        8579-!-item-!-187;#058&006012
What is the value of y ?

(1) y is an odd integer between 28 and 34.

(2) 31 < y < 36
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【答案】C
【思路】在AT數學中，所有的未知數都只有唯一解，無法解出唯一解的條件，不能算充分條件。
本題極簡單，在考場上碰到絕對是進入低分區的證明。

(1) y可能是29, 31, or 33
(2) y可能是32,33,34,35
(1)+(2) y=33
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95.        8634-!-item-!-187;#058&006039
Is    < b - a ?

(1) a < b.

(2) 1 < .
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【答案】A
【思路】在AT數學中，所有的未知數都只有唯一解，無法解出唯一解的條件，不算作充分條件。

(1) 可知a-b是負數，b-a是正數，因此  < b – a，條件充分。
(2) 可知 a-b > 1 或 a-b < -1。
    若 a-b > 1： b-a < -1，因此  > b – a。
    若 a-b < -1： b-a > 1，因此  < b – a。
    沒有唯一解答，因此條件不充分。
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96.        8688-!-item-!-187;#058&006049
If x and y are integers greater than 1, is x a multiple of y ?

(1)  3y2 + 7y = x.

(2)  x2 – x is a multiple of y.
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【答案】A
【思路】文字題，注意文字敘述上的陷阱。

(1) y(3y+7) = x ：x是y的倍數。
(2) x(x-1) = y：y是x的倍數。
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97.        8745-!-item-!-187;#058&006054
Is   = 3 - x

(1) x   3

(2) –x >0
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【答案】B
【思路】欲成立  = 3 – x 必須 x=0 或 x-3 < 0   x = 0 or x < 3

(1) 很明顯不充分。
(2) – x   > 0  從中可得知 x < 0 (故x < 3) 的結論，條件充分。
   負負正    正
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98.        8799-!-item-!-187;#058&006082
What is the value of 5x2 + 4x - 1 ?

(1) x(x + 2) = 0

(2) x = 0
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【答案】B
【思路】在AT數學中，所有的未知數都只有唯一解，無法解出唯一解的條件，不算作充分條件。
(1) x = 0 or -2；x = 0 時原式 = -1，x = -2 時原式 = 11，非唯一解。
(2) x = 0：原式 = -1，條件充分。
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99.        9270-!-item-!-187;#058&006322
If m > 0 and n > 0, is  >  ?

(1) m < n

(2) x > 0
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【答案】C
【思路】分母分子同加同減一個數，其大小關係的改變跟該分數為真分數或假分數有關。舉例如下：
        真分數：    同加一數值會越來越大，同減一數值會越來越小
        假分數：    同加一數值會越來越小，同減一數值會越來越大

(1) 只能得知 是真分數。
(2) 只能得知分子分母要同加一數。
(1)+(2) 可知是真分數的分子分母要同加一數，所以  > 。
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100.        9327-!-item-!-187;#058&006480



In the rectangular coordinate system shown above, does the line k (not shown) intersect quadrant II ?

(1)  The slope of k is -

(2)  The y-intercept of k is -6.
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【答案】A
【思路】AT數學中的斜率與截距定義為 y = mx + b 中的m值與b值，有正負。本題實際畫圖解較快。

(1) 斜率為負的直線一定經過二、四象限，條件充分。

(2) 只給了截距，不知斜率，無法判定。

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101.        9432-!-item-!-187;#058&006634
If the sum of three different numbers is 54, what is the largest number?

(1)  The largest number is twice the smallest number.

(2)  The sum of the two smaller numbers is 30.
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【答案】B
【思路】小＋中＋大=54，本題應該也是低分區的證明。

(1) 不知道中間的數字是多少。
(2) 可知最大數為24。
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102.        9486-!-item-!-187;#058&006637
If M is a finite set of negative integers, is the total number of integers in M an odd number?

(1)  The product of all the integers in M is odd.

(2)  The product of all the integers in M is negative.
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【答案】B
【思路】注意讀題，題目問集合M包含的數字個數是不是奇數，不是問集合M中的奇數有幾個。

(1) 只能判斷出M裡的數字全是奇數。剛才若誤解題意就會卡在這，典型的迷惑選項。
(2) 條件充分。因為奇數個負數相乘的乘積才會是負數。
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103.        9540-!-item-!-187;#058&006638
A certain bag contains red, blue, and green marbles.  What is the ratio of the number of green marbles to the number of red marbles in the bag?

(1)  The number of blue marbles in the bag is 2 times the number of green marbles in the bag.

(2)  The number of blue marbles in the bag is 3 times the number of red marbles in the bag.
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【答案】C
【思路】求綠球跟紅球的比例。

(1) 藍球：綠球 = 2 : 1
(2) 藍球：紅球 = 3 : 1
(1)+(2) 可知 綠球：紅球 = (1/2) : (1/3)
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104.        9643-!-item-!-187;#058&006681
What is the value of x ?

(1) y - x = y - 6

(2) x + 2y = 10
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【答案】A
【思路】又是個典型低分區考題。

(1) 同減y，解得 x = 6
(2) 顯然不充分，因為不知y值。
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105.        9789-!-item-!-187;#058&006817
What is the number of members of Club X who are at least 35 years of age?

(1) Exactly   of the members of Club X are under 35 years of age.

(2) The 64 women in Club X constitute 40 percent of the club’s membership.
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【答案】C
【思路】這種題目不必細看，凡是題目求數字條件卻只給比例的，都錯。

(1) 只能判斷出35歲以上的成員佔了 。
(2) 只能判斷出總人數 = 64/0.4 = 160。
(1)+(2) 可知35歲以下人數 = 160 x   = 40。
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106.        9843-!-item-!-187;#058&006850
What was Jean's insurance premium in 1995 ?

(1) The ratio of Jean’s insurance premium in 1995 to her insurance premium in 1994 was  .

(2) Jean’s insurance premium in 1995 was 20 percent more than her insurance premium in 1994.
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【答案】E
【思路】同上題，凡是題目求數字條件只給比例的，都錯。

(1) 只給比例推不出金額。
(2) 只給比例推不出金額。
(1)+(2) 兩個比例還是推不出金額。
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107.        9897-!-item-!-187;#058&006904
A combined total of 55 lightbulbs are stored in two boxes; of these, a total of 7 are broken.  If there are exactly 2 broken bulbs in the first box, what is the number of bulbs in the second box that are not broken?

(1)  In the first box, the number of bulbs that are not broken is 15 times the number of broken bulbs.

(2)  The total number of bulbs in the first box is 9 more than the total number of bulbs in the second box.
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【答案】D
【思路】先列表記下已知條件以助判斷，只要x能算出唯一解，條件就充分：
        破        沒破
箱子一        2        48-x
箱子二        5        x

(1) 48 – x = (15)(2)，條件充分。
(2) [2 + (48 - x)] = (5 + x) + 9，條件充分。
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108.        10385-!-item-!-187;#058&007204
Three friends rented a car for a week and divided the cost equally.  What was the total cost of renting the car?

(1) If the three friends had kept the car for a second week, they could have obtained the two-week rate, which was 1.5 times the cost of a one-week rental.

(2) If a fourth friend had joined the three friends and the cost had been divided equally among the four friends, the cost to each of the original three would have been reduced by $15.
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【答案】B
【思路】凡是題目求數字條件只給比例的，都錯。

(1) 只給比例推不出金額。
(2) 假設3個人時每個人要負擔x元，可知 4(x - 15) = 3x，條件充分。
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109.        10439-!-item-!-187;#058&007237
If R = 1 + 2xy + x2y2, what is the value of xy ?

(1) R = 0

(2) x > 0
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【答案】A
【思路】

(1) 可知   xy = -1
(2) 顯然不充分，不知y值。
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110.        10493-!-item-!-187;#058&007240
If zn = 1, what is the value of z ?

(1) n is a nonzero integer.

(2) z > 0
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【答案】C
【思路】第一，在AT數學中，所有的未知數都是實數；大家忘了棣美弗定理這檔事吧。
        第二，在AT中所有的未知數都必須是唯一解。
        在實數的前提下，答案只剩三種可能：
(A) n=0而z是任意數  (B) n是≠0的任意數而z＝1  (C) n是≠0的偶數而z＝-1

(1) 只排除了可能性(A)
(2) 只排除了可能性(C)
(1)+(2) 保證了z=1是唯一解，條件充分。
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111.        10884-!-item-!-187;#058&007442
If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6

(2) z = 3
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【答案】E
【思路】x, y, z 其實就是百位、十位與個位數。7並沒有只看部分位數就能判斷倍數的法則。

(1) 只知道十位數。
(2) 只知道個位數。
(1)+(2) 只知道個個位數與十位數。
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112.        11176-!-item-!-187;#058&007606
Each person attending a fund-raising party for a certain club was charged the same admission fee.  How many people attended the party?

(1) If the admission fee had been $0.75 less and 100 more people had attended, the club would have received the same amount in admission fees.

(2) If the admission fee had been $1.50 more and 100 fewer people had attended, the club would have received the same amount in admission fees.
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【答案】C
【思路】總入場費 = (入場費) x (進場人數)。 設入場費為$x，進場人數為n。

(1) 只能推出 (x - 0.75)(n + 100) = xn
(2) 只能推出 (x + 1.5)(n -100) = xn
(1)+(2) 兩條件式解兩未知數，條件充分。
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113.        11760-!-item-!-187;#058&007855
For a certain set of n numbers, where n > 1, is the average (arithmetic mean) equal to the median?

(1) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2.

(2) The range of the n numbers in the set is 2(n - 1).
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【答案】A
【思路】求平均數須知道每個數字的大小；求中位數則須知道數字間的相對大小。

(1) 這n個數字是一個公差為2的等差數列，可知平均數與中位數相同，即 (首項+末項) / 2。
(2) 只知道全距，條件明顯不充分。
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114.        11816-!-item-!-187;#058&007868
Henry purchased 3 items during a sale.  He received a 20 percent discount off the regular price of the most expensive item and a 10 percent discount off the regular price of each of the other 2 items.  Was the total amount of the 3 discounts greater than 15 percent of the sum of the regular prices of the 3 items?

(1)  The regular price of the most expensive item was $50, and the regular price of the next most expensive item was $20.

(2)  The regular price of the least expensive item was $15.
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【答案】A
【思路】題目問總折扣率是否超過15%，因此必須先知道這三個item的價格。
        總折扣率 = (總折扣金額) / (三item總價格)
(1) 設第三個item的價格為x，可知若總折扣率大於15%，則會滿足 的關係。
    解得 x < 30，原題又告訴我們X是三者中價格最低者故x < 20，總折扣率必定大於15%。
(2) 只知道最低item價格，條件不充分。
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115.        11870-!-item-!-187;#058&007880
If xy = -18, is x less than y ?

(1) x < 0

(2) y < 10
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【答案】 A
【思路】

(1) 條件充足，因可知x為負，y為正。
(2) 無法得知x的大小。
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116.        11924-!-item-!-187;#058&007881
An insurance company has a contract with a medical laboratory to pay a discounted price for a certain medical test performed on patients referred to the laboratory by the insurance company.  If the laboratory's original bill for this medical test on a patient referred by the insurance company is $230, what is the percent discount specified by the contract between the laboratory and the insurance company?

(1)  The insurance company is required to pay only 20 percent of the original bill for the test.

(2)  The insurance company is required to pay $46 for the test.
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【答案】D
【思路】問保險公司對medical test費用的discount為多少？

(1) 直接說出了答案。
(2) 可知 discount為 46 / 230 = 20%。
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117.        11978-!-item-!-187;#058&007947
If P, Q, and R are points on the number line, what is the distance between P and R ?

(1) Q is between P and R.

(2)  The distance between P and Q is 5.
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【答案】E
【思路】

(1) 沒提供跟距離有關的資訊。
(2) 沒提供跟R有關的資訊。
(1)+(2) 只能得到P跟R距離大於5的結論。
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118.        12032-!-item-!-187;#058&007948
If x is positive, what is the value of y ?

(1) 5x = 15

(2) xy + y = 18
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【答案】C
【思路】

(1) 只知x = 3，沒提供跟y有關的資訊。
(2) 顯然y無法得出唯一解。
(1)+(2) 將條件一帶入條件二，可知 x = 3， y = 4.5。
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119.        12086-!-item-!-187;#058&008079
A contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight.  What is the value of x ?

(1) y = 10

(2) z = 16
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【答案】D
【思路】x噸10%溶液與y噸2%溶液混合，變成了z噸5%溶液。
        兩溶液混合為一時，總質量根據質量守恆原理是不會改變的，由此可知 z = x + y  …(A)
        z中溶質的質量是x與y溶質質量的總和，故 0.1x + 0.02y = 0.05(x + y)  …(B)

(1) 顯然y代入(B)式即可解x
(2) z代入(A)式可知y = 16 – x，代回(B)式變成 0.1x + 0.02(16 - x) = 0.05(16)，可解x
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120.        12140-!-item-!-187;#058&008146
How many of the students in a certain class are taking both a history and a science course?

(1)  Of all the students in the class, 50 are taking a history course.

(2)  Of all the students in the class, 70 are taking a science course.
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【答案】E
【思路】
(1) 未提供科學課相關資訊。
(2) 未提供歷史課相關資訊。
(1)+(2) 仍然不知道同時修歷史與科學的人數。
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121.        12194-!-item-!-187;#058&008148
Pat bought 5 pounds of apples.  How many pounds of pears could Pat have bought for the same amount of money?

(1) One pound of pears costs $0.50 more than one pound of apples.
(2) One pound of pears costs 1  times as much as one pound of apples.
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【答案】B
【思路】
P=Pear price, Qp=Pears重量
A=Apple price, QA=Apples重量=5 pounds
Price × Q =TP
TP/price=Q
Qp × P=QA × A=TP
(1) P=A(1) +0.5
Qp × P = QA × A
Qp × (A+0.5) = 5 × A
QP =

上式無法將未知數A消除,無法求得QP,故insufficient
(2) P=1.5A
Qp × P = QA × A
Qp × 1.5A = 5 × A
     QP =   =   sufficient
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122.        12343-!-item-!-187;#058&008383
During a 40-mile trip, Marla traveled at an average speed of x miles per hour for the first y miles of the trip and at an average speed of 1.25x miles per hour for the last 40 - y miles of the trip.  The time that Marla took to travel the 40 miles was what percent of the time it would have taken her if she had traveled at an average speed of x miles per hour for the entire trip?

(1) x = 48
(2) y = 20
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【答案】B
【思路】題目是問: 全程如果用x mile所花的時間: 全程用題目的數據所花的時間

距離/速度=時間
   你發現到x會被消掉,主導權在y而已
(1) Y是未知數,insufficient
(2) X被消除了,y=20, sufficient
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123.        12586-!-item-!-187;#058&008871
If integer p is greater than 1, is p a prime number?
If integer p is greater than 1
(1) p is odd.

(2) The only positive factors of p are 1 and p.
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【答案】B
【思路】
Def: A positive integer p greater than 1 is called prime (質數) if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite (合成數).
例： 2, 3, 5, 7, 9, 11, 13, 17, 19, and 23 are prime.
(1) 奇數與是否為prime number沒有關係. insufficient
(2) prime number的定義. sufficient
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124.        12829-!-item-!-187;#058&008976
A certain company divides its total advertising budget into television, radio, newspaper, and magazine budgets in the ratio of 8 : 7 : 3 : 2, respectively.  How many dollars are in the radio budget?

(1)  The television budget is $18,750 more than the newspaper budget.

(2)  The magazine budget is $7,500.
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【答案】D
【思路】
(1) T = N+18750
    T =   N
      N – N = 18750
      N – N = 18750
    N = 11250
    R =11250 x   = 26250……sufficient
(2) M=7500
  x 7 =   x 7 = 26250……sufficient
其實都是比例問題 所以方法很多
Statement 1:

8x-3x = 18750
=> 5x=18750
=> x=3750
Radio Budget = 7x = 7*3750=26250---Sufficient

Statement 2:

2x=7500
=> x= 3750
Radio Budget = 7x = 7*3750=26250---Suffcient
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125.        12932-!-item-!-187;#058&009020
The sequence a1, a2, a3, . . . , an of n  integers is such that ak = k if k is odd and ak = -ak - 1 if k is even.  Is the sum of the terms in the sequence positive?

(1)  n is odd.
(2)  an is positive.
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【答案】D
【思路】
當k奇數時，ak=k
當k偶數時，ak=-ak-1

If k = odd        ak = k        a1 = 1        a3 = 3        a5 = 5        a7 = 7
                                       
If k = even        ak = -ak-1        a2 = -1                a4 = -3        a6 = -5        a8 = -7

(1) If n is odd:
sum(ak) = (a1 - a1) + (a3- a3) + ... + (an-2 - an-2) + an = an
    見上表aodd=even, 數列總和=第n個數(因為前面兩兩一組總和為0)，所以總和一定為正, sufficient
(2) If a(n) is positive, then n has to be odd, 第n個數為正，也就是n is odd, 結果跟(1)一樣，總和一定為正, sufficient
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126.        13035-!-item-!-187;#058&009047
S is a finite set of numbers.  Does S contain more negative numbers than positive numbers?

(1)  The product of all the numbers in S is -1,200.

(2)  There are 6 numbers in S .
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【答案】E
【思路】
(1) 只告訴有一些正負的數字,insufficient
    表示一定要有負數
(2) 六個數字並不能表達什麼
(1)+(2)
因為S contain more negative numbers than positive numbers
所以至少要有四個負數以上,但一定要奇數的負數,
The product of all the numbers in S 才會是負數
因此會有5個負數,就可以成立more negative numbers than positive numbers
(-)(-)(-)(-)(-)(+)=(-) 一正五負
但是也可能S contain less negative numbers
(-)(+)(+)(+)(+)(+)=(-) 一負五正
所以答案為E
舉例：
1.
-1200=-2*15*40 Num of +ve nums greater than the num of -ve nums.
-1200=-2*-15*-40 Num of +ve nums less than the num of -ve nums.
So Insuff

2.Insuff.Those 6 nums can include any number.

Combining 1 and 2,
the 6 nums whose product is -1200 can be -1,2,3,4,5 and 10 or -1,-2,-3,4,5 and 10.So Insuff.
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127.        13089-!-item-!-187;#058&009187
On the number line, if the number k is to the left of the number t, is the product kt  to the right of t ?

(1) t < 0
(2) k < 1
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【答案】A
【思路】

------k------t------
(1) t < 0
------k------t------0------
k < t < 0
since the number k is to the left of t, k must also be negative.
kt = (-) × (-) = (+) ,  the product of two negative number is positive.
hence kt > t,
k × t < t, kt is “always” less than value of t, sufficient

(2) k < 1
k<1 means k could be positive(0,1), 0 or negative.
t ≧ 1
------k---------1------t------
◎        Case 1: if k is positive , t is also positive. so kt is positive and hence to the right of t.
----0--k----t----1-------------

◎        Case 2: k=0. since k is to the left of t, t is positive, but kt =0 which is on the left of t(since t is positive).
◎        ------k(0)------t------1----------------
Or
------k(0)-------------1--------t-------

◎        Case 3: if k is negative, t could be negative, zero or positive which means kt could be positive(to right of t), zero(on t), or negative(to the left of t).
------k---------0---------1------t------
Or
------k----t----0---------1-------------
Or
------k---------0----t----1-------------

So statement (2) is not sufficient to answer the question.

也可直接把數值帶入來解題
k<1
If k=-3 or K=0.3,
If k =0.3 and t i=4 , kt = 1.2, kt is to the left of t.

If k = -3 and t = 0 , kt = 0

If k=-2 and t=-1, kt = 2, kt is on the right of of t
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128.        13143-!-item-!-187;#058&009201
The points A, B, C, and D are on a number line, not necessarily in that order.  If the distance between A and B is 18 and the distance between C and D is 8, what is the distance between B and D ?

(1) The distance between C and A is the same as the distance between C and B.
(2) A is to the left of D on the number line.
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【答案】E
【思路】

(1) 一定要當場做圖
因CA and CB等距, AB=18, 因此 AB = CB = 9, This tells us that C is on the mid-point between A and B.
----A----9----C----9----B---- 是固定的，But D could be to the right or left of C
D can be either 8 units to the right of C or eight units to the left
possible 1  ------A----------9---------C-----8------D--1--B----------- Line BD長=1
possible 2  ------A--1--D------8------C---------9---------B----------- Line BD常=17
insufficient
(2) D可以在A右邊的任何位置，例：見上圖
(1)+(2) 如上圖，we don't know which side of C D is on，無法確定
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129.        13198-!-item-!-187;#058&009205
Is   < 0?

(1) -1 < x < 1

(2) x2 - 4 < 0
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【答案】A
【思路】
  可看成(x+1)×(x-3)< 0，
(x+1) × (x-3)< 0 又可以看成 x+1>0 且x-3<0 ,或者  x+1<0 且 x-3>0
可以解得
-1<x<3, OK
x<-1 , x>3 不可能
綜合上式-1<x<3
(1) -1 < x < 1含蓋在-1<x<3裡面, sufficient
(2)   
2 > x > -2 未含蓋在-1<x<3, NOT sufficient
另解
(1)先推出 0<x+1<2 , -4<x-3<-2
然後相除就會變成 -1<(x+1)/(x-3)<0 (記得相除要找最小&最大),所以確定小於0,充分
(2) (x+2)(x-2)<0 =>-2<x<2
推出 -1<x+1<3 , -5<x-3<-1
相除以後變成 -3<(x+1)/(x-3)<1,可能小於0或大於0(沒說是整數),不充分
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130.        13252-!-item-!-187;#058&009207
In the sequence of positive numbers x1, x2, x3, . . . , what is the value of x1 ?

(1) xl =   for all integers l > 1

(2) x5 =  
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【答案】C
【思路】
(1) xl =    insufficient
(2) x5 =   we can just find that relation between x5 and x4 , insufficient
(1)+(2)聯立即可求得答案
x4/2 = x4/( x4+1)
  - x4 = 0
x4 (x4-1)=0
x4= 0, 1 and x4 is positive i > 1, then x4 = 1
substituting values > 1 in xl =  
x5=x4/2=1/2
x4=x3/2→ x3= x4 x 2 = 2
x3=x2/2→ x2= x3 x 2 =4
x2=x1/2→ x1= x2 x 2 =8

x1=2(2(2x4))) = 8 Answer: C
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131.        13306-!-item-!-187;#058&009213
The positive integer k has exactly two positive prime factors, 3 and 7.  If k has a total of 6 positive factors, including 1 and k, what is the value of k ?

(1) 32 is a factor of k.

(2) 72 is not a factor of k.
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【答案】D
【思路】
對任意正整數 n，先定義因子個數函數 τ(n)。可以證明 τ 是一個 multiplicative 函數，意思是說 τ(mn) = τ(m)τ(n)。m, n 互質，才有 τ(mn) = τ(m)τ(n)。而對於質數 p，很明顯我們有：τ( ) = (r + 1)。
舉例：用這公式算一下 90 的因子個數，τ( ) = (r + 1)，
τ(90) = τ[(2)(  ) (5)] = τ(2)τ( )τ(5) = (1+1)(2+1)(1+1) =  (2)(3)(2) = 12個
再舉120為例：120=2^3*3^1*5^1 因子個數為 (3+1)(1+1)(1+1)=16 個

題目說 , 共6個因子
因此(x+1)(y+1)=6,則共有(2,3)(3.2)-->(x,y) 有 either(1,2)or(2,1) 的組合
所以 ,  , , 的互相乘積皆是可能factors
(1) 32 is a factor of k.
(3^0)(7^0) = 1
(3^0)(7^1) = 7
(3^1)(7^0) = 3
(3^1)(7^1) = 21
(3^2)(7^0) = 9
(3^2)(7^1) = 63 (k, itself) (x≧2, so it must be true that x=2 and y=1)
由條件得知k's positive factors:1,3,7,9,21,k
k=63 .........sufficient
(2) 72 is not a factor of k.
k有6個positive factors,且只有3,7兩個質因子
表示k可因式分解為 ×7or 3×
此條件排除了第二個可能，y<2, so x=2 and y=1.
所以k=63 ...............sufficient
練習題：
現在要求在 τ(n) = 20 中最小的 n？
首先，τ(n)= 20 = 10 x 2＝ 5 x 4 = 5 x 2 x 2
(1) 假如 n 只有一個質因子，那麼要求 n 為最小時此質因子一定是 2，於是 n = 219，明顯超大的。
(2) 假如 n 有兩個質因子，那麼一個是 2，另一個是 3，且把大的次方數放在 2 那兒，就會令 n 最小，所以 n = (2)4(3)3 = 432
或是另一個可能 n = (2)9(3)，這個不用算，超大的。
(3) 假如 n 有三個質因子，那令 n 最小的一定是 2, 3, 5，又依序把較小次方數的放在較大的質因數，所以得 n = (2)4(3)(5) = 240。
很明顯答案就是 240 了。
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132.        13360-!-item-!-187;#058&009215
If l and k are lines in the xy-plane, is the product of the slopes of l and k equal to -1 ?

(1)  Line l passes through the origin and the point (1, 2).
(2)  Line k has x-intercept 4 and y-intercept 2.
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【答案】C
【思路】
設y=ax+b
Formula for slope is (y1-y2)/(x1-x2)
(1) we dont know anything about the other line k
but we can find the slope of line l = 2... insufficient
(2) we dont know anything about the other line l
but we can find the slope of line k = -1/2.
    (4,0) and (0,2) --> (0-2)/(4-0) --> -2/4 --> -1/2………..insufficient
(1)+(2):
(1) & (2) taken together we can see if the product of the slopes is -1 or not.
(2)×(-1/2)= -1.........sufficient
But you could just realize that 1 and 2 are suff together b/c we know the slopes we are going to get a YES OR A NO NOT BOTH. so u could save some time.
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133.        13609-!-item-!-187;#058&009567
At a certain company, the average (arithmetic mean) number of years of experience is 9.8 years for the male employees and 9.1 years for the female employees.  What is the ratio of the number of the company's male employees to the number of the company's female employees?

(1)  There are 52 male employees at the company.
(2)  The average number of years of experience for the company's male and female employees combined is 9.3 years.
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【答案】B
【思路】
(1) M:F?
    F?.....insufficient
(2) M:F?
     = 9.3
9.8M + 9.1F = 9.3M + 9.3F
   0.5M = 0.2F
   M:F = 2/5……..sufficient
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134.        13755-!-item-!-187;#058&009724
An attorney charged a fee for estate planning services for a certain estate. The attorney's fee was what percent of the assessed value of the estate?

(1)  The assessed value of the estate was $1.2 million.
(2)  The attorney charged $2,400 for the estate planning services.
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【答案】C
【思路】
(1) only gives the information for the estate value.
(2) only gives the amount that was charged.
(1) + (2)

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135.        13860-!-item-!-187;#058&009754
During a one-day sale, a store sold each sweater of a certain type for $30 more than the store's cost to purchase each sweater.  How many of these sweaters were sold during the sale?

(1) During the sale, the total revenue from the sale of these sweaters was $270.

(2) During the sale, the store sold each of these sweaters at a price that was 50 percent greater than the store's cost to purchase each sweater.
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【答案】C
【思路】

S = 150C
(1) (C + 30) × N = 270……只能求出 C, insufficient
(2) C + 30 = (1 + 50%) × C = 1.5C
C = 30 × 2 = 60…………只能求出 C, insufficient
(1)+(2)
(C + 30) × N = 270
N = 270/90 = 3…………………sufficient
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136.        13960-!-item-!-187;#058&009782
An integer greater than 1 that is not prime is called composite. If the two-digit integer n is greater than 20, is n composite?

(1)  The tens digit of n is a factor of the units digit of n.

(2)        The tens digit of n is 2.
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【答案】A
【思路】
Keep in mind that they are not asking for the actual value of n but rather whether n is composite
(1)
        For every prime numbers, tens digit is not a factor of unit digits. 23,29,31,37,41,47... If it's a factor..it's definitely composite number
        For every prime no#, tens digit is not a factor of unit digits.
23,29,31,37,41,47... If it's a factor..it's definitely composite number.
        This means n will be a product of its tens digit and the two-digit number with tens digit 1 and unit digit (n's unit digit)/(n's tens digit). For example, 26, 2 is a factor of 6, so 26 is a product of 2 and 13.
        n = AB = 10xA + AC = A(10+C).. So A is factor, it cannot be prime.

(2)  not sufficient, because there are two digit prime numbers that have 2 as their tens ditig: 23, 29.

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137.        14062-!-item-!-187;#058&009802
If x is an integer, is (x2 + 1)(x + 5) an even number?

(1)  x is an odd number.

(2)        Each prime factor of x2 is greater than 7.
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【答案】D
【思路】
(1) X is an odd number
Square of an odd number will always be an odd number mean ( +1) we always be even.
adding 5 to an odd number will always result in even number.
偶×偶=偶…..Sufficient
(2) As each prime factor is>7 mean   could be 121,169,20449 etc.
Each prime factor of   > 7. That means  cannot be divided by 2. Therefore x cannot be divided by 2. All the prime factors > 7 are odd. X got be an odd number.
代表每個質數因數必是奇數
so 奇×奇=奇
so   is odd mean  +1 is even.)
adding 5 to an odd number will always result in even number.
偶×偶=偶……sufficient

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138.        14162-!-item-!-187;#058&009822
A certain theater has a total of 884 seats, of which 500 are orchestra seats and the rest are balcony seats.  When tickets for all the seats in the theater are sold, the total revenue from ticket sales is $34,600.  What was the theater's total revenue from ticket sales for last night's performance?

(1) The price of an orchestra seat ticket is twice the price of a balcony seat ticket.

(2) For last night's performance, tickets for all the balcony seats were sold, but only 80 percent of the tickets for the orchestra seats were sold.
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【答案】C
【思路】
o = the price of an orchestra seat,
b = the price of a balcony seat.
500o + 384b = 36,400
(1)
o = 2b
that's a substitution, which we can use in the equation above, and which will therefore yield a unique solution (o, b) for the prices of the 2 different types of tickets.
however, this statement tells us nothing about last night's performance, so it is insufficient.
(2)  
we don't have price points for the two different types of seats, so we can't figure this one out. (the more that the orchestra seats cost, relative to the balcony seats, the farther short of $34,600 the revenue will fall.)
(1)+(2)
statement (1) we can figure the price of each type of ticket, so we know 'o' and 'b'. using statement (2), we know that the revenue is 384b + 400o. plug in and find out the revenue.
sufficient
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139.        14263-!-item-!-187;#058&009908
If it took Carlos   hour to cycle from his house to the library yesterday, was the distance that he cycled greater than 6 miles? (Note:  1 mile = 5,280 feet)

(1)  The average speed at which Carlos cycled from his house to the library yesterday was greater than 16 feet per second.

(2)  The average speed at which Carlos cycled from his house to the library yesterday was less than 18 feet per second.
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【答案】E
【思路】
average speed × time = distance
(1)   
the distance that he cycled is greater than 5.45.
We cannot know whether it greater than 6.
(2)  
the distance that he cycled is less than 6.14.
We cannot know whether it greater than 6.
(1)+(2)
(1) & (2), we need value for average speed and both of them do not specify an absolute value.
Together they don't either, as the value can be a fraction……insufficient
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140.        14319-!-item-!-187;#058&009938
At a certain pet shop,   of the pets are dogs and   of the pets are birds.  How many of the pets are dogs?

(1)  There are 30 birds at the pet shop.
(2)  There are 20 more dogs than birds at the pet shop.
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【答案】D
【思路】
(1) 30 × 5 = 150
You can calculate number of dogs = 150/3 = 50…..sufficient
(2) Dogs = Birds + 20
x/3 = x/5 + 20
x/3 - x/5 = 20
x = 150
You can calculate number of dogs = 150/3 = 50…..sufficient
(Don't have to actually find the number)
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141.        14373-!-item-!-187;#058&009939
To fill an order on schedule, a manufacturer had to produce 1,000 tools per day for n days.  What is the value of n ?

(1) Because of production problems, the manufacturer produced only 600 tools per day during the first 5 days.
(2) Because of production problems, the manufacturer had to produce 1,500 tools per day on each of the last 4 days in order to meet the schedule.
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【答案】E
【思路】
(1)前五天

一個方程式,兩個未知數,無法求解…..insufficient
(2)後四天

一個方程式,兩個未知數,無法求解…..insufficient
(1)+(2)
沒有講中間幾天…..insufficient
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142.        14473-!-item-!-187;#058&010003
If x is a positive integer, what is the least common multiple of x, 6, and 9 ?

(1)  The least common multiple of x and 6 is 30.
(2)  The least common multiple of x and 9 is 45.
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【答案】D
【思路】
(1)

x=5…….sufficient
(2)

x=5…….sufficient
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143.        14534-!-item-!-187;#058&010011


In the figure shown, the measure of angle PRS is how many degrees greater than the measure of angle PQR ?

(1)  The measure of angle QPR is 30°

(2)  The sum of the measures of angles PQR and PRQ is 150°
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【答案】D
【思路】
PRS - PQR = ?
PQR + QPR + PRQ = 180 (三角形PQR)
PRS+RPS+90=180 (三角形RPS) =>PRS+RPS=90
180 = PRQ+PRS
(1) QPR = 30
1. PQR + QPR + PRQ = 180 (三角形PQR)
=> PQR+PRQ = 150 (=180-30)
2, 180 = PRQ+PRS
1,2 互相結合
PQR + QPR + PRQ= PRQ + PRS
=>PQR + QPR = PRS
PRS - PQR = QPR = 30…..sufficient

(2) PQR + PRQ = 150
PQR + QPR + PRQ = 180 (三角形PQR)
QPR = 180 – PQR – PRQ = 180 – 150 = 30
其他解題同(1)
故PRS - PQR = QPR = 30…..sufficient
法2 另一種想法
Extend SQ out and that external angle is 180-PQR, which must be equal to the two internal angles, one of which is 30 degrees given in one and two and the other is 180-PRS. Using algebra the difference is 30 degrees.
PRQ=(180-PQR)-QPR
   =180-PQR-30
=>PQR+PRQ=150
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144.        14635-!-item-!-187;#058&010061
If $1,000 is deposited in a certain bank account and remains in the account along with any accumulated interest, the dollar amount of interest, I , earned by the deposit in the first n years is given by the formula I =1000〔(1 +  )n - 1〕  , where r percent is the annual interest rate paid by the bank.  Is the annual interest rate paid by the bank greater than 8 percent?

(1)  The deposit earns a total of $210 in interest in the first two years.
(2)  ( 1 +   )2 > 1.15
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【答案】A
【思路】
(1)
210 = 1000  
0.21=1+ + -1
0.21= +
求得166.4….. The deposit earns a total of $210 in interest in the first two years.
是的…….sufficient
(2)
>1.15
=1+2 + >1.15
=10000+200r+r2>11500
= >11500
=r+100>105(大約值)
所以 r 可能大於也可能小於 8……insufficient
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145.        14689-!-item-!-187;#058&010088
On her way home from work, Janet drives through several tollbooths.  Is there a pair of these tollbooths that are less than 10 miles apart?

(1)  The first tollbooth and the last tollbooth are 25 miles apart.
(2)  Janet drives through 4 tollbooths on her way home from work.
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【答案】C
【思路】tricky
這題是問眾多的收費站中,其中的兩個收費站距離有沒有少於10哩?
(1) 只說第一個與最後一個的距離是25哩, 但不知中間到底有幾個, 可能很多個, 可能一個都沒有, 所以不充份
(2) 只說有4個收費站, 但不知總距離為何, 所以不充份
(1)+(2)
四個tollbooths, 三個區間
|_____|_____|_____|
A　　B　　C　D
假如AB = 10, BC = 10, CD = 5 一對低於10 miles
假如AB = 5, BC = 5, CD = 20 兩隊低於10 miles
假如AB = 10, BC = 10, CD = 10 ; 破30, 爆了
因此怎麼看都不可能全部都是超過10 miles的距離, 所以一定有至少一組是少於10 miles.

條件一及條件二合起來就充份了, 因為總長25哩, 有4個收費站, 如果每個收費站間的距離是均等的話, 收費站之間則會是25/3<10, 但如果不均等的話, 則其中有距離較長的及距離較短的, 題目只要求其中兩個收費站的距於有沒有少於10哩, 所以最少一定會有一個間距少於10哩
所以選項(C)充份
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146.        14746-!-item-!-187;#058&010097
If the symbol △ represents either addition or multiplication, which operation does it represent?

(1)  a △ b = b △ a for all numbers a and b.
(2)  a △ ( b – c ) = ( a △ b ) – ( a △ c )for all numbers a, b, and c.
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【答案】B
【思路】
(1)
if △= +:  a + b = a - b
if △= ×:  a × b= b × a
△可表示×或+…….insufficient
(2)
if △= +:  a + b – c ≠ a + b – a - c
if △= ×:  ab – ac = ab – ac
只有△= × …….sufficient
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147.        14800-!-item-!-187;#058&010134
Each employee on a certain task force is either a manager or a director.  What percent of the employees on the task force are directors?

(1) The average (arithmetic mean) salary of the managers on the task force is $5,000 less than the average salary of all employees on the task force.
(2) The average (arithmetic mean) salary of the directors on the task force is $15,000 greater than the average salary of all employees on the task force.
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【答案】C
【思路】
令MANAGER人數=X, Director人數=Y
M=Manger平均工資¸D=Director平均工資
(1) M+5000=(MX+DY)/(X+Y)
(2) D-15000=(MX+DY)/(X+Y)
(1)+(2)
M+5000=(MX+DY)/(X+Y)
D-15000=(MX+DY)/(X+Y)
則先求出
M+5000= D-15000
M=D-20000
代入第一個方程式，
(D-20000)+5000=[(D-20000)X+DY]/(X+Y)
=>D-15000=(DX-20000X+DY)/(X+Y)
=>DX+DY-15000Y-15000X= DX-20000X+DY
=>-15000Y-15000X= -20000X
=>3Y+3X=4X
=>3(X+Y)=4X
X/(X+Y)=3/4=75%.....sufficient
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148.        14854-!-item-!-187;#058&010192
In the xy-plane, does the line with equation y = 3x + 2 contain the point (r, s) ?

(1)  (3r + 2 - s)(4r + 9 - s) = 0
(2)  (4r - 6 - s)(3r + 2 - s) = 0
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【答案】C
【思路】
r is the x coordinate, s is the y, so we want to know whether:
s=3r+2
(1) Essentially, one equation or the other must be equal to zero (if xy = 0, the x = 0 or y = 0).
either  3r+2-s=0 or 4r+9-s=0,
so:      s=3r+2 or s=4r+9
If it is true that 3r+2-x = 0, then we see that (r.s) must be included in 3r+2 = x, since we have that exact equation. However, it is possible that 4r+9 =s, in which case our equation doesn't have to be 3r+2 = s
In the first case, (r.s) falls in the line. In the second case, (r,s) does not fall in the line.
無法確定(r,S)會落在哪一條線…..insufficient because it could be the second case
(2)
either  4r-6-s=0 or 3r+2-s=0,
so:      s=4r-6 or s=3r+2
In the first case, (r.s) falls in the line. In the second case, (r,s) does not fall in the line.
無法確定(r,S)會落在哪一條線…..insufficient because it could be the second case
(1)+(2)
y=3x+2 will contain (r,s) only if s=3r+2.
(1) s=3r+2 OR 4r+9. Insuff.
(2) s=4r-6 OR 3r+2. Insuff.
Combining: s=3r+2. Suff. C
當(1)、(2)同時存在時，會出現交集，此交集上的任何一點皆可以滿足題目s=3r+2，故選C。
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149.        14908-!-item-!-187;#058&010196
If m, r, x, and y are positive, is the ratio of m to r equal to the ratio of x to y ?

(1)  The ratio of m to y is equal to the ratio of x to r.

(2)  The ratio of m + x to r + y is equal to the ratio of x to y.

--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
問 m:r =x:y-->r×x=m×y?
(1) m/y=x/r ------->x×y=m×r ..........insufficient
(2) (m+x)/(r+y)=x/y -------> r×x + x×y = m×y + x×y (x,y皆為正數,x×y對消)
故 rx=my.........sufficient
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150.        15103-!-item-!-187;#058&010253
If a, b, k, and m are positive integers, is ak a factor of bm ?

(1) a is a factor of b.
(2) k   m
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【答案】C
【思路】
問題是要問 bm / ak 是否可以沒有餘數remainder
(1) b/a 沒有餘數是不夠的
例如b=ax (x being an positive integer)
bm = (ax)m = amxm
bm/ak=(amxm)/ak,
a = 2 and b = 10
2 has one factor 2 and say k = 2 and m = 2
In this case 22 = 4 is a factor of 100, However, if m = 1 then 4 is not a factor of 10.
we don't know the value of k & m.
(2) m-k≧0
No information about a & b.
(1)+(2)
bm / ak = (amxm)/ak = xm a (m-k)
= (integer) × (integer≧ 1)
So ans = C
變形題（幾乎一模一樣）
If a, b, k, and m are positive integers, is ak a factor of bm?
(1) a is a factor of b.
(2) k = m
Answer: C
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151.        15157-!-item-!-187;#058&010256
Is x > 0.05 ?

(1) x >  

(2) x is greater than 3 percent of 50.
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【答案】D

【思路】(1) 因為 X>0.075 , 所以X>0.05 , 故條件充分
        (2) 因為 X>0.06 , 所以X>0.05 , 故條件充分
(1) (2) 皆充分
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152.        15303-!-item-!-187;#058&010299
If xyz > 0, is x > 0 ?

(1) xy > 0

(2) xz > 0
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【答案】C

【思路】(1) xy>0 , x= + or - , y=+ or - , if x= - , y= - , xy =+ , 故不充份
        (2) xz>0 , x= + or - , y=+ or - , if x= - , y= - , xz =+ , 故不充份
xyz > 0 , 結合(1) (2) x>0 ,y >0 , z>0 , 則 xyz >0
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153.        15354-!-item-!-187;#058&010304
A wooden rod is cut into two pieces.  What is the length of the longer piece?

(1) One of the pieces is 20 inches longer than the other piece.

(2) The length of the shorter piece is   the length of the longer piece.
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【答案】C

【思路】假設 longer piece = x
        (1) longer piece = x , short piece = x – 20 , 無法得知 length of the longer piece , 故不充份
        (2) shorter piece =  x , , 故不充份
結合(1) (2) short piece = x – 20 = x , x = 30 , 充份
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154.        15408-!-item-!-187;#058&010332–
The sum of positive integers x and y is 77.  What is the value of xy ?

(1) x = y + 1

(2) x and y have the same tens digit.
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【答案】D

【思路】(1) 題目 x + y = 77 , (1) x = y + 1 ,   x = 39 , y = 38 , xy =1482 , 充份
        (2) 題目 x + y = 77 , x , y有共同的10位數 , 只有38 , 39符合 , 充份
(1) (2)皆充份 , 故選D
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155.        15462-!-item-!-187;#058&010358
Did it take Pei more than 2 hours to walk a distance of 10 miles along a certain trail?  (1 mile = 1.6 kilometers, rounded to the nearest tenth.)

(1)  Pei walked this distance at an average rate of less than 6.4 kilometers per hour.

(2)        On average, it took Pei more than 9 minutes per kilometer to walk this distance.
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【答案】D

【思路】10 miles = 16 kilometers
        (1) average rate <6.4 , 2小時所行走距離<12.8 , 所以12.8 < 16 ,則要超過兩個小時才能走16 kilometers ,充份
        (2) per kilometer花費9分鐘,16 kilometers花費144分,所以超過2小時,充份
(1) (2)皆充份 , 故選D
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156.        15516-!-item-!-187;#058&010359
A store sells milk in cartons of two sizes, one-half gallon and one-quarter gallon.  If the store sold a total of 300 cartons of milk yesterday, how many gallons of milk did it sell yesterday?

(1)  Yesterday the store sold 120 one-quarter-gallon cartons of milk.

(2)        Yesterday the store sold 90 gallons of milk in one-half-gallon cartons.
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【答案】D

【思路】題目問: h+q=300

(i) q = 120
Therefore, h=180. Total milk sold = 0.5*180+0.25*120 (SUFF)
(ii) 這邊的題目要注意,有小變化(90 gallons of milk in one-half-gallon cartons)
h = 90*2 = 180
Therefore, q=120. Total milk sold = 0.5*180+0.25*120 (SUFF)
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157.        15663-!-item-!-187;#058&010465
If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r ?

(1) n + 1 is divisible by 3.

(2) n > 20
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【答案】A

【思路】(1) n + 1 是3的倍數 , 則n=2,5,8……,帶入 ,則r=0 ,故充份
        (2) n>20 , n=21,22,23…. ,帶入 ,則r=3,6……,所以不充份
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158.        15810-!-item-!-187;#058&010518
Each of the 25 balls in a certain box is either red, blue, or white and has a number from 1 to 10 painted on it.  If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it?

(1)  The probability that the ball will both be white and have an even number painted on it is 0.

(2)        The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2.
--------------------------------------------------------------------------------------------------------------------------------------------------------
【答案】E

【思路】問P (either white or even)= P (white)+P(even)- P(both)
(1)只知 P(both) ,不充分
(2)只知P(white)-P(even) ,不充分
(1)+(2)也不充分
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159.        15957-!-item-!-187;#058&010660
If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r ?

(1) n is not divisible by 2.

(2) n is not divisible by 3.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】 =
        (1) n = 3 , 5 , 7 ,9…… ,則r=8 , 1 ……,不充份
      (2)n=1 , 2 , 4,….,r=2 ,3………,不充份

(1) + (2) 不能被2 , 3整除 ,2 ,3 為 24 的公因數 ,所以n排除為24因數後,r=0
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160.        16057-!-item-!-187;#058&010664
If Company M ordered a total of 50 computers and printers and Company N ordered a total of 60 computers and printers, how many computers did Company M order?

(1)  Company M and Company N ordered the same number of computers.

(2)  Company N ordered 10 more printers than Company M.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E

【思路】(1) company M 和 company N 訂購同樣數量的電腦,但題目並未告知companyM ,N 訂購 computers和printers,不充份
        (2)假設訂購printers=x , x , x-10為M , N訂購printers , 總數量為110 ,但並不是printers總數 ,所以不充份

(1) +(2)還是無法得知電腦或是printers的總數 ,故選E
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161.        16203-!-item-!-187;#058&010722
If the average (arithmetic mean) of the five numbers x, 7, 2, 16, and 11 is equal to the median of the five numbers, what is the value of x ?

(1) 7 < x < 11

(2) x is the median of the five numbers.
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【答案】D

【思路】(1) 7 < x < 11 , 則數列為 2 , 7 , x , 11 , 16 , m=  , 中位數為x , x=  ,x=9 ,充份
        (2)x為中位數, 數列為 2 , 7 , x , 11 , 16 , x=  ,x=9 ,充份

(1) (2)皆充份 ,故選D
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162.        16257-!-item-!-187;#058&010730
What fraction of this year's graduating students at a certain college are males?

(1)  Of this year's graduating students, 33 percent of the males and 20 percent of the females transferred from another college.

(2)  Of this year's graduating students, 25 percent transferred from another college.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】(1) 0.33M+0.2F=transfer 不充份
        (2) 0.25（M+F)=transfer 不充份

(1)+ (2) M=5/8F 充份 選E
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163.        16312-!-item-!-187;#058&010731
Linda put an amount of money into each of two new investments, A and B, that pay simple annual interest.  If the annual interest rate of investment B is 1.5 times that of investment A, what amount did Linda put into investment A ?

(1)  The interest for 1 year is $50 for investment A and $150 for investment B.

(2)  The amount that Linda put into investment B is twice the amount that she put into investment A.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E

【思路】假設 investmentsA=X , investmentsB=y , interest rateA=X1, interest rateB=Y1 , 1.5X1=Y1
    (1) X(X1)=50 , Y(Y1)=150 , Y(1.5X1)=150 , 無法得investment ,不充份
    (2) Y=2X 無法得investment ,不充份
(1)+(2) 2X(1,5X1)=150 , Y(1.5X1)=150 無法得 investment 故選E
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164.        16412-!-item-!-187;#058&010774
At a two-day seminar, 90 percent of those registered attended the seminar on the first day.  What percent of those registered did not attend the seminar on either day?

(1)  A total of 1,000 people registered for the two-day seminar.

(2)  Of those registered, 80 percent attended the seminar on the second day.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E

【思路】(1)只知全部總數為1000, 90 percent參加第一天, 10 percent不知是否參加別天或是沒參加 ,故不充份
        (2)得知80 percent參加第2天 ,但有可能和第一天重複的人數或是只參加第二天的, 故不充份
(1)＋(2) 全部人數為1000 ,可是沒資料得知重複人數,所以無法得知兩天都沒參加的人 故選E
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165.        16466-!-item-!-187;#058&010795
Of the 1,400 college teachers surveyed, 42 percent said that they considered engaging in research an essential goal.  How many of the college teachers surveyed were women?

(1) In the survey, 36 percent of the men and 50 percent of the women said that they considered engaging in research an essential goal.

(2) In the survey, 288 men said that they considered engaging in research an essential goal.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A

【思路】(1) total（1400) x  =588(參加人數） , 588 x women比例（ )=參加女人的比例 , 充份
        (2)原題問surveyed werer women , 包含 engage 和 not engage , so we only can get engage women , 不充份
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166.        16522-!-item-!-187;#058&010838
In the xy-coordinate plane, the slope of line l is  .  Does line l pass through the point (- , )?

(1)  Line l passes through the point (4,4).

(2)  Line l passes through the point (-4,-2).
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D

【思路】同斜率所通過的點相同 , 所以帶入（Y2-Y1）/（X2-X1） , 得(1)(2)皆可通過(- , ) ,故選D
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167.        16626-!-item-!-187;#058&010859
If x, y, and z are integers and xy + z is an odd integer, is x an even integer?

(1) xy + xz is an even integer.

(2) y + xz is an odd integer.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A

【思路】xy + z = odd integer , xy(odd)+z(even)= odd integer , xy(even)+z(odd)=odd integer
  (1) xy + xz = even , xy(even)+xz(even)=even , so z 應該為odd , x為even , xz才為even , xy(odd)+xz(odd)=even ,
z 為even , xz不等於odd , 充份
  (2) y + xz = odd integer , y(odd)+xz(even)=odd , z=even ,x=even , y(even)+xz(odd)=odd ,z=odd,x=odd , 所以不充份
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168.        16680-!-item-!-187;#058&010861
On the number line, the distance between x and y is greater than the distance between x and z.  Does z lie between x and y on the number line?

(1) xyz < 0

(2) xy < 0
--------------------------------------------------------------------------------------------------------------------------------------------------------
【答案】E

【思路】(1) xyz < 0 , 可能為 + - + , + + - , - + + , - - - , 所以不充份
        (2)xy < 0 , - + , + - , 不充份
(1)＋(2) = - + + , + - + ,不充份 選E
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169.        16735-!-item-!-187;#058&010875
If the integers a and n are greater than 1 and the product of the first 8 positive integers is a multiple of an, what is the value of a ?

(1)  an = 64

(2)  n = 6
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B

【思路】(1)   = 64 ,  =64 ,  =64 , not sufficient
        (2) n = 6 ,   = 64 , sufficient
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170.        16927-!-item-!-187;#058&010946
If n and t are positive integers, what is the greatest prime factor of the product nt ?

(1)  The greatest common factor of n and t is 5.

(2)  The least common multiple of n and t is 105.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B

【思路】(1)最大公因數為5 , 但其質因數有多組 ,不充份
        (2)105=(5)(3)(7) , 故其最質因數為7 , 充份
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171.        16981-!-item-!-187;#058&010960
Is |x - y | > |x | - |y | ?

(1)  y < x

(2)  xy < 0
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】 B

【思路】(1)  y < x , (y= - , x=+) , (y=+ , x=+) , (y=- , x=-) , 當 x=+ - , y=- - , |x - y | > |x | - |y | , 當 (y=+ , x=+)時, |x - y | = |x | - |y | ,所以不充份
        (2) xy < 0 , (y= - , x=+) or (y= + , x=- ) , |x - y | > |x | - |y | , 充份
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172.        17130-!-item-!-187;#058&011022
If m, k, x, and y are positive numbers, is mx + ky > kx + my ?

(1) m > k

(2) x > y
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】(1) m > k , 另外兩變數 x y 不知大小 , 所以不充份
        (2) x > y , m, k 不知 , 故不充份
(1)＋(2)= m>k , x>y ,  m , x 兩個最大變數相乘 ,恆為最大,則 mx + ky > kx + my 成立
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173.        17281-!-item-!-187;#058&011070
If N is a positive integer, is the units digit of N equal to zero?

(1) 14 and 35 are factors of N.

(2) N = (25)(32)(57)(76)
--------------------------------------------------------------------------------------------------------------------------------------------------------
【答案】D

【思路】(1)14和35為N的因數,70為最小公倍數,N為70倍數,充份
        (2) N = (25)(32)(57)(76) , 充份
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174.        17336-!-item-!-187;#058&011114
In the xy-plane, what is the y-intercept of line l

(1)  The slope of line l is 3 times its y-intercept.

(2)  The x-intercept of line l is - .
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E

【思路】(1) 直線方程式為 y=mx+b ,m為斜率 ,b為截距 , 通過（0 ,Y1) , Y1=3b(x)+b , Y1=b, 不充份
        (2) y=mx+b , (-1/3, 0) 帶入 ,b= m ,不充份
(1)＋(2) 還是不充份 故選E
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175.        17390-!-item-!-187;#058&011123
The sum of the integers in list S is the same as the sum of the integers in list T.  Does S contain more integers than T ?

(1)  The average (arithmetic mean) of the integers in S is less than the average of the integers in T.

(2)  The median of the integers in S is greater than the median of the integers in T.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A

【思路】(1) mean=   ,當樣本越大 ,m越小 ,所以得知S的average小於T時 ,其樣本大於T ,充份
        (2) median of the integers 的大小不影響數列樣本的多少 ,不充份
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176.        17491-!-item-!-187;#058&011140
If x and y are positive integers, is the product xy even?

(1)  5x - 4y is even.

(2)  6x + 7y is even.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D

【思路】xy=even , 有三種情況 ,(x=even . y=even) ,(x=even ,y=odd) , (x=odd ,y=even)
        (1) 5x - 4y is even , even  - even =even , 所以xy=even ,充份
        (2) 6x + 7y is even , even  + even =even , 所以xy=even ,充份
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177.        17595-!-item-!-187;#058&011159
If x and y are positive integers, what is the value of xy ?

(1)  The greatest common factor of x and y is 10.

(2)  The least common multiple of x and y is 180.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】(1) The greatest common factor of x and y is 10 ,其資料不充份
        (2) 若兩數互質 , 則其最小公倍數為其兩者相乘 ,有許多組合 ,所以不充份
(1)＋(2)=(a,b)[a,b]=ab 所以選C
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178.        17649-!-item-!-187;#058&011192
If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0 ?

(1)  The product of any two numbers in the list is equal to 0.

(2)  The sum of any two numbers in the list is equal to 0.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B

【思路】(1)任何數相乘只要數列有一個0,乘積為0,但不代表數列均為0,不充份
        (2)任何數相加為0,其必為兩個0,充份
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179.        17750-!-item-!-187;#058&011285
A manufacturer conducted a survey to determine how many people buy products P and Q.  What fraction of the people surveyed said that they buy neither product P nor product Q ?

(1)    of the people surveyed said that they buy product P but not product Q.

(2)    of the people surveyed said that they buy product Q.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】(1)  買P沒買Q , 資料不充份
        (2)  買Q  , 資料不充份
(1)+ (2) 1-  -  =no P no Q , 充份 故選C
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180.        17851-!-item-!-187;#058&011342
Are x and y both positive?

(1)  2x – 2y = 1

(2)    > 1
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C

【思路】(1) 2x – 2y = 1 , x=+ ,y=- or x=+ , y=+ ,or x=- y=- ,不充份
        (2)   > 1 , 則x= - , y= - ,x < y or x=+ ,y=+ , x>y ,不充份
(1)＋(2)= x=+ ,y=+ , 選C
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181.        17908-!-item-!-187;#058&011358


In the figure shown, what is the value of x ?

(1)  The length of line segment QR is equal to the length of line segment RS.

(2)  The length of line segment ST is equal to the length of line segment TU.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) QR=RS 不能判斷x
(2) ST=TU 不能判斷x
(1)+(2)  RSQ= RQS &&  TSU= SUT
SUP=x+ TSU
SQP=x+ RSQ
SUP+ SQP=180+x
QPU+180+2x = 360 (四邊形內角和)
2x=90, x=45
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182.        17963-!-item-!-187;#058&011379
In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries.  If Mary's annual salary was the highest of the 3 people, what was the average (arithmetic mean) annual salary of the 3 people that year?

(1)  Jim's annual salary was $30,000 that year.

(2)  Kate's annual salary was $40,000 that year.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】Ｂ
【思路】
由題意得知M-J=2(M-K)
M+J=2K　代入 = =K
(1) 有2個未知
(2) 代入上面簡化的算式得到.
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183.        18017-!-item-!-187;#058&011390
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.  Is z even?

(1) x is even.

(2) y is even.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
由題意得知 y=m*x and z=n*y therefore z=n*m*x
(1) z=(n*m)*x 偶數乘不等於0的整數=偶數
(2) z=n*y 偶數乘不等於0的整數=偶數
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184.        18071-!-item-!-187;#058&011405
Jack and Mark both received hourly wage increases of 6 percent.  After the wage increases, Jack's hourly wage was how many dollars per hour more than Mark's?

(1)  Before the wage increases, Jack's hourly wage was $5.00 per hour more than Mark's.

(2)  Before the wage increases, the ratio of Jack's hourly wage to Mark's hourly wage was 4 to 3.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】
(1) J-M=5 1.06(J-M)=5.3
(2) J:M=4:3 無法得知時薪增加多少
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185.        18511-!-item-!-187;#058&011604
Is x between 0 and 1 ?

(1) x is between -  and  

(2)   is   more than x.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) - < x <  x可能小於0或大於0
(2) x =   -  = 2
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186.        18566-!-item-!-187;#058&011610
If 0 < x < y, what is the value of  ?

(1) x2 + y2 = 3xy

(2) xy = 3
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】
(1)  = =5
(2)  　不知道x,y的值
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187.        18620-!-item-!-187;#058&011611
If x, y, and z are integers greater than 1, what is the value of x + y + z ?

(1) xyz = 70

(2)   =  
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】
(1) xyz = 70 = 1x2x5x7 or 1x7x10(不合題目條件)
   (x,y,z)={2,5,7} x+y+z=14
(2) x 可以是7或14等..
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188.        18674-!-item-!-187;#058&011626
Is p + pz = p ?

(1) p = 0

(2) z = 0
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
p+pz = p(1+z)
(1) 0x(1+z) = 0
(2) p(1+0) = p
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189.        18728-!-item-!-187;#058&011689
If the operation △ is one of the four arithmetic operations addition, subtraction, multiplication, and division, is (6 △ 2) △ 4 = 6 △ (2 △ 4) ?

(1) 3 △ 2 > 3

(2) 3 △ 1 = 3
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】
(6 △ 2) △ 4 = 6 △ (2 △ 4) 乘或加,兩式相等. 除或減,兩式不相等
(1) △可以是乘或加
(2) △可以是乘或除
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190.        18785-!-item-!-187;#058&011838



On the number line shown, is zero halfway between r and s ?

(1) s is to the right of zero.

(2) The distance between t and r is the same as the distance between t and -s.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) r 可以是正或負數
(2) s 可以是正或負數
(1)+(2)
t and s are positive. t+|r|= t+|-s|
r=s or –s. r is not s (from graph), so r=-s
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191.        18839-!-item-!-187;#058&011851
If the product of the three digits of the positive integer k is 14, what is the value of k ?

(1) k is an odd integer.

(1)        k < 700
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E
【思路】
14=1x2x7
(1) {271, 721, 127, 217}
(2) {172, 127, 271, 217}  
(1)+(2) {271, 127, 127}
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192.        18939-!-item-!-187;#058&011984
Of the 75 houses in a certain community, 48 have a patio.  How many of the houses in the community have a swimming pool?

(1) 38 of the houses in the community have a patio but do not have a swimming pool.

(2) The number of houses in the community that have a patio and a swimming pool is equal to the number of houses in the community that have neither a swimming pool nor a patio.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) 得知10戶有陽台和游泳池,但是不知道沒有陽台有游泳池的.
(2) 75-48 =28 (沒有陽台)
   28- (沒有陽台沒有游泳池) = 沒有陽台有游泳池的
   有陽台和游泳池的=沒有陽台沒有游泳池
   有游泳池的 = 沒有陽台有游泳池的+有陽台和游泳池的 =28- (沒有陽台沒有游泳池) +(沒有陽台沒有游泳池)=28
--------------------------------------------------------------------------------------------------------------------------------------------------------
193.        18993-!-item-!-187;#058&012050
The attendees at a certain convention purchased a total of 15,000 books.  How many of these attendees were female?

(1) There was a total of 4,000 attendees at the convention.

(2) The male attendees purchased an average (arithmetic mean) of 3 books each, and the female attendees purchased an average of 5 books each.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) Male有x人,Female有y人, x+y=4000
(2) 3x+5y=15000　
(1)+(2)   
3x+5y=15000, x+y=4000　兩式聯立解y, y=1500
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194.        19047-!-item-!-187;#058&012065
If x and y are positive integers, what is the value of x + y ?

(1)  2x3y = 72

(2)  2x2y = 32
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
(1) 72 = 23 x 32
(2) 32 = 25 = 2x2y
   x+y=5
   (x,y) = {(1,4) (2,3) (4,1) (3,2)}
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195.        19101-!-item-!-187;#058&012103
What is the value of h ?

(1) h2 = 36

(2) h2 + 12h = -36
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) h = -6 or 6
(2) h2 + 12h + 36 = (h+6) 2= 0  h=-6
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196.        19155-!-item-!-187;#058&012165
How many odd integers are greater than the integer x and less than the integer y ?

(1)  There are 12 even integers greater than x and less than y.

(2)  There are 24 integers greater than x and less than y.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) 不知道有幾個數是介於x,y. 有可能奇數比偶數多1或少1
(2) 有一半是奇數
--------------------------------------------------------------------------------------------------------------------------------------------------------
197.        19209-!-item-!-187;#058&012193
Jason's salary and Karen's salary were each p percent greater in 1998 than in 1995.  What is the value of p ?

(1)  In 1995 Karen's salary was $2,000 greater than Jason's.

(2)  In 1998 Karen's salary was $2,440 greater than Jason's.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) K-J=2000 不能得知p
(2) (1+p)*K-(1+p)*J = 2440
   (1+p)(K-J) = 2440
(1)+(2)
(1+p)(K-J) – (K-J) = (K-J)p =440
p = 440/2000 = 22%
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198.        19263-!-item-!-187;#058&012213
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

(1)  The 3 members sold a total of 6 raffle tickets yesterday.

(2)  No 2 of the members sold the same number of raffle tickets yesterday.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
(1) 設3人賣的數量為a,b,c
(a,b,c)={(6,0,0) (1,1,4) (1,2,3) (2,2,2)…}
(2) (a,b,c)={(0,1,2) (1,2,3),(3,4,5)…}　　
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199.        19366-!-item-!-187;#058&012225
If s and t are two different numbers on the number line, is s + t equal to 0 ?

(1)  The distance between s and 0 is the same as the distance between t and 0.

(2)  0 is between s and t.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】
(1) 兩點不重疊且與0的距離相等,這兩點在0的左右邊
(2) 如果與0的距離不相等,相加不等於0
--------------------------------------------------------------------------------------------------------------------------------------------------------
200.        19420-!-item-!-187;#058&012241
Of the 1,000 companies responding to a certain survey, what percent indicated that they had a business recovery plan?

(1) 200 of the companies did not indicate that they had a business recovery plan.

(2) The number of companies that indicated that they had a business recovery plan was 4 times the number that did not indicate that they had a business recovery plan.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
(1) 1000-200 = 800, 800/1000=80%
(2) 1000*(4/5) = 800, 800/1000=80%
--------------------------------------------------------------------------------------------------------------------------------------------------------
201.        19474-!-item-!-187;#058&012274
Martha bought an armchair and a coffee table at an auction and sold both items at her store.  Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?

(1)  Martha paid 10 percent more for the armchair than for the coffee table.

(2)  Martha sold the armchair for 20 percent more than she sold the coffee table.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E
【思路】
gross profit=sales-cost

(1) Martha paid 10 percent more for the armchair than for the coffee table.
設coffee table' cost=c
則armchair's cost=1.1c

(2) Martha sold the armchair for 20 percent more than she sold the coffee table.
設coffee table' price=s
則armchair's price=1.2s

題目問
兩個未知數，故insufficient
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202.        19528-!-item-!-187;#058&012279
Each employee of Company Z is an employee of either Division X or Division Y, but not both.  If each division has some part-time employees, is the ratio of the number of full-time employees to the number of part-time employees greater for Division X than for Company Z ?

(1) The ratio of the number of full-time employees to the number of part-time employees is less for Division Y than for Company Z.

(2) More than half of the full-time employees of Company Z are employees of Division X, and more than half of the part-time employees of Company Z are employees of Division Y.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
以下是eced大大的有算式驗證法: (推薦)
F(x)/P(x) > F(z)/P(z) ? F(z)=F(x)+F(y) ; P(z)=P(x)+P(y)
(1) F(y)/P(y) < F(z)/P(z)
=>[ F(z)-F(x)] / [ P(z)-P(x)] < F(z)/P(z)
=> P(z)F(z)-P(z)F(x) < F(z)P(z)-F(z)P(x)
=>F(x)/P(x) > F(z)/P(z)
sufficient

(2) F(x) > F(y); P(y) > P(x)
F(x)/P(x) > F(y)/P(y)=[F(z)-F(x)]/[P(z)-P(x)]
F(x)P(z)-F(x)P(x) > P(x)F(z)-P(x)F(x)
F(x)/P(x) > F(z)/P(z)
sufficient

以下是我的 two cents:
(1) X部門和Y部門相加之後的full-time/part-time比是公司整體的full-time/part-time比例.
    把公司整體的full-time/part-time比看作是1的話,不難看出以下的關悉
(2) 這句話就是說X部門有高於公司整體的full-time/part-time比例. 由(1)的公式,把公司整體的full-time/part-time比例看作1,再代入假設的數字驗算一下. ex, 1/1<0.6/0.4.
--------------------------------------------------------------------------------------------------------------------------------------------------------

203.        19582-!-item-!-187;#058&012370
What is the remainder when the positive integer x is divided by 6?

(1)  When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0.

(2)  When x is divided by 12, the remainder is 3.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
(1) x = 3(2n+1) = 6n+3
(2) x = 12m+3 = 3(4m)+3 = 6k+3
--------------------------------------------------------------------------------------------------------------------------------------------------------
204.        19682-!-item-!-187;#058&012407
On a certain sight-seeing tour, the ratio of the number of women to the number of children was 5 to 2.  What was the number of men on the sight-seeing tour?

(1)  On the sight-seeing tour, the ratio of the number of children to the number of men was 5 to 11.

(2)  The number of women on the sight-seeing tour was less than 30.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) C:M=5:11 問有多少男的,題目條件都沒給數字
(2) W<30 沒給women和man的比例
(1)+(2)
  &   
女男的比例,女的少於30個只可能是25,男的22個.
--------------------------------------------------------------------------------------------------------------------------------------------------------
205.        19736-!-item-!-187;#058&012415
Ellen can purchase a certain computer at a local store at the price of p dollars and pay a 6 percent sales tax.  Alternatively, Ellen can purchase the same computer from a catalog for a total of q dollars, including all taxes and shipping costs.  Will it cost more for Ellen to purchase the computer from the local store than from the catalog?

(1) q - p < 50

(2) q = 1,150
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) 不知道p的6%是否大於50
(2) 沒有相對於p的條件
(1)+(2) (1150-50)x(6/100)=66 (銷售稅大於差價)
[另解]
C1 = p + 0.06p, C2 = q
(1) q - p < 50 not sufficient
(2) p = 1,150 --> C1 = 1,219 not sufficient
(1) + (2) C2 - C1 = q - ( p + 0.06p) = q - p - 0.06 x 1,219 = (<50) - 73.14 < 0
so it costs more for Ellen to purchase the computer from the local store than from the catalog
Ans C
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206. 19977-!-item-!-187;#058&012599
If p is a positive integer, what is the value of p ?
(1) p/4 is a prime number
(2) p is divisible by 3.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
(1) p=4,12….. insufficient
(2) p=3,6,9,12….. insufficient
故，(1)+(2) p = 12 sufficient --------------------------------------------------------------------------------------------------------------------------------------------------------
207. 20123-!-item-!-187;#058&012633
Is y < 2x ?
(1)   
(2)  
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
(1) 2y<4x - sufficient
(2) y-2x<0 - sufficient
(1)+(2) - NO!
--------------------------------------------------------------------------------------------------------------------------------------------------------
208. 20177-!-item-!-187;#058&012683
Is x > y ?
(1) x + y < 0
(2) x - y > 0
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) x<-y - insufficient
(2) x>y - sufficient
(1)+(2) - NO!
--------------------------------------------------------------------------------------------------------------------------------------------------------
209. 20231-!-item-!-187;#058&012732
For each month of next year, Company R's monthly revenue target is x dollars greater than its monthly revenue target for the preceding month. What is Company R's revenue target for March of next year?
(1) Company R's revenue target for December of next year is $310,000.
(2) Company R's revenue target for September of next year is $30,000 greater than its revenue target for June of next year.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】if its monthly revenue target for the preceding month is Y, then monthly revenue target is X+Y
(1) 明年12月營業額目標310,000
(2) 給出6-9月增加數字
(1)+(2) 明年三月營業額目標=310,000-(30,000/3)*9
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210. 20285-!-item-!-187;#058&012741
Is m + z > 0 ?
(1) m - 3z > 0
(2) 4z - m > 0
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】
遇到這種不知怎解最好辦法就是在座標軸上劃出他們的線圖和所在區塊來處理
單純看(1)或(2)都知不能解答
只有(1)+(2)來討論
兩者相加 z>0
(1) m>3z
(2)4z>m
=> 4z>m>3z
==>5z>m+z> 4z
因為z>0 所以m+z>0
--------------------------------------------------------------------------------------------------------------------------------------------------------
211. 20437-!-item-!-187;#058&012817
If k is a positive integer, then 20k is divisible by how many different positive integers?
(1) k is prime.
(2) k = 7
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) 20*2=40 - 8 factors / 20*3=60 - 12 factors - insufficient
(2) 20*7=140 - 12 factors - sufficient
(1)+(2) - NO!
--------------------------------------------------------------------------------------------------------------------------------------------------------
212. 20491-!-item-!-187;#058&012823
This morning, a certain sugar container was full. Since then some of the sugar from this container was used to make cookies. If no other sugar was removed from or added to the container, by what percent did the amount of sugar in the container decrease?
(1) The amount of sugar in the container after making the cookies would need to be increased by 30 percent to fill the container.
(2) Six cups of sugar from the container were used to make the cookies.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A
【思路】Full - 100%
(1)+30% → 100%, 70% is the answer - sufficient
(2) Six cups of sugar are quantity, not percentage. - insufficient
(1)+(2) - NO!
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213. 20735-!-item-!-187;#058&013187
If a and b are positive numbers, what are the coordinates of the midpoint of line segment CD in the xy-plane?
(1) The coordinates of C are (a, 1 - b).
(2) The coordinates of D are (1 - a, b).
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C
【思路】ask the coordinates of the midpoint of line segment CD - use intuition
求兩點的中點座標，關鍵是知道兩點的座標，無所謂這兩點的座標具體是啥，只要有兩點座標就能求出來，其他條件都屬於擾亂思維的，我覺得這個是ets的花招，好多題寫的老長老長的，其實都是廢話，你用數學語言總結出來就是一個簡單的公式，中點就是（1/2,1/2），即兩點的橫縱坐標和分別除以2.所以選擇C

--------------------------------------------------------------------------------------------------------------------------------------------------------
214. 20789-!-item-!-187;#058&013258
The cost of a square slab is proportional to its thickness and also proportional to the square of its length. What is the cost of a square slab that is 3 meters long and 0.1 meter thick?
(1) The cost of a square slab that is 2 meters long and 0.2 meter thick is $160 more than the cost of a square slab that is 2 meters long and 0.1 meter thick. 4y2x 32xy
(2) The cost of a square slab that is 3 meters long and 0.1 meter thick is $200 more than the cost of a square slab that is 2 meters long and 0.1 meter thick.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D
【思路】
根據這題意 :
因為是square, 所以long and width是一樣的.
1. a square slab is proportional to its thickness
假設thickness = T => T * k1
2. also proportional to the square of its length
假設length = L => L * k2
成本C = T * k1 * L^2 * k2^2
=> C = T * L^2 * (k1*k2^2) 由於不管length or thickness怎樣變動, 常數(constant)都不會變, 因為他是成比例的, 因此簡化為 (k1*k2^2) = K
=> T * L^2 * K

C=cost, k=constant, T=thickness, L =length
C= k * T * L^2

so when T=0.1, L=3, C=?
(we need to know the value of k)

(1) The cost of a square slab that is 2 meters long and 0.2 meter thick
--> C= k * 0.2 * 2^2
the cost of a square slab that is 2 meters long and 0.1 meter thick
--> C= k * 0.1 * 2^2
difference of two costs=160 (given)
--> (k * 0.2 * 2^2) - (k * 0.1 * 2^2) =160
so you can get the value of k=400 ---> sufficient
(2) The cost of a square slab that is 3 meters long and 0.1 meter thick
--> C= k * 0.1 * 3^2
the cost of a square slab that is 2 meters long and 0.1 meter thick
--> C= k * 0.1 * 2^2
difference of two costs=200 (given)
--> (k * 0.1 * 3^2) - (k * 0.1 * 2^2) =200
so you can get the value of k=400 ---> sufficient
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215. 20843-!-item-!-187;#058&013306
If Bob produces 36 or fewer items in a week, he is paid x dollars per item. If Bob produces more than 36 items in a week, he is paid x dollars per item for the first 36 items and 1.5 times that amount for each additional item. How many items did Bob produce last week?
(1) Last week Bob was paid a total of $480 for the items that he produced that week.
(2) This week Bob produced 2 items more than last week and was paid a total of $510 for the items that he produced this week.
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E
【思路】
此題需要把低於36個 或多於36個想進去

(1) Last week Bob was paid a total of $480 for the items that he produced that week.
設last week =y item
if y<36 -->xy=480
if y>36--> 36x+1.5x(y-36)=480

(2) This week Bob produced 2 items more than last week and was paid a total of $510 for the items that he produced this week.
This week =y+2
If y+2<36--> x(y+2)=510
if y+2>36--> 36x+1.5x(y-34)=510

(1)+(2)
If y<36 combine (1)(2) x=15 y=32
If y>36 combine (1)(2) x=10 y=44
--------------------------------------------------------------------------------------------------------------------------------------------------------
216. 20897-!-item-!-187;#058&013307
Is x < y ?
(1) 2x < 3y
(2) xy > 0
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E
【思路】intuition
(1) insufficient
(2) insufficient
(1)+(2) - NO!
--------------------------------------------------------------------------------------------------------------------------------------------------------
217. 21093-!-item-!-187;#058&013542
a1 , a2 , a3 , . . . , a15 In the sequence shown, an = an - 1 + k, where   and k is a nonzero constant. How many of the terms in the sequence are greater than 10 ?
(1) a1 = 24
(2) a8 = 10
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】In the sequence shown, an = an - 1 + k, where 2<=n<=15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10 ?

(1) a1 = 24
a2 = a1 + k => a2 = 24 + k
a3 = a2 + k => a3 = 24 + 2k
a4 = a3 + k => a4 = 24 + 3k
不知到k到底是positive or negative 也不知到 k的大小, 通通會影響>10的個數 insufficient
比方說若k=-15, 那麼全都<10
(2) a8 = 10
a7 = a6 + k => a6 = 10 - 2k
a8 = a7 + k => a7 = 10 - k
a9 = a8 + k => a9 = 10 + k
a10 = a9 + k => a10 = 10 + 2k
a11 = a10 + k => a11 = 10 + 3k
由於已知的數據是10, k不管大小, 都會影響10變成大於10或小於10, 可以推論
由此可見假若k>0, 那麼從a9 ~ a15有七個大於10 ; 反之, 假若k<0, a8 ~ a2有七個大於10;充分
--------------------------------------------------------------------------------------------------------------------------------------------------------
218. 21147-!-item-!-187;#058&013552
Is 4z > -6 ?
(1) z < 7
(2) z > -1
--------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B
【思路】
(1) insufficient
(2) sufficient
(1)+(2) - NO!
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