[原创]2006年6月Math机经讨论稿第4篇(57-78) 6.16日更新

洛拉

Note: The interpretation here isn't quite correct.  You can skip this and go straight to the 22nd floor for the final solution if you'd like.

以下是引用yaoyao99在2006-6-16 11:06:00的发言：

Ahhh.. thank you 洛拉!  That makes much more sense.

So when x < 1000, interest is about 0.02x and when x > 1000, interest is about 0.02 * 1000 - 0.025 * (x-1000) = 0.025x - 5

1) x = 2000 because 0.025 * (x-1000) = 25 => interest from 2% = 20 less than interest from 2.5%

2) when x < 1000, there's only interest from 2%

when x > 1000, interst is about 0.02*3*1000 from 2% and 0.025*3*(x-1000) from 2.5%

interest from 2% = 60 whereas interest from 2.5% = 0.075x - 75 (inconclusive without the value of x)

Answer is A?

这道题我算完觉得如果2%时利息是25元，那么选C，如果是65元，就得选E

因为过程实在太长，我只写个大概意思，分情况讨论，不是复利息嘛，就分成第一种情况：x *（1+2%）只出现了1年（由25元得出一个本金1，代入蓝色的式子里），后面2年因为都超过了1000元所以用2.5%的利率，即{[x* （1+2%）] * （1+2.5%) ^2} - [x *(1+2)]），第二种情况：x *（1+2%）出现了2年（由25元得出本金2代入红色的式子里），只有最后1年是2.5%的利率, 即{[x *（1+2%）] ^2} * (1+2.5%) -  [x *（1+2%）] ^2), 得出两种情况的利息看看是否都一致地大于或者小于25元，如果一致则选C，如果不一致，就选E无法判定。如果是按天计算复利息也同理分成类似上述两种情况就行了。

版主看看对不对？

yaoyao99

, I see what you mean now.  What about this:

2) Interest from year 1 = x * 0.02 (assume x < 1000), let the total $ in the bank at the end of year 2 = x2 and the total at the end of year 3 = x3

	year 2	total interest at 2nd year end	year 3	total interest at 3rd year end	total interest from 2%	total interest from 2.5%
1	x2 < 1000	x * 1.02 * 1.02 – x	x3 > 1000	x * 1.02 * 1.02 * 1.025 - x	0.04x	0.066x – 0.04x = 0.026x < 0.04x
2	x2 > 1000	x * 1.02 * 1.025 - x	x3 > 1000	x * 1.02 * 1.025 * 1.025 - x	0.02x	0.071x – 0.02x = 0.051x > 0.04x

Now let's look at condition 1)

If interest from 2% is $65, then both case 1 and 2 are invalid (x > 1000).  => invalid condition?

If
   interest from 2.5% is $25, then it's also an invalid condition:

case 1: 0.026x = 25 => x = $960 (invalid) => x3 < 1000 and case 2: 0.051x = 25 => x =  $490 (invalid) => x3 < 1000

If interest from 2% is $25, then:

case 1: 0.04x = 25 => x = $625 (valid) => we know that interest from 2.5% < interest from 2%

case 2: 0.02x = 25 => x =  $1250 (invalid, we assumed x, the principal, to be less than 1000 at the beginning of the first year) 

answer C (only interest from 2% is $25 is a valid condition)

Note, (1+n%)^m 约等于 (1+n%*m), based on Taylor's Expansion (for (1 + alpha)^x, when alpha is reasonably small, the value is approximately 1 + alpha*x).  This should save you some time on the test even if it says interest compounded daily/month/whatever.  In case the numbers are different, you can calculate the interest as compounded annually even if it says compounded several times during the year.

Simple interest: interest = principal * annual interest rate * num of years

Interst compounded annually: interest = principal * (1 + annual interest rate) ^ number of years - principal

Interst compounded several times during the year:

interest = principal * (1 + annual interest rate / num of times interest is compounded) ^ (num of times interest is compounded * num of years) - principal

[此贴子已经被作者于2006-6-16 22:18:16编辑过]

洛拉

Note: The final solution is summarized in the 22nd floor. 

以下是引用yaoyao99在2006-6-16 12:30:00的发言：

, I see what you mean now.  What about this:

2) Interest from year 1 = x * 0.02 (assume x < 1000)

case	year 2	total interest at year end	year 3	total interest at year end
1	x2 < 1000	x* 1.02*1.02 – x	x3 > 1000	x* 1.02*1.02 * 1.025 – x
2	x2 > 1000	x *1.02 * 1.025 - x	x3 > 1000	x * 1.02 * 1.025 * 1.025 – x

 

At the end of year 3

Case 1: interest from 2% is about 0.04x and interest from 2.5% is about:

0.066x – 0.04x = 0.026x < 0.04x

Case 2: interest from 2% is about 0.02x and interest from 2.5% is about:

0.071x – 0.02x = 0.051x > 0.02x

 

Now back to 1)

If interest from 2% is $25, then

case 1: 0.04x = 25 => x = $625 (valid) => interest from 2.5% < interest from 2%

case 2: 0.02x = 25 => x =  $1250 (invalid, we assumed the principal to be less than 1000)

 

C

版主到底是版主啊！能把乱七八糟的一片列成这么清晰的表！

我再写清楚点儿：设一开始的本金是x

第一种情况：2%的利率只用了1年，x*( 1+2%)的积就已经大于1000元了，此时， x*2%=25， x=1250，于是，1250 * （1 + 2%）* （1 + 2.5%) ^2 - 1250 * ( 1+ 2%) 就等于用2.5% 时产生的利息啦！等于64.546875. 注意这个地方要不是减去最初的本金x，而是减去1250 * （1 + 2%），因为这是适用2.5%利率时的“新的本金”了嘛！我的意思也不知道说清楚没有？这道题出现2个复利息的利率，就相对应2个本金啊，一个是初始本金，一个是初始本金按2%利率计算以后因为超出1000元了就构成了“适用2.5%利率的基础，也就是新本金”，等于让你算，新本金在2.5%的利率情况下产生的利息多呢？还是原始本金在2%的利率情况下产生的利息25元多？

第二种情况：2%的利率用了2年，连本带利才超过1000元能用2.5%，此时 x * （1+2%）^2 - x = 25，x约等于618.8，于是，618.8 * (1 + 2%)^2 * ( 1+ 2.5%)  - 618.8 * (1 + 2%）^2 ，约等于16.094988

多谢多谢版主啊！我这次为了准确用计算器算，才发现自己之前手算有错误，真不好意思误导了你吧？一种情况大于25，一种情况小于25，所以如果题目是25，那也是E了（应是C了，1250那种情况无效）

唉，这种题，算错一点都白费

辛苦版主了！再看看对吗？

yaoyao99

Hee hee, I just fixed my answer in 第 22 楼 and added the various versions of condition 1.  What do you think?

Here's the table again, summarizing what you wrote:

	year 2	total interest at year end	year 3	total interest at year end	Interest from 2%	Interest from 2.5%
情况2	x2 < 1000	x* 1.02*1.02 – x	x3 > 1000	x* 1.02*1.02 * 1.025 - x	0.04x	0.066x – 0.04x = 0.026x
情况1	x2 > 1000	x *1.02 * 1.025 - x	x3 > 1000	x * 1.02 * 1.025 * 1.025 - x	0.02x	0.071x – 0.02x = 0.051x

Note: x=1250 isn't a valid answer because we assumed x < 1000

BTW, (1+n%)^m 约等于 (1+n%*m), based on Taylor's Expansion.  That should save you some time on the test, in case the numbers are different. 

[此贴子已经被作者于2006-6-16 14:00:15编辑过]

洛拉

Note: The final solution is summarized in the 22nd floor. 

以下是引用yaoyao99在2006-6-16 13:42:00的发言：

Hee hee, I just fixed my answer in 第 22 楼 and added the various versions of condition 1.  What do you think?

Here's the table again, summarizing what you wrote:

	year 2	total interest at year end	year 3	total interest at year end	Interest from 2%	Interest from 2.5%
情况2	x2 < 1000	x* 1.02*1.02 – x	x3 > 1000	x* 1.02*1.02 * 1.025 - x	0.04x	0.066x-0.04x = 0.026x
情况1	x2 > 1000	x *1.02 * 1.025  - x	x3 > 1000	x * 1.02 * 1.025 * 1.025 - x	0.02x	0.071x – 0.02x = 0.051x

Note: x=1250 isn't a valid answer because we assumed x < 1000

BTW, (1+n%)^m 约等于 (1+n%*m), based on Taylor's Expansion.  That should save you some time on the test, in case the numbers are different. 

版主，应该保留你的原表，我改动你的表时顾此失彼了。你列表的方法很值得学习！感谢帮助！！

洛拉

真希望考试时候不要遇到这种题！否则像我这种学文科的，计算能力又差，算不了几步就晕菜了！

yaoyao99

Note: The final solution is summarized in the 22nd floor. 

But x * 1.02 * 1.025 - x * 1.02 gives you the interest from the 2nd year ONLY.  I'm doing accummulated interest where the "total interest at 2nd year end" = interest from year 1 + interest from year 2 and "total interest at 3rd year end" = TOTAL interest from year 1 to 3

I don't understand how you got to the equations in the final column (interest from 2.5%).

Your equation x* 1.02*1.02 * 1.025 - x*(1.02)*(1.02) already gave you interest from 2.5% (interest from the 3rd year ONLY) which is the same as what I have 1.066x - 1.04x = 0.026x. 

Similarly, your equation  x * 1.02 * 1.025 * 1.025 - x*(1.02) gave you interest from 2.5% in 情况1, which is the same as what I have as well, 1.071x - 1.02x = 0.051x.

Anyway, I think the answer is C because only 第二种情况 is valid so we know interest from 2% is greater than interest from 2.5%.  Assuming interest from 2% is $25 and we're calculating interests from a 3 year period. 

[此贴子已经被作者于2006-6-17 22:29:06编辑过]

洛拉

Note: The final solution is summarized in the 22nd floor. 

以下是引用yaoyao99在2006-6-16 14:35:00的发言：

But x * 1.02 * 1.025 - x * 1.02 gives you the interest from the 2nd year ONLY.  I'm doing accummulated interest where the "total interest at 2nd year end" = interest from year 1 + interest from year 2 and "total interest at 3rd year end" = TOTAL interest from year 1 to 3

I don't understand how you got to the equations in the final column (interest from 2.5%).

Your equation x* 1.02*1.02 * 1.025 - x*(1.02)*(1.02) already gave you interest from 2.5% (interest from the 3rd year ONLY) which is the same as what I have 1.066x - 1.04x = 0.026x. 

Similarly, your equation  x * 1.02 * 1.025 * 1.025 - x*(1.02) gave you interest from 2.5% in 情况1, which is the same as what I have as well, 1.071x - 1.02x = 0.051x.

Anyway, I think the answer is C because only 第二种情况 is valid so we know interest from 2% is greater than interest from 2.5%.  Assuming interest from 2% is $25 and we're calculating interests from a 3 year period. 

感谢版主指出我的纰漏！讨论还是给我很大帮助的！

洛拉

我忽略了1250无效的情况，同意选C。

LilyJune

   70. 告诉一点（4，5）问它关于X轴对称点的坐标，注意reflection.一开始我以为是映射呢结果没答案，结果是对称点
答案：（4，-5）

[討論]這是以x轴為中線對摺後得的結果.

映射是虾米东东？应该以前高中学过，但是过了太久了，偶只是听过完全没有概念了。。。

谢谢~