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John invests $x at the semi-annual constant compounded rate of 2 perce

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John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 13 May 2016, 05:07
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John invests $x at the semi-annual constant compounded rate of 2 percent and also does $10,000 at the quarterly constant compounded rate of 4 percent. If the interests are the same after 1 year, what is the value of x??
A. $20,001
B. $20,101
C. $20,201
D. $20,301
E. $20,401

* A solution will be posted in two days.
[Reveal] Spoiler: OA

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Re: John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 15 May 2016, 21:13
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From x(1+2%/2)^2-x=10000(1+4%/4)^4-10000, we get x(1.1^2-1)=10000(1.1^4-1)=10000(1.1^2-1)(1.1^2+1). (1.1^2-1) can be canceled out in either sides. So we get x=10000(1.1^2+1)=20,201. Hence, the correct answer is C.
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Re: John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 15 May 2016, 23:31
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A = P(1+r/n)^nt

A= total amount accrued
P = principal deposited
r = rate of interest in decimal form
n = number of times per year, interest compounded
t = time in number of years.
.
x(1+0.02/2)^2 - x = 10,000(1+0.04/4)^4 - 10,000 [ when the principal is subtracted from the total amount accrued , the resulting difference is the interest portion and question states interests are equal)
=> x[(1.01)^2 - 1] = 10,000[(1.01)^4 - 1]
=> x[(1.01)^2 - 1] = 10,000[(1.01)^2+1][(1.01)^2-1] --> Using a^2-b^2 = a+b X a-b formula and cancel common expression on both sides
=> x = 10,000(1.0201+1) = 20,201.
.
Hence answer is C.

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John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 16 May 2016, 02:53
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Maybe I was just lucky on this one but here's how I solved it in 2,25 minutes without complicated math:

This will come down to decimals:

Second investment
4% per annum = 1% per quarter
$10000 * 101% = $10100
$10100 * 101% = $10201
$10201 * 101% = $10303.01
$10303.01 * 101% = $10306.0401

First investment
2% per annum = 1% per 6 month period
So the same pattern as above wil repeat.
To get a decimal of .0401 after 2 periods we need 2 in the hundreds digit.
Only (C) satisfies this constraint.

(This only works because both were compounded at the same rate - 1% per period)

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Re: John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 30 Jan 2017, 23:01
MathRevolution wrote:
John invests $x at the semi-annual constant compounded rate of 2 percent and also does $10,000 at the quarterly constant compounded rate of 4 percent. If the interests are the same after 1 year, what is the value of x??
A. $20,001
B. $20,101
C. $20,201
D. $20,301
E. $20,401

* A solution will be posted in two days.



May I am probably the only one who didnt understand the question right. I interpreted that 2% is the semi-nannual interest compounded twice each year. This way we will have final amount of x(1.02)^2 by the end of one year. However, I see that the correct solution involves x(1+(.02/2))^2. This approach tells me that the annual rate was 2% which was compounded semi-annually.

How likely is that the interpretation can be first one. I know it is the words, but is it the latter way always?

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Re: John invests $x at the semi-annual constant compounded rate of 2 perce [#permalink]

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New post 27 Aug 2017, 01:41
4% compounding quarterly, i.e. 1 % per quarter
\(Q_1\) :1% of 10000 = 100
\(Q_2\) :1% of 10000 + 1% of 100 (interest for\(Q_1\)) = 100 + 1=101
\(Q_3\) :1% of 10000 + 1% of 100 (interest for\(Q_1\)) + 1% of 101 (interest for \(Q_2\)) = 100 + 1 + 1.01 = 102.01
\(Q_4\) :1% of 10000 + 1% of 100 (interest for\(Q_1\)) + 1% of 101 (interest for \(Q_2\)) + 1% of 102.01 (interest for \(Q_3\)) = 100 + 1 + 1.01 + 1.0201 = 103.0301
Total interest = 100 + 101 + 102.01 + 103.0301 = 406.0401

Now using option
Option A: 20001 at 2% semiannually
1%of 20001= 200.01
1%of 20001 + 1%of 200.01 = 200.01 + 2.0001 = 202.0101
Total interest = 200.01 + 202.0101 = 402.0201 which is not equal to 406.0401

Option B: 20101
Using the same pattern total interest is 201.01 + 201.01 + 2.0101 which is not equal to 406.0401

Option C: 20201
Using the same pattern total interest is 202.01 + 202.01 + 2.0201 which is equal to 406.0401

Thus answer is Option C
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Re: John invests $x at the semi-annual constant compounded rate of 2 perce   [#permalink] 27 Aug 2017, 01:41
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