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Carl invested a certain amount of money for one year, some of which ea

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yrozenblum
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Carl invested a certain amount of money for one year, some of which ea [#permalink] New post   06 Nov 2023, 09:43
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Carl invested a certain amount of money for one year, some of which earned interest at a simple annual interest rate of 2.2 percent and the rest of which earned interest at a simple annual rate of 2.5 percent. If the amount of money that Carl invested for the year earnead a total of $680 in interest, which of the following could have been the total amount of money that he invested for the year?

I. $24,708
II. $27,896
III. $31,284

A. None
B. II only
C. III only
D. I and II
E. II and III­


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B
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Re: Carl invested a certain amount of money for one year, some of which ea [#permalink] New post   06 Nov 2023, 10:15
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yrozenblum wrote:
Carl invested a certain amount of money for one year, some of which earned interest at a simple annual interest rate of 2.2 percent and the rest of which earned interest at a simple annual rate of 2.5 percent. If the amount of money that Carl invested for the year earnead a total of $680 in interest, which of the following could have been the total amount of money that he invested for the year?

I. $24,708
II. $27,896
III. $31,284

A. None
B. II only
C. III only
D. I and II
E. II and III


The minimum principle value is when all the amount is invested at a 2.5% rate.

\(\frac{p_{\text{min}} * 2.5 * 1}{100} = 680\)

\(p_{\text{min}} = 680*\frac{100}{2.5} = 27200\)

The maximum principle value is when all the amount is invested at a 2.2% rate.

\(\frac{p_{\text{max}} * 2.2 * 1}{100} = 680\)

\(p_{\text{max}} = 680*\frac{100}{2.2} \approx 30900\)

Only one value lies in this range i.e. $27,896

Option B
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Re: Carl invested a certain amount of money for one year, some of which ea [#permalink] New post   06 Nov 2023, 10:09
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1) 24708*2.5%=617.7 : thus not possible to reach 680

2) 27896 *2.5 =697.4 and 27896*2.2 = 613.7 . Since required value is in between, possible.

3) 31284 * 2.2 = 688.2. Thus both value would be above 680, hence value of 680 not possible.

Hence answer is B.
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Re: Carl invested a certain amount of money for one year, some of which ea [#permalink] New post   09 Apr 2024, 01:55
Let’s have A total amount invested
X amount of A at a 2.2 rate and A-X at 2.5 rate

Interest 1 = X*2.2/100 and interest 2 = (A-X)*2.5/100

Total interest earned = X*2.2/100 + (A-X)*2.5/100 =680

—->> 22X + 25(A-X) =680.000
25 A -3X = 680.000

—->> A = [680.000+3X]/25
——>> A= 27.200 + 3X/25

We noticed that A should be greater than 27.200
We have to consider B and C
But knowing that, X <<<A let’s verify the worst case for X=27200
3*27200/25 = 3264

27200+3264= 30464 == max value

C can’t be our answer

Hence B

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Re: Carl invested a certain amount of money for one year, some of which ea [#permalink]
09 Apr 2024, 01:55
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