M37-08

Question

Quentin deposited an amount of money into each of two new investments, X and Y. Investment X pays $x\%$ simple annual interest, and investment Y pays $y\%$ simple annual interest. After 1 year, the total interest earned was $7.6\%$ of the total investment. If $x$ and $y$ are consecutive positive integers, in that order, what fraction of the total amount was deposited into investment X?

A. $\frac{1}{5}$
B. $\frac{1}{3}$
C. $\frac{2}{5}$
D. $\frac{3}{5}$
E. $\frac{2}{3}$

Show Answer

Official Answer

C

A.  \frac{1}{5} 
B.  \frac{1}{3} 
C.  \frac{2}{5} 
D.  \frac{3}{5} 
E.  \frac{2}{3}

Bunuel · Answer

Official Solution:

Quentin deposited an amount of money into each of two new investments, X and Y. Investment X pays $x\%$ simple annual interest, and investment Y pays $y\%$ simple annual interest. After 1 year, the total interest earned was $7.6\%$ of the total investment. If $x$ and $y$ are consecutive positive integers, in that order, what fraction of the total amount was deposited into investment X?

A. $\frac{1}{5}$
B. $\frac{1}{3}$
C. $\frac{2}{5}$
D. $\frac{3}{5}$
E. $\frac{2}{3}$

Show Answer

Official Answer

C

A.  \frac{1}{5} 
B.  \frac{1}{3} 
C.  \frac{2}{5} 
D.  \frac{3}{5} 
E.  \frac{2}{3}

Since  x  and  y  are consecutive positive integers, in that order, and the total interest earned was  7.6\%  of the total investment, then  x=7\%  and  y=8\% .
 
 \frac{7}{100}*a+\frac{8}{100}*b=\frac{7.6}{100}(a+b) , where  a  and  b  are amounts invested in X and Y, respectively.
 
 \frac{a}{b}=\frac{2}{3} ;
 
 \frac{a}{a+b}=\frac{2}{2+3}=\frac{2}{5} .

Answer: C

r_putra_rp · Answer

hi  Bunuel

Can you explain why did you assume that  x = 7\%  and  y = 8\%  given that their average is  7.6\%

The question gives us that the  7.6\%  is the average of the  interest  not the average of the  interest rate . If so, shouldn't it be a weighted average ie  \frac{(x * 7\%) + (y * 8\%)}{(x + y)} = 7.6\% ?
I mean, even if we take  \frac{(8+7)}{2} * 100\%  the result is quite significantly different;  7.5\%

please advise

Bunuel · Answer

Here's what the question implies:

$200 was invested in X and earned 7%, therefore $14.
$300 was invested in Y and earned 8%, therefore $24.

A total of $500 was invested and earned 7.6%, therefore $38. The average interest earned is (14 + 24)/500 = 0.076.

However, to avoid ambiguity, I revised the text. I hope it's clearer now. Thank you!

btheche · Answer

Hi Bunuel,

I apologise, but I have hard time to see how the statement implies that 200$ was invested in X and 300$ in Y according to your previous reply.

I tried to solve the problem algebragically but I did not managed. I spent quite a lot of time to understand the logic. There is an alternative aproach to solve this problem?

Bunuel · Answer

You can check alternative solutions here: https://gmatclub.com/forum/quentin-put- ... 44829.html Hope it helps.

btheche · Answer

Thank you for clarification. Now it make sense.

SaiJ1011 · Answer

I am unable to understand why x is 7% and y is 8%.
Can someone explain the weighted average method for this.
Thanks

Bunuel · Answer

Since x and y are consecutive positive integers and the overall average interest rate is 7.6%, the only pair that fits is 7% and 8%. That’s because whatever the percentages are, the weighted average must lie between them. So if the average is 7.6%, it must fall between 7% and 8%, meaning x = 7 and y = 8. For alternative approaches check this topic: https://gmatclub.com/forum/quentin-put- ... 44829.html Hope it helps.

Goldenfuture · Answer

Scale method - distance b/w 7 -------(0.6)......7.6...................0.4...... 8
7's weight is 4/10

modisint · Answer

I like the solution - it’s helpful. Another way to solve this using algebra

Let the amounts invested in X and Y be X and Y.

Since the total interest after 1 year is 7.6% of the total investment:

7.6(X+Y) = Xx + Yy
Since x and y are consecutive: 7.6X + 7.6Y = Xx + Y(x+1)
7.6X + 7.6Y = Xx + Yx + Y
7.6X − Xx = Yx + Y − 7.6Y
X(7.6−x) = Y(x−6.6)
Y/X = (7.6−x)/(x−6.6)

Since X/Y must be positive and x is a positive integer, both numerator and denominator must be positive:

x − 6.6 > 0 and 7.6 − x > 0
6.6 < x < 7.6

x=7

Y/X = (7.6 − 7)/ (7 − 6.6) = 0.6/0.4 = 3/2

X/(X+Y) = 2/5

M37-08

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