Question

A $\$500$ investment and a \[\$1500\] investment have a combined yearly return of 8.5 percent of the total of the two investments. If $\$500$ investment has a yearly return of 7 percent, what percent yearly return does the \[\$1500\] investment have?
(a) 9
(b) 10
(c) 11
(d) $11\dfrac{1}{9}$

Answer 1

Hint: Assume that the \[\$1500\] investment has a yearly return of ‘x’ percent. The combined yearly return of \[\$500\] investment and \[\$1500\] investment is the sum of individual yearly returns of \[\$500\] investment and \[\$1500\] investment. Therefore, the yearly return of \[\$2000\] = yearly return of \[\$500\] + yearly return of \[\$1500\]. So, use the relation: \[8.5\%\text{ of }\left( \text{ }$\text{1500+}$\text{500}\right)=7\%\text{ of }$\text{500}+x\%\text{ of }$\text{1500}\]. Solve this equation to find the value of ‘x’ which is our answer.

Complete step-by-step answer:
Let us assume that the \[\$1500\] investment has a yearly return of ‘x’ percent. Therefore, required yearly return of \[\$1500\]
$=x\%\text{ of }\left( \text{ }\!\!\$\text{1500}\right)$.
$=\dfrac{x}{100}\times \left( \text{ }\!\!\$\text{1500}\right)$.
It is given that the \[\$500\] investment and \[\$1500\] investment have a combined yearly return of 8.5 percent of the total of the two investments. Therefore, the combined yearly return of these investments
\[=8.5\%\text{ of }\left( \text{ }\!\!\$\text{1500+ }\!\!\$\text{500}\right)\].
\[=\dfrac{8.5}{100}\times \left( \text{ }\!\!\$\text{2000}\right)\].
Now, we have been provided that, \[\$500\] investment has a yearly return of 7 percent. Therefore, required yearly return of \[\$500\]
$=7\%\text{ of }\left( \text{ }\!\!\$\text{500}\right)$.
$=\dfrac{7}{100}\times \left( \text{ }\!\!\$\text{500}\right)$.
We know that the combined yearly return of \[\$500\] investment and \[\$1500\] investment is the sum of individual yearly returns of \[\$500\] investment and \[\$1500\] investment. So, using the relation: \[8.5\%\text{ of }\left( \text{ }\!\!\$\text{1500+}\!\!\$\text{500}\right)=7\%\text{ of }$\text{500}+x\%\text{ of }$\text{1500}\], we get,
\[\Rightarrow \dfrac{8.5}{100}\times (\$2000)=\dfrac{7}{100}\times(\text{ }\!\!\$\text{500)}+\dfrac{x}{100}\times(\text{ }\!\!\$\text{1500)}\].
\[\Rightarrow 8.5\times 20=\left( 7\times 5 \right)+\left( x\times 15 \right)\].
\[\Rightarrow 170=35+15x\].
\[\Rightarrow 15x=170-35\].
\[\Rightarrow 15x=135\].
\[\Rightarrow x=\dfrac{135}{15}\].
\[\Rightarrow x=9\].
So, the correct answer is “Option (a)”.

Note: One may note that there are two investments of different amounts, so we get different yearly returns for these amounts. We need to remember that if the amounts were equal then the percentage return would have been the same. In the above solution, we have taken the sum of individual returns of both the investments because both these investments contribute to overall yearly returns. We should not make the difference in place of the sum as you will get the wrong answer.