Question

A man has part of ₹$4500$invested at $4\% $ and the rest at $6\% $. If the annual return on each investment is same, the average rate of interest which he realizes on the ₹ $4500$ is
(A) $5\% $
(B) $4.8\% $
(C) $5.2\% $
(D) $4.6\% $
(E) None of these.

Answer 1

Hint: Assume one part to be $y$ and find the second part in terms of $y$. Then calculate the annual return on each part using the information about the rate of interest given in the question. Use that to find the total interest. This information about total interest will then be used to find the average rate of interest.

Complete Step by Step Solution:
It is given that a man invests ₹ 4500 in two parts.
Let the amount invested at $4\% $ be $y$
Total amount$ = 4500$
$\therefore $Amount invested at $6\% $ is $(4500 - y)$
Annual return on each investment is the same.
The annual return can be calculated by using the formula of simple interest as
$S.I. = \dfrac{{PRT}}{{100}}$
Where,
$S.I.$ is the simple interest
$P$ is the principle amount
$R$ is the rate of interest
$T$ is the time period in years.
Therefore, we can calculate the simple interest for both the investments for a year as
$S.I. = \dfrac{{4y}}{{100}}$ and $S.I. = \dfrac{{6(4500 - y)}}{{100}}$
Now, it is given to us that the return on investment is the same. Therefore, we can write
$\dfrac{4}{{100}}y = \dfrac{6}{{100}}(4500 - y)$
By cancelling the common terms, we get
$400y = 600(4500 - y)$
Opening the brackets, we get
$400y = 2700000 - 600y$
Rearranging it we can write
$1000y = 2700000$
$ \Rightarrow y = \dfrac{{2700000}}{{1000}}$
$ \Rightarrow y = 2700$
$\therefore $The amount invested at $4\% $ is ₹ $2700.$
$\therefore $The amount invested at $6\% $ is $[4500 - 2700] = $ ₹ $1800$
$\therefore $Total interest is
$S.I{._T} = \dfrac{4}{{100}}(2700) + \dfrac{6}{{100}}(1800)$
$ = 4 \times 27 + 6 \times 18$
$ = 108 + 108 = 216.$
$\therefore $The total interest is $216$
Therefore, the average rate of interest the man is getting is,
$\dfrac{{216}}{{4500}} \times 100 = 4.8\% $

Hence, the average rate of interest the man is realizing on the investment of ₹ $4500$ is $4.8\% $

Note:
Do not just take the average of two given rates of interest to find the average rate of interest. That could work if the amount invested is the same. But in this question, the amount invested is different. So for this question, we need to calculate the amount of interest separately. Then add it to find the total interest earned and hence the average rate with which the interest is earned.