Question

A man invests a certain sum of money at 6% p.a. simple interest and another sum at 7% p.a. simple interest. His income from interest after 2 years was Rs.354. One-fourth of the first sum is equal to one-fifth of the second sum. The total sum invested was
A. Rs. 1500
B. Rs. 1200
C. Rs. 2700
D. Rs. 5400

Answer 1

Hint: We will use the formula of Simple Interest to solve this question. The formula to calculate SI is given by, \[SI=\dfrac{PRT}{100}\], where P is the principal at the starting, r is the rate of interest and t is the time duration.

Complete Step-by-Step solution:
Given, a man invests a certain sum of money at 6% p.a. simple interest and another sum at 7% p.a. simple interest. His income from interest after 2 years was Rs.354. One-fourth of the first sum is equal to one-fifth of the second sum. We have to find the total sum invested.
Let us assume the first sum to be equal to \[x\] and the second sum to be equal to \[y\], then according to the given conditions in the question we have, he invests a certain sum of money at 6% p.a. simple interest and another sum at 7% p.a. simple interest and his income from the interest in two years becomes equal to 354 rupees.
We have the formula of Simple interest as,
\[SI=\dfrac{PRT}{100}\], where P is the principal, r is the rate of interest and t is the time duration.
Here we have P in the first sum as x rate as 6% and time as two years which gives SI for the first sum as,
\[\dfrac{x(6)(2)}{100}\]
And similarly, for the second sum as,
 \[y(7)\dfrac{2}{100}\].
Adding both the interest becomes equal to Rs. 354.
Hence, adding both the above Simple interests we have,

\[\begin{align}
  & \dfrac{x(6)(2)}{100}+y(7)\dfrac{2}{100}=354 \\
 & \Rightarrow 12x+14y=35400 \\
 & \Rightarrow 6x+7y=17700.........(i) \\
\end{align}\]
 Now given that the one-fourth of the first sum is equal to the one-fifth of the second sum,
Then above implies,
 \[\dfrac{x}{4}=\dfrac{y}{5}\]
\[\begin{align}
  & \Rightarrow 5x=4y \\
 & \Rightarrow 5x-4y=0.......(ii) \\
\end{align}\]
Multiply equation (i) by 4
\[\Rightarrow 24x+28y=70800......(iii)\]
Multiply equation (ii) by 7
\[\Rightarrow 53x-28y=0.........(iv)\]
Add equation (iii) and (iv)
\[\Rightarrow 59x=70800\]
\[\Rightarrow x=1200\]
Put the value of \[x\] in equation (ii)
\[\begin{align}
  & \Rightarrow 5(1200)-4y=0 \\
 & \Rightarrow -4y=-6000 \\
 & \Rightarrow y=1500 \\
\end{align}\]
Total sum =x + y = \[1200+1500=2700rs\].
Hence, we obtain the total sum as Rs.2700.

Note: The possibility of error in this question is assuming the same variable for the different given sum which is wrong because assuming the same variable means that both the sum are equal which is not the case here.