Shan has Rs. 1218 with him. He divided it amongst his sons Piyush and Manoj and asked them to invest it at 10% rate of interest compounded annually. It was seen that Piyush and Manoj got same amount after 18 and 19 years respectively. How much (in Rs. ) did Shan give to Manoj?

Correct option is D

Given:

Total amount Shan has: ₹1218

Piyush invests for 18 years, and Manoj invests for 19 years at a 10% annual compound interest rate.

Both Piyush and Manoj end up with the same amount after their respective periods of investment.

Formula Used:
The formula for compound interest is:

​$$A = P \left( 1 + \frac{r}{100} \right)^t$$​​

Where:

A is the amount after t years,

P is the principal (initial amount),

r is the rate of interest 

t is the time in years.

Solution:

Let the amount given to Piyush be $$P_1$$ and the amount given to Manoj be $$P_2$$​. Therefore, 

​$$P_1 + P_2 = 1218$$​​

For Piyush:

​$$A_1 = P_1 \left( 1 + \frac{10}{100} \right)^{18} = P_1 \times 1.1^{18}$$​​

For Manoj:

​$$A_2 = P_2 \left( 1 + \frac{10}{100} \right)^{19} = P_2 \times 1.1^{19}$$​​

Given that $$A_1 = A_2:$$​​

​$$P_1 \times 1.1^{18} = P_2 \times 1.1^{19}$$​​

​$$P_1 = 1.1 \times P_2$$​​

Substituting $$P_1 = 1.1 \times P_2$$​ into $$P_1 + P_2 = 1218$$​​

​$$1.1 \times P_2 + P_2 = 1218$$​​

​$$P_2 = \frac{1218}{2.1} = 580$$​​