Question

A sum of Rs 16896 is to be divided between Reena and Meena who are respectively 18 and 19 years old, in such a way that if their shares be invested at 6.25% per annum at compound interest, they will receive equal amounts on attaining the age of 21 years. The present share (in Rs) of Reena is:

• A. 4096
• B. 8921
• C. 8192
• D. 2040

Hint:

When two shares are invested at the same compound interest rate $r$ for time periods differing by 1 year, the ratio of the share of the younger person to that of the older person is always: \[ \frac{\text{Younger Share}}{\text{Older Share}} = \frac{1}{1 + \frac{r}{100}} \] Since the rate is $6.25\% = \frac{1}{16}$, the ratio is directly $\frac{16}{17}$.
This reduces calculation time.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem is based on compound interest and the division of a sum between two individuals such that their future values after different periods of time are equal.
Reena is 18 years old, and Meena is 19 years old.
Both will receive their respective accumulated amounts when they reach 21 years of age.
This means Reena's share is invested for $21 - 18 = 3$ years, while Meena's share is invested for $21 - 19 = 2$ years.

Step 2: Key Formula or Approach: The future value under compound interest is given by: \[ A = P \left(1 + \frac{R}{100}\right)^T \] Let Reena's present share be $R_p$ and Meena's present share be $M_p$.
The total sum is: \[ R_p + M_p = 16896 \] Since the final amounts they receive at age 21 are equal: \[ R_p \left(1 + \frac{r}{100}\right)^3 = M_p \left(1 + \frac{r}{100}\right)^2 \]

Step 3: Detailed Explanation:

• Simplify the Equal Amount Equation: Divide both sides by $\left(1 + \frac{r}{100}\right)^2$: \[ R_p \left(1 + \frac{r}{100}\right) = M_p \] \[ \frac{R_p}{M_p} = \frac{1}{1 + \frac{r}{100}} \]

• Substitute the Interest Rate ($r$): Given, $r = 6.25\% = 6\frac{1}{4}\% = \frac{25}{4}\%$. Therefore: \[ \frac{r}{100} = \frac{25}{400} = \frac{1}{16} \] Substitute this into the ratio equation: \[ \frac{R_p}{M_p} = \frac{1}{1 + \frac{1}{16}} = \frac{1}{\frac{17}{16}} = \frac{16}{17} \] So, the ratio of Reena's share to Meena's share is $16 : 17$.

• Divide the Total Sum (Rs 16896) in the Ratio $16 : 17$: The sum of the ratio parts is: \[ 16 + 17 = 33 \] Therefore, Reena's present share is: \[ R_p = \frac{16}{33} \times 16896 \] Let us divide 16896 by 33: \[ 16896 \div 3 = 5632 \] Now, divide 5632 by 11: \[ 5632 \div 11 = 512 \] So: \[ R_p = 16 \times 512 = 8192 \] Thus, Reena's present share is Rs 8192.

Step 4: Final Answer:
Reena's present share is Rs 8192.
Therefore, the correct option is (C).