A man divides ₹16850 between his two sons in such a way that after 20 years both of his sons receive the same money. If the ages of the sons are 12 years and 16 years, and the rate... A man divides ₹16850 between his two sons in such a way that after 20 years both of his sons receive the same money. If the ages of the sons are 12 years and 16 years, and the rate of compound interest is 33 1/3%, find the share of each son. Read more

解决步骤

1. Define variables

Let $x$ be the amount given to the elder son (16 years old) and $y$ be the amount given to the younger son (12 years old). The total amount is 16850 Rs, so we have: $$x + y = 16850$$

2. Calculate the investment periods

The investment period for the elder son is $20 - 16 = 4$ years. The investment period for the younger son is $20 - 12 = 8$ years.

3. Calculate the compound interest rate

The rate of compound interest is $33\frac{1}{3}% = \frac{100}{3}% = \frac{100}{3 \times 100} = \frac{1}{3}$.

4. Set up the equation for equal returns

The amount each son receives after the investment period can be calculated using the compound interest formula: $Amount = Principal \times (1 + rate)^{time}$. Since both sons receive the same amount after their respective investment periods, we can set up the following equation:

$$x \left(1 + \frac{1}{3}\right)^4 = y \left(1 + \frac{1}{3}\right)^8$$ $$x \left(\frac{4}{3}\right)^4 = y \left(\frac{4}{3}\right)^8$$

5. Simplify the equation Divide both sides by $\left(\frac{4}{3}\right)^4$: $$x = y \left(\frac{4}{3}\right)^4$$ $$x = y \left(\frac{256}{81}\right)$$

6. Solve the system of equations

We have two equations: $$x + y = 16850$$ $$x = \frac{256}{81}y$$ Substitute the second equation into the first: $$\frac{256}{81}y + y = 16850$$ $$\frac{256y + 81y}{81} = 16850$$ $$\frac{337y}{81} = 16850$$ $$y = \frac{16850 \times 81}{337}$$ $$y = 50 \times 81 = 4050$$

7. Calculate the value of x

Now substitute the value of $y$ back into the equation $x + y = 16850$: $$x = 16850 - y$$ $$x = 16850 - 4050$$ $$x = 12800$$

8. Final Answer

The share of the elder son is 12800 Rs, and the share of the younger son is 4050 Rs.