How should a sum of rupees 25000 be divided between two boys aged 15 years and 10 years now, so that at the age of 25 they should get equal amount, interest being charged at 8% per annum

The problem asks to divide a sum of ₹25,000 between two boys aged 15 and 10 years, so that both boys will have an equal amount when they reach the age of 25. The interest rate is 8% per annum, and it is assumed that the interest is compounded annually.

### Steps to Solve:

1. **Time until age 25:**
   - For the 15-year-old boy, he will reach 25 in \( 25 - 15 = 10 \) years.
   - For the 10-year-old boy, he will reach 25 in \( 25 - 10 = 15 \) years.

2. **Let the share of the 15-year-old be \( x \) rupees**, so the share of the 10-year-old will be \( 25000 - x \).

3. **Future Value Formula:**
   The future value \( A \) of an investment under compound interest is given by:
   \[
   A = P \left( 1 + \frac{r}{100} \right)^t
   \]
   Where:
   - \( P \) is the principal (initial amount),
   - \( r \) is the rate of interest (8%),
   - \( t \) is the time in years,
   - \( A \) is the amount after time \( t \).

4. **Equating the future values**:
   The amount the 15-year-old will have at age 25 is:
   \[
   A_{15} = x \left( 1 + \frac{8}{100} \right)^{10}
   \]
   The amount the 10-year-old will have at age 25 is:
   \[
   A_{10} = (25000 - x) \left( 1 + \frac{8}{100} \right)^{15}
   \]
   Since both boys should have the same amount when they turn 25, we equate the two future values:
   \[
   x \left( 1.08 \right)^{10} = (25000 - x) \left( 1.08 \right)^{15}
   \]

5. **Solving for \( x \):**
   Simplifying the equation:
   \[
   x (1.08)^{10} = (25000 - x)(1.08)^{15}
   \]
   Divide both sides by \( (1.08)^{10} \):
   \[
   x = (25000 - x) \times (1.08)^5
   \]
   Now solve for \( x \):
   \[
   x = (25000 - x) \times 1.4693
   \]
   \[
   x = 25000 \times 1.4693 - x \times 1.4693
   \]
   \[
   x + x \times 1.4693 = 25000 \times 1.4693
   \]
   \[
   x (1 + 1.4693) = 36732.5
   \]
   \[
   x \times 2.4693 = 36732.5
   \]
   \[
   x = \frac{36732.5}{2.4693}
   \]
   \[
   x \approx 14877.91
   \]

So, the 15-year-old boy should receive approximately **₹14,877.91**, and the 10-year-old boy should receive **₹10,122.09** (which is \( 25000 - 14877.91 \)).

### Conclusion:
- The sum of ₹25,000 should be divided such that the 15-year-old gets ₹14,877.91 and the 10-year-old gets ₹10,122.09 to ensure both have equal amounts at the age of 25.

Would you like more details on any step or further clarifications?

### Related Questions:
1. How does compound interest differ from simple interest in this scenario?
2. What would the division be if the interest rate were 10% instead of 8%?
3. How would the result change if the interest were compounded quarterly?
4. How do we calculate the future value of an investment?
5. What if the sum to be divided were ₹50,000 instead of ₹25,000?

### Tip:
When dealing with compound interest problems, always ensure you're clear about the compounding frequency (annually, quarterly, etc.) as it significantly affects the final outcome.

How should a sum of rupees 25000 be divided between two boys aged 15 years and 10 years now, so that at the age of 25 they should get equal amount, interest being charged at 8% per annum?

Learn how to divide ₹25,000 between a 15-year-old and a 10-year-old, ensuring both receive equal amounts at age 25 with compound interest at 8% per annum. Step-by-step solution provided, covering future value calculations and algebraic equations. Suitable for Grades 10-12.

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