Math Problem Statement

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

Solution

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 2515=1025 - 15 = 10 years.
    • For the 10-year-old boy, he will reach 25 in 2510=1525 - 10 = 15 years.
  2. Let the share of the 15-year-old be xx rupees, so the share of the 10-year-old will be 25000x25000 - x.

  3. Future Value Formula: The future value AA of an investment under compound interest is given by: A=P(1+r100)tA = P \left( 1 + \frac{r}{100} \right)^t Where:

    • PP is the principal (initial amount),
    • rr is the rate of interest (8%),
    • tt is the time in years,
    • AA is the amount after time tt.
  4. Equating the future values: The amount the 15-year-old will have at age 25 is: A15=x(1+8100)10A_{15} = x \left( 1 + \frac{8}{100} \right)^{10} The amount the 10-year-old will have at age 25 is: A10=(25000x)(1+8100)15A_{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(1.08)10=(25000x)(1.08)15x \left( 1.08 \right)^{10} = (25000 - x) \left( 1.08 \right)^{15}

  5. Solving for xx: Simplifying the equation: x(1.08)10=(25000x)(1.08)15x (1.08)^{10} = (25000 - x)(1.08)^{15} Divide both sides by (1.08)10(1.08)^{10}: x=(25000x)×(1.08)5x = (25000 - x) \times (1.08)^5 Now solve for xx: x=(25000x)×1.4693x = (25000 - x) \times 1.4693 x=25000×1.4693x×1.4693x = 25000 \times 1.4693 - x \times 1.4693 x+x×1.4693=25000×1.4693x + x \times 1.4693 = 25000 \times 1.4693 x(1+1.4693)=36732.5x (1 + 1.4693) = 36732.5 x×2.4693=36732.5x \times 2.4693 = 36732.5 x=36732.52.4693x = \frac{36732.5}{2.4693} x14877.91x \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 2500014877.9125000 - 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.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Algebra
Future Value

Formulas

Future Value Formula: A = P (1 + r/100)^t
Equation: x (1.08)^10 = (25000 - x) (1.08)^15

Theorems

Compound Interest Theorem

Suitable Grade Level

Grades 10-12