Solving Compound Interest Problems: Finding the Deposit Amounts for Two Sons

Compound interest problems not only challenge our mathematical skills but also help us in understanding the real-world applications of financial mathematics. Let's delve into a classic example: How much was deposited for one son, if the total deposit is 2,523, with different durations and interest rates?

The Problem Statement

A person, wishing to provide future financial support to his two sons, A and B, deposited a total of 2,523 in two parts. Son A's deposit of PA is invested for 3 years at a 5% compound interest rate, while son B's deposit of PB is invested for 5 years under the same rate. The key condition is that both sons receive the same amount at the end of their respective investment periods.

Setting Up the Equations

We know that:

The total deposit: PA PB 2,523 Son A's amount after 3 years: AA PA(1 5/100)3 PA1.053 Son B's amount after 5 years: AB PB(1 5/100)5 PB1.055

Since both sons receive the same amount, we can set up the equation:

PA1.053 PB1.055

Rearranging for PB, we get:

PB PA1.05-2 PA/1.052

Solving for PA

Substituting PB into the total deposit equation:

PA PA/1.052 2,523

Factoring out PA and simplifying:

PA(1 1/1.1025) 2,523

The value of 1.10252 is 1.05, and 1/1.1025 ≈ 0.907. Substituting this, we get:

PA0.907 2,523

Solving for PA:

PA 2,523/0.907 ≈ 1,323.48

Rounding to the nearest whole number, we get:

PA ≈ 1,323

Conclusion

Thus, the sum deposited for son A is approximately 1,323. This solution is derived using the principles of compound interest, a fundamental concept in financial mathematics. Problems like these not only test our understanding of compound interest but also help in practical money management and planning.

Additional Solutions

The given problem can have multiple solutions, and here are a couple of alternative approaches:

Solution 1

Using the same method as above, but simplifying the fractions:

P 1,323 (For son A) Q 1,200 (For son B)

Here, the total money deposited is 2,523, and P Q 2,523. The ratios are consistent with the compound interest conditions.

Solution 2

Using an alternative method where the parts are directly determined:

A's part: 10 B's part: 2,523 - 10 A's amount in 3 years: 10(1 5/100)3 1.1576x B's amount in 5 years: (1 5/100)5(2,523 - 10) 1.276y Setting these equal, we find: 1.1576x 1.276y 0.11025 1.276/1.1576 1.102 Thus, x 1.276/1.1576 y 1.102y 2.102y 2,523 y 1,200 x 1,323

These steps clearly illustrate the process of solving compound interest problems and highlight the importance of mathematical logic in financial planning and analysis.