When Will Their Amounts Be Equal: A Simple Interest Calculation Example

This article explores a problem of simple interest calculation where two individuals, A and B, borrow different amounts at different interest rates. We will derive the formula to find the time (T) when their total outstanding amounts will be equal. We will then solve for T using both traditional and derived formulas for simple interest.

Scenario and Problem Statement

Let us consider the scenario where:

A borrows Rs. 8000 at 12% per annum simple interest. B borrows Rs. 9100 at 10% per annum simple interest.

The question is: In how many years will the outstanding amounts of A and B be equal?

Formulation and Solution Using Simple Interest Formula

Traditional Simple Interest Formula

Mathematically, the equation for simple interest is expressed as:

A P (PRT/100)

Where:

A: Total amount of money after time T years. P: Principal amount. R: Rate of interest per annum. T: Time in years.

Applying the Formula to the Scenario

For A: A 8000 (8000 × 12 × T / 100) This simplifies to: A 8000 960T For B: A 9100 (9100 × 10 × T / 100) This simplifies to: A 9100 910T Setting the two equations equal to each other to find T: 8000 960T 9100 910T 960T - 910T 9100 - 8000 50T 1100 T 1100 / 50 22 years

Thus, in 22 years, the outstanding amounts of A and B will be equal.

Alternative Representation and Explanation

Another Example and Solution

Let us consider a similar scenario where:

A borrows Rs. 1600 at 12% per annum simple interest. B borrows Rs. 1820 at 10% per annum simple interest.

We need to find T, the number of years after which both will have the same debt.

The formula for the total amount after T years becomes:

A 1600 (1600 × 0.12 × T) A 1600 192T B 1820 (1820 × 0.10 × T) B 1820 182T

Setting the amounts equal to find T:

1600 192T 1820 182T 192T - 182T 1820 - 1600 10T 220 T 220 / 10 22 years

Once again, in 22 years, the outstanding debts of A and B will be equal.

General Formula and Explanation

The general formula for simple interest can be expressed as:

Amount owed by A: A_A P_A (P_A × R_A × T) / 100 Amount owed by B: A_B P_B (P_B × R_B × T) / 100

Where:

P_A, P_B: Principal amounts for A and B. R_A, R_B: Interest rates for A and B. T: Time in years.

Setting A_A equal to A_B to find T:

P_A (P_A × R_A × T / 100) P_B (P_B × R_B × T / 100)

Rearranging, we find:

P_A × R_A × T - P_B × R_B × T P_B - P_A T (P_B - P_A) / (P_A × R_A - P_B × R_B) × 100

Using this formula, we can compute the time to equalize the amounts for any given principal and interest rates.

In conclusion, both examples illustrate how to calculate the amount of time needed for the outstanding debts of two individuals with different principal amounts and interest rates to become equal through the application of the simple interest formula. The concept is relevant in financial planning and interest rate calculations.