Deducing a Formula for Compound Interest

Zubeda asked her teacher, ‘Is there an easier way to find compound interest?’ The teacher said ‘There is a shorter way of finding compound interest. Let us try to find it.’

Suppose P1 is the sum on which interest is compounded annually at a rate of R% per annum.

Let P1 = ₹ 5000 and R = 5. Then by the steps mentioned above

(1) SI1 = ₹ ×5×1100 (or) SI1 = ₹ ( P1×R×1)/100

So, A1 = ₹ + 5000×5×1100 (or) A1 = P1 + SI1 = P1 +(P1R)/100

= ₹ 5000 1+100 = P2 (or) P11+R100 = P2

(2) SI2 = ₹ 5000 1+5100××1100

SI2 = ( P2 × R × 1)/100

= ₹ ×1001+5100 (or) P11+R100×R100

P1R100 1+R100

A2 = ₹ 1+5100 + ₹ 5000×51001+5100

A2 = P2 + SI2

₹ 5000 1+5100 1+

= P1 1+R100 + P1 R100+ 1+R100

= P1 1+R1001+R100

= P1 1+R1002= P3

Proceeding in this way the amount at the end of n years will be

An = P11+R100n

A = P1+R100n

So, Zubeda said, but using this we get only the formula for the amount to be paid at the end of n years, and not the formula for compound interest.

Aruna at once said that we know CI = A – P, so we can easily find the compound interest too.

Example 8: Find CI on ₹ 12600 for 2 years at 10% per annum compounded annually.

Solution: We have

A = P1+R100n, where Principal (P) = ₹ , Rate (R) = , Number of years (n) =

₹ 12600 1+101002 = ₹ 1260011102

₹ 12600 1110×1110 = ₹

CI = A - P = ₹ - ₹ = 26462062

TRY THESE

1. Find CI on a sum of Rs. 8000 for 2 years at 5% per annum compounded annually.

To find the compound interest (CI) on a sum, we can use the formula:

CI = P × 1+r100t1−r100t -

Where: P is the principal amount (Rs. ), r is the rate of interest ( %), t is the time period ( years).

First, calculate the amount after 2 years using the formula:

A = P × 1+r100t = × 1+51002 = × 1.0520.952

= 8000 × =

Now, find the compound interest:

CI = A − P = − =

So, the compound interest on Rs. 8000 for 2 years at 5% per annum is Rs. 820.