Simple and Compound Interest: Formulas and Worked Examples

Simple interest

Under simple interest (SI) the interest is charged on the original principal only, so it is the same every year.

• SI = P × R × T ÷ 100, where P is the principal, R the rate percent per annum and T the time in years.
• Amount A = P + SI.
• Rearranged: R = 100 × SI ÷ (P × T) and T = 100 × SI ÷ (P × R).
• If a sum becomes n times itself in T years at SI, R = 100 × (n - 1) ÷ T.

Compound interest

Under compound interest (CI) the interest of each period is added to the principal, so later interest is earned on interest.

• Annual compounding: A = P × (1 + R/100)^T and CI = A - P.
• Half-yearly compounding: use half the rate and twice the number of periods, A = P × (1 + R/200)^(2T).
• Quarterly compounding: use R/4 and 4T periods.
• Principal from amount: P = A ÷ (1 + R/100)^T.
• If the amounts at the end of two consecutive years are known, the rate is (later amount - earlier amount) ÷ earlier amount × 100.

Difference between CI and SI

For the first year CI and SI are equal. After that CI is larger.

Effective rate

The effective annual rate is the simple annual rate that gives the same interest as a nominal rate compounded more than once a year.

Effective rate = (1 + r/n)^n - 1, where r is the nominal rate as a decimal and n is the number of compoundings per year. For 12% compounded half-yearly: 1.06² - 1 = 0.1236, that is 12.36%.

Equal instalments

When a loan is repaid by equal annual instalments at compound interest, the present values of the instalments add up to the loan. For two instalments of x each: x ÷ (1 + r) + x ÷ (1 + r)² = P.

Worked examples

1. SI. Rs 12,000 at 8% for 3 years: SI = 12,000 × 8 × 3 ÷ 100 = Rs 2,880. Amount = Rs 14,880.
2. Half-yearly CI. Rs 20,000 at 10% per annum for 1 year compounded half-yearly: rate per half-year 5%, two periods. A = 20,000 × 1.05² = Rs 22,050, so CI = Rs 2,050. SI would be Rs 2,000.
3. Difference for two years. Rs 25,000 at 8%: difference = 25,000 × 0.08² = Rs 160. Check: SI = Rs 4,000 and CI = 25,000 × 1.1664 - 25,000 = Rs 4,160.
4. Instalment. Rs 21,000 at 10% repaid in two equal annual instalments: x × (1/1.1 + 1/1.21) = 21,000, so x × 2.1 ÷ 1.21 = 21,000 and x = Rs 12,100. Check: 21,000 × 1.1 = 23,100; less 12,100 leaves 11,000; 11,000 × 1.1 = 12,100, cleared by the second instalment.

Exam pointers

• Read whether the question asks for the interest or the amount; both usually appear among the options.
• For half-yearly compounding halve the rate and double the periods. Doing only one of the two is the standard trap.
• The shortcut P × (R/100)² works only for two years; for three years multiply by (3 + R/100).
• With amounts for consecutive years under CI, find the rate from their ratio first, then discount back to the principal.
• A sum doubles at SI when the total interest equals the principal, so R × T = 100.
• An equal instalment is always more than the loan divided by the number of instalments, because interest is included.