Simple and Compound Interest

Interest is the extra money paid for using someone's money. Simple interest and compound interest are tested in almost every aptitude paper, and they are also useful in real life when you compare loans and savings. The ideas are simple. The usual errors come from mixing up the formulas or the time units.

Terms

• Principal (P): the money borrowed or invested at the start.
• Rate (R): the interest per 100 rupees per year, written R% per annum (p.a.).
• Time (T): the number of years.
• Amount (A): principal plus interest.

Simple interest (SI)

In simple interest the interest is calculated only on the original principal, so it is the same every year.

SI = P × R × T / 100, and A = P + SI.

Example: the SI on 8,000 at 7.5% p.a. for 4 years is 8,000 × 7.5 × 4 / 100 = 2,400. The amount is 10,400.

Rearranged, the formulas give each unknown: P = SI × 100 / (R × T), R = SI × 100 / (P × T), and T = SI × 100 / (P × R).

If the time is in months, divide by 12 first. Two years and six months is 2.5 years.

Money that doubles

At simple interest, a sum doubles when the interest equals the principal. That means R × T = 100. At 10% p.a. it takes 10 years to double, and at 8% it takes 12.5 years.

Compound interest (CI)

In compound interest, the interest is added to the principal at the end of each period, and the next period's interest is calculated on the larger amount. This is "interest on interest".

A = P × (1 + R/100)ⁿ, and CI = A − P.

Example: 10,000 at 10% p.a. compounded yearly for 2 years gives A = 10,000 × 1.1 × 1.1 = 12,100. The CI is 2,100, whereas the SI would have been only 2,000.

Compounding more than once a year

For half-yearly compounding, halve the rate and double the number of periods. For quarterly compounding, divide the rate by 4 and multiply the periods by 4.

Example: 4,000 at 10% p.a. compounded half-yearly for 1 year. The rate per half-year is 5%, and there are 2 half-years. A = 4,000 × 1.05 × 1.05 = 4,410.

The difference between CI and SI

For the first year, CI and SI are the same. From the second year, the extra interest on interest appears.

• For 2 years: CI − SI = P × (R/100)².
• For 3 years: CI − SI = P × (R/100)² × (3 + R/100).

Example: P = 5,000 and R = 8%. The difference for 2 years is 5,000 × 0.08 × 0.08 = 32.

Finding the principal from two amounts (SI)

If a sum becomes A₁ after t₁ years and A₂ after t₂ years at simple interest, then the interest per year is (A₂ − A₁) / (t₂ − t₁). The principal is A₁ minus t₁ times that yearly interest.

Worked examples

Example 1. At what rate does 4,500 earn 810 SI in 3 years? R = 810 × 100 / (4,500 × 3) = 6% p.a.

Example 2. What is the CI on 6,000 at 5% p.a. for 2 years? A = 6,000 × 1.05 × 1.05 = 6,615, so CI = 615.

Example 3. A sum earns 1,200 SI in 2 years at 10% p.a. What is the sum? P = 1,200 × 100 / (10 × 2) = 6,000.

Common mistakes

• Using the SI formula for a compound problem, or the reverse.
• Forgetting to subtract the principal at the end to get the interest from the amount.
• Not halving the rate and doubling the periods for half-yearly compounding.
• Entering months as years, or the rate as a decimal inside the SI formula.

Exam tips

1. For 2 or 3 years of compound interest, multiply year by year rather than using a calculator-style power.
2. For a rate of 10%, multiply by 1.1, for 5% by 1.05, for 20% by 1.2.
3. Always check the sign: CI is never smaller than SI for the same rate and time.

Try the quiz and compare your working to the explanations.