Simple and compound interest
Interest is the fee paid for using money. At simple interest the same amount is added every year, so growth is a straight line. At compound interest the interest is added to the principal at each interval and earns interest itself, so growth speeds up.
Formulas and when they apply
- Simple interest SI = P × R × T / 100Applies when: R is the rate per year in %, T is the time in years
- Amount A = P + SI = P × (1 + R × T/100)Applies when: simple interest
- Amount A = P × (1 + R/100)ⁿ, and compound interest CI = A − PApplies when: interest added once a year, n is the number of years
- Half-yearly: A = P × (1 + R/200)^(2T). Quarterly: A = P × (1 + R/400)^(4T)Applies when: R is still the rate per year
- For 2 years, CI − SI = P × (R/100)²Applies when: yearly compounding, exactly 2 years
- Two successive growth rates a% and b% combine to a + b + ab/100 %Applies when: compound interest over 2 years uses a = b = R
- The time taken to grow n times at simple interest is (n − 1) × 100 / R yearsApplies when: simple interest only
Worked examples
Example 1. Find the compound interest on ₹8,000 for 2 years at 5% per year, compounded yearly.
Solution. Amount = 8,000 × 1.05 × 1.05 = 8,000 × 1.1025 = ₹8,820. Compound interest = 8,820 − 8,000 = ₹820.
Example 2. A sum of ₹6,000 earns ₹1,350 simple interest in 3 years. Find the rate per cent per year.
Solution. SI = P × R × T/100, so 1,350 = 6,000 × R × 3/100 = 180 × R. R = 1,350 ÷ 180 = 7.5% per year.
Example 3. What is the amount on ₹10,000 for 6 months at 12% per year, compounded quarterly?
Solution. Quarterly rate = 12 ÷ 4 = 3%, and 6 months is 2 quarters. Amount = 10,000 × 1.03 × 1.03 = 10,000 × 1.0609 = ₹10,609.
Shortcuts and their limits
- For 2 years, CI − SI = P × (R/100)². With P = ₹25,000 and R = 8% the difference is 25,000 × 0.0064 = ₹160.Does not apply when: Only for exactly 2 years with yearly compounding. For 3 years the difference is P × (R/100)² × (3 + R/100).
- Use fractions for neat rates: 10% growth multiplies by 11/10, 20% by 6/5, 5% by 21/20. If the principal is a multiple of the denominators, there is no decimal work.Does not apply when: Helpful only when the rate gives a simple fraction and the principal divides cleanly.
- At simple interest, if a sum becomes n times itself in t years, it becomes m times in (m − 1)/(n − 1) × t years.Does not apply when: Does not apply to compound interest, where each year's growth is a multiple of the amount so far.
Common mistakes
- Giving the interest when the question asks for the amount, or the amount when it asks for the interest.
- Using R as the rate per period for half-yearly or quarterly compounding. Divide the rate and multiply the number of periods.
- Treating a growth in the third year as a fixed yearly interest when the interest is compound.
- Using the rate as a decimal in one place and as a percentage in another, such as 8 instead of 0.08.
- Mixing months and years: 18 months is 1.5 years for T, but is 3 half-years for half-yearly compounding.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.