Simple and Compound Interest: Formulas and Worked Examples

Interest questions appear regularly in the Lok Sewa maths and mental ability paper, and they are among the easiest marks to win once the formulas are clear. Almost every question is a small change on one main idea: interest is money paid for using someone else’s money for some time. This note explains simple interest first, then compound interest, with worked examples in rupees for every formula.

1. Basic terms

So A = P + I and I = A − P.

• Example: Sita borrows Rs 20,000 and pays back Rs 23,500 in total. The amount is Rs 23,500 and the interest is 23,500 − 20,000 = Rs 3,500.

2. Simple interest

In simple interest (SI), interest is charged only on the original principal. The interest is the same every year.

SI = P × T × R / 100

Amount at SI = P + SI = P × (100 + R × T)/100

Worked examples:

• Rs 5,000 at 8% per annum for 3 years. SI = 5,000 × 3 × 8/100 = Rs 1,200. Amount = Rs 6,200.
• Rs 16,000 at 12.5% per annum for 2 years. SI = 16,000 × 2 × 12.5/100 = Rs 4,000. Amount = Rs 20,000.

Because the interest each year is equal, SI for 3 years is simply three times the SI for one year.

3. Finding P, T or R

Rearrange the same formula to find the missing value.

Worked examples:

• SI = Rs 1,800 in 3 years at 12%. P = 1,800 × 100/(3 × 12) = Rs 5,000.
• Rs 8,000 earns Rs 2,400 in 5 years. R = 2,400 × 100/(8,000 × 5) = 6%.
• Rs 6,000 at 10% earns Rs 1,500. T = 1,500 × 100/(6,000 × 10) = 2.5 years, that is 2 years 6 months.
• A sum amounts to Rs 7,800 in 3 years at 10%. P = 7,800 × 100/(100 + 30) = Rs 6,000.

Two amounts trick: if a sum amounts to Rs 5,400 in 2 years and Rs 6,000 in 4 years at SI, then the interest for the extra 2 years is Rs 600, so the interest for one year is Rs 300. The principal = 5,400 − 2 × 300 = Rs 4,800, and the rate = 300 × 100/4,800 = 6.25%.

4. When money doubles or triples at SI

A sum becomes n times itself when the interest equals (n − 1) times the principal. This gives a quick formula:

T = (n − 1) × 100 / R and, the other way round, R = (n − 1) × 100 / T

Worked examples:

• A sum doubles in 8 years at SI. Rate = 1 × 100/8 = 12.5%.
• A sum doubles in 5 years. The rate is 20%, so it becomes 3 times in 2 × 100/20 = 10 years.

5. Interest for months and days

The rate is per year, so time must also be in years.

• Months: T = number of months / 12.
• Days: T = number of days / 365.

Worked examples:

• Rs 12,000 at 10% for 9 months. T = 9/12. SI = 12,000 × 9/12 × 10/100 = Rs 900.
• Rs 7,300 at 10% for 73 days. T = 73/365 = 1/5. SI = 7,300 × 1/5 × 10/100 = Rs 146.
• Rs 4,000 at 9% for 1 year 8 months. T = 20/12 = 5/3 years. SI = 4,000 × 5/3 × 9/100 = Rs 600.

6. Compound interest (yearly)

In compound interest (CI), the interest of each year is added to the principal, and the next year’s interest is charged on this bigger sum. So the interest grows every year.

A = P × (1 + R/100) to the power T, and CI = A − P

• For 2 years: A = P × (1 + R/100)²
• For 3 years: A = P × (1 + R/100)³

Worked example (2 years): Rs 10,000 at 10% compounded yearly.

By formula: A = 10,000 × 1.1 × 1.1 = Rs 12,100, so CI = Rs 2,100. The SI for the same case is only Rs 2,000.

Worked examples (3 years):

• Rs 8,000 at 5% for 3 years. A = 8,000 × 1.05 × 1.05 × 1.05 = Rs 9,261. CI = Rs 1,261.
• Rs 20,000 at 10% for 3 years. A = 20,000 × 1.331 = Rs 26,620. CI = Rs 6,620.

Finding the rate: Rs 10,000 amounts to Rs 12,544 in 2 years at CI. Then (1 + R/100)² = 12,544/10,000 = 1.2544 = 1.12 × 1.12, so R = 12%.

Note: for the first year, CI and SI are always equal. The difference starts from the second year, because only CI charges interest on the first year’s interest.

7. Half-yearly compounding

Sometimes interest is added to the principal every six months. The rate is still given per annum, so make two changes:

• Rate per period = R/2 (half of the yearly rate)
• Number of periods = 2 × T (twice the number of years)

A = P × (1 + R/200) to the power 2T

Worked examples:

• Rs 10,000 at 10% per annum, compounded half-yearly, for 1 year. Rate per half-year = 5%, periods = 2. A = 10,000 × 1.05 × 1.05 = Rs 11,025. CI = Rs 1,025, which is Rs 25 more than yearly compounding gives.
• Rs 40,000 at 10% per annum, compounded half-yearly, for 1.5 years. Periods = 3. A = 40,000 × 1.05³ = 40,000 × 1.157625 = Rs 46,305.

The more often interest is compounded, the larger the amount at the same yearly rate.

8. Difference between CI and SI for 2 years

For the same P and R over 2 years, the extra interest under CI is the interest on the first year’s interest. This gives a fast shortcut:

CI − SI (2 years) = P × (R/100)²

Worked examples:

• Rs 10,000 at 10%: difference = 10,000 × (10/100)² = 10,000 × 1/100 = Rs 100. (Check: CI 2,100 − SI 2,000 = 100.)
• Rs 30,000 at 5%: difference = 30,000 × 1/400 = Rs 75.
• Reverse: the difference between CI and SI on a sum for 2 years at 4% is Rs 40. P = 40 ÷ (4/100)² = 40 × 625 = Rs 25,000.

9. Step-by-step method

1. Write down P, R, T and decide whether the question asks for SI or CI.
2. Convert months or days into years.
3. For SI, use P × T × R/100; for CI, turn the rate into a factor such as 1.1 for 10% and multiply once for each year.
4. For half-yearly compounding, halve the rate and double the periods.
5. Read the question again: does it ask for the interest or the amount?