Compound Interest Exit
Quantitative aptitude

Compound Interest

CI is SI on a growing base. For 2 years the only extra over SI is interest on the first year's interest.

Start with this

With compound interest, last year's interest starts earning interest too. Rs 1,000 at 10% becomes 1,100, then 1,210, then 1,331.

Compound interest questions test the multiplier (1 + R/100)n, the difference between CI and SI, and half-yearly or quarterly compounding.

Interest on interest

Amount after n years: A = P (1 + R/100)n. CI = A - P.

Year by year it is successive percentage increases: each year multiplies by (1 + R/100).

yr 0yr 1yr 2yr 3simple 13,000compound 13,310
Rs 10000 at 10%: compound pulls ahead because interest earns interest

Interest on interest: the rules

A = P (1 + R/100)n
CI = A - P
Half-yearly: rate R/2, periods 2n
Quarterly: R/4 and 4n.

Worked example

Find the compound interest on Rs 10,000 at 10% a year for 3 years.

  1. A = 10000 × 1.13.
    Why: Multiply by 1.1 each year.
  2. 1.13 = 1.331, so A = 13,310.
  3. CI = 3,310.

Answer: 3310

Worked example

Rs 8,000 at 10% a year compounded half-yearly for 1 year. Amount?

  1. Half-yearly: rate 5% per half-year, 2 periods.
    Why: Halve the rate, double the periods.
  2. A = 8000 × 1.052 = 8000 × 1.1025 = 8,820.

Answer: 8820

CI minus SI

For 2 years: CI - SI = P (R/100)2. That is the 'interest on the first year's interest'.

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

Worked example

The difference between CI and SI on a sum at 10% for 2 years is Rs 50. Find the sum.

  1. CI - SI = P × (10/100)2 = P/100.
    Why: Two-year formula.
  2. P/100 = 50, so P = 5,000.

Answer: 5000

Doubling and the rule of 72

At compound interest, if a sum doubles in n years it becomes 4 times in 2n, 8 times in 3n. Powers, not multiples.

Rough rule: money doubles in about 72/R years. At 8%, about 9 years.

Worked example

A sum doubles in 5 years at compound interest. In how many years will it be 8 times?

  1. 8 = 23.
    Why: Three doublings.
  2. 3 × 5 = 15 years.

Answer: 15

Shortcut: 10% CI by hand

Use it when: Build the year-by-year amounts; it is faster than powers.

  1. Year 1: add 10%. Year 2: add 10% of the new amount.

10% CI by hand: try it

CI on 10,000 at 10% for 3 years

  1. 11000, 12100, 13310.
  2. CI = 3310.

Answer: Rs 3,310

Common mistakes

Doubling in 5 years means 4 times in 10 years at SI.
Only at CI. At SI, 4 times takes 15 years.

Your turn medium

Difference between CI and SI on Rs 10,000 for 3 years at 10%?

  1. 10000 × 100 × 310 / 106.

Answer: Rs 310

Your turn medium

CI on Rs 16,000 at 20% p.a. compounded quarterly for 9 months?

  1. 5% per quarter, 3 quarters: 16000 × 1.157625 = 18522.

Answer: Rs 2,522

Your turn hard

A sum at CI becomes Rs 6,050 in 2 years and Rs 6,655 in 3 years. Principal?

  1. Rate = 605/6050 = 10%.
  2. P = 6050 / 1.21.

Answer: Rs 5,000

Your turn hard

CI - SI on a sum for 2 years at 5% is Rs 25. The sum?

  1. P × 0.052 = 25.

Answer: Rs 10,000

Recap

Now practise

11 questions with full solutions.

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