Lessons / Quantitative aptitude
Quantitative aptitude · Day 3

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.

Practise (11 questions) Present as slides

Why this topic matters

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
A = P (1 + R/100)n
CI = A - P
Half-yearly: rate R/2, periods 2n
Quarterly: R/4 and 4n.
Example 1Find the compound interest on Rs 10,000 at 10% a year for 3 years.
  1. A = 10000 × 1.13.Multiply by 1.1 each year.
  2. 1.13 = 1.331, so A = 13,310.
  3. CI = 3,310.
Example 2Rs 8,000 at 10% a year compounded half-yearly for 1 year. Amount?
  1. Half-yearly: rate 5% per half-year, 2 periods.Halve the rate, double the periods.
  2. A = 8000 × 1.052 = 8000 × 1.1025 = 8,820.

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).

Example 1The 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.Two-year formula.
  2. P/100 = 50, so P = 5,000.

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.

Example 1A sum doubles in 5 years at compound interest. In how many years will it be 8 times?
  1. 8 = 23.Three doublings.
  2. 3 × 5 = 15 years.

Formula summary

A = P(1 + R/100)n
Half-yearly
Rate R/2, periods 2n. Quarterly: R/4 and 4n. Monthly: R/12 and 12n.
CI - SI, 2 years
P(R/100)2.
CI - SI, 3 years
P R2 (300 + R) / 1003.
Doubles in n years
Becomes 4x in 2n, 8x in 3n.
Two amounts, consecutive years
Rate = (later - earlier)/earlier × 100.

Shortcuts and tricks

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.
ExampleCI on 10,000 at 10% for 3 years
  1. 11000, 12100, 13310.
  2. CI = 3310.

Common mistakes

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

Before you move on, you should be able to...

  • compute CI with the multiplier
  • adjust rate and periods for half-yearly and quarterly
  • use CI - SI = P(R/100)2 for two years

Practice questions (11)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy Difference between CI and SI on Rs 15,000 for 2 years at 8%?
  1. 15000 × 0.0064.

Answer: Rs 96

easy CI on Rs 8,000 at 10% p.a. compounded half-yearly for 1 year?
  1. 8000 × 1.052 = 8820.

Answer: Rs 820

easy A sum doubles in 5 years at CI. In how many years does it become 8 times?
  1. 23 = 8 → 3 × 5.

Answer: 15 years

easy Find the amount on Rs 5,000 at 20% a year compounded yearly for 2 years.
  1. 5000 × 1.2 × 1.2.Multiply by 1.2 twice.
  2. = 7,200.

Answer: 7200

medium Difference between CI and SI on Rs 10,000 for 3 years at 10%?
  1. 10000 × 100 × 310 / 106.

Answer: Rs 310

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

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

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

hard Rs 8,000 is borrowed at 10% CI. Rs 4,400 is repaid after year 1. Amount to clear it after year 2?
  1. Year 1: 8800 - 4400 = 4400.
  2. Year 2: 4400 × 1.1.

Answer: Rs 4,840

hard What sum becomes Rs 4,913 in 3 years at 6.25% CI?
  1. 6.25% = 1/16, so x (17/16)3.
  2. 4913/4913 × 4096.

Answer: Rs 4,096

hard A loan of Rs 2,100 is repaid in 2 equal yearly instalments at 10% CI. Each instalment?
  1. x/1.1 + x/1.21 = 2100.
  2. x (2.1/1.21) = 2100 → x = 1210.

Answer: Rs 1,210

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
15 minYear-by-year table vs SI (the chart).
25 minA = P(1 + R/100)n, CI.
20 minHalf-yearly and quarterly.
20 minCI - SI for 2 and 3 years.
10 minDoubling powers; rule of 72.
15 minPractice set.
5 minRecap.
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