Compound Interest

CI, compounding annually/half-yearly

105 practice questionsPage updated 1 Sept 2026Reviewed by AISEA Editorial DeskPractice questions

Compound interest is interest calculated on the accumulated amount, so each period can earn interest on earlier interest as well as on the original principal. In aptitude exams, first identify the principal, nominal annual rate, compounding frequency and number of periods. Convert the rate and time to one compounding period, apply the growth factor repeatedly, and subtract the principal only when the question asks for compound interest rather than the final amount.

Key formulas and rules

Annual compounding: A = P × (1 + R / 100) raised to the power T, where T is the number of years.

Compounding n times per year: A = P × (1 + R / (100 × n)) raised to the power n × T.

Compound interest: CI = A − P.

Present principal: P = A divided by the applicable compound growth factor.

Effective annual rate for nominal R compounded n times = [(1 + R / (100 × n)) raised to the power n − 1] × 100.

Depreciation at annual rate R: final value = initial value × (1 − R / 100) raised to the power T.

For two years at annual rate R, CI − SI = P × (R / 100)².

For one compounding period, interest = amount at the start of that period × rate per period.

Key concepts

Separate principal, amount and interest

The principal P is the starting sum. The amount A is the principal plus all accumulated interest. Compound interest CI is A − P. Many mistakes come from calculating A correctly and then reporting it as CI, or from subtracting the principal twice.

Convert the nominal rate and frequency

If an annual rate R is compounded twice a year, each half-year uses R / 2 and there are 2 periods per year. Quarterly uses R / 4 and four periods per year; monthly uses R / 12 and twelve periods per year. The rate per period and the number of periods must change together.

Worked examples

Example 1

Find the compound interest on ₹10,000 at 10% per annum for 2 years, compounded annually.

Solution

Step 1: A = 10,000 × (1 + 10 / 100) raised to the power 2 = 10,000 × 1.1 × 1.1 = ₹12,100.
Step 2: CI = A − P = ₹12,100 − ₹10,000 = ₹2,100.
Answer: ₹2,100.

Example 2

Find the difference between compound interest and simple interest on ₹5,000 for 2 years at 8% per annum.

Solution

Step 1: For two years, CI − SI = P × (R / 100)².
Step 2: Difference = 5,000 × (8 / 100)² = 5,000 × 0.0064 = ₹32.
Step 3: Direct check: CI = ₹5,000 × 1.08 × 1.08 − ₹5,000 = ₹832; SI = ₹800; difference = ₹32.
Answer: ₹32.

Example 3

In how many years will ₹8,000 amount to ₹9,261 at 5% per annum compound interest?

Solution

Step 1: Growth factor = 9,261 / 8,000 = 1.157625.
Step 2: At 5% per year, the factor after 3 years is 1.05 × 1.05 × 1.05 = 1.157625.
Step 3: Therefore the required time is 3 years.
Answer: 3 years.

Example 4

What is the effective annual rate for a nominal rate of 20% per annum compounded half-yearly?

Solution

Step 1: Rate per half-year = 20% / 2 = 10%.
Step 2: One-year growth factor = 1.1 × 1.1 = 1.21.
Step 3: Effective annual rate = (1.21 − 1) × 100 = 21%.
Answer: 21% per year.

Example 5

₹12,500 amounts to ₹15,125 in 2 years at compound interest compounded annually. Find the annual rate.

Solution

Step 1: Growth factor for two years = 15,125 / 12,500 = 1.21.
Step 2: The positive square root of 1.21 is 1.1, so the one-year growth factor is 1.1.
Step 3: Rate = (1.1 − 1) × 100 = 10%.
Answer: 10% per annum.

Example 6

Find the amount when ₹20,000 is invested at 8% per annum compounded quarterly for 1 year.

Solution

Step 1: Rate per quarter = 8% / 4 = 2%.
Step 2: Number of quarters = 4 × 1 = 4.
Step 3: A = 20,000 × 1.02 × 1.02 × 1.02 × 1.02 = ₹21,648.64.
Answer: ₹21,648.64.

Example 7

Find the compound interest on ₹25,000 for 1.5 years at 10% per annum compounded half-yearly.

Solution

Step 1: Rate per half-year = 10% / 2 = 5%.
Step 2: Number of half-year periods = 1.5 × 2 = 3.
Step 3: A = 25,000 × 1.05 × 1.05 × 1.05 = ₹28,940.625.
Step 4: CI = ₹28,940.625 − ₹25,000 = ₹3,940.625.
Answer: ₹3,940.625, or ₹3,940.63 to two decimal places.

Example 8

The compound interest on a sum for 2 years at 5% per annum is ₹512.50. Find the principal.

Solution

Step 1: Two-year growth factor at 5% = 1.05 × 1.05 = 1.1025.
Step 2: CI = P × (1.1025 − 1) = 0.1025P.
Step 3: 512.50 = 0.1025P, so P = 512.50 / 0.1025 = ₹5,000.
Answer: ₹5,000.

Example 9

A machine worth ₹100,000 depreciates at 10% per annum. What will be its value after 2 years?

Solution

Step 1: Retention factor each year = 1 − 10 / 100 = 0.9.
Step 2: Value after two years = 100,000 × 0.9 × 0.9 = ₹81,000.
Answer: ₹81,000.

Example 10

A population increases by 10% in the first year and decreases by 10% in the second year. If it starts at 100,000, what is the population after 2 years?

Solution

Step 1: First-year factor = 1.10, so population becomes 100,000 × 1.10 = 110,000.
Step 2: Second-year factor = 0.90, so population becomes 110,000 × 0.90 = 99,000.
Answer: 99,000. The equal percentage increase and decrease do not cancel because the second percentage uses a different base.

Example 11

₹6,000 is invested for 2 years at 10% per annum. How much larger is the compound interest when compounded half-yearly rather than annually?

Solution

Step 1: Annually: A = 6,000 × 1.1 × 1.1 = ₹7,260, so CI = ₹1,260.
Step 2: Half-yearly: rate per period = 5% and there are 4 periods. A = 6,000 × 1.05 raised through four periods = ₹7,293.0375, so CI = ₹1,293.0375.
Step 3: Difference = ₹1,293.0375 − ₹1,260 = ₹33.0375.
Answer: Approximately ₹33.04.

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Compound Interest

Key formulas and rules

Key concepts

Separate principal, amount and interest

Convert the nominal rate and frequency

Use the compound growth factor

Effective annual rate versus nominal rate

Solve for time or rate from powers

Handle partial years explicitly

Find interest for a particular period

Compare compound and simple interest

Apply successive increases and decreases

Do not confuse deposits with instalments

Check units and rounding

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Key formulas and rules

Key concepts

Separate principal, amount and interest

Convert the nominal rate and frequency

Use the compound growth factor

Effective annual rate versus nominal rate

Solve for time or rate from powers

Handle partial years explicitly

Find interest for a particular period

Compare compound and simple interest

Apply successive increases and decreases

Do not confuse deposits with instalments

Check units and rounding

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.