Compound Interest: Formula, Shortcuts and Solved Questions

Compound Interest is calculated on the principal and the interest accumulated in earlier periods. This page covers the standard compound interest formula, amount calculation, half-yearly and quarterly compounding, depreciation, rate changes, useful CI tricks and solved numerical examples for quantitative aptitude.

On this page

What is Compound Interest?

Compound Interest is the interest calculated on the principal plus the interest accumulated during previous compounding periods. The total amount is calculated first, and the compound interest is obtained by subtracting the principal.

For principal P, annual rate R% and time T years with annual compounding, Amount = P(1 + R/100)^T and Compound Interest = Amount − P. If interest is compounded more than once a year, the rate and number of periods must be adjusted accordingly. For example, ₹10,000 at 10% per annum for 2 years gives A = 10,000(1.10)^2 = ₹12,100, so CI = ₹2,100.

Compound Interest Formula & Tricks

Important Formulas

Annual compounding
A = P(1 + R/100)^T; CI = A − P

P is the principal, R is the annual rate in percent, T is time in years, A is the final amount and CI is compound interest.

Half-yearly compounding
A = P(1 + R/200)^(2T)

For half-yearly compounding, the rate for each period is R/2% and the number of periods is 2T.

Quarterly compounding
A = P(1 + R/400)^(4T)

For quarterly compounding, the rate for each quarter is R/4% and the number of periods is 4T.

Different annual rates
A = P(1 + R₁/100)(1 + R₂/100) ... (1 + Rₙ/100)

Use this product when different rates apply in successive years or periods.

Difference between CI and SI for two years
CI − SI = P(R/100)^2

This shortcut applies when the same annual rate R% is used for exactly two years with annual compounding.

Quick Tricks

Use amount factors

Replace each interest rate by its growth factor. At 10%, the factor is 1.10; at 20%, it is 1.20. Multiply the factors and then multiply by the principal.

Example: For ₹5,000 at 10% for 2 years, A = 5,000 × 1.1 × 1.1 = ₹6,050 and CI = ₹1,050.
Use the two-year CI difference shortcut

For two years at the same annual rate, directly use CI − SI = P(R/100)^2 instead of calculating both amounts.

Example: For P = ₹8,000 and R = 5%, the difference is 8,000 × (5/100)^2 = ₹20.
Convert the period before applying the formula

For half-yearly compounding, divide the annual rate by 2 and multiply the time by 2. For quarterly compounding, divide the rate by 4 and multiply the time by 4.

Example: At 12% per annum compounded half-yearly for 1 year, use 6% for 2 periods: A = P(1.06)^2.
Calculate depreciation as compound reduction

For a value that decreases by R% per period, multiply by (1 − R/100) for every period.

Example: A machine worth ₹20,000 depreciating by 10% annually for 2 years has value 20,000 × 0.9² = ₹16,200.

Compound Interest Concepts

Calculation of Amount and Compound Interest

The amount is the principal after all compound growth, while compound interest is the excess of the amount over the principal.

For annual compounding, use A = P(1 + R/100)^T and then CI = A − P. The exponent T represents the number of annual compounding periods.

Example: If P = ₹4,000, R = 5% and T = 2 years, A = 4,000 × 1.05² = ₹4,410. Therefore, CI = ₹4,410 − ₹4,000 = ₹410.

Half-Yearly and Quarterly Compounding

When interest is compounded several times in a year, divide the annual rate by the number of compounding periods per year and multiply the time by that number.

For half-yearly compounding, use rate R/2% and periods 2T. For quarterly compounding, use rate R/4% and periods 4T. Thus, half-yearly amount is P(1 + R/200)^(2T), and quarterly amount is P(1 + R/400)^(4T).

Example: For ₹10,000 at 8% per annum compounded quarterly for 1 year, A = 10,000(1 + 8/400)^4 = 10,000(1.02)^4 = ₹10,824.32.

Difference Between Simple and Compound Interest

For two years at the same annual rate, compound interest exceeds simple interest by P(R/100)^2.

Simple interest is calculated only on the original principal, whereas compound interest also includes earlier interest. For more than two years, expand the compound amount or calculate CI and SI separately.

Example: For ₹12,000 at 10% for 2 years, CI − SI = 12,000 × (10/100)^2 = ₹120.

Compound Interest with Changing Rates

When the rate changes from one period to another, multiply the principal by the growth factor for each individual period.

For successive rates R₁%, R₂% and R₃%, the amount is P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100). The rates must not be added unless the calculation specifically uses an equivalent-rate method.

Example: ₹10,000 invested for two years at 10% in the first year and 20% in the second year gives A = 10,000 × 1.10 × 1.20 = ₹13,200.

Compound Depreciation

For a value reduced by R% in each period, the remaining value after T periods is P(1 − R/100)^T.

The reduction factor is 1 − R/100 because each period retains only the stated percentage of the previous value. The total depreciation is the original value minus the final value.

Example: A car valued at ₹5,00,000 depreciates by 20% annually for 2 years. Its value is 5,00,000 × 0.8² = ₹3,20,000, so depreciation is ₹1,80,000.

Compound Interest Video Lessons

Watch short topic-wise lessons for quick revision.

6 Lessons
Lesson 1 of 6 Quick Revision

Compound Interest Annual Compounding Formula

Learn the compound interest formula for annual compounding, including how to calculate the final amount from the principal, rate, and time.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Compound Interest Lessons Scroll to explore →

Practice Compound Interest Questions

Practise published questions related to this topic.

1Find the value of x in the following expression. (0.36)³ × (0.216)³ ÷ (0.1296)³ = (60 ÷ 100)^(x+1)→ 2The cost price of an article is Rs. 10000. It is sold at a profit of 40 percent. A discount of 30 percent was given. What is the marked price of the article?→ 3A shopkeeper earns a profit of 10% after allowing a discount of 20% on the marked price. The cost price of the article whose marked price is ₹ 880 is:→ 4If √x = 80/√4900, then what is the value of x?→ 5The LCM of the numbers 3, 7.2 and 0.64 is:→ 6Find the rate of discount when the marked price is ₹11,280 and the selling price is ₹9,588.→ 7Ravi buys an article with a discount of 25% on its marked price. He makes a profit of 10% by selling it at ₹1,320. The marked price of the article was:→ 8Successive discounts of 20% and 29% are equivalent to a single discount of:→ 9Marked price of an article is Rs. X. A person purchased it at a discount of 40 percent. He then sold it a price of 10 percent more than X. What is his profit percentage?→ 10During the New Year sale, a company is offering a 'Buy 9 sweaters, get 3 free' deal to customers. What is the percentage discount offered in this deal?→

Compound Interest Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Compound Interest Revision Points

Remember these formulas and calculation rules for compound interest questions.

  • Annual amount: A = P(1 + R/100)^T.
  • Compound interest: CI = A − P.
  • For half-yearly compounding, use R/2% per period and 2T periods.
  • For quarterly compounding, use R/4% per period and 4T periods.
  • For two years, CI − SI = P(R/100)^2.
  • For successive rates, multiply the separate factors instead of adding the rates.
  • For depreciation, use P(1 − R/100)^T.
  • If the final amount and principal are known, CI equals A − P.

Compound Interest FAQs

What is the compound interest formula for annual compounding?

The formula is A = P(1 + R/100)^T, where A is the amount. Compound interest is CI = A − P.

How is compound interest calculated for half-yearly compounding?

Use A = P(1 + R/200)^(2T). For example, at 12% per annum for 1 year, use 6% per half-year for 2 periods.

How is compound interest calculated for quarterly compounding?

Use A = P(1 + R/400)^(4T). The annual rate is divided by 4, and the number of years is multiplied by 4.

What is the difference between CI and SI for two years?

For principal P and annual rate R%, CI − SI = P(R/100)^2. For ₹5,000 at 10%, the difference is ₹5,000 × 0.1² = ₹50.

What is the amount of ₹10,000 at 10% compound interest for 2 years?

A = 10,000(1.10)^2 = ₹12,100. Therefore, the compound interest is ₹2,100.

How do you calculate compound interest when the rate changes each year?

Multiply the corresponding growth factors. At 10% for the first year and 20% for the second year, A = P × 1.10 × 1.20.

Continue learning Compound Interest on PrepShots

Continue on PrepShots