Interest: Simple and Compound Interest Formulas

Interest is the additional amount paid or earned on a principal over time. This chapter covers simple interest, compound interest, amount, rate, time, changing compounding periods and the difference between SI and CI. It also includes formulas, shortcuts and numerical methods for solving interest questions accurately.

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What Is Interest?

Interest is the money charged or earned on a principal for using it over a specified period. The total value after adding interest to the principal is called the amount.

If P is the principal, I is the interest and A is the amount, then A = P + I. Interest may be calculated directly on the original principal, as in simple interest, or on the accumulated amount after each period, as in compound interest. The rate is generally expressed as an annual percentage.

Interest Formula & Tricks

Important Formulas

Simple Interest
SI = (P × R × T) / 100

P is the principal, R is the annual rate in percent and T is the time in years.

Amount in Simple Interest
A = P + SI = P(1 + RT/100)

The amount is the principal plus the simple interest earned or charged.

Compound Amount
A = P(1 + R/100)^n

For annual compounding, n is the number of years and R is the annual rate.

Compound Interest
CI = A - P = P[(1 + R/100)^n - 1]

Compound interest is calculated by subtracting the original principal from the compound amount.

Half-Yearly Compounding
A = P(1 + R/200)^(2T)

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

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

For quarterly compounding, the rate per period is R/4 and the number of periods is 4T.

Difference Between CI and SI for Two Years
CI - SI = P(R/100)^2 = PR^2/10,000

This shortcut applies when the principal, annual rate and time are fixed for exactly two years with annual compounding.

Difference Between CI and SI for Three Years
CI - SI = P[3(R/100)^2 + (R/100)^3]

This applies for exactly three years with annual compounding and a constant annual rate.

Quick Tricks

Convert the Time Unit Before Applying the Formula

Use years in the standard SI formula. Convert months into years by dividing by 12 and days into years according to the convention given in the question.

Example: For P = ₹5,000, R = 8% and T = 9 months, SI = 5000 × 8 × (9/12) / 100 = ₹300.
Use the Two-Year CI-SI Shortcut

For two years of annual compounding, calculate the difference directly as PR²/10,000 instead of finding both interests separately.

Example: For P = ₹10,000 and R = 10%, CI - SI = 10,000 × 10² / 10,000 = ₹100.
Adjust Rate and Number of Periods Together

When interest is compounded more than once a year, divide the annual rate by the number of compounding periods per year and multiply the time by the same number.

Example: At 12% per year compounded quarterly for 2 years, use 3% per quarter and 8 quarters: A = P(1.03)^8.
Use Successive Growth Factors

For different rates in successive years, multiply the separate growth factors rather than using one average rate.

Example: At 10% in the first year and 20% in the second year, A = P × 1.10 × 1.20 = 1.32P.

Interest Concepts

Simple Interest and Amount

Simple interest is calculated only on the original principal throughout the entire period.

The formula is SI = PRT/100, and the amount is A = P + SI. Since the interest for every equal time period remains constant, simple interest increases linearly with time. For example, on ₹8,000 at 5% per annum for 3 years, SI = 8000 × 5 × 3 / 100 = ₹1,200 and A = ₹9,200.

Example: If ₹6,000 earns simple interest at 7% per annum for 2 years, SI = ₹840 and the amount is ₹6,840.

Compound Interest with Annual Compounding

Compound interest is calculated on the principal plus the interest accumulated in earlier periods.

For annual compounding, A = P(1 + R/100)^n and CI = A - P. The interest is added to the principal at the end of each year. For ₹10,000 at 10% per annum for 2 years, A = 10,000 × 1.1² = ₹12,100, so CI = ₹2,100.

Example: The first year's interest is ₹1,000. The second year's interest is 10% of ₹11,000, or ₹1,100.

Compounding Half-Yearly and Quarterly

For multiple compounding periods in a year, use the rate and number of periods for each compounding interval.

If compounding occurs m times per year, the amount is A = P(1 + R/(100m))^(mT). Thus, half-yearly compounding uses rate R/2 per half-year and 2T periods; quarterly compounding uses rate R/4 per quarter and 4T periods.

Example: For ₹10,000 at 8% per annum compounded half-yearly for 1 year, A = 10,000(1.04)² = ₹10,816 and CI = ₹816.

Difference Between Simple and Compound Interest

For the same principal, rate and time, compound interest is greater than simple interest when compounding occurs at least once before the final calculation.

For two years with annual compounding, the difference is CI - SI = PR²/10,000. For three years, the difference is P[3(R/100)² + (R/100)³]. For ₹10,000 at 10% for 2 years, the difference is ₹100; SI is ₹2,000 and CI is ₹2,100.

Example: The two-year shortcut gives 10,000 × 10² / 10,000 = ₹100.

Changing Rates and Depreciation

When rates change in different periods, multiply the corresponding growth or reduction factors for each period.

For successive increases of r1%, r2% and so on, A = P(1 + r1/100)(1 + r2/100) ... . For depreciation at rates d1%, d2% and so on, A = P(1 - d1/100)(1 - d2/100) ... . A decrease of 10% followed by a decrease of 20% leaves 0.9 × 0.8 = 0.72 of the original value, which is a total decrease of 28%.

Example: If ₹20,000 grows by 10% and then by 5%, the final value is 20,000 × 1.10 × 1.05 = ₹23,100.

Interest Video Lessons

Watch short topic-wise lessons for quick revision.

6 Lessons
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Simple Interest Formula: Find P, R, T

Learn how to use the simple interest formula to calculate Principal (P), Rate (R), or Time (T) when the other values are given.

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Practice Interest Questions

Practise published questions related to this topic.

1A dealer marks his goods at 25% above the cost price and allows a discount of 28% on the marked price. What is his gain or loss percentage?→ 2An officer computes (√50 + √8).Simplify.→ 3A shopkeeper fixes the marked price of an item 120% above its cost price. What percentage of discount on marked price should be allowed to gain 43%?→ 4An article is sold for 30 percent loss. The value of discount given is equal to the (7/10) of the selling price. Cost price of the article is what percent of its marked price?→ 5If the ratio of two numbers is 8 : 3, and their HCF is 8, then their LCM is:→ 6The marked price of an article is 50 percent more than its cost price. What maximum discount percentage can be offered on this article so that there is no loss?→ 7An officer checks the cost ratio: √18 ÷ √2 .Simplify.→ 8A sofa set marked at Rs. 78,000 is sold at Rs. 65,000. What is the percentage of discount?→ 9The HCF and LCM of two numbers are 27 and 324, respectively. If one number is 81, find the other number.→ 10A clerk simplifies (√20)(√5) for budget entries. The result is:→

Interest Quick Quiz

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Quick Revision Notes

Interest Revision Points

Use these formulas and rules for quick revision of interest questions.

  • SI = PRT/100 and A = P + SI.
  • For annual compound interest, A = P(1 + R/100)^n and CI = A - P.
  • For m compounding periods per year, use A = P(1 + R/(100m))^(mT).
  • Convert months into years before using the simple interest formula.
  • For two years, CI - SI = PR²/10,000.
  • For three years, CI - SI = P[3(R/100)² + (R/100)³].
  • For changing rates, multiply the separate growth factors.
  • For depreciation, replace each growth factor 1 + r/100 with 1 - r/100.

Interest FAQs

What is the formula for simple interest?

Simple interest is SI = (P × R × T)/100, where P is the principal, R is the annual rate in percent and T is time in years.

How is compound interest calculated annually?

Use A = P(1 + R/100)^n and then CI = A - P. Here, n is the number of years.

What is the amount on ₹5,000 at 6% simple interest for 4 years?

SI = 5,000 × 6 × 4 / 100 = ₹1,200. Therefore, the amount is ₹5,000 + ₹1,200 = ₹6,200.

How does half-yearly compounding change the interest formula?

Use A = P(1 + R/200)^(2T). The annual rate is divided by 2, and the number of years is multiplied by 2.

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

For annual compounding, CI - SI = PR²/10,000. For P = ₹20,000 and R = 5%, the difference is ₹20,000 × 25 / 10,000 = ₹50.

Can the average of two yearly rates be used for successive compound interest?

No. Multiply the growth factors separately. Rates of 10% and 20% give a factor of 1.10 × 1.20 = 1.32, not a factor based on a simple average rate.

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