Compound Interest | ICSE Class 9 Maths Notes

This note covers principal and interest, simple and compound interest, annual and half-yearly compounding, repeated interest calculations, the amount formula, finding unknown quantities, the difference between simple and compound interest, population growth and depreciation.

What do principal, interest and amount mean?

Principal is the starting sum borrowed or invested. Interest is the extra money paid on borrowed money or received on deposited money. The amount is the principal together with the interest accumulated over the stated time.

Use the symbol P for the original principal, A for the final amount and I for the interest. Their relationship is A = P + I. Consequently, interest is found by subtraction: I = A − P.

How is the rate expressed?

A percentage expresses a quantity per hundred; the symbol % means per cent. An interest rate of 15% per annum means ₹15 interest on each ₹100 for one year. The symbol ₹ denotes rupees, and per annum means per year.

Let r be the numerical annual percentage rate and n the time in years. For a rate of 15% per annum, r = 15, while the fraction used in multiplication is 15/100. These are two ways of expressing the same rate.

Definition: Simple interest, abbreviated SI, is calculated on the original principal throughout the stated period. Compound interest, abbreviated CI, includes interest calculated on interest already added to the principal.

Compounding means adding interest to the current sum before calculating interest for the next period. The compounding period is the interval between these additions. Annual compounding uses one-year periods; half-yearly compounding uses six-month periods.

Read the question to identify which sum is given. A final amount already includes the principal. It cannot be treated as the interest alone. Equally, a stated interest figure must be added to the principal before it can serve as the final amount.

How is simple interest used as the starting point?

Result: Simple interest and total amount

When P is the principal, r the annual percentage rate and n the number of years, SI = Prn/100. Here adjacent letters mean multiplication. The amount under simple interest is found by adding this interest to P.

For one year the interest is Pr/100. If the principal and rate remain unchanged, the same interest is earned or charged in each further year. Multiplying the one-year interest by the number of years therefore gives the total simple interest.

Worked example 1. A sum of ₹10,000 is borrowed at 15% per annum for two years. Find the simple interest and the amount payable.

Answer: Interest for one year = ₹10,000 × 15/100 = ₹1,500. Interest for two years = ₹1,500 × 2 = ₹3,000. The amount payable is ₹10,000 + ₹3,000 = ₹13,000.

Why does this help with compound interest?

Compound interest can be calculated as a sequence of simple-interest calculations, each covering one compounding period. The difference is that the principal is updated between periods. The arithmetic within an individual period remains a percentage calculation on that period's opening sum.

In a year-by-year solution, calculate one year's interest first. Add it to that year's principal. Carry the resulting amount into the next year as the new principal. Repeat this procedure until the required time has elapsed.

The formula SI = Prn/100 should therefore not be applied to the original principal for the whole duration when annual compound interest is required. That would omit the interest earned on earlier interest. It is valid for each separate one-year step with the appropriate updated principal.

How do you calculate compound interest year by year?

Under annual compounding, the amount at the end of one year becomes the principal for the next year. A constant percentage is applied to a changing sum. For a positive interest rate and no intervening payments, the interest consequently grows from one year to the next.

Worked example 2. Heena borrows ₹20,000 for two years at 8% per annum compounded annually. Find the amount and compound interest.

1. Start with the first year's principal, ₹20,000. The first year's interest is ₹20,000 × 8/100 = ₹1,600.
2. Add this interest to obtain the first year's amount: ₹20,000 + ₹1,600 = ₹21,600.
3. Use ₹21,600 as the second year's principal. Its interest is ₹21,600 × 8/100 = ₹1,728.
4. Add again: the final amount is ₹21,600 + ₹1,728 = ₹23,328.

Answer: The amount is ₹23,328. The compound interest is ₹23,328 − ₹20,000 = ₹3,328, also equal to ₹1,600 + ₹1,728.

Which interest is being requested?

The second year's interest is ₹1,728. The total interest for two years is ₹3,328. These answer different questions. Subtracting the original principal from the final amount gives the total interest, whereas subtracting the previous year's amount gives the latest year's interest.

Notice that the rate remains 8% in both years. The larger second-year interest does not require an increase in the rate. It follows from calculating that rate on ₹21,600 instead of ₹20,000.

A useful check is to find total interest in both ways: add the separate annual interest figures, then subtract the original principal from the final amount. The answers should agree. This also checks that every year's interest has been included once.

How does compounding continue over three years?

The same procedure extends to a third year: calculate interest on the second year's closing amount and add it to that amount. It is important to distinguish the original principal from the current principal, meaning the sum used to calculate the next period's interest.

Worked example 3. Compare ₹100 at 10% per annum over three years under simple interest and annual compound interest.

Answer: Under simple interest the yearly interest is ₹10.00, giving an amount of ₹130 and total interest of ₹30 after three years. Under compound interest, the yearly interest figures are ₹10.00, ₹11.00 and ₹12.10, giving an amount of ₹133.10 and total interest of ₹33.10.

What changes in the annual record?

The simple-interest column separates the unchanged interest-bearing principal from the growing total amount. The compound-interest column uses the entire previous amount as the new interest-bearing principal. Reading these two columns carefully explains why equal rates need not produce equal final amounts.

The first year's calculations agree because both begin with the same principal. The difference starts in the second year, when compound interest is calculated on a sum that includes the first year's interest. The third year repeats that process on the further increased sum.

How is the compound amount formula obtained and used?

Result: The annual compound amount formula

For an original principal P at a constant annual rate r%, compounded annually for n years, A = P(1 + r/100)ⁿ. The raised n is an exponent: for a whole number of years, it tells how many times the bracketed factor is multiplied.

The expression 1 + r/100 is the growth factor, meaning the multiplier that includes both the starting sum and its percentage increase. The 1 retains the whole opening sum; r/100 adds the interest fraction.

1. After the first year, amount = P + Pr/100 = P(1 + r/100).
2. For the second year, multiply the whole first-year amount by the same factor, giving P(1 + r/100)².
3. For the third year, multiply once more, giving P(1 + r/100)³.
4. Continuing the same operation for n complete annual periods gives A = P(1 + r/100)ⁿ.

The formula calculates amount. To obtain compound interest, use CI = A − P. Combining the two statements gives CI = P[(1 + r/100)ⁿ − 1]. The square brackets group the full expression from which the original principal is removed.

Worked example 4. Find compound interest on ₹12,600 for two years at 10% per annum compounded annually.

Answer: A = ₹12,600 × (1 + 10/100)² = ₹12,600 × 11/10 × 11/10 = ₹15,246. Therefore CI = ₹15,246 − ₹12,600 = ₹2,646.

Worked example 5. Find compound interest on ₹8,000 for two years at 5% per annum compounded annually.

Answer: A = ₹8,000 × (21/20)² = ₹8,820. Hence CI = ₹8,820 − ₹8,000 = ₹820. A year-by-year check gives first-year interest ₹400 and second-year interest ₹420, whose sum is ₹820.

Keep the whole bracket under the exponent. Squaring the growth factor means multiplying the factor by itself; it does not mean doubling the percentage rate. Fractions such as 11/10 and 21/20 can keep these calculations exact.

What changes when interest is compounded half-yearly?

With half-yearly compounding, interest is added after each six-month interval. A rate quoted per annum still refers to one year. The rate used for one half-year is therefore half the annual percentage rate, while the number of compounding periods is twice the number of years.

Result: Match the rate to the period

Let h be the number of completed half-year periods. If n is the duration in years, h = 2n. The percentage rate for each half-year is r/2, and its fraction is r/200. The amount is A = P(1 + r/200)ʰ.

For two years, apply the half-year factor four times. For three years, apply it six times. These counts come from the two half-years in each year. Subtract the original principal afterwards to obtain compound interest, just as for annual compounding.

How should the calculation be organised?

First write the quoted annual rate and duration. Next convert both to half-year units. Then multiply the original principal by the half-year growth factor once for every completed half-year. Finally state whether the result is the amount or the interest.

Note: Halving the rate without doubling the period count gives too few additions of interest. Doubling the period count while retaining the annual rate applies a full year's percentage in every half-year.

The method still represents repeated simple-interest calculations on a growing principal. The shorter interval changes when interest is added, so an annual-compounding answer cannot be transferred unchanged to a half-yearly question with the same principal, annual rate and duration.

How can the formula find an unknown principal, rate or time?

An unknown quantity is a value the question asks you to find. Begin by identifying the known quantities and the compounding period. The relationship A = P(1 + r/100)ⁿ can then be rearranged, meaning rewritten without changing its equality, to isolate the unknown.

How do you work backwards from an amount?

If A, r and n are known for annual compounding, divide by the complete growth factor: P = A/(1 + r/100)ⁿ. Division reverses the repeated multiplication that originally produced the amount. Reducing A by the same percentage is a different operation.

If CI and P are known, first use A = P + CI. If A and CI are known, use P = A − CI. These relationships use the total interest accumulated over the stated duration, not the interest for the last year alone.

If CI, r and n are known, the combined formula gives P = CI/[(1 + r/100)ⁿ − 1]. The subtraction belongs inside the denominator because CI is the increase above the original principal. This expression assumes the denominator is non-zero.

How do you recover rate and time?

For a two-year annual-compounding problem, A/P = (1 + r/100)². The symbol √ denotes the positive square root, the positive number whose square is the given value. Thus r = 100[√(A/P) − 1] for a non-negative interest rate.

For an unknown number of years, compare A/P with repeated powers of the known annual factor. The number of factors needed gives n. If the question uses half-yearly compounding, count half-years first and divide that count by two to express the duration in years.

If SI, P and r are supplied, SI = Prn/100 gives n = 100SI/(Pr). If SI, P and n are supplied instead, it gives r = 100SI/(Pn). Use the recovered time or rate in the compound amount formula, keeping the units consistent.

Check a recovered value by substituting it into the original relationship. A principal found by division must reproduce the given amount when multiplied forward. This checks both the algebra and whether the interest period has been interpreted correctly.

How are simple and compound interest compared?

A valid comparison uses the same original principal, annual rate and duration, with the compound-interest frequency specified. Simple interest leaves the interest-bearing principal unchanged. Compound interest adds each period's interest before calculating the interest for the following period.

Worked example 6. For ₹20,000 at 8% per annum over two years, find how much annual compound interest exceeds simple interest.

Answer: SI = ₹20,000 × 8 × 2/100 = ₹3,200. Annual compounding gives CI = ₹3,328. The difference is ₹3,328 − ₹3,200 = ₹128. This equals 8% of the first year's interest, ₹1,600.

Result: The two-year difference

Let D denote CI − SI for two years with annual compounding at a constant rate. The first year's interest is Pr/100. In the second year, compound interest includes an extra r% on that first year's interest.

Therefore D = P(r/100)². This is a two-year result with annual compounding. It should not be used unchanged for three years or for half-yearly compounding, because those cases involve a different number of interest additions.

If D and r are known, P = D/(r/100)². If D and P are known, r = 100√(D/P), taking a non-negative rate. These are reverse forms of the same relationship, so the conditions remain the same.

How can simple interest help identify the difference?

For these same two years, SI = 2Pr/100. Dividing the difference by SI gives D/SI = r/200. Hence r = 200D/SI when SI is non-zero. The principal can then be found from the simple-interest formula.

For a longer comparison, calculate the two interest totals separately using their respective formulas and then subtract. This makes the duration explicit and avoids extending the two-year shortcut beyond the situation for which it was derived.

How does the formula describe population growth?

Growth is an increase in a quantity. When the same percentage increase is applied successively to the latest quantity, the calculation has the form of compound interest. The starting population takes the place of principal, and the final population takes the place of amount.

For a constant annual growth rate r%, use A = P(1 + r/100)ⁿ, with P now meaning the starting population and A the population after n years. These symbols represent numbers of people here, so the answer is not expressed in rupees.

Worked example 7. A city's population was 20,000 at the end of 1997. With an increase of 5% per annum, estimate its population at the end of 2000.

Answer: The three annual increases give 20,000 × (21/20)³ = 23,152.5. The estimated population is therefore 23,153. The successive populations before the final increase are 21,000 and 22,050.

Why is the answer an estimate?

The percentage calculation produces 23,152.5, but a population is expressed as a whole number of people. State the rounded result as an estimated population. Keep the unrounded calculation visible so that the rounding can be checked.

Each increase uses the latest population: 5% of 20,000 is 1,000; 5% of 21,000 is 1,050; 5% of 22,050 is 1,102.5. The percentage stays constant even though the number added changes.

To find an earlier population from a later one, divide by the growth factor for the intervening years. To find a later population, multiply by it. The direction of time determines whether the operation builds up the quantity or reverses that build-up.

The period must still match the rate. A growth rate stated per hour belongs with a count of hours; a rate stated per annum belongs with a count of years. The repeated percentage idea is the same, but the unit of time changes.

How is depreciation calculated?

Depreciation means reduction in an item's value due to use and age. When an item loses a fixed percentage of its current value each year, each reduction is calculated on the value remaining at the beginning of that year.

Result: The repeated decrease formula

Let P represent the initial value, A the value after n years and r the annual percentage depreciation rate. Then A = P(1 − r/100)ⁿ. The expression 1 − r/100 is the retention factor, the fraction of the current value left after one reduction.

The minus sign distinguishes a percentage decrease from a percentage increase. Find the final value using the retention factor; find the total loss in value by subtracting that final value from the initial value, P − A.

Worked example 8. A television is bought for ₹21,000 and depreciates by 5% after one year. Find its value after that year.

Answer: Reduction = ₹21,000 × 5/100 = ₹1,050. The remaining value is ₹21,000 − ₹1,050 = ₹19,950. Equivalently, ₹21,000 × (1 − 5/100) = ₹21,000 × 19/20 = ₹19,950.

What is the base for each reduction?

For a further year at the same percentage depreciation rate, the new calculation would begin with the remaining value, not the original purchase price. A fixed percentage reduction is therefore different from subtracting the same fixed sum each year.

To recover an initial value from a later value, divide by the complete retention factor for the stated number of years. Do not merely increase the later value by the depreciation percentage: that percentage would then be calculated on a smaller base.

Before calculating, identify whether the question asks for the remaining value or the depreciation itself. The television's value is ₹19,950, while its depreciation is ₹1,050. Both quantities follow from the same calculation, but they describe different parts of the original ₹21,000.

Glossary

• Principal — The starting sum borrowed or invested, before interest is added to it.
• Interest — Extra money paid on a loan or received on money deposited for a stated time.
• Amount — The total of the original principal and the interest accumulated over the stated period.
• Simple interest — Interest calculated on the original principal throughout the whole stated period.
• Compound interest — Interest accumulated when earlier interest is added before interest for the next period is calculated.
• Per annum — Per year, indicating that a quoted rate applies to a period of one year.
• Compounding period — The interval after which interest is added before the next interest calculation begins.
• Annual compounding — Adding interest once each year, so the closing amount becomes the next year's principal.
• Half-yearly compounding — Adding interest after each six-month period, using half the quoted annual percentage rate.
• Growth factor — The multiplier that includes the current quantity together with its increase for one period.
• Exponent — A raised number or symbol indicating a power, such as repeated multiplication for a positive whole-number power.
• Depreciation — A reduction in an item's value due to its use and age.
• Retention factor — The fraction of a quantity remaining after a specified percentage decrease has been applied.

Common errors and misconceptions

• Misconception: The compound amount formula directly gives compound interest. Correct: It gives the final amount, including the original principal. Subtract that principal to obtain the compound interest for the full period.
• Misconception: Compound interest for the second year is calculated on the original sum. Correct: With annual compounding, use the first year's closing amount, including the interest already added.
• Misconception: Doubling an annual rate gives the compound increase over two years. Correct: Multiply by the annual growth factor twice. The second multiplication also acts on the first year's interest.
• Misconception: Half-yearly compounding changes the rate but leaves the period count unchanged. Correct: Halve the annual percentage rate and double the number of years to count the completed half-years.
• Misconception: P(r/100)² gives the difference between compound and simple interest for any duration. Correct: This difference formula applies to two years with annual compounding at the same constant rate.
• Misconception: An earlier value can be recovered by subtracting the growth percentage from the later value. Correct: Divide by the complete growth factor, because the original increase was calculated on the earlier base.
• Misconception: Depreciation is the value left after a reduction. Correct: Depreciation is the loss in value. Subtract that loss from the starting value to find the remaining value.

Exam-style questions with model answers

Q1. What are principal and compound interest? [2 marks]

1. Principal is the starting sum borrowed or invested before interest is added.
2. Compound interest is accumulated interest calculated by adding each period's interest before calculating interest for the next period.

Q2. A sum of ₹10,000 is borrowed at 15% per annum simple interest for two years. Find the interest and final amount. [3 marks]

1. The principal is ₹10,000 and the annual rate is 15%. Therefore the interest for one year is ₹10,000 × 15/100 = ₹1,500.
2. Simple interest uses the same principal in each year. The total interest for two years is therefore ₹1,500 × 2 = ₹3,000.
3. The final amount includes the principal and the interest. It is ₹10,000 + ₹3,000 = ₹13,000.

Q3. Heena borrows ₹20,000 at 8% per annum compounded annually for two years. Calculate the amount year by year and find the compound interest. [5 marks]

1. The first year's interest is calculated on the original principal, ₹20,000. At 8% for one year it is ₹20,000 × 8/100 = ₹1,600.
2. Add this interest to the principal. The amount at the end of the first year is ₹20,000 + ₹1,600 = ₹21,600.
3. This amount becomes the second year's principal. The second year's interest is therefore ₹21,600 × 8/100 = ₹1,728.
4. Add the second year's interest to its opening principal. The amount at the end of two years is ₹21,600 + ₹1,728 = ₹23,328.
5. Compound interest is the final amount less the original principal: ₹23,328 − ₹20,000 = ₹3,328. This also equals the sum of the two annual interest figures.

Q4. Find the compound interest on ₹12,600 at 10% per annum compounded annually for two years. [3 marks]

1. Use the annual amount formula, with principal ₹12,600, annual percentage rate 10 and time two years: A = ₹12,600 × (1 + 10/100)², where A is the final amount.
2. The annual growth factor is 11/10. Thus A = ₹12,600 × 11/10 × 11/10 = ₹15,246.
3. Subtract the original principal from the amount to obtain compound interest: ₹15,246 − ₹12,600 = ₹2,646.

Q5. For ₹20,000 at 8% per annum over two years, calculate simple interest and annual compound interest, then find their difference. [5 marks]

1. Simple interest is calculated on the original principal for the full duration. It equals ₹20,000 × 8 × 2/100 = ₹3,200.
2. For annual compounding, the first year's interest is ₹20,000 × 8/100 = ₹1,600, giving a first-year amount of ₹21,600.
3. The second year's interest is calculated on ₹21,600. It equals ₹21,600 × 8/100 = ₹1,728, giving a final amount of ₹23,328.
4. The compound interest for both years together is the final amount less the original principal: ₹23,328 − ₹20,000 = ₹3,328.
5. The required difference is ₹3,328 − ₹3,200 = ₹128. Compound interest exceeds simple interest by the interest earned on the first year's ₹1,600.

Q6. A principal P is invested at an annual percentage rate r for n years, where n is a positive whole number. Interest is compounded half-yearly. State the period rate, period count and formulas for amount and compound interest. [4 marks]

1. The rate per half-year is r/2 per cent, because each compounding period lasts half a year and r is the annual percentage rate.
2. There are two half-years in each year, so the total number of compounding periods is 2n. Denote this count by h.
3. The amount, denoted by A, is A = P(1 + r/200)ʰ, because the half-year interest fraction is r/200.
4. The compound interest, denoted by CI, is CI = A − P, removing the original principal from the final amount.

Q7. A city had a population of 20,000 at the end of 1997. It increases at 5% per annum. Estimate its population at the end of 2000, showing the successive annual changes. [5 marks]

1. The duration is three years. Apply the annual percentage increase successively, using the latest population as the base for each new year's calculation.
2. The first increase is 5% of 20,000, which is 1,000. The population after the first annual increase is therefore 21,000.
3. The next increase is 5% of 21,000, which is 1,050. The population after the second annual increase is therefore 22,050.
4. The final increase is 5% of 22,050, which is 1,102.5. The calculated population at the end of 2000 is 23,152.5.
5. Express the population as a whole number of people. Rounding the calculation gives an estimated population of 23,153 at the end of 2000.

Q8. A television bought for ₹21,000 depreciates by 5% after one year. Find its depreciation and its remaining value, and verify the value using a multiplier. [3 marks]

1. Depreciation is the reduction in value due to use and age. The reduction for this year is 5% of ₹21,000, equal to ₹1,050.
2. Subtract the reduction from the original value. The television's value after one year is ₹21,000 − ₹1,050 = ₹19,950.
3. The fraction remaining is 1 − 5/100 = 19/20. Multiplication confirms the value: ₹21,000 × 19/20 = ₹19,950.

Key takeaways

• Principal is the original sum; amount includes that sum and interest. Subtract principal from amount to find total interest.
• Simple interest uses the original principal throughout; compound interest uses an updated principal after each addition of interest.
• Annual compounding multiplies the principal by the annual growth factor once for each completed year.
• The compound amount formula gives the final amount. A further subtraction is needed to find compound interest.
• Half-yearly compounding uses half the annual percentage rate and twice the number of years as its period count.
• The two-year difference between annual compound interest and simple interest is the interest on the first year's interest.
• Work backwards to an original quantity by dividing by its accumulated factor, with the correct number of periods.
• Population growth uses repeated percentage increases, while depreciation uses repeated percentage decreases on the current value.

Test yourself

What sum becomes the next year's principal under annual compounding?

The previous year's closing amount becomes the next year's principal, including the interest just added.

Why can annual interest increase even when the percentage rate stays unchanged?

The same percentage is applied to a larger principal after earlier interest has been added.

An original principal of ₹12,600 becomes ₹15,246. What is the total compound interest?

Subtract the original principal: ₹15,246 − ₹12,600 = ₹2,646 in total compound interest.

How many compounding periods occur in three years with half-yearly compounding?

There are six compounding periods, because each year contains two half-year periods.

For what duration does the annual-compounding difference formula P(r/100)² apply, with P the principal and r the annual percentage rate?

It gives compound interest minus simple interest over two years, using the same principal and constant rate.

Why is an earlier population found by division when the later population and growth rate are known?

Division by the accumulated growth factor reverses the repeated multiplication that produced the later population.

A television starts at ₹21,000 and loses ₹1,050 in value. What remains?

The remaining value is ₹21,000 − ₹1,050 = ₹19,950 after the stated reduction.

What does the minus sign in a depreciation factor represent?

It subtracts the fraction lost from the whole, leaving the fraction of value retained.