The Extra Money Is Interest Earned on Last Year's Interest
Work compound interest out year by year on a growing principal, handle a rate that changes each year, find exactly why compound interest beats simple interest, and work backwards to the principal or rate.
Where exactly does the extra money in compound interest come from?
Put ₹6000 in an account at 10% a year for two years.
Simple interest gives ₹600 a year, twice, so ₹1200 in all.
Compound interest gives ₹1260 — sixty rupees more.
That ₹60 is not a mysterious bonus, and it is not a different rate. It is interest on the first year's interest.
At the end of the first year you had ₹600 of interest sitting in the account. In the second year that ₹600 earns 10% like everything else, and . That is the entire difference, exactly.
So the whole idea of compound interest is one sentence: the interest is added to the principal, and the next year's interest is calculated on the larger amount. Simple interest always uses the original sum; compound interest uses a growing one.
This page works it out that way — year by year, with the simple interest formula applied to a principal that changes — which is slower than the formula but shows exactly what is happening at each step.
This page covers the ICSE Class 9 Mathematics chapter on compound interest without the formula: the year-by-year method, a rate that changes each year, the difference between compound and simple interest, and finding the principal, rate or time from given information.
Simple interest gives ₹600 a year, twice, so ₹1200 in all.
Compound interest gives ₹1260 — sixty rupees more.
That ₹60 is not a mysterious bonus, and it is not a different rate. It is interest on the first year's interest.
At the end of the first year you had ₹600 of interest sitting in the account. In the second year that ₹600 earns 10% like everything else, and . That is the entire difference, exactly.
So the whole idea of compound interest is one sentence: the interest is added to the principal, and the next year's interest is calculated on the larger amount. Simple interest always uses the original sum; compound interest uses a growing one.
This page works it out that way — year by year, with the simple interest formula applied to a principal that changes — which is slower than the formula but shows exactly what is happening at each step.
This page covers the ICSE Class 9 Mathematics chapter on compound interest without the formula: the year-by-year method, a rate that changes each year, the difference between compound and simple interest, and finding the principal, rate or time from given information.
How do you work out compound interest year by year?
Find the interest for one year with the simple interest formula, add it to the principal, and use that new amount as the principal for the next year.
The only formula needed is the simple interest one, applied one year at a time:
Worked example 1. Find the compound interest on ₹8000 at 10% per annum for 2 years.
Year 1. The principal is ₹8000.
Year 2. The principal is now ₹8800, not ₹8000.
So the compound interest is
Worked example 2, over three years. Find the amount and the compound interest on ₹12500 at 8% per annum for 3 years.
Year 1. , so the amount is .
Year 2. , so the amount is .
Year 3. , so the amount is .
Notice that the yearly interest rises every year — ₹1000, then ₹1080, then ₹1166.40 — even though the rate never changed. That is the whole effect: the rate is fixed and the principal it acts on is growing.
And notice that the three interests add up to the compound interest: , which matches. That addition is a free check on every year-by-year calculation, and it catches an arithmetic slip in any of the three rows.
A point of method worth fixing now. Write the working as three short blocks, each with the principal, the interest and the new amount on separate lines. A single long chain of arithmetic is where marks are lost, because if one number is wrong the examiner cannot see which step failed.
The only formula needed is the simple interest one, applied one year at a time:
Worked example 1. Find the compound interest on ₹8000 at 10% per annum for 2 years.
Year 1. The principal is ₹8000.
Year 2. The principal is now ₹8800, not ₹8000.
So the compound interest is
Worked example 2, over three years. Find the amount and the compound interest on ₹12500 at 8% per annum for 3 years.
Year 1. , so the amount is .
Year 2. , so the amount is .
Year 3. , so the amount is .
Notice that the yearly interest rises every year — ₹1000, then ₹1080, then ₹1166.40 — even though the rate never changed. That is the whole effect: the rate is fixed and the principal it acts on is growing.
And notice that the three interests add up to the compound interest: , which matches. That addition is a free check on every year-by-year calculation, and it catches an arithmetic slip in any of the three rows.
A point of method worth fixing now. Write the working as three short blocks, each with the principal, the interest and the new amount on separate lines. A single long chain of arithmetic is where marks are lost, because if one number is wrong the examiner cannot see which step failed.
What if the rate of interest is different in each year?
Nothing changes in the method — use each year's own rate on that year's own principal.
The year-by-year method handles a changing rate without any extra work, which is exactly why it is worth learning before the formula.
Worked example. Find the amount and the compound interest on ₹10000 if the rate is 5% for the first year, 6% for the second year and 8% for the third year.
Year 1, at 5% on ₹10000:
Year 2, at 6% on ₹10500:
Year 3, at 8% on ₹11130:
Check by adding the three interests: .
Now something genuinely surprising about this calculation. Suppose the rates had come in a different order — 8% first, then 6%, then 5%. Work it through and the final amount is exactly the same, ₹12020.40.
Why? Each year multiplies the amount by a factor: , then , then . The final amount is
and multiplication does not care about order. Rearranging the three factors gives the same product, so it gives the same amount.
So the order of the rates makes no difference to the final amount — though the interest earned in each individual year certainly does change. A question that gives you three rates and asks for the final amount can therefore be done in whichever order is easiest to multiply, and a question asking for the second year's interest cannot.
A common variant worth expecting. Sometimes the rate is given as "5% for the first year and 6% thereafter", or a sum is deposited for two years at one rate and then withdrawn and redeposited at another. In every case the rule is the same: whatever the amount is at the start of a year, that is the principal for that year.
The year-by-year method handles a changing rate without any extra work, which is exactly why it is worth learning before the formula.
Worked example. Find the amount and the compound interest on ₹10000 if the rate is 5% for the first year, 6% for the second year and 8% for the third year.
Year 1, at 5% on ₹10000:
Year 2, at 6% on ₹10500:
Year 3, at 8% on ₹11130:
Check by adding the three interests: .
Now something genuinely surprising about this calculation. Suppose the rates had come in a different order — 8% first, then 6%, then 5%. Work it through and the final amount is exactly the same, ₹12020.40.
Why? Each year multiplies the amount by a factor: , then , then . The final amount is
and multiplication does not care about order. Rearranging the three factors gives the same product, so it gives the same amount.
So the order of the rates makes no difference to the final amount — though the interest earned in each individual year certainly does change. A question that gives you three rates and asks for the final amount can therefore be done in whichever order is easiest to multiply, and a question asking for the second year's interest cannot.
A common variant worth expecting. Sometimes the rate is given as "5% for the first year and 6% thereafter", or a sum is deposited for two years at one rate and then withdrawn and redeposited at another. In every case the rule is the same: whatever the amount is at the start of a year, that is the principal for that year.
Why is compound interest always more than simple interest?
Because simple interest never earns anything itself, and compound interest does — and for two years the whole difference is the interest on the first year's interest.
Worked example 1, over two years. Find the difference between the compound interest and the simple interest on ₹6000 at 10% per annum for 2 years.
Simple interest:
Compound interest, year by year:
Year 1: , amount .
Year 2: , amount .
Difference:
Now see where the ₹60 is, because this is the idea of the whole chapter. The first year's interest was ₹600. In the second year, that ₹600 sat in the account and earned interest of its own:
Exactly the difference. So for two years:
Check that shortcut on this problem: . It agrees.
Worked example 2, over three years. Find the difference for ₹8000 at 5% per annum for 3 years.
Compound, year by year: , amount ₹8400; , amount ₹8820; , amount ₹9261.
Notice that the two-year shortcut does not apply here. For three years the difference is ₹61, while — nothing like it.
**So is the difference for TWO years only. For three years there is interest on interest in the second year and in the third, and interest on that interest as well, so no such short formula exists at this level.
That is worth knowing precisely, because it is the standard trap. A question that says "for 2 years" may be done with the shortcut; a question that says "for 3 years" must be worked out in full. Checking which one you have been given takes two seconds and saves the whole answer.**
Worked example 1, over two years. Find the difference between the compound interest and the simple interest on ₹6000 at 10% per annum for 2 years.
Simple interest:
Compound interest, year by year:
Year 1: , amount .
Year 2: , amount .
Difference:
Now see where the ₹60 is, because this is the idea of the whole chapter. The first year's interest was ₹600. In the second year, that ₹600 sat in the account and earned interest of its own:
Exactly the difference. So for two years:
Check that shortcut on this problem: . It agrees.
Worked example 2, over three years. Find the difference for ₹8000 at 5% per annum for 3 years.
Compound, year by year: , amount ₹8400; , amount ₹8820; , amount ₹9261.
Notice that the two-year shortcut does not apply here. For three years the difference is ₹61, while — nothing like it.
**So is the difference for TWO years only. For three years there is interest on interest in the second year and in the third, and interest on that interest as well, so no such short formula exists at this level.
That is worth knowing precisely, because it is the standard trap. A question that says "for 2 years" may be done with the shortcut; a question that says "for 3 years" must be worked out in full. Checking which one you have been given takes two seconds and saves the whole answer.**
Formula
How do you find the principal or the rate when the interest is given?
Write the two-year amount as a multiple of the principal, then solve the resulting simple equation.
For two years at rate , each year multiplies the principal by , so
Every backwards question is this one relation with a different unknown.
Worked example 1 — find the principal. The compound interest on a sum for 2 years at 10% per annum is ₹1050. Find the sum.
At 10%, each year multiplies by , so over two years the factor is . Hence
Check it year by year. Year 1: , amount ₹5500. Year 2: , amount ₹6050. So . Correct.
Worked example 2 — find the rate. A sum of ₹6250 amounts to ₹7290 in 2 years, interest compounded annually. Find the rate.
Check it. , and . Correct.
Worked example 3 — find the time. In what time will ₹5000 amount to ₹5832 at 8% per annum, compounded annually?
and since , the time is years.
Check it. , and . Correct.
**Notice that examples 2 and 3 produced the same number, **, from completely different data. That is not a coincidence — it is the same growth factor appearing in both, once with the rate unknown and once with the time unknown.
So all three backwards problems are the same equation rearranged, and recognising the growth factor is what solves them:
- Divide by to get the growth factor for the whole period
- If the rate is wanted, take the square root of that factor (for 2 years) and subtract
- If the time is wanted, see what power of the factor is
- If the principal is wanted, divide the given interest by (factor )
One practical warning. The square root step only gives a clean answer because examiners choose numbers that work — is , is , is . **If your square root is not coming out neatly, check the division first**, because that is almost always where the slip is.
For two years at rate , each year multiplies the principal by , so
Every backwards question is this one relation with a different unknown.
Worked example 1 — find the principal. The compound interest on a sum for 2 years at 10% per annum is ₹1050. Find the sum.
At 10%, each year multiplies by , so over two years the factor is . Hence
Check it year by year. Year 1: , amount ₹5500. Year 2: , amount ₹6050. So . Correct.
Worked example 2 — find the rate. A sum of ₹6250 amounts to ₹7290 in 2 years, interest compounded annually. Find the rate.
Check it. , and . Correct.
Worked example 3 — find the time. In what time will ₹5000 amount to ₹5832 at 8% per annum, compounded annually?
and since , the time is years.
Check it. , and . Correct.
**Notice that examples 2 and 3 produced the same number, **, from completely different data. That is not a coincidence — it is the same growth factor appearing in both, once with the rate unknown and once with the time unknown.
So all three backwards problems are the same equation rearranged, and recognising the growth factor is what solves them:
- Divide by to get the growth factor for the whole period
- If the rate is wanted, take the square root of that factor (for 2 years) and subtract
- If the time is wanted, see what power of the factor is
- If the principal is wanted, divide the given interest by (factor )
One practical warning. The square root step only gives a clean answer because examiners choose numbers that work — is , is , is . **If your square root is not coming out neatly, check the division first**, because that is almost always where the slip is.
Exam tip
Exam tip: write each year as its own block and add the interests to check
Lay each year out in three lines — principal, interest, new amount. One long chain of arithmetic hides the error and loses the method marks.
The principal for year 2 is the AMOUNT at the end of year 1, never the original sum. Write it down explicitly.
Use the simple interest formula for one year at a time: .
Check by adding the yearly interests. They must total the compound interest — . This one line catches almost every slip.
For a changing rate, use each year's own rate on that year's own principal. Nothing else changes.
The order of the rates does not affect the final amount — the factors multiply — but it does affect each individual year's interest.
** is for TWO YEARS ONLY. Check the time before using it; for three years work it out in full.
Explain the two-year difference as interest on the first year's interest — that sentence is often worth a mark and it proves you understand rather than remember.
For a backwards problem, compute the growth factor first.** Then take a square root for the rate, or match a power for the time, or divide by (factor ) for the principal.
Learn the common factors on sight: , , , , .
Keep paise to two decimal places and write the rupee sign in the final answer.
And always verify a backwards answer by working the interest forward again — it takes three lines and confirms the whole solution.
The principal for year 2 is the AMOUNT at the end of year 1, never the original sum. Write it down explicitly.
Use the simple interest formula for one year at a time: .
Check by adding the yearly interests. They must total the compound interest — . This one line catches almost every slip.
For a changing rate, use each year's own rate on that year's own principal. Nothing else changes.
The order of the rates does not affect the final amount — the factors multiply — but it does affect each individual year's interest.
** is for TWO YEARS ONLY. Check the time before using it; for three years work it out in full.
Explain the two-year difference as interest on the first year's interest — that sentence is often worth a mark and it proves you understand rather than remember.
For a backwards problem, compute the growth factor first.** Then take a square root for the rate, or match a power for the time, or divide by (factor ) for the principal.
Learn the common factors on sight: , , , , .
Keep paise to two decimal places and write the rupee sign in the final answer.
And always verify a backwards answer by working the interest forward again — it takes three lines and confirms the whole solution.
Did you know
Why the second year's interest is the only honest measure of growth
Look at the three yearly interests in the ₹12500 example: ₹1000, then ₹1080, then ₹1166.40.
The rate never changed. It was 8% throughout. Yet each year's interest is bigger than the last.
This is worth dwelling on, because it is the only place in school arithmetic where a quantity grows by a fixed percentage rather than a fixed amount — and the two behave completely differently.
A fixed amount added each year gives a straight line. ₹1000 a year for ten years is ₹10000, and the tenth year looks exactly like the first.
A fixed percentage gives something that curves upwards. Each year's addition is larger than the last, because it is a percentage of a larger number. And the gap widens: over ten years at 8%, the interest in the tenth year is roughly twice what it was in the first.
That is why compound growth is so easy to underestimate. A person asked to guess what ₹12500 becomes after twenty years at 8% will usually guess far too low, because the mind naturally adds rather than multiplies.
And the same shape appears far away from money. A population growing at a fixed percentage, a bacterial culture doubling at intervals, the value of a machine falling by a fixed percentage a year, the number of people who have heard a rumour — all of them multiply rather than add, and all of them produce the same upward-bending curve.
So the last section of the next chapter, on population growth and depreciation, is not a separate topic bolted onto compound interest. It is compound interest with the word "rupees" removed, and once you have understood why ₹1080 follows ₹1000, you have understood every one of them.
Which is also why the year-by-year method is worth learning even though the formula is faster. The formula gives you the answer; the three blocks of working show you where it came from.
The rate never changed. It was 8% throughout. Yet each year's interest is bigger than the last.
This is worth dwelling on, because it is the only place in school arithmetic where a quantity grows by a fixed percentage rather than a fixed amount — and the two behave completely differently.
A fixed amount added each year gives a straight line. ₹1000 a year for ten years is ₹10000, and the tenth year looks exactly like the first.
A fixed percentage gives something that curves upwards. Each year's addition is larger than the last, because it is a percentage of a larger number. And the gap widens: over ten years at 8%, the interest in the tenth year is roughly twice what it was in the first.
That is why compound growth is so easy to underestimate. A person asked to guess what ₹12500 becomes after twenty years at 8% will usually guess far too low, because the mind naturally adds rather than multiplies.
And the same shape appears far away from money. A population growing at a fixed percentage, a bacterial culture doubling at intervals, the value of a machine falling by a fixed percentage a year, the number of people who have heard a rumour — all of them multiply rather than add, and all of them produce the same upward-bending curve.
So the last section of the next chapter, on population growth and depreciation, is not a separate topic bolted onto compound interest. It is compound interest with the word "rupees" removed, and once you have understood why ₹1080 follows ₹1000, you have understood every one of them.
Which is also why the year-by-year method is worth learning even though the formula is faster. The formula gives you the answer; the three blocks of working show you where it came from.
Exam relevance
Why does JEE Main care about compound growth?
Because a quantity that multiplies by a fixed factor each period is a geometric progression, and geometric progressions are examined throughout Class 11.
This is the foundation for Class 11 Mathematics Sequences and Series, examined in JEE Main. The amounts , , , form a geometric progression with first term and common ratio . So the compound interest table on this page is a GP written out term by term, and the th-term formula is what the compound interest formula becomes in that notation.
The GP sum formula answers questions this chapter cannot. An annuity — a fixed sum deposited every year, each instalment compounding for a different number of years — is summed with . Instalment, annuity and sinking-fund problems appear in JEE Main, and they are compound interest applied to a series of deposits rather than one.
Growth and decay become exponential functions. Class 11 Relations and Functions and Class 12 Application of Derivatives treat as an exponential function, and Class 12 Differential Equations derives it from the statement that the rate of growth is proportional to the amount present. The population-growth and depreciation problems of the next chapter are the elementary form of that differential equation, and its solution is examined directly.
Logarithms solve for the time. Finding by matching a power works only when examiners choose convenient numbers, as this page notes. Class 11 replaces the guess with , and questions asking in how many years a sum will double are standard — the answer coming out as a logarithm rather than a whole number.
The binomial theorem explains the difference between CI and SI. Class 11 expands The first two terms give exactly the simple interest, and every term after them is the compound excess. For that excess is , which is the shortcut of this page — now derived rather than remembered, and extendable to three years and beyond.
Approximation questions use the same expansion. When is small, , which is why simple interest is a decent approximation over a short period and a poor one over a long one. JEE Main sets approximation questions of exactly this kind.
What the questions look like. For board work, expect find the compound interest year by year for 2 or 3 years, compute the amount when the rate changes each year, find the difference between compound and simple interest, and find the principal, rate or time from given data. Every year's working must be shown. For JEE Main, expect GP terms and sums, annuity problems, logarithms for the time, and binomial approximations.
How board and competitive emphasis differ. A board paper rewards the three-line-per-year layout with the check. A competitive paper assumes the arithmetic and asks for the th term, the sum of a series of deposits, or the time as a logarithm.
The single trap that costs the most marks. Using for a three-year difference. That shortcut counts one piece of interest-on-interest — the second year's — and for three years there are three such pieces plus interest on them. On the ₹8000 example the true difference is ₹61 and the shortcut gives ₹20. The defence is to read the number of years before reaching for any shortcut, and to remember what the shortcut actually represents: for two years the entire excess is the interest on the first year's interest, and that sentence is only true for two years.
This is the foundation for Class 11 Mathematics Sequences and Series, examined in JEE Main. The amounts , , , form a geometric progression with first term and common ratio . So the compound interest table on this page is a GP written out term by term, and the th-term formula is what the compound interest formula becomes in that notation.
The GP sum formula answers questions this chapter cannot. An annuity — a fixed sum deposited every year, each instalment compounding for a different number of years — is summed with . Instalment, annuity and sinking-fund problems appear in JEE Main, and they are compound interest applied to a series of deposits rather than one.
Growth and decay become exponential functions. Class 11 Relations and Functions and Class 12 Application of Derivatives treat as an exponential function, and Class 12 Differential Equations derives it from the statement that the rate of growth is proportional to the amount present. The population-growth and depreciation problems of the next chapter are the elementary form of that differential equation, and its solution is examined directly.
Logarithms solve for the time. Finding by matching a power works only when examiners choose convenient numbers, as this page notes. Class 11 replaces the guess with , and questions asking in how many years a sum will double are standard — the answer coming out as a logarithm rather than a whole number.
The binomial theorem explains the difference between CI and SI. Class 11 expands The first two terms give exactly the simple interest, and every term after them is the compound excess. For that excess is , which is the shortcut of this page — now derived rather than remembered, and extendable to three years and beyond.
Approximation questions use the same expansion. When is small, , which is why simple interest is a decent approximation over a short period and a poor one over a long one. JEE Main sets approximation questions of exactly this kind.
What the questions look like. For board work, expect find the compound interest year by year for 2 or 3 years, compute the amount when the rate changes each year, find the difference between compound and simple interest, and find the principal, rate or time from given data. Every year's working must be shown. For JEE Main, expect GP terms and sums, annuity problems, logarithms for the time, and binomial approximations.
How board and competitive emphasis differ. A board paper rewards the three-line-per-year layout with the check. A competitive paper assumes the arithmetic and asks for the th term, the sum of a series of deposits, or the time as a logarithm.
The single trap that costs the most marks. Using for a three-year difference. That shortcut counts one piece of interest-on-interest — the second year's — and for three years there are three such pieces plus interest on them. On the ₹8000 example the true difference is ₹61 and the shortcut gives ₹20. The defence is to read the number of years before reaching for any shortcut, and to remember what the shortcut actually represents: for two years the entire excess is the interest on the first year's interest, and that sentence is only true for two years.
Key takeaways
Compound interest by the year-by-year method: quick revision
- Simple interest always uses the original principal; compound interest adds each year's interest to the principal and uses the larger amount next year.
- One year at a time: , then new principal .
- ₹8000 at 10% for 2 years: , amount ₹8800; , amount ₹9680; .
- ₹12500 at 8% for 3 years: (₹13500); (₹14580); (₹15746.40); .
- The yearly interests rise even at a fixed rate, because the principal grows. Their total must equal the CI — — which is a free check.
- Changing rate: use each year's own rate on that year's own principal. ₹10000 at 5%, 6%, 8% gives (₹10500), (₹11130), (₹12020.40), so .
- The order of the rates does not change the final amount, since is the same product however arranged — but each year's own interest does change.
- ₹6000 at 10% for 2 years: ; ; difference .
- For 2 years the whole difference is the interest on the first year's interest — — so .
- ₹8000 at 5% for 3 years: ; ; difference . The two-year shortcut does NOT apply — it would give ₹20.
- Backwards problems all use .
- **Find **: at 10% for 2 years. Factor , so , giving .
- **Find **: ₹6250 becomes ₹7290 in 2 years. , and , so .
- **Find **: ₹5000 becomes ₹5832 at 8%. , so years.
- Method for any backwards problem: compute the growth factor , then take a root for the rate, match a power for the time, or divide the interest by (factor ) for the principal.
- Useful factors on sight: , , , , .
Take any principal and rate, work two years out by hand, then check that the difference from simple interest equals one year's interest on the first year's interest — if it does, your method is sound.
- One year at a time: , then new principal .
- ₹8000 at 10% for 2 years: , amount ₹8800; , amount ₹9680; .
- ₹12500 at 8% for 3 years: (₹13500); (₹14580); (₹15746.40); .
- The yearly interests rise even at a fixed rate, because the principal grows. Their total must equal the CI — — which is a free check.
- Changing rate: use each year's own rate on that year's own principal. ₹10000 at 5%, 6%, 8% gives (₹10500), (₹11130), (₹12020.40), so .
- The order of the rates does not change the final amount, since is the same product however arranged — but each year's own interest does change.
- ₹6000 at 10% for 2 years: ; ; difference .
- For 2 years the whole difference is the interest on the first year's interest — — so .
- ₹8000 at 5% for 3 years: ; ; difference . The two-year shortcut does NOT apply — it would give ₹20.
- Backwards problems all use .
- **Find **: at 10% for 2 years. Factor , so , giving .
- **Find **: ₹6250 becomes ₹7290 in 2 years. , and , so .
- **Find **: ₹5000 becomes ₹5832 at 8%. , so years.
- Method for any backwards problem: compute the growth factor , then take a root for the rate, match a power for the time, or divide the interest by (factor ) for the principal.
- Useful factors on sight: , , , , .
Take any principal and rate, work two years out by hand, then check that the difference from simple interest equals one year's interest on the first year's interest — if it does, your method is sound.