How do you calculate a percentage increase?

Multiply by 1 plus the percentage as a decimal. A 15% rise on 240 is 240 x 1.15 = 276. The multiplier form is worth learning because it composes: two successive rises of 10% are 1.10 x 1.10 = 1.21, a 21% increase, not 20%. Adding percentages instead of multiplying multipliers is the most common error in this whole topic.

What is a reverse percentage?

It is finding the original amount when you only know the amount after a change. Divide rather than multiply. If a coat costs 68 after 15% off, then 68 is 85% of the original, so the original is 68 divided by 0.85 = 80. Taking 15% of 68 and adding it back gives 78.20, which is wrong, because the discount was never a percentage of the sale price.

What is the difference between simple and compound interest?

Simple interest is always calculated on the original amount; compound interest is calculated on the balance, so past interest earns interest too. On 1,000 at 5% for 3 years, simple interest pays 150 while compound pays 157.63. The gap is small early and widens without limit: simple growth is a straight line, compound growth is a curve.

What is the compound interest formula?

The final amount is the starting amount times (1 + r) to the power n, where r is the rate per period as a decimal and n is the number of periods. For 1,000 at 5% for 3 years: 1000 x 1.05 cubed = 1,157.63. The power is just the multiplier applied once per period, so the formula is repeated multiplication written compactly.

Why is money today worth more than the same money next year?

Because money held today can earn a return, so a pound now and a pound in a year are not the same asset. To compare amounts at different dates you discount the later one: at 5%, 1,050 received in a year is worth 1,050 divided by 1.05, which is 1,000 today. Inflation compounds the effect by reducing what the later money buys.

How do you divide by a decimal?

Multiply both numbers by ten until the divisor is a whole number, then divide as normal. 12 divided by 0.4 becomes 120 divided by 4, which is 30. Scaling both by the same factor leaves the answer untouched, because the question 'how many 0.4s fit in 12' is the same question as 'how many 4s fit in 120'.

Decimals, Percentages and Interest

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Updated 7 October 2026

Decimals, percentages and interest are one subject wearing three hats. A decimal is a fraction whose bottom is a power of ten; a percentage is a fraction whose bottom everyone has agreed will be a hundred; and interest is what happens when you apply a percentage over and over.

This page runs that ladder end to end, and spends most of its length on the far end — percentage change, reverse percentages, compounding and the arithmetic of loans — because that is where the money is and where the mistakes are expensive. If you want the direct how-to for decimal arithmetic, the decimals guide and the percentages guide answer those questions on their own.

What the decimal places mean

Every step right of the point is ten times smaller, exactly as every step left of it is ten times bigger. Decimals introduces tenths, Hundredths the next column, and Thousandths the one after — by which point the pattern is doing the teaching.

$0.25 = \frac{25}{100} and 0.3 = \frac{30}{100}, 25 < 30$

the tenth from 0.2 to 0.3 holds ten hundredths, and every number inside it is < 0.3, however large its last digit looks: 0.25 = 25/100 < 30/100

Find 0.27 between 0.2 and 0.3

3.47 = 3 ones + 4 tenths + 7 hundredths.

After the dot, digits count smaller steps: 2.3 is two wholes and three tenths. Full lesson: Decimals

Comparing Decimals settles the mistake that outlives all the others: more digits does not mean bigger. 0.9 > 0.35, because nine tenths beats three and a half tenths. Padding to equal length — 0.90 against 0.35 — makes it obvious. And 0.5 Is a Half anchors the handful worth knowing by sight, which removes the arithmetic from most everyday questions.

0.35 is less than 0.40. Full lesson: Comparing Decimals

Multiplying and Dividing Decimals by 10 and 100 is the lesson that replaces "add a zero" with something true: the digits shift and the point stays put. That distinction is invisible for whole numbers and decisive the moment a point appears.

The four operations

Lining Up the Dot is the rule for addition, and Adding in Columns and Subtracting in Columns carry it into written work. 3.7 + 1.25 → 3.70 + 1.25 = 4.95. Lining up the points puts tenths over tenths; lining up the right-hand ends adds tenths to hundredths.

Counting the Decimals is multiplication: ignore the points, multiply, then count the decimal places in both factors and put the point that far in. 0.4 × 0.3 → 4 × 3 = 12 → two places → 0.12.

0.4 × 0.3 is (4/10) × (3/10) = 12/100. Each factor brings its own division by ten and the answer keeps both, which is exactly what "count the places" is counting. It is also why multiplying by a number below one makes things smaller — you are taking four tenths of something.

0.3 has one decimal place, and so does 0.4. That is 2 decimal places in total. Full lesson: Counting the Decimals

Dividing a Decimal handles division by a whole number, where the point rises straight up into the answer. Dividing by a Decimal handles the other case by scaling both numbers until the divisor is whole: 12 ÷ 0.4 → 120 ÷ 4 = 30. The answer being larger than 12 is correct — lots of small pieces fit into a thing.

Rounding Decimals to One and Two Decimal Places closes the arithmetic: one digit decides, the one immediately after the place you are keeping.

3.47 sits above the halfway mark of 3.45 and rounds up to 3.5 Full lesson: Rounding Decimals to One and Two Decimal Places

Now you

Round 2.52 to one decimal place

Round 6.65 to one decimal place

What is a percentage?

A percentage is a number of hundredths: the word percent means per hundred, so 30% is 30/100. Percentage starts there, and Fractions, Decimals and Percents links the three ways of writing the same amount. To find a percentage of an amount, find 10% by dividing by ten and build from there. For 30% of 70, 10% is 7, so 30% is three times as much, 21. The same sum written with the fraction is 30/100 × 70 = 21. A fixed bottom of 100 makes percentages easy to compare. A score of 13 out of 20 becomes 13/20 = 65/100 = 65%, which compares directly with 60% on another test. A percentage can be more than 100 when the part is bigger than the original: 150% of 40 = 60. Compare with half to check: 30% is less than 50%, so the answer must be less than half of 70, and 21 is less than 35.

$40% \times 240 = 96$

a percent is a hundredth, so n% of a quantity is that quantity × n/100

Make a quarter

Cut the whole into 100 pieces and color 40 in: 40/100, written 40%. Full lesson: Percentage

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
1/5	0.2	20%
1/8	0.125	12.5%
1/3	0.333…	33.3%

Finding a Percent is the core move, and the mental route is to build the awkward ones from the easy ones: 10% by dividing by ten, 5% by halving that, 1% by dividing by a hundred. 35% of 240 → 24 + 24 + 24 + 12 = 84. Comparing with Percent is why percentages exist at all — 18 out of 25 and 26 out of 40 are hard to rank until they become 72% and 65%.

Split 120 into ten equal parts: 120 ÷ 10 = 12. The lower scale reads 10%. Full lesson: Finding a Percent

Now you

90% of 90

40% of 150

How do you work out a percentage change?

To change an amount by a percentage, multiply it by a single number. A rise of 15% leaves 100% + 15% = 115% of the original, so the multiplier is × 1.15. A fall of 15% leaves 85%, so the multiplier is × 0.85. Percentage Increase and Taking a Percent Off both use this method. For a price of 60 that rises by 15%, the new price is 60 × 115/100 = 69. The multiplier finds the new amount in one step instead of two, where the part is found first and then added on. To go the other way and find the percentage change from two amounts, divide the change by the original amount, never by the new one. The price went from 60 to 69, a change of 9, and 9/60 = 15%. Dividing by 69 instead would give a smaller, wrong percentage. Working the two-step way once confirms the answer: 15% of 60 = 9, and 60 + 9 = 69.

Why the multipliers compose and the percentages do not, and how to undo a change, are worked through in how to work out percentage change.

Depreciation applies the same multiplier repeatedly downwards. A car losing 18% a year is worth × 0.82 each year, so after three years it retains 0.82³ ≈ 0.551 — about 55% of its price, not the 46% that subtracting 18% three times would suggest.

Simple interest or compound interest?

Simple interest is always worked out on the original amount, while compound interest is worked out on the balance, so the interest earns interest of its own. Simple Interest shows the straight-line case. For 2,000 saved at 10% a year for 3 years, simple interest pays 10% of 2,000 = 200 every year, so 200 × 3 = 600 in total. With compound interest, each year's interest is added before the next year's is worked out. The balance grows from 2,000 to 2,200, then 10% of 2,200 = 220 makes 2,420, then 10% of 2,420 = 242 makes 2,662. That is 662 of interest, 62 more than the simple total. The difference comes entirely from interest on earlier interest, which is why the gap is small at first and large over many years. In the first year the two methods pay the same. After that, compound interest should always come out ahead, which is a quick test of any answer.

$100 (1 + 0.08)^{16} = 343100 (1 + 0.08 \times 16) = 228$

simple interest adds 8% of 100 = 8 every year, a straight line; compound interest adds 8% of a growing balance, so it curves away — the gap this year is 115

Find the first year the balance has doubled

One calculation covers it: principal × rate × years ÷ 100. Here 200 × 5 × 3 ÷ 100 = 30. Full lesson: Simple Interest

Compound Interest is the multiplier applied once per period: 1,000 × 1.05³ = 1,157.63, so 157.63 of interest. Simple versus Compound Interest puts the two side by side, and the honest summary is that the gap is unimpressive early and enormous late — over 30 years at 5%, simple pays 1,500 while compound pays 3,322.

The formula is not a new idea, it is repeated multiplication written compactly. Multiplying by 1.05 three times is multiplying by 1.05³, and the exponent is only counting how many times the period has passed. Anyone who can see that can also see why the curve bends upward: each year's multiplier acts on a larger number than the last one did.

Each year multiplies the balance by 1.1 — so after n years it is the start times 1.1ⁿ. Full lesson: Compound Interest

Compounding More Than Once a Year answers the next question. A rate of 6% compounded monthly means 0.5% twelve times, which is 1.005¹² = 1.0617 — an effective 6.17%. More frequent compounding pays more, but the gain shrinks quickly and approaches a ceiling rather than growing without bound.

Now you

$1000 is invested at 10% simple interest per year. What is the balance after 3 years?

$2000 is invested at 10% a year for two years. How much more does compound interest pay than simple interest?

How does the arithmetic of borrowing work?

Borrowing is compound interest seen from a different direction. The Time Value of Money sets out the principle: money you have now can earn interest, so amounts on different dates cannot be compared until one is moved to the other's date. At 5% a year, 1,000 today grows to 1,000 × 105/100 = 1,050 in a year. Dividing moves money back, so 1,050 due in a year is worth 1,050 ÷ 1.05 = 1,000 today. A loan grows the same way. Borrow 500 at 10% a year for two years, and you owe 500 × 110/100 × 110/100 = 605. When a loan is repaid in installments, each payment covers the interest first and only the rest reduces the balance. Owing 1,000 at 10%, a payment of 300 at the end of the year leaves 1,000 + 100 − 300 = 800. The cost of a loan is everything repaid minus what was borrowed: 605 − 500 = 105 for the two-year loan.

$\frac{1100}{1 + 0.05} = 1048 today, 1000 \times (1 + 0.05) = 1050 next year$

B = 1100 in a year is worth 1048 today, more than 1000; and 1000 today grows to 1050, less than 1100: the comparison comes out the same on either date

Find the B that ties with 1000 today at the rate shown

$1000 due in five years is worth $747.26 today at 6% — that is the time value. Full lesson: The Time Value of Money

Real Value After Inflation applies the same discount to purchasing power. Savings paying 3% while prices rise 4% are losing about 1% a year in real terms, which is the number that matters and the one headline rates hide.

Amortizing a Loan follows that schedule to the last payment. Early payments are mostly interest; late payments are mostly principal; the payment itself never changes. The Value of an Annuity is the same stream of payments valued as a whole, and it is the tool for comparing a lump sum against an income — a pension, a lease, or a settlement offer.

The payment never changes, but the gold interest slice shrinks as the debt falls. Full lesson: Amortizing a Loan

Now you

The payments on a loan are all equal. What happens to the split inside them?

A $150000 loan at 0.5% a month is down to $70000. How much of the next $1200 payment is interest?

One question, worked end to end

A shop lists a jacket at 96 after a 20% sale reduction, and the price includes 9% tax. What did the shop set as the pre-tax list price?

Work outwards from the number you were given, undoing each change with a division rather than a subtraction.

Remove the tax. 96 is 109% of the pre-tax sale price, so that price is 96 ÷ 1.09 = 88.07.
Remove the discount. 88.07 is 80% of the list price, so the list price is 88.07 ÷ 0.80 = 110.09.
Check forwards. 110.09 × 0.80 × 1.09 = 96. The multipliers compose in either order, which is the reason the check is quick.

Doing this by subtracting 9% and then adding 20% back gives 104.83, and the error is not small. Every step of a percentage chain is a multiplication, so every step of undoing one is a division.

After a 20% rise the price is $60, so $60 is 120% of the old price. Full lesson: Reverse Percentages

The four mistakes worth naming

Four mistakes come up again and again: adding successive percentages, reversing a discount by adding it back, dividing by the new amount in a percentage change, and confusing a percentage with a percentage point. Each is worked through, with its figure, in the mistakes worth naming.

Where this leads next

Decimals are fractions in another notation, so the fractions ladder is the same material one step earlier. Comparing two quantities rather than a part against a whole is ratio, rates and proportion. And a compound-interest multiplier applied n times is an exponential function, which is where place value finally hands over to algebra.

Your turn

Three to try — tap what you get.

0.35 as a percentage

$200 at 5% simple interest earns, in one year,

20% off $50 — what do you pay?

Questions and answers

How do you calculate a percentage increase?: Multiply by 1 plus the percentage as a decimal. A 15% rise on 240 is 240 x 1.15 = 276. The multiplier form is worth learning because it composes: two successive rises of 10% are 1.10 x 1.10 = 1.21, a 21% increase, not 20%. Adding percentages instead of multiplying multipliers is the most common error in this whole topic.
What is a reverse percentage?: It is finding the original amount when you only know the amount after a change. Divide rather than multiply. If a coat costs 68 after 15% off, then 68 is 85% of the original, so the original is 68 divided by 0.85 = 80. Taking 15% of 68 and adding it back gives 78.20, which is wrong, because the discount was never a percentage of the sale price.
What is the difference between simple and compound interest?: Simple interest is always calculated on the original amount; compound interest is calculated on the balance, so past interest earns interest too. On 1,000 at 5% for 3 years, simple interest pays 150 while compound pays 157.63. The gap is small early and widens without limit: simple growth is a straight line, compound growth is a curve.
What is the compound interest formula?: The final amount is the starting amount times (1 + r) to the power n, where r is the rate per period as a decimal and n is the number of periods. For 1,000 at 5% for 3 years: 1000 x 1.05 cubed = 1,157.63. The power is just the multiplier applied once per period, so the formula is repeated multiplication written compactly.
Why is money today worth more than the same money next year?: Because money held today can earn a return, so a pound now and a pound in a year are not the same asset. To compare amounts at different dates you discount the later one: at 5%, 1,050 received in a year is worth 1,050 divided by 1.05, which is 1,000 today. Inflation compounds the effect by reducing what the later money buys.
How do you divide by a decimal?: Multiply both numbers by ten until the divisor is a whole number, then divide as normal. 12 divided by 0.4 becomes 120 divided by 4, which is 30. Scaling both by the same factor leaves the answer untouched, because the question 'how many 0.4s fit in 12' is the same question as 'how many 4s fit in 120'.

Mr. Chalk