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 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

Q: 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.

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.

3.47 = 3 ones + 4 tenths + 7 hundredths.

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.

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.

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.

What is a percentage?

A percentage is a fraction with the bottom fixed at 100, which is what makes any two of them comparable at a glance. Percentage starts there, and Fractions, Decimals and Percents is the conversion table that ties this page's three notations together.

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%.

How do you work out a percentage change?

Multiply by a single number. A rise of 15% is × 1.15; a fall of 15% is × 0.85. Percentage Increase and Taking a Percent Off both make the case for the multiplier over the two-step "find the part, then add or subtract it".

240 after a 15% rise → 240 × 1.15 = 276. 80 after 15% off → 80 × 0.85 = 68.

The multiplier is worth the small effort of learning because it composes and the two-step method does not. Two rises of 10% are 1.10 × 1.10 = 1.21 — a 21% increase, not 20%. A 20% rise followed by a 20% fall is 1.2 × 0.8 = 0.96, so you end 4% down, not level. Percentages do not add; their multipliers multiply.

Percentage Change runs it in reverse, working out the percentage from the two amounts: divide the change by the original, never by the new value. 40 → 50 is 10/40 = 25% up; 50 → 40 is 10/50 = 20% down. The same ten, two different percentages, because the reference moved.

Reverse percentages

Reverse Percentages is the one worth slowing down for, because the wrong method gets close enough to look right. If a coat cost 68 after 15% off, then 68 is 85% of the original, so the original is 68 ÷ 0.85 = 80. Adding 15% of 68 gives 78.20 — wrong, because the discount was never a percentage of the sale price.

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 calculated on the original amount; compound interest is calculated on the balance, so interest earns interest. Simple Interest gives the straight-line case: 1,000 at 5% for 3 years pays 50 × 3 = 150.

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.

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.

How does the arithmetic of borrowing work?

Every question below is compound interest asked from a different direction. The Time Value of Money establishes the principle: money now can earn a return, so amounts at different dates are different things and cannot be compared until one is moved to the other's date. Discounting is compounding run backwards — 1,050 in a year at 5% is worth 1,050 ÷ 1.05 = 1,000 today.

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 explains the payment schedule that surprises most first-time borrowers: each payment covers the interest accrued since the last one, and only what is left over reduces the balance. 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.

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.

The four mistakes worth naming

Adding successive percentages. 20% off then 10% off is 28% off, not 30%. Multiply the multipliers.
Reversing a discount by adding it back. Divide by the multiplier instead; the discount was a percentage of the original, not of the sale price.
Dividing by the new amount in a percentage change. The original is the reference, always.
Confusing a percentage with a percentage point. A rate moving from 4% to 5% is up one percentage point — and up 25%.

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.

Practise it in the game

Math Challenge teaches every rung above as an illustrated lesson, from the tenths grid to the amortization schedule, inside a catalog of 800+ lessons. Each has a worked derivation with pictures and try-it problems that re-teach the exact step you missed.

The Percentages topic drills the core move — a percentage of a number — starting at 10%, 25% and 50% of friendly numbers and widening as you get them right.

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?

Practise percentages free →