Fractions, Decimals And Percentages Explained: IGCSE Mathematics Complete Breakdown

Three representations, one underlying value

Fractions, decimals and percentages are three ways of expressing the same quantity. A student who can move fluently between all three holds a significant advantage across every IGCSE Mathematics paper, because Cambridge examiners routinely set questions that require conversion from one form to another mid-calculation. This guide covers each form, the techniques for converting between them, and the exam-level skills that distinguish strong candidates from the rest.

Key facts

• A fraction represents a part of a whole: numerator over denominator.
• A decimal is a fraction expressed in base ten.
• A percentage is a fraction with denominator 100.
• Converting between the three is a core skill tested on every IGCSE Mathematics paper.
• Extended candidates must also convert recurring decimals to fractions algebraically.
• Percentage problems (increase, decrease, reverse, compound interest) carry some of the highest mark allocations on Paper 4.

Conversion between forms

The table below is your reference. Memorise the common equivalences; derive the rest using the methods that follow.

Fraction to decimal

Divide the numerator by the denominator. That's it. 3/8 means 3 divided by 8, which gives 0.375. On a calculator paper, this is straightforward. On a non-calculator paper, use short division or recognise the fraction from the table above.

Decimal to percentage

Multiply by 100. So 0.375 becomes 37.5%. The reverse (percentage to decimal) divides by 100: 37.5% becomes 0.375.

Fraction to percentage

Two routes. Either convert the fraction to a decimal first, then multiply by 100. Or multiply the fraction by 100 directly: (3/8) x 100 = 300/8 = 37.5%.

Percentage to fraction

Write the percentage over 100 and simplify. 37.5% = 37.5/100 = 375/1000 = 3/8. Always reduce to lowest terms.

Ordering fractions

Cambridge frequently asks candidates to arrange a set of fractions, decimals and percentages in order of size. The cleanest method: convert everything to decimals, then compare.

An alternative for fractions only: find a common denominator. For mixed sets containing decimals and percentages, the decimal route is faster and less prone to error.

The four operations with fractions

Addition and subtraction

Find the lowest common denominator (LCD), rewrite each fraction with that denominator, then add or subtract the numerators.

Common error: adding numerators and denominators directly (writing 2/3 + 3/5 = 5/8). This produces a completely wrong answer and earns zero marks.

Multiplication

Multiply numerators together, multiply denominators together, then simplify. Cancel common factors before multiplying to keep numbers small.

Division

Multiply by the reciprocal of the divisor. "Keep, change, flip" is the mnemonic many students use.

Mixed numbers

Convert to improper fractions before performing any operation. 2 1/3 becomes 7/3. Perform the calculation, then convert back to a mixed number if the question requires it.

Recurring decimals (Extended tier)

IGCSE Extended candidates must convert recurring decimals to fractions using algebra. The technique is systematic.

1. Let x equal the recurring decimal.
2. Multiply x by a power of 10 that shifts the repeating block one full cycle to the left of the decimal point.
3. Subtract the original equation from the new one to eliminate the recurring part.
4. Solve for x and simplify.

For decimals where only part of the digits recur (e.g. 0.1666...), you need two multiplications. Multiply to shift the non-recurring part past the decimal, then shift again to align the recurring blocks.

Percentage increase and decrease

The multiplier method is the most efficient approach and the one Cambridge mark schemes reward.

Why multipliers? They compress the calculation into a single step, reduce arithmetic errors, and extend naturally into compound problems. An increase of 12% means the new value is 112% of the original, which is 1.12 as a decimal multiplier. A decrease of 12% means the new value is 88% of the original: 0.88.

Successive percentage changes

Multiply the individual multipliers together. A 10% increase followed by a 20% decrease: 1.10 x 0.80 = 0.88, which is a net 12% decrease. Note that a 10% increase and a 10% decrease do not cancel out. 1.10 x 0.90 = 0.99, giving a net 1% decrease.

Reverse percentages

Reverse percentage questions appear frequently on IGCSE papers. They give the value after a percentage change and ask for the original. The key insight: the given value corresponds to the multiplier, not to 100%.

The most common error: calculating 15% of $68 and adding it back. This gives $78.20, which is wrong because 15% of $68 is not the same as 15% of the original price. The multiplier method avoids this trap entirely.

Simple and compound interest

Simple interest

Interest is calculated on the original principal only. The formula:

I = PRT/100

where P is the principal, R is the annual rate (%), and T is the time in years.

Compound interest

Interest is calculated on the principal plus all accumulated interest. The formula:

A = P(1 + r/100)^n

where A is the final amount, P is the principal, r is the annual rate, and n is the number of compounding periods.

Compare: simple interest yields $240 over 3 years; compound interest yields $249.73. The gap widens dramatically over longer periods and higher rates.

Depreciation

Depreciation uses the compound interest formula with a decrease multiplier. A car worth $15,000 depreciating at 8% per year for 5 years:

A = 15000 x (0.92)^5 = 15000 x 0.65908... = $9886.22 (to nearest cent).

The multiplier 0.92 reflects the 8% annual loss. The structure is identical to compound interest - only the multiplier sits below 1 instead of above it.

Common exam pitfalls

Self-check questions

1. Convert 7/8 to a decimal and a percentage.
2. Arrange in ascending order: 5/12, 0.42, 41%.
3. Calculate 2/3 + 5/9. Give your answer as a fraction in its simplest form.
4. A shop increases its prices by 15%. A shirt now costs $57.50. What was the original price?
5. Convert 0.363636... to a fraction in its simplest form.
6. $5000 is invested at 3.5% compound interest for 4 years. Find the total amount at the end of the 4 years. Give your answer correct to 2 decimal places.
7. A car depreciates by 12% each year. It is currently worth $22,000. What will it be worth after 3 years? Give your answer to the nearest dollar.

Answers

1. 0.875 and 87.5%
2. 41%, 5/12 (= 0.4166...), 0.42
3. 2/3 + 5/9 = 6/9 + 5/9 = 11/9 = 1 2/9
4. 57.50 / 1.15 = $50
5. x = 0.363636..., 100x = 36.3636..., 99x = 36, x = 36/99 = 4/11
6. 5000 x (1.035)^4 = $5738.44
7. 22000 x (0.88)^3 = $14,997 (to nearest dollar)