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Cambridge O Level · Mathematics 4024

Fractions, decimals and percentages

Core section 1.4 practice covers fraction, decimal and percentage representations, equivalence, simplest form and recurring-decimal conversions. Ordering and arithmetic are clearly marked as supporting practice from sections 1.5 and 1.6, while percentage of a quantity and one quantity as a percentage are basic supporting practice from section 1.13; percentage growth and reverse percentages remain in their separate topic.

2025 / 2026 / 2027 · Academic review not recorded · Published 2026-09-08

AI-assisted practice — not independently academically reviewed. Answers may contain errors; check important results against your course materials.

Guided tutorial

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18 lesson sections

What you will learn

  • Interpret proper fractions, improper fractions, mixed numbers, decimals and percentages as representations of quantities.
  • Generate equivalent fractions and write fractions in their simplest form.
  • Convert between mixed numbers and improper fractions.
  • Convert among fractions, terminating decimals and percentages.
  • Convert recurring decimals to fractions and fractions to recurring decimals.
  • Order positive and negative fractions, decimals and percentages by magnitude.
  • Use the four operations accurately with fractions and decimals, including brackets and correct operation order.
  • Calculate a basic percentage of a quantity and express one quantity as a percentage of another.

Explanation

One quantity, several representations

Core syllabus section 1.4 concerns the language and notation of proper fractions, improper fractions, mixed numbers, decimals and percentages, together with equivalence, conversion, simplest form and recurring decimals. This pack also includes explicitly labelled supporting practice: ordering from section 1.5, the four operations from section 1.6, and the two basic percentage skills from section 1.13.

A fraction, decimal and percentage can describe the same part of a whole. For example, 3/4 = 0.75 = 75%. The representation changes, but the quantity does not.

A fraction a/b represents a ÷ b, where b is not zero. A decimal records place value in tenths, hundredths and beyond. A percentage is a number of parts per hundred, so 28% means 28/100.

Explanation

Proper, improper and mixed fractions

In a proper fraction, the numerator has smaller magnitude than the positive denominator, so a positive proper fraction lies between 0 and 1. In an improper fraction, the numerator is at least as large as the denominator, so its positive value is at least 1.

A mixed number combines a whole number and a proper fraction. The notation 3 2/5 means 3 + 2/5, not 3 × 2/5. Improper fractions are usually easier for calculations; mixed numbers are often easier to interpret.

Worked example

Worked example: convert between mixed and improper forms

Convert 4 3/7 to an improper fraction. Four wholes contain 4 × 7 = 28 sevenths. Add the remaining 3 sevenths: 4 3/7 = (28 + 3)/7 = 31/7.

Convert 38/9 to a mixed number. Since 38 ÷ 9 = 4 remainder 2, 38/9 = 4 2/9. Check by reversing the process: (4 × 9 + 2)/9 = 38/9.

Explanation

Equivalent fractions and simplest form

Multiplying or dividing the numerator and denominator by the same non-zero number does not change a fraction's value. Thus 3/5 = 6/10 = 21/35.

A fraction is in simplest form when numerator and denominator have no common factor greater than 1. Divide both by their highest common factor when it is known, and keep a negative sign in the numerator or in front of the fraction rather than in both places.

Worked example

Worked example: simplify a fraction

Simplify 126/198. The highest common factor of 126 and 198 is 18. Divide both parts by 18: 126/198 = 7/11.

The result is fully simplified because 7 and 11 have no common factor greater than 1. A multiplication check gives 7 × 18 = 126 and 11 × 18 = 198.

Explanation

Terminating decimals and fractions

To turn a terminating decimal into a fraction, use its place value and simplify. For example, 0.045 = 45/1000 = 9/200. Include any whole-number part if the decimal is greater than 1.

To turn a fraction into a decimal, divide the numerator by the denominator. Alternatively, make an equivalent fraction with denominator 10, 100 or 1000 when possible.

Worked example

Worked example: convert a terminating decimal and percentage

Convert 0.375 to a fraction. It is 375 thousandths, so 0.375 = 375/1000. Dividing numerator and denominator by 125 gives 3/8.

As a percentage, multiply the decimal by 100: 0.375 × 100% = 37.5%. Therefore 3/8 = 0.375 = 37.5%.

Explanation

Percentages as fractions and decimals

To convert a percentage to a decimal, divide by 100; to convert a decimal to a percentage, multiply by 100. Thus 7.2% = 0.072 and 1.35 = 135%.

To convert a percentage to a fraction, write it over 100 and simplify. For example, 45% = 45/100 = 9/20. Percentages can exceed 100%: 125% = 1.25 = 5/4.

Explanation

Recurring decimals

A recurring decimal has a digit or block of digits that repeats forever. This lesson uses brackets to make the repeating part clear on screen: 0.(3) means 0.333… and 0.1(27) means 0.1272727… . The brackets contain only the recurring block; they do not mean multiplication here.

School dot notation places dots above the first and last digits of a recurring block. A dot above one digit means that digit repeats; dots above the first and last digits mark the whole block between them. Thus the school-dot forms corresponding to 0.(27) and 0.1(27) have dots above 2 and 7, with the initial 1 in the second number left undotted.

Recurring decimals are exact values, not rounded measurements. Fractions whose simplified denominators contain prime factors other than 2 and 5 produce recurring rather than terminating decimal expansions.

Worked example

Worked example: one recurring digit to a fraction

In this pack, 0.(7) means 0.777… . Let x = 0.(7). Multiplying by 10 shifts the recurring decimal one place: 10x = 7.(7). Subtract the original equation: 10x − x = 7, so 9x = 7.

Therefore x = 7/9. The subtraction removes the identical infinite recurring tails, and dividing 7 by 9 reproduces 0.777… .

For the reverse direction, convert 5/12 by long division. Twelve goes into 50 four times, leaving remainder 2; into 20 once, leaving remainder 8; and into 80 six times, again leaving remainder 8. That repeated remainder makes every following digit 6, so 5/12 = 0.41666… = 0.41(6), where only the 6 recurs.

Worked example

Worked example: a recurring block after a non-recurring digit

The notation 0.1(6) means 0.1666…: the digit 1 is a non-recurring prefix and only 6 repeats. Let x = 0.1(6). First shift past the one-digit prefix: 10x = 1.(6). Then shift one full recurring digit further: 100x = 16.(6).

Subtract the aligned equations: 100x − 10x = 16.(6) − 1.(6) = 15. Hence 90x = 15 and x = 15/90 = 1/6. In general, first multiply past the non-recurring prefix, then multiply by an additional power of 10 equal to the length of the recurring block.

Explanation

Supporting practice from section 1.5: ordering different forms

Convert all quantities to one convenient form before comparing. Decimals are often quickest: 5/8 = 0.625, 61% = 0.61 and 0.603 stays 0.603, so 0.603 < 61% < 5/8.

For negative values, the number farther left on the number line is smaller. For example, −0.7 < −2/3 because −0.7 is more negative than −0.666… . Keep exact fractions where rounding could hide a small difference.

Explanation

Supporting practice from section 1.6: adding and subtracting fractions

Fractions can be added or subtracted only after their parts have a common size, so use a common denominator. For 2/3 + 5/8, denominator 24 gives 16/24 + 15/24 = 31/24 = 1 7/24.

Convert mixed numbers to improper fractions when borrowing would be awkward. Simplify the final answer, and convert an improper result to a mixed number if that better suits the context.

Worked example

Worked example: subtract mixed numbers

Calculate 5 1/4 − 2 2/3. Convert first: 5 1/4 = 21/4 and 2 2/3 = 8/3. The lowest common denominator is 12.

Then 21/4 − 8/3 = 63/12 − 32/12 = 31/12 = 2 7/12. Adding 2 2/3 back to 2 7/12 returns 5 1/4, which checks the subtraction.

Explanation

Supporting practice from section 1.6: multiplying and dividing fractions

To multiply fractions, multiply numerators and multiply denominators. Cancel common factors across numerator and denominator before multiplying to keep numbers small. For division, multiply by the reciprocal of the divisor.

Never invert the first fraction by mistake. In a calculation such as 3/5 ÷ 9/10, keep 3/5 and multiply by 10/9, giving 30/45 = 2/3. Division by zero is undefined.

Worked example

Supporting worked example from section 1.6: four operations with decimals

Calculate 4.8 − 1.35 + 0.6 × 2.5. Multiplication comes first: 0.6 × 2.5 = 1.5. Then work left to right: 4.8 − 1.35 + 1.5 = 3.45 + 1.5 = 4.95.

For decimal multiplication, estimate the size and then place the decimal point. For decimal division, multiplying both dividend and divisor by the same power of 10 can make the divisor a whole number without changing the quotient.

Explanation

Supporting practice from section 1.13: basic percentage calculations

To find p% of a quantity Q, calculate (p/100) × Q. Useful mental anchors include 10% = one tenth, 5% = half of 10%, 1% = one hundredth and 50% = one half.

To express quantity A as a percentage of quantity B, calculate (A/B) × 100%, using matching units first. This pack covers these two foundation skills only; percentage increase, decrease, repeated change and reverse percentage belong to the separate percentage-change topic.

Worked example

Worked examples: percentage of and percentage comparison

Find 32% of 450. Calculate 0.32 × 450 = 144. A mental check is 30% of 450 = 135 and 2% = 9, giving 144.

Express 36 cm as a percentage of 80 cm. The units already match, so calculate (36/80) × 100% = 45%. The wording matters: 36 is the compared amount and 80 is the reference whole.

Original practice, not an official examination paper. Readnary is not affiliated with the awarding body. Prepared with AI assistance.