Fractions: Division, Operators and Simplest Form

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

Key words

1. A fraction is a division

The fraction line means divide. So 3/4 means 3 ÷ 4, and 3 ÷ 4 can be written as 3/4. The numerator is the number being shared; the denominator is the number of equal shares.

Improper fractions to mixed numbers. Because 17/5 means 17 ÷ 5, divide: 17 ÷ 5 = 3 remainder 2. The 3 is the whole number and the remainder 2 is still to be shared by 5, so 17/5 = 3 2/5 (three and two-fifths).

Mixed numbers to improper fractions. Reverse it: 3 2/5 = (3 × 5 + 2)/5 = 17/5.

Remainders as fractions. Sharing 22 metres of rope between 5 people: 22 ÷ 5 = 22/5 = 4 2/5 metres each. The remainder 2 is divided by 5 too.

2. Fractions as operators: "a fraction of a number"

In maths, of means multiply. 3/5 of 40 means 3/5 × 40. The fraction is an operator — it does something to the number.

1. Divide the number by the denominator to find one part (40 ÷ 5 = 8).
2. Multiply by the numerator to find the parts you need (8 × 3 = 24).

Improper fractions as operators work the same way, but the answer is bigger than the starting number, because the fraction is more than 1 whole.

Working backwards. If 2/3 of a number is 18, then 1/3 of it is 18 ÷ 2 = 9, and the whole number (3/3) is 9 × 3 = 27. Divide by the numerator to find one part, then multiply by the denominator.

3. Equivalent fractions and simplest form

Equivalent fractions have the same value. You make them by multiplying (or dividing) the numerator and the denominator by the same number.

3/4 = 6/8 (× 2 top and bottom) = 9/12 (× 3 top and bottom). Going the other way, dividing top and bottom by a common factor, simplifies the fraction.

Method — writing a fraction in its simplest form:

1. Find a common factor of the numerator and denominator (try 2, 3, 5 … or use your times tables).
2. Divide both by it.
3. Repeat until the only common factor is 1. (Quicker: divide by the highest common factor in one step.)
4. Check: can any number except 1 divide both? If not, you are finished.

Missing numbers. 4/5 = ☐/35: the denominator was multiplied by 7 (5 × 7 = 35), so multiply the numerator by 7 too: 4 × 7 = 28. So 4/5 = 28/35.

Worked examples

Worked example 1 — 4 friends share 3 pizzas equally. What fraction of a pizza does each friend get? [1 mark]

Sharing means dividing: 3 ÷ 4 = 3/4 of a pizza. Common mistake: writing 4/3 — the number being shared (3 pizzas) goes on top.

Worked example 2 — Write 23/6 as a mixed number. [1 mark]

23 ÷ 6 = 3 remainder 5 (because 3 × 6 = 18 and 23 − 18 = 5). The remainder is still divided by 6: 3 5/6. Check: 3 × 6 + 5 = 23 ✓.

Worked example 3 — Work out (a) 5/8 of 56 (b) 9/4 of 20. [2 marks]

1. (a) 56 ÷ 8 = 7, then 7 × 5 = 35.
2. (b) 20 ÷ 4 = 5, then 5 × 9 = 45. The answer is more than 20 because 9/4 is more than 1 whole.

Worked example 4 — Write 42/56 in its simplest form. [1 mark]

Both are in the 7 times table: 42 ÷ 7 = 6 and 56 ÷ 7 = 8, giving 6/8. 6 and 8 are both even: ÷ 2 gives 3/4. (The HCF of 42 and 56 is 14, so ÷ 14 does it in one step.) Common mistake: stopping at 6/8 — check again for a common factor.

Worked example 5 — Hana has 60 stickers. She gives 2/5 of them to her brother. Then she gives 1/4 of the stickers she has left to her friend. How many stickers does Hana keep? [2 marks]

1. 2/5 of 60: 60 ÷ 5 = 12, 12 × 2 = 24 given to her brother.
2. Left: 60 − 24 = 36.
3. 1/4 of 36 = 36 ÷ 4 = 9 given to her friend.
4. Keeps 36 − 9 = 27 stickers.

Common mistake: finding 1/4 of 60 instead of 1/4 of the stickers that are left. Underline "the rest" or "left" in the question. (A method mark is usually given for 24 or 36 seen.)

Worked example 6 — 3/4 of a number is 27. What is the number? [1 mark]

3 quarters = 27, so 1 quarter = 27 ÷ 3 = 9. The whole = 4 quarters = 9 × 4 = 36. Check: 3/4 of 36 = 36 ÷ 4 × 3 = 27 ✓.

Practice

All pages in this course

Start here

• Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

• Counting On and Back and Negative Numbers
• Sequences, Letters and Position-to-Term Rules
• Order of Operations, Brackets and Laws of Arithmetic
• Multiplying Whole Numbers (up to 10 000 × 2-digit)
• Dividing Whole Numbers (up to 1000 ÷ 2-digit)
• Factors, Multiples and Tests of Divisibility
• Square Numbers and Cube Numbers

Number: place value, decimals, fractions, percentages and ratio

• Place Value and Decimals to Thousandths
• Rounding Decimals
• Fractions: Division, Operators and Simplest Form
• Adding and Subtracting Fractions
• Multiplying and Dividing Fractions by Whole Numbers
• Fractions, Decimals and Percentages: Equivalence and Ordering
• Adding, Subtracting, Multiplying and Dividing Decimals
• Ratio and Direct Proportion

Geometry and measure

• Time: Decimals and Mixed Units
• Quadrilaterals: Properties and Classification
• Circles: Parts and Constructing Circles
• Angles: Measuring, Drawing and Angles in a Triangle
• Area of Rectangles and Right-angled Triangles
• 3D Shapes, Compound Shapes and Nets
• Capacity, Volume and Surface Area
• Coordinates in Four Quadrants
• Rotational Symmetry, Translation, Reflection and Rotation

Statistics and probability

• Collecting and Representing Data
• Mode, Median, Mean and Range, and Interpreting Data
• Probability

Practice tests

• Checkpoint 6 Maths — Practice Test 1
• Checkpoint 6 Maths — Practice Test 2
• Checkpoint 6 Maths — Practice Test 3