MATHEMATICSNumber & Arithmetic

CURRENT TOPICFractions

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● Master Topic Guide Verified 11+ Curriculum

Fractions Master Guide 🍕

Master definitions, equivalent fractions, addition, subtraction, multiplication, division, mixed numbers, top-heavy fractions, and fraction word problems for 11+ exams!

Practice Levels:Warm-Up Core Mode Pro Mode

Practice Papers

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FRACTIONS OVERVIEW

Fractions show parts of a whole, a group or a quantity. A fraction has two parts: the numerator (top) and the denominator (bottom).

1. WHAT IS A FRACTION?

3 Numerator (parts you have)

4 Denominator (equal parts total)

This shows 3 quarters (3/4)

2. EQUAL PARTS

The parts must be equal for a fraction to be fair.

✔️ Equal parts

❌ Not equal parts

3. FRACTIONS OF A SET

Fractions can show parts of a group. What fraction is shaded?

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3 out of 5 apples are shaded = 3/5

4. UNIT FRACTIONS

A unit fraction has a numerator of 1. It shows one equal part of a whole.

1/2one half

1/3one third

1/4one quarter

1/5one fifth

5. COMPARING FRACTIONS

Same denominator:compare numerators: 1/5 < 3/5 < 4/5

Same numerator:larger denominator = smaller fraction: 1/2 > 1/3 > 1/4

6. EQUIVALENT FRACTIONS

Fractions equal in value but with different numerators and denominators.

1/2 ═ 2/4 ═ 3/6 ═ 4/8

7. ADDING FRACTIONS

Same denominator:
2/5 + 1/5 = 3/5

Different denominator:
1/2 + 1/4 = 2/4 + 1/4 = 3/4

8. SUBTRACTING FRACTIONS

Same denominator:
4/5 − 1/5 = 3/5

Different denominator:
3/4 − 1/2 = 3/4 − 2/4 = 1/4

9. MULTIPLY & 10. NUMBER LINE

2/3 × 3/4 = 6/12 = 1/2

Fractions on a number line:
0 ── 1/4 ── 1/2 ── 3/4 ── 1

COMMON FRACTIONS VISUAL QUICK REFERENCE

1/2one half

1/3one third

1/4one quarter

2/3two thirds

3/4three quarters

2/5two fifths

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Read. Understand. Plan. Solve. Check.

FRACTION WORD PROBLEMS

Fractions show parts of a whole. Word problems describe real-life situations. Follow the steps and use your fraction skills to find the answer!

TYPES OF FRACTION PROBLEMS

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Finding Fractions of AmountsFind a part (fraction) of a whole amount.

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Comparing FractionsFind which fraction is greater, smaller or equal.

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Mixed OperationsUse +, −, ×, ÷ with fractions.

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Multi-Step ReasoningSolve problems with two or more steps. Think carefully!

HOW TO SOLVE FRACTION WORD PROBLEMS

1READ:Read carefully. What is asked?

Tip: Underline key numbers

2UNDERSTAND:What is the whole?

Tip: Draw a picture/diagram

3PLAN:Choose needed operation(s).

Tip: Write number sentence

4SOLVE:Carry out calculation carefully.

Tip: Show your working

5CHECK:Does your answer make sense?

Tip: Check correct units

EXAMPLE PROBLEMS & PRACTICE

1. FINDING FRACTIONS OF AMOUNTS

A recipe uses 48 g of sugar. Nicole uses 3/4 of the sugar. How much sugar does she use?

Plan: 3/4 of 48

Solve: (3/4) × 48 ═ (3 × 48)/4 ═ 144/4 ═ 36

Answer: 36 g of sugar

Try one!2/5 of 75 m ribbon is used. How much ribbon is used?Ans: 30 m

2. COMPARING FRACTIONS

Compare 3/5 and 7/10. Which fraction is greater?

Plan: Use common denominator

Solve: 3/5 ═ (3×2)/(5×2) ═ 6/10

Compare: 6/10 vs 7/10

Answer: 7/10 is greater than 3/5

Try one!Compare 2/3 and 4/6. Which is greater?Ans: Equal! (2/3 = 4/6)

3. MIXED OPERATIONS

2/3 of 27 apples are red, 1/9 of 27 are green. How many more red than green?

Red: (2/3) × 27 ═ 18

Green: (1/9) × 27 ═ 3

Diff: 18 − 3 ═ 15

Answer: 15 more apples are red

Try one!1/4 of a number is 12. Take away 3/8 of same number. Remaining?Ans: 30

4. MULTI-STEP REASONING

Emily has 120 stickers. Gives 1/9 to friend. Uses 2/5 of remaining in scrapbook. Left?

Step 1: (1/9)×120 ═ 13 ⅓ ➔ 106 ⅔ left

Step 2: (2/5)×106 ⅔ ═ 42 ⅔

Left: 106 ⅔ − 42 ⅔ ═ 64

Answer: 64 stickers left

Try one!Tank has 80 L water. 3/5 used in morning, 1/4 of rest in afternoon. Tank left?Ans: 24 L

USEFUL REMINDERS

• Fraction = equal parts of whole
• Simplify fractions where possible
• Show your working clearly
• Practice makes progress!

KEY FRACTION FACTS

• 1/2 ═ 2/4 ═ 3/6
• 1/3 ═ 2/6, 1/2 ═ 3/6 (1/2 > 1/3)
• (a/b) of N ═ (a/b) × N
• 2 ⅓ ═ 7/3, 1 ¾ ═ 7/4

WORDS MEANING FRACTIONS

of, per, out of, part of, a quarter of, a third of, twice as much as, half of, remaining, left, shared equally, each

CHECK YOUR ANSWER

• Is the answer reasonable?
• Answer question that was asked?
• Have you used correct units?

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FRACTION ADDITION & SUBTRACTION

Fractions show parts of a whole or a set. To add or subtract fractions, the pieces must be the SAME SIZE!

ADDING FRACTIONS

1Same denominators:

Add numerators. Keep denominator the same.

2/7 + 3/7 ═ 5/7

2Different denominators:

Find equivalent fractions with same denominator!

1/3 + 1/6 ═ 2/6 + 1/6 ═ 3/6 ═ 1/2

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3Simplify if possible:

3/6 ═ 1/2

SUBTRACTING FRACTIONS

1Same denominators:

Subtract numerators. Keep denominator the same.

5/8 − 2/8 ═ 3/8

2Different denominators:

Find equivalent fractions with same denominator!

3/4 − 1/6 ═ 9/12 − 2/12 ═ 7/12

3Simplify if possible:

7/12 is already in simplest form!

MIXED WORKED EXAMPLES

Add: 2/5 + 1/10

2/5 ═ 4/10

4/10 + 1/10 ═ 5/10 ═ 1/2

Subtract: 7/10 − 3/5

3/5 ═ 6/10

7/10 − 6/10 ═ 1/10

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MULTIPLICATION & DIVISION

Fractions show parts of a whole or a set. When multiplying or dividing fractions, there are simple rules to follow!

FRACTION MULTIPLICATION

Multiply the numerators together and the denominators together.

Rule: (a/b) × (c/d) ═ (a × c) / (b × d)

Example 1:

2/3 × 3/4 ═ (2×3)/(3×4) ═ 6/12 ═ 1/2

Example 2:

3/5 × 4/7 ═ (3×4)/(5×7) ═ 12/35

FRACTION DIVISION (KCF)

Keep first fraction, change ÷ to ×, flip second fraction (reciprocal).

Rule: (a/b) ÷ (c/d) ═ (a/b) × (d/c)

Example 1:

2/3 ÷ 3/4 ═ 2/3 × 4/3 ═ (2×4)/(3×3) ═ 8/9

Example 2:

3/5 ÷ 2/7 ═ 3/5 × 7/2 ═ (3×7)/(5×2) ═ 21/10

REAL-LIFE 11+ EXAM EXAMPLES

MULTIPLICATION: Recipe flour calculation

You use 2/3 cup flour in a recipe. You make 3/4 of recipe.

2/3 × 3/4 ═ 6/12 ═ 1/2 cup flour

DIVISION: Ribbon cutting problem

A 2-metre ribbon is cut into pieces that are 2/5 metre long.

2 ÷ 2/5 ═ 2 × 5/2 ═ 10/2 ═ 5 pieces

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MIXED NUMBERS & IMPROPER FRACTIONS

We can write the exact same amount as a mixed number or an improper fraction.

1. MIXED NUMBER

Has a whole number and a proper fraction part ($2\frac34$).

2 ¾

2. IMPROPER FRACTION

Numerator is greater than or equal to denominator ($11/4$).

11/4

3. CONVERT MIXED TO IMPROPER

1. Multiply whole number by denominator: 2 × 4 = 8
2. Add numerator: 8 + 3 = 11
3. Keep denominator: 11/4

2 ¾ ═ 11/4

4. CONVERT IMPROPER TO MIXED

1. Divide numerator by denominator: 11 ÷ 4 = 2 rem 3
2. Whole number = 2
3. Remainder = 3
4. Keep denominator: 2 ¾

11/4 ═ 2 ¾

INTERACTIVE FRACTION CHAMPION QUIZ11+ Exam Practice

1. Simplify 12/16 to its lowest terms:

2. Convert 2 3/5 to an improper fraction:

3. Calculate: 3/8 + 2/8:

4. Calculate: 2/3 × 3/4 (in lowest terms):

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