EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Fractions
Fractions, ratio and percentages · Lesson 1 of 5
Before We Start
Answer each one, then check.
1. Simplify 12/18.
⅔
2. What is the lowest common multiple of 4 and 6?
12
3. Work out ¾ of 20.
15
4. Write 2⅓ as an improper fraction.
7/3
Learning Objectives
1. Add and subtract fractions and mixed numbers.
2. Multiply and divide fractions and mixed numbers.
3. Find a fraction of an amount, and write one quantity as a fraction of another.
4. Find the whole amount when you know a fraction of it.
Equivalent Fractions and Mixed Numbers
Multiplying or dividing the top and bottom by the same number gives an equivalent fraction.
▸ Simplify. Divide the top and bottom by their HCF: 12/18 = ⅔ (dividing by 6).
▸ Mixed to improper. 2⅓ = (2 × 3 + 1)/3 = 7/3.
▸ Improper to mixed. 17/5 = 32/5, because 17 ÷ 5 = 3 remainder 2.
▸ Always finish simply. Give answers in their simplest form, as a mixed number if the question uses them.
Adding and Subtracting
Fractions can only be added or subtracted when their denominators are the same.
▸ Common denominator. Use the lowest common multiple of the denominators: for ¼ + 1/6, use 12.
▸ Change each fraction. ¼ = 3/12 and 1/6 = 2/12, so the answer is 5/12.
▸ Mixed numbers. Change them to improper fractions first - it avoids borrowing mistakes.
▸ Never add the bottoms. ¼ + 1/6 is NOT 2/10.
Adding Mixed Numbers
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Work out 2¾ + 15/6 |
1. Change to improper fractions
11/4 + 11/6
2. Common denominator: the LCM of 4 and 6 is 12
33/12 + 22/12
3. Add the numerators
55/12
4. Change back to a mixed number
55 ÷ 12 = 4 remainder 7
Answer: 47/12
Subtracting Mixed Numbers
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Work out 3⅕ − 1⅔ |
1. Change to improper fractions
16/5 − 5/3
2. Common denominator 15
48/15 − 25/15
3. Subtract the numerators
23/15
4. Change back
18/15
Answer: 18/15
Multiplying and Dividing
No common denominator is needed - but mixed numbers must be made improper first.
▸ Multiply. Multiply the tops and multiply the bottoms: ⅔ × ¾ = 6/12 = ½.
▸ Cancel first. Divide a top and a bottom by a common factor before multiplying to keep numbers small.
▸ Reciprocal. Flip a fraction to get its reciprocal: the reciprocal of 4/5 is 5/4.
▸ Divide. Dividing by a fraction is the same as multiplying by its reciprocal: keep, flip, change.
Why Multiplying Fractions Works
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"Of" means multiply. Take ¾ of the square, then ⅔ of that, and the overlap is 6 of the 12 small rectangles - exactly ⅔ × ¾ = 6/12 = ½. |
⅔ of ¾ covers 6 of the 12 small rectangles: ½. |
Multiplying Mixed Numbers
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Work out 2¼ × 1⅓ |
1. Change to improper fractions
9/4 × 4/3
2. Cancel: the 4s cancel, and 9 and 3 share a factor of 3
3/1 × 1/1
3. Multiply
3
Answer: 3
Dividing Mixed Numbers
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Work out 3⅓ ÷ 1¼ |
1. Change to improper fractions
10/3 ÷ 5/4
2. Keep, flip, change
10/3 × 4/5
3. Multiply
40/15 = 8/3
4. Change back
2⅔
Answer: 2⅔
Fractions of Amounts
Divide by the bottom, multiply by the top.
▸ A fraction of an amount. 3/5 of 40: 40 ÷ 5 = 8, 8 × 3 = 24.
▸ One amount as a fraction of another. 15 out of 40 is 15/40 = ⅜. Make sure both are in the same units first.
▸ Finding the whole. If 3/5 of a number is 24, then ⅕ is 8, so the whole is 5 × 8 = 40.
▸ The rest. If 2/5 is spent, 1 − 2/5 = 3/5 is left.
Key Terms
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Numerator The top number of a fraction. |
Denominator The bottom number of a fraction. |
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Improper fraction A fraction whose numerator is bigger than its denominator, e.g. 7/3. |
Mixed number A whole number and a fraction, e.g. 2⅓. |
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Reciprocal 1 divided by a number; for a fraction, the fraction flipped. |
Common denominator A denominator shared by two fractions, so they can be added or subtracted. |
Your Task: Fraction Target
10 minutes
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Using each of the digits 1, 2, 3 and 4 once, make two fractions a/b and c/d whose sum is as close to 1 as possible. Then find the pair whose product is as small as possible. 1. Try several combinations. 2. Work out each sum exactly. 3. Explain why your best answer is best. |
A good answer shows: Closest sum to 1: ¼ + ⅔ = 11/12 (or ⅓ + 2/4 = 5/6, further away). Smallest product: ⅓ × 2/4 = 1/6 or ¼ × ⅔ = 1/6.
Can I...?
☐ Simplify fractions.
☐ Convert between mixed numbers and improper fractions.
☐ Add and subtract fractions.
☐ Add and subtract mixed numbers.
☐ Multiply fractions and mixed numbers.
☐ Divide fractions and mixed numbers.
☐ Find a fraction of an amount.
☐ Find the whole from a fraction of it.
Summary
✓ Add and subtract: common denominator first; mixed numbers become improper.
✓ Multiply: tops times tops, bottoms times bottoms; cancel first.
✓ Divide: keep, flip, change.
✓ Fraction of an amount: divide by the bottom, multiply by the top.
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EXAM FOCUS Work out 1¾ ÷ 2⅓. Give your answer in its simplest form. (3 marks) Show every step - the improper fractions, the flipped fraction, the multiplication. On the non-calculator paper each step can earn a method mark. |