Fractions
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is a Fraction?
A fraction represents a part of a whole, written as numerator/denominator.
- Numerator: the top number - how many parts you have.
- Denominator: the bottom number - how many equal parts the whole is divided into.
- Example: in ³⁄₄, the numerator is 3 and the denominator is 4.
3/4 means 3 shaded parts out of 4 equal parts
Types of Fractions
| Type | Description | Example |
|---|---|---|
| Proper Fraction | Numerator < Denominator (less than 1) | 3/4, 2/5, 7/9 |
| Improper Fraction | Numerator ≥ Denominator (1 or more) | 5/4, 7/3, 11/5 |
| Mixed Number | Whole number + proper fraction | 1¼, 2⅓, 3⅖ |
| Unit Fraction | Numerator is 1 | 1/2, 1/3, 1/10 |
Equivalent Fractions
Equivalent fractions represent the same value but look different - created by multiplying or dividing both numerator and denominator by the same number.
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Question: Find three fractions equivalent to 1/2.
- Step 1: 1/2 = (1×2)/(2×2) = 2/4
- Step 2: 1/2 = (1×3)/(2×3) = 3/6
- Step 3: 1/2 = (1×4)/(2×4) = 4/8
- Answer: 2/4, 3/6, 4/8 (all equal to 1/2)
Simplifying Fractions
Divide both the numerator and denominator by their Highest Common Factor (HCF) to write a fraction in its simplest form.
-
Question: Simplify 12/18.
- Step 1: HCF of 12 and 18 = 6
- Step 2: 12 ÷ 6 = 2, and 18 ÷ 6 = 3
- Answer: 2/3
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Question: Simplify 45/60.
- Step 1: HCF of 45 and 60 = 15
- Step 2: 45 ÷ 15 = 3, and 60 ÷ 15 = 4
- Answer: 3/4
Converting Between Mixed Numbers and Improper Fractions
Mixed Number → Improper Fraction
(Whole × Denominator) + Numerator, all over Denominator
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Question: Convert 2¾ to an improper fraction.
- Step 1: (2 × 4) + 3 = 8 + 3 = 11
- Step 2: Put over the denominator: 11/4
- Answer: 11/4
Improper Fraction → Mixed Number
Divide the numerator by the denominator: the whole-number part of the answer becomes the whole part, and the remainder becomes the new numerator.
-
Question: Convert 17/5 to a mixed number.
- Step 1: 17 ÷ 5 = 3 remainder 2
- Step 2: Whole number = 3, remainder = 2
- Answer: 3⅖
Adding and Subtracting Fractions
Before calculating exactly, it's good practice to estimate first - round each fraction to the nearest half or whole to check your final answer is reasonable.
Same Denominator
Add or subtract the numerators, keeping the denominator the same.
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Question: Calculate 3/7 + 2/7.
- Step 1: Add the numerators: 3 + 2 = 5
- Step 2: The denominator stays 7
- Answer: 5/7
Different Denominators
- Find the Lowest Common Multiple (LCM) of the denominators.
- Convert each fraction to an equivalent fraction with the LCM as the denominator.
- Add or subtract the numerators, then simplify if needed.
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Question: Calculate 1/3 + 1/4.
- Step 1: LCM of 3 and 4 = 12
- Step 2: 1/3 = 4/12 (× 4/4), and 1/4 = 3/12 (× 3/3)
- Step 3: 4/12 + 3/12 = 7/12
- Answer: 7/12
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Question: Calculate 5/6 − 1/4.
- Step 1: LCM of 6 and 4 = 12
- Step 2: 5/6 = 10/12, and 1/4 = 3/12
- Step 3: 10/12 − 3/12 = 7/12
- Answer: 7/12
Mixed Numbers
-
Question: Calculate 2⅓ + 1¾ by converting to improper fractions first.
- Step 1: 2⅓ = 7/3, and 1¾ = 7/4
- Step 2: LCM of 3 and 4 = 12, so 7/3 = 28/12 and 7/4 = 21/12
- Step 3: 28/12 + 21/12 = 49/12 = 4 1/12
- Answer: 4 1/12
Multiplying Fractions
a/b × c/d = (a×c)/(b×d)
Multiply the numerators together, multiply the denominators together, then simplify.
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Question: Calculate 2/3 × 3/5.
- Step 1: Numerators: 2 × 3 = 6. Denominators: 3 × 5 = 15
- Step 2: Simplify 6/15 to 2/5
- Answer: 2/5
Cancelling Before Multiplying
Cancelling common factors before multiplying keeps the numbers smaller and avoids simplifying a large fraction at the end.
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Question: Calculate 4/9 × 3/8.
- Step 1: Cancel 4 and 8 (divide both by 4): 1/9 × 3/2
- Step 2: Cancel 3 and 9 (divide both by 3): 1/3 × 1/2
- Step 3: 1/3 × 1/2 = 1/6
- Answer: 1/6
Dividing Fractions
a/b ÷ c/d = a/b × d/c
Dividing by a fraction is the same as multiplying by its multiplicative inverse (reciprocal) - flip the second fraction, then multiply.
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Question: Calculate 3/4 ÷ 2/5.
- Step 1: Flip the second fraction: 2/5 becomes its reciprocal, 5/2
- Step 2: Multiply: 3/4 × 5/2 = (3×5)/(4×2) = 15/8
- Step 3: Convert to a mixed number: 15/8 = 1⅞
- Answer: 1⅞
-
Question: Calculate 2/3 ÷ 4.
- Step 1: Write 4 as a fraction: 4/1
- Step 2: Flip to its reciprocal: 1/4
- Step 3: Multiply: 2/3 × 1/4 = 2/12 = 1/6
- Answer: 1/6
Combining Operations with Fractions (Order of Operations)
When an expression mixes several fraction operations together, the usual order of operations still applies: brackets first, then multiplication/division, then addition/subtraction.
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