1. What a fraction is
Numerator and denominator
- A fraction expresses one integer divided by another and is written with one number above the other
- The lower number, the denominator, states how many equal parts one whole has been split into
- The upper number, the numerator, states how many of those parts are being counted
- In 3/5, each whole is cut into five parts and three of them are taken
- A proper fraction has a numerator smaller than its denominator, so its value is below 1
- An improper fraction, also called a top-heavy fraction, has a numerator at least as large as its denominator, so its value is 1 or more
2. Equivalent fractions and simplifying
Equivalent fractions
- Equivalent fractions look different but represent the same amount
- Multiplying numerator and denominator by the same non-zero number leaves the value unchanged
- 4/7 = 8/14 = 12/21 = 16/28
- Every fraction therefore has infinitely many equivalent forms
- Equivalence is easiest to see by regrouping: 21 small squares arranged as 7 columns of 3 means shading 15 squares shades exactly 5 whole columns, so 15/21 and 5/7 cover the same area
Simplifying
- Dividing numerator and denominator by a common factor produces a smaller equivalent fraction, a process also called cancelling
- A fraction is in its simplest form when the only common factor left is 1
- Dividing by the highest common factor reaches simplest form in one step
- 24/36 has highest common factor 12, so 24/36 = 2/3
- 30/48 has highest common factor 6, so 30/48 = 5/8
- Repeated cancelling by smaller factors reaches the same place, only more slowly
Worked example
Simplifying a fraction fully
Write 84/126 in its simplest form.
Solution:
- Look for a common factor of the numerator and the denominator
- Both are even, so divide each by 2: 84/126 = 42/63
- Both digit sums are multiples of 3 (4 + 2 = 6 and 6 + 3 = 9), so divide by 3: 42/63 = 14/21
- Both are multiples of 7: 14/21 = 2/3
- Check nothing is left: 2 and 3 share no factor above 1, so this is fully simplified
- Dividing by the highest common factor of 42 in one step gives the same answer, and either route earns the mark
3. Finding a fraction of an amount
Three routes to the same answer
- Dividing by the denominator and then multiplying by the numerator keeps every number whole for as long as possible
- For 3/8 of 56: 56 ÷ 8 = 7, then 7 × 3 = 21
- Converting the fraction to a decimal and multiplying suits fractions with a short decimal form
- For 3/4 of 26: 0.75 × 26 = 19.5
- Writing the amount over a denominator of 1 turns the problem into a fraction multiplication
- For 2/9 of 45: 2/9 × 45/1 = 90/9 = 10
- The word "of" between a fraction and a quantity always means multiply
Shading a fraction of a shape
- Shading works the same way: count the parts, divide by the denominator, then shade as many groups as the numerator asks for
- To shade 5/7 of a grid of 21 squares, 21 ÷ 7 = 3, and 3 × 5 = 15 squares are shaded
- Which particular squares are shaded does not matter, though grouping them together makes the fraction easier to read
- Where the shape is not already divided, cut it into the number of equal parts the denominator names first
Worked example
Finding a fraction of a quantity
A theatre has successfully sold 5/6 of its 342 seats. Work out how many seats were sold.
Solution:
- Divide by the denominator first, since 342 divides exactly by 6
- 342 ÷ 6 = 57, so one sixth is 57 seats
- Multiply by the numerator: 57 × 5 = 285
- 285 seats were sold
4. Mixed numbers and improper fractions
Recognising each form
- A mixed number is written as a whole number beside a proper fraction
- 4 2/9 means four wholes together with two ninths
- Any value above 1 can be written either as a mixed number or as an equivalent improper fraction
Converting a mixed number to an improper fraction
- Multiply the whole number by the denominator, then add the numerator
- Write the result over the original denominator
- For 4 2/9: 4 × 9 = 36, and 36 + 2 = 38, giving 38/9
Converting an improper fraction to a mixed number
- Work out how many whole denominators fit inside the numerator
- That count becomes the whole part, and whatever is left over sits above the original denominator
- For 29/6: 29 ÷ 6 = 4 remainder 5, giving 4 5/6
Worked example
Converting both ways
Write 4 3/5 as an improper fraction, and write 47/6 as a mixed number.
Solution:
- For the mixed number, multiply the whole part by the denominator and add the numerator
- 4 × 5 = 20, then 20 + 3 = 23
- The denominator does not change, so 4 3/5 = 23/5
- For the improper fraction, divide the numerator by the denominator
- 47 ÷ 6 = 7 remainder 5
- The whole part is the 7 and the remainder becomes the new numerator, so 47/6 = 7 5/6
- Check by reversing: 7 × 6 = 42, and 42 + 5 = 47
5. Adding and subtracting fractions
Working with a common denominator
- Fractions can only be added or subtracted once they share a denominator
- Find the lowest common denominator, which is the lowest common multiple of the two denominators
- Scale each fraction up so that both sit over that shared denominator
- Combine the numerators and keep the denominator unchanged
- Denominators are never added together
- Simplify the result, returning it to mixed-number form where the question demands that
When a mixed number appears
- Convert every mixed number to an improper fraction before starting
- Carry out the addition or subtraction as usual, then convert back at the end
Worked example
Subtracting mixed numbers
Work out 4 1/6 − 1 3/4, giving the answer as a mixed number.
Solution:
- Convert both to improper fractions: 4 1/6 = 25/6 and 1 3/4 = 7/4
- The lowest common multiple of 6 and 4 is 12
- Rewrite over 12: 25/6 = 50/12 and 7/4 = 21/12
- Subtract the numerators: 50 − 21 = 29, giving 29/12
- Convert back: 29 ÷ 12 = 2 remainder 5, so the answer is 2 5/12
Exam tip
Write the common-denominator line down
Before you add or subtract fractions, write both of them over a common denominator on a line of their own. The mark schemes give a method mark for that line alone, so it still scores if the arithmetic after it slips. The denominator need not be the lowest: 25/6 − 7/4 written as 100/24 − 42/24 earns the mark just as 50/12 − 21/12 does.
6. Multiplying and dividing fractions
Multiplying
- Cancel any common factor between a numerator and a denominator before multiplying, taking factors from either fraction
- Multiply the numerators together, then multiply the denominators together
- Cancelling first keeps the numbers small, which matters on the non-calculator papers
Dividing
- Dividing by a fraction is the same as multiplying by its reciprocal, so invert the second fraction and change the sign to multiplication
- Cancel and multiply exactly as above
- Only the second fraction is inverted, and only after the division sign has been changed
Mixed numbers
- Convert any mixed number to an improper fraction before multiplying or dividing
- Attempting to multiply the whole parts and the fraction parts separately does not work
Worked example
Multiplying fractions by cancelling first
Work out 9/20 × 8/15, giving the answer in its simplest form.
Solution:
- Cancel 9 against 15 using their common factor 3, leaving 3/20 × 8/5
- Cancel 8 against 20 using their common factor 4, leaving 3/5 × 2/5
- Multiply the numerators: 3 × 2 = 6
- Multiply the denominators: 5 × 5 = 25
- The answer is 6/25, which has no remaining common factor
Worked example
Dividing by a fraction
Work out 2 1/3 ÷ 7/9.
Solution:
- Convert the mixed number: 2 1/3 = 7/3
- Invert the second fraction and change the operation: 7/3 × 9/7
- Cancel 7 against 7, leaving 1/3 × 9/1
- Cancel 9 against 3, leaving 1/1 × 3/1
- The answer is 3