Fractions, Decimals, Percentages and Ratio
Section: Number | Syllabus: Cambridge Primary Mathematics (0845)
Fractions as Division and as Operators
A fraction has two closely connected meanings: it can be read as a division, and it can be used to find a portion of an amount.
- A fraction can be understood as the numerator divided by the denominator
- 3/4 means 3 ÷ 4
- This works for both proper fractions (numerator smaller than denominator, e.g. 3/4) and improper fractions (numerator equal to or bigger than the denominator, e.g. 7/4)
- A fraction can also act as an operator - a fraction "of" an amount
- Finding 2/3 of 18 means dividing 18 by 3, then multiplying by 2: 18 ÷ 3 × 2 = 12
- This works for improper fractions too: finding 5/4 of 40 means 40 ÷ 4 × 5 = 50 - since 5/4 is greater than 1, the answer is bigger than the original amount
3 out of 4 equal parts shaded = 3/4
Worked Example: A Fraction as an Operator
- Question: Find 3/5 of 40.
- Step 1: Divide by the denominator: 40 ÷ 5 = 8
- Step 2: Multiply by the numerator: 8 × 3 = 24
- Answer: 24
Common Mistakes
MistakeThinking a fraction "of" an amount means adding the fraction on, e.g. finding 3/4 of 40 as 40 + 3/4
Fix"of" means multiply - divide by the denominator, then multiply by the numerator, to find a portion of the amount
Simplifying Fractions
A fraction can be written in many equivalent ways without changing its value. Simplifying means finding the version with the smallest possible numerator and denominator.
- A fraction is in its simplest form when the numerator and denominator have no common factor other than 1
- Divide the numerator and denominator by the same common factor, repeating until no common factor remains
18/24 and 3/4 are the same size - only the number of equal parts has changed
Worked Example: Simplifying a Fraction
- Question: Simplify 18/24.
- Step 1: Find a common factor of 18 and 24 - both share a factor of 6
- Step 2: 18 ÷ 6 = 3, and 24 ÷ 6 = 4
- Answer: 18/24 = 3/4 in its simplest form
Common Mistakes
MistakeDividing only the numerator or only the denominator by the common factor
Fixwhatever is done to the numerator must be done to the denominator too, so the fraction's value stays the same
Fraction, Decimal and Percentage Equivalence
The same value can be written as a fraction, a decimal or a percentage - recognising these equivalences makes it much easier to compare and calculate with them.
- Fractions, decimals (with one or two decimal places) and percentages can all represent the same value
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
Worked Example: Converting Between Forms
- Question: Write 3/8 as a decimal and as a percentage.
- Step 1: Divide to convert to a decimal: 3 ÷ 8 = 0.375
- Step 2: Multiply the decimal by 100 to convert to a percentage: 0.375 × 100 = 37.5%
- Answer: 3/8 = 0.375 = 37.5%
Common Mistakes
MistakeTreating 0.5 as 5% instead of 50% when converting a decimal to a percentage
Fixmultiply the decimal by 100 to convert to a percentage: 0.5 × 100 = 50%
Calculating with Fractions
Fractions can only be added or subtracted directly once they share the same denominator. Multiplying or dividing a fraction by a whole number changes how many equal parts there are, or how big each part is.
- To add or subtract fractions with different denominators, first change them to equivalent fractions with the same denominator
- To multiply a proper fraction by a whole number, multiply the numerator by the whole number - there are now more parts of the same size
- To divide a proper fraction by a whole number, multiply the denominator by the whole number - each part is shared into smaller pieces, so there are more of them, each one smaller
- Before calculating exactly, you can estimate by comparing each fraction to a benchmark such as 0, 1/2 or 1 - for example, 1/3 and 1/4 are both less than 1/2, so their total must be less than 1
1/3 lines up with 4/12, and 1/4 lines up with 3/12 - splitting both bars into twelfths makes the parts the same size, so they can be added
Worked Example: Adding Fractions with Different Denominators
- Question: Work out 1/3 + 1/4.
- Step 1: Find a common denominator of 3 and 4: 12
- Step 2: Convert: 1/3 = 4/12, and 1/4 = 3/12
- Step 3: Add the numerators only: 4/12 + 3/12 = 7/12
- Answer: 7/12
Worked Example: Subtracting Fractions with Different Denominators
- Question: Work out 3/4 − 1/6.
- Step 1: Find a common denominator of 4 and 6: 12
- Step 2: Convert: 3/4 = 9/12, and 1/6 = 2/12
- Step 3: Subtract the numerators only: 9/12 − 2/12 = 7/12
- Answer: 7/12
Worked Example: Multiplying and Dividing a Fraction by a Whole Number
- 2/5 × 3 = 6/5 (multiply the numerator by 3 - three times as many fifths)
- 2/5 ÷ 3 = 2/15 (multiply the denominator by 3 - each fifth split into 3 smaller pieces)
Common Mistakes
MistakeAdding fractions by adding the numerators and denominators separately (e.g. 1/2 + 1/3 = 2/5)
Fixchange both fractions to a common denominator first, then add the numerators only
Percentages of Amounts
Percentages of whole numbers and percentages of shapes work the same way - 1% is one hundredth of the whole, so any percentage can be built from steps of 1% or 10%.
- Percentages of shapes and whole numbers can be found for 1%, and multiples of 5% up to 100%
- Find 10% by dividing by 10, then scale up or down to reach the percentage needed
- Find 1% by dividing by 100, then multiply to reach the percentage needed
Worked Example: Finding a Percentage of an Amount
- Question: Find 35% of 240.
- Step 1: Find 10%: 240 ÷ 10 = 24
- Step 2: Find 30%: 24 × 3 = 72
- Step 3: Find 5%: 24 ÷ 2 = 12
- Step 4: Add 30% and 5%: 72 + 12 = 84
- Answer: 35% of 240 = 84
35% of the bar is shaded - the same proportion as 35% of 240 being 84
20 out of 100 equal squares shaded = 20% of the shape
Common Mistakes
MistakeFinding a percentage by guessing or moving the decimal point at random
Fixfind 10% (÷10) or 1% (÷100) first, then scale up or down to the percentage needed
Comparing and Ordering Numbers
Fractions, decimals and percentages can look very different but still represent comparable sizes. Converting them all to the same form - often decimals - makes it possible to compare them using =, > and <.
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