Fraction basics
Simplify, find equivalent fractions, and compare them.
Best in your head · no pen, no calculator why?
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the idea
A fraction is a division: 3/4 means 3 ÷ 4. The bottom number, the denominator, says how many equal parts make a whole. The top number, the numerator, says how many of those parts you have.
Multiply or divide the top and bottom by the same number and you get an equivalent fraction: the same amount, written differently. So 3/4, 6/8 and 15/20 are all equal. A fraction is in lowest terms when no number bigger than 1 divides both its top and its bottom.
Bigger numbers don't make a bigger fraction. To compare fractions, put them on one scale.
techniques
Divide by a common factor
- Find a number that divides both the top and the bottom.
- Divide both by it. The biggest such number, the greatest common factor, finishes in one step.
- Check again: if a number bigger than 1 still divides both, keep going.
worked example
Simplify 24/36 to lowest terms.
- 12 divides both: 24 ÷ 12 = 2, and 36 ÷ 12 = 3.
- Only 1 divides both 2 and 3, so stop.
Answer: 2/3
Scale top and bottom together
- Find the pair you know on both sides: the two tops or the two bottoms.
- Work out what the first was multiplied by to make the second.
- Multiply the other number by the same amount.
worked example
Fill in the missing number: 2/3 = ?/18
- 3 became 18 by multiplying by 6.
- Multiply the top by 6 too: 2 × 6 = 12.
Answer: 12
Put them on one scale
- Turn each fraction into a decimal by dividing the top by the bottom.
- Three decimal places are plenty.
- Compare the decimals, and ignore how big the numbers in the fractions look.
worked example
Which fraction is largest: 7/12, 3/4, 5/8 or 2/3?
- 7/12 ≈ 0.583, 3/4 = 0.75, 5/8 = 0.625, 2/3 ≈ 0.667.
- The largest is 3/4.
Answer: 3/4
watch out for
- Stopping before lowest terms. Halving 18/24 gives 9/12, but 3 still divides both, so keep going to 3/4.
- Adding instead of multiplying. In 3/4 = ?/20 the bottom was multiplied by 5, so the top is 3 × 5, not 3 + 16.
- Picking the fraction with the biggest numbers. 7/12 has bigger numbers than 2/3 but is the smaller amount.
practice
Simplify a fraction
worked example
Simplify 24/28 to lowest terms.
Answer: 6/7
- Divide top and bottom by their greatest common factor, 4: 24 ÷ 4 = 6 and 28 ÷ 4 = 7, so 6/7.
Equivalent fractions
worked example
Fill in the missing number: 2/7 = ?/70
Answer: 20
- 7 became 70 by multiplying by 10, so multiply the top by 10 too: 2 × 10 = 20.
Which is largest?
worked example
Which fraction is largest: 3/4, 9/10, 9/11 or 5/7?
- 3/4
- 9/10
- 5/7
- 9/11
Answer: 9/10
- Put them on one scale: 3/4 = 0.75, 9/10 = 0.9, 9/11 ≈ 0.818, 5/7 ≈ 0.714.
- The largest is 9/10.