Fraction basics
Simplify, find equivalent fractions, and compare them.
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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.
more of the idea
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.
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.
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.
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
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Simplify a fraction
worked example
Simplify 24/28 to lowest terms.
- Divide top and bottom by their greatest common factor, 4: 24 ÷ 4 = 6 and 28 ÷ 4 = 7, so 6/7.
Answer: 6/7
Equivalent fractions
worked example
Fill in the missing number: 2/7 = ?/70
- 7 became 70 by multiplying by 10, so multiply the top by 10 too: 2 × 10 = 20.
Answer: 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
- 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.
Answer: 9/10