How to compare fractions
Is 2/3 of a pizza more than 3/4 of it? Fractions are hard to compare by looking at the numbers, because a bigger number can mean a smaller piece. A few simple checks, done in the right order, settle it every time.
The rule
If the bottom numbers are the same, the fraction with the bigger top is bigger. If the top numbers are the same, the fraction with the smaller bottom is bigger, because its pieces are bigger. If neither is the same, rewrite both with the same bottom number first.
How to do it
- Look at the bottom numbers. If they are the same, the pieces are the same size, so the fraction with more pieces, the bigger top number, is bigger.
- If the top numbers are the same instead, you have the same number of pieces. The smaller bottom number means bigger pieces, so that fraction is bigger.
- If neither matches, find a number that both bottom numbers divide into, and turn each fraction into an equal fraction with that bottom. Then compare the tops.
- Write the answer with a sign: > means "is bigger than" and < means "is smaller than". The open side of the sign faces the bigger fraction.
Worked examples
Which is bigger, 3/5 or 4/5?
4/5 > 3/5
Both are in fifths, so the pieces are the same size. 4 fifths is more than 3 fifths, so 4/5 is bigger.
Which is bigger, 1/3 or 1/4?
1/3 > 1/4
Both have one piece. Cutting the bar into 3 makes bigger pieces than cutting it into 4, so 1/3 is bigger, even though 3 is smaller than 4.
Which is bigger, 2/3 or 3/4?
2/3 = 8/12 (× 4 on top and bottom)
3/4 = 9/12 (× 3 on top and bottom)
9/12 > 8/12, so 3/4 > 2/3
Thirds and quarters are different sizes, so turn both into twelfths, because 3 and 4 both divide into 12. Now the pieces match: 9 twelfths is more than 8, so 3/4 is bigger, by 1/12.
Which is bigger, 3/8 or 4/7?
half of 8 is 4, and 3 < 4
half of 7 is 3½, and 4 > 3½
No need for a common bottom here. Half of 8 is 4, and 3 is less, so 3/8 is less than a half. Half of 7 is 3½, and 4 is more, so 4/7 is more than a half. The dashed line marks a half: 3/8 stops before it and 4/7 goes past it, so 4/7 is bigger.
Cross-multiplying, a shortcut for older children
To compare 2/3 and 3/4, multiply each top number by the other fraction's bottom number. 2 × 4 = 8 belongs to 2/3, and 3 × 3 = 9 belongs to 3/4. 9 is bigger, so 3/4 is bigger.
It is quick, and it always works, but it hides why. The 8 and the 9 are exactly the top numbers you get by writing both fractions in twelfths, as in the third example. Once a child can do that, the shortcut is just a faster way of writing it.
How close to a whole?
For fractions just under 1, look at what is missing. 7/8 is one eighth short of a whole and 6/7 is one seventh short. An eighth is the smaller gap, so 7/8 is closer to 1, and 7/8 > 6/7.
Lowest common denominator
The shared bottom number is called a common denominator. Any number both bottoms divide into will do, and multiplying the two bottoms together always gives one. The smallest is their lowest common multiple, and it keeps the numbers small.
Why a bigger bottom means a smaller piece
Share one cake between 3 people and each gets a third. Share the same cake between 5 and each gets a fifth, which is less. The bottom number is how many people share the cake, so the bigger it is, the smaller each piece.
Is 2 big bags of sweets more than 3 small bags? You cannot tell until the bags are the same size. Fractions with different bottoms are like that. Making the bottoms the same cuts both wholes into pieces of the same size, and then you only have to count them.
Rewriting a fraction with a new bottom does not change how much it is. 2/3 and 8/12 cover the same amount of bar, cut finer. Equivalent fractions shows why.
Where mistakes happen
- Thinking the bigger bottom number wins. 1/8 is smaller than 1/4, not bigger, because eighths are smaller pieces than quarters.
- Comparing only the top numbers when the bottoms are different. In 3/8 and 2/3 the 3 is bigger than the 2, but 2/3 is the bigger fraction: it is more than a half, and 3/8 is less.
- Comparing what is missing without looking at its size. 2/3 and 3/4 are each one piece short of a whole, but a third is a bigger piece than a quarter, so 2/3 is further from 1 and is the smaller fraction.
- Changing only the bottom number. To write 2/3 in twelfths, multiply the top by 4 as well: 8/12, not 2/12.
Questions people ask
Which way round do > and < go?
The wide, open side faces the bigger number and the point faces the smaller one. 3/4 > 2/3 reads "3/4 is bigger than 2/3", and 2/3 < 3/4 says the same thing the other way round.
What if the two fractions are the same size?
Then write = between them. 2/4 and 3/6 are both a half, so 2/4 = 3/6. Making the bottoms the same shows it: both are 6/12.
How do I put three or more fractions in order?
Write them all with the same bottom number, then order the tops. For 1/2, 2/3 and 3/4, use twelfths: 6/12, 8/12 and 9/12, so the order from smallest is 1/2, 2/3, 3/4.
Is it quicker to change them into decimals?
Sometimes, and it works with a calculator. 3/8 is 0.375 and 4/7 is about 0.571. But for most fractions children meet, a common bottom number or a comparison with a half is quicker, and it does not need a calculator.