Comparing fractions (age 10+)

Comparing fractions means deciding which one is bigger.

1. Look at the bottom numbers, called denominators. Are they the same? If they are, the fraction with the bigger top number wins. Comparing 5/8 and 3/8 is easy: both are eighths, and 5 is more than 3, so 5/8 is bigger.

2. If the bottom numbers are different, change the fractions so the bottoms match. Find a number that both denominators can divide into. For 2/3 and 3/4, both 3 and 4 go into 12.

3. Rewrite each fraction. 2/3 becomes 8/12, and 3/4 becomes 9/12.

4. Now the pieces are the same size. Compare the tops: 9 beats 8, so 3/4 is bigger than 2/3.

A handy shortcut: if the top numbers are the same, the fraction with the smaller bottom number is bigger. 3 pieces of something cut into 5 parts are bigger than 3 pieces cut into 7 parts, because fifths are larger than sevenths.

Try it yourself: which is bigger, 4/9 or 5/6? Give them both denominator 18 and see!

Ordering a list of fractions works just like comparing two of them, one step at a time.

1. Find the least common multiple of all the denominators. That is the smallest number every denominator fits into. For 2/3, 5/6, and 7/10, the denominators are 3, 6, and 10, and they all fit into 30.

2. Rewrite every fraction with that shared denominator. 2/3 becomes 20/30, 5/6 becomes 25/30, and 7/10 becomes 21/30.

3. Now ordering is simple. Just read the top numbers from smallest to biggest: 20, then 21, then 25.

4. Write the original fractions in that same order: 2/3, 7/10, 5/6.

You can also use shortcuts. If one fraction is less than 1 and another is more than 1, the bigger-than-1 fraction always goes last. And if two fractions have the same top number, the one with the smaller bottom number is bigger.

Fractions greater than 1 always have a top number bigger than the bottom number. Examples are 7/4, 9/6, and 11/8.

You can also write them as mixed numbers, which mix a whole number with a fraction:

- 7/4 = 4/4 + 3/4 = 1 3/4 - 9/6 = 6/6 + 3/6 = 1 1/2

Mixed numbers make comparing easy. Compare 7/4 and 9/6 by first writing them as 1 3/4 and 1 1/2. Both have 1 whole, so you only need to compare the leftover parts: 3/4 is bigger than 1/2, so 7/4 is bigger.

Another trick is to use benchmarks like 1 and 1/2. Ask yourself: is this fraction less than 1/2, between 1/2 and 1, or greater than 1? A fraction greater than 1 always beats a proper fraction.

When the leftover parts are hard to compare, switch back to common denominators. To compare 1 3/4 and 1 2/3, turn 3/4 into 9/12 and 2/3 into 8/12. Since 9/12 is bigger, 1 3/4 wins.

When fractions have the same denominator, we say they have common denominators. Common denominators make comparing, adding, and subtracting fractions much easier, because every piece is the same size.

How to find a common denominator:

List the multiples of each denominator until you spot a number they share. That shared number is your common denominator.

Example: find a common denominator for 7/8 and 3/10.

- Multiples of 8: 8, 16, 24, 32, 40, 48 - Multiples of 10: 10, 20, 30, 40, 50

The smallest number on both lists is 40. Now rewrite each fraction with 40 on the bottom. 7/8 becomes 35/40, and 3/10 becomes 12/40. Now it is easy to see that 7/8 is much bigger.

You can always use a bigger common denominator if you like. For 7/8 and 3/10 you could also use 80. But the smallest one keeps the numbers friendly and quick to compare.

A tape diagram is a long rectangle cut into equal parts. It is a great picture for comparing fractions, because you can see which one covers more of the same whole.

When two fractions have the same numerator, they show the same number of parts, but the parts can be different sizes. The denominator tells you how many equal parts make the whole, so a bigger denominator means smaller pieces.

For example: - 2/3 means 2 parts out of 3 equal parts of the whole. - 2/5 means 2 parts out of 5 equal parts.

Draw two same-length bars. Cut the first into 3 equal pieces and shade 2 of them. Cut the second into 5 equal pieces and shade 2 of them. The shaded part of the first bar is longer, so 2/3 is greater than 2/5.

Quick rule to remember: same top number, the fraction with the smaller bottom number is bigger. So 4/7 is greater than 4/9, because sevenths are bigger pieces than ninths.

Tape diagrams are also handy for checking your answer after you compare fractions using common denominators.