Comparing and ordering fractions

Is 5/8 more than 3/5? Cross multiplying gives 25 against 24, so yes, by exactly 1/40.

Why comparing fractions needs care

With whole numbers, the bigger-looking number is the bigger number. With fractions that shortcut fails. 1/8 has a bigger denominator than 1/3, yet it is much smaller, because the whole was cut into more pieces. To compare fractions reliably you need to look at both parts together. This unit gives four methods, from the quickest special cases to one that always works.

Same denominator or same numerator

When the denominators are the same, the pieces are the same size, so the fraction with more pieces is bigger: 7/12 > 5/12. When the numerators are the same, you have the same number of pieces, so the fraction with bigger pieces is bigger, and bigger pieces come from a smaller denominator: 4/7 > 4/9, and 3/4 > 3/5. Many people get this second case backwards, so it is worth picturing: 3 slices from a pizza cut in 4 are larger than 3 slices from a pizza cut in 5.

Benchmarks

Sometimes a landmark settles the question at once. Compare 4/9 and 5/8. Half of 9 is 4.5, so 4/9 is a little less than 1/2. Half of 8 is 4, so 5/8 is more than 1/2. One is below the halfway mark and the other is above it, so 5/8 > 4/9 with no calculation. The same idea works near 1: 7/8 is 1/8 short of 1 and 9/10 is 1/10 short, and since 1/10 is the smaller gap, 9/10 is bigger.

A common denominator

The general method is to rewrite both fractions with the same denominator and then compare numerators. For 2/3 and 3/4, a denominator of 12 works for both: 2/3 = 8/12 and 3/4 = 9/12, so 3/4 is bigger by 1/12. Any common multiple of the two denominators will do. The smallest one keeps the numbers small, but multiplying the two denominators together always works.

Cross multiplication

Cross multiplication is the common-denominator method with the denominator left out. To compare a/b with c/d, compare a × d with c × b. For 5/8 and 3/5: 5 × 5 = 25 and 3 × 8 = 24. Since 25 > 24, 5/8 > 3/5. Those two products are the numerators you would get with the shared denominator 8 × 5 = 40, which also tells you the difference: 25/40 − 24/40 = 1/40.

Cross products also work for fractions that look almost identical. Which is larger, 99/100 or 100/101? 99 × 101 = 9,999 and 100 × 100 = 10,000, so 100/101 is larger. Each fraction is one piece short of 1, and the pieces of 101 are smaller than the pieces of 100.

Decimals

You can also divide each numerator by its denominator and compare the decimals. 17/20 = 0.85 and 21/25 = 0.84, so a score of 17 out of 20 is slightly better than 21 out of 25. This is handy with a calculator, but repeating decimals such as 1/3 = 0.333... need enough digits to be sure.

Putting several fractions in order

To order a list, rewrite every fraction with one common denominator. Order 1/2, 5/12, 2/3, 3/4 and 7/12. All the denominators divide 12, so write them as 6/12, 5/12, 8/12, 9/12 and 7/12. Now read the numerators from smallest to largest: 5/12, 1/2, 7/12, 2/3, 3/4. With a mixed list it also helps to sort first by benchmark: everything below 1/2, then everything between 1/2 and 1, then everything above 1.

Example. A recipe calls for 2/3 cup of milk and you have a measuring cup marked in quarters. Is 3/4 cup enough? Cross multiply: 3 × 3 = 9 and 2 × 4 = 8, so 3/4 > 2/3 and the 3/4 mark holds a little more than you need, by 1/12 of a cup.