Comparing and ordering fractions and decimals — Mathematics, 11–13
Pick a smart strategy to say which number is greater, and place numbers on the number line.
Greater means further right
On the number line, numbers grow to the right: the one further right is greater. To compare fractions you have strategies: with the same denominator, the greater numerator wins; with the same numerator, the fraction with the smaller denominator wins, because its parts are bigger; if one denominator is a multiple of the other, rewrite both with the larger one; and you can use 1/2 or 1 as a reference. With decimals, compare from left to right, place by place.
Choosing a path that saves work
Comparing is not just finding which is greater: it is choosing a path that saves work. If you split a bar into 8 parts, each part is smaller than if you split it into 5, so 3/8 is less than 3/5 with no calculation at all. Two people can reach the same answer by different paths; comparing the paths shows which is faster and when one of them fails.
Ordering 3/4, 1/2 and 5/8
Order 3/4, 1/2 and 5/8 from least to greatest. Step 1: the denominators are 4, 2 and 8, and 8 is a multiple of both 4 and 2; use 8. Step 2: 3/4 = 6/8 (× 2) and 1/2 = 4/8 (× 4); 5/8 stays as it is. Step 3: with the same denominator, compare the numerators: 4 < 5 < 6. Step 4: so 1/2 < 5/8 < 3/4. On the number line from 0 to 1 split into 8 parts, they sit at marks 4, 5 and 6.
More digits does not mean greater
When comparing 2.514 and 2.63, many people say 2.514 is greater, because “514 is greater than 63”. But decimals are compared place by place, from left to right: the units are equal (2) and, in the tenths, 5 < 6; so 2.514 < 2.63. It helps to match the places: 2.63 = 2.630 and 514 < 630. In increasing order: 2.49 < 2.51 < 2.514 < 2.63 < 2.72. On the line, 2.514 sits just after 2.51 and before 2.6.
Race times and recipes
Ordering numbers shows up in many places. In a race, 12.38 s is better than 12.4 s, because 12.4 = 12.40 and 12.38 < 12.40 (less time). In a recipe, 3/4 of a cup is more than 5/8 of a cup, because 3/4 = 6/8. And when you read a table of results, knowing how to order decimals stops you from putting 2.514 ahead of 2.63.
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