Comparing and Ordering Fractions

Comparing fractions is easy in one situation: when the denominators already match. Look at three eighths and five eighths. Both strips are cut into eight identical pieces, so the pieces are exactly the same size and all you are doing is counting how many of them you have. Five is more than three, so five eighths is greater than three eighths. That is it. Sorting two ninths, five ninths and seven ninths is no harder than sorting 2, 5 and 7. A quick reminder about the signs: the less-than sign and the greater-than sign both point their narrow end at the smaller number, and their wide open end at the bigger one. Picture a mouth that always wants to eat the larger amount.

Now look at the opposite situation, where the numerators match but the denominators do not. Three fifths and three eighths both have three pieces shaded, but the fifths are much fatter than the eighths, so three fifths is clearly the larger amount. The principle is that the more pieces you cut something into, the smaller each piece becomes. So when the numerators are equal, the smaller denominator gives the bigger fraction. This is the hardest habit to break in the whole topic, because with whole numbers 8 beats 5, and your instinct says three eighths must be bigger. Think of pizza instead: one pizza shared between 5 people gives you a much better slice than the same pizza shared between 8.

For any two fractions the reliable method is to give them a common denominator. Compare three quarters and five sixths. First find the lowest common multiple of 4 and 6. The multiples of 4 are 4, 8, 12, 16 and so on, and the multiples of 6 are 6, 12, 18, so the lowest they share is 12. Now rewrite both fractions over 12. Three quarters times three over three is nine twelfths, and five sixths times two over two is ten twelfths. The pieces are now identical in size, so it is simply 9 against 10, and ten twelfths wins. That means five sixths is bigger than three quarters. One habit worth having: give your answer using the original fractions, so say five sixths is bigger, not ten twelfths.

Sometimes you can settle a comparison without any common denominator, by using a half as a benchmark. Here is the test: double the numerator and compare it with the denominator. For five ninths, double 5 is 10, which is more than 9, so five ninths sits above a half. For three eighths, double 3 is 6, which is less than 8, so it sits below. One is above and one is below, so five ninths must be the bigger one — done, with no conversion. The same trick handles seven twelfths against four ninths. A second benchmark is 1: nine tenths is one tenth short of a whole while seven eighths is one eighth short, and a tenth is the smaller gap, so nine tenths is larger.

To order a whole list, put every fraction over one common denominator. Take two thirds, three fifths, seven tenths and a half. The denominators are 3, 5, 10 and 2, and all four divide into 30, so 30 is your common denominator. Two thirds times 10 over 10 is twenty thirtieths. Three fifths times 6 is eighteen thirtieths. Seven tenths times 3 is twenty-one thirtieths. And a half times 15 is fifteen thirtieths. Now just sort the numerators: 15, 18, 20 and 21. Translating back, the order from smallest to largest is one half, three fifths, two thirds, seven tenths. Two habits matter here. Write the answer using the original fractions, and check whether the question wanted smallest to largest or the other way round.