CTK Insights

Two fractions a/b and c/d are equal (or often equivalent) only if ad = bc. The same cross-multiplication helps compare two fractions. For simplicity I shall assume all integers positive. Thus, a/b > c/d if and only if ad > bc. This is a simple and, for small numbers, not an overly difficult test. However, there are many other ways which may require a lesser effort to come to a conclusion.

For starters, if a > b then naturally a/c > b/c. Also, for c < d, a/c > a/d. This follows from the interpretation of fraction x/y as designating x parts of a whole divided into y parts. The fraction grows as the number of parts x grows or when the number of subdivisions y decreases.

These two simple rules can be combined. For example, let's compare 8/7 and 9/5: 8/7 < 9/7 < 9/5 so that 8/7 < 9/5.

The example suggests another approach: it is often possible to discern a fraction in-between the two given ones to which both are easily compared. For example, 8/7 > 7/8 because one is greater than 1, the other is less: 8/7 > 1 > 7/8. Also, 5/9 > 10/21 because one is greater than 1/2, the other is less: 5/9 > 1/2 > 10/21.

There are additional tricks.

Assume a > b, i.e., assume that fraction a/b is improper. Then a/b < (a - 1)/(b - 1) and (a + 1)/(b + 1) < a/b. For example, 5/4 < 4/3 because 5/4 = 1 + 1/4 while 4/3 = 1 + 1/3 and 1/4 < 1/3.

For the proper functions, the situation is reversed: a/b > (a - 1)/(b - 1) and (a + 1)/(b + 1) > a/b. For example, 4/5 > 3/4 which could be read as 1 - 1/5 > 1 - 1/4 which is true because 1/5 < 1/4.

Observe that 1 can be added to, or subtracted from, the numerators and denominators repeatedly. Compare 17/19 and 11/13. 17/19 > 11/13 because 11 = 17 - 6 and 13 = 19 - 6.

When looking for a fraction included between the given two, the simplest one to compute is their mediant. The mediant fraction is obtained as the result of the wrong addition of fractions which is frequently employed by students.

By definition, m/n is the mediant of a/b and c/d if m/n = (a + c)/(b + d). (Let's emphasize once more that (a + c)/(b + d) ≠ a/b + c/d unless, of course, one of the fractions is 0.) If, say, a/b > c/d then also

For the sake of verification, the mediant of, say, 9/8 and 8/9 is 1 and, obviously, 9/8 > 1 > 8/9. Also, 5/11 < 3/5 because 6/11 < 8/16 = 1/2 < 3/5, 8/16 being the mediant of 5/11 and 3/5.

Compare 2/7 and 4/11. The mediant is 6/18 = 1/3. So, let's compare separately 2/7 and 4/11 with 1/3. 2/7 < 2/6 = 1/3. 4/11 > 4/12 = 1/3. It follows that 2/7 < 1/3 < 4/11. (Of course, in this case, cross-multiplication is quite straightforward: 2·11 < 4·7.)

At the Interactive Mathematics Miscellany and Puzzles there are two interactive tools to practice fraction comparison: Compare Fractions: Interactive Practice and Fraction Comparison Sped up.

Related posts:

1. Dividing Apples as a Motivation for Fraction Addition
2. Fraction Bloopers