Level 4 · Ages 13–15
Read it again any time, including while you work through the exercises.
Who eats more cake: the one who takes 5/6 or the one who takes 7/9? Impossible to answer at first glance... and that is exactly what this lesson will change. Comparing fractions means deciding which one represents the bigger amount, and there are several strategies depending on the situation.
Strategy 1: same denominator. If the parts have the same size, just count the parts: 5/7 > 3/7, as simple as that. When the denominators differ, you can rewrite the fractions over the same denominator using equal fractions: 5/6 = 15/18 and 7/9 = 14/18, so 5/6 > 7/9. Mystery solved!
Strategy 2: same numerator. If you take the same number of parts, the bigger fraction is the one with the bigger parts, so the one with the SMALL denominator: 4/5 > 4/9, because fifths are bigger than ninths. This is the most common trap: a big denominator does not mean a big fraction, it is actually the opposite!
Strategy 3: benchmarks and decimals. Compare each fraction to simple benchmarks like 1/2 or 1: a fraction is greater than 1 when its numerator is bigger than its denominator. And when in doubt, division gives the decimal form: 7/8 = 0.875, easy to compare with 0.9. With these three tools, you can order any list of fractions from smallest to largest or the other way round.
Compare 5/6 and 7/9.
6 × 3 = 18 et 9 × 2 = 18
I look for a common denominator: 18 is a multiple of both 6 and 9.
5/6 = 15/18 et 7/9 = 14/18
I convert each fraction: 5 × 3 = 15 and 7 × 2 = 14.
15/18 > 14/18
Same parts (eighteenths): I compare the numerators, 15 > 14.
Result: 5/6 > 7/9
Order from smallest to largest: 2/3, 3/4 and 7/12.
dénominateur commun : 12
12 is a multiple of 3, 4 and 12: I rewrite everything in twelfths.
2/3 = 8/12, 3/4 = 9/12, 7/12 = 7/12
I convert: 2 × 4 = 8 and 3 × 3 = 9.
7/12 < 8/12 < 9/12
I order the numerators: 7, then 8, then 9.
Result: 7/12 < 2/3 < 3/4