Students seem to love using cross multiplication to compare fractions and check for equivalence; this simple, mysterious procedure will assure them that one fraction is indeed larger than another or that two fractions are equivalent. But ask them why it works, and they likely will respond with “it just does!” This is because they have not explored and reasoned that cross multiplication is just a way of making fractions into equivalent fractions with the same denominator so they are easier to compare.
Suppose you want to compare 3/8 and 2/5. A common denominator for 8 and 5 is 40, since both 8 and 5 can divide evenly into 40.
3/8 = (3 x 5) / (8 x 5) = 15/40
2/5 = (2 x 8) / (5 x 8) = 16/40
Now that the denominators are the same, we can just compare the numerators. 3/8 is less than 2/5 because 15/40 is less than 16/40.
Now look at the numerators in your new fractions. You got them by multiplying the old numerators times the denominator of the other fraction.
Cross multiplication is therefore just a shortcut to find those new numerators. We are basically changing the given fractions to equivalent fractions with the same denominator – the product of the two denominators – and comparing the numerators. Since we know the denominators will be the same, we ignore them and use a shortcut to just find the new numerators by multiplying each old numerator by the other fraction’s denominator.
So to compare 3/8 and 2/5 : 3 x 5 is less than 2 x 8, just like we saw above. Therefore 3/8 is less than 2/5.
In Teaching Fractions According to the Common Core Standards, mathematician Hung-Hsi Wu states:
Let’s all work towards eliminating “tricks” from our math instruction and help students make sense of the math they are learning.