Which is greater?
Most people would approach this question by converting the fractions into decimals or putting both the fractions to a common denominator. While there’s nothing wrong with those methods, there’s a simpler and more elegant way of solving this problem.
The secret is cross-multiplication! Just multiply the left numerator with the right denominator and multiply the right numerator with the left denominator. Afterwards, compare the two products.
The side of has a larger product (40) than the side of (39). We can now conclude that is greater than . In decimal, = 0.375 and = 0.385.
That’s it! You don’t have to do anything else.
I promise that this method works with any two fractions. However, note that you can only use this method for comparing two fractions. Furthermore, this method also works for negative fractions.
Here are some exercises for you to practice your new-found skill.
Which is greater?
For those of you who want to see the proof before trusting me, here it is.
Theorem: Given integers p, q, r and s, where q and s are positive, we have,
If and only if ps < rq
The statement remains true if ‘<‘ is replaced with ‘>’ or ‘=’.
Proof: First, we write,
where ε < 0, = 0 or > 0 according to whether the inequality is ‘<‘, ‘>’ or ‘=’ respectively.
Multiplying both sides by qs, we get,
ps = qr + ε1 where ε1 = εqs Eq. 2
ε1 has the same sign as ε, since qs is positive.
To prove the converse, we start with equation 2. Since q and s are positive, qs is positive so we can divide Eq. 2 through by qs (or equivalently, multiply by the reciprocal of qs). This gives Eq. 1. Again, ε has the same sign as ε1, since qs is positive.
Don’t worry if you don’t understand the proof. You can use this trick with no problem without knowing how it works.