GMAT Math Pro speed tip: Which fraction is greater?

Whenever someone asks me which of two fractions is greater, and I am, for whatever reason, obliged to take their query seriously, I’m always hopeful that the two fractions will turn out to be something like 999/1000 and 1/1000000. This is rarely the case, but I’m almost as grateful to get something like 2/21 and 3/19. In this case I can reason, without tedious calculation, that \frac{3}{19}, with its bigger numerator and smaller denominator, certainly has the greater value. But what if the fractions turn out to be something like \frac{2}{29} and \frac{3}{43}. Now it is not quite so easy. \frac{3}{43} has the bigger numerator, but it also has a bigger denominator. Is the increase in the denominator enough to offset the increase in the numerator? It’s difficult to say. In the old days your only option would be to get out your abacus, learn how to use an abacus, and compute the equivalent form of each fraction over a common denominator. \frac{2}{29} becomes \frac{86}{1297} and¬†\frac{3}{43} becomes¬†\frac{87}{1247}. So¬†\frac{3}{43} wins by a nose. The increased speed comes in realizing that there is no point in computing the actual value of the denominator. You KNOW the denominators will be the same because that’s the way you’re constructing them. The whole comparison depends on the numerators. So why not skip straight to them?

The fastest way to do it is to think of it as being akin to cross-multiplication. You’re comparing \frac{2}{29} to¬†\frac{3}{43}. Start with the bottom right number, 43, and draw a diagonal to the top left number, 2. Multiply along this diagonal to get 86, and write this to the left of¬†\frac{2}{29}. Then start with the bottom left number, 29, and draw a diagonal up to the top right number, 3, and multiply along this diagonal to get 87. Write this number to the right of¬†\frac{3}{43}. 87 is the bigger number, and it is next to¬†\frac{3}{43}, so¬†\frac{3}{43} is the bigger fraction.

To summarize, if you want to compare the the sizes of two fractions,¬†\frac{a}{b} and¬†\frac{c}{d}, if ad>bc, then¬†\frac{a}{b} is bigger and if bc>ad then¬†\frac{c}{d} is bigger. But don’t think of it this abstractly. If it comes up, your eyes should immediately be moving along those diagonals from bottom to top to see which side has the bigger product.

Now if someone asks you to compare the sizes of two fractions, whether it be a friend, neighbor, or the computerized testing agent of an organization that controls your future, you can confidently answer in a matter of seconds.

–GMAT Math Pro
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