GRE Quantitative Strategy: Everything Is Easier with Integers!

For many of my GRE students, manipulating and simplifying fractions is the bane of their GRE Quant preparation. Conceptually, fractions aren’t too difficult, but once it’s a matter of multiplying by reciprocals, rationalizing the denominator, taking fractions of fractions, and so on, many of my students struggling with the math tend to look like a deer-in-the-headlights. Though fractions are often unavoidable, there are strategies that can help simplify the process of dealing with them. One strategy concerns using integers when a question seems to involve an intimidating number of fractions.

Specifically, if you’re given a fraction question with unspecified amounts and asked to solve for a relationship between these amounts, the best strategy is to choose values that satisfy the relationships in the question. By doing this, you’ll put the question in a more real-world context by using integers, which we use in our day-to-day lives, instead of fractions. When plugging in numbers for fraction questions with no specified amounts, keep the following tips in mind:

1) When choosing a value, choose a value for the whole. For example, if you’re told that 1/3 of the employees in a company are administrators and that ¾ of the administrators are men, then choose a value for the number employees (the whole), and not the administrators or men.

2) Choose a value for the whole that will be divisible by the denominators of all the fractions in the question. For example, if you’re told that 1/3 of the employees in a company are administrators and that ¾ of the administrators are men, then the value you choose for the number of employees should be divisible by 3 (the denominator of the first fraction) and 4 (the denominator of the second fraction). Good numbers here would be multiples of 12. Why is this strategy important? Because, to the extent that’s possible, you want to work with integers. If the number that you choose is not divisible by 3 or 4, then the value for one of the parts will end up being a fraction, where the math will almost certainly be messier.

Let’s look at an example:

2/3 of the cars in a lot are sedans, and the rest are trucks. If 9/10 of the sedans are new and 3/8 of the trucks are new, what fraction of the cars in the lot are used?

Since the question does not provide any amounts, you should choose numbers to determine values for the total number of cars and the number of used cars. Based on tip #1, you should choose a value for the whole, which in this case is the total number of cars. Based on tip #2, the value you choose should be divisible by 3, 8, and 10. The most obvious value here is 3 x 8 x 10 = 240. If there are 240 cars, then 2/3(240) = 160 are sedans, and the other 80 are trucks. Now determine how many sedans and how many trucks are used. If 9/10 of the sedans are new, then the remaining 1/10 are used. 1/10 of 160 = 16, so there are 16 used sedans. If 3/8 of the trucks are new, then 5/8 are used. 5/8 of 80 = 50, so 50 of the trucks are used. In total, there are 50 + 16 = 66 used cars. The final fraction of used cars/total cars = 66/240. Divide numerator and denominator by 6 and arrive at: 11/40.