Quantitative Comparison Questions for GRE Preparation

Strategies when the numbers are huge Let us try a tricky quantitative comparison question. Let’s see if you can get it in less than one minutes!

Quantity-A	Quantity-B
10^7	7^10

One way of solving this problem is to do the tedious calculations. However, doing so is a time-consuming, error-prone process, and though it may work, the GRE is not testing your ability to do laborious mathematics. Rather, it is testing the way you think, your logical approach to a complex problem. One useful approach is to approximate. For instance if we take 7^² we get 49. Notice 49 is very close to 50. So let’s say 7^² is equal to 50. Therefore (7^²)^² is equal to 50^² or 2500. 2500 can also be represented as 2.5 x 10^³. So when we take 2500^² or 7^⁸ we can convert 2500 into 2.5 x 10^3 and we have (2.5 x 10^³)^². Again, because we are approximating let’s round 2.5 down to 2. Now we have (2 x 10^³)^² = 4 x 10^⁶ = 7^⁸ . This figure is almost as large as 10^7. Since we are comparing 7^¹⁰ to 10^⁷ you can see that 7^¹⁰ will be much larger. Granted, this method still involves a decent amount of calculation. Another technique we can apply is pattern recognition. For example, we can use smaller numbers that adhere to the pattern above. So let’s compare 2^⁵ to 5^². Notice that I’ve kept a similar pattern to the original question: x^^y vs. y^^x. , where y = x + 3. In this case we get 2^⁵ = 32 and 5^² = 25. Notice that the number with the greater exponent is greater. We should try one more set of numbers just to be sure, 3^{^6} vs. 6^³. Notice that this time the difference between 3^⁶ and 6^³ is even greater than that between 2^⁵ and 5^². That is, the difference between x^{^y} and y^^x where y = x + 3 increases the larger the numbers we plug in. Therefore 7^¹⁰ will be much greater than 10^⁷. The answer is (B).