## Friday, January 3, 2014

### 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).
This question is by no means an easy one. The key is to not spend three or four minutes calculating both columns. Remember, when you see huge numbers or ones that seem very difficult to calculate come up with a logical approach. Let’s try a similar problem. Try to use the logical approach vs. solving approach. Okay, here is another QC question. Set your timers at 1 minutes. Good luck!
 Quantity-A Quantity-B 10^-100 + 99^100 9^-99 + 100^99
The numbers here are so astronomical as to preclude calculation. The first thing we want to do is step back from the problem and see if we can simplify anything. One helpful strategy in QC is to see if you can remove a sum or value from both sides. That is, if the values are very similar on both sides take them out. In this case we have 10^-100 in Quantity A and 9^-99 in column. The outcome of both of these is going to be numbers that are so tiny as to not make any difference in the overall sum of the product. So, instead of even trying to figure out which one is bigger remove both of them. Now let’s focus on the 99^100 and the 100^99. We don’t want to calculate the sum; we simply want to find out which side is bigger. Drawing from what we learned in Part 1 let’s test the relationship using small numbers. I’ll try 2^3 vs 3^2. (Note how 3 and 2 differ by one just like 100 and 99.) In this case we get 8 and 9 and the number with the smaller exponent is bigger. But as soon as we try 3^4 vs 4^3 (81 and 64, respectively) the relationship changes– the number with the larger exponent is bigger. We could try 4^5 and 5^4 (1,024 vs 625) just to be safe, but remember not to spend too much time. Anyways, when numbers are really small with exponents (1^2 and 2^1) the relationship with much larger exponents can’t always be correctly inferred. But notice that 21 is twice as much as 1^2, and 2^3 is almost the same as 3^2. In essence the number with the higher exponent “catches up” by the time we reach 3^4 vs 4^3. The higher is the exponent, the greater is the difference between Quantity A and B. Therefore by the time you get to 99^100 Quantity A will be far greater than column B. So the answer is (A). Keep visiting TCYonline.com, for more tips and tricks on GRE. Remember, we here at TCY are committed to your success.