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Brian Galvin is the Director of Academic Programs at Veritas Prep, where he oversees all of the company’s GMAT preparation courses.
Jaime Escalante, the famed calculus teacher and subject of the movie Stand and Deliver, famously told his students “calculus doesn’t need to be made easy; it’s easy already”. To paraphrase the late icon, there are no “tricks” to success on the GMAT; math in itself is clever enough.
Sure, there are countless ways to save time and cut corners on the GMAT, but none is inherently “tricky”. Each of these methods just takes advantage of the beauty of math. As you attempt to maximize your efficiency and accuracy on the GMAT, and ultimately your score, embrace these shortcuts for what they are, but also learn to appreciate the logic behind them. Knowing that each of these tricks has a logical foundation in math, you can use that understanding to apply that logic to solve several problems. Consider an example of a pretty elegant “trick” that could help you on the GMAT:
Say you needed to find the square of 31. Rather than stack up 31 *31 and plunge through the math, you could use a fairly clever “trick” to calculate it quite efficiently. Because 30*30 is 900, you can simply add 30 and 31 to determine that 31-squared is 961. Why?
If you have 30*30 and need to get to 31*31, you might first start by adding 30 to that initial 30*30. Adding an extra 30 means that you have 31 30s, or 31*30. You could also, then, consider 31*30 to mean that you have 30 31s. To get another – to have 31 31s, or 31*31 – you’d need to add one more 31. So, you’d add 31 to get there, and you’d have 31*31.
Reiterating those steps, you took 30*30, then added 30 (itself) and 31 (the next integer) to it to determine that 31-squared is 961. Essentially, the rule for finding the next square in the sequence is:
Take the previous square, then add its square root plus that square root + 1.
Or, more theoretically, n-squared = (n-1)-squared + (n-1) + n.
This property may well save you some time and arduous calculation in a situation like 31-squared, but its application may prove even more helpful (and definitely more fascinating):
The squares of integers increase by adding consecutive odd integers.
Think about it; to replicate the pattern of squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
You add:
3 (to get from 1 to 4), 5 (to get from 4 to 9), 7 (9 to 16), 9 (16 to 25), 11, 13, 15, 17, and 19
Why is this true?
Let’s use 25, or 5-squared, as our starting point. To get to 6-squared, we need to add 5, then add 6, which means we add 11. To get to 7-squared, we’ll add 6 again (which is now our “base” number, or n), then add n+1, which is 7. 6+7 is 13. Each time we do this, we’re adding two consecutive integers (and the sum of consecutive integers is always odd), and we’re just moving up each integer by one, so we’re adding another 1+1 to what we previously added. That pattern will go on forever; to get to 27-squared, we can start by knowing that 25-squared is 625. To get to 26-squared, we’d add 25+26, or 51, then to go up one more square, to 27-squared, we’d just add another 53. Essentially, we’re taking 625 and adding 51, then 53; so, we can take 625, add 104, and get 729 as 27-squared.
This “trick” can be a time-saver on test day, but if you have to think about it to know why it works, you also put yourself in a position to better attack the Number Properties and Sequences concepts that the GMAT tends to use as the bases of some of its tougher questions. Several GMAT questions will ask you to “spot the pattern” – either explicitly or as a way to more efficiently tackle a seemingly labor-intensive problem – and there are few better ways to develop that skill than by checking each “trick” you come across to see why such a pattern holds.
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