Mental Calculations on the GMAT

One of the reasons calculators aren’t allowed on the GMAT is to ensure that people are really thinking about the numbers they are using to solve problems. Being at ease with mental math is a skill that has been slowly eroded since the advent and subsequent ubiquity of the calculator in the education process (sadly my frequent calls to bring back the abacus have gone unheeded). Too often, people mindlessly type in numbers, and don’t even notice if they hit the wrong number or a button gets pressed twice. Of course 5*6=45, the machine told me so! (Dependence on machines also eventually leads to Skynet) However, being good at mental math can be helped along if you already have a good idea which numbers you might expect to see on test day.

On the GMAT, the absence of calculators makes longhand or mental math much more prevalent. Being able to quickly calculate relatively easy numbers in your head can be a very impressive tool in your arsenal as a business student and as a person who has to deal with money and time every day (all of which are just numbers). When I was a lowly pizza delivery boy working weekends in college, I’d always be able to calculate the amount owed to customers a split second after they paid me. It’s not particularly difficult to eventually determine that you owe the customer 14.38$ for a 25.62$ order that was paid for with two twenties, but telling the customer quickly that you owe them 14.38$ made it a lot easier for them to say “give me back 10$” then fumbling for change on the front steps. The secret to my success was that many orders (2 large pizzas) came to the same exact price. In short, it was a number I was expecting to see.

On the GMAT, there are also numbers you’re expecting to see regularly, so it makes sense to know and understand them ahead of time. I tell my students from the very first class that knowing the multiplication table forwards and backwards will save them significant time over the course of the exam. My (completely unscientific) observations indicate that certain numbers such as 56 (7×8) and 96 (8×12) come up in a lot of questions, but really any numbers in the 12×12 grid could show up, so they’re all worth knowing. The most common numbers that will show up will be entries on the standard multiplication table you learned in grade school.

However, other numbers show up a lot that aren’t necessarily the product of two numbers less than or equal to twelve. Do the numbers 128 or 256 seem familiar? How about 243 or 729? These are all multiples along the 2^x and 3^x lines, respectively, and it wouldn’t be a bad idea to know both of these lines up to at least the power of 5. Powers of 2 give a fairly easy streak from 2 to 1024 (2^10), numbers that become very useful in binary decisions and coin flip probability. Powers of 3 go from 3 to 9 to 27 to 81 to 243. 729 is a little ambitious, but you can figure it out very quickly with 243 if you need to (or it’s 27^2, obviously!).

Why should we bother with these kinds of numbers? Because the GMAT tends to give you questions like this:

If a and b are integers and (a*b)^5 = 96y, y could be:

(A)     5
(B)      9
(C)      27
(D)     81
(E)      125

Now, the math on this question is going to get very messy (think fifth root of 96×5), so we’ll have to look for some kind of logic behind the number.  The total number must have a fifth root that is an integer, as both a and b are integers and therefore their product must be an integer as well. The only numbers that you could conceivably see on the GMAT to the fifth power are 2 and 3, with 2^5 being 32 and 3^5 being 243. Anything more than that and you’re getting a little outlandish for what the test can realistically ask you to know or calculate in less than 2 minutes.

If you look at 96y, the number 32 should jump out at you pretty quick. If either a or b were 2, then 32 could be easily factored out of 96y leaving 3y. If 3y had to be equivalent to some number (a or b, whichever is left over) to the power of 5, then the obvious candidate is 3. Assuming b is left over then b^5=3*y. Replacing b by 3, we get 3^5=3^1 * y, so y must be 3^4. Answer choice D is exactly 3^4 (or 81). None of the other answer choices could have fifth roots that are integers. This question could have multiple answers, but they must follow very precise criteria and none of the other choices presented even come close to working.

The correct answer is (D). 

Doing the math longhand we find that 6^5 is actually 7776, highlighting the fact that the exam will never ask you to calculate that. Not having a calculator is a blessing in disguise. You may not be able to easily calculate any number in a few hurried seconds, but you are not responsible for the entire gamut of real numbers either. Some numbers (like the price of two large pizzas) can be expected to show up on test day, and 2^5 is the type of number you should always have in your back pocket, just in case you need to show the GMAT how prepared you are for their predictable traps.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.