Avoid Tedious Calculations on the Quant Section

by on March 27th, 2015

One of the hardest things for people to get used to on the GMAT is that there is no calculator for the quantitative section. The reasoning behind this is simple: human beings will not be faster than machines at pure calculations. Human beings, however, will be better at logic, reasoning and deduction than a machine.

The GMAT wants to determine how good the test taker is at solving problems through logic and analytical reasoning, not brute strength. Despite this stated goal, the GMAT frequently features questions that can turn students into mindless calculators. The goal is to avoid falling for this sinister trap and solving the problem with sound strategy and logical applications of mathematical theory.

The quintessential large calculation will be something like “Multiply all the integers from 1 to 10” (or more succinctly, find 10!).  Now, such a calculation is possible within the 2 minutes we typically have to solve a question, but even when you get the result, there is often another portion to the question that must be solved. Even if the end goal is just to find one number, the brute force approach is time-consuming and error-prone. You are much better off approaching the problem using either order of magnitude or unit digit properties.

Generally speaking, asking someone to compute 10! can be tedious. However, the GMAT is in fact asking you which of the five choices provided is 10! The answer choices provided are typically fairly far apart, so an approach that cares only about the order of magnitude of the answer will help narrow down the possibilities tremendously. Sometimes there may still be two contending answer choices, and additional calculations may be required to confirm which one is correct.

For 10!, we can calculate the small numbers easily and approximate the rest. You can get much more detailed than this, but 5! = 120, and then multiplying by 6 and then 7 is like multiplying by 5 and then 5 again, so 120*25 = approximately 3,000. Multiplying by 8, 9, and then 10 is like multiplying by 10 thrice, so the answer will be somewhere around 3,000,000. I approximated a couple of numbers up and a couple of them down to somewhat balance out. You can approximate more closely to reality but you should still get an answer in the same ballpark (actual retail price: 3,628,800).

Another potential shortcut is to consider only the unit digit. The answer choices will tend to be far apart and have different unit digits, so if you can calculate which number should be the unit digit, you can eliminate several answers quickly. In our case, we know we’re multiplying by 10, so the unit digit will be zero. Furthermore, we are multiplying by 5 as well, and there are many 2’s (including 2 itself), so there will be a second zero as a tens digit. In this case knowing factors simplified the process, but even trying to figure out the unit digit of 2^88 is simply an exercise in pattern recognition.

The above example may have been somewhat abstract as there were no answer choices to compare, so let’s look at an actual GMAT question and apply these same strategies:

A small cubical aquarium has a depth of 1 foot. In the small aquarium there is a big fish with volume 44 cubic inches. A big cubical aquarium has a depth of 2 feet and 88 fish, each with a volume of 2 cubic inches. What is the difference in the amount of water between the two aquariums if they are both completely filled? (Note: 1 foot = 12 inches)

A) 246 cubic inches

B) 300 cubic inches

C) 11,964 cubic inches

D) 13,824 cubic inches

E) 16,348 cubic inches

This question is considered geometry because it is dealing with a 3-dimensional shape, but the question is primarily concerned with converting cubic feet to cubic inches. As such, the question is really asking for a laborious calculation. Therefore, we need to find a shortcut to avoid spending the rest of the hour calculating cubic inches in our aquarium.

A cubical aquarium with three sides of 1 foot is 1 cubic foot (or foot^3), but that doesn’t help much in terms of cubic inches. The easiest thing is to convert to inches from the get go, which leaves us with a cube that has height, width and depth of 12 inches. Since the formula for the volume of a cube is side^3, we know that the volume of the aquarium is 12^3 cubic inches. 12^2 is easy, so now we must multiply 144 by 12. It might take a few seconds, but we can break it down to 144*10 + 144*2, which yields a total of 1,728 cubic inches.

At this point, a lot of people would think about removing the volume that is being filled by the big fish. While this is technically correct, if we’re considering this problem from an order of magnitude point of view, it will be a drop in the bucket, taking the total volume from 1,728 down to 1,684 if you subtract 44 cubic inches. Both of these numbers are essentially 1,700, so there’s not much value in taking the time to remove the fish. I’m more concerned with shortcutting the calculation for the big aquarium.

The big aquarium has sides of 2 feet, or 24 inches. This means that it will be twice as wide, twice as tall and twice as deep as the small aquarium, leading to an overall eight-fold increase from the original aquarium. This means that I can take the 1,728 I calculated earlier and multiply it by 8 to get the total volume (sans fish) of the big aquarium. However, the problem eventually asks for the difference in water between the two aquariums , which means I’ll have to take the 8Y volume and subtract the original Y volume. This means we’re better off shortcutting the calculation and just multiplying the original volume by 7. It’s tantamount to saying I’ll lend you 100$ then you lend me 20$. I think we can just make one transaction for 80$ and call it a day.

Multiplying 1,728 by 7 isn’t necessarily trivial, but remember that we are  mostly interested in the order of magnitude of the answer. This means we can ignore some digits and think of it as approximately 1,700*7, which is (1,000 x 7 =7000 + (700*7=) 4,900, yielding a total of about 11,900. It should be a little higher than this because we rounded 1,728 downward. This is almost exactly answer choice C, with answer choice D looking about 1,700 bigger and thus likely the volume of the bigger aquarium only. The other three answer choices are way off.

At this point we are essentially done, but you can confirm the number, particularly with the consideration of the fish (plural but hard to tell). The volume of the small aquarium is 1,728, and of the big aquarium is indeed 13,824 cubic inches. If we subtract the 44 cubic inch fish from the small aquarium, we get 1,684. If we subtract the 176 cubic inches (88*2) of the big aquarium fish, we get 13,648. Finding the difference of these two numbers yields exactly 11,964 cubic inches. Answer choice C is correct, but you don’t have to meticulously calculate every element in order to know it given the five choices provided.

It is  worth noting that unit digits don’t help much on this problem. The smaller aquarium has a volume of 12^3, and the 2^3 unit digit will yield an 8. Subtracting the 44 cubic inches for the fish (which we must do if we’re being precise), the water in the small aquarium should end with a 4. The big aquarium has a total volume of 24^3, which will give a unit digit of 4. Subtracting the 176 cubic inches for the fish leaves us with a unit digit of 8. Finally, subtracting the 4 of the little aquarium from the 8 of the big aquarium means the answer choice must end with a 4. Despite all that abstract and confusing math, we still can’t choose between answer choices C and D, and must therefore perform additional calculations.

Sometimes the GMAT likes to ask questions that would take 15 seconds if you had a calculator, but 5 minutes if you stubbornly decided to use an inflexible brute force approach. Sometimes unit digits will be faster, and sometimes order of magnitude will be faster, but both have their place in your tool belt. Each question on the GMAT is like a door, and you may be able to knock down the door with brute strength, but you’ll go faster with a deft touch.

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