Everyday Ways To Improve Your Mental Math Skills
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In a world where we are often carrying at least one device, if not multiple devices, that can complete calculations, there is little need to do calculations manually. For this reason, the lack of a calculator on the GMAT Quantitative section is a significant point of concern—or perhaps even fear—to many test takers, even some with strong quantitative skills. That brings me to some good and some bad news for prospective GMAT takers.
The bad news: At some point during the GMAT, you will find yourself using the pen and pad provided to crank out some long division or complex multiplication.
The good news: If you take some time to sharpen your mental math skills, you can significantly lessen the need for hand calculations and prevent some mistakes.
Mental math skills can be helpful in a variety of ways.
- Allowing you to do an exact calculation in your head, thus saving time
- Enabling you to make an estimation that may allow you to get close enough to pick the correct answer (or at least eliminate some answers)
- Providing you the ability to do a ballpark calculation in your head that will allow you to catch major mistakes you might make in pen and paper calculations
How to Practice Your Mental Math Skills
Although we do not deal with all aspects of GMAT math frequently (I am still waiting for my knowledge of the Pythagorean Theorem and trapezoid area calculations to enable me to MacGyver my way out of an extremely tight spot), most of us do interact with two types of quantitative information everyday: time and money. These interactions provide great opportunities to practice your mental math skills. Several examples are provided below.
1) What’s the total? (addition and multiplication)
If you are going into the grocery store or pharmacy to buy a few items, try to keep a running total of the cost of your purchases. If you have trouble keeping an exact tab, try to estimate a value you know your total will fall above or below. For example, I buy one item that cost $6.80, one that costs $3.83 and on for $4.25. I know the total (before tax) will be less than 7 + 4 + 4 = $15 because in the estimation I rounded up more than I rounded down. As your skills improve, challenge yourself to find ways to make closer approximations. Also, challenge yourself to keep the total on larger shopping trips when you purchase more items. You can check the accuracy of your estimate against the receipt.
You can also practice multiplication using this tool if you are buying several of the same items. One good opportunity occurs when filling your car up with gas. For example, I know it usually takes about 12 gallons to fill up my car when I am running on almost empty. Gas today cost $3.87 per gallon. My first estimate for my total was $4 * 12 gallons = $48. I then challenged myself to get closer. For each gallon I was paying $0.13 less than the full $4, so for every 8 gallons I would actually pay about $1 less than my initial estimate. With a total of 12 gallons (1.5 times 8), I would estimate the total to be closer to $48 – $1.50 = $46.50.
2) What’s the unit cost? (division)
When purchasing a product that is available in multiple different sizes, try to compare the price per unit to determine which size is a better value. For example, if a 14 oz package of a product costs $1.80, the price is slightly less than $0.13 per ounce. To make this calculation, I determined that at $0.10 per ounce the package would cost $1.40, leaving an additional $0.40—adding about another $0.03 per ounce (3 * 14 = 42). If a 9 oz package costs $1.30, the price per ounce is more than $0.14 ($0.10 per ounce would be $0.90 leaving an additional $0.40 to be divided among 9 oz). Many stores provide per unit costs on product signs, so you may have the opportunity to compare your calculations.
You can also find creative ways to compare prices without taking calculations all the way down to the per unit cost. Using the products in the previous example, the price of the 9 oz package is more than 2/3 the price of the larger package ($1.30 > $1.20). The 9 oz package is less than 2/3 the size of the 14 oz package (14 = 42/3 and 2/3 of this value is 28/3 or 9 and 1/3). If the smaller package is more than 2/3 the price and less than 2/3 the size, it is more expensive per unit.
3) What’s the tax? (percentages)
If you are just purchasing a single item, try to calculate the total cost with tax. Again, as you get some practice, challenge yourself to make your estimations more accurate. For example, if buying an item that costs $5.85 in a state with a 7% sales tax, a good start might be to find 7% of $6, $0.42, and add that amount to the original price to get $6.27. If you want to get more exact, you actually should pay about $0.01 less in tax because the price was $0.15 (more than 1/7 of a dollar) short of the $6 mark. This calculation would lead to a revised estimate of $6.26. (Warning: You will get strange looks from cashiers if you give them exact change before they ring up your total). You can also practice the same calculations with tips or percentage discounts.
4) How much time is left? (fractions and percentages)
Any event that you expect to run for a specific amount of time provides another practice opportunity. I tend to do this practice when running on the treadmill in hopes that the strain of running at 7.5 mph simulates the duress of taking the GMAT. A boring meeting or conference call, however, would work just as well. When you have a given amount of time left, try to find a relatively simple fraction or percentage that would provide a good estimate of the amount of time remaining or time completed. For example, let’s say I plan to run for 45 minutes and have currently run for 34 minutes. I have 11 minutes remaining. As a fraction, this value is of course equivalent to 11/45, but it is not possible to simplify that fraction further. Thus, I try to find a close estimate of my original fraction that can be simplified, such as 11/44. Once the fraction is simplified, I can see I have about ¼ or 25% (you should be able to convert thirds, fourths, fifths, sixths, and eighths into percentages) of my run remaining. I should also realize that I actually have a little less than ¼ remaining, because to create my easier fraction I lowered the denominator while leaving the numerator untouched.
Another variation on this exercise is to figure out when you will have completed a certain fraction or percentage of the event. For example, I could try to figure out when I will have 85% of my 45 minute run completed. When I have completed 85% of my run, I will have 15% of my run remaining. I can calculate 15% by taking 10% of 45 (4.5 minutes) and adding 5% (half of 4.5 = 2.25): 4.5 + 2.25 = 6.75. So I will have completed 85% of my run when I have done 45 – 6.75 = 38.25 minutes, or 38 minutes and 15 seconds (using fractions of minutes and converting to seconds is a nice extra challenge once you get more comfortable with calculations). You can also perform this exercise with distances (e.g. what percentage of your bike ride have you completed?).
These are far from the only opportunities to practice mental math skills in daily life. Get creative and look for other opportunities (and please share any good ones in the comments). Also, don’t be afraid to get creative in how you attempt your estimations. While I provided examples of my approaches to each calculation above, there are a myriad of ways to attempt each one and you need to find those that work best for you. As your mental math skills improve, challenge yourself to make closer approximations and take on harder calculations. In my experience, doing just two to three calculations a day for about a month will significantly increase your comfort and facility with these math skills. Even if you are only able to use these new skills once or twice on the GMAT, they could save you valuable time or lead to an additional correct answer—results that are likely well worth the effort. ?
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Andrea Pawliczek is a Manhattan Prep instructor based in Denver, Colorado. Andrea is the owner of a perfect 800 GMAT score, a summa cum laude graduate of Emory University with a degree in economics and chemistry, an MBA from Duke’s Fuqua School of Business, and is currently a Ph.D. student in accounting at the University of Colorado – Boulder. Andrea began tutoring during her time at Emory and further honed her teaching skills at Fuqua, where she served as a teaching assistant for accounting and finance courses. Now, she continues to teach financial accounting to CU undergraduates and the GMAT with Manhattan Prep. Check out her upcoming GMAT courses here.