Simple calculation tricks

Much of the Quant section involves calculations, and as calculators are not allowed, we need to be able to add, subtract, multiply and divide numbers quickly and accurately. A surprising number of errors are made on easier problems simply because of computational error and, interestingly, many trap answers on the exam come from common errors made in computations!

 

1) The Distributive Property is Your Friend

Many of us recall the basic properties of operations (associative property, commutative property, distributive property and identity property), but rarely do we consciously use them except in some algebraic manipulations. However the Distributive Property is also very useful in quickly and easily multiplying larger numbers.

For example, take the following: 163 x 30. Multiplying this out long-hand is not overly difficult, however, since we do not do this in our everyday lives, we are prone to errors. Using the Distributive Property here can help.  Intuitively, many of us would see that 163 x 30 can be broken up into 163 x 10 three times – THAT is the Distributive Property. We can then re-write the expression as 163 x (10 x 3), and then “distribute” the operands into (163 x 10) x 3

Now, couple the Distributive Property with some other tricks that many of you already use, and this tip becomes even more valuable.  Take, for instance, a more difficult calculation to “distribute” – say, 163 x 48.  You could “distribute” this into 163 x (40 + 8) and get 163 x 40 (an easy calculation) + 163 x 8.

A common trick that is used when multiplying by 8 or 9 is multiply the number by 10 and then subtract out “extra.” In this case, we would multiply 163 by 10 and then subtract out the 2 additional 163’s. If we combine these steps on the front end, we could say 163 x 48 can be re-written as 163 x (50 – 2) and get (163 x 50) – (163 x 2). 1633 x 50 can be further broken down as 163 x 10 x 5, 1630 x 5. Putting it all together, we get 163 x 50 = 163 x 10 x 5 = 1,630 x 5 = ½ of 16300 = 8,150.

8,150 – (163 x 2) = 8, 150 – 326 = 7, 824

This is a good trick to help make long-hand multiplication simpler. Be careful in breaking numbers down into too many pieces as this can overly complicate the process and lead to errors.

2) Estimation and Proportionality

Many times this is straightforward, but other times it can be quite vexing. For instance, a problem may tell you that the ratio of boys to girls in a particular class is 4:9 and ask what percentage of the class is boys. Obviously, we can see that the TRAP answer would be 44% (4/9 = 44%). But, how can we easily and accurately calculate, or estimate, the correct answer?

Here is a trick to help with that: because we have ratios down pat, we know that a ratio of 4:9 tells us that there are 4 boys out of a class of 13, so the proper fraction would be 4/13. Now the tough part is turning this into a percent. Since percents simply are fractions with a denominator of 100, we can set up an algebraic equation. In this case, we have 4/13 = x/100 and using cross-multiplication we see the answer is 400 ÷ 13, 30 and 10/13, or a little less than 31%.

An easier way, might be to use proportionality to estimate the answer. Here is how that would work:  proportionally, our fraction of 4/13 needs to be converted to a fraction over 100. To increase our denominator of 13 to 100, we need to multiply it by a bit over 7 ½. To keep the fraction in its original proportion, we will also need to multiply the numerator by the same (a bit over 7 ½), giving us a value of a little over 30%.

The takeaway from this is that by sharpening your computational skills, you can save time, improve accuracy and even minimize your effort on the exam. This can translate to higher scores by eliminating silly mistakes and saving brain power for the more difficult questions.