When To Actually “Do the Math” On Data Sufficiency

by on December 8th, 2012

Today’s article comes courtesy of Veritas Prep GMAT instructor and BTG expert, David Newland.

I have read many variations of the statement – “And on Data Sufficiency you don’t even have to do the math!” But when do you really get to skip “the math” and when do have to do the math just to be sure?

High jumpers and weight lifters have this curious way of going about their competitions. Top competitors will often wait several rounds before they take their first jump or make their first lift.  An Olympic high jumper might sit out many heights and wait until after 2.2 meters to take his first jump. (Americans – that is an impressive 7 feet 2.5 inches).

How does an athlete decide which rounds to sit out and when to finally jump? Basically, the competitor will skip any height that he (or she) is certain that he can clear. The competitor will save energy and focus by not attempting anything that he knows he can do. He will only start jumping when the bar gets high enough that there is at least some doubt as to whether he can clear it.

That is the standard for Data Sufficiency as well. You can “skip” the math only if you are certain that you will get result that you predict. For example, if you know that you will get a single value for the variable “x” then you know the statement is sufficient and you know there is no need to do the math. But if there is any doubt as to the result then, like the Olympic high jumper, you need to prove it.

When you can safely skip the Math

You can safely skip the math when you are certain that the statement will be sufficient or not sufficient. Here are some examples:

Example A:

Is x > 5?

1) 2x - 15 = 17(x - 5) + 175

You can see that statement 1 here will give you a single value for the variable x. This is a linear equation with a single variable. The only way that this will fail to yield a single value is if you have the same coefficient for x on both sides. Since this is not the case here you are safe to skip the math and declare that this statement is sufficient. You do not need to do the math because the single value that you get will be either greater than 5 (Yes) or not (No): in either case you have a consistent answer and the statement is sufficient. This statement would, of course, also be clearly sufficient if the question asked for a single value for x. So there is no need to do the math when you have a linear equation that will absolutely yield a single value.

Example B:  

What is the value of x?

1) x^2=100 + (88/3)* (3/2)

It is again pretty clear that statement 1 will give you 2 values for x a positive and a negative. The math here is not difficult but there is no need to do the math. You know that you will get two values – the positive root and the negative root – so this is immediately not sufficient. You can skip the math when the statement is clearly not sufficient as well as when it is clearly sufficient.

Example C:      

Is y a positive number?

1) 2x + y > 27

This is an automatic “not sufficient.” Statement 1 gives you two variables and no way to fix a value for either one. Y can be a positive number, or if x is a large enough number y could be negative or zero.  There really is very little “math” that you could do here anyway. Not sufficient.

When you need to do the math

If you are not certain that you will get the result that you expect, then you had better do the math – at least until the point where you are certain what the result will be.

Returning to the above examples, let’s try statement 2 for each and see why sometimes you need to do the math.

Example A:  

Is x > 5?

1) Discussed above

2) 2x-5 + 3y = 5 - 2 (19 + x - 3/2 y)

Statement 2 has two variables and only one equation and does not seem likely to yield a value for x. However, if you do the math you see that the y’s will actually zero out and you will get a single value for x.

Begin by multiplying – 2 across the quantity and the equation becomes
2x - 5 + 3y = 5 - 38 -2x + 3y

Since 3y is on both sides of the equation you can subtract 3y from both sides and the y variable disappears. The coefficient of x is 2 however it is 2x on the right side of the equation and -2x on the left side.

Simplifying the equation becomes “4x = -28“ and “x = -7.”  The last two steps are not actually necessary since the equation has shown that it is actually a single variable linear equation. However, it is essential to do enough math to make sure that the statement is indeed sufficient or not sufficient. In this case it is sufficient.

Example B:

What is the value of x?

1) Discussed above

2) x^2-19x = 5x-144

Returning to example B we go to statement 2 and find a quadratic equation. This will clearly give two different values for B right? So you can skip this one? Maybe we better just do the math to see if we get two different values for x. In particular that 144 could be a problem since that is a perfect square.

Subtract 5x from both sides = x^2- 24x = - 144.

Add 144 to both sides = x^2-24x + 144 = 0.

Factor = (x - 12) ( x - 12) = 0.

Therefore x must equal 12. Meaning that statement 2 is sufficient. It was important to just do enough “math” to check to see if we would in fact get two different values for x. The GMAT does not reward you for making assumptions on Data Sufficiency!

Example C:    

Is y a positive number?

1) 2x + y > 27″ title=”2x + y > 27″/></p>
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Returning to example C, we now have a second statement that is also clearly not sufficient alone. As with statement 1, there are two variables in the inequality and no way to fix a value for either of them. Statement 2 also allows y to positive, negative and zero.

But what about the two statements together? It is pretty clear that you will need to do some math here. But how much?

Begin by multiplying the second statement by  - 2.

It becomes “-2x + 6y > – 48″.

Statement 1 is “2x + y > 27“

You might be tempted to stop here. The x’s will zero out and you will be left with a single-variable inequality. This would seem to be enough to answer the question of whether y is a positive number. However, inequalities can be particularly tricky in this regard. You had better finish the math to ensure that you get a consistent answer to this yes/no question.

Adding the two statements zeros out the x’s and leaves you with 7y > -21 or y > -3. This means that y could be “-2″ or “-1″ which is negative and of course y could be any positive number as well.  This is not sufficient since the answer is not consistent. Even with both statements, y could still be positive or negative. Not sufficient, the answer is E.

If you want to find success in GMAT Data Sufficiency then compete like an Olympic high jumper – save your energy by not attempting anything that you are absolutely sure of, but be sure to do the work any time there is a doubt about the outcome.

Challenge Problem

The above examples were all taken from the new Veritas Prep Data Sufficiency Book (being used in classes now). The following challenge problem is from that book as well. You can post your explanation in the comments section below:

The ratio of television sets to radios at an electronic store before a new shipment arrives is 12:7. If no other televisions sets or radios leave the store, and the only television sets and radios to arrive are in the new shipment, what is the ratio of television sets to radios after the new shipment arrives?

  1. The new shipment contains 132 television sets.
  2. The new shipment contains 77 radios. 

11 comments

  • In the second use of example C, statement 2 is not listed - - it should be "x - 3y - 48"

    I hope to get this corrected soon. I swear that statement 2 was in original article!

  • its either both of them or none of them, because with each individual one, we simply have no idea how much of the other is added, and these directly influence the ratio.

    In fact, the answer is E. You cant add numbers to a ratio to get a ration, because the higher you get, a million radios and tvs, the ratio CHANGES to a greater amount.

    • Thanks for the reply!

      I certainly agree that adding to a ratio (as opposed to say multiplying) rarely yields a set new ratio.

      However, what is the one instance in which adding to the ratio still leaves us certain of the new ratio - even though we do not know what the beginning numbers are?
      Hint: think about cooking and what people do with recipes.

  • I spent the last 2 hours thinking...NO IDEA. I tried 1:1, fractions, and with respect to the recipes we'll do 3X, 2X, or cut the proportions in half..but those are multiplying!
    Let me get this straight. You're saying there is a ratio, say bats to balls, where we don't know the actual number of original bats and balls, but when we add a KNOWN number of bats and balls, we can know the new ratio? I just want to make sure I understand the question, and if I do thats a great question.

    • I will put it to you this way...the answer to the TV and Radios question is not E. Does that help? Find a way for it to not be E!

    • okay so i just saw that the ratio of the new radios and tvs are the same as the initial ratio, that is 12 to 7. I'm still confused as to why this works though. (i just checked the calculations and saw that it did). The new ratio here is just equal to the initial one. But when we add equal fractions, 1/2 + 3/6, we dont get 1/2, we get 1. So its not an addition rule of ratios...What rule of ratios is this?

  • Try it with some numbers - what if you have 12 tvs and 7 radios and then you add the new 132 tvs and 77 radios. So what do you have 144: 84 which reduces to 12 : 7 right?

    What if you have 48 tvs and 28 radios and you add the 132 tvs and 77 radios? That is 180 tvs and 105 radios. If you divide each of these numbers by 15 then you get 12 and 7.

    Do you see the pattern. If you add numbers at the same ratio you maintain the original ratio -- and it does not matter what numbers you started with...so no matter what the original numbers, if we add TVs and radios at 12: 7 we maintain that ratio. Therefore the answer is C.

  • Hey this is a nice info. Only when nos. are added in the SAME RATIO as the original, the result is the same. I was taking different nos.(ratios) to check so was also getting ans. E. Thanks David for the explanation!

    • How come you can only get a ratio if you add the same ratio?

      Say if you had 12 TV's and 7 Radios Radios. If you added say 8 TVs and 13 Radios you would have a 1:1 Ratio. This still gives you a ratio...which is what the question is asking. As long as you know the number of tvs and the number of radios added shouldn't you be able to get a ratio regardless of the ratio of the tvs and radios added?

      What am I missing here? Anybody care to explain?

    • Nick -

      You have made the assumption that you get to start with exactly 12 TVs and 7 Radios. But you only have the ratio of 12:7. So that could mean that you start with 24 and 14 or 48 and 28 or any other actual numbers, so long as they are in the ratio of 12:7.

      So you can see that adding 8 TVs and 13 Radios would yield a different ratio if you started from 12 TVs and 7 Radios rather than from 24 TVs and 14 radios. 

      Yet if you add the TVs and Radios in exactly the same ratio as is currently true, then you will continue to have that same ratio, no matter what the actual starting number.

  • When someone writes an piece of writing he/she maintains the thought of a user in his/her mind that how a user can understand
    it. So that's why this post is great. Thanks!

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