Common Data Sufficiency Mistakes - Part III

Welcome to part three of a three-part series on common mistakes that students make when answering GMAT Data Sufficiency questions.

Mistake #1 – Assuming that 2 equations with 2 variables is always sufficient

Consider this question:

What is the value of x?

(1) 2x - 4y = 8

(2) y + 2 = 0.5x

Clearly, statements 1 and 2 are insufficient on their own, but are the combined statements sufficient?

No.

Here’s why. The equations 2x – 4y = 8 and y + 2 = 0.5x are actually equivalent. To see this, first take the equation y + 2 = 0.5x and multiply both sides by 4 to get 4y + 8 = 2x. From here, we subtract 4y from both sides, we get 8 = 2x – 4y.

So, while the statements do, indeed, provide two equations with two variables, the two equations are actually equivalent, which means the combined statements do not provide enough information to determine the value of x. In other words, the correct answer here is E.

The takeaway here is that, in most* cases (*more later), if the two given equations are equivalent, we cannot use them to solve for one of the variables.

But what if we have two equations with two variables, AND the equations are not equivalent? Do we then have enough information to solve for one of the variables?

Not necessarily.

Consider this question:

What is the value of x?

(1) x + y = 5

(2) xy = 6

Once again, statements 1 and 2 are insufficient on their own, but are the combined statements sufficient?

No.

Even though we have two equations with two variables, AND the two equations are not equivalent, there are still two possible solutions to this system of equations. It could be the case that x = 2 and y = 3, or it could be the case that x = 3 and y = 2. Since we cannot determine the value of x, the correct answer here is E.

So, when exactly does a system of two equations with two variables provide sufficient information in a Data Sufficiency question?

To provide sufficient information, the two equations must be non-equivalent equations and that are both linear equations.

Aside: A linear equation can be written in the form Ax + By = C, for some values of A, B and C.

Notice that in the second example, the equation xy = 6 is not a linear equation.

Okay, now that we have a nice rule to help us determine when a system of two equations with two variables provides sufficient information, we may be prone to making a different kind of mistake.

Mistake #2 – Assuming that 1 equation with 2 variables is never sufficient

To set this up, please consider the following partial question:

Marta bought some apples and some bananas. If each apple costs 15 cents, and each banana costs 20 cents, how many apples did Marta buy?

(1) Marta spent a total of 85 cents

If we let A = the number of apples purchased, and let B = the number of bananas purchased, we can use statement 1 to write the equation 15A + 20B = 85. Does this two-variable equation provide sufficient information to find the value of A?

Yes it does.

Here’s why. Since this is a real world question, A and B must have integer values. Furthermore, since we’re told that Marta bought some apples and some bananas, we know that she bought at least one of each fruit. In other words, A and B cannot equal zero.

With all of this in mind, we can simplify the equation 15A + 20B = 85 by dividing both sides by 5, to get 3A + 4B = 17. Since A and B are restricted to positive integers, there’s exactly one solution to this equation: A = 3 and B = 2. In other words, statement 1 is sufficient.

So, even though the equation 15A + 20B = 85 has an infinite number of solutions when A and B can have any values, the same equation has only one solution when we restrict the values of A and B to positive integers.

The big takeaway here is that, before you apply any rules regarding the sufficiency of equations, look for restrictions on the variables that may render those rules invalid. Oh, and also watch out for equivalent equations.

Now it’s time to test your skills. Try answering the following questions:

You should also read Ian Stewart’s great post concerning systems of equations: http://www.beatthegmat.com/n-variables-n-distinct-equations-rule-t20728.html