Data Sufficiency Strategies: Pieces of the Puzzle

Even though Data Sufficiency questions test the same math content as Problem Solving, most GMAT students find that they can’t use the same strategies on both question types. One of the common errors students make in their approach to Data Sufficiency questions is jumping to the two statements without analyzing the question first and figuring out what information is missing.  You wouldn’t start hunting for a jigsaw puzzle piece without first looking at the shape of the hole you’re trying to fill, would you? Most Data Sufficiency questions can be simplified with the Pieces of the Puzzle approach, and some would be nearly impossible to solve without it.

There are a two principles that underlie the Pieces of the Puzzle strategy:

1. It’s easier to do one thing at a time than to do two things simultaneously.
2. It’s easier to find something when you know what you’re looking for.

Let’s look at some examples.

A bicycle rider rides four stages of a race in 2 hours and 45 minutes. The four stages are named with four different colors: blue, yellow, red, and white. What is the rider’s average speed for the whole race?

1) The red and blue stages combined are 55 miles long.
2) The blue and yellow stages combined are 42 miles long.

Before looking closely at the statements, we should consider the question and determine what we’re looking for. The question asks for an average speed, so this is a question about rates and we should write down the rate formula: Rate = Distance / Time. The time is given to us, so the only thing we need to know in order to find the rate is the distance. That’s the missing piece of this puzzle. This means that when we turn to the statements we’re no longer thinking about rate, we’re thinking about distance. If the statement allows us to find the total distance of the race then it is sufficient. If not, it’s insufficient.

Statement 1 tells us the combined distance of two of the stages, but that isn’t enough to know the distance of the whole race. Thus Statement 1 is insufficient.

Statement 2 also tells us the combined distance of only two stages, and is therefore insufficient for the same reason as Statement 1. When we combine the statements we have to be a bit careful. It may seem that now we have all four stages, but in fact the blue stage was mentioned twice, and nowhere do we have any information about the white stage.

There is no way to know the total distance of the race, and the answer is (E).

Let’s look at another example:

If x is an integer, is  a factor of 15! ?

1) x is the sum of two distinct single-digit prime numbers.
2) 0 < x < 6

This is a much more complicated question than the first one, and we really have to do some thinking up front before we worry about the statements. First, this is a Yes/No question, so we need to be clear that it doesn’t matter whether the answer is yes or no, whether is a factor of 15! or not. All that matters is that we know for certain one way or the other. So how will we know?

 describes a certain number of 3s multiplied together. If  is a factor of 15! then all those 3s divide evenly into 15! with nothing left over (if z is a factor of some number then that number is divisible by z). The question becomes, “How many factors of 3 are there in 15! ?” Write out 15! and count the factors of 3.

15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

There’s one in the 15 (3 x 5), one in the 12 (3 x 4), two in the 9 (3 x 3), one in the 6 (3 x 2) and one in the 3 (3 x 1). That’s a total of six 3s. So now we can finally say what the missing piece of the puzzle is. Since 15! contains six factors of 3, if x = 6 or anything less than 6 then will be a factor of 15! and the answer will be Yes. If x is anything larger than 6 then  will not be a factor of 15! because it will have too many 3s. The answer will be No. So the key question we’re concerned with as we turn to the statements is do we know whether or so we have answered our question definitively. Statement 2 is sufficient and the answer to the question is (B).

That last question was a lot of work! But we were able to get the answer because we knew what we were looking for when we turned to the statements. Without that initial investment in analyzing the question and figuring out the missing piece of the puzzle, we would have had a hard time seeing anything useful in the statements. Remember to use the Pieces of the Puzzle approach on Data Sufficiency questions, and you’ll find yourself solving them more quickly and more accurately.

Attempt a few Data Sufficiency Questions.