GMAT Tip: The Weighted-Average Tug of War
This tip on improving your GMAT score was provided by Brian Galvin at Veritas Prep.
Does your heart stop when you see problems that begin with phrasing such as “Mixture X is 30 percent vinegar and Mixture Y is 60 percent vinegar?” Don’t worry, you’re not alone. When Mixture X is 30 percent of anything, 70 percent of GMAT students become composed of nearly 100 percent fear. But there’s an easy solution to beat those tricky solution problems—and it dates back more to third-grade gym class than to sixth-grade algebra class.
Here’s how it works. Most solution problems are just weighted-average problems. And in weighted-average problems both items are essentially “pulling” the weighted average in their direction. In the above setup, if you were to combine Mixture X with Mixture Y, how much vinegar would be in the combined solution? You can’t tell without knowing how much of each mixture you’d have, but you can know for certain that the vinegar would be between 30 percent and 60 percent. And the more of X you’d have, the closer the weighted average would be to 30 percent. (X is, in a way, “pulling harder” if there’s more of it.) And the more of Y you’d have, the closer it would be to 60 percent.
So weighted-average/mixture problems, if you boil them down, typically come down to a game of tug of war between the two individual numbers, which each pull proportionately to their weight. If you have one 30 and one 60, the average will be 45 because each number pulls the same weight, so the average will be immediately in between. But if you had one 30 and two 60s, the 60s will pull at a 2:1 rate, so the weighted average will be twice as close to 60 as it is to 30. Since the two numbers are 30 apart, the weighted average will be 10 away from 30 and 20 away from 60—that weighted average will be 40.
How does this help you? With weighted-average or solution problems, it’s often most efficient to draw out the tug of war between the two weights to determine the ratio of one to the other. For example, consider this question:
When x liters of Mixture X, which is 30 percent vinegar, are mixed with y liters of Mixture Y, which is 60 percent vinegar, the result is 15 liters of a solution that is 40 percent vinegar. How much of Mixture X is used to form that solution?
Here, the tug of war strategy will give you the ratio of X to Y. If you draw out the two individual averages, 30 and 60, with the weighted average of 40 on the number line, you have:
Then note the distances. It’s 10 from X to the weighted average and 20 from 60 to the weighted average. So you know that:
• The ratio is 2 to 1.
• There is more of solution X than of Y, since the weighted average is closer to X.
Therefore, there are 2 parts of X for every 1 of Y, meaning that X makes up 2/3 of the solution. Since there are 15 liters total, that means that 10 of them are solution X.
Let’s summarize. When you’re dealing with two-item-solution or weighted-average problems, you can draw out the tug of war map to determine the ratio between the two, a strategy that is typically much quicker than doing the algebra. Just remember the most common mistake with this strategy: The distances will be inverse to the weight, since the smaller distance (between one component and the weighted average) will belong to the component with the greater weight. So do these problems in steps. First, find the ratio, then logically fit the ratio to the weight. And there is the solution to your solution problem woes.
Brian Galvin has studied the GMAT full time since 2006 as the director of academic programs for Veritas Prep. He received a Masters in Education from the University of Michigan and is the proud owner of a 99th percentile GMAT score.