Most people don’t like weighted averages, and for good reason. The formula is complicated, and these often come in the form of story problems, which are hard to set up. We’re going to talk today about a couple of great little techniques to make these fast and easy well, easier anyway!

First, try this GMATPrep problem. Set your timer for 2 minutes. and GO!

*  A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit’s diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture?

 

(A) 100

(B) 140

(C) 150

(D) 160

(E) 200

Wow. I’m glad I don’t have to feed this rabbit. This sounds annoying. : )
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In an earlier post, we tackled a medium-level GMATPrep weighted average question; click here to read that article before reading this one. This week, we’re trying a harder GMATPrep  weighted average question in order to test whether you learned the concept as well as you thought you did. : )

As we discussed earlier, every weighted average problem I’ve seen (so far!) on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. Last week, we focused on understanding how weighted averages work via writing some equations. We’ll try to apply that understanding to our harder problem this week, along with a more efficient solution method.

Let’s start with a sample problem. Set your timer for 2 minutes. and GO!

* A contractor combined x tons of gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight. What is the value of x?

(1) y = 10

(2) z = 16

There are two kinds of gravel: 10% gravel and 2% gravel. These are our two sub-groups. When the two are combined (in some unknown “ for now! “ amounts), we get a 3^rd kind:5% gravel. The number of tons of 10% gravel (x) and the number of tons of 2% gravel (y) will add up to the number of tons of 5% gravel (z), or x + y = z. We need to find the number of tons of 10% gravel used in the mixture.

The problem this week throws in a new wrinkle: we’re not just trying to calculate a ratio this time. We have to have enough info to calculate the actual amount of 10% gravel used. Last week, we never had to worry about the actual number of employees. We’ll have to keep that in mind to see how things might change.

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This week, we’re going to tackle a GMATPrep question from the quant side of things. We’ll tackle a medium-level question this week in order to learn how to master weighted average questions in general, and in the next article, we’ll try a very hard one “ just to see whether you learned the concept as well as you thought you did. : )

Before we begin, I want to mention that every weighted average problem I’ve seen on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. So we’re going to work on that conceptual understanding today and then we’ll discuss a neat calculation shortcut next week (built on the same principles!), just in case we do need to solve.

Let’s start with a sample problem. Set your timer for 2 minutes. and GO!
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If you’ve read my previous post you know I got married very recently. When I asked my new wife the other day to name her favorite celebrity, she said Ryan Gosling; unfortunately I look nothing like him “ so I’m not quite sure where that leaves me. As a form of revenge I’ve decided to use Mr. Gosling to demonstrate some key insights in the commonly misunderstood topic of Weighted Average. Ryan will never forgive me!

For the purpose of this blog post let’s assume that Ryan Gosling made $10M per movie in 80% of his movies and $20M per movie in 20% of his movies. His average paycheck would have been $15M if his salary were distributed evenly between $10M and $20M “ but an 80-20 distribution means we’ll have to put a little more thought into the situation. If we want to know how much Mr. Gosling made on average per movie, we have no choice but to calculate the weighted average.

Some math lovers might use an algebraic formula to calculate the weighted average, but I believe using a visual approach for this calculation will drive a deeper level of understanding for us regular folks.

Use your intuition and try a visual approach

If I asked you for a range of the weighted average of Ryan Gosling’s paychecks, your intuition would probably suggest between $10M and $20M. You might even propose that the weighted average be closer to $10M than to $20M (since $10M has a heavier weight “ 80% vs. 20%). You would be absolutely correct!

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