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 8020 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!
Now ask yourself: how much closer should the weighted average be to $10M than to $20M? The answer is determined by the ratio of the weights.
The ratio of 80% to 20% is 4:1. That means the weighted average is 4 times closer to $10M than it is to $20M. Since the total difference between those numbers is $10M ($20M minus $10M), we must split that difference in a 4:1 ratio.
Note that when we want to get a 4:1 split we divide the number line into five equal units; this is because we have a total of five ratio units (4+1).
You might also notice that the shorter distance ended up on the same side as the heavier weight: this happens because the weighted average must be closer to the heavier weight, and closer=shorter distance. If we insert the ratio into the diagram (naturally making it a bit messier), we must flip the sides of that ratio at some point to account for this phenomenon (see red arrows):
Test your understanding (Adapted from #224 in the Official Guide for GMAT Review, 13^{th} Edition)
A banana protein shake is 25 percent protein, 30 percent water, and 45 percent banana by weight; a strawberry protein shake is 5 percent protein, 35 percent water, and 60 percent strawberry by weight. If a mixture of both protein shakes contains 20 percent protein, what percent of the weight of the mixture is the strawberry protein shake?
The first step is identification: the word mixture is a strong hint to the fact that we’re dealing with an average; the fact that we’re only given the protein percentage of the mixture and the fact that each type of shake consists of different protein percentages imply that we’re looking at a weighted average of protein content. In this question we are given the weighted average (a mixture of both protein shakes contains 20 percent protein) and we need to use that to determine the weight (as a percentage of the whole mixture) of the strawberry protein shake. Let’s take a second to eyeball the situation and see which shake is closer to the average
I’ll wait a couple of minutes, try to solve this on your own by drawing a diagram.
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The banana shake has 25% protein and the strawberry shake has 5% protein; the mixture is 20% protein, which is closer to the banana shake; therefore, the banana shake has a heavier weight. Since they are asking about the strawberry shake, it must be less than 50% (cross out any answer choices of 50% or more). The next step is to draw a number line and find the ratio of the distances (see diagram).
The average is located 15 away from strawberry and 5 away from banana, so the ratio is 15:5 or 3:1. Therefore the weights must also have a ratio of 1:3 (notice how I flipped the ratio again when converting from distance ratio to weight ratio)!
The question was what percent of the weight of the mixture is the strawberry protein shake?, and so we have one final step in the solution: with a ratio of 1:3 (strawberry:banana), the total mixture is worth 4 ratio units (1+3). Since the total weight of the mixture is 100%, each ratio unit must be worth 25% (100/4) “ and strawberry must account for exactly 25% because it’s worth one ratio unit!
Key Takeaways:
 Identification “ when a question mentions a weighted average (either explicitly or implicitly): pause for a second and ask yourself which side the average will be closer to (when weights are given) or which side has the heavier weight (when the weighted average is given).
 Draw a number line for the range in question (between the two amounts).
 When weights are given:
 Break down the number line into equal sized pieces “ these will be our ratio units; remember: the number of pieces will equal the sum of the ratio units.
 Flip the ratio of the weights and apply to those pieces on the number line “ the weighted average is located where the two sides meet.
 When weighted average is given:

 What are the distances between the weighted average and each side? Take the ratio of those distances, flip it, and you’ll get the weights!
 When searching for percentage of one weight out of total weights, take the desired weight and divide by total weights; in our example with a ratio of 1:3 we calculated 1/(1+3) = 25%
 Ryan Gosling is overpaid!
I quite enjoyed applying your technique. I tried doing this question using algebra too and here’s how I worked out the solution:
Let x be the total volume of banana mixture
Let y be the total volume of strawberry mixture
Therefore:
(25% of x) + (5% of y) = 20% of (x+y)
5x + y = 4x + 4y
x = 3y
So if x = 100 ml y should be 300 ml
Thus strawberry shake should be (100/400)*100 = 25%
But algebraic working can become ugly if tougher problem is presented. So, I’m going to try applying allegationlike techniques discussed above.
Thanks again!
Hello.
What I don’t understand is how you drew the number line with “four” pieces.!
I got really confused in the process. Could you please explain a little more.
Thanks.
Hi Tanvi,
The reason I chose to draw the number line with four equal pieces is that with a ratio of 1:3 (strawberry:banana), the total mixture is worth 4 ratio units (1+3).
In general, the number of pieces you should break the number line into is going to be the sum of the ratio units (but you should always fully reduce the ratio first, to keep the numbers nice and small).
Hi Avi
I had got confused too but then I realized that if you do split like below its even easier.
Banana Strawberry
25 5
20
15 5
This this would give ratio 3:1 without splitting. The key was to subtract banana and strawberry from 20% and write them diagonally. This way I can avoid using line altogether.
What do you think?
Pankil
Hi Pankil,
Writing the numbers diagonally is a neat trick, but I prefer using the number line because I get to ‘touch and feel’ the numbers and the situation. Seeing the logic behind what I’m doing greatly reduces my chance of error. Sometimes we take shortcuts so far that we lose sight of the original logic.
For example, you can solve 147/3 using long division, where you blindly follow an algorithm developed by someone hundreds of years ago, or you can solve using logic/algebra:
147/3 = (1503)/3 = 150/3 – 3/3 = 50 – 1 = 49
Let me know what you think
Avi
Excellent article.really helped me understand weighted average .Thanks!
Please explain how you calculate the ratio of 4:1 to the first problem and 3:1 this is new for me. Thanks!
Hi Juan,
It’s actually simpler than it looks – all you have to do is reduce the fractions…
80%/20% = 4:1 (divide both numerator and denominator by 20%)
75%/25% = 3:1 (divide both numerator and denominator by 25%)
Hi Avi,
Very useful article !
I just wanted to understand a point in the first “Paycheck Problem”.
Distance between $10M and the wt. avg should be = 4/5 and the distance between $20M and the wt.avg should be = 1/5 Am i correct?
If that is the case , then we can find the wt. avg by just adding the distance.
Please let me know whether i am missing any point.
Hi Avi,
Very useful article !
I just wanted to understand a point in the “Paycheck Problem”.
Distance between $10M and the wt. avg should be = 4/5 and the distance between $20M and the wt.avg should be = 1/5 Am i correct?
If that is the case , then we can find the wt. avg by just adding the distance.
Please let me know whether i am missing any point.
Hi Monika,
It’s actually the other way round: since the $10M has a heavier weight (80%) it must be closer to the weighted average than $20M. So the distance between $10M and the weighted average will be only 1/5 out of the total distance and the distance between $20M and the weighted average will be 4/5 of the total distance.
Remember, the HEAVIER WEIGHT goes with the SHORTER DISTANCE, which is why this phenomenon of FLIPPING SIDES occurs.
If you find this confusing, try reading the post again, perhaps at a slower pace.
Best, Avi
Got it now.
Many Thanks, Avi !
You’re welcome!!
Here’s a site with some good practice problems. Some may require a calculator (these are not GMAT problems) but most can be solved easily with the method I discussed in the blog post.
http://www.ixl.com/math/algebra1/weightedaverageswordproblems