Quarter Wit, Quarter Wisdom: Heavily Weighted Weighted Averages

GMAT PrepToday, I will delve into one of the most important topics (ubiquitous application) that are tested on GMAT. It is also one of the topics that will appear time and again during MBA e.g. in Corporate finance, you might be taught how to find ‘Weighted Average Cost of Capital’. So it will be highly beneficial if you have a feel for weighted average concepts.

The first question is – What is Weighted Average? Let me explain with an example.

A boy’s age is 17 years and a girl’s age is 20 years. What is their average age?

Simple enough, isn’t it? Average age = (17 + 20)/2 = 18.5

It is the number that lies in the middle of 17 and 20. (Another method of arriving at this number would be to find the difference between them, 3, and divide it into 2 equal parts, 1.5 each. Now add 1.5 to the smaller number, 17, to get the average age of 18.5 years. Or subtract 1.5 from the greater number, 20, to get the average age of 18.5 years. But I digress. I will take averages later since it is just a special case of weighted averages.)

Now let me change the question a little.

There are 10 boys and 20 girls in a group. Average age of boys is 17 years and average age of girls is 20 years. What is the average age of the group?

Many people will be able to arrive at the following:

Average Age = (17*10 + 20*20)/(10 + 20) = 19 years

Average age will be total number of years in the age of everyone in the group divided by total number of people in the group. Since the average age of boys is 17, so total number of years in the 10 boys’ ages is 17*10. Since the average age of girls is 20, the total number of years in the 20 girls’ ages is 20*20. The total number of boys and girls is 10 + 20. Hence you use the expression given above to find the average age. I hope we are good up till now.

To establish a general formula, let me restate this question using variables and then we will just plug in the variables in place of the actual numbers above (Yes, it is opposite of what you would normally do when you have the formula and you plug in numbers. Our aim here is to deduce a generic formula from a specific example because the calculation above is intuitive to many of you but the formula is a little intimidating.)

There are w1 boys and w2 girls in a group. Average age of boys is A1 years and average age of girls is A2 years. What is the average age of the group?

Average Age = (A1*w1 + A2*w2)/(w1 + w2)

This is weighted average. Here we are not finding the average age of 1 boy and 1 girl. Instead we are finding the average age of 10 boys and 20 girls so their average age will not be 18.5 years. Boys have been given less weightage in the calculation of average because there are only 10 boys as compared to 20 girls. So the average has been found after accounting for the weightage (or ‘importance’ in regular English) given to boys and girls depending on how many boys and how many girls there are. Notice that the weighted average is 19 years which is closer to the average age of girls than to the average age of boys. This is because there are more girls so they ‘pull’ the average towards their own age i.e. 20 years.

Now that you know what weighted average is and also that you always knew the weighted average formula intuitively, let’s move on to making things easier for you (Tougher, you say? Actually, once people know the scale method that I am going to discuss right now (It has been discussed in our Statistics and Problem Solving book too), they just love it!)

So, Average Age, Aavg = (A1*w1 + A2+w2)/(w1 + w2)

Now if we re-arrange this formula, we get, w1/w2 = (A2 – Aavg)/(Aavg – A1)

So we have got the ratio of weights w1 and w2 (the number of boys and the number of girls). How does it help us? Knowing this ratio, we can directly get the answer. Another example will make this clear.

John pays 30% tax and Ingrid pays 40% tax. Their combined tax rate is 37%. If John’s gross salary is $54000, what is Ingrid’s gross salary?

Here, we have the tax rate of John and Ingrid and their average tax rate. A1 = 30%, A2 = 40% and Aavg = 37%. The weights are their gross salaries – $54,000 for John and w2 for Ingrid. From here on, there are two ways to find the answer. Either plug in the values in the formula above or use the scale method. We will take a look at both.

1. Plug in the formula

w1/w2 = (A2 – Aavg)/(Aavg – A1) = (40 – 37)/(37 – 30) = 3/7

Since A1 is John’s tax rate and A2 is Ingrid’s tax rate, w1 is John’s salary and w2 is Ingrid’s salary

w1/w2 = John’s Salary/Ingrid’s Salary = 3/7 = 54,000/Ingrid’s Salary

So Ingrid’s Salary = $126,000

It should be obvious that either John or Ingrid could be A1 (and the other would be A2). For ease, it a good idea to denote the larger number as A2 and the smaller as A1 (even if you do the other way around, you will still get the same answer)

2. Scale Method

On the number line, put the smaller number on the left side and the greater number on the right side (since it is intuitive that way). Put the average in the middle.

The distance between 30 and 37 is 7 and the distance between 37 and 40 is 3 so w1:w2 = 3:7 (As seen by the formula, the ratio is flipped).

Since w1 = 54,000, w2 will be 126,000

So Ingrid’s salary is $126,000.

This method is especially useful when you have the average and need to find the ratio of weights. Check out next week’s post for some 700 level examples of weighted average.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!

16 Responses

  1. Francesco Tuzzolino says:

    Hi Karishma,
    thanks for this post. it is great!!:)
    little typo on the formula you mention,

    it reads = “…So, Average Age, Aavg = (A1*w1 + A2+w2)/(w1 + w2)…”
    However, it should read Average Age, Aavg = (A1*w1 + A2*w2)/(w1 + w2),
    All the best and good job..:)

  2. Francesco Tuzzolino says:

    Last things.. I got all the full set of veritas prep and I love them..:)

  3. Karishma says:

    Hey Francesco,

    Yeah, you are right! Thanks for catching that typo.

    It should be Aavg = (A1*w1 + A2*w2)/(w1 + w2)

    I am glad you liked the books!

    Best,
    Karishma

  4. rob says:

    This does not make sense. The correct answer should be 130,000.6666 for Ingrid. When you plug that value in along with 56k for John you get 37 percent for the answer. Using 126,000 as her salary and 56k you get 36.92 percent.

    30J + 40i/(56000 + i) = 37

    Plug in 56,000 for J

    (30 * 56000) + 40i)/56000 + i) = 37

    Now just solve for i.

    • Karishma says:

      That’s another typo in this post! John’s gross salary is $54,000 not $56,000. The question incorrectly mentions it as 56,000. If you notice, the entire solution uses $54,000. I will update the post.

      And yes, the following straight forward calculations can be done to reach the answer.

      Total tax they pay = (30/100)*54,000 + (40/100)*x = (37/100)*(54,000 + x)

      However, my aim in using Weighted averages here is to minimize calculations and save time.

  5. rob says:

    also another way to do this is

    subtracts Johns average from combined average and subtract Ingrids average from total and set them equal to each other

    3i=7(56000)

    and solve for i

  6. Pearl says:

    Such an awesome blog! Love it! Does Veritas Prep have the same methods to solve questions like the ones on your blog?

  7. Karishma says:

    Thanks Pearl!
    We ensure that we discuss all the methods you need/want for a 99% through the various study sources we provide you – study guides, on-demand lessons, practice tests, in-person classes, blog posts, forum access etc. Every method may not be in each source but we do discuss everything through the various sources (our study guides are pretty comprehensive. We discuss the scale method given above in them). The focus is on strategic thinking and developing reasoning skills. The intent is to make you realize that people who score 99%ile are just those who know their core concepts really well. This blog is a part of our effort toward that end.

  8. GG says:

    Could you explain more why (A2 – Aavg)w1 = (Aavg – A1)w2 in the Scale Method? Does that contains a ratio concept or something? I just can’t figure it out. Thank you so much!

    • Karishma says:

      The formula is just a re-arrangement of the standard:
      Average Age = Aavg = (A1*w1 + A2*w2)/(w1 + w2)

      Aavg(w1 + w2) = (A1*w1 + A2*w2)
      Aavg*w1 + Aavg*w2 = A1*w1 + A2*w2
      (Aavg – A1)*w1 = (A2 – Aavg)*w2

      So basically it is the same formula. I just re-arranged it to this format.

      • vijay says:

        i am a bit confused..can you please explain how we reached
        (A2 – Aavg)w1 = (Aavg – A1)w2
        i understood the above (Aavg – A1)*w1 = (A2 – Aavg)*w2

        in short,i didnt understand the flipping of the ratios?

  9. Rohan Ahmed says:

    Hi Karishma,
    Could it be possible to solve the below math with the above mentioned weighted average way?

    A driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour for the entire 40-mile trip? (Assume that the driver did not make any stops during the 40-mile trip.)

    Thanks

    • Karishma says:

      Yes you can. But you need to be very careful. When we talk about speed, the weight is the time taken. It is not the distance traveled. Hence, here the weights will not be 20 and 20. The weights will be the time taken i.e. 20/50 and 20/x (where x is the speed for the other 20 mile trip).

      Cavg = (C1*w1 + C2*w2)/(w1 + w2)
      60 = [50*(20/50) + x(20/x)]/[20/50 + 20/x]
      60 = 40/[20/50 + 20/x]
      [10/50 + 10/x] = 1/3
      10/x = 2/15
      x = 75 mph

      So instead, you can simply use
      Average Speed = Total Distance/Total time
      or
      Average Speed in case distance traveled is the same in the two cases = 2*a*b/(a+b)
      (a is the speed for half the distance and b is the speed for the other half of the distance)

      • Vaishno says:

        Great explanation Karishma!

        Would you be able to exhibit the above speed question using the scale method?

        Is it something similar to this?

        50—————–60——————–x

        • Karishma says:

          Yes, absolutely. Anything that you can solve using the weighted average formula, you can solve using the scale method because the scale method represents the same formula. When you separate out w1 and w2 from
          Cavg = (C1*w1 + C2*w2)/(w1 + w2)
          you get
          w1/w2 = (C2 – Cavg)/(Cavg – C1)

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