Everyone knows how to find an average, but the power of this formula is often underestimated. We know:

average = (sum of the items)/(number of items)

Notice, we can also write this as:

sum of items = (average)*(number of items)

This latter form can be powerful. For example, if we add or subtract one item from a set, we can easily figure out how that changes the sum, and that can allow us to calculate the new average. Also, if we are combining two groups of different sizes, we can’t add averages, but we can add sums.

## Practice Questions: Averages

1) There are 17 students in a certain class. On the day the test was given, Taqeesha was absent. The other 16 students took the test, and their average was 77. The next day, Taqeesha took the test, and with her grade included, the new average is 78. What is Taqeesha’s grade on the test?

(A) 78

(B) 80

(C) 87

(D) 91

(E) 94

2) A company has 15 managers and 75 associates. The 15 managers have an average salary of $120,000. The 75 associates have an average salary of $30,000. What is the average salary for the company?

(A) $35,000

(B) $45,000

(C) $55,000

(D) $65,000

(E) $75,000

## Answers and Explanations

1) The average of the first 16 students is 77. This means, the sum of these 16 scores is

sum = (average)*(number of scores) = 77*16 = 1232

Once Taqeesha takes her test, the average of all 17 scores is 78. This means, the sum of these 17 scores is:

sum = (average)*(number of scores) =78*17 = 1326

Once we had the sum of the 16 scores, all we had to do was add Taqeesha’s score to that total to get the sum of all 17. Therefore, the difference in these two sums is Taqeesha’s score. 1326 – 1232 = 94.

**Answer: E.**

2) The 15 managers have an average salary of $120,000. The sum of their salaries is:

sum = (average)*(number of salaries) = $120,000*15 = $1,800,000

The 75 associates have an average salary of $30,000. The sum of their salaries is:

sum = (average)*(number of salaries) = $30,000*75 = $2,250,000

When we add those two sums, we get the total payroll of all 90 employees.

$1,800,000 + $2,250,000 = $4,050,000

So, we have 90 employees, and together they earn $4,050,000, so the average is

average = $4,050,000 ÷90 = $45,000

**Answer: B.**

We can think of this a mixture problem, you mix manager’s and associate’s salary together in the ratio of ( 15:75 or 1:5). So the resultant mixture will have 1/6th of managers’: 20000 and 5/6th of associates( 25000) with a total of 45,000 giving the average salary.

It is my understanding that the qualification w ≤ x ≤ y would not allow the set (16, 16, 28). Please confirm because that would change the answer, correct??

This great simple information!

I’m glad you found it helpful! Thanks for your feedback!

Mike 🙂

For the second question here if you want to avoid multiplying big numbers, just take the ratio of Managers and associates (1:5). Now you have to multiply smaller numbers and the average of this will be same as the average calculated by using the figures 15 managers and 75 associates.

[($120,000 x 1) + ($30,000 x 5)]/6 = $ 45,000

Arman:

That’s a great simplifying insight! That’s a trick that might not occur to everyone in the test conditions, but it’s a great strategy to have up your sleeve. Thank you very much for sharing this!

Mike 🙂

Additionally, the calculation can be speeded up by assuming 30000 as one unit or in other words, factoring 30000 in the equation (12000 is a multiple of 30000)

That is, (4*1+1*5)/6=1.5

As one unit represents 30000, hence 1.5 means 45000.

Thanks Mike for helpful posts.

Arun,

Yes, for folks skilled in proportional reasoning, this can be a huge short-cut, one of many folks like that will find on the GMAT Quant. The trouble is — some folks are just wrestling with the basic arithmetic & what an average is in the first place, so leaps of proportional reasoning are totally beyond where they are. For those who can get it, it’s a great trick.

Mike 🙂

For the first question, for advanced users, following explanation should also help.

When Taqeesha took the test, the average increased from 77 to 78. That means, Taqeesha scored 78 plus 1 mark for each one of the other students. i.e., 78 + 16 = 94.

Bala — yes, for folks with good number sense, that’s a slick way to approach this problem. I explained the slightly longer method above because some folks don’t have the facility with number that you apparently do. Best of luck to you.

Mike 🙂

Indeed very helpful. Thanks 🙂

You are quite welcome.

Mike 🙂