ACT Math: Working with Averages

Averages are one of the concepts that the ACT includes in the “Pre-Algebra & Elementary Algebra” category. So what could be hard about that? We all know how to compute an average, right? Just add up all the numbers and divide by however many there are.

But the thing is the ACT finds clever ways to test the “easy” concepts. If you look at the answer key for any of the ACT practice tests, you’ll see Pre-Algebra & Elementary Algebra questions at the end of the test among the harder questions.

So let’s take a look at averages and how they work. The first thing that’s important to understand is that average problems are very often about the total. If, for instance, you are given a problem in which the average age of five people has to be 30, what do you know about the total of their ages? That total has to be 5 x 30 = 150.

Let’s see what this looks like in a problem.

On his first three tests of the marking period, John has gotten scores of 83, 91 and 87. What does he need to get on the fourth test to have his average be exactly 90?

A.  87
B.  90
C.  93
D.  95
E.  99

In order to have an average of 90, John’s four tests have to add up to 4 x 90 = 360. On the first three tests, he earned 83 + 91 + 87 = 261 points. Therefore, on the fourth test he needs to get a 360 – 261 = 99. The correct answer is Choice E.

Here’s a second problem, which is a little bit trickier.

Kala’s parents will give her a reward if she has an average of 92 or higher for all her quizzes over a two-week span. She has had three math quizzes and got an average of 91 on those. She took two quizzes in science and got an average of 95 on those. She has one quiz scheduled in social studies. If these are the only quizzes over the two-week span, what minimum score must Kala get on the social studies quiz to earn the reward?

F.  88
G.  89
H.  90
J.   91
K.  93

In order to have an average of 92 on the 6 quizzes, Kala needs to earn a total of 6 x 92 = 552 points. Let’s see what she has so far.

Math:  3 x 91 = 273
Science:  2 x 95 = 190
Social Studies: ?

So far Kala has earned 273 + 190 = 463 points on her first 5 quizzes. In order to have an average of 92 or higher, she needs to get at least a 552 – 463 = 89 on the social studies quiz. The correct answer is Choice G.

Another type of problem might involve what are sometimes called weighted averages. These can be approached in a very similar way.

The average age of the 20 children in Mrs. Rankin’s class is 8 years old and the average age of the 30 children in Mr. Rodriguez’s class is 10 years old. What is the average age of the 50 students in both their classes?

A.  8.8
B.  9.0
C.  9.1
D.  9.2
E.  9.4

This problem would appear near the end of the test. If you’re looking at it and thinking that all you have to do is average 8 and 10, hopefully you’ll pause for a moment and think “hey! that’s way too easy for a number 57!” And it is. If you’re thinking the answer is simply 9, you’re missing something. As a matter of fact, if I had to guess on this one, the first answer I’d eliminate is 9.

So what do we know on this problem? Remember how you find an average: it’s the sum of the ages divided by the number of children.

Mrs. Rankin’s class: 20 x 8 = 160
Mr. Rodriguez’s class: 30 x 10 = 300
Total of all the ages: 160 + 300 = 460
Total number of students: 20 + 30 = 50
Average age of the students: 460/50 = 9.2

Here are a couple of problems that you can try on your own. The answers appear at the bottom of the page.

1.  Sarah had 3 tests in her US History class in April and she got an average of 92 on those tests. In May she had only 1 test and got an 86 on that test. What must she average on her 2 tests in June in order to end up with an average of 90 on the 6 tests for the marking period?

A.  86
B.  88
C.  89
D.  90
E.  92

2.  The average height of the 12 boys on the boys basketball team is 76 inches, and the average height of the 18 girls on the girls soccer team is 66 inches. What is the average height, in inches, of all the players on these two teams?

F.  68
G.  69
H.  70
J.   71
K.  72

If you have questions about these problems or anything else to do with the ACT, send us an email at info@cardinalec.com

Solutions:

1.  C
2.  H

 

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