ACT Math: Finding Averages

ACT questions testing you on averages usually require you to realize that if the average of x numbers is n, then the sum of the x numbers is nx. To illustrate with a numerical example: if the average age of 5 girls is 8, then the sum of their ages must be 5*8=40.
Knowing this principle is extremely important. Let’s try and do a few questions together so that you can see how it works.
 

If the average grade of k students is 51 and the average grade of m students is 64 and the average of all the students is 54. Then find k/m

1. if the average of k students is 51, the total score of the k students = 51k
2. if the average of m students is 64, the total score of the m students = 64m
3. if the average score of all the students is 54, the total score of (k + m) students = 54(k + m)
4. since the total score of k students + total score of m students = total score of all the students, then

51k + 64m = 54(k + m)
Most of you might get here and worry about getting k/m. It’s actually not that difficult. If you expand out the equation you get 51k + 64m = 54k + 54m. Putting the k’s on one side and the m’s on the other, you get 10m = 3k. Dividing both sides by m, then dividing by 3, you get k/m = 10/3

If a question tells you that in finding the average of a group of things, some values were measured wrongly, then here are the steps you should take.

1. Given the average, find the total by multiplying the average by the number of things in the group.
2. Subtract or add the incorrect amount from this total
3. Divide by the number of things to find the actual average.

For example, if the average of John, Bob and Sally’s backpacks is 60lbs and the person weighing the bags forgot to set it to zero and added 10lbs to each of the measurements, what is the real average of the three bags?

1. Find the total: 60lbs x 3 = 180 lbs
2. Subtract the false weight that was added, which was 10lbs per bag, so 180lbs – 30lbs = 150lbs
3. Divide 150lbs by 3 to find the actual average: 150 lbs / 3 = 50 lbs

The last type of averages question usually wants you to find a new average when the number of items has changed. In this case, you have to find the total of all the items given, subtract the value of the items they don’t want, and divide by the remaining number of items to get the new average.
For example, if Adam, Bob and Chris has an average age of 12 and Chris is 8 years old, what’s the average age of Adam and Bob. You would take 12*3 = 36 to find the total age of Adam, Bob and Chris. Subtract Chris’ age 8 to get 28 and divide that by 2 (because its only Adam and Bob now) to get the new average of 14.
Conversely, the question could also look like this: Adam and Bob’s average age is 14. When Chris’ age is added, the average is now 12. How old is Chris?

• Adam + Bob + Chris’ age = 12*3 = 36
• Adam + Bob’s age = 14*2 = 28
• So Chris’ age is 36 – 28 = 8

Remember, when dealing with averages, unless it’s a straightforward question that applies the definition of average, you usually need to find the total first, adjust it to what the question needs and then find the new average.