ACT Math: Averages
Averages, or arithmetic means, are likely to show up on the ACT Math section. Most of us know how to find the average, but the test will probably present average questions in a more complicated way.
Rather than present you with all the numbers in a set and ask you to find the average of those numbers, the ACT average problems will present you with various combinations of known and unknown information.
Before we begin, let’s go over some basic rules for finding averages. There are 3 numbers you want to know; they are related by the formula A= T / n, where A is average, T is the total sum of values, and n is the number of figures in a set.
Questions and Strategies
Here are some possibilities for average questions and the strategies to solve them:
Finding the total
IF you know the average and the number of figures/items (n) in a set, then simply multiply the average and the number of figures to find the sum total (T).
Example 1: John caught 14 fish after a long day of fishing. After weighing all of them together, he calculated the average weight of the fish to be 4.7 lbs. What is the total weight, in pounds, of all the fish?
Answer: Simply multiply 14 and 4.7.
Example 2: Throughout the year, Janet took 8 math tests; her average score was 83. If her average score after the first three five tests was 89, what was the average of her last three tests?
Here, we have to find two totals before we can calculate the average of the final three tests.
8*83= 664, the total number of percentage points on all the tests
5*89= 445, the total number of percentage points on the first five tests.
With this information, we know that the total number of percentage points on the last three tests must be the total of all the tests minus the total of the first five tests:
664-445 = 219
Now, we have the info we need to find the average in question.
219 (total) / 3 (number of figures) = 73
Example 3: If the average of 34, 44, 28, and x is 35, what is the value of x?
Remember that Average = Total / Number of figures.
All you have to do is set up an equation with the information you know. Don’t forget that ‘x’ counts as a number in the list, so our total number of figures is 4.
35= ( 34+44+28+x) / 4
35= 106 + x / 4
4 (35) = 106 + x
x = 4(35) – 106
x = 34
Average Speed = total distance / total time
The formula for average speed is quite simple and intuitive, but many overlook the formula when approaching average speed problems. Remember, we need both the total distance and the total time to calculate average speed.
Example 4: In traveling from city A to city B, John drove for 1 hour at 50 mph and for 3 hours at 60 mph. What was his average speed for the whole trip?
First, let’s figure out the total distance. 1 hour at 50 mph would be 50 miles, and 3 hours at 60 mph would be 180 miles. Our total distance is 180 + 50 = 230 miles. The total time is 3+1 = 4 hours.
Average Speed = total distance / total time = 230 / 4 = 57.5
Note: the average speed is not merely the average of 50 and 60–that is a mistake that many students make. If John traveled a greater distance at 60 mph, it wouldn’t make sense for the average speed to lie right in the middle of 50 and 60. Rather, the average speed should be closer to 60.
Always remember: when in doubt, go back to the formula A=T / n. Even the most complicated average problems stem from the formula.