Test makers like to be tricky, because sneaky questions separate the students who comprehend a concept and those who have just memorized a formula they do not understand. We see examples of these tricky questions on the ACT and SAT in hidden triangles, "fake" quadratic equations, systems of equations, and many more concepts, including today's topic: disguised average questions. These popular ACT and SAT Math questions involve averages, but they do not appear to be about averages at all!
Let's look at an example:
The sum of 7 consecutive even integers is 224. What integer has the least value in the list?
(A) 16
(B) 18
(C) 26
(D) 29
(E) 32
Notice that the word “average” is never used in the question; however, the presence of “consecutive” and “sum” indicates an average question. Set up the question using your formula for averages:
The average of the 7 integers is 32. Since the average and the median are the same value in consecutive sets of odd numbers, you find the other numbers in the list by counting down and up from 32:
The number with the least value is 26, so answer (C) is correct.
Can you guess the most common wrong answer? It’s (D). Some students forget that the list is made up of consecutive even integers, and they therefore simply find the consecutive list: 29, 30, 31, 32, 33, 34, 35. This is incorrect, though, due to the wording of the question. After solving a disguised average question, be sure to reread it to confirm that it requires consecutive, consecutive even, or consecutive odd numbers.
Remember, the easiest way to identify these questions is by the words “consecutive” and “sum.” If they appear together in a question, it’s most likely a disguised average question!
PowerScore Practice:
Can you solve the following ACT and SAT math questions?

The sum of seven integers is 406. If the numbers are consecutive integers and arranged from least to greatest, what is the value of the second integer on the list?

What is the greatest of 5 consecutive odd integers if the sum of these integers is 255?

A list of five numbers is made up of consecutive multiples of 5. If the sum of these five numbers is 675, what is the number with the least value in the list?
And if you need more math tips like these, be sure to check out our ACT courses!
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Answers: 1) 56 2) 55 3) 125