If a ≤ b ≤ c ≤ d ≤ e ≤ 110 and the average (arithmetic mean) of a, b, c, d

If $a≤b≤c≤d≤e≤110$ and the average (arithmetic mean) of $a, b, c, d, \and e$ is $100$, what is the least possible value of $a$?

  1. $0$
  2. $20$
  3. $40$
  4. $60$
  5. $80$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 17 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of Numerical Methods for Describing Data can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

The question mentions the average of a set of numbers, which is part of the Numerical Methods for Describing Numbers GRE math skill. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. We want to find the least possible value of $a$
  2. $a≤b≤c≤d≤e≤110$
  3. We know that the average of $a, b, c, d, \and e$ is $100$

 

Develop a Plan

The question asks about averages, so let’s start by recalling our equation for calculating the average of a group of numbers.

$$\Average = {\Sum \of \Values}/{\Number \of \Values}$$

So for example, if the sum of $10$ numbers is $200$, then the average of those $10$ numbers is $200/10$, or $20$. Viewed a slightly different way, if the average of $10$ numbers is $20$, then the sum of those $10$ numbers is $200$. We should keep our minds flexible when it comes to sums and averages. Some questions it’ll be easier to work with sums, whereas for other questions it’ll be easier to work with averages.

Solve the Question

Let’s try working with sums for this question. We know that the average of the five variables is $100$. So their sum must be $5·100$, or $500$. We want the least possible value for $a$. Since the sum of the five variables must be $500$,
$a$ should have its least value when the other variables are as large as possible. From our inequality, the largest the variables can be is $110$. So let’s set $b, c, d, \and e$ equal to $110$, then calculate the value of $a$

$a$ $=$ $500-b-c-d-e$
$ $ $ $
$a$ $=$ $500-110-110-110-110$
$ $ $ $
$a$ $=$ $60$

The correct answer is D, $60$.

What Did We Learn

For questions involving an average, we need to keep our minds flexible. There’s a good chance that the question might be more easily solved if we convert the average to a sum. Let’s remember that just because a question is worded a certain way, that doesn’t pigeonhole us into only being able to think of the question that way.

 

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