# For the 3 numbers in a list, the average (arithmetic mean) and the

For the \$3\$ numbers in a list, the average (arithmetic mean) and the median are equal to \$8\$. If the greatest number in the list is \$10\$ greater than the least number, what is the greatest number in the list?

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 13 of Section 6 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.

This question contains info about the mean and median of a list of three numbers. These two topics are covered in the Numerical Methods for Describing Data math section of the GRE. 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 have a list of three numbers
2. The mean and median are both \$8\$
3. We’re given a sentence relating the greatest and least numbers to each other
4. We want to find the greatest number in our list

## Develop a Plan

Our list has three numbers in it: Least, Middle, Greatest.

\$\$\Least, \Middle, \Greatest\$\$

We know that the median of a list is the number in the very middle of the list once the numbers have been ordered from least to greatest. Thus, since the median is \$8\$, the Middle number is also \$8\$.

\$\$\Least, 8, \Greatest\$\$

Next, we’re given two more clues to figure out the least and the greatest values in the list. First, we know that the average of all three numbers is \$8\$. We remember the equation for average as:

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

Translating this for our question, we get:

\$\${\Least + 8 + \Greatest}/3 =8\$\$

The question also states, “the greatest number in the list is \$10\$ greater than the least number.” Translated into a math equation, this tells us:

\$\$\Greatest =\Least +10\$\$

Solving this equation for “Least”, we get:

\$\$\Least = \Greatest – 10\$\$

Let’s solve these two equations simultaneously to get the value of the Greatest number.

## Solve the Question

Let’s start with our equation for the average of the three numbers and then plug in “\$\Greatest – 10\$” for “Least”

 \${\Least + 8 + \Greatest}/3\$ \$=\$ \$8\$ \$ \$ \$ \$ \${\Greatest – 10 + 8 + \Greatest}/3\$ \$=\$ \$8\$ \$ \$ \$ \$ \${2·\Greatest – 2}/3\$ \$=\$ \$8\$

Very good so far. Let’s continue solving for “Greatest” by multiplying both sides by \$3\$

 \${2·\Greatest – 2}/3\$ \$=\$ \$8\$ \$ \$ \$ \$ \${2·\Greatest – 2}\$ \$=\$ \$24\$ \$ \$ \$ \$ \${2·\Greatest}\$ \$=\$ \$26\$ \$ \$ \$ \$ \$\Greatest\$ \$=\$ \$13\$

Excellent! So the greatest number in this list is \$13\$. The correct answer is \$13\$.

## What Did We Learn

Math is like a language. Well, actually, math is a language. And learning how to translate words in English into a math equation is not only a valuable skill to have, but we should expect the GRE quantitative section to test us on it as well.
This question definitely became very manageable once we wrote out our equations and solved them. Let’s not try to leap to the answer. Slow and steady wins the race.

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