The average (arithmetic mean) of the 11 numbers in a list is 14. If

The average (arithmetic mean) of the $11$ numbers in a list is $14$. If the average of $9$ of the numbers in the list is $9$, what is the average of the other $2$ numbers?

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 20 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 revolves around the average of a list of numbers, so it likely tests our Numerical Methods for Describing Data 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. The average of $11$ numbers is $14$
  2. The average of $9$ of these numbers is $9$
  3. We want to know the average of the other $2$ numbers

 

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.

Questions that give us an average of a set of numbers and an average of a subset of the bigger set of numbers tend to be easier to solve if we think of sums instead of averages for those numbers. Here, we know that if we have the sum of the “other $2$ numbers,” then we could just divide that sum by $2$ to get the average of those $2$ numbers. And if we look closely, we have enough information to figure out the sum of the other $2$ numbers, since:

$$\Sum \of 9 \Numbers + \Sum \of \Other 2 \Numbers = \Sum \of 11 \Numbers$$

From our equation with sum and average, we know that the sum of a set of numbers is the number of values multiplied by the average. So from this we get:

$$\Sum \of 9 \Numbers = \;9·9 \;\;= 81$$
$$\;\Sum \of 11 \Numbers = 11·14 = 154$$

Let’s use these two sums to find the sum, and then the average, of the other $2$ numbers.

Solve the Question

So if all $11$ numbers sum up to $154$ and $9$ of the numbers sum up to $81$, then the other $2$ numbers must add up to the difference between these two sums:

$$\Sum \of \Other 2 \Numbers = 154-81=73$$

And if the two numbers have a sum of $73$, then their average must be $73/2=36.5$. So the correct answer is $36.5$.

What Did We Learn

For questions involving averages, we need to keep our minds flexible. There’s a good probability that the question might be more easily solved if we convert the averages to sums. 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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