# Word Problems Involving the Mean of a Data Set

The
mean
of a set of numbers, sometimes simply called
**
the average
**
, is the sum of the data divided by the total number of data.

**
Example 1
**
:

When a number $x$ is added to the data set $\text{4,8,20,25,32,}$ the new mean is $15$ . Find the value of $x$ .

Including $x$ , there are 6 numbers in the set. Write an equation for calculating the mean with the unknown value.

$\frac{4\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8\text{\hspace{0.17em}}+\text{\hspace{0.17em}}20\text{\hspace{0.17em}}+\text{\hspace{0.17em}}25\text{\hspace{0.17em}}+\text{\hspace{0.17em}}32\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x}{6}=15$

Simplify.

$\frac{89\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x}{6}=15$

Multiply both sides by $6$ .

$89+x=90$

Subtract $89$ from both sides.

$x=1$

**
Example 2
**
:

Kalief's scores on his first four History tests were $\text{80,85,88and95}$ .

To get an $A$ in the class, he needs to have an average of $90$ or better.

What score must he make on the fifth test to get an $A$ ?

Let $x$ be the required score on the fifth test. Write an equation for calculating the mean with the unknown value.

$\frac{80\text{\hspace{0.17em}}+\text{\hspace{0.17em}}85\text{\hspace{0.17em}}+\text{\hspace{0.17em}}88\text{\hspace{0.17em}}+\text{\hspace{0.17em}}95\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x}{5}=90$

Simplify.

$\frac{348\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x}{5}=90$

Multiply both sides by $5$ .

$348+x=450$

Subtract $348$ from both sides.

$x=98$

He needs to score a $98$ or higher in order to get an $A$ .