Means in Word Problems

Finding a mean or an average in a word problem is often quite simple.  

To find a mean, you add up the data points and divide by the number of datapoints.  Because people often find means in real life, it's easy to do word problems of this type.

The simplest type of mean or average word problem simply requires you to read a word problem to find the data, and then find the mean (or average).

Example: You and your siblings went strawberry picking.  You picked 20 strawberries.  Your big brother picked 28 strawberries and your baby sister picked 4.  What is the average number of strawberries that the children in your family picked?

Start with the formula (this may seem silly, but it's gives you a good starting point for any word problem that uses means):

$\dfrac{\text{Sum of datapoints}}{\text{Number of datapoints}}=\text{Mean of data}$

Plug in your data:

$$\eqalign{\dfrac{20+28+4}{3}&=\text{Mean of data}\\\dfrac{52}{3}&=17.333}$$

You and your siblings picked an average of $17.33$ strawberries each.

However, word problems sometimes don't just give you the datapoints.  In a word problem, you can be given the average and asked to find a datapoint (or the sum of the datapoints).

To do any of these problems, you want to remember the formula for finding a mean:

$\dfrac{\text{Sum of datapoints}}{\text{Number of datapoints}}=\text{Mean of data}$

Then, you read the word problem and plug the information that the problem gives you into the formula.

Example: You have taken four math tests and received the following scores: 8, 9, 7, 10, 7.  If, after you take a 6th test, the mean of your scores is 8, what was your 6th test score?

Start with the formula:

$\dfrac{\text{Sum of scores}}{\text{Number of scores}}=\text{Mean of scores}$

What do you know?

Sum of scores: $8+9+7+10+7+x$ (x is the 6th score that you don't know)

Number of scores: $6$

Mean of scores: $8$

Plug what you know into the equation:

$$\eqalign{\dfrac{\text{Sum of scores}}{\text{Number of scores}}&=\text{Mean of scores}\\\\\dfrac{8+9+7+10+7+x}{6}&=8}$$

Then solve the equation:

$$\eqalign{\dfrac{8+9+7+10+7+x}{6}&=&&8\\\dfrac{41+x}{6}&=&&8\\6\times(\dfrac{41+x}{6})&=&&8\times 6\\41+x&=&&48\\-41\text{     }&\text{}&&-41\\x&=&&7}$$

Your final score was $7$.

Overall, whenever you face a word problem that uses means or averages, start with the formula, plug in what you know, and do the math.  You'll solve the problem!