Lesson 28: Two-Step Word Problems, Average, Part 1

New Concept

The average, or mean, is a value that represents a group of numbers. We find the mean of a set of numbers by adding the numbers and then dividing the sum by the number of numbers.

What’s next

This is a foundational concept for data analysis. Next, you will master this two-step process through worked examples involving everyday situations and different sets of numbers.

Property

Word problems often require more than one step to solve. These problems involve two or more themes, where the answer to the first step is necessary to complete the second step and find the final solution.

Examples

• Julie has 20 dollars and buys 8 cans of food at 0.67 dollars each. First, find total cost: $8 \times 0.67 \text{ dollars} = 5.36 \text{ dollars}$. Then, find change: $20.00 \text{ dollars} - 5.36 \text{ dollars} = 14.64 \text{ dollars}$.
• Three-eighths of 32 ducks are wood ducks and the rest are mallards. How many are mallards? First, find wood ducks: $\frac{3}{8} \times 32 = 12$. Then, find mallards: $32 - 12 = 20$ mallards.

Explanation

Think of these problems like a fun, two-part mission! You can't just jump to the final answer. First, you must complete a preliminary objective, like figuring out the total amount of money spent. Only after you have that crucial piece of information can you proceed to the main goal, such as calculating the change. It's all about finding the hidden first step!

Property

To find the average of a group of numbers, combine the numbers by finding their sum, and then divide the sum by the count of the numbers. The formula is:

Examples

• Find the average of 3, 8, and 10 people in three rows. First, add them up: $3 + 8 + 10 = 21$ people. Then, divide by the number of rows: $\frac{21}{3} = 7$ people per row.
• What is the average of 40 and 70? First, add the numbers together: $40 + 70 = 110$. Then, divide by the count of numbers, which is 2: $\frac{110}{2} = 55$.

Explanation

Imagine you and your friends dump all your snacks into one big pile. The 'average' is what happens when you redistribute them so everyone gets an identical share. It’s the ultimate move for fairness! You first combine everything by adding, then you divide that total by the number of friends to find that perfectly balanced, equal amount for everyone.

Property

Another name for the average is the mean. We find the mean of a set of numbers by adding the numbers and then dividing the sum by the number of numbers. It represents the central value of a set.

Examples

• What is the mean of 46, 37, 34, 31, 29, and 24? First, find the sum: $46 + 37 + 34 + 31 + 29 + 24 = 201$. Then, divide by the 6 numbers: $\frac{201}{6} = 33.5$.
• A team's scores are 82, 85, and 91. What is their mean score? First, add them: $82 + 85 + 91 = 258$. Then, divide by the 3 scores: $\frac{258}{3} = 86$.

Explanation

Don't get thrown off by this fancy word! 'Mean' is just the mathematical codename for 'average'—they are exactly the same concept. It's not a 'mean' number, it's the central or typical value in a dataset. Just like finding an average, you calculate it by adding all the numbers together and then dividing that sum by how many numbers you added.