Finding the Average of Three Numbers

Let's break down this word problem step by step to find the average of the three numbers. We are given the total sum of the three numbers and how the second and third numbers relate to the first number.

Setting up the Relationship Between the Numbers

We need to represent the three unknown numbers using variables based on the information given:

• Let the first number be represented by the variable \(x\).
• The second number is four times the first number, so the second number is \(4 \times x = 4x\).
• The third number is three times the first number, so the third number is \(3 \times x = 3x\).

Using the Total Sum to Find the Value of the First Number

The total of the three numbers is given as 240. We can write this as an equation:

First number + Second number + Third number = Total

\(x + 4x + 3x = 240\)

Now, we combine the terms with \(x\):

\((1 + 4 + 3)x = 240\)

\(8x = 240\)

To find the value of \(x\), we divide both sides of the equation by 8:

\(x = \frac{240}{8}\)

\(x = 30\)

So, the first number is 30.

Calculating the Value of Each Number

Now that we know the first number is 30, we can find the values of the second and third numbers:

• First number \( = x = 30\)
• Second number \( = 4x = 4 \times 30 = 120\)
• Third number \( = 3x = 3 \times 30 = 90\)

Let's quickly check if their sum is indeed 240:

\(30 + 120 + 90 = 150 + 90 = 240\)

The sum is correct.

Calculating the Average of the Three Numbers

The average of a set of numbers is found by dividing the total sum by the count of numbers. In this case, we have three numbers.

Average = \(\frac{\text{Total sum of the numbers}}{\text{Number of numbers}}\)

We are given that the total sum is 240, and there are three numbers.

Average = \(\frac{240}{3}\)

Average = 80

Therefore, the average of the three numbers is 80.

Revision Table: Key Concepts Reviewed

Additional Information: Understanding Averages

The average, also known as the arithmetic mean, is a fundamental concept in statistics. It gives us a single value that summarizes the center of a dataset. It's calculated by adding up all the numbers in a set and then dividing by the count of those numbers.

In this problem, the numbers (30, 120, 90) vary quite a bit, but their average (80) gives us a single number that is representative of the group as a whole. Averages are useful for comparing different sets of data or tracking changes over time.

Other types of averages exist, such as the median (the middle value when numbers are ordered) and the mode (the number that appears most frequently), but the term "average" usually refers to the arithmetic mean.