Finding the Included Number for Average Change

This problem asks us to find a new number that, when added to a set of existing numbers, changes their average to a specific value. Let's break it down using the concept of average.

The average of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers.

\text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}

Calculating the Sum of the Original Numbers

We are given that the average of 5 numbers is 15. We can use the average formula to find the sum of these 5 numbers.

• Average of 5 numbers = 15
• Count of numbers = 5

Using the formula:

\text{Sum of 5 numbers} = \text{Average} \times \text{Count of numbers}

\text{Sum of 5 numbers} = 15 \times 5

\text{Sum of 5 numbers} = 75

So, the sum of the original 5 numbers is 75.

Finding the New Number to Include

Now, we want to include one more number in this set. This means the total count of numbers will become 5 + 1 = 6. The new average is required to be 16.

Let the new number to be included be \(x\).

The sum of the new set of 6 numbers will be the sum of the original 5 numbers plus the new number \(x\):

\text{Sum of 6 numbers} = 75 + x

The new average is given as 16, and the count of numbers is now 6. Using the average formula again:

\text{New Average} = \frac{\text{Sum of 6 numbers}}{\text{Count of numbers}}

Substitute the values we know:

16 = \frac{75 + x}{6}

Solving for the Unknown Number

Now, we need to solve this equation for \(x\):

Multiply both sides by 6 to isolate the term with \(x\):

16 \times 6 = 75 + x

Calculate the left side:

96 = 75 + x

Subtract 75 from both sides to find the value of \(x\):

x = 96 - 75

x = 21

Therefore, the number that should be included is 21.

Verification

Let's check if including 21 makes the average 16.

• Original sum = 75
• New number = 21
• New sum = 75 + 21 = 96
• New count = 6

New Average = \frac{96}{6} = 16

This matches the required average. So, the number is indeed 21.

Comparing this with the given options, the number 21 corresponds to option 3.

Revision Table: Average Calculation

Understanding the relationship between average, sum, and count is crucial for solving such problems. Here's a quick summary:

• Definition: Average = Sum / Count
• Finding Sum: Sum = Average × Count
• Finding Count: Count = Sum / Average

These formulas can be rearranged based on what information you have and what you need to find.

Additional Information: Average Problems

Problems involving averages often require you to work backward from the average to find the total sum, or to calculate how adding or removing numbers affects the sum and subsequently the average. Key steps often include:

• Calculating the initial sum based on the initial average and count.
• Determining the new count after adding or removing numbers.
• Setting up an equation for the new average using the new sum (which includes the unknown number) and the new count.
• Solving the equation to find the unknown number.

Practice with different variations of these problems, such as removing a number or changing the average by a specific amount, will help solidify your understanding.