The mean of the numbers 13, 16, x, 18, 21, 2x, 35, is 22. Find the value of x

Analysis

To find the value of $x$, we need to use the definition of the mean (average). The mean is calculated by finding the sum of all the numbers in a set and dividing that sum by the total count of the numbers.

Concept

The formula for the mean is:

Given:

• The numbers are: $13, 16, x, 18, 21, 2x, 35$
• The total count of numbers ($n$) = $7$
• The mean = $22$

Step-by-Step Solution

Step 1: Set up the equation.

Step 2: Simplify the numerator by adding the constants and the $x$ terms separately.

• Constants: $13 + 16 + 18 + 21 + 35 = 103$
• $x$ terms: $x + 2x = 3x$

So, the equation becomes:

Step 3: Solve for $x$. Multiply both sides by $7$ to clear the fraction:

Subtract $103$ from both sides:

Divide by $3$:

Conclusion

By substituting $x = 17$ back into the set, we can verify the mean: $(13 + 16 + 17 + 18 + 21 + 34 + 35) / 7 = 154 / 7 = 22$.

Exam Tip

When dealing with mean questions, always count the number of items carefully. A common mistake is to miscount the items in the list (e.g., forgetting to count $x$ or $2x$ as individual entries). Also, remember that $2x$ is a single term in the sum, not two separate numbers.