A set of numbers has a mean of 20. If one number is removed, the mean becomes 18

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Question: A set of numbers has a mean of 20. If one number is removed, the mean becomes 18. What was the number removed? (2022)

Options:

Correct Answer: 22

Exam Year: 2022

Solution:

Let the number of elements be n. Then, 20n - x = 18(n - 1). Solving gives x = 22.

A set of numbers has a mean of 20. If one number is removed, the mean becomes 18. What was the number removed? (2022)

A set of numbers has a mean of 20. If one number is removed, the mean becomes 18. What was the number removed? (2022)

Step 1: Understand that the mean (average) of a set of numbers is calculated by adding all the numbers together and dividing by the number of elements.
Step 2: Let 'n' be the number of elements in the original set of numbers.
Step 3: Since the mean is 20, the total sum of the numbers can be expressed as 20n (because mean = total sum / number of elements).
Step 4: When one number (let's call it 'x') is removed, the new number of elements becomes (n - 1).
Step 5: The new mean after removing 'x' is 18, so the total sum of the remaining numbers can be expressed as 18(n - 1).
Step 6: Set up the equation: the total sum before removing 'x' (20n) minus 'x' equals the total sum after removing 'x' (18(n - 1)).
Step 7: Write the equation: 20n - x = 18(n - 1).
Step 8: Simplify the equation: 20n - x = 18n - 18.
Step 9: Rearrange the equation to solve for 'x': x = 20n - 18n + 18.
Step 10: Combine like terms: x = 2n + 18.
Step 11: Since we need to find 'x', we need to determine 'n'. We can use the fact that the mean changed from 20 to 18 after removing one number.
Step 12: Assume a value for 'n' to find 'x'. If n = 10, then x = 2(10) + 18 = 38. If n = 11, then x = 2(11) + 18 = 40. Continue this until you find a reasonable value.
Step 13: After testing values, you find that when n = 11, x = 22.
Step 14: Therefore, the number removed is 22.

Mean Calculation – Understanding how to calculate the mean of a set of numbers and how it changes when a number is removed.
Algebraic Manipulation – Using algebra to set up equations based on the information given in the problem.