The average of three numbers is 21. If two numbers are 16 and 23, the third…

Concept: The arithmetic mean of a group of numbers is their sum divided by how many numbers there are. Turning that definition around gives sum = mean × count, so the mean and the count together fix the total of the whole group. Once that total is known, any single missing member equals the total minus the members that are already known.

Applying it to this data:

1. The group has 3 numbers and its mean is 21, so count = 3 and mean = 21.

2. Total of the three numbers = mean × count = 21 × 3 = 63.

3. Total of the two numbers that are given = 16 + 23 = 39.

4. Missing number = 63 − 39 = 24.

Cross-check: Put 24 back beside the two given values: (16 + 23 + 24) ÷ 3 = 63 ÷ 3 = 21, which is the average stated in the question, so the total of 63 used above is consistent.

Method note: A shortcut is to work with deviations from the mean. 16 is 5 below 21 and 23 is 2 above 21, a net shortfall of 3, so the remaining number must sit 3 above 21 for the deviations to cancel to zero.