If the mean of the observations x, x + 3, x + 5, x + 7, x + 10 is 9, then mean…

Concept

The mean of a set of observations equals the sum of the observations divided by their count. So once the mean and the count are known, the sum can be recovered as mean times count, and any unknown term inside that sum can then be solved for.

Applying it here

1. The mean of the five observations x, x + 3, x + 5, x + 7, x + 10 is given as 9, so their sum equals 5 times 9 = 45.

2. Adding the five expressions gives x + (x + 3) + (x + 5) + (x + 7) + (x + 10) = 5x + 25.

3. Equating the two sums: 5x + 25 = 45.

4. Solving for x: 5x = 20, so x = 4.

5. The last three observations are x + 5, x + 7 and x + 10, which evaluate to 9, 11 and 14 respectively.

6. Their mean is (9 + 11 + 14) divided by 3 = 34/3, written as the mixed number 11 1/3.

Cross-check

Substituting x = 4 back into all five original observations gives 4, 7, 9, 11 and 14; their sum is 45 and their mean is 45 divided by 5 = 9, matching the given data. This confirms x = 4 is correct, and hence that the mean of the last three observations, 11 1/3, is correct.