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1.7.1 Exercises

Evaluate the expressions using the correct order of operations.

1. 6 + 16 ÷ 4

2. 16 · 4 − 48

3. 15 − 9 − 4

√
4. 2 · 6+ 2( 36 − 1)

5. 4 × 3 × 2 ÷ 8 − 3

6. (2 · 5) 2
√
7. 21 − 30 ÷ 6

8. [18 ÷ (9 ÷ 3)] 2

9. 2 + 2 × 8 − (4 + 4 × 3)

1.8 Average

The average of 2 numbers is their sum, divided by 2. The average of 3 numbers is their sum, divided
by 3. In general, the average of n numbers is their sum, divided by n.
Example 32. Find the average of each of the following multi-sets of numbers.(A multi-set is a set in
which the same number can appear more than once.) (a) {10, 12};(b) {5, 6, 13};(c) {8, 12, 9, 7, 14}.

Solution. In each case, we take the sum of all the numbers in the multi-set,and then (following the
order of operations), divide by number of numbers:

(a) (10 + 12) ÷ 2= 11; (b) (5 + 6 + 13) ÷ 3= 8; (c) (8 + 12 + 9 + 7 + 14) ÷ 5= 10.

The average of a multi-set of numbers is a description of the whole multi-set, in terms of only one
number. If all the numbers in the multi-set were the same, the average would be that number. For
example, the average of the multi-set {5, 5, 5} is

(5 + 5 + 5) ÷ 3= 5,

and the average of the multi-set {2, 2, 2, 2, 2, 2} is

(2 + 2 + 2 + 2 + 2 + 2) ÷ 6= 2.

This gives us another way to define the average of a multi-set of n numbers: it is that number which,
added to itself n times, gives the same total sum as the sum of all the original numbers in the multi-set.

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