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4. additions and subtractions (in order of appearance) last.

“In order of appearance” means in order from left to right. Thus in the expression

2+ 5 − 3,

the addition comes first, so it is evaluated first,

2+ 5 − 3= 7 − 3= 4,

while in the expression
8 − 6+ 11,

the subtraction is done first because it comes first,

8 − 6+ 11 = 2 + 11 = 13.

Example 27. Evaluate 6 · 5 − 4 ÷ 2+ 2.

Solution. All four operations appear here. There are no grouping symbols, exponents or roots. Following
the order of operations, as well as the order of appearance, we do the computations in the following
order: multiplication, division, subtraction, addition. The computations are as follows:

6 · 5 − 4 ÷ 2+ 2 =
30 − 4 ÷ 2+ 2 =
30 − 2+ 2 =
28 + 2 =
30

In the next two examples, we use the same numbers and the same operations, but we insert grouping
symbols to change the order of operations. As expected, this changes the final result.
Example 28. Evaluate 6 · (5 − 4) ÷ 2+ 2.

Solution. The grouping symbols force us to do the subtraction first. After that, the usual order of
operations is followed.

6 · (5 − 4) ÷ 2+ 2 =

6 · 1 ÷ 2+ 2 =
6 ÷ 2+ 2 =
3+ 2 =
5

Example 29. Evaluate 6 · 5 − 4 ÷ [2 + 2].

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