3. Order of Operations
Problem: There are 2 ways to answer this question involving numbers:
`2 + 3 xx 7`
We can either
- Add the 2 and 3 first and then multiply the result by 7 (we would get 35), OR
- We could multiply the 3 and 7 then add the 2 (giving us 23).
Which is correct?
To reduce confusion and so that we all get the same answer when combining numbers using the operations of adding, subtracting, multiplying or dividing, the following convention is used.
BODMAS
BODMAS is the agreed convention for answering problems that are a mixture of adding, subtracting, multiplying and dividing. It stands for:
B: Do what is inside brackets first O: Order is important in mathematics! DM: Do divisions and multiplications as you come to them, from left to right AS: Finally, do additions and subtractions as you come to them, from left to right.
So for our problem above, the correct answer would be:
`2 + 3 xx 7`
`= 2 + 21`
`= 23`
Example 1
Evaluate 10 − 2 × 3
Answer
10 − 2 × 3
= 10 − 6
= 4
Example 2
Evaluate (10 − 2) × 3
Answer
This time we have the same numbers as the previous example, but the brackets tell us to do it differently:
(10 − 2) × 3
= 8 × 3
= 24
Always be careful with brackets. As you can see, they make a big difference.
Example 3
Simplify −4 + 5 × (7 + 3)
Answer
We do the operations in brackets first:
(7 + 3) = 10
So the original problem is now
−4 + 5 × 10
We do the multiplication next (before even looking at the addition):
5 × 10 = 50
So the original problem now has become:
−4 + 50
And this equals
46
So −4 + 5 × (7 + 3) = 46
Example 4
Simplify 12(5 + 6 × 3) ÷ (4 + 2)
Answer
The first bracket:
(5 + 6 × 3) = 5 + 18 = 23
The second bracket:
(4 + 2) = 6
So we have, since we need to work from left to right:
12 × 23 ÷ 6 = 46
So
12(5 + 6 × 3) ÷ (4 + 2) = 46
Example 5
Simplify −2(x + 5 × 7 + 3)
[Note the difference between the letter "x" and the multiplication operator "×".]
Answer
Here, we consider the bracket first:
(x + 5 × 7 + 3)
We do the multiplication inside the brackets:
5 × 7 = 35
So the whole bracket will be:
x + 35 + 3 = x + 38
Now for the -2 out the front:
−2(x + 38)
= −2x −76