8.1 Rules of Precedence

Addition and subtraction
How would you do the following calculations, according to model a or b?

Fifteen minus seven plus three.
15 – 7 + 3 = ?

a) 15 – 7 + 3 = 8 + 3 = 11

b) 15 – 7 + 3 = 15 – 10 = 5

Remember that the sign which belongs to the number is always in front of it.

The answer to the question becomes 15 - 7 + 3 = 8 + 3 = 11

You could say that in addition and subtraction you can calculate the numbers in the order that they stand, from left to right.

Multiplication and division
 Mary is going to buy a chocolate bar for 12 kr and two lollipops for 5 kr each. How much should she pay?

 

Foto: Fredrik Enander

Write out the calculation (an expression) for Mary’s purchase.

Answer: 12 kr + 2 · 5 kr = 12 kr + 10 kr = 22 kr

The example above is an expression which includes two arithmetic operators: addition and multiplication. Actually, it may not be quite clear as to how to do the calculations by simply looking at the expression. Which is actually the correct way?

a) 12 + 2 · 5 = 14 · 5 = 72

b) 12 + 2 · 5 = 12 + 10 = 22

You could think like this:

You will be buying the following:
  • Chocolate for 12 kr
  • Two lollipops for 5 kr each

The expression becomes: 12 kr + 5 kr + 5 kr = 12 kr + 2 · 5 kr = 22 kr

The answer 22 is then the right one!

a) 12 + 2 ·· 5 = 14 · 5 = 72

b) 12 + 2 · 5 = 12 + 10 = 22

Depending on if you have a simple or more advanced calculator you will get different answers for the above exercise. The more advanced calculators calculated according to the correct orders of precedence while the more simple ones only in the order the numbers are entered. Check to see how your calculator prioritizes.

What do you think applies for division and subtraction?

How do you calculate “thirty six minus half of twelve”?

36 – 12 ÷ 2 = ?

Which of the alternatives do you think is correct?

Here division should be done before subtraction.

The answer becomes:

36 – 12 ÷ 2 = 36 – 6 = 30

Powers and Roots
What rules apply concerning powers and roots? We will show you a few examples, can you figure out what rule applies?

a) 3 + 2² = 3 + 2² = 3 + (2 · 2) = 3 + 4 = 7

b) 4 · 3² = 4 · (3 · 3) = 4 · 9 = 36

c) 8 + = 8 + 3 = 11

d) 8 ÷ = 8 ÷ 4 = 2

Parenthesis
There is a large difference between how parenthesis are used in texts compared to how they are used in mathematics. In mathematics parenthesis denote precedence.

We will use our earlier examples concerning addition and subtraction.  Try to calculate:

a) 15 – 7 + 3 = ?

b) 15 – (7 + 3) = ?

The answer becomes:
Express without parenthesis are calculated from left to right.
a) 15 – 7 + 3 = 11

The parenthesis are calculated first.
b) 15 – (7 + 3) = 5

The same thing applies to multiplication and division:

(12 + 3) · 4 - 6 ÷ 2 =

15 · 4 – 3 =

60 – 3 = 57

The expression in the parenthesis is always calculated first.