5 Order of operations
Since many everyday problems that you encounter require the use of more than one operation, you need to make sure you know how to correctly proceed, write and carry out the calculation.
The five operations you have looked at so far are joined by brackets when considering order of operation.
Brackets indicate the highest priority, followed by any exponents (powers). Then carry out the multiplication and division, and finally any addition and subtraction. If part of the calculation involves only multiplication and division or only addition and subtraction, work through from left to right.
The correct order to carry out the operations can be summarised by using the mnemonic BEDMAS, where the letters stand for Brackets, Exponents, Division, Multiplication, Addition and Subtraction.
The following everyday problem involving more than one operation demonstrates the importance of BEDMAS.
Let’s suppose you purchased four very large pepperoni pizzas that cost £15.99 each and you want to split the total cost among six people. One way to do this is by using addition and division.
You might think that you can enter this into your calculator as:
15.99 + 15.99 + 15.99 + 15.99 ÷ 6.
However, this will give you £50.635 as the answer! That is clearly not correct as each pizza only costs £15.99.
The calculator has actually worked out:
15.99 + 15.99 + 15.99 + (15.99 ÷ 6) = 47.97 + 2.665 = 50.635. This is because it follows BEDMAS, dividing only the last £15.99, not the total, by six. To obtain the correct answer you must use brackets:
(15.99 + 15.99 + 15.99 + 15.99) ÷ 6 = 10.66
Now the total cost of the pizzas is divided by 6 (remember, B comes before D in BEDMAS), to give a much more sensible answer of £10.66.
You can now see how important it is to get the correct order for calculations.
Now, it is your turn to have a go. Try applying this rule in the next activity.
Before you start this activity you need to ensure that your calculator does know the BEDMAS rules. So, check this by working out 3 + 5 × 2 , without pressing the equals sign until after the final 2. If your calculator knows the rules the answer will be 13, if it doesn’t it will give you 16.
But don’t worry! Just be aware of the rules and work through the calculations carefully yourself.
Activity 5 BEDMAS on paper versus the calculator
First of all, try these examples on paper without using a calculator. Then, check your answer using a calculator.
(a) (3 + 4) × 2
Hint: Remember BEDMAS. Are there brackets? If so, then do the calculation inside those first. Next, look for exponents, then multiplication, division, addition, and subtraction.
Answer
Carry out the calculation in brackets first: 3 + 4 = 7. Now multiply by two and the answer is 14.
(b) 2 + 32
Answer
No brackets this time, so start with the exponent: 32 = 9. Add the two and you get 11.
(c) (2 + 3)2
Answer
This looks like part (b), but this time there are brackets, so you must do the calculation inside the brackets first: 2 + 3 = 5. Now square the result, and the answer is 25.
(d) 32 + 42
Answer
Work out the exponents first: 32 = 9 and 42 = 16. Finally add 9 + 16 = 25.
(e) 2 + 3 × 4 + 5. Take care with this one!
Answer
It can help to make calculations like this easier to read if you put brackets around the part that you need to do first.
This time, you have addition and multiplication, so you must do the multiplication first: 3 × 4 = 12.
So now, the calculation is 2 + 12 + 5 = 19.
Using brackets you would write 2 + (3 × 4) + 5.
This will give you the same answer, but just might make your job easier!
Hopefully you got the same answers using your calculator.
The last few sections have been activities that were not related to any real-world problems so could seem a little abstract in nature. Next, you will apply this maths to more real-world problems and also continue your work on problem-solving skills from Week 1.