3.4 Order of calculations
You may have noticed that sometimes the order in which calculations are carried out seems to matter and sometimes it does not. When using a calculator, it is very important to know the order in which it will do calculations. It is not always the order in which you enter them.
Although written English is read from left to right, this is not the case for all written languages (Chinese is read top to bottom, right to left). With mathematics, the order of the written operations does not always indicate the order in which they should be carried out.
Example 14
(a) Multiply 3 by 365. How many days are there in 3 years?
(b) Divide 366 by 3 to find out how many days there are per term, in a 3term leap year.
(c) Does it matter if you interchange the numbers in each of the sums you did above?
Answer

(a) Multiplying by 365 is quite hard work, whereas multiplying by 3 is comparatively easy. Since 3 lots of 365 are the same as 365 lots of 3, it makes no difference whether you multiply 3 by 365 or multiply 365 by 3.
3 × 365 = 365 × 3 = 1095.
Since 3 × 365 = 1095, there are 1095 days in 3 years, provided one is not a leap year. If it were there would be 1096.
(b) 366 ÷ 3 = 122. So there will be 122 days.
(c) 3 × 365 = 365 × 3
So you can interchange the numbers in (a).
But 366 ÷ 3 does not give the same answer as 3 ÷ 366.
So you cannot interchange the numbers in (b).
To see that dividing 3 by 366 is not the same as dividing 366 by 3 note the following:
gives a value of over a hundred, whereas is much less than 1.
When you multiply two numbers together, it makes no difference which number you write first.
When you divide one number by another, it does matter which number you write first.
If you add two numbers, the order does not matter,
3 + 2 = 2 + 3,
but the same is not true with subtraction,
3 − 2 ≠ 2 − 3.
(≠ means ‘is not equal to’.)
When adding two numbers together, it makes no difference which number you write first.
When subtracting one number from another, it does matter which number you write first.