Try some yourself
Activity 31
Insert brackets in the following calculations to emphasise the order in which a scientific calculator would perform them, then do the calculations by hand and on your calculator, with and without the brackets, as a check.
(a) 3 × 60 + 70.
(b) 10 − 15 ÷ 5.
(c) 20 − 2 × 8.
(d) 3 + 16 − 10.
(e) 3 × 10 ÷ 5.
Answer
(a) 3 × 60 + 70 = (3 × 60) + 70 = 180 + 70 = 250.
(b) 10 − 15 ÷ 5 = 10 − (15 ÷ 5) = 10 − 3 = 7.
(c) 20 − 2 × 8 = 20 − (2 × 8) = 20 − 16 = 4.
(d) 3 + 16 − 10 = (3 + 16) − 10 = 19 − 10 = 9.
(e) 3 × 10 ÷ 5 = (3 × 10) ÷ 5 = 30 ÷ 5 = 6.
Activity 32
In which of the calculations in Activity 31 does the order of the calculations make a difference?
Answer
The order makes a difference in calculations (a), (b) and (c) of Activity 31.
Activity 33
(a) Calculate
(b) Does the calculation in part (a) represent an exception to the ‘division before addition’ rule?
Answer
(a)
= 24 ÷ 8 = 3
(b) No, because there are implicit brackets around each of the top and bottom line of the fraction. When you write one number over another, you should treat the expressions above and below the line as if they were each in brackets. So the fact that the additions in part (a) were carried out before the division does not represent an exception to the ‘division before addition’ rule; it simply follows the ‘brackets before division’ rule.
Activity 34
The target score in a game of darts is obtained by subtracting the score of the three darts thrown in a player’s turn from the current target score on the board. In the game of ‘301’, the target score starts at 301. If a player’s first three darts are a double 19, a single 20 and a treble 17, what is the new target score?
Answer
The player scores 2 × 19, 20 and 3 × 17, which gives 38 + 20 + 51 = 109.
301 − 109 gives a new target score of 192.
Activity 35
Are the following statements true or false?
(a) When carrying out an addition and a multiplication, such as 2 + 3 × 4, it doesn’t matter whether you do the addition or the multiplication first.
(b) Putting brackets into an expression can change the order of the calculations.
Answer
(a) False. Carrying out the addition first gives 2 + 3 = 5, then multiplying by 4 gives 20. Carrying out the multiplication first gives 3 × 4 = 12, then adding to 2 gives 14 (which is the correct procedure here).
(b) True. For example writing brackets around 2 + 3 in the expression 2 + 3 × 4 gives (2 + 3) × 4 = 20, whereas without the brackets you should carry out the multiplication first to give 14.