EXERCISES AND PROBLEMS

31 Calculate.

a) (–2) · [(+3) · (–2)] b) [(+5) · (–3)] · (+2) c) (+6) : [(–30) : (–15)] d) [(+40) : (– 4)] : (–5) e) (–5) · [(–18) : (– 6)] f ) [(–8) · (+3)] : (– 4) g) [(–21) : 7] · [8 : (– 4)] h) [6 · (–10)] : [(–5) · 6] a) 17 – 6 · 2 b) (17 – 6) · 2 c) (–10) – 2 · (–3) d) [(–10) – 2] · (–3) e) (–3) · (+5) + (–2) f ) (–3) · [(+5) + (–2)] a) 5 · [11 – 4 · (11 – 7)] b) (– 4) · [12 + 3 · (5 – 8)] c) 6 · [18 + (– 4) · (9 – 4)] – 13 d) 4 – (–2) · [–8 – 3 · (5 – 7)] e) 6 · (7 – 11) + (–5) · [5 · (8 – 2) – 4 · (9 – 4)] a) (x 2)5 b) (m 4)3 c) [a 10 : a 6]2 d) (a · a 3)3 e) (x 5 : x 2) · x 4 f ) (x 6 · x 4) : x 7

32 Calculate and note that the result changes depending on the position of the brackets.

33 Calculate.

34 Reduce to a single power.

35 Look at the example and reduce as shown. • () xx x · 63 23 2 == = x 3 a) () x 22 b) () m 32 c) () a 42 d) x 4 e) m 6 f ) a 8

CONSIDER, DECIDE AND APPLY

36 Find a divisor of 427 with two digits.

37 A number less than 50 is a multiple of 6 and 7. Which number is it?

38 A group of 20 people can be arranged into an exact number of rows and columns. For example, four rows and five columns. However, we cannot do the same thing with a group of 13 people. We can only arrange them into a single row.

A three-digit number is a multiple of 150 and a divisor of 2100. Which number could it be?

Solve Simple Problems

Problems with natural numbers

39 The WHO recommends that a 14 year old sleep between 9 and 11 hours a day, and a 40 year old between 7 and 9 hours a day. What is the difference between the annual hours of sleep of a 14 year old and a 40 year old?

40 A dance company with 156 dancers is rehearsing a routine that involves rows and columns. In one row there are 20 more dancers than in a column. How many rows and columns are there?

41 You want to divide a piece of card measuring 50 cm × 65 cm into the smallest squares possible. What is the length of the sides of each square?

42 Wooden cubes weighing 30 grams are placed on one side of a weighing scale and glass marbles weighing 36 grams on the other. If the scale is balanced and there is a total of 15 cubes and marbles: a) How much does each side weigh? b) How many wooden cubes and how many marbles are there?

43 Two lorries transporting identical fridges leave a factory. The first is carrying a load of 481 kilos, and the second 555 kilos. How much does each fridge weigh and how many fridges are there in each lorry?

44 A roll of cable is longer than 150 m, but shorter than 200 m. Exactly, how long is the cable if it can be divided into sections of 15 m, and also into sections of 18 m, without any cable left over?

45 The council is offering people allotment sites. For this purpose, a square piece of land has been divided into 15 m × 20 m rectangular plots. If there are almost 50 plots, what were the measurements of the piece of land?

Find all the numbers between 150 and 170 that can only be arranged in a single row.

46 This week, a bakery wants to sell 2 400 muffins and 2 640 biscuits in bags with the same number of units, but without mixing the products. How many muffins or how many biscuits can be put into each bag, taking into account that the number must be larger than 10 but smaller than 15?

Problems with integers

47 Draw a coordinate system and represent the points A (–2, 0) and B (4, 2).

Draw all the squares that have their vertices at these points (there are three different squares).

Finally, write the coordinates of the vertices of each of the squares.

Brain Teasers

50 The sum of two integers is 3 and their difference 7. What are the two numbers?

51 The sum of two integers is –22, and the sum of their absolute values is 70. What are the two numbers?

52 A local bakery makes cupcakes every morning. They pack them into bags of half a dozen cupcakes, with two left over.

If they packed them into bags of 5, 3 cupcakes would be left over. If they packed them into bags of 8, no cupcakes would be left over.

48 If you write all the integers from –50 to +50, how many times will you use the number 7? And 5? And 5? And 3?

49

Problem Solved

The sum of two integers is (–5) and their difference is (+19). What are the two numbers?

Let’s try with a very simple example

We take the numbers 6 and 4:

64 10 64 2 –+= = 3

8 10 – 2 = 8 8 8 : 2 = 4 6 (the smaller number)

If we subtract the difference from the sum of the two numbers, we get double the smaller number.

Now let’s solve the original problem

– The sum is (–5) and the difference is (+19).

– The sum minus the difference is double the smaller number:

(–5) – (+19) = –5 – 19 = –24

(double the smaller number)

The smaller number is: (–24) : 2 = –12

The larger number is: –12 + 19 = 7

Now check the answer.

If we know that they pack just over 40 bags, how many cupcakes do they make?

53 The members of an athletics club have decided to give their coach a stopwatch that costs €130.

‘It is a shame the shot put, discus and javelin throwers are not taking part,’ says the team captain, ‘with three more people we would each pay €3 less.’

How many people are contributing to the gift for the coach if each person pays an exact amount in Euros?

54 I am going with my brother to buy a gift that we have chosen for our mother. My brother says that after paying for his share of the gift, he still has €10. I borrow €5 from him to pay my share.

How much is the gift if we have €85 between us?

55 I have two accounts at the same bank. In the first one there are €200 more than in the second one. If I transfer money from one to the other so they both have the same balance, there will be €20 in each account.

How much money is there in each account?

You can refer to this graph:

You can also look back to problem 50 and ask: What is the sum of both accounts and what is the difference?