Applications of Mathematics (Intermediate 1) - Newbattle Community ...

APPLICATIONS OF MATHEMATICS (INT 1) – STUDENT MATERIALS 

CONTENTS 

Straightforward Calculations in a Social Context 

A. Basic Pay 5 

B. Overtime, Bonus and Gross Pay 6 

C. Deductions and Net Pay 7 

D. Paying Back a Loan 9 

Checkup 12 

Logic Diagrams 

A. Network Diagrams 15 

B. Tree Diagrams 19 

C. Shortest Path Problems 22 

D. Flowcharts 27 

E. Spreadsheets 31 

Checkup 38 

Scale Drawings and Surface Areas of Solids 

A. Points of the Compass 43 

B. Three-Figure Bearings – Recognising and Measuring 44 

C. Three-Figure Bearings – Drawing 46 

D. Scale Drawing 49 

E. Surface Area of a Triangular Prism 53 

F. Surface Area of a Cylinder 55 

Checkup 57 

Statistical Assignment 

A. A Statistical Assignment 61 

B. Quartiles and Semi-interquartile Range 62 

C. The Boxplot 63 

Specimen Assessment Questions 67 

Answers 71 

Mathematics: Applications of Mathematics (Int 1) – Student Materials 1


STRAIGHTFORWARD CALCULATIONS IN A SOCIAL CONTEXT 

By the end of this set of exercises, you should be able to: 

(a) 

(b) 

(c) 

carry out calculations using the terms: gross pay, net pay, basic pay, 

overtime at ‘double time’ and ‘time and a half’, bonus, annual salary. 

calculate total deductions, gross pay and net pay. 

understand APR and know the meaning of the term ‘with (without) loan 

protection’. Also, calculate total repayments and the cost of the loan. 



A. Basic Pay 

Exercise 1 

1. Tanya is paid a weekly wage of £147. Calculate her yearly earnings. 

2. Peter is paid £810 per month. How much is his annual salary? 

3. Abed earns £949 per month. 

Anya earns £219 per week. 

Calculate the annual earnings for both, and state who earns the greater annual salary. 

4. Brian got the job as a trainee accountant. 

TRAINEE ACCOUNTANT 

His wages were paid to him at the end of each 

Starting Salary 

month. 

£11880 per annum 

(a) Calculate his monthly pay. 

(b) After being in the post for a year, Brian was 

given an increase of £720 in his salary. 

Calculate his new: (i) annual salary; (ii) monthly pay. 

5. Janice is a part-time nurse in a health centre. She works for 22 weeks of the year and 

earns £2652·10. 

How much does she get paid weekly? 

6. Tommy is paid £4·25 per hour and he works for 35 hours each week. 

Harold works for 40 hours each week and gets paid £3·80 per hour. 

Who earns more and by how much? 

7. A building firm pays its workers £12 per hour. A bricklayer is expected to work 5 days a 

week from 8 a.m. until 5 p.m. each day (including a 1 hour unpaid lunch break). 

(a) For how many hours, each day, is a bricklayer paid? 

(b) How many paid hours does a bricklayer work in a week? 

(c) What is a bricklayer’s gross pay for a week? 

8. Freda is a college student. She works part-time in the college library to earn some extra 

cash. 

Her hours each week are: 

Mon. 6 p.m. – 8 p.m. 

Wed. 6.30 p.m. – 8 p.m. 

Fri. 6 p.m. – 9 p.m. 

Sat. 9 a.m. – 12.30 p.m. 

If her rate of pay is £5 per hour, how much does she earn each week? 


B. Overtime, Bonus and Gross Pay 

Exercise 2 

1. Shaminder is a typist in her local council offices. Her basic wage is £10 per hour, but for 

any overtime she works she gets paid double time. 

(a) What is her overtime rate of pay per hour? 

(b) One week she worked 5 hours overtime. How much did she get paid for this? 

2. Melanie’s bosses ask her to work 6 hours overtime. She will get paid time and a half. 

If her usual hourly rate is £6·80, how much will Melanie get for doing the overtime? 

3. Alison is a filing clerk in the same offices as Shaminder. 

Her basic wage is £9 per hour and her overtime is at the rate of time and a half. 

(a) What is her overtime rate of pay per hour? 

(b) One week she worked 4 hours overtime. How much did she get paid for this? 

4. Colin is a computer operator. He works a basic 40-hour week and is paid £8·60 per hour. 

Overtime is paid at double time. One week Colin worked a total of 50 hours. 

Find: 

(a) his basic pay 

(b) his overtime pay 

(c) his total gross pay. 

5. Rena is a nursery nurse. Her basic rate is £11·50 per hour for a 38-hour week, but at 

weekends she is paid an overtime rate of double time for looking after the children. 

Calculate her gross pay for a basic week, plus 6 hours on a Saturday and 4 hours on a 

Sunday. 

6. Irene is a plumber’s assistant. She works a 36-hour week at a rate of £6·50 per hour plus 

any amount of overtime at time and a half. 

Calculate her gross pay for a week in which she works her normal hours, plus 10 hours 

overtime. 

7. A hotel porter is paid an hourly rate of £5·16 for a basic 35-hour week. 

Overtime rates are time and a half for weekdays and double time at weekends. 

Calculate the gross wage for a porter who works a basic week as well as the following 

overtime: Tuesday 5 hours; Thursday 1 hour; Sunday 6 hours. 

8. Fred Davis applied for and got this job. 

Calculate: 

(a) how many hours per week he works 

(b) how much he earns per week 

(c) how much he earns altogether in a 

week in which he gets a £20 bonus. 

PART-TIME DELIVERY PERSON 

(6 p.m.–11.30 p.m.) 6 DAYS PER WEEK 

£4·50 per hour + BONUS 


9. Mary Baker works in a bakery in charge of making birthday cakes. 

She works a basic 36 hour week for £4·80 per hour and receives a bonus of £1·50 per cake 

for each cake over the normal 200 she makes in a week. 

(a) What is her basic pay for a week? 

(b) How much of a bonus does she receive if she bakes 250 cakes in a week? 

(c) What would her gross pay be for such a week? 

C. Deductions and Net Pay 

Some Deductions: 

National Insurance – safeguard against illness or job loss 

Superannuation – an additional pension scheme on top of state pension 

Pension Fund – for your old age (not compulsory) 

Income Tax – deducted by the Government for defence, public services etc. 

Others – may include Union Dues, Council Tax etc. 

NET PAY = Gross Pay – Deductions 

Exercise 3 

Examine the payslips and answer the questions. 

1. 

Name Employee No. Week No. Tax Code NI Number 

June Thomson 325417 06 341L AB4182 C 

Basic Pay Overtime Bonus Extra Payment Gross Pay 

£280 £20·00 £0·00 £0·00 

Nat. Insurance Income Tax Pension Other Total Deductions 

£22 £43 £6 £0·00 

Net Pay 

(a) Whose payslip is this? 

(b) What is her tax code? 

(c) Calculate her Gross Pay. 

(d) How much did Mrs Thomson pay towards her pension? 

(e) Write down her Total Deductions. 

(f) Calculate her Net Pay. 


2. 

Name Employee No. Week No. Tax Code NI Number 

Tom Sim 347419 27 356L D4252 E 


£403·25 £0·00 £60·00 £8·00 


£34 £62·40 £22·54 £10·00 

Net Pay 

Calculate Tom’s: 

(a) Gross Pay. 

(b) Total Deductions. 

(c) Net Pay. 

3. 

Name Employee No. Month Tax Code NI Number 

Omar Sheriff 313487 4 525H YX3541 T 


£1425·50 £110·00 £80·20 – 


£121·10 £221·54 £72 

Net Pay 

Calculate Omar’s: 

(a) Gross Pay. 

(b) Total Deductions. 

(c) Net Pay. 

4. 


Penny Law 574815 6 527H YX2147 L 


£2340 £400·00 – – 


£230 £420 ? – 

Net Pay 

£1990 

(a) Use Penny’s Gross Pay and Net Pay to calculate her Total Deductions. 

(b) Use your answer to (a) to find what Penny had paid towards her pension. 


D. Paying Back a Loan 

Exercise 4 

The diagram below is a typical repayment table for paying back a loan from a Building 

Society. 

• The amounts shown in the boxes are your repayments each month. 

• Those shown in italics (£179·80) and with the letter p alongside indicate the amount you 

will pay back if you take the payment protection scheme. 

• Those shown as normal (£164·79) indicate the amount you will pay back if you have not 

taken payment protection. 

For example: if I borrow £2000 over 36 months I will have to pay back: 

£73·80 per month with payment protection or £66·38 without payment protection. 

Amount of 12 months 24 months 36 months 48 months 60 months 

Loan (£) APR 12·4% APR 12·7% APR 12·6% APR 12·4% APR 12·3% 

1000 

p 

95·34 88·75 

p 

51·37 47·08 

p 

36·90 33·19 

p 

29·66 26·25 

p 

25·40 22·08 

1500 

p 

143·02 133·12 

p 

77·05 70·62 

p 

55·36 49·79 

p 

44·50 39·37 

p 

38·10 33·12 

2000 

p 

190·69 177·50 

p 

102·74 94·16 

p 

73·80 66·38 

p 

59·34 52·50 

p 

50·80 44·16 

2500 

p 

238·36 221·87 

p 

128·42 117·70 

p 

92·26 82·98 

p 

74·17 65·62 

p 

63·51 55·20 

3000 

p 

286·04 266·25 

p 

154·11 141·25 

p 

110·72 99·58 

p 

89·00 78·75 

p 

76·21 66·25 

3500 

p 

333·71 310·62 

p 

179·80 164·79 

p 

129·17 116·18 

p 

103·84 91·87 

p 

88·91 77·29 

4000 

p 

381·39 355·00 

p 

205·48 188·33 

p 

147·62 132·77 

p 

118·68 105·00 p 101·61 88·33 

4500 

p 

429·06 399·37 

p 

231·17 211·87 

p 

166·08 149·37 

p 

133·51 118·12 p 114·31 99·37 

5000 

p 

476·73 443·75 

p 

256·85 235·41 

p 

184·53 165·97 

p 

148·34 131·25 p 127·02 110·41 

6000 

p 

572·08 532·50 

p 

308·22 282·50 

p 

221·44 199·16 

p 

178·01 157·50 p 152·42 132·50 

7000 

p 

667·43 621·25 

p 

359·60 329·58 

p 

258·35 232·36 

p 

207·68 183·75 p 177·83 154·58 

8000 

p 

762·78 710·00 

p 

410·97 376·66 

p 

259·25 265·55 

p 

237·35 210·00 p 203·23 176·66 

9000 

p 

858·12 798·75 

p 

426·34 423·75 

p 

332·16 298·75 

p 

267·02 236·25 p 228·63 198·75 

10000 

p 

953·47 887·50 

p 

513·71 470·83 

p 

369·07 331·94 

p 

296·69 262·50 p 254·04 220·83 

1. From the table, write down how much your monthly repayment would be if you 

borrowed: 

(a) 

(b) 

(c) 

(d) 

£4000 over 12 months without payment protection. 

£3000 over 48 months without payment protection. 

£8000 over 24 months with payment protection. 

£10 000 over 60 months with payment protection. 


2. Calculate the total amount to be repaid if you borrowed: 

(a) £3500 over 12 months without payment protection. 

(b) £5000 over 36 months without payment protection. 

(c) £1500 over 60 months with payment protection. 

(d) £9000 over 24 months with payment protection. 

3. In the following, calculate how much extra the loan has cost you. 

(a) £1500 borrowed over 24 months without payment protection. 

(b) £7000 over 60 months without payment protection. 

(c) £4500 over 36 months with payment protection. 

(d) £6000 over 48 months with payment protection. 

Exercise 5 

This exercise follows the same idea as exercise 4, but the Royal Bank of Scotia prefers to 

have 2 tables: one With Loanguard, the other Without Loanguard (the same as with/ 

without payment protection). 

THE ROYAL BANK OF SCOTIA 

Customer Loan Repayment Tables 

WITH LOANGUARD 

Loan Period Amount of Loan Monthly Payment 

12 months £1000 £102·76 

£2000 £199·10 

24 months £2000 £109·67 

£3000 £162·07 

£5000 £268·84 

36 months £3000 £118·16 

£5000 £195·62 

£10 000 £379·24 

60 months £5000 £138·06 

£7500 £198·24 

£10 000 £263·88 

£15 000 £395·16 

WITHOUT LOANGUARD 


12 months £1000 £94·22 

£2000 £182·97 

24 months £2000 £98·81 

£3000 £146·19 

£5000 £242·61 

36 months £3000 £104·11 

£5000 £172·47 

£10 000 £335·27 

60 months £5000 £116·36 

£7500 £167·98 

£10 000 £223·61 

£15 000 £334·86 

1. From the table, write down how much you would have to pay back per month if you 

borrowed: 

(a) £1000 over 12 months with Loanguard. 

(b) £1000 over 12 months without Loanguard. 

Bank 

(c) £5000 over 24 months with Loanguard. 

(d) £3000 over 36 months without Loanguard. 

SCOTIA 

(e) £7500 over 60 months with Loanguard. 

(f) £15 000 over 60 months without Loanguard. 


2. What is the difference in your monthly payment if you borrowed £5000 over 36 months 

without Loanguard instead of with Loanguard. 

3. Work out the total amount to be repaid in the following examples: 

(a) £2000 over 12 months with Loanguard. 

(b) £2000 over 12 months without Loanguard. 

(c) £3000 over 24 months with Loanguard. 

(d) £10 000 over 36 months without Loanguard. 

(e) £10 000 over 60 months with Loanguard. 

(f) £15 000 over 60 months without Loanguard. 

4. The Hendry family borrowed £3000 (without Loanguard) from the Scotia Bank over a 

period of 36 months. 

(a) What was their monthly repayment? 

(b) What was the total amount that they had to repay over the 36 months? 

(c) What was the extra cost of the loan to the Hendry family? 

5. The Hendersons also borrowed £3000 from the Scotia Bank over a period of 36 months, 

but they took the loan with Loanguard. 

(a) What was their monthly repayment? 

(b) What was the total amount that they had to repay over the 36 months? 

(c) What was the cost of the loan to the Hendersons? 

(d) How much more did this method of payment cost the Hendersons, compared with the 

Hendry family? 

6. In the same way, work out the difference in the final cost of these loans: 

Loan A 

Mr Chic Old borrowed £10 000 over 60 months with Loanguard 

Loan B 

Mrs H. Nutt borrowed £15 000 over 60 months without Loanguard 

7. Now, examine the large tables in Exercise 4 and in Exercise 5 to solve this problem: 

Hamish wants to borrow £5000 over 3 years to help pay for a second-hand sports car. 

Naturally he does not want his monthly repayments to be high, so he decides not to take 

out any repayment protection plan. 

You have to advise Hamish, giving him reasons, as to whether he should go to the 

Building Society (page 9) or the Scotia Bank (page 10) for his loan. 


APPLICATIONS OF MATHEMATICS (INTERMEDIATE 1) 

Checkup Exercise for Straightforward Calculations in a Social Context 

1. Mr Samar earns £1260 per month. 

Mrs Ramas earns £291 per week. 

Calculate the annual earnings for both, and state who earns the greater annual salary. 

2. Isobel is a fashion designer. She works a long, basic 50-hour week at a rate of £68 per 

hour plus a limited amount of overtime at time and a half. 

Calculate her gross pay for a week in which she works her normal 50 hours, plus 20 hours 

overtime. 

3. Look at the payslip for Miss Dina Grubb and answer the questions which follow: 


Dina Grubb 657841 7 545H TT3575 X 


£980·50 £70·00 £25 £17 


£96 £150·54 £42 – 

Net Pay 

(a) If April was regarded as month 1, what month of the year was this payslip for? 

(b) Calculate: 

(i) Dina’s Gross Pay. 

(ii) Dina’s Total Deductions. 

(iii) Dina’s Net Pay. 

4. The McCann family borrowed £5000 from a Loan Company over two years with payment 

protection. 

Loan Company – Repayment Table 

WITH PROTECTAPLAN 

WITHOUT PROTECTAPLAN 

Loan Period Amount of Loan Monthly Payment Loan Period Amount of Loan Monthly Payment 

12 months £1000 £103·56 12 months £1000 £95·22 

£2000 £194·10 

£2000 £183·96 

24 months £2000 £110·69 24 months £2000 £99·81 

£3000 £163·08 

£3000 £145·19 

£5000 £267·74 

£5000 £241·60 

(a) How much was their monthly repayment? 

(b) What was the total amount that they had to repay over the 2 years? 

(c) What was the extra cost of the loan to the McCann family? 


LOGIC DIAGRAMS 


(a) 

(b) 

(c) 

(d) 

(e) 

(f) 

know the meaning of the terms: vertices (nodes) and arcs in a network 

diagram. 

interpret a simple network diagram. 

recognise statement boxes and decision boxes in a flowchart. 

interpret and use simple flowcharts which contain at least one decision box. 

enter given data and simple formulae into a spreadsheet as well as the 

replication of formulae and use the functions SUM and AVERAGE in a 

spreadsheet. 

design a simple spreadsheet. 



A. Network Diagrams 

The drawing on the right is called a network diagram. 

– the dots are called vertices or nodes. 

– the lines joining them are called arcs. 

Introductory Question Try to draw the network diagram (fig. 1) 

sticking to the following rules: 

(a) You must not lift your pen or pencil from start to finish. 

(b) You must not go over any line more than once. 

(c) You may go through a vertex (node) more than once. 

(d) You do not have to end up where you started from. 

Can it be done? 

Exercise 1 

node 

arc 

fig. 1 

Take your time with each network and be as careful as you can. 

You will be asked to return to it, so keep your sketches neat and handy. 

1. Which of the following networks can be drawn using the above four rules (a) to (d)? 

(a) (b) (c) 

(d) (e) (f) 

(g) (h) (i) 

(j) (k) (l) 


(m) (n) (o) 

(p) (q) (r) 

Make a list of all those networks which can be drawn. 

Make a list of all those networks which cannot be drawn. 

Now check your answers with those at the back – make any corrections needed to your 

list. 

5 

2. Now make a neat sketch of all the network diagrams (q) 

from question 1. 

Beside each node, neatly write its order. 

(This means ‘the number of lines coming from each point’) 

Question 1 (q) has been done for you. 

3. (a) Look back at the list of all the network diagrams from question 1 which you found 

could be drawn using the four rules. 

Can you make a guess as to the connection between those which can be done and the 

order of their nodes? 

(b) Now look at those which could not be drawn. What do you notice about their nodes? 

4. Copy and complete this statement: 

‘A network diagram can be drawn according to the four rules as long as every one of its 

nodes is ........’ 

5. Copy and complete this statement: 

‘A network diagram looks as if it cannot be drawn if every one of its nodes is ........’ 

(we shall see soon that this is not strictly true). 

6. (a) Can this simple network be drawn? 

(b) Make a sketch of it and fill in the order of its nodes. 

(c) What does this mean about your answer to question 5? 

In all the questions in Exercise 1 you may have noticed that all of the nodes were either odd 

or all of them were even. 

In Exercise 2, you are going to study networks which have a mixture of odd and even nodes. 

3 

3 

5 

1 

5 

This means 5 

lines come 

from this point 


Exercise 2 

1. Many people have four clothes poles in their back 

garden set out in the shape of a square. 

The clothes rope can be represented as shown 

in the network diagram. 

3 

3 

Is it possible to use one single piece of rope 

to form the pattern shown, without duplicating 

any part of itself? 

(i.e. can it be drawn using the four rules?) 

2. Look at this diagram: 

4 

3 3 

A 

Make a sketch of it, filling in the order of each node. 

C 

B 

(a) Try drawing it, using the 4 rules, by starting at 

(i) A (ii) B or C (iii) D 

Can it be drawn from any of these points? 

F 

D 

E 

(b) Try drawing it by starting at E or F. 

Can it now be done? 

3. Each of the following networks can be drawn, but to do so you must be careful which 

node you start from. 

Try them, listing the order of the nodes you had to start from. 

(a) (b) (c) 

3 

3 

3 

4 

4 

4 

2 2 1 

4 

4 

2 2 

(d) (e) (f) 

3 

1 

1 

3 

6 

3 

4 

3 

2 

2 

5 

2 

5 

4 

4 4 

2 


4. Look at the six networks in question 3. 

(a) How many odd nodes were there in each network? 

(b) Copy and complete the following statements: 

‘If a network has odd nodes, it can still be drawn as long as there are only ..... of them’. 

‘To draw a network with exactly .... odd nodes you must begin at ....................... and 

you will always end up at the ...........’ 

Check the answer given at the back. If you did not get it, write it down now. 

You now have a rule to decide if a network can be drawn, and if it can, where to start from. 

(These rules are given at the bottom of the page.) 

5. Make sketches of several network diagrams and decide, in advance, whether they can be 

drawn or not. 

You may like to work in pairs and check your answers. 

6. The Königsberg Bridges problem 

In the city of Königsberg, in the eighteenth century, there was a central island called 

Kneiphof, around which the River Preigel flowed before dividing in two. 

This divided the city into four parts (A, B, C and D) which were connected by seven 

bridges. 

KÖNIGSBERG 

C 

River Preigel 

Kneiphof 

A 

B 

D 

The residents of Königsberg spent their spare time trying to find a route which would 

involve them crossing each bridge exactly once. 

(a) Can it be done? If so, where should they start from and where would they finish? 

(b) Try using your rules given below to help decide if it can be done or not. 

(Don’t just draw it or say it can’t be done – give a reason.) 

Rule A A network can always be drawn if all of the nodes are even. You can start at any node and you 

will always end up back at your starting node. 

Rule B If there are only 2 odd nodes the network can still be drawn. You must start at one of the odd 

nodes and you will always end up at the other. 

Rule C If the network has more than 2 odd nodes, it can never be drawn, no matter where you try to 

start from. 


B. Tree Diagrams 

A tree diagram is a way of listing the various possibilities of events happening. 

For example, if you toss a coin twice, you can see, using the tree diagram, that there are four 

possibilities: 

(head,head) 

(head,tail) 

(tail,head) 

(tail,tail) 

N E 

O 

1 

P 

E N 

N 

Y 

First 

Toss 

Head 

Second 

Toss 

Head 

Tail 

Possibilities 

Head, Head 

Head, Tail 

Tail 

Head 

Tail, Head 

Exercise 3 

Tail 

Tail, Tail 

1. Of the four possibilities when tossing a coin twice, how many ways did: 

(a) only one head occur 

(c) two tails occur 

(b) at least one head occur 

(d) both tosses produced the same result? 

2. (a) Draw a tree diagram to show what happens when a coin is tossed three times. 

(b) How many different possibilities are there? 

(Note: (head, head, tail) is different from (head, tail, head)). 

(c) Of all the possibilities, how many ways could the following occur: 

(i) three heads (ii) two heads (iii) at least two tails (iv) all three the same? 

3. When offering someone coffee you could ask your guest two questions. 

‘Do you wish sugar or not?’ and ‘do you wish cream, milk or neither?’ 

(a) Show the possibilities by copying and completing this tree diagram. 

First 

Choice 

Second 

Choice 

cream 

Possibilities 

sugar, cream 

milk 

sugar 

neither 

no sugar 

(b) How many possibilities are there? 


4. A boy chooses from one of three red cards, then from one of two yellow ones. 

i o a n t 

Red cards 

Yellow cards 

His choices can be shown using a tree diagram as follows; 

Red Card Yellow Card Word? 

i 

i 

n 

n 

i n 

(a) Copy and complete the diagram. 

(b) How many possible ‘two-letter’ combinations are there? 

(c) How many of them read as sensible two-letter words? 

5. A girl chose one of two circular numbered discs, then one of three square ones, followed 

by one of two triangular ones. 

1 2 4 5 

6 8 9 

(a) Copy and complete this tree diagram to show all the possible three-digit numbers she 

could make from her choices. 

circular 

disc 

1 1 

(b) How many possible three-digit numbers can be formed? 

(Stick to the order, circular –> square –> triangular.) 

(c) Of all the possibilities, how many of the three-digit numbers are greater than 249? 

6. A three-sided spinner with the numbers 1, 2 and 3 on it 

is spun twice and the digit is recorded each time. 

(a) Use a tree diagram to show all the possible 

two-digit numbers which can be formed. 

3 

1 

2 


(b) How many possibilities are there? 

(c) How many of the possible two-digit numbers are EVEN? 

7. Repeat question 6, parts (a), (b) and (c), but this time 

using a four-digit spinner, spun twice. 

3 

4 

2 

1 

8. Mr Jones leaves his house and walks to the bank, via the Cross. 

New Street 

The 

Cross 

Abbey Way 

Upper High Street 

Lower High Street 

Mr Jones’ 

house 

South Drive 

Brier Road 

Bank 

(a) Show all the possible routes he can take using a tree diagram. 

New St 

House 

(b) How many ways are there altogether? 

(c) How many routes are there which: 

(i) pass along Abbey Way 

Startley 

Ardeer 

Coldrill 

(ii) do not go along New Street? 

9. This map shows several ways of driving from Startley to Endpoint through one of three 

towns: Ardeer, Brewster or Coldrill. 

A46 

(23) 

A19 

(27) 

A14 

(20) 

B111 

(15) 

M77 

(17) 

Brewster 

A72 

(15) 

A213 

(13) 

B52 

(17) 

B171 

(22) 

M74 

(12) 

B65 

(16) 

The distances between the towns are shown in brackets (in kilometres). 

Endpoint 


(a) Show the various routes using the tree diagram started below: 

A 

E 

S 

A14 

B111 

M77 

A46 

A19 

A 

B 

C 

C 

(b) How many different routes are there? 

(c) Which route is the (i) shortest? (ii) longest? 

C. Shortest Path Problems 

Exercise 4 

1. The Travelling Salesman problem. 

This map shows four towns with the roads 

and the distances between the towns given. 

A salesman leaves his office in Arluss and has 

to visit customers in the other three towns. 

He does not want to go through any town more 

than once and does not need to return to Arluss. 

A tree diagram can be drawn to study the various 

routes he can take: 

Arluss 

8 

Brierlee 

Drylie 

11 

Arluss 

Canrool 

7 

Drylie 

14 

8 

Brierlee 

11 

12 

14 Canrool 

(distances shown in miles) 

Drylie 

(a) Copy and complete the diagram showing all the routes he could take. 

(b) Which route is the shortest? 

(c) Which is the longest? 

(d) If he wished to return to the depot after visiting the customers, could he do it without 

going through any town more than once? 

2. Repeat all four parts of question 1 but this time start the journey from Brierlee. 


3. A pilot leaves Playville airport and has to drop passengers off at three surrounding 

airports, Queensridge, Ranloch and Stewarton. 

Ranloch 

128 km 

64 km 

56 km 

Stewarton 

71 km 

52 km 

Playville 

Queensridge 

95 km 

(a) Draw a tree diagram to show all the possible routes leaving from Playville and passing 

through the other three airports. 

(b) What is the best route to keep the distance travelled, and hence the fuel cost, as low as 

possible? 

19 

4. In an orienteering competition, contestants start at the 

2 

point marked S on the map shown opposite. 

They have to get their cards stamped at the four 1 

20 

checkpoints numbered 1, 2, 3 and 4 but can do so in 

21 

any order. 

22 

George has prepared in advance for the competition 14 

by running between every pair of points and he has 

17 4 

3 

estimated the time, in minutes, each part takes. 

16 

(These times are shown also on the map). 

One route is S –> 1 –> 2 –> 3 –> 4 (76 minutes) 

S 

18 

(a) Copy and complete the tree diagram to show all possible routes. 

He does not need to return to his starting point, S. 

S 

14 

16 

1 

4 

18 

3 

(b) There is a problem with one route. Which one is it not possible to do without going 

through one of the checkpoints more than once? 

(c) Which route should George take if he wants to try to win the competition? 


5. A van leaves a depot at Pudsley and makes deliveries at Quenton, Robbart, Sneddon and 

Tyrole before returning to the depot at Pudsley. 

Quenton 

4 km 

Robbart 

3 km 

Pudsley 

1 km 

2 km 

3 km 

2 km 

Tyrole 

5 km 

Sneddon 

What is the ideal route to take to make the distance travelled, in kilometres, the shortest 

possible? 

The driver may go through a town more than once if he finds it helps. 

(A tree diagram will help.) 

6. Four towns are situated along the 

Dudley Exham Fortnam Gourley 

main A12 road. They are Dudley, 

Exham, Fortnam and Gourley. The 

map shows the distances between them. 

18 11 

(distances in kilometres) 

23 

(a) How far is it from 

(i) Dudley to Fortnam? 

(ii) Exham to Gourley? 

(b) Copy and complete this distance chart, 

showing the distances between each pair of towns. 

18 

Dudley 

Exham 

Fortnam 

Gourley 

7. This distance chart indicates how far it is between a series of four towns, joined by six 

roads. 

Hegton 

Iswell 

22 

35 14 

18 26 20 

Jarrow 

Kilday 

Hegton 

22 

Iswell 

Kilday 

20 

(distances in kilometres) 

Jarrow 

(a) Copy the map and fill in the missing distances using the distance chart. 

(b) How much farther is it from Hegton to Jarrow via Kilday than via Iswell? 


8. This chart shows distances between four local mountain peaks in kilometres. 

(a) Copy and complete the map below showing the distances. 

(Use the distance table to help.) 

Hightops 

The Knot 

3·2 

4·1 

Hightops 

6·0 

The Knot 

5·4 3·7 1·9 

Drymun 

Kingsfoot 

Kingsfoot 

Drymun 

(b) A hill walker climbs to the top of Hightops mountain. From there he intends to climb 

the other three. 

Draw a tree diagram to show the various routes and determine the ideal route to take 

in order that the climber takes the shortest possible route. 

(He obviously does not want to climb any mountain more than once!) 

Decision Trees (or Keys) are used mainly in science to determine the species of plants or 

animals, or to decide on the type of rock found, based on a variety of yes/no questions. 

9. Use the key to decide what kind of 

plants the following properties relate 

to: 

Six related plants 

(a) It has white flowers and thorns. 

Thorns 

No thorns 

(b) It has no thorns and four petals. 

(c) It has five petals, no thorns and 

white flowers. 

White 

flowers 

Trailing 

rose 

White flowers 

tinged with pink 

Dog rose 

Five 

petals 

Four petals 

Common 

tormentil 

(d) It has thorns and its white 

flowers are tinged with pink. 

White flower 

White flower tinged 

with pink 

Yellow flower 

(e) It has no thorns, its flowers are 

white tinged with pink, and it 

has five petals. 

Pear 

Crab apple 

Silver weed 


10. A student made some sketches of insects on a field study trip. 

Use the key to determine the species of each. 

Start here 

(a) 

legs not jointed 

caterpillar 

all legs jointed 

4 pairs of legs 3 pairs of legs 

body in 2 parts 

body in 1 part 

(b) 

6 legs 

jointed 

spider 

legs longer 

than body 

harvestman 

legs shorter 

than body 

mite 

(c) 

body striped 

body not striped 

back end of 

body pointed 

back end of 

body not pointed 

body small 

and green 

body not small 

and green 

(d) 

wasp 

bee 

greenfly 

body round 

and spotted 

body long 

and not spotted 

(e) 

(f) 

6 legs 

jointed 

ladybird 

(g) (h) (i) 

wings narrow 

cranefly 

wings wide 

feelers 

club-shaped 

butterfly 

feelers not 

club-shaped 

moth 


D. Flowcharts (Flow Diagrams) 

Exercise 5 

1. This flowchart tells a builder’s merchant how to calculate how 

much VAT to add on to your bill for the item(s) you have just 

purchased. 

Calculate the extra VAT to be added when you buy: 

(a) a load of bricks priced £60 

(b) a lorry load of slates priced £640 

(c) roofing timber at £370. 

START 

Multiply cost by 7 

Divide answer by 40 

STOP 

2. This simple flowchart is used to changefrom degrees Fahrenheit 

START 

to degrees Celsius. Use the flowchart to change these 

temperatures from °F to °C: 

Subtract 32 


(a) 59°F (b) 32°F 

Multiply by 5 

STOP 

(c) 212°F (d) –4°F 

(e) –40°F. 

3. This flowchart tells you how to 

calculate, approximately, the distance a 

lightning strike is from your position. 

Use the flowchart to calculate how far 

away, in metres, a lightning bolt is if 

the number of seconds between the 

flash and the peal of thunder is: 

(a) 3 seconds 

(b) 4·5 seconds 

(c) 6·2 seconds. 

START 

Start your stopwatch when you see the flash 

Stop your stopwatch when you hear the thunder 

Multiply the number of seconds by 330 

Read off the distance in metres 

STOP 

4. Put these boxes together in the correct order to make a flowchart which shows how to 

phone a plumber to get a repair done. 

STOP 

Replace receiver 

Lift receiver 

Look up plumber’s number 

Discuss the problem on phone 

START 

Arrange for plumber to comeDial number 


Exercise 6 

1. This flowchart shows the 

cost of buying tickets for a 

show, depending on whether 

it is a group of adults or a 

group of children. 

Use the flowchart to 

calculate the cost of the 

following: 

No 

START 

Number of tickets 

Is it 

a group of 

adults? 

Yes 

Multiply by £2·50 Multiply by £3·75 

(a) a group of six adults 

(b) a group of four children. 

STOP 

2. This one shows how to work out the cost 

of hiring a van for a number of hours. 

Use it to work out the cost of hiring the 

van for: 

(a) three hours on a Sunday 

Yes 

START 

Is it 

the weekend? 

No 

(b) five hours on a Tuesday. 

Multiply hours 

by £8·50 

Multiply hours 

by £6·50 

Add on £10·00 

STOP 

3. This flowchart shows how to work out the 

cost of hiring a cement mixer for several 

days. 

How much would it cost to hire the cement 

mixer for: 

(a) 3 days 

(b) 5 days 

No 

Multiply days 

by £11·50 

START 

Is it 

for more than 5 

days? 

Yes 

Multiply days 

by £9·50 

(c) 10 days? 

Add on £6·75 Add on £4·50 

STOP 


4. This flowchart is used to calculate the 

commission and final salary due to a 

salesperson depending on his or her sales 

in a year. 

Commission = 

0·08 × Sales 

No 

START 

Are the 

annual sales over 

£50 000? 

Yes 

Commission = £4 000 + 

0·10 × (Sales –£50 000) 

Use it to calculate the salary due to: 

(a) Alistair, who sold £42 000 of goods. 

(b) Claire, who sold £50 000 of goods. 

(c) Cheryl, who sold £62 000 of goods. 

Add on £17 500 

STOP 

5. This more complicated flowchart shows 

how much extra a driver has to pay in car 

insurance. 

Use the flowchart to calculate the final 

insurance premium due by: 

(a) Andrew, aged 17, having had no 

accidents at all and whose basic 

premium was £480 per year. 

START 

Are you 

under 18 years 

of age? 

No 

Yes 

Add 30% to 

basic rate 

(b) Sheila, aged 24, with no previous 

accidents. Her basic premium was 

£280 per year. 

(c) Donald, aged 30, with one accident 

6 months ago. His basic premium 

was £180 per year. 

Are you 

under 25 years 

of age? 

No 

Yes 

Add 20% to 

basic rate 

(d) Julian, aged 22, with one accident 

15 months ago. His basic premium 

for his sports car was £880. 

Any 

accidents in last 2 

years? 

No 

Yes 

Add a further 25% to 

basic rate 

STOP 


6. Look at this flowchart. 

It is used to calculate the averages of a 

set of values. 

(a) Use it to calculate the average of: 

23, 54, 61, 18, 44 

(b) Use it to calculate the average of: 

4·7, 3·8, 6·2, 8·1. 

START 

Set 

TOTAL = 0 

Read first 

value 

Add this value 

to TOTAL 

Read next 

value 

Are 

there any more 

values? 

Yes 

No 

Divide by 

number of values 

STOP 

7. This flowchart is used to decide 

a pupil’s Standard Grade award 

in their prelim exam, depending 

on their mark. 

The Maths exam was out of 80. 

Use the flowchart to calculate 

the grades of the following: 

(a) Ted scored 36 

(b) Bill scored 52 

(c) Ruth scored 72 

(d) Iain scored 40 

START 

Read in name and mark 

Multiply mark by 5 


Is 

score over 

75? 

Yes 

No 

Is 

score over 

55? 

Yes 

No 

Grade = 3 Grade = 4 Grade = 7 

Read next 

name and 

mark 

(e) Susie scored 44 

(f) John scored 48. 

Any 

more names 

on list? 

No 

Yes 

STOP 


8. This flowchart is used to decide if a particular 

year is a leap year or not. 

START 

Use it to decide which of the following are 

leap years: 

Enter the year 

(a) 1966 

(b) 1972 

(c) 1800 

(d) 1940 

No 

Is it 

divisible 

by 4? 

Yes 

Is it 

divisible by 

100? 

No 

(e) 1900 

Yes 

(f) 2000. 

This is a 

leap year 

Yes 

Is it 

divisible by 

400? 

No 

This is not a 

leap year 

STOP 

E. Spreadsheets 

apple File Edit Format Calculate Option View ? 

Entering Data 

The following exercise assumes you are 

working individually, but you may try it 

as a group or as a whole class. 

Exercise 7 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

A B C D E F G 

1. A group of five First Year pupils are comparing their marks in three of their maths tests. 

David Smith scored 62%, 81% and 79%. Brian Jones scored 63%, 59% and 91%. 

Alan Young scored 71%, 83% and 65% Iain Taylor scored 73%, 76% and 79%. 

John Hughes scored 59%, 62% and 71%. 

(a) Open up a new spreadsheet and type in the following headings: 

cell A1 – 1st name cell B1 – 2nd name 

cell C1 – Test 1 cell D1 – Test 2 cell E1 – Test 3 

1 

2 

A B C D E 

1st name 2nd name Test 1 Test 2 Test 3 


(b) Fill in the five names along with the marks, starting in cell A3. 

1 

2 

3 

A B C D E 


David Smith 62 81 79 

(c) Save this on to disc for use later. Call it Ex7, Qu1. 

(d) Take a printout of your spreadsheet and keep it. 

2. A glazier is cutting a series of rectangular panes of glass for six customers. He is going to 

calculate the area and cost of each piece eventually. 

(a) Open a new spreadsheet with the headings: 

cell A1 – Customer cell B1 – length (cm) cell C1 – breadth (cm) 

(b) Fill in the following customer details, starting in cell A3: 

Mr Davies – 60 cm by 80 cm Mrs White – 90 cm by 120 cm 

Mr Gordon – 210 cm by 160 cm Mrs Wylie – 75 cm by 160 cm 

Mr Rivers – 130 cm by 110 cm Mrs Jones – 80 cm by 150 cm 

(c) Your spreadsheet should look like this. 

(d) Save this for later under Ex7, Qu2. 

(e) Take a printout of your spreadsheet 

and keep it. 

1 

2 

3 

A B C 

Customer length breadth 

Mr Davies 60 80 

3. A fruit shop computerises its pricing. 

(a) Open a new spreadsheet and enter the headings: 

cell A1 – fruit 

cell B1 – weight (kg) 

cell C1 – cost / kg 

(b) Fill in Mr Stevenson’s fruit order, starting at cell A3. 

Mr Stevenson: 2·5 kg of apples 

1·5 kg of oranges 

0·5 kg of grapes 

3 kg of bananas 

0 kg of pears 

0·75 kg of peaches. 

(c) Save it under Ex7, Qu3. 

(d) Take a printout and keep it. 

1 

2 

3 

fruit cost / kg 

apples £1·52 

oranges £1·85 

grapes £1·68 

bananas £1·20 

pears £1·46 

peaches £1·92 

A B C 

fruit (kg) weight cost / kg 

apples 2.5 £ 1.52 


4. Bloggs Engineers employ six workers. They wish to calculate the weekly wages of the 

workforce. 

Fred works a basic 40 hours and 6 hours overtime. His basic rate of pay is £6·50 / hour. 

Tom works a basic 38 hours and 4 hours overtime. His basic rate of pay is £6·20 / hour. 

Gina works a basic 36 hours and 5 hours overtime. Her basic rate of pay is £4·80 / hour. 

Alex works a basic 39 hours and 4 hours overtime. His basic rate of pay is £5·10 / hour. 

Sara works a basic 40 hours and 2 hours overtime. Her basic rate of pay is £6·40 / hour. 

Dave works a basic 32 hours and 0 hours overtime. His basic rate of pay is £5·30 / hour. 

(a) Open a new spreadsheet and enter the headings: 

in cell A1 – Name 

in cell C1 – Basic Hrs 

in cell B1 – Rate of pay 

in cell D1 – Overtime Hrs 

(b) Now enter the details for the six workers starting at cell A3. 

(c) Save it under Ex7, Qu4 and take a printout of it. 

1 

2 

3 

A B C D 

Name Rate of Pay Basic Hrs Overtime Hrs 

Fred £ 6.50 4 0 6 

Formulae and Calculations in Spreadsheets. 

Exercise 8 

1. Open your spreadsheet for Ex7, Qu1. It should look like this. 

1 

2 

3 

4 

5 

6 

7 

A B C D E F G 


David Smith 62 81 79 

Brian Jones 63 59 91 

Alan Young 71 83 64 

Iain Taylor 73 76 79 

John Hughes 59 62 71 

(a) Put two new headings in: cell F1 – TOTAL 

cell G1 – AVERAGE. 

(b) Tap on cell F3 and type in the formula: = C3 + D3 + E3 (return) 

(cell F3 should now show 222). 

(c) Copy the formula in cell F3 down into cells F4, F5, F6 and F7. 

(Check that these totals are correct). 

(d) Tap on cell G3 and type in the formula: = F3 / 3 (return) (use ÷ or /) 

(cell G3 should now read 74). 

(e) Copy the formula from cell G3 down onto cells G4, G5, G6 and G7. 


(f) Print it out and use SAVE AS to save this under the new name Ex8, Qu1. 

This means you keep the original as Ex7, Qu1. 

(g) Try playing around with your spreadsheet by changing some of the test marks or by 

adding on one or two extra pupils’ names and scores. You will have to duplicate (copy 

down) the formulae if you wish the calculations done. 

2. Open up your spreadsheet, Ex7, Qu2. 

It should look like this. 

(a) Enter the heading AREA in cell D1, 

and the heading Cost (£) in cell E1. 

(b) In cell D3, enter a formula which 

will calculate the area of glass using 

cells B3 and C3 (return). 

(c) Copy this formula down into cells 

D4 to D8. 

1 

2 

3 

4 

5 

6 

7 

8 

A B C D E 


Mr Davies 60 80 

Mrs White 90 120 

Mr Gordon 210 160 

Mrs Wylie 75 160 

Mr Rivers 130 110 

Mrs Jones 80 150 

(d) The glass costs £0·005 per cm 2 (i.e. 1 /2 a pence). In cell E3, enter a formula, using 

cell D3, to calculate the cost of the glass for Mr Davies, and copy this formula down 

onto cells E4 to E8. 

(e) In cell D10, type in TOTAL =, and in cell E10 enter a formula which will give the 

total of all the monies in cells E3 to E8. 

(f) Print it out and use SAVE AS to save your new spreadsheet under Ex8, Qu2. 

3. Open up your spreadsheet, Ex7, Qu3. It should look like this: 

1 

2 

3 

4 

5 

6 

7 

8 

A B C D 

fruit weight (kg) cost/kg 

apples 2.5 £ 1.52 

oranges 1.5 £ 1.86 

grapes 0.5 £ 1.68 

bananas 3 £ 1.20 

pears 0 £ 1.46 

peaches 0.75 £ 1.92 

(a) In cell D1, type in the heading COST. 

(b) Enter a formula in cell D3 to calculate the cost of the apples and copy this 

down into cells D4 to D8. 

(c) As in Question 2, use cells C10 and D10 to calculate the TOTAL bill. 

(d) Print your spreadsheet and save it under Ex8, Qu3. 

(e) Try changing weights and costs and see the results. 


4. Open up your spreadsheet, Ex7, Qu4. It should look like this: 

1 

2 

3 

4 

5 

6 

7 

8 

A B C D E F G 

Name Rate of Pay Basic Hrs Overtime Hrs 

Fred £ 6.50 4 0 6 

Tom £ 6.20 3 8 4 

Gina £ 4.80 3 6 5 

Alex £ 5.10 3 9 4 

Sara £ 6.40 4 0 2 

Dave £ 5.30 3 2 0 

(a) In cells E1, F1 and G1, type in the headings, Basic Pay, Overtime Pay, Gross Pay. 

(b) In cell E3, enter a formula to calculate Fred’s basic pay and copy it down to cells E4 to 

E8. 

(c) In cell F3, enter a formula to calculate Fred’s overtime pay and copy it down to cells 

F4 to F8. (Overtime is paid at ‘time and a half’ – i.e multiply it by 1·5.) 

(d) In cell G3, enter a formula to calculate Fred’s total gross pay, using cells E3 and F3 

and copy it down to cells G4 to G8. 

(e) Use cells F10 and F11 to calculate the TOTAL Wage Bill for Bloggs. 

(f) Save your spreadsheet under Ex8, Qu4 and take a printout of it. 

Special Formulae – SUM and AVERAGE – and designing a spreadsheet. 

Exercise 9 

1. Open up your original spreadsheet from Exercise 7 (Ex7, Qu1) 

(a) This time, type the heading AVERAGE in cell F1. 

(b) Use the AVERAGE function in cell F3, to find the average of the numbers in cells C3, 

D3 and E3. 

(In Claris, it is in the menu bar under Edit – Paste Function). 

(c) Copy this formula down into cells F4 to F8, and check it gives the correct answers. 

(d) Save it, using SAVE AS, under Ex9, Qu1, and take a printout of it. 

2. (a) Open up a new spreadsheet. Somewhere in the spreadsheet, not necessarily starting at 

cell A1, type in the 5 headings: 

Customer, Item 1, Item 2, Item 3, and Total. 

(b) Fill in the following three customers’ details in an appropriate place: 

Mr Jones – £3·85, £9·62, £4·75 

Mrs Paton – £6·94, £5·73, £11·64 

Mr Wilson – £9·85, £7·24, £1·68 


(c) Use the SUM function, (again in the menu bar under Edit – Paste Function), to 

calculate the total for Mr Jones’ three items and copy this formula down to include the 

other two customers. 

(d) In an appropriate cell, type in the heading TOTAL =, and in the cell next to it use the 

SUM function again to find the TOTAL of the above three Totals. 

(e) Save your new spreadsheet, using SAVE AS, under the new name, Ex9, Qu2, and take 

a printout of it. 

3. (a) Open up a new spreadsheet and enter a line of 4 or 5 numbers starting at cell A1. 

(b) Try some of the various ‘functions’, from the Paste Function in the Edit menu to see 

what they do to your numbers, and how to use them. 

(c) There is no need to save this spreadsheet unless you wish to do so. 

4. Go back and open the spreadsheet from Exercise 8, (Ex8, Qu4). 

(a) Extend the spreadsheet to include two new headings, 

‘Deductions’ and ‘Net Pay’. 

(b) In the appropriate cells, add in the six employees’ details: 

Employee Fred Tom Gina Alex Sara Dave 

Deductions £92·40 £75·30 £57·79 £52·72 £68·77 £31·42 

(c) Use a standard function, or devise your own to calculate Fred’s net pay from his gross 

pay and his deductions, and copy your formula down to find the net pay of the other 

five. 

(d) Save it under Ex9, Qu4, and take a printout of your spreadsheet. 

5. Devise a simple spreadsheet which will produce the following sets of values: 

1 

2 

3 

4 

5 

A B C 

x x squared x cubed 

1 1 1 

2 4 8 

3 9 27 

11 1 

12 2 

9 81 729 

10 100 1000 

Note: you should use some of the functions or devise your own. 


6. Imagine you were the treasurer for the school tuck shop which sells crisps (22p), Big Bars 

(31p), Tukins (28p), Cans (25p) and Juice (18p). 

Sales for Week 1 were – 63 Crisps, 22 Big Bars, 46 Tukins, 36 Cans and 29 Juice. 

(a) Design and devise a spreadsheet which works out the total income due (in £’s) for 

week 1 and take a printout of the figures. Save it under Ex9,Qu6. 

(b) Now use it to work out the money due in Week 2 when the sales were: 

31 Crisps, 42 Big Bars, 33 Tukins, 32 Cans and 24 Juice. 

7. (a) Devise a spreadsheet which will produce the ×2, ×3, ×4, ×5, ×6, ×7, ×8, ×9 and ×10 

tables. 

It should have headings and look like this: 

1 

2 

3 

4 

A B C D 

number 2 times 3 times 4 times 

1 2 3 4 

2 4 6 8 

(b) Take a printout and save it as Ex9, Qu8. 

8. Thomson’s Builders’ Merchants record their expenses and income each quarter. 

(a) Produce a spreadsheet to help them. It should look like this: 

Thomson’s Builders 

Jan Feb Mar 

EXPENSES 

Wages £11000 £11000 £12100 

Rent £250 £250 £250 

Gas £320 £280 £240 

Electricity £290 £285 £260 

Materials £2130 £1940 £2050 

Petty Cash £60 £85 £70 

TOTAL ***** ***** ***** 

INCOME (sales) 

TOTAL £18640 £17960 £20470 

BALANCE ***** ***** ***** 

Quarterly Balance ***** 

(b) Write down the formulae which will do the calculations in the cells marked *****. 

(c) Enter the formulae into the spreadsheet and make sure it is as neat and presentable as 

possible, with bold and centralised cells where appropriate. 

(c) Take a printout and save it as Ex9, Qu9. 



Checkup Exercise for Logic Diagrams 

1. Try to decide, in advance, which of the following six diagrams can be drawn. 

For those which are, state whether you can start at any node, or if not, say which node you 

must start at. 

Draw those which can be drawn and say why the others can’t be drawn. 

(a) (b) (c) 

(d) (e) (f) 

2. For the Königsberg Bridges problem (see Exercise 2, qu 6), build one extra bridge over the 

river in such a way as to solve the problem. 

(For your solution say where a person would then have to begin their walk and where they 

would end up.) 

3. A board, with a spinner on it, has five equally spaced 

sectors numbered 1, 2, 3, 4 and 5. 

1 

3 

A triangular die with the numbers 1, 2 and 3 is rolled. 

4 

2 

5 

2 

3 

1 

Complete the tree diagram which shows all the totals you can get 

when the spinner is spun, the die rolled and the values added together. 

(a) How many possibilities are there? 

(b) If you were asked to bet on an odd or an even total, which would 

you be better to choose, and why? 


4. A baker’s van leaves its depot in Coolridge 

and has to make deliveries to Duckworth, 

Edfu and Fordbank. 

Coolridge 

27 km 

Duckworth 

(a) Draw the tree diagram to show all the 

possibilities. 

24 km 

17 km 

16 km 

C 

(b) Which is the most efficient? (shortest!) 

Edfu 

19 km 

Fordbank 

5. Copy and complete the distance chart representing the map shown below. 

Ashvie 

31 

Dakeley 

Ashvie 

Cromer 

Blythe 

Dakeley 

13 km 

15 km 

16 km 

20 km 

18 km 

Cromer 

Blythe 

6. This flowchart shows how the supermarket chain 

SAFEDA works out the hourly rate of its part– 

time workers. 

(a) What would the hourly rate be for: 

(i) Bob who is 16 

(ii) Jenny who is 19 

(iii) Ted who is 24? 

(b) George is 20 years old and 

worked 12 hours for 

SAFEDA last week. 

What should his pay be? 

Yes 

START 

Note the age 

Is 

worker under 

17? 

No 

Is 

worker under 

21? 

No 

Yes 

Hourly rate is £3·20 

+ (age – 16) x 0·20 

Hourly rate is £4·20 

+ (age – 21) x 0·30 

Hourly rate is 

£3·20 

STOP 


7. 

Item Cost (before VAT) 

Bag of cement £3·60 

100 common bricks £9·20 

Bag of sand £2·40 

100 facing bricks £19·60 

Pack of 20 slates £16·40 

The spreadsheet below has been designed to calculate the VAT (at 17·5%) and the final 

price of the above items sold at a builder’s merchant. 

1 

2 

3 

4 

A B C D 

Item Basic Cost V.A.T. Final Cost 

Cement £ 3.60 

C Bricks £ 9.20 

(a) Write down the formula which would go into cell C3 to calculate the VAT and the 

formula for cell D3 which gives the final cost. 

(b) Optional: 

If you have access to a computer you could produce and type up the spreadsheet and 

take a printout of it. 

(i) In your spreadsheet, put the headings Item, Basic Cost, VAT and Final Cost in 

Cells A1, B1, C1 and D1. 

(ii) In Cells A3 to A7, type in the Item Names. 

(iii) In Cells B3 to B7 put in the item costs. 

(iv) Devise a formula for calculating the VAT and enter it in Cell C3. 

Copy this down into Cells C4 to C7. 

(v) In Cell D3 enter a formula to calculate the final cost with the VAT added on. 

Copy this down into Cells D4 to D7. 

(vi) Save your spreadsheet and take a printout of it. 


SCALE DRAWINGS AND SURFACE AREAS OF SOLIDS 


(a) 

(b) 

(c) 

(d) 

(e) 

(f) 

(g) 

recognise the eight main compass points. 

know and understand three-figure bearings. 

measure bearings in scale drawings. 

plot points using three-figure bearings. 

construct and interpret scale drawings. 

recognise cylinders and triangular prisms from their nets. 

calculate surface areas of triangular prisms and cylinders. 



A. Points of the Compass 

N 

Exercise 1 

1. Copy and complete the eight point compass. 

W 

SE 

2. You are at the entrance to the Elephant House in a zoo. 

Bears 

Lions 

N 

Chimps 

Giraffes 

Elephant 

House 

Elephant 

House 

Camels 

Goats 

Penguins 

Seals 

(a) Which animal would you see if you faced: 

(i) North (ii) West (iii) South-West (iv) North-East? 

(b) From the Elephant House, in which direction are the: 

(i) penguins (ii) camels (iii) bears? 

3. The map shows some of the villages on RAINBOW Island. 

Each village is represented by a capital letter. 

(a) If you travel South-East from H, which 

place do you reach? 

(b) If you travel from D to M, in which 

direction are you moving? 

(c) Which place is North-West of M? 

(d) Which place is East of W and North 

West of C? 

(e) Which two places lie due West of C? 

W 

T 

B 

M 

RAINBOW 

Island 

H 

D 

E 

P 

R S C 


B. Three-Figure Bearings – Recognising and Measuring 

Exercise 2 

1. Three-figure bearings are measured 

clockwise from the North. 

Kildour 

N 

Write down the three-figure bearing of each 

town: 

e.g. Easton is on a bearing of 087° 

100° 

87° 

45° 

Easton 

108° 

Aberdoch 

2. Write down the three-figure bearing for: 

(a) East (b) West (c) South 

(d) South-East (e) North-East (f) South-West. 

Prestwich 

3. What compass direction do these three-figure bearings represent: 

(a) 045° (b) 180° (c) 315° (d) 000°? 

4. A ship is sailing northwards. What will its new compass direction be after a turn of: 

(a) 180° (b) 45° clockwise (c) 225° clockwise 

(d) 45° anti-clockwise 

(e) 225° anti-clockwise? 

5. The picture (not drawn to scale) shows three lighthouses. 

Calculate and write down the bearings of the following: 

(a) Pott Island from Muir Head 

(b) Wind Point from Pott Island 

(c) Wind Point from Muir 

Head 

(d) Pott Island from Wind 

Point. 

Muir 

Head 

N 

65° 

33° 

N 

Pott 

Island 

150° 

N 

(Do NOT use a protractor.) 

30° 

52° 

Wind 

Point 


6. YOU MAY USE A PROTRACTOR IN THIS QUESTION. 

(Do not mark the drawing in any way.) 

Invernorth 

(a) Which town is due North of 

Middleville? 

Wesbay 

N 

Eastcorner 

(b) Measure and write down the threefigure 

bearing of Eastcorner from 

Middleville. 

MIDDLEVILLE 

(c) Measure and write down the threefigure 

bearing of: 

Southfork 

(i) Wesbay from Middleville 

(ii) End Point from Middleville. 

End Point 

Southview 

7. Trevor is taking part in an orienteering competition. The map shows his course. 

N 

N 

N 

1 

N 

3 

N 

2 

Start 

Finish 

4 

(a) COPY this table. 

Stage 

Start –> 1 

1 –> 2 

2 –> 3 

3 –> 4 

4 –> Finish 

Bearing 

(b) Measure each bearing with a PROTRACTOR and put the results in your table. 


8. For each of these golf holes: 

(a) Use a protractor to measure the bearing of each green from the tee. 

(b) Find the actual length of the hole if the scale of the drawings is – 

‘1 centimetre represents 50 metres’. 

N 

Tee 

Hole 2 

N 

Hole 1 

Tee 

C. Three-Figure Bearings – Drawing (YOU NEED A PROTRACTOR) 

Exercise 3 

1. Draw three North lines, like this, spaced out across a page of your jotter. 

Use these North lines to draw: a bearing of 070°, a bearing of 225° and 

a bearing of 330°. 

N 

2. Two yachts are sighted off the coast of Pembroke Isle. The Marie Rose is on a bearing of 

075° and is a distance of 80 km from Pembroke Isle. The Petal bears 090° and is also 80 

km from Pembroke Isle. 

N 

Marie Rose 

Pembroke Isle 

Petal 

(a) In the middle of your page: use a scale of 1 cm to 10 km to make a scale drawing. 

(b) How far apart are the two yachts? (Give your answer in kilometres.) 


3. A plane flies on a bearing of 115° for 150 km. 

(a) Use a scale of 1 cm to 15 km to make a 

scale drawing. 

(b) Find from your drawing how far SOUTH 

the plane is from the airport. 

Airport 

? 

N 

150 km 

4. The early morning shuttle flies from London to Glasgow via Edinburgh. 

The diagram below shows the direct route for the Edinburgh to Glasgow part of the 

journey. The distance is 70 km, and the bearing is 260°. 

N 

Glasgow 

70 km 

Edinburgh 

260° 

(a) Use a scale of 1 cm to 10 km to make a scale drawing. 

(b) Measure the bearing from Glasgow to Edinburgh. 

5. The Sannox sails 18 km on a bearing of 070°, 

then 12 km on a bearing of 160°. 

N 

N 

(a) Use a scale of 1 cm to 2 km to make a scale 

drawing. 

(b) How far is the Sannox from its starting 

point in kilometres? 

N 

18 km 

12 km 

Finish 

(c) What bearing should it follow to return 

home? 

Start 

6. 

Devil’s Island 

N 

N 

A 

060° 

B 

315° 

Town B is due East of town A. 

Devil’s Island is on a bearing of 060° from A and a bearing of 315° from B. 

Use this information to make your own scale drawing showing where Devil’s Island lies. 


7. Control tower D is 100 km due West of control tower C. 

The bearing of a hot air balloon is 155° from C and 235° from D. 

N 

N 

C 

D 

(a) Use a scale of 1 cm to 10 km to make a scale drawing, showing the position of the 

balloon. 

(b) How far is the balloon from point C (in km)? 

(c) How far is the balloon from point D (in km)? 

(d) What is the bearing of point C from the balloon? 

8. Another hot air balloon is tethered by two ropes attached to two points, P and Q, which are 

60 metres apart. Q is due East of P. 

The balloon is on a bearing of 050° from P and 300° from Q. 

N 

B 

N 

P 

60 m 

Q 

(a) Make a scale drawing of the balloon and the ropes. 

(Use a scale of 1 cm to 10 m) 

(b) Use your scale drawing to calculate how far the balloon is from point Q. 

9. The diagram shows two coast guard stations. The one at Barnes Point (B) is 60 kilometres 

due East of the one at Able Key (A). 

A boat at P is in trouble and sends out a distress signal. 

The boat is found to be on a bearing of 115° from 

Able Key. 

At the same time the boat is on a bearing 230° 

from Barnes Point. 

N 

N 

(a) Use a ruler and protractor to make a scale drawing. 

Use a scale of 1 centimetre to 5 kilometres. 

A 

60 km 

B 

(b) How far is the boat from Able Key? 

P 


D. Scale Drawing 

Exercise 4 

1. This map shows a part of the town of Adensfield. 

Church 

Garage 

Traffic 

Island 

Nursery 

School 

Roundabout 

Bank 

Scale: 1 centimetre represents 30 metres 

(a) Measure the distance from the traffic island to the roundabout in centimetres. 

(b) Find the actual distance from the traffic island to the centre of the roundabout. 

2. The scale of this drawing of a garage is: 1 centimetre represents 0·5 metres. 

? metres 

GARAGE 

? metres 

What are the real length and breadth of the garage in metres? 


3. Shown is a scale drawing of a living room. 

50 cm 

(a) Use your ruler to measure the length of AB (in cm). 

(b) Write down what the scale must be. 

8 metres 

(c) How many of these square carpet tiles are needed to 

cover the floor? 

50 cm 

50 cm 

A 

10 metres 

B 

4. Shown below is a scale drawing of Mrs Jamieson’s garden. 

The scale is 1 centimetre = 2 metres. 

Hut 

Flower bed 

Pond 

Vegetable plot 

Garden wall 

(a) Measure the distance from the tree to the door of the hut (in cm). 

(b) Write down the real distance from the tree to the door of the hut in metres. 

(c) Find the real length and breadth of the vegetable plot in metres. 

(d) Find the real length and breadth of the flower bed in metres. 

(e) Find the length of the pond in metres. 

(f) How long is the garden wall in metres? 


5. Make scale drawings of the following and use them to find the height of the buildings: 

(a) 

(b) 

Scale 1 cm to 20 m 


100 m 

40° 

50° 

80 m 

(c) 

(d) 



25° 

150 m 

75° 

40 m 

6. A building casts a shadow 25 metres long when the angle of 

elevation of the sun is 38°. 

(a) Using a suitable scale, make a scale drawing. 

38° 

(b) Find the height of the building. 

7. The flag pole casts a shadow 12 metres long when the angle of 

elevation of the sun is 41°. 

Make a scale drawing with a scale of 1 cm to 0·5 m to find the 

height of the flag pole. 

41° 

12 m 

8. The dragon looks down on the bones from a window which 

is 5 metres above the ground. 

The angle of depression is 70°. 

5 m 

70° 

Bones 

Using a scale of 1 cm to 1 m, make a scale drawing and find 

how far the bones are from the foot of the tower. 


9. A ship leaves Port P on a bearing 058° for 30 km to 

Island I. 

A storm is forecast, so at I the captain changes 

course and sails on a bearing of 118° for 20 km to 

Port Q. 

I 

N 

118° 

N 

(a) Make a scale drawing. 

(b) Calculate the distance 

between ports P and Q. 

P 

N 

58° 

30 km 

58° 62° 

20 km 

Q 

10. (a) Make a scale drawing of the following 

sailboat journey: 

1st Leg – Boat leaves the Marina and 

sails on a bearing of 115° for 

12 km to Buoy 1. 

2nd Leg – Boat leaves Buoy 1 and sails 

for 8 km on a bearing 225° for 

Buoy 2. 

3rd Leg – Sails from Buoy 2 back to the 

Marina. 

(b) Calculate the length (in kilometres) of the 3rd leg of the journey by using your scale 

drawing. 

(c) Measure and write down the bearings of the boat as it left Buoy 2 to sail to the Marina. 

11. Using suitable scales, make a scale drawing of each of these and answer the question 

which follows: 

N 

(a) Cliff top 

(b) 

N 

Marina 

N 

Buoy 2 

N 

Buoy 1 

90 m 

Sea 

? 

80 km 

063° 

? 

360 m 

Find the size of the angle of elevation of 

the cliff top from the boat. 

Airport 

A plane flies on a bearing of 063°. From 

your drawing, find how far it has flown 

when it is 80 km north of the airport. 


E. Surface Area of a Triangular Prism 

Exercise 5 

1. The NET of the triangular prism has been put together again. 

6 cm 

10 cm 

16 cm 

8 cm 

6 cm 

10 cm 

a 

b 

c 

i 

h 

e 

d 

g 

f 

Write down what length each of the letters a, b, c, d, e, f, g, h and i stands for. 

2. A NET of the triangular prism has been partially drawn. 

Copy this part of the net, showing exact measurements, and complete the net. 

5 cm 

2 cm 

2 cm 

2 cm 

2 cm 

5 cm 

2 cm 

5 cm 

2 cm 

3. Here again is the net of the triangular prism from Q1. 

Calculate: 

16 cm 

(a) The area of each triangle. 

A 

8 cm 

10 cm 

(b) The areas of rectangles A, B and C. 

(c) The area of the whole net. 

6 cm 

B 

6 cm 

10 cm 

C 


4. Here is a NET of a triangular prism similar to Q2. 

14 cm 

Calculate: 

(a) the area of each rectangle. 

(b) the area of each triangle. 

(c) the area of the whole net. 

5 cm 

6 cm 

4 cm 

5. For both these triangular prisms, calculate the total surface area. 

(a) 

(b) 

1·8 m 

60 mm 

1·5 m 

3 m 

48 mm 

90 mm 

2 m 

36 mm 

6. Florence loved to make jelly .... especially making jelly houses in moulds like the one 

shown below. Her special mould had two square ends with triangles on top, a rectangular 

back and front, a rectangular base, and two sloping rectangular roofs. 

3·6 cm 

3 cm 

4 cm 

12 cm 

Calculate the areas of the two squares, the two triangles and the five rectangles to obtain 

the total surface area of Florence’s favourite jelly mould. 


F. Surface Area of a Cylinder 

Exercise 6 

1. Here is a NET of a cylinder. 

4 cm 

Draw a sketch of the actual cylinder, 

marking in the relevant sizes. 

10 cm 

2. Here is another NET of a cylinder 

5 cm 

20 cm 

Draw a sketch of the actual cylinder, marking in the relevant size. 

3. Find the areas of these shapes: 

(a) 

(b) 

10 cm 

30 cm 

(c) 

40 cm 

(d) 

15 cm 

4. Calculate the Curved Surface Area of the following cylinders using the formula: 

A = 2πrh or A = πdh, 

(a) (b) (c) 

Curved 

Surface 

Area 

2 m 

Curved Surface 

Area 

1·5 m 

radius = 1 cm 

Curved 

Surface 

Area 

85 cm 

radius of base = 10 cm 

height of cylinder = 25 cm 


5. Calculate the total surface area of these cylinders: 

(a) (b) 

12 m 

radius of base 20 cm 

height 5 cm 

14·5 m 

(c) 

4 cm 

(d) 

7 cm 

5 cm 

9 cm 

6. A toy mug is 5 cm high and the diameter of its base is 3 cm. 

Calculate: 

(a) the area of its base. 

(b) its curved surface area. 

(c) its total surface area. 

7. A can of juice is 15 cm high and has a base with radius 4 cm. 

Calculate the total surface area of the can. 

JUICE 



Checkup Exercise for Scale Drawings and Surface Areas of Solids 

1. In a computer game ‘Aircraft Fire’, the idea is to shoot down as many aircraft as you can 

over a certain number of seconds. The tank always faces North at the start. 

Tiger Lily 

Blue Dragon 

N 

Gypsy Moth 

N 

Killer Dolphin 

TANK 

White Bird 

W 

E 

S 

Yellow Peril 

Black Belle 

Red Devil 

Which aircraft is: 

(a) due East of the tank 

(c) North-West of the tank 

(e) on a bearing of 180° from the tank 

(b) South-East of the tank 

(d) on a bearing of 045° from the tank 

(f) on a bearing of 225° from the tank? 

2. 

Ben Crowe 

N 

Ben Avich 

N 

Ben Bovis 

W 

E 

Roue Hill 

S 

LOCH 

Ben Always 

Clyde Hill 

(a) Use a protractor to find the bearing of Ben Avich from Ben Bovis. 

(b) (i) Measure the distance on the map from Ben Avich to Clyde Hill. 

(ii) The scale of the map is: 2 centimetres represents 1 kilometre. 

How far is it from Ben Avich to Clyde Hill? 


3. A satellite (S) is ‘picked up’ on radar screens at two stations 80 kilometres apart. 

The satellite is on a bearing of 060° from Q and 290° from R. 

N 

S 

N 

Q 

R 

80 km 

(a) Make a scale drawing of the picture. (Use a scale of 1 cm to 10 km.) 

(b) Use your scale drawing to calculate how far the satellite is from Q. 

4. Make a scale drawing showing the mountain and the angle of elevation of its top from the 

tree. 

Use your drawing to calculate the height of the mountain. 


20° 

8000 m 

5. Calculate how much cardboard will be needed to make the cardboard cover for this 

chocolate bar. 

1 cm 

1·5 cm 

Lobertone Choc Bar 

6 cm 

1·25 cm 

6. The diameter of the circular end of this can of Coci 

Loca is 6 cm. 

The can is also 10 cm high. 

Calculate its total surface area. 

Coci Loca 

10 cm 

6 cm 


STATISTICAL ASSIGNMENT 

By the end of this set of exercises, you should be able to 

(a) 

(b) 

(c) 

(d) 

(e) 

collect data and illustrate it using a boxplot. 

prepare a numerical summary of a data set consisting of minimum, 

maximum, quartiles and range. 

calculate the range and semi-interquartile range for a data set. 

use a numerical summary to construct a boxplot. 

compare two or more data sets by constructing and interpreting multiple 

boxplots. 



A. A Statistical Assignment 

Here are the ages, heights and weights of the girls and boys in a youth club. 

The task is to compare the ages, 

heights and weights of the girls 

against the boys of the youth 

club. 

Age Comparison 

To do this you will require to 

find for girls and boys: 

• the minimum age 

• the maximum age 

• the range of ages 

• the mean age 

• the modal age 

• the median age 

• the quartiles 

• the semi-interquartile range 

You will also need to be able to 

draw boxplots. 

You will be led through this 

assignment, but first 

you are required to attempt 

Exercises 1 and 2 over the page. 

There you will learn how to find 

quartiles, the semi-interquartile 

range and how to draw 

boxplots. 

You will then be able to 

complete the age comparison. 

Name Age (yrs) Height (cm) Weight (kg) 

Angela 1 1 138 4 0 

Brenda 1 8 148 5 5 

Claire 12 140 45 

Denise 1 8 1 5 4 5 0 

Eve 17 160 50 

Gina 1 7 1 4 8 4 5 

Helen 1 3 1 4 2 4 0 

Jane 1 5 146 4 5 

Kate 1 1 140 4 5 

Lisa 1 7 150 5 5 

Maureen 1 8 155 5 0 

Nina 1 3 1 4 0 5 0 

Albert 14 146 50 

Brian 15 148 55 

Charlie 1 6 150 7 0 

Dean 1 4 1 5 4 6 0 

Frank 1 6 160 6 0 

George 1 5 1 6 2 6 5 

Henry 30 170 80 

Jim 15 146 50 

Ken 17 155 65 

Len 16 145 60 

Mark 14 140 60 

Norrie 14 146 55 

Pete 1 6 155 6 5 

Richard 1 5 154 6 0 

Russell 1 6 138 5 5 

Stuart 14 146 55 

Tom 16 140 65 

William 15 148 70 


B. Quartiles and Semi-interquartile Range 

Quartiles Just as the median divides a set of scores into two equal sets, the quartiles divide 

the numbers into four equal sets. 

There are three quartiles: 

The lower quartile, or Q1. 

The middle quartile, or Q2 (the median). 

The upper quartile, or Q3 

Semi-Interquartiles Range 

The interquartile range is simply Q3 – Q1 

and the semi-interquartile range is: 

SIQR = 1 /2 (Q3 – Q1 ) 

Exercise 1 

1. Twelve college students were given a psychology test mark, out of 20, for an experiment 

they carried out. The scores were: 

8, 8, 9, 10, 11, 12, 14, 14, 14, 14, 16, 19 

Find the three quartiles and the semi-interquartile range. 

2. Find the range and semi-interquartile range for each of the following sets of scores: 

(a) 21, 19, 17, 24, 20, 22 

(b) 9, 10, 11, 13, 5, 4, 3, 8, 12, 6, 7 

(c) 118, 118, 120, 111, 120, 121, 114, 114, 115, 117 

(d) 1004,1005, 1005, 1001, 1001, 1002, 1008, 1008, 1009, 1009, 1008, 1007, 1002. 

3. A group of people measured the lengths of their index fingers. The lengths in centimetres 

were: 

7·0, 6·9, 7·5, 7·9, 7·8, 8·7, 8·7, 8·1, 6·9, 8·1, 8·5, 7·0, 7·8, 7·1 

Find the median and semi-interquartile range. 

4. The ages (in years) of a group of workmen on a building site were recorded: 

24 31 25 25 26 29 26 27 28 27 27 27 28 

27 28 28 28 28 26 29 29 29 30 30 24 

Find the median and semi-interquartile range. 


C. The Boxplot 

A new way of representing the information just calculated in exercise 1 is by means of a 

boxplot. 

Example: 

Find the median and SIQR for the following set of numbers and draw a boxplot to 

show the results. 

2, 3, 4, 6, 7, 8, 11, 11, 12, 15, 15, 19 

Solution: Can you see that Q2 = 1 /2 (8 + 11) = 9·5 

Q1 = 1 /2 (4 + 6) = 5 

Q3 = 1 /2 (12 + 15) = 13·5 SIQR = 4 . 25 

The Boxplot: 

lowest 

value 

lower 

quartile 

Q1 

median 

Q2 

upper 

quartile 

Q3 

highest 

value 

interquartile range 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 

The boxplot is a quick, neat way of showing the highest and lowest score as well as the 

median, quartiles and interquartile range. 

It is also very useful when comparing two different sets of scores. 

Exercise 2 

1. A father timed how long his daughter was on the phone during 10 phone calls over a two 

night period. Here are the times in minutes: 

2, 5, 6, 8, 10, 10, 10, 12, 17, 22 

(a) Find the median and upper and lower quartiles. 

(b) Show the information as a boxplot. 

2. The daily rainfall (in millimetres) was measured on the roof of the meteorological offices 

from 10 March to 23 March: 

0, 3, 5, 6, 9, 15, 12, 5, 2, 0, 8, 12, 5, 8 

(a) Rearrange the measurements in order and find the medians and quartiles. 

(b) Show the results as a boxplot. 


3. A man timed himself over a three week period on how long it took him to drive to work in 

the morning. The times, in minutes, were: 

(a) Find the median and quartiles. 

(b) Show your results as a boxplot. 

15, 17, 20, 23, 29, 32, 30, 29, 25, 23, 18, 29, 15, 17, 23, 20. 

4. A factory manager noted the number of absences, because of ‘illness’, both the men and 

women had during 1996. 

men 2, 2, 2, 3, 3, 3, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 8, 10,11,11, 12, 12, 13, 17 

women 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 6, 6, 7, 8, 8, 8, 8, 8, 9, 9. 

(a) Find the median and quartiles for each set of data. 

(b) On the same diagram, draw the two boxplots to represent the two sets of data. 

(c) Make two or three observations about your results. 

5. A group of senior citizens were asked at what age they first realised that they were going 

grey. The results were: 

men 31 31 32 35 35 38 40 42 42 44 45 45 47 48 48 51 51 52 55 60 61 65 67 

women 38 38 39 40 42 44 44 45 47 48 50 51 51 55 56 60 62 63 68 71 

(a) Find Q1, Q2 and Q3 for each sex. 

(b) On a single diagram, show the two boxplots representing this information. 

(c) Make one or two comments about the results. 


A. A Statistical Assignment (continued) 

Age Comparison 

Now that you know how to find the minimum age, the maximum age, the range of ages, the 

mean age, the modal age, the median age, the quartiles and the semi-interquartile range, plus 

you can now draw boxplots, you will be able to understand being led through the assignment 

on the comparison of ages between girls and boys in the youth club. 

You will also be able to write a short report on any conclusions which could be drawn from 

the calculations. 

Height Comparison and Weight Comparison 

It is now your job to undertake assignments on: 

(a) the comparison of heights between girls and boys in the youth club. 

(b) the comparison of weights between girls and boys in the youth club. 

1. You should find the minimums, the maximums, the ranges, the modes, the medians, the 

quartiles and the semi-interquartile ranges. 

2. Draw boxplots. 

3. Write short reports on conclusions. 

A Statistical Assignment of Your Own 

When the tasks above have been completed, you have to undertake one similar statistical 

assignment of your own by collecting data from two or more groups. 

Some suggestions: Shoe Sizes, Number of Children in a Family, Pocket Money, 

OR Any idea of your own! 



APPLICATIONS OF MATHEMATICS (INT 1) 

Specimen Assessment Questions 

1. The weekly wage of Mr Herbie Pirie, the head greenkeeper at St Roberts Golf Club, is 

made up as follows: 

Basic Weekly Wage £338 (his basic hourly rate is £8·00) 

WAGES Overtime 12 hours at double time 

Bonus for good greens £20 

National Insurance £36 

DEDUCTIONS Income Tax £58 

Pension £15 

Calculate Mr Pirie’s: 

(a) gross weekly wage (b) total weekly deductions (c) net weekly wage. 

2. Mr Pirie decides to put double glazing into his house. He borrows £5000 over 36 months 

for this, choosing to take the loan protect scheme as he is unsure of his job at St Roberts. 

WITH LOAN PROTECT 


24 months £2000 £109·67 

£3000 £162·07 

£5000 £268·84 

36 months £3000 £118·16 

£5000 £195·62 

£10 000 £379·24 

WITHOUT LOAN PROTECT 


24 months £2000 £98·81 

£3000 £146·19 

£5000 £242·61 

36 months £3000 £104·11 

£5000 £172·47 

£10 000 £335·27 

Use the loan repayment table to work out how much: 

(a) his monthly payments are 

(b) he will repay in total over the 36 months 

(c) his loan, above the £5000 he borrowed, will actually cost him. 

3. One dark rain cloud appeared in the sky over 

the town of Palma Nova. The angle of 

elevation of the cloud from the weather station 

was 50°. 


Make a scale drawing to find out how far the 

cloud was above the ground. 

50° 

Weather 

Station 

1000 m 


4. Only one of these two networks can be drawn using the rules given in the unit: 

A 

e 

d 

B 

g 

f 

g 

h 

c 

f 

e 

d 

a 

b 

(a) Say which one can be drawn and state which node you should start from and where 

you would end. 

(b) Say why the other network could not possibly be drawn. 

5. North 

A goods train leaves Central Station and has to 

Station 

call in to the other three stations to deliver mail. 

a 

b 

c 

Central 

Station 

16 km 

18 km 

19 km 

13 km 

16 km 

(a) Draw a tree diagram to show all the possible 

routes. 

Highbury 

Station 

Central 

North 

South 

Station 

(b) Which route is the shortest? 

6. This flowchart shows how much a butcher 

has to pay the farmer for his lamb. 

START 

Age of lamb 

Cost = £1·75 per kg 

Yes 

Is the 

lamb under 9 

months 

old? 

No 

Cost = £1·45 per kg 

Use the flowchart to calculate the cost of: 

(a) a six-month-old lamb weighing 15 kilograms 

(b) a one-year-old lamb weighing 32 kilograms. 

STOP 


7. A van driver wishes to record the weight of 

the vegetables he delivers in his van each 

day. 

Tuesday 15 January 

Deliveries: 

6 sacks of potatoes 

8 sacks of turnips 

3 sacks of cabbages 

4 sacks of carrots 

He knows the weights of each sack of vegetables: a sack of potatoes weights 25 kilograms 

a sack of turnips weights 20 kilograms 

a sack of cabbages weights 15 kilograms 

a sack of carrots weights 12 kilograms. 

(a) Design a spreadsheet to help him record and calculate the total weight of the 

vegetables. 

1 

2 

3 

It should have the above headings. 

(b) Enter the data for the four vegetables in the first three columns of your spreadsheet. 

(c) Write down the formula which you would enter for calculating the total weight of the 

potatoes. 

(d) Enter the formulae into your spreadsheet to calculate the total weight of each of the 

four types of vegetables. 

(e) Enter a formula in a suitable cell to calculate the total overall weight of vegetables in 

the van. 

(f) Save your spreadsheet and take a printout if necessary. 

8. The picture (not drawn to scale) shows 

three coastlines. 

Use your drawing to measure and 

calculate the bearings of: 

(a) Sma’ Island from Fass Point 

(b) Breaker Point from Sma’ Island 

(c) Breaker Point from Fass Point 

(d) Fass Point from Breaker Point. 

Vegetable 

Deliveries 

A B C D 

Vegetable Weight / sack No. of sacks Total Weight 

Potatoes 

Fass 

Point 

N 

75° 

44° 

110° 

N 

N 

Breaker Point 

Sma’ 

Island 

200° 


9. (a) Name this shape. 

(b) Make a copy of it in your answer book and mark in the sizes of its nine edges, using 

its net (shown below) to guide you. 

(c) Calculate the total surface area of the shape. 

4 cm 

5 cm 

3 cm 

10 cm 

3 cm 

5 cm 


ANSWERS TO APPLICATIONS OF MATHEMATICS (INT 1) 

Straightforward Calculations in a Social Context 

Exercise 1 

1. £7644 2. £9720 3. Both £11 388 Same! 

4. (a) £990 (b) (i) £12 600 (ii) £1050 

5. £120·55 6. Harold by £3·25 

7. (a) 8 hrs (b) 40 hrs (c) £480 

8. £50 

Exercise 2 

1. (a) £20 per hr (b) £100 

2. (a) £13·50 per hr (b) £54 

3. (a) £344 (b) £172 (c) £516 

4. £667 5. £61·20 6. £331·50 7. £288·96 

8. (a) 33 hrs (b) £148·50 (c) £168·50 

9. (a) £172·80 (b) £75 (c) £247·80 

Exercise 3 

1. (a) June Thomson (b) 341L (c) £300 

(d) £6 (e) £71 (f) £229 

2. (a) £471·25 (b) £128·94 (c) £342·31 

3. (a) £1615·70 (b) £414·64 (c) £1201·06 

4. (a) £750 (b) £100 

Exercise 4 

1. (a) £355 (b) £78·75 (c) £410·97 (d) £254·04 

2. (a) £3727·44 (b) £5974·92 (c) £2286 (d) £10 232·16 

3. (a) £194·88 (b) £2274·80 (c) £1478·88 (d) £2544·48 

Exercise 5 

1. (a) £102·76 (b) £94·22 (c) £268·84 

(d) £104·11 (e) £198·24 (f) £334·86 

2. £23·15 

3. (a) £2389·20 (b) £2195·64 (c) £3889·68 

(d) £12 069·72 (e) £15 832·80 (f) £20 091·60 

4. (a) £104·11 (b) £3747·96 (c) £747·96 

5. (a) £118·16 (b) £4253·76 (c) £1253·76 (d) £505·80 

6. £4258·80 

7. Building Soc. cheaper by £234. 


Checkup 

1. £15 120 £15 132 Mrs Ramas by £12. 2. £5440 

3. (a) October (b) (i) £1092·50 (ii) £288·54 (iii) £803·96 

4. (a) £267·74 (b) £6425·76 (c) £1425·76 


Logic Diagrams 

Exercise 1 

1. a, c, e, g, i, k, m, o can be drawn. 

2. See sketches with nodes. In a, c, e, g, i, k, m, o all the nodes are even. In others, all odd. 

3. (a) Those with even nodes can be drawn. (b) They were all odd nodes. 

4. ‘Even’. 5. ‘Odd’. 

6. (a) Yes. (b) 3 3 (c) Looks like it is not a true rule. 

Exercise 2 

1. No 2 (a) Can’t be drawn from A, B, C or D. (b) Can be drawn from E or F. 

3. Each time, you must start at one of the 2 odd nodes and you will end at the other one. 

4. (a) ... only 2 of them. (b) ... 2 odd nodes ..... one of them .... the other one. 

5. Various. 

6. (a) No. 

C 

(b) All 4 nodes are odd 

(see rule C). 

Exercise 3 

A 

D 

B 

1. (a) 2 (b) 3 (c) 1 (d) 2 

2. (a) See tree diagram showing 8 possibilities. 

(b) (HHH), (HHT), (HTH), (THH), (HTT), (THT), (TTH), (TTT) 

(c) (i) 1 (ii) 3 (iii) 4 (iv) 2 

3. (a) Tree diagram showing 6 possibilities. 

(b) (s,c), (s,m), (s,n), (no s,c), (no s,m), (no s,n) 


(b) in, it, on, ot, an, at (c) 5 of them (in, it, on, an, at). 

5. (a) Tree diagram showing 2 × 3 × 2 = 12 choices. 

(b) 148, 149, 158, 159, 168, 169, 248, 249, 258, 259, 268, 269 (c) 4 


(b) 11, 12, 13, 21, 22, 23, 31, 32, 33 (c) 3 


(b) 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44 (c) 8 



(b) 9 routes. (c) (i) 3 (ii) 6 

9. (a) Tree diagram showing 9 routes. (b) 9 

(c) (i) S —> B —> E (29 km) (ii) S —> C (A19) —> E (B171) (49 km) 

Exercise 4 

1. (a) Tree diagram showing all routes. 

(b) A —> D —> B —> C (30 miles) 

(c) A —> B —> D —> C (34 miles) 

(d) Yes, A —> B —> C —> D —> A, or A —> D —> C —> B —> A (40 miles) 

2. (a) Tree diagram. 

(b) Shortest B —> A —> D —> C (29) 

(c) Longest B —> C —> D —> A (32) 

(d) Use B —> A —> D —> C —> B , or B —> C —> D —> A —> B 

3. (a) Tree diagram showing all 6 routes. 

(b) P —>R —> Q —> S (191) 

4. (a) Tree diagram showing all 10 routes. 

(b) Can’t go S —> 4 —> 2, or you end up having to pass through 2 checkpoints. 

(c) S —> 1 —> 4 —> 2 —> 3 (72 miles) 

5. (a) Tree diagram showing the only 5 possibilities (P-Q-R-S-T-P) (P-Q-T-R-S-T-P) 

(P-Q-T-S-R-T-P) (P-T-Q-R-S-T-P) (P-T-S-R-Q-T-P). 

(b) Either (P-Q-T-R-S-T-P) or P-Q-T-S-R-T-P) (16 km) 

6 (a) (i) 29 km (b) 

18 

(ii) 34 km 

29 11 

7. (a) 

52 34 23 

H 

22 

I 

8. (a) Hightops 

3·2 The Knot 

14 

5·4 3·7 

6·0 

18 26 

4·1 

35 

Drymun 

K 20 

J 

Kingsfoot 1·9 

(b) 2 km 

(b) Tree diagram showing 6 routes. 

(c) H –> The K –> K –> D (8·8 km) 

9. (a) Trailing rose (b) Common tormentil (c) Pear (d) Dog rose (d) Crab apple. 

10. (a) spider (b) butterfly (c) ladybird (d) wasp 

(e) caterpillar (f) moth (g) mite (h) harvestman (i) cranefly. 

Exercise 5 

1. (a) £10·50 (b)£112 (c) £64·75 

2. (a) 15°C (b)0°C (c) 100°C (d) –20°C (e) –40°C 

3. 990 m (b) 1485 m (c) 2046 m 


4. Start –> Look up plumber’s number –> Lift the receiver –> Dial Number –> Discuss the 

problem on the phone –> Arrange for plumber to come –> Replace receiver –> Stop 

Exercise 6 

1. (a) £22·50 (b) £10·00 

2. (a) £35·50 (b) £42·50 

3. (a) £41·25 (b) £64·25 (c) £99·50 

4. (a) £3360 (b) £4000 (c) £5200 

5. (a) £624 (b) £336 (c) £225 (d) £1276 

6. (a) 40 (b) 5·7 

7. (a) Grade 7 (b) Grade 4 (c) Grade 3 (d) Grade 7 

(e) Grade 7 (f) Grade 4 

8. (a) No (b) Yes (c) No (d) Yes 

(e) No 

(f) Yes 

Exercise 7 

1. 2. 

1 

2 

3 

4 

5 

6 

7 

A B C D E 

1st name2nd name Test 1 Test 2 Test 3 

David Smith 6 2 8 1 7 9 

Brian Jones 6 3 5 9 9 1 

Alan Young 7 1 8 3 6 5 

Iain Taylor 7 3 7 6 7 9 

John Hughes 5 9 6 2 7 1 

1 

2 

3 

4 

5 

6 

7 

8 

A B C 


Davies 6 0 8 0 

White 9 0 1 2 0 

Gordon 2 1 0 1 6 0 

Wylie 7 5 1 6 0 

Rivers 1 3 0 1 1 0 

Jones 8 0 1 5 0 

3. A B C 

4. 

1 

2 

3 

4 

5 

6 

7 

8 

fruit (kg) weight cost/kg 

apples 2.5 £ 1.52 

oranges 1.5 £ 1.85 

grapes 0.5 £ 1.68 

bananas 3 £ 1.20 

pears 0 £ 1.46 

peaches 0.75 £ 1.92 

1 

2 

3 

4 

5 

6 

7 

8 

A B C D 

Name Rate of Pay Basic Hrs O'time hrs 

Fred £ 6.50 4 0 6 

Tom £ 6.20 3 8 4 

Gina £ 4.80 3 6 5 

Alex £ 5.10 3 9 4 

Sara £ 6.40 4 0 2 

Dave £ 5.30 3 2 0 

Exercise 8 

1. 

1 

2 

3 

4 

5 

6 

7 

A B C D E F G 

1st name2nd name Test 1 Test 2 Test 3 TOTAL AVERAGE 

David Smith 6 2 8 1 7 9 2 2 2 7 4 

Brian Jones 6 3 5 9 9 1 2 1 3 7 1 

Alan Young 7 1 8 3 6 5 2 1 9 7 3 

Iain Taylor 7 3 7 6 7 9 2 2 8 7 6 

John Hughes 5 9 6 2 7 1 1 9 2 6 4 


2. 3. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

A B C D E 

Customer length breadth AREA Cost 

Davies 6 0 8 0 4800 £ 24.00 

White 9 0 1 2 0 10800 £ 54.00 

Gordon 2 1 0 1 6 0 33600 £ 168.00 

Wylie 7 5 1 6 0 12000 £ 60.00 

Rivers 1 3 0 1 1 0 14300 £ 71.50 

Jones 8 0 1 5 0 12000 £ 60.00 

TOTAL= £ 437.50 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

A B C D 

fruit (kg) weight cost/kg COST 

apples 2.5 £ 1.52 £ 3.80 

oranges 1.5 £ 1.85 £ 2.78 

grapes 0.5 £ 1.68 £ 0.84 

bananas 3 £ 1.20 £ 3.60 

pears 0 £ 1.46 £ 0.00 

peaches 0.75 £ 1.92 £ 1.44 

TOTAL= £ 12.46 

4. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

A B C D E F G 

Name Rate of Pay Basic Hrs O'time hrs Basic Pay O'Time Pay Gross Pay 

Fred £ 6.50 4 0 6 £ 260.00 £ 58.50 £ 318.50 

Tom £ 6.20 3 8 4 £ 235.60 £ 37.20 £ 272.80 

Gina £ 4.80 3 6 5 £ 172.80 £ 36.00 £ 208.80 

Alex £ 5.10 3 9 4 £ 198.90 £ 30.60 £ 229.50 

Sara £ 6.40 4 0 2 £ 256.00 £ 19.20 £ 275.20 

Dave £ 5.30 3 2 0 £ 169.60 £ 0.00 £ 169.60 

TOTAL= £ 1474.40 

Exercise 9 

1. Same as answer to Exercise 8, qu 1. 

2. Same as answer to Exercise 8, qu2. 

3. Various. 

4. 

1 

2 

3 

4 

5 

6 

7 

8 

A B C D E F G H I 

Name Rate of Pay Basic Hrs O'time hrs Basic Pay O'Time Pay Gross Pay Deductions Net Pay 

Fred £ 6.50 4 0 6 £ 260.00 £ 58.50 £ 318.50 £ 92.40 £ 226.10 

Tom £ 6.20 3 8 4 £ 235.60 £ 37.20 £ 272.80 £ 75.30 £ 197.50 

Gina £ 4.80 3 6 5 £ 172.80 £ 36.00 £ 208.80 £ 57.79 £ 151.01 

Alex £ 5.10 3 9 4 £ 198.90 £ 30.60 £ 229.50 £ 52.72 £ 176.78 

Sara £ 6.40 4 0 2 £ 256.00 £ 19.20 £ 275.20 £ 68.77 £ 206.43 

Dave £ 5.30 3 2 0 £ 169.60 £ 0.00 £ 169.60 £ 31.42 £ 138.18 

5. 6. (a) 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

A B C 

x x squared x cubed 

1 1 1 

2 4 8 

3 9 27 

4 16 64 

5 25 125 

6 36 216 

7 49 343 

8 64 512 

9 81 729 

10 100 1000 

1 

2 

3 

4 

5 

6 

7 

8 

9 

A B C D 

Item Cost (each) No. sold Amount 

Crisps £ 0.22 63 £ 13.86 

Big Bars £ 0.31 22 £ 6.82 

Tukins £ 0.28 46 £ 12.88 

Cans £ 0.25 36 £ 9.00 

Juice £ 0.18 29 £ 5.22 

(b) Week 2 total = £41·40. 

Week 1 TOTAL= £ 47.78 


7. 

A B C D E F G H I J 

Number 2 times 3 times 4 times 5 times 6 times 7 times 8 times 9 times 10 times 

1 2 3 4 5 6 7 8 9 10 

2 4 6 8 10 12 14 16 18 20 

3 6 9 12 15 18 21 24 27 30 

4 8 12 16 20 24 28 32 36 40 

5 10 15 20 25 30 35 40 45 50 

6 12 18 24 30 36 42 48 54 60 

7 14 21 28 35 42 49 56 63 70 

8 16 24 32 40 48 56 64 72 80 

9 18 27 36 45 54 63 72 81 90 

10 20 30 40 50 60 70 80 90 100 

8. 

A B C D 

1 Thomson's Builders 

2 

3 

Jan Feb Mar 

4 

5 

EXPENSES 

Wages £ 11000.00 £ 11000.00 £ 12100.00 

6 Rent £ 250.00 £ 250.00 £ 250.00 

7 Gas £ 320.00 £ 280.00 £ 240.00 

8 Electricity £ 290.00 £ 285.00 £ 260.00 

9 Materials £ 2130.00 £ 1940.00 £ 2050.00 

10 Petty Cash £ 60.00 £ 85.00 £ 70.00 

11 TOTAL £ 14050.00 £ 13840.00 £ 14970.00 

12 

13 

14 

INCOME 

TOTAL £ 18640.00 £ 17960.00 £ 20470.00 

15 

16 BALANCE £ 4590.00 £ 4120.00 £ 5500.00 

17 

18 

Quarterly Balance = £ 14210.00 

Checkup Exercise 

1. (a) Can be done if you start at top or right node. 

(c) Can be done starting at any node. 

(e) Can be done as long as you don’t start at top or bottom node. 

(f) Can be done starting at any node. (b) and (d) Cannot be done. 

2. Various solutions. Whatever sections the new bridge joins, start at either of the other two 

and you will end up at the other one. 

3. (a) 5 × 3 = 15 possibilities (1,1),(1,2),(1,3),(1,4),(1,5), 

(2,1),(2,2),(2,3),(2,4),(2,5), 

(b) Bet on EVEN (3,1),(3,2),(3,3),(3,4),(3,5) 

8 out of 15 are even. 

4. (a) Tree diagram to show all 4 possibilities (CDFE), (CDEF), (CEDF), (CEFD). 

(b) C –> E –> D –> F (57 km) is shortest. 

5. 

13 

31 

18 

15 16 20 


6. (a) (i) £3·20 (ii) £3·80 (iii) £5·10 

(b) 12 × £4·00 = £48·00 

7. (a) cell C3 = cell B3 × 0·175, cell D3 = cell B3 + cell C3 

(b) Optional. 

1 

2 

3 

4 

5 

6 

7 

A B C D 

Item Cost (before V.A.T) V.A.T. Net Cost 

Bag of Cement £ 3.60 £ 0.63 £ 4.23 

100 Common Bricks £ 9.20 £ 1.61 £ 10.81 

Bag of Sand £ 2.40 £ 0.42 £ 2.82 

100 Facing Bricks £ 19.60 £ 3.43 £ 23.03 

Pack of 20 slates £ 16.40 £ 2.87 £ 19.27 


Scale Drawings and Surface Areas of Solids 

Exercise 1 

1. 2. (a) Lions, Giraffes, Goats, Chimps. 

NW 

N 

NE 

W 

E 

SE 

SW 

(b) SE, East, NW 

3. (a) C (b) SW (c) T 

(d) H (e) R and S. 

S 

Exercise 2 

1. Easton 087°, Prestwich 132°, Aberdoch 240°, Kildour 340°. 

2. (a) 090° (b) 270° (c) 180° (d) 135° (e) 045° (f) 225° 

3. (a) NE (b) South (c) NW (d) North 

4. (a) South (b) NE (c) SW (d) NW (e) SE 

5. (a) 065° (b) 150° (c) 098° (d) 330° 

6. (a) Invernorth (b) 045° (c) (i) 315° (ii) 225° 

7. (b) 045° 150° 070° 210° 270° 

8. Hole 1 135° and 450 m long (approx.). Hole 2 310° and 325 m long (approx.). 

Exercise 3 

1. Check Drawings 

2. (a) Drawing (b) 20 km approx. 3. (a) Drawing (b) 63 km (60 km) 

4. (a) Drawing (b) 080° 

5. (a) Drawing (b) 21–22 km approx. (c) 280–285° approx. 

6. Various (remember to state scale). 

7. (a) Drawing (b) 58 km (c) 92 km (d) 335° 

8. (a) Drawing (b) 41 m 

9. (a) Drawing (b) 28 km 


Exercise 4 

1. (a) 9·5 cm (b) 285 m 2. 4·5 m by 2·5 m 

3. (a) 5 cm (b) 1 cm to 2 m (c) 320 

4. (a) 6·5 cm (b) 13 m (c) 8 m by 6 m (d) 12 m by 6 m 

(e) 16 m (f) 11 m 

5. (a) 84 m (b) 95 m (c) 70 m (d) 150 m 

6. (a) Various (b) 20 m 7. 10–11 m 8. About 2 m 

9. (a) Drawing (b) 43 - 44 km 

10. (a) Drawing (b) 12 km (c) 333° approx. 

11. (a) 14° (b) 176 km 

Exercise 5 

1. a = 8 b = 6 c = 10 d = 6 e = 8 f = 10 g = 16 h = 16 i = 16 

2. 

5 cm 

3. (a) 24 cm 2 each 

2 cm 2 cm (b) 128 cm 2 , 96 cm 2 , 160 cm 2 

(c) 432 cm 2 

2 cm 

5 cm 

2 cm 

2 cm 

2 cm 

2 cm 

2 cm 

5 cm 

2 cm 2 cm 

6. 274·4 cm 2 

Exercise 6 

5 cm 

5 cm 

1. 4 cm 2. 3. (a) 314 cm 2 

10 cm 

20 cm 

4. (a) 70 cm 2 , 70 cm 2 , 84 cm 2 

(b) 12 cm 2 each 

(c) 248 cm 2 

5. (a) 19·8 m 2 

(b) 14688 mm 2 

(b) 706·5 cm 2 

(c) 628 cm 2 

(d) 177 cm 2 

4. (a) 1570 cm 2 (b) 9·42 m 2 (c) 534 cm 2 

5. (a) 3140 cm 2 (b) 772 m 2 (c) 113 cm 2 (d) 181 cm 2 

6. (a) 7·065 cm 2 (b) 47·1 m 2 (c) 54·2 cm 2 

7. 477 cm 2 

Checkup 

1. (a) White Bird (b) Black Belle (c) Blue Dragon (d) Gypsy Moth 

(e) Red Devil (f) Yellow Peril 


2. (a) 070° (approx). (b) 5.5 cm 2·75 km 

3. (a) Drawing (b) 36 km 

4. 2900 m 

5. 25·5 cm 2 

6. 245 cm 2 

Statistical Assignment 

Exercise 1 

1. lower quartile = 9·5,median = 13, SIQR = 2·25 

2. (a) Range = 7 SIQR = 1·5 (b) Range = 10 SIQR = 3 

(c) Range = 10 SIQR = 3 (d) Range = 8 SIQR = 3 

3. median = 7·8 SIQR = 0·55 

4. median = 28 SIQR = 1·5 

Exercise 2 

1. (a) median = 10 (b) 2 6 10 12 22 

lower quartile = 6 

upper quartile = 12 

2. (a) median = 5·5 (b) 0 3 5·5 9 15 

lower quartile = 3 


3. (a) median = 23 (b) 15 17·5 23 29 32 

lower quartile = 17·5 


4. (a) median = MEN – 7 WOMEN – 4 

lower quartile = MEN – 4 WOMEN – 1 

upper quartile = MEN – 10·5 WOMEN – 8 

(b) 

0 5 10 15 20 

(c) Range of women’s absences smaller, median lower, so generally fewer absences than 

men. 

5. (a) MEN Q1 = 38, Q2 = 45, Q3 = 52. 

(b) WOMEN Q1 = 43, Q2 = 49, Q3 = 58. 

30 40 50 60 70 

(c) Women go greyer later in life, median age is later in life (or women lie more!) 


Answers to Height Comparison Assignment 

Girls 

Boys 

Max. height 160 170 

Min. height 138 138 

Range 22 32 

Mean 146·75 150 

Mode 140 146 

Lower quartile 140 146 

Median 147 148 

Upper quartile 152 155 

SIQR 6 4·5 

138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 

Boys’ wider range, again because of the taller 30-year-old! 

Generally boys taller, though not much difference in medians. 

Boys’ median better average than their mean. 

Mode? Girls’ mode not near other averages. 

More young, smaller girls. 

Answers to Weight Comparison Assignment 

Girls 

Boys 

Max. weight 55 80 

Min. weight 40 50 

Range 15 30 

Mean 47·5 61 

Mode none (4@45, 4@50) 60 

Lower quartile 45 55 

Median 47·5 60 

Upper quartile 50 65 

SIQR 2·5 5 

40 45 50 55 60 65 70 75 80 

Boys’ wider range, this time not just because of 30-year-old. 

Generally boys much heavier, fair difference in medians. 

Girls’ median and mean same – boys’ mean, median, mode not much difference. 

Younger, smaller girls are lighter. 



Answers to Specimen Questions 

1. (a) £550 (b) £109 (c) £441 

2. (a) £195·62 (b) £7042·32 (c) £2042·32 

3. Around 1200 metres. 

4. (a) B can be drawn, only if you start at b or g and finish at g or b. 

(b) Four of the nodes, (c, d, e and f) are odd. Can’t be drawn. 

5. (a) Highbury 

North 

South 

(b) C —> S —> H —> N is shortest (45 km) 

Highbury 

6. (a) £26·25 (b) £46·40 

7. 

1 

2 

3 

4 

5 

6 

7 

8 

Central 

8. (a) 075° (b) 200° (c) 119° (d) 280° 

9. (a) Triangular prism. (c) 132 cm 2 

(b) 

10 cm 

South 

South 

North 

Highbury 

Highbury 

North 

A B C D 

Vegetable Weight / sack No. of sacks Total Weight 

Potatoes 2 5 6 1 5 0 

Turnips 2 0 8 1 6 0 

Cabbages 1 5 3 4 5 

Carrots 1 2 4 4 8 

Total Weight = 403 

4 cm 

4 cm 

5 

c 

cm 

10 cm 

3 cm 

5 cm 

3 cm 

10 cm

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