Solution - Education Scotland

Core Skills

Numeracy

Outcome 4

Applying Numerical Skills

[INTERMEDIATE 1]



© Learning and Teaching Scotland 2004

This publication may be reproduced in whole or in part for educational purposes by educational establishments in Scotland provided that no profit accrues at any stage. ii NUMERACY: OUTCOME 4 (INTERMEDIATE 1) TEXT VERSION


CONTENTS

Section 1 Working with Whole Numbers and Decimals

1.1 Whole number

1.2 Decimals1

1.3 The four rules

Section 2 Rounding

2.1 The rule of five

2.2 Extending the use of the rule of five

2.3 Rounding to two decimal places

2.4 Rounding in a money calculation

2.5 The problems of real life

2.6 Significant figures

4.2 Finding a percentage by using a calculator

4.3 Percentage profit and loss

4.4 Percentage errors, increases and decreases

Section 5 Earnings

5.1 Earning a wage

5.2 Pay increases

5.3 Dealing with income tax

5.4 Higher tax rate

1

3

Section 3 Fractions

3.1 Quick and easy fractions

3.2 Fractions to percentages

15

18

Section 4 Percentages

4.1 Finding a percentage without using a calculator 25

9

9

11

6

8

8

26

27

28

33

36

37

42

Section 6 Working with money

6.1 Working with VAT

6.2 Practical problems

6.3 Hire purchase

Section 7 Calculations with time

7.1 Lengths of time

7.2 Calculating speed

7.3 Calculating distance

7.4 Calculating time

47

49

50

53

58

59

60

NUMERACY: OUTCOME 4 (INTERMEDIATE 1) TEXT VERSION iii


CONTENTS

Section 8 Ratio

8.1 Ratio as comparison

8.2 Mixed units

8.3 Proportional division

8.4 Direct proportion

Section 9 Formulas

9.1 The four rules of formulas

9.2 Two- and three-step formulas

63

67

68

70

75

77 iv NUMERACY: OUTCOME 4 (INTERMEDIATE 1) TEXT VERSION


WORKING WITH WHOLE NUMBERS AND DECIMALS

SECTION 1

1.1 Whole numbers

Count from 1 to 10 in your head. The numbers you just used are called whole numbers.

Whole numbers are just the numbers that we use for counting whole things – no halves or quarters or decimals.

But – there is another whole number that everyone forgets about.

Suppose that you have fifty vouchers to hand out to customers as they come in to a shop. How many do you have left after giving them all away?

None. The number of vouchers that you have left is zero . Zero is also a whole number.

Definition

Whole numbers are the counting numbers and the number zero:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and so on.

1.2 Decimals

When we use decimals we are almost always working with money or measurements.

£5.68 per hour 1.78 metres tall 6.4 square metres of carpet

NUMERACY: OUTCOME 4 (INTERMEDIATE 1) TEXT VERSION 1



Activity 1

Here are four simple questions. Work out the answer and then see if you can tell what made each question different from the other.

Q1

Day

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Cars sold

2

2

5

4

1

8

6

How many cars were sold in total?

Q2 A car odometer reads 24 500 miles at the start of a journey. It reads 24 660 miles at the end of the journey. How far was the journey?

Q3 Shirley is paid £8 per hour. How much does she earn for a 6-hour day?

Q4 A £12 million lottery pay-out is shared between two winners. How much does each receive?

2 NUMERACY: OUTCOME 4 (INTERMEDIATE 1) TEXT VERSION



1.3 The four rules

Addition, subtraction, multiplication and division are known as the four rules of numeracy.

You need to know how to use all the four rules. However, it is more important that you know when to use them.

If you knew which rule to use in the four questions above then you are well on your way to success in numeracy.

Questions 1.3

For each of the following questions, say which of the four rules you would use. (You may calculate the answer if you wish, but you need to respond by saying which rule you used.)

1. Joyce earns £280 per week. Steve earns £246. How much more does Joyce earn than Steve?

2. There are six eggs in a box. How many eggs are in 20 boxes?

3. There are 284 pupils in one primary school and 311 in another.

How many pupils are there altogether?

4. There are 10 chairs in a row and 16 rows. How many chairs are there altogether?

5. If £12 is shared equally among three people, how much does each get?

6. Hilary is 148 centimetres tall. Joseph is 122 centimetres tall.

How much taller is Hilary than Joseph?

7. Norrie’s dog eats 180g of dog food each day. What weight of dog food does it eat in 4 days?

8. The distance driven by a rep one morning is 40 miles. The distance driven in the afternoon is 83 miles. What is the total distance driven that day?

Knowing what to do is as important as knowing how to do it.




Section 1: Answers

Activity 1 Answers

Q1 28 cars

Q2 160 miles

Q3 £48

Q4 £6 million

Answers 1.3

1. Subtraction

2. Multiplication

3. Addition

4. Multiplication

5. Division

6. Subtraction

7. Multiplication

8. Addition addition subtraction multiplication division

2 + 2 + 5 + 4 + 1 + 8 + 6

24660 – 24500

£8 x 6

£12 million ÷ 2



ROUNDING

SECTION 2

You will need a calculator for most of this section.

2.1 The rule of five

Example 2.1a

Calculate 171 ÷ 25

171 ÷ 25 = 6.84

6.84 has two decimal places – two figures after the decimal point.

We can give an approximate answer by rounding 6.84 to only one decimal place.

Keep the 6 and the 8. Get rid of the 4.

171 ÷ 25 = 6.8 (to 1 decimal place) (1dp for short)

It is okay to throw away a small amount. This is a little like saying

£6.84 is close to £6.80 – we may lose the 4p without much worry.

Example 2.1b

Calculate 172 ÷ 25

172 ÷ 25 = 6.88

6.88 can also be rounded to one decimal place.

Keep the 6 and the 8. Get rid of the second 8.

Think money again. £6.88 is not like £6.84 – it is more like £6.90

Before we can get rid of the 8 we must compensate for its size. It is a large amount to throw away, so we change the 6.8 to 6.9



ROUNDING

The rule of five

To round to one decimal place, look at the second decimal place. If it is a 5 or more, then round up. Otherwise round down.

6.80 becomes 6.8

6.81 becomes 6.8

6.82 becomes 6.8

6.83 becomes 6.8

6.84 becomes 6.8

6.85 becomes 6.9

6.86 becomes 6.9

6.87 becomes 6.9

6.88 becomes 6.9

6.89 becomes 6.9

Questions 2.1

Round the following numbers to 1 decimal place.

(a) 8.37 (b) 5.91

(d)

(g)

3.64

24.17

(e) 4.75

(h) 63.89

(c) 2.36

(f) 5.55

(i) 12.15



ROUNDING

2.2 Extending the use of the rule of five

When rounding to one decimal place, we always consider the second decimal place and use the rule of five.

Examples 2.2

14.372 = 14.4 to 1 dp

22.316 = 22.3 to 1 dp

57.853 = 57.9 to 1dp

(the 7 is bigger than 5)

(the 3 is smaller than 5)

(the second decimal place is a 5 and 5 or

Questions 2.2

more rounds up)

Round the following numbers to 1 decimal place.

(a) 23.146

(d) 47.352

(b) 42.484

(e) 86.159

(c) 37.168

(f) 77.552

(g) 126.118 (h) 204.301

2.3 Rounding to two decimal places

(i) 222.888

When rounding to two decimal places, we always consider the third decimal place and use the rule of five.

Examples 2.3

14.372 = 14.38 to 1 dp (the 2 is smaller than 5)

22.316 = 22.32 to 1 dp (the 6 is larger than 5)

57.815 = 57.82 to 1dp (the third decimal place is a 5 and 5 or more rounds up)

Questions 2.3

Round the following numbers to 2 decimal places.

(a) 23.171 (b) 14.283

(d) 16.414

(g) 29.915

(e) 73.843

(h) 63.885

(c) 25.317

(f) 15.427

(i) 40.404



ROUNDING

2.4 Rounding in a money calculation

Example 2.4

Share £1264 among 7 people.

£1264 ÷ 7 = £ 180.5714286 This rounds to £180.57 (the 1 is smaller than 5)

Questions 2.4

(a) Share £5647 among 7 people.

(b) Share £15661 among 13 people.

(c) Share £2568 among 11 people.

2.5 The problems of real life

Example 2.5a

£100 is shared between five people. How much does each receive?

£100 ÷ 5 = £20

Example 2.5b

£100 is shared between seven people. How much does each receive?

£100 ÷ 7 = £14.28571429 which rounds to £14.29

Problem! If 7 people all get £14.29 then we need

7 x £14.29 = £100.03

But we only have £100.



ROUNDING

If a fixed sum of money is to be shared out then the answer must always be rounded down. This makes sure that there is enough money to go round. There may be a small amount left over.

Questions 2.5

(a) Share £2263 among 7 people.

(b) Share £6638 among 12 people.

(c) Share £4493 among 7 people.

(d)

Share £15800 among 12 people.

(e) Share £265390 among 13 people.



ROUNDING

2.6 Significant figures

“20 000 attend cup tie!”

Sources within United reported that 20,000 attended last Tuesday’s cup tie…

It is highly unlikely that exactly 20 000 spectators were in the crowd.

The headline tells us that close to 20 000 were there. Closer to 20 000 than 10 000 or 30 000.

The significant (important) figure is the 2 – it is in the tens of thousands column.

Suppose the exact figure was 18 638. This could be rounded to

20 000 showing 1 significant figure

19 000 showing 2 significant figures



18 638 has 5 significant figures

Significant figures are used to give an impression of the size of a large number by rounding. They are counted from the left. The rule of five still applies when rounding.

Examples 2.6

1 Round 8432 to 2 significant figures

We keep the thousands and hundreds. The 3 in the tens column tells us to round down (smaller than a 5).

Answer 8400



ROUNDING

2 Round 7172 to 2 significant figures

We keep the thousands and the hundreds. The 7 in the tens column tells us to round up (larger than a 5).

Answer 7200

3 Round 6312 to 1 significant figure

We keep the thousands. The 3 in the hundreds column tells us to round down (smaller than a 5).

Answer 6000

4 Round 48 617 to 3 significant figures

We keep the tens of thousands, the thousands and the hundreds.

The 1 in the tens column tells us to round down (smaller than a

5).

Answer 48 600

Questions 2.6

Round each of the following numbers to the number of significant figures shown in the brackets.

(a) 732 (2) (b) 8196 (2) (c) 3481 (2)

(d)

(g)

(j)

593 (1)

7341 (3)

21839 (4)

(e) 211 (1)

(h) 15684 (3)

(k) 21839 (3)

(f) 7619 (1)

(i) 27215 (3)

(l) 21839 (2)



SAQ ANSWERS


Answers 2.1

(a) 8.4

(d) 3.6

(g) 24.2

Answers 2.2

(a) 23.1

(d) 47.4

(g) 126.1

Answers 2.3

(a) 23.17

(d) 16.41

(g) 29.92

Answers 2.4

(a) £806.71 each

(b) £1204.69 each

(c) £233.45 each

(b) 5.9

(e) 4.8

(h) 63.9

(b) 42.5

(e) 86.2

(h) 204.3

(b) 14.28

(e) 73.84

(h) 63.89

(c) 2.4

(f) 5.6

(i) 12.1

(c) 37.2

(f) 77.6

(i) 222.9

(c) 25.32

(f) 15.43

(i) 40.40



SAQ ANSWERS

Answers 2.5

(a) £323.28 each

(b) £553.16 each

(c) £641.85 each

(d) £1316.66 each

(e) 20414.61 each

Answers 2.6

(a) 730

(d) 600

(g) 7340

(j) 21840

(b) 8200

(e) 200

(h) 15700

(k) 21800

(c) 3500

(f) 8000

(i) 27200

(l) 22000



FRACTIONS

SECTION 3

3.1 Quick and easy fractions

The title ‘quick and easy’ isn’t supposed to make you think of boil-inthebag rice. It’s just that in the world of fractions (which most people hate) there are some very quick and easy results. We can all understand them. And we should all be able to store them in our brains.

Understanding fractions

1. The symbol for division is just a picture of a fraction.



something



over



something

2. Think about one half. If you take one chocolate bar and divide it between two children then each gets a half. A half is the answer to ‘one divided by two’.

1

2







one

divided by

two



FRACTIONS

Activity 2

1. Since

1 means one divided by two, what do these mean…?

2

(a)

(b)

(c)

1

3

1

4

1

5

(d)

(e)

1

10

1

100

2.

Try a harder activity.

What do you think

2

3

means?



FRACTIONS

Activity 2 - Answers

1. one divided by three one divided by four one divided by five one divided by ten one divided by a hundred

2. 2 divided by 3.

Look at these chocolate bars.

Divide the two chocolate bars equally between three people.

Each person got

2

3

of a chocolate bar.

Two bars were divided among three people.

2 ÷ 3 =

2

3

.



FRACTIONS

Fractions are the numbers we use to describe the results of sharing.

Questions 3.1

Give a fraction answer to each of the following

1. one divided by four

3. two divided by five

5. seven divided by eight

2. one divided by six

4. three divided by four

6. three divided by ten



FRACTIONS

3.2 Fractions to percentages

So you’ve heard yourself say it many times before: ‘OK, you can each have half the chocolate bar’. That’s one bar divided between two.

I guess you could say (though it’s a bit unnatural): ‘You can have 50% of a chocolate bar each.’

Unnatural, but true. (Is that a definition of mathematics?)

1

= 50%

2

We also know that

75% =

3

4

1

4

= 25%

Do we know any others?

How about 75%

75% =

3

4

1

Similarly

10

= 10%

1

Also

100

= 1%



FRACTIONS

Definition

‘per cent’ means per hundred

Results

1

100

= 1%

1

10

= 10%

1

4

= 25%

1

2

= 50%

3

4

= 75% one whole = 100%

More fractions



What about

3

10

? Since

1

10

= 10%, can test this by using a calculator.

3

We know that

3

10

should be 30%. We means 3 ÷ 10. So do 3 ÷ 10 on your calculator.

10

You should get 0.3 This is not exactly 30!

So what is the problem?



FRACTIONS

Slick trick

Multiplying by 1 doesn’t change things.

Check it out…

7



1 = 7

15



1 = 15

3



1 = 3

999



1 = 999

But we know that 1 (one whole) = 100%.

So… (this is the tricky bit)… multiplying by 100 % (that’s 100% and not

100 the number) is the same as multiplying by 1 and doesn’t change the answer.

Read that ag ain, if you’re not still with me.

3

Back to

10

then.

3 ÷ 10 = 0.3 (a decimal)

3 ÷ 10 

100% must equal the same thing because we are only multiplying by one (whole). 

On your calculator do

3 ÷ 10 

100 =

You should get 30.

Important note

Your calculator doesn’t show you that it means 30%. You have to know that for yourself.

Activity

7

Check that

10

= 70% by doing 7 ÷ 10 

100 = on your calculator.



FRACTIONS

Questions 3.2

Use your calculator to convert the following fractio ns to percentages.

Use the routine

TOP ÷ BOTTOM 

100 = every time.

Some answers are whole numbers. Some are decimals, e.g.

3

16

= 18.75%

1.

4.

7.

10.

13.

16.

19.

3

5

7

10

1

5

11

20

1

40

5

16

14

25

2.

5.

8.

11.

14.

17.

20.

1

8

4

5

7

8

9

40

11

50

9

16

4

25

3.

6.

9.

12.

15.

18.

21.

3

10

5

8

9

20

17

40

41

50

33

50

9

25




Answers 3.1

1.

4.

1

4

3

4

Answers 3.2

1. 60%

4. 70%

7. 20%

10. 55%

13. 2.5%

16. 31.25%

19. 56%

2.

5.

1

6

7

8

2. 12.5%

5. 80%

8. 87.5%

11. 22.5%

14. 22%

17. 56.25%

20. 16%

3.

6.

2

5

3

10

3. 30%

6. 62.5%

9. 45%

12. 42.5%

15. 82%

18. 66%

21. 36%

FRACTIONS



PERCENTAGES

SECTION 4

4.1 Finding a percentage without using a calculator

1

Remember that: 10% =

10

(one whole divided by 10)

1

1% =

100

(one whole divided by 100)

Example 4.1a

10% of £60 = £6 (just £60 ÷ 10)

(0.4 = 4 ÷ 10) 10% of 4 = 0.4

Example 4.1b

1% of £6 = 1% of 600p

= 6p

1% of 1573 = 15.73

Tricks of the trade

(600 ÷ 100)

(15.73 ÷ 100)

1. To find 5%, find 10% and half it.

2. To find 3%, find 1% and treble it.

3. To find 15%, find 10% then half that answer to find 5%. Then add your two answers together.

4. To find 2½%, find 5% then half the answer to get 2½%.

Example 4.1c

Find 5% of 80

Example 4.1d

10% of 80 = 8

5% of 80 = 4

1

2

of 8 = 4

Find 15% of 80

Example 4.1e

Find 3% of 220

10% of 80 = 8

5% of 80 = 4

15% of 80 = 12

1% of 220 = 2.2

3% of 220 = 6.6

8 + 4 = 12

2.2



3

6.6



PERCENTAGES

Example 4.1f

Find 40% of 160

Questions 4.1

10% of 160 = 16

40% of 160 = 64

16



4

64

Calculate the following percentages. (Do not use a calculator.)

1. 5% of 60 2. 15% of 64 3. 20% of 160

4. 25% of 800

7. 7% of 300

10. 40% of £400

13.

2½% of 36

16. 17½% of 80

5. 60% of 500

8. 9 0% of £20

11. 75% of £240

14.

7½% of 48

6. 80% of 540

9. 8% of 120

12. 70% of 620

15. 150% of 62

4.2 Finding a percentage by using a calculator

Trick of the trade

So long as we can find 1%, we can find any percentage. Fo r example,

17% is 17 times as much as 1%.

Example 4.2a

Find 17% of £300.

Solution

(First find 1% by dividing by 100. Then multiply by 17).

£300 ÷ 100 = £3

£3  17 = £51



PERCENTAGES

Questions 4.2

Using a calculator work out the following:

1. 24% of 200 2. 16% of 400

4. 30% of 180

7. 95% of 400

10. 16% of 4000

5.

8.

16% of 2000

6% of 360

11. 12.5% of 880

13. 99% of 2000

16. 125% of 600

14. 40% of 36

4.3 Percentage profit and loss

3. 55% of 750

6.

9.

80% of 1600

28% of 480

12. 15% of 96

15. 17.5% of 400

Percentage profit =

Profit

Cost price



100%

Loss

Percentage loss =

Cost price



100%

Note: the cost price is t he amount of money paid by the ‘shopkeeper’ for the item being sold. Sometimes it is called t he ‘Buying Price’.

Example 4.3a

An antique dealer buys a vase of £60 and sells it for £72. Calculate his percentage profit.

Profit = £72 – £60

= £12

Percentage profit =

Profit

Cost price



100%

12

= 100%

60



= 20%



PERCENTAGES

Example 4.3b

A shop buys coke cans for 24p each and sells them for 27 p each.

Calculate the percentage profit on each can of coke sold.

Profit = 27p –24p = 3p

Percentage profit =

Profit

Cost price



100%

3

= 100%

24



= 12.5%

Questions 4.3

For each of the following, calculate the actual profit or loss and the percentage profit or loss.

Cost price Selling price

1. £2.50

2.

£20

3. £15

4. £4.30

5. £24.80

6. £6.30

7. £15

8. £5.20

9. £126

10. £9.40

£2.70

£23.60

£11.25

£3.44

£31

£8.19

£17.55

£4.42

£157.50

£12.69



PERCENTAGES

4.4 Percentage errors, increases and decreases

Error

Percentage error =

Actual amount



100%

Increase

Percentage increase =

Original amount



100%

Decrease

Percentage decrease =

Original amount



100%

Example 4.4

In an experiment Peter calculated the weight of carbon to be 15g instead of 12g. Calculate his percentage error.

Error = 15 – 12 = 3g

Percentage error =

Error

Actual amount



100%

=

3



100%

12

= 25%

Questions 4.4

1. When measuring up a room for the new carpet, Ross measured the length of the room as 3.78m instead of 3.6m.

(a) What was the actual error in his measurement?

(b) What was the percentage error in his measurement?

2. A box of matches should contain 48 matches.

One box is selected at random and found to contain only 45 matches.

(a) What is the actual error in the number of matches contained in the box?

(b) What is the percentage error in the number of matches contained in the box?



PERCENTAGES

3.

Helen’s salary increases from £18000 to £19080.

Calculate:

(a) the actual increase in her salary.

(b) the percentage increase in her salary.

4. A newspaper’s readership increases from 64000 to 66000.

Calculate:

(a) the actual increase in readership.

(b) the percentage increase in readership.

5. In order to help support a military campaign, the Chancellor of th e

Exchequer reduced the spending of the education department from £800,000,000 to £720,000,000.

Calculate:

(a) the actual reduction in spending.

(b) the percentage reduction in spending.

6. In a sale, the price of a dress decreases by £12. If its original price was £80 then calculate the percentage reduction.



SAQ ANSWERS


Answers 4.1

1. 3

4. 200

7. 21

10. £160

13. 0.9

16. 14

Answers 4.2

1. 48

4. 54

7. 380

10. 640

13. 1980

16. 750

Answers 4.3

Actual profit/loss

1. £0.20 profit

2. £3.60 profit

3.

£3.75 loss

4. £0.86 loss

5. £6.20 profit

6. £1.89 profit

7. £2.55 profit

8. £0.78 loss

9. £31.50 profit

10. £3.29 profit

2. 9.6

5. 300

8. £18

11. £180

14. 3.6

2. 64

5. 320

8. 21.6

11. 110

14. 14.4

Percentage

8%

18%

25%

20%

25%

30%

17%

15%

25%

35%

3. 32

6. 432

9. 9.6

12. 434

15. 93

3.

6.

412.5

1280

9. 134.4

12. 14.4

15. 70



SAQ ANSWERS

Answers 4.4

1. (a) 0.18m

2. (a) 3 matches

3. (a) £1080

4. (a) 2000

5. (a) £80,000,000

6. 15%

(b) 5%

(b) 6.25%

(b) 6%

(b) 3.125%

(b) 10%



EARNINGS

SECTION 5

This section applies all the skills of the previous sections to our earnings.

It is vital that we are able to do calculations relating to the money we earn. This is part of budgeting.

5.1 Earning a wage

Here, we will look at many of the key calculations that are used in working out how much we should be paid. There are many specialist terms that we need to get to know. These will be defined as we go along.

Example 5.1a

Sukhjinder works 38 hours per week and is paid £6.30 per hour.

Calculate his gross pay.

Solution

38 x £6.30 = £239.40 per week.

Gross pay is money earned before deductions.

Deductions are amounts of money taken off before we receive our pay, e.g. income tax or company pension scheme payment.

Example 5.1b

Stephanie works 40 hours at £7 per hour and 3 hours overtime at time and a half . Calculate her pay for the week.

Solution

Overtime rate = 1.5 x £7 = £10.50 (time and a half = 1.5 x normal rate)

Basic pay: 40 x £7 = £280

Overtime: 3 x £10.50 = £ 31.50

Total pay £311.50



EARNINGS

Overtime means you are paid more per hour for each extra hour you work.

Example 5.1c

Colin works 36 hours at £8.40 per hour and 9 hours overtime at time and a quarter . Calculate his earnings for the week.

Solution

Overtime rate = 1.25 x £8.40 = £10.50

36 x £8.40 = £302.40

9 x £10.50 = £ 94.50

Total pay = £396.90

£8.40 ÷ 4 = £2.10 £8.40 + £2.10 = £10.50

Example 5.1d

Janice is a sales assistant, selling fashion sweaters. She earns a basic wage of £120 per week plus £3 for every sweater she sells. One week she sells 23 sweaters. Calculate her earnings for that week.

Solution

23 x £3 = £69

Example 5.1e

£120 + £69 = £189

Alan has a job as a car salesman. The garage pays him no basic wage at all. He only earns money if he sells cars and he is paid 4% of the value of his sales. In one year he sells cars to a total value of £560 000.

Calculate his earnings for the year.

Solution

£560 000 ÷ 100 x 4 = £22 400



EARNINGS

Examples 5.1d and 5.1e involved commission .

Commission is the money a sales person earns from selling. It may be just part of your earnings – or all of it.

Example 5.1f

Shamshad has a part-time job in a factory. She earns a basic wage of

£63 per week. She also earns money for the components she assembles.

However this part of her wage is only paid for the number of components above 200 that she assembles. Calculate her wage in a week when she assembles 580 components.

Solution

Basic wage £63.00

380 x £0.03 £11.40

Total wage £74.40

Note:

Enter 3p as 0.03 on the calculator (otherwise the calculator reads it as

£3)

Being paid for each piece of work that you do is called piecework .



EARNINGS

Questions 5.1

1. Kit earns £7.20 per hour. Calculate his gross pay for a 36-hour week plus 5 hours’ overtime at time and a quarter.

2. Laura earns £6.50 per hour for a 35-hour basic working week.

Calculate her gross pay in a week when she works 37 hours.

(Overtime is paid at time and a half.)

3. Brian earns £12 per hour. He normally works 36 hours per week.

The first two hours of any overtime is paid at normal rate. After that, any more hours are paid at double time. Calculate his gross pay for a 43-hour week.

4. Jimmy works on the assembly line of a factory. He is expected to assemble at least 800 components per week. He earns 4p for every component assembled above this number. He also earns a basic wage of £81 per week. Calculate his gross pay in a week when he assembles 2130 components.

5. Jenna sells weapons to the armed forces. She is paid 1.2% of the total value of her sales. Calculate her annual salary in a year when she sells £4,600,000 of weapons.

6. Andrew earns £1350 per month plus 3% of his sales. Calculate his gross pay in a month where his sales total £19000.

7. Kris earns £6.80 per hour for a 36-hour week. He receives a 5% bonus on his basic pay. He also w orks 7 hours’ overtime at time and a quarter. Calculate his gross pay for that week.



EARNINGS

5.2 Pay increases

Pay increases are often announced on the news as percentages. It is important that we are able to work out what that means for us.

Example 5.2a

Martin earns £23430 per year as a fire-fighter. He is awarded a 4.2% pay increase. Calculate his new annual salary.

Solution

£23430 ÷ 100 x 4.2 = £984.06 (this would be rounded to £984 for an annual salary)

£23430 + £984 = £24414

Example 5.2b

Work ou t what Martin’s old and new pay per month will be. Also work out the monthly increase in his wages.

Solution

£23430 ÷ 12 = £1952.50 per month

£24414 ÷ 12 = £2034.50 per month

Monthly increase = £2034.50 – £1952.50 = £82

Questions 5.2

1. Jonathon earns £21750 per year. Calculate his new annual salary after a 2.3% pay increase.

2. Ross earns £31320 per annum (per year). Calculate his monthly increase if he gets a pay rise of 3.4%.

3. Emma earns £5.60 per hour. She is also paid 20p for every component she assembles.

(a) Calculate her new hourly rate after an increase of 5%.

(b) What should she be paid for each component assembled if her piecework rate is also increased by 5%?.



EARNINGS

5.3 Dealing with income tax

The whole area of income tax is very complex. Some accountants are specialists in income tax. We will study the basics of the system, but we will need to take account of many different situations.

For many people, calculating how much income tax they should pay can be quite straightforward. However, if someone has a few jobs, with different methods of payment, or perks (e.g. a company car) or is married then the system can be more complicated.

Some things you should know:

Superannuation

If you pay into a superannuated pension scheme then you do not pa y income tax on the contribution.

Tax allowance

Everyone is allowed to earn a certain amount of money that is not taxed – called your Personal allowance. The amount depends on your circumstances. For example, pensioners are entitled to a higher Personal allowance.

Tax code

Your tax code is shown on your pay slip and is usually made up of three figures and a letter, e.g. 379L or 401P. An L indicates the basic Personal allowance while P signifies the full Personal allowance for those aged 65 to 74. The 3 79 and 401 mean that the people may earn £3790 and £4010 respectively before tax, i.e. this is their tax allowance.

Up-to-date tax allowances can be found in Inland Revenue publications and at www.inlandrevenue.gov.uk/leaflets/p3.htm#e.

In 2004 the Per sonal allowance basic amount is £4745.

Tax allowances change every year in the budget. There are other types of allowances depending on people’s jobs and circumstances.

Tax rate

Percentage tax rates also change in the budget. The rates we will use are:

Starting rate

Basic rate

Higher rate

10%

22%

40%.



EARNINGS

Taxable income

This is equal to your gross pay less your tax allowances. It is this amount which the Inland Revenue taxes (not your whole salary).

Examples

£4745.

Example 5.3a

These examples use the basic Personal allowance of

Nicola earns £10 985 p.a.* and has an additional personal allowance of

£530. Calculate her taxable income.

*p.a. stands for per annum and means ‘in each year’

Solution

Allowances 4745

+530

5275

Taxable Income = £5710

Salary 10985

–5275

5710

Example 5.3b

Mairi works parttime and earns £6545 each year. Calculate the tax she pays.

Solution

Salary

Tax allowance

6545

–4745

Taxable income 1800

Her taxable income is less than £1960 so the Starting rate of 10% applies.

Tax paid = £1800 ÷ 100 x 10 = £180



EARNINGS

Example 5.3c

Mark earns £14200 p.a. Calculate the tax he pays.

Solution

Salary

Tax allowance

14200

–4745

9455

Taxable income 9455

Threshold for Basic rate –1960

To be taxed at Basic rate 7495 Taxable income

Tax payable = 10% of £1960 + 22% of £7495

= £196 + £1649

= £1845

Example 5.3d

James earns £10833 p.a. Calculate the tax he must pay per month.

Solution

Annual salary

Tax allowance

10833

–4745

Taxable income (for whole year) 6088

Threshold for Basic rate

To be taxed at Basic rate

–1960

4128

£1960 ÷ 100 x 10 + £4128 ÷ 100 x 22 = £196 + £908 = £1104

Tax paid per month = £1104 ÷ 12 = £92

Example 5.3d

Rhona earns £15000 p.a. and pays 6% of her salary in superannuation

(superannuation is non-taxable).

Calculate:

(a) her taxable income

(b) her take-home pay.

Solution

(a) 6% of £15000 = £900, Tax allowance = £4745

900

+4745

15000

–5645

Total non-taxed income £5645 Taxable income = £9355



EARNINGS

(b)

Taxable income

Threshold for Basic rate

9355

–1960

£7395 To be taxed at Basic rate

£1960 ÷ 100 x 10 + £7395 ÷ 100 x 22 = £196 + £1627 = £1823

Takehome pay = £15000 – £900 – £1823 = £12277

Questions 5.3

Refer to the tax allowances mentioned in the notes.

1. Mo earns £18200 per annum. She has an additional personal tax allowance of £250.

Calculate:


(b) her tax paid

(c) her take-home pay for the year.

2. Jennifer earns £27000 per annum. She pays 5% of this as superannuation (before tax).

Calculate:


(b)

(c) her tax paid her take-home pay for the year.

3.

Scott earns £13410 per year. He pays £660 per year into a pension fund (before tax).

Calculate

(a) his taxable income

(b) his tax paid

(c) his annual take-home pay

(d) his weekly take-home pay.



EARNINGS

5.4 Higher tax rate

If your taxable income is above a certain amount (called the threshold ) then you move into a higher tax band. Only the difference is taxed at the higher rate. In 2004 the higher rate tax threshold was £30500 and the higher rate was 40%.

On this basis a person with a taxable income of £35000 would be taxed as follows:

10% of £1960

22% of £28540 (£30500 – £1960 = £28540)

40% of £4500 (£35000 – £30500 = £4500)

Example 5.4a

Calculate the takehome salary of someone earning £38000 p.a.

Solution

Salary

Tax allowance

38000

–4745

Taxable income

Higher threshold

33255

–30500

Taxable income 33255 To be taxed at higher rate 2755

Tax payable = 10% of £1960 + 22% of £28540 + 40% of £2755

= £196 + £6279 + £1102

= £7577

Takehome salary = £38000 – £7577 = £30423

Example 5.4b

A man earns an annual salary of £36705. He has an additional personal tax allowance of £210.

Calculate

(a) his taxable income

(b) his tax paid in one year

(c) his net monthly salary.



EARNINGS

Solution

(a) Personal allowance

Total tax allowance

4745

Additional personal allowance +210

36705

–4955

4955 Taxable income £31750

(b) Tax paid = 10% of £1960 + 22% of £28540 + 40% of £1250

= £196 + £6279 + £500

= £6975

(c) Net annual salary = Gross annual salary – Tax paid

= 36705 – 6975

= £29730

Net monthly salary

Questions 5.4

= £29730 ÷ 12

= £2477.50

1. Calculate the annual takehome salary of someone earning £37200 per annum.

2. Calculate the takehome salary of a man earning £43200 per annum.

3. A woman earns £38700 per annum. She pays 6% superannuation

(before tax).

Calculate

(a)

(b)

(c) her taxable income her tax paid her take-home pay per month.



EARNINGS


Answers 5.1

1. £304.20

4. £134.20

7. £316.54

2. £247

5. £55200

Answers 5.2

1. £22250 (to the nearest £)

2. £88.74

3. (a)

£5.88

(b) 21p

Answers 5.3

1. (a) £13205

2. (a) £20905

3. (a) £8005

(d) £215.85

(b) £2669.90

(b) £4363.90

(b) £1525.90

3. £576

6. £1920

(c) £15530.10

(c) £21286.10

(c) £11224.10



Answers 5.4

1. £37200 – £4745 = £34455 (taxable income)

£34455 – £30500 = £1955

10% of £1960 = £196

22% of £28540 = £6279

40% of £1955 = £782

Tax paid = £196 + £6279 + £782 = £7257

£37200 – £7257 = £29943

2. £43200 – £4745 = £38455

£38455 – £30500 = £7955

10% of £1960 = £196

22% of £28540 = £6279

40% of £7955 = £3182

Tax paid = £196 + £6279 + £3182 = £9657

£43200 – £9657 = £33543

3. (a) £31633

(b)

£6928 (to the nearest £)

(c) £2454.15

EARNINGS



WORKING WITH MONEY

SECTION 6

Working with money is one of the most important skills we can learn.

Very often we will have the use of a calculator, a computer or a checkout till to help us. However, knowing what calculations to do and how to do them is key to many jobs.

6.1 Working with VAT

In this section we will be using multiplication and percentages mostly. We will often be adding value added tax (VAT) to amounts. When we do this, we will always use 17.5% for the VAT.

When adding VAT to a number of items it is always quicker to add up the total first. Then we need only calculate the VAT once, at the end.

It is important to remember that most prices displayed in shops already include VAT. However, some do not. This is often the case when ordering goods over the internet or by post.

Example 6.1a

A computer costs £980 + VAT. Calculate:

(a) the VAT to be paid

(b) the total price

Solution

VAT = £980 ÷ 100 x 17.5 = £171.50

Total price = £980 + £171.50 = £1151.50




Example 6.1b

A stereo costs £350 + VAT. Post and packing is added at a cost of £8.95.

Calculate:

(a) the VAT to be paid

(b) the total price

Solution

(a) VAT = £350 ÷ 100 x 17.5 = £61.25

(b) Total price = £350 + £61.25 + £8.95 = £420.20

Note: VAT is not charged on the post and packing, only on the goods.

Questions 6.1

In questions 1 to 6, calculate (a) the VAT and (b) the total price.

1. £200

4. £26

2. £750

5. £102

3. £840

6. £12.40

7. Goods valued at £103, £88, £14 and £23 are bought.

Calculate the total bill after VAT has been added.

8. A scanner costs £98 + VAT + £3.50 post and packing.

Calculate the total cost to be paid when ordering the scanner.




6.2 Practical problems

Everyday life and work is full of practical problems involving four rules calculations. We need to look at problems involving lengths, areas and volumes, as well as weight and temperature. Of course, we will also come across money calculations in all this as well.

Example 6.2a

A room is 4m long by 3.6m broad. Calculate:

(a) the area of the floor

(b) the cost of carpeting the room with carpet priced at £8.65 per square metre.

Solution

(a) 4m x 3.6m = 14.4 m 2

(b) 14.4 x £8.65 = £124.56

Example 6.2b

A room is 4m long by 3.6m broad. It is 2.8m high. Calculate:

(a) the perimeter of the floor.

(b) the area of all the walls

(c) the number of litres of paint needed to cover all the walls if one litre covers 15m 2 . (Don’t worry about the windows and doors!)

(d) the number of tins to be bought if paint comes in 2 litre tins

(e) the cost of the paint if each tin costs £9.55

Solution

(a) 4m + 3.6m + 4m + 3.6m = 15.2m

(b) 15.2m x 2.8m = 42.56 m 2 (round this up to 43 m 2 )

(c) 43 ÷ 15 = 2.866 … round this up to 3 litres

(d) 2 tins

(e) £9.55 x 2 = £19.10




Questions 6.2

1. A room is 3.4m long by 3.8m broad.

(a) Calculate the area of the floor.

(b) Calculate the cost of carpeting the room with carpet costing

£9.50 per m 2 .

(c) How much more would it cost to carpet the room with carpet costing £11.65 per m 2 ?

2. A classroom is 8m long by 7.2m broad. It is 2.9m high.

Calculate:

(a)

(b) the perimeter of the floor the total area of all the walls

(c) the number of litres of paint to be bought if four such classrooms are to be painted and one litre covers 22m 2 .

(d) the number of tins to be bought if paint comes in 5 litre tins

(e) the cost of the paint if each tin costs £18.50.

6.3 Hire purchase

Hire purchase agreements are sometimes called:



paying it up



buying on the never-never!

Definitions

Deposit an initial payment of part of the price of the goods

Instalments weekly or monthly payments made to ‘pay up’ the rest of the price

Cost of credit how much more the HP payment method costs, above the cash price




Example 6.3a

A sound system costs £349 cash OR a deposit of £50 and 12 monthly instalments of £29.

Calculate:

(a) the total HP price

(b) the cost of credit

Solution

(a) Deposit £50.00

12 x £29 £348.00

Total HP price £398.00

Example 6.3b

(b) £398 – £349 = £49

A computer costs £995 cash OR a 10% deposit and 24 monthly instalments of £41.

Calculate:

(a) the total HP price

(b) the cost of credit

Solution

(a) Deposit

24 x £41

£99.50

£984.00

Total HP price £1083.50

(b) £1083.50 – £995 = £88.50

(£995 ÷ 100 x 10)




Questions 6.3

1. Work out the HP costs of items bought under the following terms:

(a) Deposit £40, 12 instalments of £18.

(b) Deposit £250, 24 instalments of £40.

(c) Deposit £8, 52 instalments of £3.50.

(d) Cash price £260. Deposit 10% of cash price and 6 instalments of £45.

(e) Cash price £3250. Deposit 20% of cash price and 24 instalments of £120.

2. How much cheaper is the cash price than the HP price in questions

1(d) and 1(e)?

3. A dishwasher can be purchased for £390 cash. On HP a deposit of

10% is required followed by 12 monthly instalments of £29.25.

(a) Calculate the total HP price.

(b) Comment on your answer.



SAQ ANSWERS


Answers 6.1

1. (a) £35

2. (a) £131.25

3. (a) £147

4. (a) £4.55

5. (a) 17.85

6. (a) £2.17

7. £267.90

8. £118.65

(b) £235

(b) £881.25

(b) £987

(b) £30.55

(b) £119.85

(b) £14.57

Answers 6.2

1. (a) 12.92 m 2

(b)

£122.74

(c) £27.78

2. (a) 30.4m

(b) 88.16 m 2

(c) 16.03 litres

(d) 4 tins

(e)

£74

Answers 6.3

1. (a) £256

(b) £1210

(c) £1.90

(d) £296

(e)

£3530

2. £36, £280

3. (a) £390

(b) HP price is the same as cash price. (This is sometimes called

0% interest.)



CALCULATIONS WITH TIME

SECTION 7

7.1 Lengths of time

‘How long does the film last?’

‘How many hours have I worked?’

We all need to be able to calculate lengths of time. This s ection is concerned with making awkward calculations simpler. We won’t be dealing with questions like, ‘How long from 10 am to 1 pm?’ That’s too easy.

Example 7.1a

Dyce

Aberdeen

Portlethen

Stonehaven

Montrose

Arbroath

Carnoustie

Monifieth

Broughty Ferry

Dundee

Invergowrie

Perth

Gleneagles

Dunblane

Stirling

Larbert

Lenzie

Glasgow Queen St

0650

0707

…

0723

0745

0759

0806

…

…

0820

…

0842

…

…

…

…

…

0938

0908

0939

…

0955

1017

1031

1038

…

1044

1051

…

1113

1148

1201

1143

1158

1216

1215

How long is the 09.08 journey from Dyce to Glasgow Queen Street?




Solution

Method 1 (Count it out)

09.08 to 09.15 = 7 mins

09.15 to 12.15 = 3 hours

Total time = 3 hours 7 mins

Method 2 (Calculation)

Take the start time away from the finish time.

12.15

– 09.08

3.07 3 hours 7 mins

Note

Never give an answer to this kind of question as 03.07.

03.07 is a time of day

3 hours 7 minutes is a length of time

Example 7.1b (trickier)

How long is the 06.50 journey from Dyce to Glasgow Queen Street?

Solution

Method 1 (Count it out)

06.50 to 07.00 = 10 mins

07.00 to 09.00 = 2 hours

09.00 to 09.38 = 38 mins

Total time = 2 hours 48 minutes




Method 2 (Calculation)

Take the start time away from the finish time.

09.38

– 06.50

Problem! You can’t take 50 from 38.

Here, we use a clever trick. We do the minutes ’ calculation off to one side, but first we ‘steal’ an hour.

Changes to 08 09.38 + 60 mins

– 06.50

98

– 50

48

08

– 06

02

Answer 2 hours 48 minutes

Questions 7.1

Calculate these lengths of time.

1. From 07.13 to 11.48

2. From 13.22 to 20.55

3. From 03.46 to 08.29

4. From 17.23 to 23.03

5. From 10.27 to 16.11

6. From 18.51 to 22.17

7. From 06.29 to 21.13

8. From 21.13 to 06.29 next day

Hint for question 8. pretend that 06.29 next day is 30.29 today (!), i.e. add 24 hours to the 6 hours.




7.2 Calculating speed

Speed = Distance ÷ Time

Example 7.2a

Sam drove 189 miles in 4½ hours. Calculate her average speed.

Solution


= 189 ÷ 4.5

= 42 mph

N otice that the ‘units’ all tie up.

DISTANCE

SPEED miles hours miles per hour

Example 7.2b

Geoff runs 100m in 12.5 seconds. Calculate his speed.

Solution


= 100 ÷ 12.5

= 8 m/sec

TIME




Questions 7.2

Calculate the speed of each of these journeys.

1. 78 km in 6 h

4. 60 km in 1½ h

7. 1500 km in 1¼ h

2. 133 km in 2 h

5. 180 km in 2¼ h

8. 750 km in 1½ h

7.3 Calculating distance

3. 187 km in 5 h

6. 280 km in 2h 30 min

9. 560 km in 1h 45 min

Distance = Speed



Time

Example 7.3

A train travels across the American prairies at a steady speed of 85 mph.

If it travels for 9 hours non-stop, how far will it get?

Solution

Distance = Speed



Time

= 85



9

= 765 miles

Questions 7.3

Calculate the distanced travelled.

1. 6 h at 80 km/h

2. 3 h at 100 km/h

3. 5 h 30 min at 60 km/h

4. 2½ h at 78 km/h

5. 3½ h at 112 km/h

6. 7 h 45 min at 42 km/h12.

7. 7 h at 30.2 km/h

8. 4 h at 71.6 km/h

9. 5½ h at 52.8 km/h

10. 160 km/h for 2¼ h

11. 220 km/h for 1½ h

12. 300 km/h for 3 h 45 min




7.4 Calculating time

Time = Distance ÷ Speed

Example 7.4

To test the endurance of a new car, it is run on a test track at a steady 80 mph until it has covered exactly 260 miles. How long does this time trial last?

Solution

Time = Distance ÷ Speed

= 260 ÷ 80

= 3.25 hours

= 3 hours 15 minutes

Note

0.25 of an hour = ¼ of an hour = 15 minutes

Questions 7.4

Find the time taken to do these journeys.

1.

2.

3.

98 km at 14 km/h

150 km at 7.5 km/h

750 km at 12.5 km/h

4. 292.5 km at 30 km/h

5. 270 km at 45 km/h

6. 370 km at 40 km/h

7. 600 km at 80 km/h

8. 750 km at 500 km/h

9. 325.5 at 62 km/h



SAQ ANSWERS


Answers 7.1

1. 4 hours 35 minutes








Answers 7.2

1. 13 km/h

4. 40 km/h

7. 1200 km/h

Answers 7.3

1. 480 km

2. 300 km

3. 330 km

4. 195 km

5. 392 km

6. 325.5 km

Answers 7.4

1. 7 hours

2. 20 hours

3. 60 hours

4. 9 hours 45 mins

5. 6 hours

2. 66.5 km/h

5. 80 km/h

8. 500 km/h

7. 211.4 km

8. 286.4 km

9. 290.4 km

10. 360 km

11. 330 km

12. 1125 km

3. 37.4 km/h

6. 112 km/h

9. 320 km/h

6. 9 hours 15 mins

7. 7 hours 30 mins

8. 1 hour 30 mins

9. 5 hours 15 mins



RATIO

SECTION 8

8.1 Ratio as comparison

The simplest use of ratios is when comparing things.

Johnny was paid £5 per hour, as compared to Sandra who was paid

£8 per hour.

Here the total wages bill was shared out in the ratio 5 : 8.

A bag of sweets is shared between Dave and Nancy. Dave is given two sweets for every one that Nancy is given.

Here the sweets are shared out in the ratio 2 : 1.

Ratios can be used to describe unequal division.

Activity

Consider the following situations. What do they have in common?

John gets 12 sweets when Elaine gets 6.

Allan gets 8 sweets when Joyce gets 4.

Tom gets 10 sweets when Jennifer gets 5.

Angus gets 2 sweets when Pauline gets 1.

Answer

Well, they’re all about handing out sweets. The boys always get more than the girls. But, did you notice the crucial similarity? The boys always get twice as many as the girls (or the girls got half as many as the boys).

This comparison is true in all four situations.



RATIO

Boys

12

8

10

:

:

:

:

2 :

These ratios are all equal.

12 : 6 = 2 : 1 8 : 4 = 2 : 1

Girls

6

4

5

1

10 : 5 = 2 : 1

Ratios are used to make comparisons.

When we write a ratio such as 12 : 6 in the form 2 : 1, we are writing it in its simplest form .

12 : 6 becomes 2 : 1 by dividing both parts of the ratio by the largest number we can find which will work.

12 can be divided by 6

6 can be divided by 6

12

÷ 6

2

: 6

÷ 6

: 1

If you use a smaller number we get:

12

÷ 3

4

: 6

÷ 3

: 2



RATIO

But 4 : 2 is not in simplest form.

4

÷ 2

: 2

÷ 2

: 1 2

Questions 8.1

1. Write these ratios as simply as possible.

(a) black dots to white dots

(b) ticks to crosses

(c) a’s to z’s

(d) squares to circles

(e) m’s to n’s

(f) ticks to crosses

(g) £ signs to $ signs

(h) 6’s to 9’s

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£ £ £ £ £ £ £ £

6 6 6 6 6

6 6 6 6 6 n n n n

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RATIO j g d a

2. For each diagram, write as simply as possible the ratio of shaded pieces to unshaded pieces. b c e h k i l f

3. Write these ratios in their simplest forms

(a) 2 : 8 (b) 4 : 6 (c) 7 : 21

(e) 9 : 12

(i) 15 : 5

(f) 6 : 18

(j) 36 : 6

(g) 3 : 12

(k) 35 : 10

(d) 10 : 15

(h) 8 : 4

(l) 18 : 12



RATIO

8.2 Mixed units

We know that 12 : 6 = 2 : 1

So in length, 12 cm : 6 cm in weight, 12 kg : 6 kg

in capacity, 12 litres : 6 litres

= 2 : 1

= 2 : 1

= 2 : 1

Each first quantity is twice the size of each sec ond quantity.

But what about 12 m : 6 cm?

12 m is not twice the length of 6 cm.

We must make the units the same before we can make a true comparison.

12 m : 6 cm

= 1200 cm : 6 cm

÷ 6 ÷ 6

= 200

Example 8.2a

: 1

Write 5p : £1 in its simplest form.

Solution

5p : £1

= 5p : 100p

÷ 5 ÷ 5

= 1 : 20



RATIO

Example 8.2b

Write 1cm : 4mm in its simplest form.

Solution

1 cm : 4mm

= 10 mm : 4 mm

÷ 2 ÷ 2

= 5 : 2

Questions 8.2

Write these ratios in their simplest forms.

(a)

(d)

(g)

1 mm : 1 cm

2 mm : 1 cm

3 min : 1 hour

(b) 1 day : 1 week (c) 1p : £1

(e) 6 days : 2 weeks (f) 2p : £1

(h) 500 cm 3 : 1 litre

(k) 4 mm : 2 cm

(i) 100 g : 1 kg

(l) 50p : £3 (j) 250 m : 1 km

(m) 30 min : 2 hours (n) 6 days : 3 weeks (o) 100 cm 3 : 2 litres

(p) 20 min : 3 hours (q) 40p : £1.60 (r) 50p : £1.25

(s) 250 g : ½ kg (t) 8 mm : 3.6 cm (u) 64p : £2.40



RATIO

8.3 Proportional division

Here is a recipe for slimeade.

Take 3 litres of semi-skimmed milk and 2 litres of limeade.

Mix together. Chill. Enjoy.

Clearly 3 litres + 2 litres = 5 litres. This is a recipe for 5 litres of slimeade.

If you only want a taste first then use this recipe:

3 spoonfuls of semi-skimmed milk

2 spoonfuls of limeade

This makes 5 spoonfuls of slimeade.

Notice that the quantities have changed, but the proportions have not.

The ratio of semi-skimmed milk to limeade in both cases is 3 : 2.

Now we turn the problem round. Suppose we want to make 10 pints of slimeade. How much of each ingredient should we start with? This problem is addressed in the next example.

Example 8.3a

Share 10 pints in the ratio 3 : 2.

Solution

Step 1 count the ‘parts’

Step 2 divide total by number of parts

3 + 2 = 5 parts

10 pints ÷ 5 = 2 pints

Step 3 allocate the parts

3 parts semi-skimmed milk

2 parts limeade

Step 4 double check

6 pints + 4 pints = 10 pints

= 3

= 2





2 pints = 6 pints

2 pints = 4 pints



RATIO

Example 8.3b

Share £84 in the ratio 3 : 4

Solution

3 + 4 = 7 parts

£84 ÷ 7 = 12

3

 £12 = £36 4

 £12 = £48

(check = £36 + £48 = £84)

£84 shared in the ratio 3 : 4 is £36 : £48

Questions 8.3

Share the following quantities in the ratio given:

1. £1200 in the ratio 2 : 1

2. £549 in the ratio 5 : 4

3. 775 kg in the ratio 2 : 3

4. 68 m in the ratio 3 : 1

5. 102 litres in the ratio 5 : 7

6.

£225.60 in the ratio 1 : 5



RATIO

8.4 Direct proportion

Activity

Decide which is the better buy:

2 chocolate biscuits cost 24p

5 chocolate biscuits cost 60p

Answer

Did you find that neither was the better buy? Each deal was as good as the other.

If you did, you probably worked out the cost of one chocolate biscuit.

In direct proportion problems, finding the cost of one is very important.

Definition

Two quantities are in direct proportion to each other if a change in one gives the same change in the other.

[NB The ‘change’ must be multiplication or division]

Example 8.4a

If 7 CDs cost £63, find the cost of 3 CDs.

Solution

1 CD costs £63 ÷ 7 = £9

3 CDs cost £9  3 = £27



RATIO

Example 8.4b

Anne walks 8 miles in 2 hours. How far should she walk in 3 hours?

Solution

1 hour = 8 miles ÷ 2 = 4 miles

3 hours = 4 miles



3 = 12 miles

Example 8.4c

Apples cost 84p for 7. How much do 4 cost?

Solution

1 costs = 84 ÷ 7 = 12p

4 cost = 12p



4 = 48p

Questions 8.4

1. 8 books cost £52.

2.

3 cars cost £36000.

3. 36 eggs cost £5.04.

4. 12 books cost £71.40.

5. 3 oranges cost £54p.

How much will 11 cost?





6. 6 sacks of potatoes cost £46.80. How much will 1½ sacks cost?



RATIO


Answers 8.1

1. (a) 1 : 2

(e) 3 : 1

2. (a) 3 : 1

(e) 3 : 1

(i) 1 : 3

3. (a) 1 : 4

(e) 3 : 4

(i) 3 : 1

Answers 8.2

(a) 1 : 10

(d) 1 : 5

(g) 1 : 20

(j) 1 : 4

(m) 1 : 4

(p) 1 : 9

(s) 1 : 2

(b) 1 : 3

(f) 2 : 1

(b) 2 : 1

(f) 5 : 1

(j) 1 : 2

(b) 2 : 3

(f) 1 : 3

(j) 6 : 1

(b) 1 : 7

(e) 3 : 7

(h) 1 : 2

(k) 1 : 5

(n) 2 : 7

(q) 1 : 4

(t) 2 : 9

Answers 8.3

1. £800 : £400

2. £305 : £244

3. 310 kg : 465 kg

4. 51 m : 17 m

5. 42.5 litres : 59.5 litres

6. £37.6 : £188

(c) 1 : 5

(g) 4 : 1

(c) 3 : 1

(g) 1 : 3

(k) 1 : 3

(c) 1 : 3

(g) 1 : 4

(k) 7 : 2

(d) 1 : 2

(h) 5 : 1

(d) 5 : 3

(h) 1 : 5

(l) 1 : 3

(d) 2 : 3

(h) 2 : 1

(l) 3 : 2

(c) 1 : 100

(f) 1 : 50

(i) 1 : 10

(l) 1 : 6

(o) 1 : 20

(r) 2 : 5

(u) 4 : 15



RATIO

Answers 8.4

1. £71.50

2. £60000

3. £0.84

4. £29.75

5. 72p

6. £11.70



FORMULAS

SECTION 9

9.1 The four rules of formulas

Definition

A formula is a mathematical rule written using letters to stand (as code) for numbers.

We need to know the four rules.

Addition: use a ‘+’ sign

Subtraction: use a ‘–‘ sign

Multiplication: write letters next to each other

Division: write letters one over the other, just like in a fraction.

Example a a ab a b

+

– b b

(means a

 b

(means a



b)

)

Example 9.1a

The cost of paying two people is a + b , where a is the wage of the first person and b is the wage of the second person. Calculate the cost of paying one person £100 and another person £120.

Solution a + b

= 100 + 120

= 220

Answer £220



FORMULAS

You do realise, I hope, that I just made a formula up when all you had to do was add two numbers together in your head.

That’s how easy formulas can be!

But be careful. Strictly speaking we have to say what we are calculating

– in this case, the total wage bill. Let’s call it W . So that:

W = a + b

= 100 + 120

= 220

( W = 220 when a = 100 and b = 120)

Example 9.1b

Calculate e if e = f

– g and f = 75, g = 10.

Solution e = f – g

= 75 –10

= 65

Example 9.1c

Calculate p if p = qr and q = 3, r = 5.

Solution p = qr

= 3



5

= 15

(Remember that

(the answer is qr not

means q

 r )

‘35’, a ‘3’ beside a ‘5’!!!)

Example 9.1d

Calculate m if m = 4 n and n = 11.

Solution m = 4 n

= 4



11

= 44

(here the 4 in front of the n means 4

 n )



Example 9.1e

t

Calculate s if s = u

and t = 100, u = 4.

Solution s =

= t u

100

4

= 25 (100 ÷ 4 = 25)

Questions 9.1

Work out each formula for the values given.

Formula Values

1. c = m + n m = 7, n = 10

2. a = y – p y = 33, p = 3 b = 11 3. t = 4 b

4. e = w d

5. s = qv w = 16, d = 2 q = 3, v = 6

FORMULAS



FORMULAS

9.2 Two- and three-step formulas

OK, so that was quite easy. But sometimes we need to ‘manage’ more than one calculation.

Example 9.2a

Calculate a when a = 3b + 2c and b = 5, c = 6.

Solution a = 3b + 2c

= 15 + 12

= 27

Example 9.2b

Calculate t =

4 p r

when p = 9, r = 6 t

Solution

=

=

4 p r

36

6

= 6

Sort out each part of the fraction separately first.



4 p = 4

 p = 4



9 = 36

Example 9.2c

This is like the last example, but trick ier. Remember to sort out each part of the fraction first, before dividing.

Calculate c = when m = 7, p = 4

2 p

Solution c =

2 p

=

32

8

= 4

(5m – 3 = 35 – 3 = 32)

5



m = 5



7



Questions 9.2

Work our each formula for the values given.

Formula

1. 2 t + 5

Values t = 4

2. d = 4 m + 3 p m = 2, p = 7

3. e =

4. n =

4 a

3

5 – 1

2 gh

5. f = m

6. w = 8 p + 4 g – 7

7. h = 11 y – 3 k

8. t =

4 – 5 s

3 w a = 12 s = 3 g = 2, h = 16, m = 4 p = 2, g = 3 y = 4, k = 2 a = 15, s = 6, w = 2

FORMULAS



SAQ ANSWERS


Answers 9.1

1. c = 17

2. a = 30

3. t = 44

4. e = 8

5. s = 18

Answers 9.2

1. c = 13

2. d = 29

3. e = 16

4. n = 7

5. f = 8

6. w = 21

7. h = 38

8. t = 5



Solution - Education Scotland

Core Skills

Numeracy

Outcome 4

Applying Numerical Skills

CONTENTS

SECTION 1

1.1 Whole numbers

1.2 Decimals

1.3 The four rules

Section 1: Answers

SECTION 2

2.1 The rule of five

2.2 Extending the use of the rule of five

2.3 Rounding to two decimal places

2.4 Rounding in a money calculation

2.5 The problems of real life

2.6 Significant figures

“20 000 attend cup tie!”

Section 2: Answers

SECTION 3

3.1 Quick and easy fractions

3.2 Fractions to percentages

Section 3: Answers

SECTION 4

4.1 Finding a percentage without using a calculator

4.2 Finding a percentage by using a calculator

4.3 Percentage profit and loss

4.4 Percentage errors, increases and decreases

Section 4: Answers

SECTION 5

5.1 Earning a wage

5.2 Pay increases

5.3 Dealing with income tax

5.4 Higher tax rate

Section 5: Answers

SECTION 6

6.1 Working with VAT

6.2 Practical problems

6.3 Hire purchase

Section 6: Answers

SECTION 7

7.1 Lengths of time

7.2 Calculating speed

7.3 Calculating distance

7.4 Calculating time

Section 7: Answers

SECTION 8

8.1 Ratio as comparison

8.2 Mixed units

8.3 Proportional division

8.4 Direct proportion

Section 8: Answers

SECTION 9

9.1 The four rules of formulas

9.2 Two- and three-step formulas

Section 9: Answers

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