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4.1 Equivalent percentages, fractions and decimals HOMEWORK 4A Example 1

–32 –– which can be cancelled down to –8– As a fraction 32% = 100 25

Example 2

As a decimal 65% = 65 ÷ 100 = 0.65

1

2

3

G

Write each percentage as a fraction in its lowest terms. a 10% b 40% c 25% d 15% g 12% h 28% i 56% j 18%

e k

75% 42%

f l

35% 6%

Write each percentage as a decimal. a 87% b 25% c 33% g 58% h 17.5% i 8.5%

e k

1% 150%

f l

72% 132%

d j

5% 68.2%

Copy and complete the table. Percentage 10% 20% 30%

Fraction

Decimal

0.4 0.5 0.6   

FM

4

If 45% of pupils walk to school, what percentage do not walk to school?

5

If 84% of the families in a village own at least one car, what percentage of the families do not own a car?

6

In a local election, of all the people who voted, 48% voted for Mrs Slater, 29% voted for Mr Rhodes and the remainder voted for Mr Mulley. What percentage voted for Mr Mulley?

7

From his gross salary, Mr Hardy pays 20% Income Tax, 6% Superannuation and 5% National Insurance. What percentage is his net pay?

FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving

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G

PS

8

Approximately what percentage of each can is filled with oil? a b c

9

Write each fraction as a percentage. 7 – a –34 b –25 c 20

d

3 – 25

e

43 – 50

f

3 – 8

Write each decimal as a percentage. a 0.23 b 0.87 c 0.09

d

0.235

e

1.8

f

2.34

10

F

AU

11

Tom scored 68 marks out of a possible 80 marks in a geography test. a Write his score as a fraction in its simplest form. b Write his score as a decimal. c Write his score as a percentage. d His next test is of the same standard but is marked out of 50. How many marks out of 50 does he need to improve his percentage score?

4.2 Calculating a percentage of a quantity HOMEWORK 4B

Example

Calculate 12% of 54 kg. Method 1 12 ÷ 100 × 54 = 6.48 kg Method 2 Using a multiplier: 0.12 × 54 = 6.48 kg

E

1

2

3

PS

30

CORE

What multiplier is equivalent to a percentage of: a 23% b 70% c 4%

d

120%?

What percentage is equivalent to a multiplier of: a 0.38 b 0.8 c 0.07

d

1.5?

Calculate the following. a 25% of £200 b e 22% of £84 f i 6% of £42 j

d h l

75% of 84 cm 95% of 320 m 37.2% of £800

10% of £120 71% of 250 g 17.5% of £56

c g k

53% of 400 kg 24% of £3 8.5% of 160 l

4

During one week at a Test Centre, 320 people took their driving test and 65% passed. How many people passed?

5

A school has 250 pupils in each year and the attendance record on one day for each year group is shown below. Year 7 96%, Year 8 92%, Year 9 84%, Year 10 88%, Year 11 80% The school has a target of 90% attendance overall. Did the school meet its target?

6

A certain type of stainless steel consists of 84% iron, 14% chromium and 2% carbon (by weight). How much of each is in 450 tonnes of stainless steel?

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7

VAT (Value Added Tax) is a tax that the Government adds to the price of goods sold. At the moment it is 17.5%. How much VAT will be added on to the following bills: a a restaurant bill for £40 b a telephone bill for £82 c a car repair bill for £240?

FM

8

An insurance firm sells house insurance and the annual premiums are usually at a cost of 0.5% of the value of the house. What will be the annual premium for a house valued at £120 000?

9

A shop has a sale and reduces all prices by 10%. One week later the shop reduces prices by a further 10%. Have the prices reduced by 20% altogether? Show how you decide.

E

4.3 Increasing or decreasing quantities by a percentage HOMEWORK 4C Example

Increase £6 by 5%. Method 1 Find 5% of £6: (5 ÷ 100) × 6 = £0.30 Add the £0.30 to the original amount: £6 + £0.30 = £6.30 Method 2 Using a multiplier: 1.05 × 6 = £6.30

FM

1

Increase each of the following by the given percentage. (Use any method you like.) a £80 by 5% b £150 by 10% c 800 m by 15% d 320 kg by 25% e £42 by 30% f £24 by 65% g 120 cm by 18% h £32 by 46% i 550g by 85% j £72 by 72%

2

Mr Kent, who was on a salary of £32 500, was given a pay rise of 4%. What is his new salary?

3

Copy and complete this electricity bill.

Fixed charges 840 units @ 6.45 p per unit 1720 units @ 2.45 p per unit Total charges VAT @ 8% Total to pay 4

D

Total charges £13.00

A bank pays 8% simple interest on the money that each saver keeps in a savings account for a year. Miss Pettica puts £2000 in this account for three years. How much will she have in her account after: a 1 year b 2 years c 3 years?

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FM

5

VAT (Value Added Tax) is a tax that the Government adds to the price of goods sold. At the moment it is 17.5% on all goods. Mrs Dow purchased these items from a gift catalogue, after VAT of 17.5% has been added. Gift Travel alarm clock

Pre-VAT price £18.00

Ladies’ purse wallet

£15.20

Pet’s luxury towel

£12.80

Silver-plated bookmark

£6.40

She estimated that the total cost would be about £60. Was this a good estimate? Show how you decide. PS FM

6

A dining table costs £300 before the VAT is added. If the rate of VAT goes up from 15% to 20%, by how much will the cost of the dining table increase?

HOMEWORK 4D Example

Decrease £6 by 5%. Method 1 Find 5% of £6: (5 ÷ 100) × 6 = £0.30 Subtract the £0.30 from the original amount: £6 – £0.30 = £5.70 Method 2 Using a multiplier: 0.95 × 6 = £5.70

D FM

FM

1

Decrease each of the following by the given percentage. (Use any method you like.) a £20 by 10% b £150 by 20% c 90 kg by 30% d 500 m by 12% e £260 by 5% f 80 cm by 25% g 400 g by 42% h £425 by 23% i 48 kg by 75% j £63 by 37%

2

Mrs Denghali buys a new car from a garage for £8400. The garage owner tells her that the value of the car will decrease by 24% after one year. What will be the value of the car after one year?

3

The population of a village in 2006 was 2400. In 2010 the population had decreased by 12%. What was the population of the village in 2010?

4

A Travel Agent is offering a 15% discount on holidays. How much will the advertised holiday now cost? NEW YORK FOR A WEEK

£540 5

New Year’s Sale: All prices reduced by 20% Matt has £160 from Christmas presents. Can he afford to buy a shirt that normally costs £30, a suit that normally costs £130, and a pair of shoes that normally cost £42?

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6

C

A shop increases all its prices by 10%. One month later it advertises 10% off all marked prices. Are the goods cheaper, the same or more expensive than before the price increase? Show how you work out your answer.

4.4 Expressing one quantity as a percentage of another quantity HOMEWORK 4E

Example

Express £6 as a percentage of £40. –6– and multiply it by 100. 6 ÷ 40 = 15%. Set up the fraction 40

FM

1

Express each of the following as a percentage. Give your answers to one decimal place where necessary. a £8 of £40 b 20 kg of 80 kg c 5 m of 50 m d £15 of £20 e 400 g of 500 g f 23 cm of 50 cm g £12 of £36 h 18 minutes of 1 hour i £27 of £40 j 5 days of 3 weeks

2

What percentage of these shapes is shaded? a b

3

In a class of 30 pupils, 18 are girls. a What percentage of the class are girls? b What percentage of the class are boys?

4

The area of a farm is 820 hectares. The farmer uses 240 hectares for pasture. What percentage of the farm land is used for pasture? Give your answer to one decimal place.

5

Here are some retail and wholesale prices: Item a b c d

Micro Hi-Fi System CD Radio Cassette MiniDisc Player Cordless Headphones

Retail price (Selling price) £250 £90 £44.99 £29.99

D

Wholesale price (Price the shop paid) £150 £60 £30 £18

A shopkeeper wants to make over 40% profit on each item. Does he succeed at these prices? AU

6

C

Paul and Val take the same tests. Both tests are worth the same number of marks. Here are their results. Test A

Test B

Paul

30

40

Val

28

39

Whose result has the greater percentage increase from test A to test B? Show your working. CORE

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HOMEWORK 4F

F

1

Copy and complete the table Fraction a

Decimal

1/4

b

0.4

c

15%

E

2

Work out these amounts. a 15% of £42 b 12% of 300 kg c 35% of 240 ml

D

3

What percentage is: a 36 out of 50 b 17 out of 25 c 60 out of 200

4

What is the result if: a 180 is increased by 25% b 4200 is decreased by 7%

5

a b c

C

FM

a b

AU PS

34

6

CORE

7

Percentage

A window-cleaner increases his fee from £12 to £15 per house. What is the percentage increase in his fee? The number of houses on his round increases from 40 to 48. What is the percentage increase in the number of houses he cleans windows for? For cleaning the windows of a bungalow, he offers a 30% discount on his new fee. What does he charge a bungalow-owner? A new computer costs £800 at full price. In it’s winter sale, a computer shop offers a 20% discount. What is the sale price of the computer? The shop is also selling some computer software at a 15% discount. Before the sale it cost £120. Patricia decides to buy both the computer and the software in the sale. She has been saving £75 per month for a year. Does she have enough money?

A group of mothers agreed to compare the weights of their newly born babies. The mothers said their babies had a mean weight of 4 kg when they were first born. The mothers said they would compare the weights one month later to see the average amount of weight they gained. When the babies were one month old the mothers said their babies had gained an average of 25% in weight. a What was the average weight after the month? b One of the mothers realised she had misread the scale and her baby was 0.5 kg heavier than she thought. Which of these statements is true? i The mean weight gain will have increased by more than 25%. ii The mean weight gain will have stayed at 25%. iii The mean weight gain will have increased by less than 25%. iv There is not enough information to answer the question.

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4.5 Ratio HOMEWORK 4G Example 1

Simplify 5 : 20. 5 : 20 = 1 : 4 (Divide both sides of the ratio by 5.)

Example 2

Simplify 20p : £2. (Change to a common unit) 20p : 200p = 1 : 10

Example 3

A garden is divided into lawn and shrubs in the ratio 3 : 2. The lawn covers 3–5 of the garden and the shrubs cover 2–5 of the garden.

1

Express each of the following ratios in their simplest form. a 3:9 b 5 : 25 c 4 : 24 d 10 : 30 f 12 : 20 g 25 : 40 h 30 : 4 i 14 : 35

e j

E

6:9 125 : 50

2

Express each of the following ratios of quantities in their simplest form. (Remember to change to a common unit where necessary.) a £2 to £8 b £12 to £16 c 25 g to 200 g d 6 miles : 15 miles e 20 cm : 50 cm f 80p : £1.50 g 1 kg : 300g h 40 seconds : 2 minutes i 9 hours : 1 day j 4 mm : 2 cm

3

£20 is shared out between Bob and Kathryn in the ratio 1 : 3. a What fraction of the £20 does Bob receive? b What fraction of the £20 does Kathryn receive?

4

In a class of students, the ratio of boys to girls is 2 : 3. a What fraction of the class is boys? b What fraction of the class is girls?

PS FM

5

Pewter is an alloy containing lead and tin in the ratio 1 : 9. a What fraction of pewter is lead? b What fraction of pewter is tin? c A foundry has 30 tonnes of lead and 90 tonnes of tin. How much pewter can it make?

AU

6

Roy wins two-thirds of his snooker matches. He loses the rest. What is his ratio of wins to losses?

PS

7

In the 2009 Ashes cricket series, the numbers of wickets taken by Steve Harmison and Monty Panesar were in the ratio 5 : 1. The ratio of the number of wickets taken by Graham Onions to those taken by Steve Harmison was 2 : 1. What fraction of the wickets taken by these three bowlers was by Monty Panesar?

D

CORE

C

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HOMEWORK 4H

Example

Divide £40 between Peter and Hitan in the ratio 2 : 3. Changing the ratio to fractions gives Peter’s share = 2–5 and Hitan’s share = 3–5 So, Peter receives 2–5 × £40 = £16 and Hitan receives 3–5 × £40 = £24.

D

1

Divide each of the following amounts in the given ratios. a £10 in the ratio 1 : 4 b £12 in the ratio 1 : 2 c £40 in the ratio 1 : 3 d 60 g in the ratio 1 : 5 e 10 hours in the ratio 1 : 9

2

The ratio of female to male members of a sports centre is 3 : 1. The total number of members of the centre is 400. a How many members are female? b How many members are male?

3

A 20 metre length of cloth is cut into two pieces in the ratio 1 : 9. How long is each piece?

4

Divide each of the following amounts in the given ratios. a 25 kg in the ratio 2 : 3 b 30 days in the ratio 3 : 2 c 70 m in the ratio 3 : 4 d £5 in the ratio 3 : 7 e 1 day in the ratio 5 : 3

5

James collects beer mats and the ratio of British mats to foreign mats is 5 : 2. He has 1400 beer mats in his collection. How many foreign beer mats does he have?

6

Patrick and Jane share out a box of sweets in the ratio of their ages. Patrick is 9 years old and Jane is 11 years old. If there are 100 sweets in the box, how many does Patrick get?

AU

7

For her birthday Reena is given £30. She decides to spend four times as much as she saves. How much does she save?

FM

8

Mrs Megson calculates that her quarterly electric and gas bills are in the ratio 5 : 6. The total she pays for both bills is £66. She has saved £33 of gas stamps and £33 of electricity stamps to pay the bill. Will she need to pay any extra and if so, how much?

C

PS

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CORE

Gas stamps cannot be used to pay for electricity.

9

You can simplify a ratio by changing it into the form 1 : n. For example, 5 : 7 can be rewritten as 5 : 7 = 1 : 1.4 by dividing each side of the ratio by 5. Rewrite each of the following ratios in the form 1 : n. a 2:3 b 2:5 c 4:5 d 5:8 e 10 : 21

10

The amount of petrol and diesel sold at a garage is in the ratio 2 : 1. One- tenth of the diesel sold is bio-diesel. What fraction of all the fuel sold is bio-diesel?

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HOMEWORK 4I

Example

Two business partners, John and Ben, divided their total profit in the ratio 3 : 5. John received £2100. How much did Ben get? John’s £2100 was 3–8 of the total profit. So, 1–8 of the total profit = £2100 ÷ 3 = £700. Therefore, Ben’s share, which was 5–8, amounted to £700 × 5 = £3500.

1

Peter and Margaret’s ages are in the ratio 4 : 5. If Peter is 16 years old, how old is Margaret?

2

Cans of lemonade and packets of crisps were bought for the school disco in the ratio 3 : 2. The organiser bought 120 cans of lemonade. How many packets of crisps did she buy?

3

In his restaurant, Manuel is making ‘Sangria’, a drink made from red wine and iced soda water, mixed in the ratio 2 : 3. Manuel uses 10 litres of red wine. a How many litres of soda water does he use? b How many litres of Sangria does he make?

4

Cupro-nickel coins are minted by mixing copper and nickel in the ratio 4 : 1. a How much copper is needed to mix with 20 kg of nickel? b How much nickel is needed to mix with 20 kg of copper?

5

The ratio of male to female spectators at a school inter-form football match is 2 : 1. If 60 boys watched the game, how many spectators were there in total?

6

Marmalade is made from sugar and oranges in the ratio 3 : 5. A jar of ‘Savilles’ marmalade contains 120 g of sugar. a How many grams of oranges are in the jar? b How many grams of marmalade are in the jar?

AU

7

Each year Abbey School holds a sponsored walk for charity. The money raised is shared between a local charity and a national charity in the ratio 1 : 2. Last year the school gave £2000 to the local charity. a How much did the school give to the national charity? b How much did the school raise in total?

PS

8

Fred’s blackcurrant juice is made from four parts blackcurrant and one part water. Jodie’s blackcurrant juice is made from blackcurrant and water in the ratio 7 : 2. Which juice contains the greater proportion of blackcurrant? Show how you work out your answer.

FM

D C

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Functional Maths Activity The cost of going to work Miss Jones • Miss Jones is 23 years old and lives in Bramley. • She works 20 miles from home, in Aston. Rail Fares • She is a manager in a small company and earns £18 000 per year. Bramley to Aston • She works from Monday to Friday each week. 7-Day ticket £56.40 • She has four weeks holiday per year. Monthly ticket £217.35 • She always takes two of these holiday weeks 3-month ticket £652.05 in July every year. • She travels to work by train 16–25 Railcard — £26 for a whole year each day using a monthly Aged 16–25 or a full-time student aged 26 or over? ticket. • She has a 16–25 Railcard. Save 1/3 on most rail fares throughout Great Britain • In July she buys weekly 16–25 Railcard discounts now apply to all tickets. Standard and First Class Advance fares. • The journey takes 45 minutes each way. • She uses the local sandwich shop for lunch each day. Sandwich shop Small sandwiches ~ £2.50 Mr Smith Large Sandwiches ~ £3.30 • Mr Smith is 45 years old and lives in Sunnyside. Pay weekly for your sandwiches and get Friday free! • He works 10 miles from home, in Todwick. • He is a maintenance worker in the same small company and earns £12000 per year. • He works from Monday to Saturday each week. • He has six weeks holiday per year. Bus fares • He travels to work by bus each day, using a daily return ticket. Todwick to Aston Single .......... £2.35 • The journey takes 30 minutes each way. Return ......... £3.20 • He has lunch in the work canteen each day. • He always has the set meal, plus a drink and two portions of extra vegetables.

Works Canteen Set meal --- £4 Drinks --- 75p each Extra vegetables 50p per portion

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Functional Maths Activity (continued) Task 1 Answer the following questions about Miss Jones. 1 How many weeks does she work in a year? 2 How much is she paid each month? 3 How much does she pay for a monthly rail ticket? 4 Why does she buy weekly tickets in July? 5 How much does she spend on travel to work, including the cost of the Railcard, in one year? 6 How much would she pay at the sandwich shop if she pays weekly for small sandwiches? 7 What proportion of the cost of the sandwiches is she saving? 8 How much more per week would it cost her if she had large sandwiches? 9 What is the ratio of the cost of small sandwiches to large sandwiches? Give your answer in its simplest form. 10 One-third of her salary is spent in taxes. How much does she have left after tax? Task 2 Use your answers to Task 1 to help you to work out how much money Miss Jones has left after deducting taxes, travelling and meal costs from her salary. Give your answer as a monthly amount. Task 3 Work the time that Miss Jones spends travelling to and from work each year. Task 4 Work out how much money Mr Smith has left after deducting taxes, travelling and meal costs from his salary. Task 5 • Imagine that you live 15 miles from work. • Decide costs for travelling to work by train or bus. • Decide what you will eat at lunchtime. • Decide what your salary will be. • If the salary is low, deduct one-third of the salary for taxes. • If the salary is high, the ratio of tax to remaining pay is 2 : 1. • Work out how much money you will have left after deducting taxes, travelling and meal costs from your salary.

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