Colour-coded questions The questions in the exercises and the review questions are colour-coded, to show you how difficult they are. Most exercises start with more accessible questions and progress through intermediate to more challenging questions. 7.3 Fractions of quantities This section will show you how to: ●
work out a fraction of a quantity
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find one quantity as a fraction of another.
1
a 2 cm, 6 cm
e 12 days, 30 days f 50p, £3
numerator
Example 3
Write £5 as a fraction of £20. 5 £5 as a fraction of £20 is written as . 20 5 1 × 5 Note that = so you can cancel by a factor of 5 to 1. 20 4 × 5 4 So £5 is one-quarter of £20.
Sometimes you will need to work out a fraction of a quantity or amount. To do this, first divide the quantity by the denominator of the fraction you are working out. Then multiply the result by the numerator of the fraction.
Example 4
Jake earns £150. He puts £50 into his bank account and spends the rest. What fraction of his earnings does he spend? 3
CM
7
Jon earns £90 and saves £30 of it. Matt earns £100 and saves £35 of it.
CM
8
In two tests Harry gets 13 out of 20 and 16 out of 25. Which is the better mark?
PS
9
In a street of 72 dwellings, 5 are bungalows. The rest are two-storey houses. Half of 12 the bungalows are detached and 2 of the houses are detached. What fraction of the 7 72 dwellings are detached?
MR
Explain your answer.
of these pages are in colour.
PS
10
I have 24 T-shirts. 1 have logos on them; the rest are plain. 2 of the plain T-shirts are 5 6 long-sleeved. 3 of the T-shirts with logos are long sleeved. What fraction of all my 4 T-shirts are long-sleeved?
EV
11
Which is the larger fraction: 26 out of 63 or 13 out of 32? Explain how you can tell without using a calculator.
MR
12
200 ÷ 4 × 3 = 150
VAT adds 51 to the price.
4
Shop B sells the same model of bicycle for £350 (excluding VAT). VAT will add 1 to the price. 5
7.4 Adding and subtracting fractions
In which shop is the bike cheaper? Show your working. 1 of £540 is 1 × £540 = £135 4 4
This section will show you how to:
In Shop B, the VAT is 1 of £350. 5 1 of £350 is 1 × £350 = £70 5 5 So you will pay £350 + £70 = £420
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add and subtract fractions with different denominators.
Key terms equivalent fraction
lowest common denominator You can only add or subtract fractions that have the same denominator. If necessary, change one or both of them to mixed number equivalent fractions with the same denominator. Then you can add or subtract the fractions by adding or subtracting numerators.
The bike is cheaper in Shop A. 7 Number: Decimals and fractions
E valuate and interpret – your answer needs to show that you have considered the information you are given and commented upon it.
Develop your mathematical skills with detailed commentaries walking you through how to approach a range of questions.
She needs to take at least £3000 each night from ticket sales. If the ticket price must be the same for both days, how much should she charge? This is a problem-solving question that puts a real-life situation into a series of mathematical processes. Write down what you are working out. Don’t just put down a series of calculations. Tickets sold on Friday:
3 4
× 400 = 300
Work out how many seats she expects to sell on Friday.
Total tickets sold on Friday and Saturday: 300 + 400 = 700 seats sold.
Now work out how many tickets she expects to sell on the two nights together.
Needs to raise £3000 a night ∴ cost per ticket: 3000 ÷ 700 = 4.2857
Then work out the price she needs to charge for each seat.
She should charge £4.50 a ticket.
You need to interpret the result in the context of the problem. Charging £4.28 would not be sensible, so round up to a ‘normal’ price. This could be £4.50 or £5.00 Note the symbol ∴, which means ‘therefore’. It is very useful in mathematics.
2
EV
Worked exemplars
A theatre is hosting a show on Friday and Saturday night. There are 400 tickets available for each night. The box office manager expects to sell all the tickets for Saturday but only three-quarters of the tickets for Friday.
MR
roblem solving and making connections – P you need to devise a strategy to answer the question, based on the information you are given.
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Worked exemplars 1
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7.4 Adding and subtracting fractions
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PS
I can identify the value of digits in different places. I can use < for less than and > for greater than. I can add and subtract numbers with up to four digits. I can multiply and divide positive integers and decimal numbers greater than 1 by a single-digit number. I can solve problems involving multiplication and division by a single-digit number. I can add and subtract positive and negative numbers. I can use negative numbers in context and solve problems involving simple negative numbers. I can do long multiplication and long division. I can use BIDMAS/BODMAS to work out calculations in the correct order.
a Do I have enough money? b Do I have enough money if I replace the chips with beans? This is a question for which you need to use and show your mathematical reasoning skills, so set out your answer clearly.
Review questions
Show the total cost of the meal and the total of the coins in your pocket.
Total of coins in my pocket = £2.55
1
No, as £2.70 > £2.55
It is important to show a clear conclusion. Do not just say ‘No’. Use the numbers to support your answer.
b Beans are 40p cheaper and £2.30 < £2.55, so, yes I have enough money.
Explain clearly why you have enough money for this meal.
Ready to progress?
Ready to progress?
I want to buy a burger, a portion of chips and a bottle of cola for my lunch. I have the following coins in my pocket: £1, 50p, 50p, 20p, 20p, 10p, 2p, 2p, 1p.
a Total cost for a burger, chips and a cola = £2.70
athematical reasoning – you need to M apply your skills and draw conclusions from mathematical information.
CM C ommunicate mathematically – you need to show how you have arrived at your answer by using mathematical arguments.
Shop A sells a games console for £360 without VAT. Shop B sells the same games console for £480 including VAT. 1 Shop B has a sale and takes 10 off the price. In which shop is the game console cheaper?
Shop A sells a bicycle for £540 including VAT but has an offer of 1 off the selling price.
In Shop A, the offer is 1 off £540. 4 So you will pay £540 – £135 = £405
As you progress you will be expected to absorb new ways of thinking and working mathematically. Some questions are designed to help you develop a specific skill. Look for the icons:
1
In a class of 30 students, 5 are boys. 3 of the boys are left-handed. What fraction of the whole class is made up of left-handed boys?
Sometimes you will have to combine the two ideas.
Example 5
During April, it rained on 12 days. For what fraction of the month did it rain?
4
Reka wins £120 in a competition and puts £40 in a bank account. She gives 41 of what is left to her sister and then spends the rest. What fraction of her winnings did she spend?
This is 150 of the book. This cancels to 15 . 32 320
158
In a class of 30 students, 18 are boys. What fraction of the class are boys?
3
6
100 × 4 × 5 This is 200 = 320 10 × 44 × 8 5 = of the book. 8
3 of the 200 pages are in colour. 4 So 150 pages have colour illustrations.
d 5 hours, 24 hours
g 4 days, 2 weeks h 40 minutes, 2 hours
2
b What fraction of the pages of the whole book have colour illustrations?
b
c £8, £20
5
a What fraction of the pages of the whole book have illustrations?
a 200 pages have illustrations.
b 4 kg, 20 kg
Who is saving the greater proportion of his earnings?
Read through the next example carefully. A book has 320 pages. 200 of the pages have illustrations.
Write the first quantity as a fraction of the second.
denominator
Sometimes you will need to give one amount as a fraction of another amount.
3 4
Mathematical skills
Exercise 7C Key terms
It costs £7 per person to visit a show.
a 215 people attend the show on Monday.
Review questions
How much do they pay altogether?
b On Tuesday the takings were £1372. How many fewer people attended on Tuesday than on Monday? CM
2
Murray and Harry both worked out 2 + 4 × 7.
Practise what you have learnt in all of the previous chapters and put your mathematical skills to the test. Questions range from accessible through to more challenging.
Murray calculated this to be 42. Harry worked this out to be 30. Explain why they got different answers. CM31
1 Worked exemplars
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The following are two students’ attempts at working out 3 + 52 − 2. Adam:
3 + 52 − 2 = 3 + 10 − 2 = 13 − 2 = 11
Bekki:
3 + 52 − 2 = 82 − 2 = 64 − 2 = 62
a Each student has made one mistake. Explain both mistakes. b Work out the correct answer to 3 + 52 − 2.
4
Review what you have learnt from the chapter with this colour-coded summary to check you are on track throughout the course.
The temperatures of the first four days of January are given below. 1 °C, −1 °C, 0 °C and −2 °C
a Write down the four temperatures, in order, starting with the lowest. b What is the difference in Celsius degrees between the coldest and the warmest of these four days?
5
Nitrogen freezes below −210 °C and becomes a gas above −196 °C. In between these temperatures, it is liquid. Write down one possible temperature where nitrogen is:
a a liquid
32
c a solid (frozen).
1 Number: Basic number
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b a gas
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