Colour-coded questions
Example 1
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. 5
Express 25 minutes : 1 hour as a ratio in its simplest form. The units must be the same, so change 1 hour into 60 minutes.
Divide both sides by 5.
= 5 : 12
So, 25 minutes : 1 hour simplifies to 5 : 12.
Ratios as fractions
6
7 of a campsite is allocated to caravans. The rest is allocated to tents. Write the ratio 10 of space allocated in the form caravans : tents.
7
Amy gets 23 of a packet of sweets. Her sister Susan gets the rest. Work out the ratio of sweets that each sister gets. Write it in the form Amy : Susan.
8
a The recipe for a fruit punch is 1.25 litres of fruit crush to 6.25 litres of lemonade. b How much fruit crush will you need to mix with 2 litres of lemonade? Hints and tips Set up a table.
Example 2
Always cancel the ratio to its simplest form before converting it to fractions.
c You have half a litre of fruit crush. How much lemonade will you need?
A garden is divided into lawn and shrubs in the ratio 3 : 2. a lawn
9
b shrubs?
The denominator (bottom number) of the fraction is the total number of parts in the ratio each time (that is, 2 + 3 = 5). 3 5
a The 3 in the ratio becomes the numerator.
The lawn covers
b The 2 in the ratio becomes the numerator.
The shrubs cover
10
of the garden.
11
b the lions
c the chimpanzees?
MR
The recipe for a pudding is 125 g of sugar, 150 g of flour, 100 g of margarine and 175 g of fruit. What fraction of the pudding made up by each ingredient? Andy plays 16 bowls matches. He wins
3 4
of them.
He plays another x matches and wins them all.
Exercise 5A
The ratio of wins : losses is now 4 : 1. Work out the value of x.
Express each ratio in its simplest form.
2
In a safari park at feeding time, the elephants, lions and chimpanzees are given food in the ratio 10 to 7 to 3. What fraction of the total food is given to:
a the elephants
of the garden. 2 5
MR
1
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:
What fraction of the punch is each ingredient?
You can express ratios as fractions by using the total number of parts in the ratio as the denominator (bottom number) of each fraction. Then use the numbers in the ratio as the numerators. If the ratio is in its simplest form, the fractions will not cancel.
What fraction of the garden is covered by:
Mathematical skills
b What fraction of the pizza did Sue eat?
= 25 minutes : 60 minutes = 25 : 60
Dave and Sue share a pizza in the ratio of 2 : 3. They eat it all.
a What fraction of the pizza did Dave eat?
25 minutes : 1 hour
Cancel the units (minutes).
a 6 : 18
b 15 : 20
c 16 : 24
d 24 : 36
e 20 to 50
f 12 to 30
g 25 to 40
h 125 to 30
MR
12
Three brothers share some money. The ratio of Mark’s share to David’s share is 1 : 2. The ratio of David’s share to Paul’s share is 1 : 2.
Write each ratio of quantities in its simplest form
a £5 to £15
b £24 to £16
CM C ommunicate mathematically – you need to show how you have arrived at your answer by using mathematical arguments.
What is the ratio of Mark’s share to Paul’s share?
c 125 g to 300 g EV
d 40 minutes : 5 minutes
e 34 kg to 30 kg
f £2.50 to 70p
g 3 kg to 750 g
h 50 minutes to 1 hour
i 1 hour to 1 day
13
Three brothers, Jarek, Jerzy and Justyn, share a block of chocolate in the ratio of their ages. Jarek gets half of the bar. Jerzy gets
3 5
of the rest.
a Work out the ratio, in the form Jarek : Jerzy : Justyn, of how the brothers share the bar of chocolate.
Hints and tips Remember as to express both parts in a common unit before
b Justyn is 8 years old. How old is Jarek?
you simplify.
4
3
A length of wood is cut into two pieces in the ratio 3 : 7. What fraction of the original length is the longer piece?
4
Jack and Thomas find a bag of marbles. They share the marbles in the ratio of their ages. Jack is 10 years old and Thomas is 15 years old. What fraction of the marbles did Jack get?
EV
14
Three cows, Gertrude, Gladys and Henrietta, produced milk in the ratio 2 : 3 : 4. Henrietta produced 1 1 litres more than Gladys. How much milk did the three cows 2 produce altogether?
CM
15
In a garden, the area is divided into lawn, vegetables and flowers in the ratio 3 : 2 : 1. If one-third of the lawn is dug up and replaced by flowers, what is the ratio of lawn : vegetables : flowers now? Give your answer as a ratio in its simplest form.
5 Ratio and proportion: Ratio, proportion and rates of change
1
a Show that a pole that is 5.25 m long will not fit in the van if it is laid on the floor.
4.20 m
2.10 m
This is a communicating mathematics question where you have to assess the validity of an argument. a
You need to find the diagonal length of the floor in order to assess the statement given. Show the calculation using Pythagoras’ theorem and then assess the statement.
4.2
Don’t just say the statement is wrong: give a clear reason for your conclusion. 2.10
The length, c, of the diagonal of the floor is:
( 2.12
I can find the output of a function given an input.
+ 4.22 ) = 4.7 m
I can rearrange more complicated formulae where the subject may appear twice or as a power.
This is shorter than the pole, so the ladder will not fit on the floor in the van. b Let the length of the diagonal of the van be d m.
3.10
d
I can find an inverse function by rearranging. I can find a composite function by combining two functions together. I can combine and simplify algebraic fractions. I can use iteration to find a solution to an equation to an appropriate degree of accuracy.
You need to find the diagonal length of the van. Use a diagram to help identify the sides to use. Use Pythagoras’ theorem in 3D to make sure you don’t round too early. After finding the length, assess the statement and give a clear reason for your conclusion.
Review questions
floor diagonal
d = ( 4.22 + 2.12 + 3.12 )
1
f (x) = 20 – 3x2. Find the value of f (–2).
2
a Make x the subject of the formula 6x – K = a – Cx.
b Hence find the value of x when a = 5, K = –12 and C = –8.
= 5.627 m The diagonal of the van is 37.7 cm longer than the pole so the pole can be put in diagonally.
3
a Write f (x) =
21x 2 − 7 x . 9x2 − 1
Simplify fully
EV
5
The iterative formula xn + 1 = 6 xn + 13 can be used to solve the equation x = 6x + 13.
CM
6
5
5
3
3
b Find x5 correct to 2 decimal places and compare it with x4. 11 Geometry and measures: Right-angled triangles
f (x) = 3x + 8
g(x) = x3 + 2
a Find a simplified expression for fg(x). 13810_P304_343.indd 340
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7
b Using the expression from part a, verify that fg(3) = 95. Find the inverse of each function.
a f (x) = px – q MR
8
f (x) =
(
2+
a
b f (x) = a – x3
)
Review what you have learnt from the chapter with this colour-coded summary to check you are on track throughout the course.
Review questions
x 9 as a single fraction in its simplest form. − x − 3 x ( x − 3)
b Hence find the inverse function f –1(x). 4
a Starting with x1 = 2.5, find the first four iterations, all correct to 2 decimal places.
340
E valuate and interpret – your answer needs to show that you have considered the information you are given and commented upon it.
Ready to progress?
Ready to progress?
2.12 + 4.22 = c2 c=
EV
Develop your mathematical skills with detailed commentaries walking you through how to approach a range of questions.
3.10 m
back of the van.
roblem solving and making connections – P you need to devise a strategy to answer the question, based on the information you are given.
Worked exemplars
The inside of the back of a van is a cuboid that is 2.10 m wide, 4.20 m long and 3.10 m high.
b Show that a pole that is 5.25 m long can be fitted in the
PS 5
5.1 Ratio
Worked exemplars CM
athematical reasoning – you need to M apply your skills and draw conclusions from mathematical information.
c f (x) = x + c
2
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.
x .
a Find the value of: i f (0)
ii ff (0)
iii fff (0)
iv ffff (0)
v fffff (0).
b Find the nth term of the sequence given by the answers to part a. CM
9 10
Show, by iteration, that a solution of the equation x3 = 2x + 2 is given by 1.77, correct to 2 decimal places. 2 a Simplify f(x) = 2 x 2 + 3 x − 14 .
x − 5x + 6
2 g(x) = 12 − x x
b Solve gf(x) = 1.
20
206745 GCSE Maths Higher SB_Howtouse_AQA.indd 7
24 Algebra: Algebraic fractions and functions
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