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3.2 Percentages-1
A Worked Examples
Applying Percentage Change
Calculating Percentage Change
A jumper costing £45 is put into the sale and
now costs £28. Find the percentage change.
Increase 146 by 32%
There are two methods that can be used:
45 − 28 = 17
Method 1:
Find 32% of 146
32
x 146 = 46.72
100
17
x 100 = 37.78% (2 dp)
45
The jumper has been decreased by 37.78% (2 dp)
46.72 + 146 = 192.72
Method 2:
Using a multiplier – increasing by 32% means
you are finding 132% overall of 146
146 x 1.32 = 192.72
Reverse Percentages:
A jumper is reduced in price by
30% and now costs £70. What
was the original cost?
70 = £100
0.7
B Applying Percentage Change
Following the announcement of Brexit, George the shopkeeper was told there would be a change to the
price of some of the items he sells. Calculate the new prices for the following items in George’s shop:
Item
Percentage Change
Original Price
Marmite
12.5%
£2.35
Ben & Jerry’s
10%
£4.00
Pot Noodle
4%
£1.00
Beef Mince
23%
£3.86
Chicken Thighs
31%
£3.25
Shower Gel
8%
£1.24
C Calculating Percentage
Change
Write the amount of percentage
change on the arrows between
the amounts in the boxes shown
opposite:
New Price
30
40
55
50
18 GCSE MATHS GRADE BOOSTER HIGHER Revision Workshop
D Finding Profit and Loss Question
Frankie buys and sells items on eBay. She often makes a profit, but sometimes she make a loss.
Calculate the total percentage profit/loss that she makes from the following items, giving your answers to 2 decimal
places.
USB Powered Desk Fan
Polaroid Mobile Printer
Disney mug
Bought £7.40
Bought £125
Bought £27
Sold £10.99
Sold £150.50
Sold £19.25
Reverse Percentages Exam Questions
1 Shops 4 U have labelled a tea cup they are selling with a “45% off” label. It now costs £6.88.
What price was the tea cup selling for originally?
(Total for Question 1 is 2 marks)
2 Clayton bought some comic books in 2012. By 2017 these books had increased in value by 15%
and are now worth £230.
How much did Clayton spend on comic books in 2012?
(Total for Question 1 is 3 marks)
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3.3 Percentages-2
A Compound Interest
Compound interest is one way that interest can be calculated, and
is a way of calculating repeated percentage change over time. Make
sure you also know how to calculate simple interest, as you may
be asked to work this out too!
Multiplier Match Up
You need to know how multipliers work for compound interest.
Here’s a quick activity to help remind you.
Match up the description to the multiplier that would be used
in the calculation.
Increase by 6%
1.94
Decrease by 6%
1.06
Increase by 94%
0.94
Decrease by 94%
0.06
Compound Interest Question
Ann invests £65,000 at a compound interest rate of 6%.
(a) Work out how much she has in her account at the end of:
i. 3 years
ii. 15 years
iii. 6 months
(b) How many years until she has £97,000?
20 GCSE MATHS GRADE BOOSTER HIGHER Revision Workshop
B Growth & Decay
Another way that repeated percentage change can be applied involves growth and decay. You may be given
a model to work from, and asked questions based on this model.
During an experiment, Nina discovers the rate at which cells are decreasing in her sample is C n+1 = 0.81 C n
where C n is the number of cells after n minutes.
(a) If C 0 = 250, calculate:
i. C1
ii. C4
(b) Nina is predicting that after k minutes the number of cells will have decreased to less than 70.
Calculate the value of k.
Exam Question
1 Here are the interest rates for two accounts which Clare is considering to invest in.
Account A
Account B
• 3% per annum compound interest
• 2% per annum compound
interest up to £1500
• 5% per annum compound interest
on amounts above £1500
Clare has £1800 to invest for 3 years. Which account gives the best return on her investment?
(Total for Question 1 is 4 marks)
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3.4 Speed, Density and Pressure
A Formulae
You need to know these formulae, which you will not be given in the exam:
speed =
distance
time
density =
mass
volume
pressure =
force
area
B Speed, Distance and Time: The Planets
Complete the table below by calculating speed, distance and time for the selected planets, giving your answers to a sensible
degree of accuracy. Assume that the orbit of each planet is circular around the sun.
How fast are the planets?
Planet
Distance from
sun (million km)
Time for 1 orbit
of Sun (days)
Mercury
58
88
Mars
228
Saturn
1,427
Speed
(km/h)
86,885
10,760
60,200
Neptune
19,557
Exam Question
1 Luke is traveling to Alton Towers from London which is 150 miles to the nearest 5 miles. He
travels at an average speed of 65 mph, correct to the nearest 5 mph.
Will he arrive at the opening time of 10 am if he leaves London at 7:50 am?
(Total for Question 2 is 2 marks)
C Density, Mass
and Volume
Make your way around
the circuit, using the
information in the
previous 2 boxes to fill
in the missing values
(give all answers 2
significant figures).
Don't forget to write in
your units!
Mass
Volume
Density
Mass
4.72 g/cm3
375 mm3
2.5 kg
Volume
Volume
Density
Density
46 g/cm3
2 Kg/m3
Mass
Mass
Volume
250 g
Density
Mass
g
Volume
3
4m
Start
Density
Volume
0.84 m3
Density
8 g/m3
22 GCSE MATHS GRADE BOOSTER HIGHER Revision Workshop
Density
Volume
Mass
kg
1.3 m3
Exam Question
2 The two wooden boxes shown are being examined.
$
6cm
8cm
6cm
$
10cm
10cm
3
The density of the cuboid is 1.2 g/cm
The mass of the cylinder is 1.89 g
Becki says that the cuboid has a higher density than the cylinder. John says that the cylinder weighs more.
Who is correct? Show your working clearly and make sure you justify your answer.
(Total for Question 2 is 5 marks)
D Growth & Decay
Practice Questions
(a) The pressure of a concrete block which covers 220 cm2 is 150 N/m2. Calculate the force applied
(b) Arlo applies a force of 90 N to a cuboid with 80 cm lengths. Calculate the pressure.
Exam Question
3 The volume of a cube, with sides of length 40 cm, and the cuboid shown are the same.
The density of the cuboid is 10 g/m2.
10cm
6cm
$
$
$
$
$
40cm
$
8cm
When the cuboid is put onto it’s front face (shaded grey), pressure is put onto it. Find the pressure exerted
onto the cuboid.
(Total for Question 3 is 5 marks)
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3.5 Direct and Inverse Proportion
A Direct or Inverse
Identify and circle whether the following equations describe a direct proportion relationship,
or inverse proportion relationship.
5
x
Direct Inverse
d) y
√x = 8
Direct Inverse
b) y = 5x
Direct Inverse
e) y = 12x 3
Direct Inverse
6
x2
Direct Inverse
f) y =
a) y =
c) y =
2x
3
Direct Inverse
B Proportionality Statements
Complete the table below for the statements of proportionality and formule.
Proportionality in words
Statement of proportionality
Formula
r is directly proportional to the
square of y
r is directly proportional to the
root of y
r is inversely proportional to
the cube of y
r is inversely proportional to
the square of y
C Graph Matching
Match each graph to the correct relationship, a), b) or c).
A
Relationship
B
a) 𝑦 y ∞ x
C
√
b) 𝑦 y ∞ 𝑥𝑥 x
c) 𝑦 y ∞
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1
x
D Ordering Activity
Sally has brought home an ordering activity she completed in class, but it has become unordered on the way home!
Help her to put these into the correct order.
𝑃 P = kQ 𝑘𝑄
P and Q are directly proportional.
If P = 30 when Q = 6, find P when Q = 8
𝑃 P = 5Q 𝑘𝑄
𝑃 30 = k x 6𝑘𝑄
𝑃 P = 5 x 8𝑘𝑄
P ∞ Q𝑄
𝑃k = 5 𝑘𝑄
𝑃 P = 40 𝑘𝑄
𝑃 x = 4.5 𝑘𝑄
9
x
k
6 =𝑘𝑄
1.5
y =𝑘𝑄
k=9
y is inversely proportional to x.
y = 6 when x = 1.5, find x when y = 2
k
x
9
2 =𝑘𝑄
x
1
y∞
x
y =𝑘𝑄
Sally is now asked to answer the following questions, using these worked examples to guide her.
Practice Questions
1) r𝑟is directly proportional to s. If r = 21 when s = 3
Find a) r when s = 9, b) s when r = 70
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2) y𝑟is inversely proportional to the square of x. if y = 4 when x = 2
Find a) y when x = 10, b) x when y = 20
Exam Question
1 The time, t, taken in minutes for passengers to board a train is inversely proportional to the square
of the number of train carriages, c, available. It takes 12 minutes for passengers to board the train
when there are 10 carriages available.
Work out how long (to the nearest minute) it will take for all passengers to board when there are
14 carriages available.
(Total for Question 1 is 3 marks)
E Key Points
Direct Proportion
As one variable increases or decreases, the other
variable increases or decreases in the same
direction by a constant amount.
If x is directly proportional to t;
you can write x ∞ t 𝑡
to write this as an equation, x = kt 𝑘𝑡
Inverse Proportion
As one variable increases or decreases, the other
variable increases or decreases in the opposite
directionby a constant amount.
If x is inversely proportional to t;
1
you can write x ∞ 𝑡
t
to write this as an equation, x =
26 GCSE MATHS GRADE BOOSTER HIGHER Revision Workshop
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GCSE maths 2017
grade booster higher
workshops
Designed to build confidence in the essential assessment skills and provide a
clear focus for students on how to make the most effective use of their remaining
revision time for the first GCSE Maths (9-1) exams in May and June 2017. The
Workshops are for students aiming for a grade 7 or 8 and for those predicted a
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The GCSE Maths Grade Booster Workshop (Higher) combines:
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including how to respond to questions involving reasoning and problem solving
Programme
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Session Four: Shape Up!
• Calculations with fractions
• Sets, Venn diagrams and conditional
probability
• Frequency tables
• Pythagoras’ and trigonometry
• Congruence and similarity
• Vectors
Session Two: Power Up!
Session Five: No problem – get on
and revise!
• Powers, roots and indices
•Standard form
•Linear graphs and simulataneous equations
• Problem solving
•Revision advice
•Exam techniques
Session Three: Factors and Formulae
• Substitution into formulae
•Quadratics
•Rearranging formulae
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GCSE Maths Grade Booster Higher – exam workshop
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