Realise what you have done previously :
1. Simplify these fractions: a.
4
8
b.
14
24
c.
18
30
d.
42
70
2. Simplify these ratios:
a. 4 : 4
b. 14 : 10
c. 18 : 12
d. 42 : 28
Can you see the relationship between questions 1 and 2?
3. The ratio of boys to girls in a class is 5 : 4.
(a) What fraction of the students are boys?
(b) What is the number of girls as a fraction of the number of boys?
4. The ratio of dry days to rainy days in June is 4 : 2. What fraction of the days were
dry, and how many days is that?
2
3
5. of the members of a football club are male. What is the ratio of girls to boys?
2
9
6. A basketball team loses of its matches and draws
draws : losses?
1
.
9
What is the ratio of wins :
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
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Title: Ratio and proportion
Date: 20 June 2022
Learning Objective: To check our understanding on ratio and proportion
Step to Success 1
• I can recall ratio and proportion
Step to Success 2
• I can solve problems on ratio and proportion
Step to Success 3
• I can link fractions, percentages to ratios and proportions
Keyword:
Realise what you have done previously :
Sharing in a ratio
3
Jake and Kevin have earned £40 washing cars. They decide that Jake will get
5
of it and Kevin will get the rest.
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1. What fraction will Kevin get?
2. How would you write their shares as a ratio? J : K = …... : ……
We say that Jake gets …… parts and Kevin gets …… parts.
How many parts is that in total? …… + …… = ……
3. If the full amount (£40) is …… parts, how can we find 1 part?
Divide by ……
So 1 part must be £……
4. If Jake gets …… parts and Kevin gets …… parts, how much does each of
them get?
Jake: …… x £…… = £……
Kevin: …… x £…… = £……
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
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Sharing in a ratio – summary of method
Jake and Kevin have earned £40 washing cars. They
3
decide that Jake will get 5 of it and Kevin will get the
rest.
1.
What fraction will Kevin get?
2.
How would you write their shares as a ratio? J : K
= 3:2
We say that Jake gets 3 parts and Kevin gets 2 parts.
How many parts is that in total? 3 + 2 = 5
3.
If the full amount (£40) is 5 parts, how can we
find 1 part?
Divide by 5
So 1 part must be £8
4.
If Jake gets 3 parts and Kevin gets 2 parts, how
much does each of them get?
Jake: 3 x £8 = £24
Kevin: 2 x £8 = £16
1. Add up how many parts there are in
total.
3+2=5
2. Divide the amount by the number of
parts to find out how much 1 part is.
5 parts = £40
1 part = £40 ÷ 5 = £8
3. Multiply up to work out how much
each person gets.
Jake gets 3 x £8 = £24
Kevin gets 2 x £8 = £16
4. Check that your shares add up to the
right amount.
£24 + £16 = £40 ü
Now do the worksheet.
Realise what you have done previously :
Problems with sharing in a ratio
1. David and Jamie share 32 sweets in the ratio 3:5. How many does
David get? (Look back at your notes if you need to.)
2. If you were told that the ratio was 4:3 and Jamie got 15 sweets (but
not how many there were in total), how would you work out
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Answers:
1) 12
a) How many David got?
2a) ÷ 3, x 4 (=20)
b) How many there were in total?
Hint: The first question to ask is “How can I work out what 1 part is?” 2b) ÷ 3, x 7 or
add shares (=35)
VAA
Now have a go at these:
Check
3) 21
3. Ali and Ben share some marbles in the ratio 3:4. Ben gets 12 marbles. 4a) £32,000
How many were there altogether?
4b) £72,000
4. Carl, Den and Ellie inherit some money, split in the ratio 2:3:4. Den
gets £24,000.
a)
b)
How much does Ellie get?
How much was there in total?
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
Realise what you have done previously :
Trickier problems with ratio
1. Sara and Joy divide up some sweets in the ratio 7:5. Sara gets 10
more sweets than Joy. How many were there to start with?
2. The ratio of men to women working for a company is 7:4. There
are 24 more male employees than female. How many women are
there?
3. Tom makes green paint by mixing blue and yellow in the ratio 2:5.
Blue paint costs him £10 per litre and yellow £3 per litre. He sells
the paint at £8 per litre. How much profit does he make per litre? Answers:
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1. 60
Hints (try to answer the questions before you peek at these!):
2. 32
Q1 & 2: How many parts does the difference represent?
3. Cost for 7
Q3: How much does it cost to make a batch of paint, and how many litres is that?
litres = £35;
profit per
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
litre = £3
Realise what you have done previously :
Manipulating ratios
1. Simplify these ratios by putting the values into the same units and
cancelling if possible:
a. 20cm : 1m
b. 5mm : 2m
c. 40g : 6kg
d. 6 days : 4 weeks
= 20 cm : 100 cm
=1:5
= 5 mm : 2000 mm
= 1 : 400
= 40 g : 6000 g
= 1 : 150
2. Express each of these ratios in the form 1 : n
a. 3 : 12 = 1 : 4
b. 4 : 6 = 1 : 𝟔
c. 12 : 30 = 1 : 𝟑𝟎
𝟒
=1 : 1.5
𝟏𝟐
=1 : 2.5
3. The scale of an OS Landranger map is 1 : 50 000.
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= 6 days : 28 days
= 3 : 14
d. 200g : 10kg
= 200 : 10 000
= 2 : 100
= 1 : 50
a) What does this mean?
50 000 cm
1cm on the map represents ……………………………
on the ground.
b) What distance on the ground (in km) is represented by 2cm on the map?
100 000 cm = 1000 m = 1 km
c) What distance on the map would represent 5km on the ground?
2 cm represents 1 km so x 5 to get 10 cm for 5 km.
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
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Realise what you have done previously :
Proportion
Proportion is just another name for a fraction, decimal or percentage.
1. 100 students take an exam. 15 get Grade A, 21 get B and 24 get C.
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a) What proportion of students get an A?
b) What proportion get C or above?
Can you give each answer as a percentage, a decimal AND a fraction (lowest terms)?
Answers:
2. The ratio of girls to boys in a class is 2 : 3.
3
a) What proportion of the class are girls?
1a. 15%, 0.15, 20
3
b) If there are 12 girls, how many boys are there?
1b. 60%, 0.6,
5
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3. To make mortar you need 5 parts sand to 1 part cement. 2a. 2 or 40% b. 18
5
a) What proportion of the mix is sand?
b) If I use 5kg of cement, how much sand is needed?
3a.
5
6
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
b. 25 kg
Realise what you have done previously :
Using proportion
1. A recipe for 2 sponge cakes uses 240g of flour.
How much flour would you need for 1 cake?
How about 5 cakes?
This is called the unitary method (finding the value of 1 unit).
2. A school buys a bulk pack of 300 pens for £120.
How much is one pen?
If they sell the pens at cost, how much should it be for 10 pens?
Can you see another way of getting from 300 pens to 10 pens?
3. A recipe to make 16 gingerbread men requires 180g of flour and 110g
of butter. How much of each is needed to make 24 gingerbread men?
Hint: What can you divide and multiply by to get from 16 to 24?
Automaticity: The ability to use some skills with such ease as they no longer require active thinking
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Hard working time
Best buys (GCSE)
1. Brand A of Cheddar cheese costs £3.00 for a 400g pack. Brand B costs £3.50
for a 500g pack.
Which is the better buy?
Hint: Work out the cost for 100g of each. No marks if you just guess without
showing any reasoning!
2. In the supermarket you often see price comparisons done for you on the
shelf label – but they’re not always in the same units. For example:
Harvest Fresh Fruit & Fibre cereal is priced at £1.79 for a 500g pack. The label
says this is £3.58 per kg. (You should be able to work this out in your head!)
Sunny Grain’s version of the same cereal costs £1.29 for 375g. This is given as
34.4p per 100g. (You wouldn’t be expected to work this one out without a
calculator, but can you do it with one?)
Which is the better buy?
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Hard
Hardworking
workingtime
time
Ratio and algebra (GCSE)
Sometimes you will be asked to turn a ratio into an algebraic expression.
ACP
Example: If the ratio of x to y is 1 : 3, express y in terms of x.
Check
𝑥
1
𝑥
𝑦
We can write x : y = 1 : 3 in the form 𝑦 = 3 (or 1 = 3 )
then cross-multiply to get 3×𝑥 = 1×𝑦
Answer:
So the algebraic expression is y = 3x.
x:y
An easy way to do this is to write
2:5
x:y
VAA
5x = 2y
Check
1:3
Divide by 2 to get y
[Now multiply the x by the 3 and the y by the 1]
on its own:
3x = 1y
so
y = 3x
𝟓
y = 𝟐x or y = 2.5x
What would you write if the ratio x : y were 2 : 5?
Remember, “express y in terms of x” means that your final answer needs
to have the y on its own.
Hard working time
Mixed ratio and proportion questions (GCSE)
1. Sally is 140cm tall and Tara is 160cm. What is the ratio of their heights (in its lowest
terms)?
2. A pan with its lid weighs 2.6 kg and the lid alone weighs 400g. What is the ratio of the
weight of the lid to the weight of the pan (lowest terms)? (Read the question carefully!)
3. Lucy, Mary and Nora’s father leaves them an inheritance in the ratio 5 : 3 : 2. Mary gets
£4500. How much was there in total?
4. A cocktail has the ratio 2 parts orange juice to 5 parts lemonade. If you have 7 litres of
orange juice to use, how much lemonade do you need?
5. If it takes 100g flour to make 12 fairy cakes, how much flour do you need to make (a) 36
cakes? (b) 30 cakes?
6. The scale of a map is 1 : 25 000.
a)
b)
c)
What does this mean? 1cm on the map represents ………………………… on the ground.
What distance on the ground (in m) is represented by 3cm on the map?
What distance on the map would represent 5km on the ground?
7. A pack of 6 pens costs £1.99. A pack of 4 pens costs £1.30. Which gives better value for
money?
8. If the ratio of a : b is 2 : 3, use an algebraic expression to express b in terms of a.
Answers:
1. 7 : 8
2. 2 : 11
3. £15000
4. 17.5 l
5. 300g,
250g
6. 25000cm,
750m,
20cm
7. 33.17p vs
32.5p:
pack of 4
is better
3
8. b = 2a or b
= 1.5a
Using equivalent ratios (GCSE)
We need to write these as equivalent ratios
with a common value at the overlap
There are 3 black pens for every 2 red ones, So let’s do 2 x 5 = 10 for red.
and the ratio of red to green pens is 5: 4.
Looking at the B : R ratio, we’ve multiplied
the 2 by 5 so we also need to multiply the 3
(a) What is the ratio of black to green
by 5, giving 15.
pens?
Looking at the R : G ratio, we’ve multiplied
(b) What’s the minimum number of pens the 5 by 2 so we also need to multiply the 4
there could be in the box?
by 2, giving 8.
So the ratio B : R : G is 15 : 10 : 8
First, write down the info you’re given:
And the ratio B : G is 15 : 8
B : R : G
(b) Total number of parts = 15 + 10 + 8 = 33
3 : 2
Each part has to be a whole number of pens
5 : 4
So the total number of pens must be a multiple of 33
15 : 10 : 8
So 33 is the minimum possible number of pens.
A box contains black, red and green pens.
Hard working time
“Fraction of a fraction” ratio problem (GCSE)
Among the members of a club,
the ratio of boys to girls is 3 : 2.
1
4
1
3
of the boys and of the girls
own dogs.
(a) What fraction of the club’s
members are dog owners?
(b) What’s the minimum
number of members in the
club?
(
*
(a) ) of the members are boys, of whom + own dogs
* (
(
so + × ) = ,- of the club consists of boys who own dogs.
* ,
,
For girls who own dogs the proportion is ( × ) = *)
(
,
So the total proportion who own dogs is ,- + *)
.
0
𝟏𝟕
= /- + /- = 𝟔𝟎
3
2
(b) 20 and 15 must both be whole numbers of people,
but the lowest common denominator we can use for
both fractions is 60
So the minimum number of members must be 60.
(Or there could be 120, 180, etc. – any multiple of 60)
Hard working time
Further ratio problems (GCSE)
1. A factory produces 3 of widget A for every 5 of widget B, and 4 of widget
C for every widget B. If they make 18 of widget A, how many of widget C
will they make?
2. 4 out of every 7 members at a gym are male. 30% of the men who visit
the gym, and 20% of the women, use the rowing machine. What
proportion of the members use the rowing machine? Give your answer
as a fraction in its lowest terms.
3. Parliament has members from 3 parties: Waffle, Filibuster and Nonsense.
There are 2 Waffle members for every 3 Filibuster members and the ratio
of Waffle to Nonsense is 5 : 6. Given that there are between 200 and 250
seats in Parliament, what is the exact number of seats?
4. Blue and white paint are mixed in the ratio 3 : 7. Blue paint costs
£2.50/litre and white costs £1.40/litre. If a 2.5-litre can sells for £7.00,
how much will the profit be on 50 litres of paint?
Answers:
1. 120
9
2. 35
3. 222
4. £53.50
Change-of-ratio problem – Method 1 (1 part = n) (GCSE)
Bags A and B contain counters in the ratio 7 : 3.
3 counters are taken out of bag A and put in bag B. The ratio is now 5 : 3.
How many counters were in bag A initially?
Let’s call 1 part n counters
Then initially,
bag A contains 7n counters
and bag B contains 3n counters.
When 3 counters are taken out of A and put
into B we have
A:B
7n – 3 : 3n + 3
=
5 : 3
so 3(7n – 3) = 5(3n + 3)
21n – 9 = 15n + 15
6n = 24
=>
n = 4
If 1 part = 4 counters then
A initially had 7 x 4 = 28 counters
(and B initially had 3 x 4 = 12 counters.)
Use “after” ratio to check:
7(4) – 3 = 25; 3(4) + 3 = 15
25 : 15 = 5 : 3 ü
Change-of-ratio problem – Method 2(simult eqns) (GCSE)
Bags A and B contain counters in the ratio 7 : 3.
3 counters are taken out of bag A and put in bag B. The ratio is now 5 : 3.
How many counters were in bag A initially?
A : B (initial values)
7:3
so 3A = 7B
①
When 3 counters are taken out of A
and put into B we have
A–3:B+3
5 : 3
so 3(A – 3) = 5(B + 3)
3A – 9 = 5B + 15
3A = 5B + 24 ②
Solve simultaneous equations ① and ②:
7B = 5B + 24 (= 3A)
2B = 24
B = 12
B’s original 3 parts = 12 counters
so 1 part = 4 counters
so A originally had 7 x 4 = 28 counters.
Use “after” ratio to check: A – 3 = 25; B + 3 = 15
25 : 15 = 5 : 3 ü
Hard working time
Ratio problems with algebra (GCSE)
1. Jan and Kim own tropical fish in the ratio 3 : 4. Jan gains 3 more
fish and the ratio is now 9 : 10. How many fish did they each have
initially?
2. The ratio of Ann’s age to Bob’s age is 3 : 4. In 7 years’ time this Answers:
ratio will be 4 : 5.
1. J 15, K 20
a) What are their ages now?
b) How many years from now will the ratio be 5 : 6?
3. The ratio a : b is 3 : 4. What are these ratios in their simplest
forms?
a) b : 3a – b
b) 2a + b : 4a – b
2a. A 21, B 28
2b. 14
3a. 4 : 5
3b. 5 : 4
4. 6
4. The ratio (a number + 2) : (the number + 4) is the same as the ratio
(the same number + 6) : (the number + 9). What is the number?