11. RATIOS
11 - 1 Rates and ratios
2
11 - 2 The units of a ratio
3
11 - 3 Equivalent ratios
5
11 - 4 Simplifying a ratio
7
11 - 5 Comparing ratios
8
11 - 6 Scale factors
11
11 - 7 More than two numbers
13
11 - 8 Fractions and ratios
16
11 - 9 Percentages and ratios
20
11 - 10 Problems with ratios
22
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Activity 11 - 1
Rates and ratios
A rate is a change in the amount of one thing in relation to a change in
the amount of something different.
Louella says she’ll give her friends 2 jelly cups for every chocolate bar.
Louella’s exchange rate is 2 jelly cups / chocolate bar
= 1 chocolate bar / 2 jelly cups.
In each of these rates, you can tell which number refers to the jelly cups
and which number refers to the chocolate bars because the name of the
thing is written directly after the number of that thing.
Jelly cups were traded for chocolate bars in a ratio of 2 to 1.
From this sentence, you can tell that the “2” refers to the number of jelly
cups and the “1” refers to the number of chocolate bars because jelly
cups are mentioned in the sentence before chocolate bars are
mentioned.
Chocolate bars were traded for jelly cups in a ratio of 1 to 2.
Louella also prefers chocolate bars to lollies.
She says she’ll give her friends 3 lollies for every 2 chocolate bars.
1) Lollies were traded for chocolate bars in the ratio of 3:2.
2) Chocolate bars were traded for lollies in the ratio of 2:3.
Notation
1) For every 5 black jelly beans in the packet, there are 6 white ones.
The ratio of black to white jelly beans in the packet is 5:6.
2) Each time the front wheel of a tricycle makes a complete turn, its back
wheels turn 2.5 times.
The ratio of the turns of the front wheel to the turns of the back wheels
of the tricycle is 1:2.5.
3) At camp, there needs to be at least 1 adult with every 12 children.
The ratio of adults to children at camp needs to be at least 1:12.
4) To get the maximum number of calves, there should be no more than
30 cows per bull.
To get the maximum number of calves, the ratio of cows to bulls
should be no more than 30:1.
5) To make up the drink, use 1 part cordial to 4 parts water.
To make up the drink, use cordial and water in the ratio of 1:4.
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Activity 11 - 2
The units of a ratio
Group A
Alice measured 1 cup of flour and poured it into the pot.
Andy measured 2 cups of water and gradually added it to the pot.
Group B
Brianna measured 2 cups of flour and poured them into the pot.
Bill measured 1 cup of water and gradually added it to the pot.
Group C
Chloe measured 2 cups of flour and poured them into the pot.
Chris forgot his cup, so he gradually added 1 bowl of water to the pot.
Group D
Diana and Dave both forgot their cups.
Diana measured 2 bowls of flour and poured them into the pot.
Dave measured 1 bowl of water and gradually added it to the pot.
The groups that are likely to make the best dough for damper are Group
B and Group D because both these groups have mixed the flour and
water in the correct ratio. The size of the measuring units doesn’t matter
as long as both units are the same (i.e. both measurements are in cups
or both measurements are in bowls).
The groups that are likely to make the worst dough for damper are
Group A and Group C. Group A has interpreted the ratios in the wrong
order and Group C has used different measuring units.
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Instructions for this liquid fertiliser state that to
fertilise pot plants, add 3 mL to 0.5 L of water.
Calculate the ratio of fertiliser to water.
0.5 L = 500 mL
The ratio of fertiliser to water is 3 : 500
A radio station said they’d limit advertising time to
37 minutes for every 3 hours of programs.
Calculate the maximum ratio of ads to programs.
3 h = 180 min
The maximum ratio of ads to programs is 37:180
.
A serving a Weet-Bix contains 3.3 g of dietary
fibre and 400 mg of fat.
Calculate the ratio of fat to dietary fibre.
400 mg = 0.4 g
The ratio of fat to dietary fibre is 3.3 : 0.4
1 cm on this map represents 100
km on the ground.
Calculate the ratio of the map
distance to the ground distance.
100 km = 100 x 1000 x 100 cm
= 10 000 000 cm
Map distance to ground distance is
1 : 10 000 000
The answers above have been obtained by converting the units of the
second number in the ratio to those of the first number. Alternatively, you
could convert the units of the first number to those of the second. The
ratios would then be:
37 : 3
0.003 : 0.5
3300 : 400
0.00001:100
60
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Activity 11 - 3
Equivalent ratios
The graph below shows how many cups of salt need to be mixed with a
given amount of flour to make playdough.
3
1
Rise
Rise
Cups of Salt
2
Run
Run
0
1
2
3
Cups of Flour
Cups of salt
Cups of flour
Ratio of salt to flour
1
2
1:2
2
4
2:4
4
5
6
3
6
3:6
1:2 = 2:4 = 3:6
Laura usually uses 1 cup of salt to make playdough for her children.
On a special day there were lots of extra children to cater for, so when
making the playdough, Laura added an extra cup of salt to the pot.
She needed to add 2 extra cups of flour to the pot.
If Laura adds 2 more cups of salt to the pot, she needs to add 4 more
cups of flour.
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Cups of
Salt
Cups of
Flour
½
1
1
2
2½
5
1¼
2½
Ratio of Salt to Flour
Ratio notation Rate notation
½
½ :1
1
1
1:2
2
2½
2½:5
5
1¼
1¼:2½
2½
Yes. Are all the ratios in the table are equivalent.
÷3
x2
1½
=
3
x2
3
6
=
1
2
÷3
The ratio in the table is in its simplest form is 1:2.
The ratio in the table that is a unitary ratio is 1:2.
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Activity 11 - 4
Simplifying a ratio
In the choir, there are 85 children. Of these children, 51 are sopranos
and 34 are altos. So the ratio of sopranos to altos is 51:34.
If 6 more sopranos want to join the choir, he will need 4 more altos to be
able to keep the sopranos and altos in the 3:2 ratio.
If he manages to get all these additional children for the choir,
there would be 57 sopranos
and there would be 38 altos.
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Activity 11 - 5
Comparing ratios
SLOPE 1
SLOPE 2
SLOPE 3
Slope 3 appears to be the steepest.
No. You cannot tell which is steepest by comparing the vertical distance
up the side of each slope.
No. You cannot tell which is steepest by comparing the horizontal
distance along the bottom of each slope.
SLOPE 1
3:6=1:2
SLOPE 2
SLOPE 3
2:6=1:3
3:3=1:1
The simplest forms of these ratios are also unitary ratios.
From unitary ratios, you can tell which slope is the steepest by the value
of the second number in the ratio. The smaller the second number, the
steeper the slope.
When slopes are written as ratios with the same second number, you
can tell which slope is the steepest by the value of the first number. The
larger the first number, the steeper the slope.
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(a) 3 : 9
(b) 2 : 5
1:3
1 : 2½
Order: (c) (a) (e) (b) (d)
(a) 1 : 4
(b) 3 : 8
2:8
3:8
Order: (e) (a) (b) (c) (d)
(c) 0.1 : 0.4
(d) ⅓ : ⅔
(e) 0.5 : 1.3
1:2
1 : 2.6
1:4
(c) 0.4 : 1
(d) ¼ : ½
3.2 : 8
4:8
(e) 0.1 : 0.5
1.6 : 8
Note: a different second number could be used for the ratios above, but
the order should stay the same.
Percentages are usually easier to compare than fractions because the
numbers are all out of 100.
(a) 3 : 4
(b) 7 : 8
57%
87½%
Order: (a) (c) (d) (b) (e)
(c) 3 : 5
(d) 2 : 3
60%
66⅔
(e) 3 : 2
150%
This road sign in England warns
motorists that they are approaching
a slope of 16%.
The vertical green line marked on
the sign is 1.5 cm and the horizontal
grren line is 3 cm long.
The slope is a 50% slope. This is
much steeper than the number
given on the sign indicates.
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Ski slope. Gradient = 1
4
No. If it is a straight slope, it does not matter what section of the slope
you choose to measure because the gradient of a straight slope is the
same all the way along the slope.
Photograph A
Photograph B
Photograph D
Photograph C
If you make the verical measurement on each photograph 2 cm (as
shown by the triangles drawn on the photographs), the horizontal
measurements will be as follows.
Photograph A - 5 cm, Photograph B - 3 cm
Photograph C - 3.5 cm, Photograph D - 4.5 cm.
The order of the photographs from least to greatest slope: A, D, C, B
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Activity 11 - 6
Scale ratios
NEW
ORIGINAL
10 cm
The scale factor needed to
produce the new box is 2.
4 cm
4 cm
1) The height of the new box to the height of the original box.
20:10 = 2:1
2) The width of the new box to the width of the original box.
8:4 = 2:1
3) The base area of the new box to the base area of the original box.
64:16 = 4:1
4) The volume of the new box to the volume of the original box.
1280:160 = 8:1
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The scale ratio for the sampler will be 1:3.
The scale factor will be 1
3
15 cm
12 cm
From the scale factor, you can tell whether the thing produced will be
larger or smaller than the original thing. If the scale factor is greater than
1, the new thing is larger. If it is less than 1, the new thing is smaller.
The circumference of the jars:
a) the original jar
π x diameter = 12 π
b) the new jar
π x diameter = π x (12 ÷ 3)
= 4π
The ratio of the circumference of the new jar to the circumference of the
original jar is 4π : 12 π = 4 : 12 = 1: 3
The area of the base of the jars:
a) the original jar
b) the new jar
2
2
π x radius2 = π x (6 ÷ 3) 2
π x radius = π x 6
= 36 π
= 4π
The ratio of the base area of the new jar to the base area of the original
jar is 4 π: 36 π = 4: 36 = 1: 9
The volume of the jars:
a) the original jar
b) the new jar
base area x height = 36 π x 15 base area x height = 4π x 5
= 540 π
= 20 π
The ratio of the volume of the new jar to the volume of the original jar is
20 π : 540π = 20 : 540 = 1: 27
The original jar contains 810 g of sauerkraut.
The weight of sauerkraut in the new jar is 810 ÷ 27 = 30 g
Boxes
Jars
Scale ratio
2:1
Length ratio
2:1
Area ratio
4:1
Volume ratio
8:1
1:3
1:3
1:9
1:27
If you know the scale ratio:
a) the length ratio is the same as the scale ratio
b) the area ratio is the square of the scale ratio
c) the volume ratio is the cube of the scale ratio.
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Activity 11 - 7
More than two numbers
Kites
E
B
D
C
A
Margy wants to make a kite that is 700 mm long. So A = 700 mm.
A:B:C:D:E=7:2:5:2:4
= 700 : 200 : 500 : 200 : 400
The total length of rod (in mm) that Margy needs to make her kite:
Length = 2 x 700 + 2 x 200 + 500 mm
= 2300 mm
The area of the material (in m2) in Margy’s kite:
Area = 2 (½ x A x D) + A x E
=AxD+AxE
= A x (D + E)
= 700 x (200 + 400) mm2
= 420 000 mm2
= 0.42 m2
Steve has a roll of material 1200 mm wide and wants to use the whole
width to make a kite. So D = 300 mm (Hint: D + E + D = 1200 mm)
A:B:C:D:E=7:2:5:2:4
= 1050 : 300 : 750 : 300 : 600
The area of material (in m2) in Steve’s kite.
Area = A x (D + E)
= 1050 x (300 + 600) mm2
= 945 000 mm2
= 0.945 m2
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Jellybeans
Mike found the following ratios.
purple : black = 4:3
green : black = 1:3
yellow : black = 5:6
orange : black = 1:2
red : black = 3:2
pink : black = 2:3
white : black = 1:1
For every 4 purple jellybeans, 3 jellybeans are black.
For every 3 black jellybeans, 1 jellybean is green.
So for every 4 purple jellybeans, 1 is green.
purple : black : green = 4:3:1
For every 5 yellow jellybeans, 6 jellybeans are black.
For every 2 black jellybeans, 1 jellybean is orange.
So for every 6 black jellybeans, 3 are orange.
yellow : black : orange = 5:6:3
For every 3 red jellybeans, 2 jellybeans are black.
So for every 9 red jellybeans, 6 jellybeans are black.
For every 3 black jellybeans, 2 jellybeans are pink.
So for every 6 black jellybeans, 4 are pink.
red : black : pink = 9:6:4
yellow : black : pink = 5:6:4
purple : green : yellow = 8:2:5
purple : green : yellow : orange : red : pink : white : black
= 8:2:5:3:9:4:6:6
The smallest possible number of jellybeans in this jar is 43.
If there are 30 black jellybeans in the jar, the total number of jellybeans
in the jar is 43 x 5 = 215.
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Gear ratios
MIDDLE GEAR
FOLLOWER
DRIVER
16 Teeth
21 Teeth
36 Teeth
Revolutions of the middle gear for 1 revolution of the driver = 16 = 4
9
36
Ratio of revolutions of driver : middle gear = 1 : 4
9
=9:4
The follower has 21 teeth.
16
Revolutions of the follower for 1 revolution of the driver =
21
Ratio of revolutions of driver : follower = 1 : 16
21
= 21 : 16
The common gear in these two ratios is the middle gear.
driver : middle gear = 9 : 4
= 63 : 28
driver : follower = 21 : 16
= 63 : 48
So driver : middle gear : follower = 63 : 28 : 48
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Activity 11 - 8
Fractions and ratios
Marbles
There are two types of marbles in this picture.
rainbow
plain
The unshaded part of the bar represents the marbles that are not rainbow
coloured.
The fraction of the marbles that are rainbow: 6 = 3
10 5
The simplest ratio of rainbow marbles to plain marbles is 3:2.
The bar diagram shows a fraction as the number of parts shaded (or parts
unshaded) in relation to the total number of parts.
The bar diagram shows a ratio as the number of parts shaded in relation to the
number of parts unshaded (or vice versa).
Cereal packs
Kellogg’s is running a promotion for Sultana Bran.
They put a superman X-ray vision pen inside 1 in 3 packs.
without pen
with pen
2
of packs has a pen inside.
3
The ratio of packs with pens to packs without
pens is 2:1.
If you had 27 packs of Sultana Bran, you would
expect
18 to have a pen inside
9 not to have a pen inside.
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Soup
Malcolm makes up tomato soup from cans.
For every can of condensed soup mix that he puts in the pot, he adds a can that
is half-filled with water and half-filled with milk.
condensed soup mix
water
milk
The ratio of water to the other ingredients in the pot is
1:3
The fraction of the soup in the pot that is water is 1 .
4
Malcolm needs to add 0.25 L of water.
He used 0.5 L of condensed soup mix.
Malcolm has 1.5 L of soup made up in the pot.
He put in 0.75 L of condensed soup mix.
Paint
A pre-school teacher has bottles of poster paint to put into small pots.
Each bottle has a different colour of paint – one is red, one is yellow and one is
blue.
To make brown paint, she mixes red : yellow : blue in the ratio of 2:3:1.
red
yellow
blue
The brown paint is made up of:
1 red paint
3
1 yellow paint
2
1 blue paint
6
The teacher needs to make 3 pots of brown paint.
She needs 1 pot of red paint.
She needs 1½ pots of yellow paint.
She needs ½ pot of blue paint.
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Fertiliser
A bag of Vergrow fertiliser is 5-10-5.
1) The simplest ratio of:
N to P is 1:2
N to K is 1:1
P to K is 2:1
K to P is 1:2
N to P to K is 1:2:1
N (nitrogen)
P (phosphorus)
K (potassium)
2) The percentage of the fertiliser that is N is 5%.
3) The fraction of the fertiliser that is P is 1 .
10
4) If the bag contains 20 kg of fertiliser,
it contains 1 kg N.
It contains 2 kg P.
A bag of Greengrow fertilizer is 8-8-2.
5)
Greengrow has the highest fraction of N.
Vergrow has the highest percentage of K.
Vergrow has the highest percentage of the 3 major nutrients.
Greengrow has the highest ratio of N to P.
Greengrow has the highest ratio of N to K.
6) Equal amounts of both fertilisers are put on a garden.
a) The percentage of P in the total amount of fertiliser:
Fraction of P fertiliser = 10 + 8 = 18 = 9 = 9%
200
200 100
b) The ratio of N to P to K is
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8 + 5 : 10 + 8 : 5 + 2 = 13:18:7
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Using fractions to divide in a given ratio
A bag of pellets weighs 30 kg.
Sam has 1 rabbit. Holly has 2 rabbits. May has 3 rabbits.
So Sam, Holly and May divide the pellets in the ratio 1:2:3.
Sam
Holly
May
1
Sam gets of the bag of pellets.
6
Weight of pellets for Sam = 1 x 30 kg
6
= 5 kg
1
Holly gets of the bag of pellets.
3
Weight of pellets for Holly = 1 x 30 kg
3
= 10 kg
May gets 1 of the bag of pellets.
2
Weight of pellets for May = 1 x 30 kg
2
= 15 kg
A bag of pellets costs $48.
Cost for Sam = 1 x $48 = $8
6
Cost for Holly = 1 x $48 = $16
3
Cost for May = 1 x $48 = $24
2
Cost for Sam: $8
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Cost for Holly: $16
19
Cost for May: $24
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Activity 11 - 9
Percentages and ratios
Using percentages to divide in a given ratio
Last year Lindsay worked in the garden for 84 hours, Ray worked for 53
hours and Graham worked for 19 hours.
So Lindsay, Ray and Graham divide their profit in the ratio 84 : 53 : 19.
Their profit from the garden last year was $2230.
Total number of hours worked last year = 84 + 53 + 19
= 156
They wanted to know the percentage of the work hours done by each
person.
Lindsay Percent of total hours = 84
156
= 54%
53
Ray
Percent of total hours =
156
= 34%
19
Graham Percent of total hours =
156
= 12%
How the $2 230 profit should be split:
Lindsay
Payment for work = $2230 x 0.54
= $1204.20
Ray
Payment for work = $2230 x 0.34
= $758.20
Graham
Payment for work = $2 230 x 0.12
= $267.60
To check this answer: $1204.20 + $758.20 + $267.60 = $2230
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Combining ratios
Justin has pineapple juice, orange juice and mango juice.
First Justin tries to find the best combination of pineapple and orange.
He makes up juices with different percentages of each juice type.
He likes a juice that is 80% pineapple and 20% orange.
pineapple
This combination is illustrated
by this bar diagram.
The ratio of pineapple to orange in this juice is 4:1.
He also likes 60% pineapple and 40% orange.
This combination is illustrated
by this bar diagram.
The ratio of pineapple to orange in this juice is 3:2.
pineapple
orange
orange
Justin thinks the best combination would be in the middle of these two
combinations, so he adds equal quantities of each combination.
To illustrate this, combine the two bar diagrams into the one below.
The ratio of pineapple to orange in this juice is 7:3.
Next Justin tries to find the best combination of orange and mango.
He has a juice that is 50% orange and 50% mango
The ratio of orange to mango in this juice is 1:1.
He also has a juice that is 75% orange and 25% mango.
The ratio of orange to mango in this juice is 3:1.
He adds equal quantities of these two combinations.
The ratio of orange to mango in this juice is 3:5.
Justin thought that combining the ratios 1:1 and 3:1 would give a ratio of
4:2 (because 1 + 3 = 4 and 1 + 1 = 2). This is not true because each part
of the first ratio is ½ of the whole and each part of the second ratio is ¼
of the whole.
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Activity 11 - 10
Problems with ratios
A house fly has a wingspan of
about 15 mm.
The wingspan of the house fly in
the diagram is 75 mm.
A length of 1 mm on the fly is
equivalent to 5 mm in the diagram.
The scale ratio is 5:1.
WINGSPAN
At a cat show, there are 25% more
Persian cats than Siamese cats.
There are twice as many Tabby cats
as Siamese cats.
The ratio of Tabby cats to Persian
cats is 8:5.
The fly been magnified by 5 times.
The length of a fly’s front leg in real
life is 20 ÷ 5 = 4 mm.
Widescreens and standard screens
have different proportions.
For widescreens the ratio of width to
height is 16 :9.
For standard screens the ratio of width
to height is 4:3.
The ratio of the area of a widescreen
to a standard screen:
- if they have the same width is 3:4
- if they have the same height is 4:3.
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The ratio of boys to girls in a class is 4:3.
When the class photograph was being
taken, 4 boys were absent.
This made the ratio of boys to girls 1:1.
When everyone is present, there are 28
students are in the class.
This circle in this diagram just touches
the sides of the larger square.
The corners of the smaller square just
touch the circle.
Find the simplest ratio of the areas of
larger square : circle : smaller square.
(Leave π in your answer.)
Area of larger square = (2r)2
= 4r2
Area of circle = π r2
Area of smaller square = (√2r)2
= 2r2
So the ratio is 4: π :2
r
radius = r units
Pru has a 1 L bottle of guava and apple
juice mixed in a ratio of 3:1.
25% is apple juice.
Olwyn has a 1 L bottle of guava and
apple juice mixed in a ratio of 4:1.
20% is apple juice.
They pour both bottles into a bowl.
The percentage of apple juice in the bowl
is 22½%.
The ratio of guava to apple juice in the
bowl is
77½ : 22½ = 155 : 45 = 31: 9
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