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Expressing Relationships as Ratios | Sharing
Ratios |
| Map Scales | Comparing
value for money | Quick quiz |
Expressing
Relationships as Ratios
The
following examples shows how ratios can be used to express many different
relationships between a number of quantities or people.
|
Example
1:
A take away food store finds that they sell burgers in the following
ratios
Super Burgers : Basic Burgers : Vegie Burgers = 25 : 10 : 2
What
is the ratio in simplest terms for
|
| a)
Super : Basic |
b)
Basic : Vegie |
| c)
Basic : Super |
d)
Vegie : Super |
| Working
Out |
| a)
Super : Basic = 25 : 10. I could choose to simplify this ratio
further, to 5:2 |
| b)
Basic : Vegie = 10 : 2. The equivalent ratio in simplest terms
would be 5 : 1 |
c)
Basic : Super = 10 : 25. I could then simplify this ratio to
2:5 (Note that we are making a similar comparison to (a) but that
the order of the ratio has changed.) |
| d)
Vegie : Super = 25 : 2. I could choose to re-express this ratio
as 12.5 : 1 but most people prefer both to be whole numbers. |
Example
2: A childcare centre requires a minimum caregiver to child ratio
of 1 : 5. Work out how many caregivers are required, or children able
to attend, for each day of the week.
a) Monday: How many caregivers are needed if 25 children attend?
b) Tuesday: If only 4 caregivers are available how many children can
attend?
c) Wednesday: How many caregivers are needed if 33 children attend?
d) Thursday: How many caregivers are needed if 26 children attend?
e) Friday: If only 5 caregivers are available, can 30 children attend?
| Working
Out |
Thinking |
|
a)
25/5 = 5
5
caregivers are required
|
I
need a 1:5 ratio of caregiver to child, this means I need 1 caregiver
for every 5 children. To find out how many caregivers I need I
can divide the number of children by 5. |
|
b)
4 caregivers x 5 children/caregiver
20
children can attend
|
Only
4 caregivers are available. The maximum number of children each
caregiver can look after is 5, therefore 5 x 4 children can attend. |
|
c)
33/5 = 6 and 3 left over
7
caregivers are required
|
33
children divided by 5 gives 6, with 3 children left over. This
means I need an extra caregiver for the 3 extra children. |
|
d)
26/5 = 5 and 1 left over
6
caregivers are required
|
26
children divided by 5 gives 5, with 1 child left over. This means
I need an extra caregiver for the extra child. |
|
e) 5
caregivers x 5 children/caregiver
25
children can attend (30 children cannot attend)
|
Only
5 caregivers are available. The maximum number of children each
caregiver can look after is 5, therefore 5 x 5 children can attend. |
Example
3: Local primary school students were surveying who was borrowing
books at the local library. Over a 1 week period it was found that
200 people borrowed books: 110 people were women, 50 were men and
the rest were children. Express the following relationships as ratios.
| a)
the number of women (borrowers) to the number of children (borrowers) |
b)
the number of children to the number of men |
| c)
the number of men to total number of borrowers |
d)
the number of men to the number of women |
| e)
the number of women to the number of men |
f)
the number of women to total number of borrowers |
|
Thinking
|
| a)
There are 110 women. There are 200 - 110 - 50 = 40 children. The
ratio of the number of women to children is 110:40. We can simplify
this ratio to 11:4 |
| b)
There are 50 men and 40 children. The ratio of the number of men
to the number of children is 50:40. We can simplify this ratio
to 5:4. |
| c)
There are 50 men. There are 200 people in total. The ratio of
the number of men to the number of total people is 50 : 200. This
ratio can be simplified to 1:4. |
| d)
There are 110 women and 50 men. The ratio of women to men is 110
: 50. This ratio can be simplified to 11:5. |
| e)
There are 50 men and 110 women. The ratio of men to women 50 :
110. We can simplify this ratio to 5:11. |
| f)
There are 110 women. There are 200 people in total. The ratio
of the number of women to the number of total people is 110 :
200. We can simplify this ratio to 11:20. |
Sharing
Ratios
Example
4: movie - sharing ratios. A prize of $450 needs to shared in the
ratio of 1:2
Example
5: movie - sharing ratios.
Three brothers pool their money for a $100 raffle ticket for a car.
Jarrod puts in $20, Jeremy puts in $20 and Gerald puts in $60. The
prize, a four wheel drive vehicle, is worth $60 000. If they win the
car, what is the dollar value of each brothers share?
The
movie below shows a another, shorter method to calculate the answer
Map
Scales
A
very common use of ratios is for map scales. A street directory might
say it uses a scale where 1 cm represents 100 metres. To write this
as a ratio both quantities must be in the same
units.
1 cm represents 100 metres so we convert 100 metres into centimetres,
100 m x 100 = 10 000 centimetres
1 cm represents 10 000 centimetres
We
can now write the ratio as 1 : 10 000. This ratio is called
a scale factor. The scale factor is what we generally look
for when we are trying to work out what distance a portion of a map
represents.
A
scale factor of 1 : 10 000 tells us that 1 cm on the map represents
10 000 cm on the ground, 1 inch on the map represents 10 000
inches on the ground and 1 finger width on the map represents 10 000
finger widths on the ground.
Example
6: movie - map scales.
On an architect's drawing 1 cm represents 0.5 of a metre, what is
the scale factor?
What
would 1 inch on the drawing represent?
Example
7: movie - map scales.
On a map in which 1 cm represents 5 km, calculate the scale factor
(express the scale in ratio form).
What
is the real distance measured by 9 cm on the map?
The
speaker makes a mistake towards the end of this movie, can you pick
it up?
'9 cm on the map equals ..... on the ground?'
If the actual distance is 35 km, what distance is this on the map?
Comparing
value for money
A
common application of ratios is comparing prices of similar products
but different quantities at the supermarket.
Example
8: Which of these two bottles of tomato sauce is most economically
priced?
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|
|
500
ml bottle is $2.74
|
250
ml bottle is $1.55 |
Without
using a calculator we can estimate which bottle of sauce is the best
buy. The
larger bottle is twice the mass of the smaller bottle, so the mass
ratio is 2:1. The price of the larger bottle is less than twice the
price of the smaller bottle, so the larger bottle is better value
for money.
Example:
Which of these bags of sugar is most economically priced?
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|
|
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2kg
bag of sugar is
$2.75
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1 kg bag of sugar is $1.25
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500
g bag of sugar is 85 cents
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|
Thinking
|
|
The
mass ratio of the bags of sugar is 2 : 1 : 1/2
The price ratio of the bags of sugar is 2.75 : 1.25 : 0.85
We
can see that the 500g is not a bargain since the price per 1/2
kg is much more than half the price for the 1 kg bag. The price
for the 2 kg bag is more than double the price for the 1 kg
bag so it is not the cheapest either. Therefore the 1 kg bag
of sugar is the best value.
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Quick quiz
| 1. |
A
cafe finds that cappuccinos are sold in the following ratios 3
: 20 : 8 = baby size: regular size: mug size. What is the ratio
in simplest terms for, |
| a) |
baby
to mug |
| b) |
baby
to regular |
| c) |
mug to regular |
| d) |
mug
to baby |
|
|
| 2. |
The
fish pond at the back of the garden has 2 turtles, 7 orange fish,
3 mottly coloured fish and 12 tiny black fish. What the are following
ratios? |
| a) |
turtles
: fish |
| b) |
orange
fish : black fish |
| c) |
turtles
: black fish |
| d) |
mottly
fish : total number of fish |
| e) |
mottly
fish to turtles |
| f) |
black
fish to orange fish |
|
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| 3. |
A
crowd is fed with 13 pizzas and 5 lasagnas. |
| a) |
What is the ratio of lasagnas to pizzas? |
| b) |
If each pizza is cut into 5 pieces and shared equally, what is
the ratio of pieces of pizza to lasagnas? |
|
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| 4. |
Calculate
the scale factor of each of the following scales |
| a) |
1 cm represents 10 m |
| b) |
1 inch represents 300 inches |
| c) |
10
mm represents 1 km |
| d) |
15 cm represents 500 m |
|
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| 5. |
From
the scale factor given, indicate whether the diagram is larger
or smaller than the object it represents |
| a) |
1 : 30 |
| b) |
1 000 000 : 1 |
| c) |
1 : 1 000 000 |
| d) |
1 : 20 |
| e) |
2 : 5 |
| f) |
15 : 2 |
|
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| 6. |
Which
is the best price for mineral water and why?
| 600
ml 98 cents |
1.25
L $ 1.85 |
2
L $ 3.55 |
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To
view the quiz answers, click here.
'Talking
through' questions
Talking
through' questions and answers have been provided to enable you to
see how an 'expert' might tackle these types of questions.
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1.
The distance between two towns is 450 km. This distance is represented
on a map by a length of 150 mm. What is the scale of this map?
Solution
1.
Change 450 km to mm. 450 km = 450 x 1000 = 450000m = 450000
x 100 = 45000000 cm = 45000000 x 10 = 450000000 mm
150
: 450000000
15
: 45000000 (dividing by 10)
1
: 3000000 (dividing by 15)
Scale
is 1 : 3000000
2.
In a club, the ratio of netballers to swimmers to athletes is
3:7:10. If there are 240 people in the club and each member
does one and only one of the above sports, and no other sports
are practised at this club, how many swimmers are there?
Solution
3
: 7 : 10 means for every 3 netballers there are 7 swimmers,
and 10 athletes.
3 + 7 + 10 = 20
This means for every 20 people, 7 people are swimmers.
Hence,  of
the 240 people are swimmers.
of 240 = 12, so
of 240 = 7 x 12 = 84
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