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PROPORTIONS
Definition
Proportion
is a way of comparing quantities of different kinds of things. For example, if
you buy a certain kind of fruit juice powder, the label tells you that for
every tablespoon of powder you need 1 liter of water. The proportion of powder
to water is 1 table spoon to 1 liter.
We use
proportions in many everyday situations, such as in cooking, mixing building
materials or fertilizing the soil. In each case, we mix different kinds of
ingredients together. It is important to know how to increase or decrease the
ingredients in the correct proportions to get the amount of mixture we need.
EXAMPLE
1. Peter makes a herbal tea for his friends. For two people he uses 20 teaspoons
of ginseng and 500ml of boiled water. If five people were coming to visit him,
what proportions of ginseng and water must be mixed together?
Solution
USING THE UNITARY METHOD. WE NEED TO
CALCULATE THE QUANTITY OF GINSENG AND WATER NEEDED BY ONE MAN BEFORE PROCEEDING.
Here we first find how many spoons of
ginseng and how much water are needed by one person then multiply these amounts
by the who have to receive ginseng and water tea.
Using the statistics below;
Number of
people Number of teaspoons of
ginseng Amount of water
2
20
500ml
1 20÷2= 10
500ml÷2= 250ml
5 10×5=50
250ml×5=1250ml
From above,
the proportion of tea spoons of ginseng to water for 5 people is 50: 1250 or
1:25
EXAMPLE 2
Dominic buys
5l tin of paint. He needs to thin the paint with thinners. The label on the
paint says ‘use 1l of thinners for every 5l paint’. Dominic is only going to
paint a small, so he decides to use only 1/2l of the paint. How much thinner
must he add to this paint?
Solution
Let’s
convert liters (l) to milliliters (ml) for easy calculations. 1l= 1000ml
Amount
of paint Amount of thinner
5l (5000ml) 1l( 1000ml)
1l (1000ml) 1l (200ml); dividing each
of the above by 5.
1l ( 1000ml)=500ml 1l (200ml)= 100ml ;
dividing above by 2.
2 2
He mixes
thinner and paint in the Proportion of 100:500 or 1:5
TYPES OF PROPORTIONS
At this
level, we are going to study only two out of the different types of
proportions. These are: i) direct proportion and, ii) indirect proportion.
i)
DIRECT
PROPORTION. When one quantity increases as a result of an increase in the
other quantity, or decrease as a result of a decrease in another quantity, the
two quantities are said to be in a direct proportion.
EXAMPLES;
a)
The quantity of fuel a car consumes is in direct
proportion to the distance it covers
b)
The amount of money a man spends on beer is in
direct proportion to the number of bottles of beer he consumes.
c)
Electricity bill is in direct proportion to the
amount of current consumed.
d)
Water bill is in direct proportion to the amount
of water consumed.
EXERCISES
1)
A car covers a distance of 20km on 5 liters of
petrol.
i)
How far will it go on 9 liters of petrol
ii)
How much petrol will be needed to cover a
distance of 60km
Solution
i)
First
method: Calculate the distance it
covers on 1 liter of petrol then multiply by 9 to get the distance it covered
on 9 liters of petrol.
5liters
covers 20km
1liter
will cover x.
Using
ratio; 1liter: 5liters=x: 20km
1
= x
5 20km
x
= 4km
1liter covers 4km
9liters will cover 9×4km=
36km
Second
method: Combine ratio and proportion.
5 liters cover 20km
9 liters will cover x
Using ratio
and proportions
9liters: 5liters = x: 20km
9 =
x
5 20km
9×20km =x
5
9×4km = x
36km=x
Therefore 9
liters will cover 36km.
ii)
First
method.
20km
used 5 liters
1km
will use n
1km:
20km =n: 5liters
1
= n
20 5liters
n=
5 liters.
20
1
km uses 5 liters
20
60km
will use 60× 5 liters
20
60km will use 15 liters of
petrol.
Second method
Combine
ratio and proportion.
20km used 5liters
60km will
use n
60km: 20km =
n: 5liters
60
= n
20 5liters
60×5liters
= n
20
3×5liters
= n
15liters =n
n = 15liters
Hence 60km
will use 15liters of petrol.
2)
A woman pays 7200FCFA for 120 units of water
used.
a)
How much will another woman who consumes 80
units of water pay?
b)
How many units of water does a cook with bill of
14,400FCFA consume?
Solution
a)
120 units cost 7200FCFA
1 unit costs 7200 FCFA=60 FRS
120
80 units will cost 720 × 80 FRS
120
80 units will cost 4800 FRS.
b)
1unit costs 60frs
x
will cost 14400frs
x
= 14400
1
60
x = 14400 ×1unit
60
X = 240 units.
240 Will cost 14400 FRS.
Exercise for practice on direct proportion.
1)
5 cans of beer cost 120frs. What is the cost of
7 cans of beer?
2)
9 bottles of milk contain 4.5liters of milk
within them. How much do 5 bottles hold
3)
A car uses 10 liters of petrol in 75km. How far
will it go on 8 liters ?
4)
A wire 11cm long has mass 187g. What is the mass
of 7cmm long of this wire?
b)INVERSE
PROPORTION
When one
quantity decreases as a result of an increase in another quantity or increase
as a result of decrease in another quantity, then the two quantities are in
inverse proportion.
EXAMPLE
1:
>The time taken by ten people to weed a
farm is less that the time taken by two of them. (More people, less time used).
>Many people buy when prices are low while few buy when
prices are high (low price high demand and vice versa)
>One person be satisfied eating a loaf of
bread than five persons eating that same loaf of bread.
>The pressure a force exerts on an area
increases when the area decreases.(small area much pressure)
EXAMPLE 2:
i)
Two students take 15 minutes to eat a bowl of
rice.
a)
How long will it take 5 students eating at the
same rate to finish the same quantity bowl of rice?
b)
How many students will take 10 minutes eating at
the same rate to finish the same quantity of rice?
Solution
a)
2 students take 15 minutes
5
students will take x
Since proportion
is indirect, we consider ratios above in
opposite directions, that is
2students: 5students = x: 15 minutes OR
5students:2students=15minutes:x
2students
= x
5students 15minutes
2
= x
5 15minutes
2
×15minutes =x
5
6minutes =x
Therefore
5students will take 6minutes to eat the bowl of rice.
So can see that more students will take less
time to finish the bowl of rice.
b)
2 students used 15minutes
x
will use 10 minutes
Considering ratio in opposite directions
x:2students=15minutes:10minutes OR 2students
:x= 10minutes:15minutes.
x = 15
2students 10
x = 15 ×2students
10
x
=3students.
Therefore 3
students will use 10 minutes to finish the bowl of rice.
iii)
Eight men can dig a trench in 4 hours. How long
will it take 5 men to dig the same size trench?
Solution
8 men used 4
hrs
5 men will
use x
Consider ratio in opposite directions;,
and write them out as fractions.
8:5 = x: 4
8 =
x
5 4
8 ×4
= x
5
x = 32
hours
5
5 men will
take 6 hours 24 minutes.
Exercise for practice on indirect proportion
1)
3 men build a war in 10 days. How long will it
take 5 men working at the same rate?
2)
3men can dig a trench in 10 hours.How many men
will needed to dig the trench in 7hours 30minutes.
3)
A ship has sufficient food to supply 600
passengers for 3 weeks. How long will the food last for 800 people?
4)
80 machines can produce 4800 identical pens in 5
hours. At this rate,
i)
How many pens would one machine produce in 1
hour?
ii)
How many pens will 25 machines produce in 7
hours?
5)
If it takes 6 men 4days to dig a hole 3 feet
deep, how long will it take 10 men to dig a hole 7 feet deep?
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