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When two ratios are equal, they are said to be in proportion.
To verify whether two ratios are in Proportion, we simplify the two ratios first and then we
determine whether they are equal or not. If both the simplified ratios are equal, they are said to be
in proportion. If the simplified ratios are not equal, then the ratios are not in proportion. We use the
symbols " :: " or " = " to denote a proportion.
Consider two ratios in proportion, ab = cd , ( a:b :: c:d). Here, we have a x d = c x d.
In a statement of proportion, the first and fourth terms are known as extreme terms and the
second and third terms are known as middle terms. Thus, if two ratios are in proportion, the
product of the extreme terms = product of the middle terms.
Proportion Definition
Two ratios are said to be in proportion if they are equal. By definition, a, b, c and d is called a
proportion, if a:b = c:d.
Consider the number of boys and girls in a class. Let there be 30 boys and 15 girls.
The ratio of boys to girls = 30:15 = 2:1
The above two ratios are the same. We say that these ratios are in proportion. That is, 60:30 is in
proportion to 30:15.
This is written as 60 : 30 :: 30 : 15
Mean proportion
If x is the mean proportion between a and b, then the mean proportion is x = ab
Examples on Mean Proportion
Given below are some examples on mean proportion.
Example 1 : Find the mean proportion of 2.5 and 0.9
Solution :
If b is the mean proportion between a and c, then the mean proportional is b = ac
The mean proportional between 2.5 and 0.9 = (2.5x0.9) = 2.25 = 1.5
Example 2 : Find the mean proportional between 18 and 8
Solution : If b is the mean proportion between a and c, then the mean proportional b = ac
The mean proportional between 18 and 8 = (18x8) = 144 = 12
Example 3 : Find the mean proportional between 1.8 and 0.2
Solution :
If b is the mean proportion between a and c, then the mean proportional b = ac
The mean proportional between 1.8 and 0.2 = (1.8x0.2) = 0.36 = 0.6
Example 4 : Find the mean proportional between 12 and 132
Solution :
If b is the mean proportion between a and c, then the mean proportional b = ac
The mean proportional between 12 and 132 = (1/2x1/32) = 1/64 = 18
Example 5 : Find the mean proportional between 38 and 272
Solution :
If b is the mean proportion between a and c, then the mean proportional b = ac
The mean proportional between 38 and 272 = (3/8x27/2) = 81/16 = 94
Direct Proportion
If two quantities are so related to each other that an increase (or decrease) in the first causes an
increase (or decrease) in the second, then the two quantities are said to vary directly.
Examples of Direct Proportion :
* The cost of an article varies directly with the number of articles. [More cost, More articles]
* The work done directly varies with the number of men at work.
* The distance covered by a car directly varies with its speed.
* More the money deposited in the bank, the more the interest earned.
Direct Proportion Rule :
If two quantities `x' and `y' vary directly with each other then always
xy = k (Constant)
Examples on Direct Proportion
Given below are some examples on Direct Proportion.
Example 1 :
The cost of 5 meters of a particular quality of cloth is $210. Tabulate the cost of 2, 4, 10 and 13
meters of cloth of the same type.
Solution :
Suppose the length of a cloth is x meters and its cost, in is $y.
x
2
4
5
10 13
y
y2 y3 210 y4 y5
As the length of the cloth increases, the cost of the cloth also increases in the same ratio. It is a
case of direct proportion.
We make use of the relation of type x1y1 = x2y2
(i) Here x1 = 5 y1 = 210 and x2 = 2
Therefore, x1y1 = x2y2
5210 = 2y2
5y2 = 2 x 210
y2 = 4205
y2 = 64
(ii) x3 = 4, then 5210 = 4y3
5 y3 = 4 x 210
y3 = 8405 = 168
(iii) x4 = 10, then 5210 = 10y4
5 y4 = 10 x 210
y4 = 21005 = 420
(iv) x5 = 13, then 5210 = 13y5
5 y5 = 12 x 210
y5 = (13x210)5 = 546
Example 2 :
An electric pole, 14 meters high, casts a shadow of 10 meters. Find the height of a tree that casts
a shadow of 15 meters under similar conditions.
Solution :
Let the height of the tree be x meters. We form a table as shown below:
Height of the Object (in metres)
14 x
Length of the Shadow (in metres) 10 15
Note : The more the height of the object, the more the length of the shadow.
Hence, this is the case of direct proportion. That is,
x1y1 = x2y2
We have 1410 = x15
(14x15)10 = x
21 = x, Thus, height of the tree is 21 m
Example 3 : If 32 horses consume 112 kg of grass in a certain period, how much grass will be
consumed by 11 horses during the same period?
Solution : Fewer Horses, lesser the consumption of grass [Direct Variation]
32 horses consume = 112 kg
1 horse will consume = 11232 kg
11 horses will consume = (112x11)32
= 38. 5 kg.
Example 4 : If the weight of 65 coffee packets of the same size is 26 kg. What is the weight of
25 such packets?
Solution : The fewer the packets, the lesser the weight [Direct Variation]
Weight of 65 packets = 26 kg
Weight of 1 packet = 2665 kg
Therefore, the weight of 25 packets = (26x25)65 kg, = 10 kg
Indirect Proportion
If two quantities are so related to each other that an increase (or decrease) in the first causes a
decrease (or increase) in the other, then the two quantities are said to vary indirectly.
Examples of Indirect Proportion :
* Time taken by a car to cover a certain distance varies indirectly with the speed of the car.
[More the speed, the less is the time taken]
* Time taken to finish a work is inversely proportional to the number of men at work. [The more
the men, the less is the time taken to finish the work]
Indirect Proportion Rule :
If two quantities `x' and `y' inversely vary with each other then xy = k (constant)
Examples on Indirect Proportion :
Given below are some examples on indirect proportion.
Example 1 : 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5
pipes of the same type are used?
Solution : Let the desired time to fill the tank be x minutes. Thus, we have the following table.
Number of Pipes 60 5
Time in Minutes 80 x
Lesser the number of pipes the more will be the time required to fill the tank. So, this is a case of
inverse proportion.
Hence, 80 x 6 = x x 5
Therefore, x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.
Example 2 :
There are 100 students in a hostel. The Food provision available for them is for 20 days. How
long will these provisions last, if 25 more students join the group?
Solution :
Suppose the provisions last for y days when the number of students is 125.
We have the following table.
Number of Students 100 125
Time in Days
20 y
Note the more the number of students, the sooner the provisions would exhaust. Therefore, this is
a case of inverse proportion.
So, 100 x 20 = 125 x y
Therefore, y = 16
Thus, the provisions will last for 16 days, if 25 more students join the hostel.
Example 3 : If 15 workers can build a wall in 48 hours, how many workers will be required to do
the same work in 30 hours?
Solution : Let the number of workers employed to build the wall in 30 hours be y. We have the
following table.
Number of Hours
48 30
Number of workers
15 y
Obviously the more the number of workers, the faster they will build the wall. So, the number of
hours and number of workers vary in inverse proportion.
So, 48 x 15 = 30 x y
Therefore, y = 24
Therefore, to finish the work in 30 hours, 24 workers are required.
Properties of Proportion
If a, b, c and d is called a proportion, then
a/b = c/d
a/c = b/d
(a2)/(b2) = (c2)/(d2)
If 1 is added to both sides of the equation ab = cd,
We get, ab + 1 = cd + 1
(a+b)/b = (c+d)/d
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