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Direct Proportion

If the relation between two quantities is such that an increase (decrease) in one quantity results in increase (decrease) in other quantity, the quantities are said to vary directly.

Some examples of direct proportions

1. Cost of articles  is directly proportional to Number of articles.
(Ratio of their costs)::(Ratio of number of articles)
2. Work done is directly proportional to the number of workers.
(Ratio of their work done):: (Ratio of number of workers)

Indirect Proportion

If the relation between two quantities is such that a increase (decrease) in one quantity results in decrease (increase) in other quantity, the quantities are said to vary indirectly or inversely.
Some examples of direct variations
1. The time taken to cover a certain distance is inversely proportional to the Speed of a car.
(Ratio of speeds)::(Inverse ratio of time taken)
(Inverse Ratio of speeds)::(Ratio of time taken)
2. Time taken to complete the work is inversely proportional to the Number of workers.
(Ratio of men)::(Inverse ratio of time taken)
(Inverse Ratio of men)::(Ratio of time taken)

The rule of three

When two ratios are given and they are proportional such that any three of two antecedent and two consequent are known then the rule of three can be applied.

a:b = c:x then x = bc/a

If p:q is directly proportional to r:s then p:q::r:s.
If p:q is inversely proportional to r:s then [1/(p:q)]::r:s i.e. q:p::r:s.
As we know that :: is a kind of equality so :: can be replaced by =.

Examples of direct proportion

1. Cost of 5 notebooks is 50 dollars. Find the cost of 7 notebooks.
Solution:
Let the cost of 7 notebooks be x dollars.
Ratio of the Cost::Ratio of No. of notebooks
50:x::5:7
By using the rule of three we get
x = 50×7/5 = 70
Hence the cost of 7 notebooks is 70 dollars.
2. 8 men build 2 rooms in 20 days. How many men will build 4 rooms in 20 days?
Solution:
Let the number of men required be x.
Ratio of Workers:: Ratio of their work
8:x::2:4
By using the rule of three we get
x = 8×4/2 = 16
Hence the number of men required is 16 men.

Examples of Indirect proportion

1. A car covers 200 km at 100 km/h in 2 hrs. In how many hours will it cover the same distance at 150 km/h.
Solution:
Let the time taken be x hrs.
Ratio of distance::Inverse ratio of time
100:150::x:2
By using the rule of three we get
x = 100×2/150
x = 4/3
It will cover the same distance at 150 km/hr in 4/3 hrs or 1 h and 20 minutes.

2. 8 men build 2 rooms in 20 days. How many days will be required to build 2 rooms by 16 men?
Solution:
Let the number of days required be x.
Ratio of Number of men::Inverse ratio of days
8:16::x:20
By using the rule of three we get
x = 8×20/16 = 10
Hence the number of days required is 10.