WRITING AN EQUATION FOR A PROPORTIONAL RELATIONSHIP

If there is a proportional relationship between the two variables "x" and "y", we can write the relationship between them using an equation. 

There are two types of proportional relationships. 

1.  Direct proportion

2.  Inverse proportion

Direct Proportion - Concept

If x gets increased, y also gets increased

or

If x gets decreased, y also gets decreased 

Then, x is directly proportional to y. And x and y can be related by the equation given below.

y  =  kx

Inverse Proportion - Concept

If x gets increased, y gets decreased

or

If x gets decreased, y gets increased 

Then, x is inversely proportional to y. And x and y can be related by the equation given below. 

y  =  k/x

The variable k is called the constant of proportionality, and it represents the constant rate of change or constant ratio between x and y.

Example 1 :

Examine the given table and determine if the relationship is proportional. If yes, write the equation which relates hours and miles. 

Solution : 

Let us get the ratio of x and y for all the given values. 

4 / 48  =  1 / 12

7 / 84  =  1 / 12

10 / 120  =  1 / 12

When we take ratio of x and y for all the given values, we get equal value for all the ratios.

Therefore the relationship given in the table is proportional.   

When we look at the above table when x gets increased, y also gets increased, so it is direct proportion. 

Then, we have 

y  =  kx 

Substitute x = 4 and y = 48.

48  =  k(4)

12  =  k

So, the required equation is y = 12x.

Example 2 :

Examine the given table and determine if the relationship is proportional. If yes, write the equation which relates days and pages read. 

Solution : 

Let us get the ratio of x and y for all the given values. 

1 / 100  =  1 / 100

3 / 300  =  1 / 100

5 / 550  =  1 / 110

6 / 600  =  1 / 100

When we take ratio of x and y for all the given values, we don't get equal value for all the ratios.

Therefore the relationship given in the table is not proportional.  

Example 3 :

Examine the given table and determine if the relationship is proportional. If yes, write the equation which relates cups of flour and loaves of bread. 

Solution : 

Let us get the ratio of x and y for all the given values. 

2 / 1  =  2

4 / 2  =  2

8 / 4  =  2

10 / 5  =  2

When we take ratio of x and y for all the given values, we get equal value for all the ratios.

Therefore the relationship given in the table is proportional.   

When we look at the above table when x gets increased, y also gets increased, so it is direct proportion. 

Then, we have 

y  =  kx 

Substitute x = 2 and y = 1.

1  =  k(2)

1 / 2  =  k

So, the required equation is y = (1/2)x.

Example 4 :

Examine the given table and determine if the relationship is proportional. If yes, write the equation which relates number of socks and cost.

Solution : 

Let us get the ratio of x and y for all the given values. 

1 / 2  =  1 / 2

2 / 4  =  1 / 2

3 / 6  =  1 / 2

4 / 6  =  2 / 3

When we take ratio of x and y for all the given values, we don't get equal value for all the ratios.

Therefore the relationship given in the table is not proportional.

Example 5 :

Examine the given table and determine if the relationship is proportional. If yes, write the equation which relates hours worked and money earned. 

Solution : 

Let us get the ratio of x and y for all the given values. 

1 / 23  =  1 / 23

2 / 36  =  1 / 18

5 / 75  =  1 / 15

When we take ratio of x and y for all the given values, we don't get equal value for all the ratios.

Therefore the relationship given in the table is not proportional.

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