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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