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
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
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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