**Proportional relationships :**

A proportion is a statement that two rates or ratios are equivalent.

For example,

62 miles / 2 hours = 3 miles / 1 hour (or) 2/4 = 1/2

A rate of change is a rate that describes how one quantity changes in relation to another quantity. A proportional relationship between two quantities is one in which the rate of change is constant or one in which the ratio of one quantity to the other is constant.

Proportional relationships are often described using words such as per or for each.

For example, the rate $1.25 / 1 pound could be described as $1.25 per pound or $1.25 for each pound.

**Example 1 :**

Examine the given table and determine if the relationship is proportional. If yes, determine the constant of proportionality.

**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 **

Plug x = 4 and y = 48

48 = k(4)

12 = k

Hence the constant of proportionality is "12"

**Example 2 :**

Examine the given table and determine if the relationship is proportional. If yes, determine the constant of proportionality.

**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, determine the constant of proportionality.

**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

Plug x = 2 and y = 1

1 = k(2)

1 / 2 = k

Hence the constant of proportionality is "1 / 2"

**Example 4 :**

**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 :**

**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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