**Constant of proportionality plays vital role in the concept of proportion ****in math. **

**There are two types of proportion: **

**1. Direct proportion**

**2. Inverse proportion**

**If "y" is directly proportional to "x", then **

**Where "k" is the constant of proportionality**

**Where "k" is the constant of proportionality**

**If two different quantities are given, how to check whether the relationship between them is proportional ? **

**We have to get ratio of the two quantities for all the given values. **

**If all the ratios are equal, then the relationship is proportional. **

**If all the ratios are not equal, then the relationship is not proportional. **

To have better understanding on "Identifying proportional relationships" let us look at some examples.

**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. **

**Let us look at the next example on "Constant of proportionality"**

**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. **

Let us look at the next example on "Constant of proportionality"

**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. **

**Example 6 :**

**Solution : **

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

2 / 4 = 1 / 2

4 / 8 = 1 / 2

6 / 12 = 1 / 2

8 / 16 = 1 / 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 = 4

4 = k(2)

2 = k

**Hence the constant of proportionality is "2"**

**Problem 1 :**

y is directly proportional to x. Given that y = 144 and x = 12. Find the value of y when x = 7.

**Solution :**

Since "y" directly proportional to "x", we have

**y = kx **

Using y = 144 and x = 12, we have to find the constant of proportionality.

144 = 12k

12 = k

Therefore, y = 12x

Plug x = 7

y = 12(7)

y = 84

**Hence, the value of "y" is 84 when x = 7. **

**Let us look at the next problem on "Constant of proportionality"**

**Problem 2 :**

y is inversely proportional to x. Given that y = 12 and x = 6. Find the value of y when x = 8.

**Solution :**

Since "y" inversely proportional to "x", we have

**y = k / x **

Using y = 12 and x = 6, we have to find the constant of proportionality.

12 = k / 6

12 x 6 = k

72 = k

Therefore, y = 72 / x

Plug x = 8,

y = 72 / 8

y = 9

**Hence, the value of "y" is 9 when x = 8. **

**Problem 3 :**

75 basketballs cost $1,143.75. Find the cost of 26 basketballs.

**Solution :**

This is a situation of direct proportion.

Because, less number of basket balls -----> cost will be less

Let "x" be the no. of basket balls and "y" be the cost.

Since this is direct proportion, we have

**y = kx**

Plug x = 75 and y = 1143.75

1143.75 = 75k

15.25 = k

Therefore, y = 15.25x

Plug x = 26,

y = 15.25(26)

y = 396.50

**Hence, the cost of 26 basket balls is $ 396.50**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 4 :**

7 men can complete a work in 52 days. In how many days will 13 men finish the same work?

**Solution :**

This is a situation of inverse proportion.

Because, more men -----> less days

Let "x" be the no. of men and "y" be the no. of days.

Since this is inverse proportion, we have

**y = k / x**

Plug x = 7 and y = 52

52 = k / 7

364 = k

Therefore, y = 364 / x

Plug x = 13,

y = 364 / 13

y = 28

**Hence, 13 men can complete the work in 28 days**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 5 :**

If David sells 2 gallons of juice for $4, how much money will he get by selling 17 gallons of juice ?

**Solution : **

**This is a situation of direct proportion.**

**Because, more gallons of juice -----> more amount of money**

**Let "x" be the no. of gallons of juice and "y" be the cost.**

Since this is direct proportion, we have

**y = kx**

Plug x = 2 and y = 4

4 = 2k

2 = k

Therefore, y = 2x

Plug x = 17,

y = 2(17)

y = 34

**Hence, David will earn $34 by selling 17 gallons of juice**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 6 :**

A book contains 120 pages and each page has 35 lines . How many pages will the book contain if every page has 24 lines per page?

**Solution : **

**This is a situation of inverse proportion.**

**Because, less lines -----> more pages**

Let "x" be the no. of pages and "y" be the no. of lines.

Since this is inverse proportion, we have

**y = k / x**

Plug x = 120 and y = 35

35 = k / 120

4200 = k

Therefore, y = 4200 / x

Plug y = 24,

24 = 4200 / x

x = 4200 / 24

x = 175

**Hence, ****if every page has 24 lines per page, the book will contain 175 pages**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 7 :**

The cost of a taxi is $40.50 for 15 miles. Find the cost for 20 miles.

**Solution : **

**This is a situation of direct proportion.**

**Because, more miles -----> more cost**

**Let "x" be the no. of miles and "y" be the cost.**

Since this is direct proportion, we have

**y = kx**

Plug x = 15 and y = 40.50

40.50 = 15k

2.7 = k

Therefore, y = 2.7x

Plug x = 20,

y = 2.7(20)

y = 54

**Hence, the cost for 20 miles is $54**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 8 :**

A truck covers a particular distance in 3 hours with the speed of 60 miles per hour. If the speed is increased by 30 miles per hour, find the time taken by the truck to cover the same distance

**Solution : **

**This is a situation of inverse proportion.**

**Because, more speed -----> less time**

**Let "x" be the hours and "y" be the speed**

Since this is inverse proportion, we have

**y = k / x**

Plug x = 3 and y = 60

60 = k / 3

180 = k

Therefore, y = 180 / x

**If the given speed 60 mph is increased by 30 mph,**

**then the new speed = 90 mph **

**So, we have to find "x" when y = 90. **

Plug y = 90,

90 = 180 / x

x = 180 / 90

x = 2

**Hence, if the speed is increased by 30 mph, time taken by the truck is 2 hours. **

**Let us look at the next problem on "Constant of proportionality"**

**Problem 9 : **

In a business, if A can earn $7500 in 2.5 years, At the same rate, find his earning for 4 years.

**Solution : **

**This is a situation of direct proportion**

**Because, ****more time -----> more earning **

**Let "x" be the years and "y" be the earning**

Since this is direct proportion, we have

**y = kx**

Plug x = 2.5 and y = 7500

7500 = 2.5k

3000 = k

Therefore, y = 3000x

Plug x = 4,

y = 3000(4)

x = 12000

**Hence, the earning for 4 years is $12000**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 10 : **

David can complete a work in 6 days working 8 hours per day. If he works 3 hours per day, how many days will he take to complete the work ?

**Solution : **

**This is a situation of inverse proportion.**

**Because, less**** hours per day-----> more days to complete the work**

**Let "x" be the days and "y" be the hours**

Since this is inverse proportion, we have

**y = k / x**

Plug x = 6 and y = 8

8 = k / 6

48 = k

Therefore, y = 48 / x

Plug x = 3,

y = 48 / 3

x = 16

**Hence, ****David can complete the work in 16 days working 3 hours per day.**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 11 :**

In 36.5 weeks, Miguel raised $2372.50 for cancer research. How much money will he raise 20 weeks ?

**Solution :**

**This is a situation of direct proportion.**

**Because, less number of weeks ----> amount raised will be less**

**Let "x" be the weeks and "y" be the amount of money raised**

Since this is direct proportion, we have

**y = kx**

Plug x = 36.5 and y = 2372.50

2372.50 = 36.5k

2372.50 / 36.5 = k

65 = k

Therefore, y = 65x

Plug x = 20,

y = 65(20)

y = 1300

**Hence, the money raised in 20 weeks is $1300.**

**Let us look at the next problem on "Constant of proportionality"**

**Problem 12 :**

Alex takes 15 days to reduce 30 kilograms of his weight by doing 30 minutes exercise per day. If he does exercise for 1 hour 30 minutes per day, how many days will he take to reduce the same weight ?

**Solution :**

**This is a situation of inverse proportion.**

**Because, more minutes per day----> less days to reduce the weight**

**Let "x" be the minutes and "y" be the days.**

Since this is inverse proportion, we have

**y = k / x**

Plug x = 30 and y = 15

15 = k / 30

450 = k

Therefore, y = 450 / x

**1 hour 30 minutes = 90 minutes **

**So, we have to find "y" when x = 90. **

Plug x = 90,

y = 450 / 90

y = 5

**Hence, i****f Alex does exercise for 1 hour 30 minutes per day, it will take 5 days to reduce 30 kilograms of weight.**

**Let us look at the next problem on "Constant of proportionality"**

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