Inversely Proportional

Unitary method is the one where we find the value of the required number of quantity by first finding the value of the unit quantity. This can be divided into two types, which are direct variation and inverse variation.

Let us consider the real life situation. If two taps having the same capacity can fill a tank in 30 minutes, how long will it take for a single tap to fill the tank? If there are three such taps how long will it take for the three taps to fill the tank. How do we solve such type of problem? We can easily solve these types of problems if we understand the basic definition and equation of inverse proportion. We shall also discuss with some of the interesting examples on inverse proportion.

An inverse relation is indicated when two quantities are so related that,

(i) an increase in one causes a corresponding decrease in the other and vice versa.

(ii) the ratio of any two values of one quantity is the inverse of the ratio of the corresponding values of the other(or) the product of the two compared quantities remains constant. Examples:
a. Speed varies inversely as time, As speed increases, time decreases, similarly as speed decreases, time increases.
b. Number of men required to complete the work varies inversely as the time. As the number of men increases, the work can be completed in less time, if the number of men decreases the work will take more time to complete.

Let us study the following table which relates the speed and time to cover the same distance.

Speed (mph)	30	40	60	120
Time (hours)	4	3	2	1

The above table shows the speed taken to travel a given distance and the time taken to cover the same distance. We observe that as the speed increases, the time taken decreases.

Also we observe that the product of the speed and the time remains constant under each column.
(i. e), 30 x 4 = 120
40 x 3 = 120
60 x 2 = 120
120 x 1 = 120

We shall discuss with the inversely proportional formula and the inversely proportional equation in the proceeding sections.

From the definition of inversely proportional, if one quantity increases the other quantity also decreases, such that the product of the two quantities remains constant.

If we assume one quantity as x, and the other as y, such that x is inversely proportional to y, then x y = a constant, which can be written as xy = k, where k is a real constant.

Let us study the following example.

Solved Example

Question: If 20 men consume a certain quantity of rice in 14 days, then in how many days will 8 men consume the same quantity of rice. How many men are required to consume the same quantity of rice in 28 days?
Solution:

Let us prepare a table relating the number of men and the number of days required to consume the rice.

Number of men ( x )	20	8	x
Number of Days ( y )	14	y	28

Since the two quantities are inversely proportional, we have xy = constant

                         ( i. e )      x . y = k
                                  => 20 x 14 = k
                                  => 280    = k
The constant of proportionality is, k = 280

(i) According to the definition of inverse proportionality, the product of the two quantity remains constant = k.
Therefore when there are 8 men, 8 . y = 280

                                        =>         y = $\frac{280}{8}$

                                                       = 35
Therefore, 35 days are required to consume the food if there are 8 men.

(ii) Using the definition, of inverse proportionality and equating the product to the constant of proportionality k,
                                we get, x . 28 = 280

                                   => x = $\frac{280}{28}$

                                                    = 10
Therefore, 10 men are required to consume the rice in 28 days.

According to the definition of inversely proportional we observe that, the two quantities x and y are such that xy is a constant, which is written as
x y = k, where k is a real constant.

Therefore we obtain the equation, y = $\frac{k}{x}$, where x is $\neq$ 0.

For the given equation x y = k, we can find the values of y, for the given values of x and viceversa.

Solved Example

Question: If x and y are inversely proportional, find such that xy = 50, find the value of y, when x = 10. Also find the value of x, when y = 20.
Solution:

We have x y = 50
Substituting x = 10, we get,

                      10 . y = 50

           => y = $\frac{50}{10}$

                                = 5
Therefore, when x = 10, y = 5, such that 10 x 5 = 50

Substiuting y = 20 in the equation xy = 50, we get,

                            x . 20 = 50

                     =>         x = $\frac{50}{20}$ = 2.5

Therefore, when y = 20, x = 2.5, such that 20 x 2.5 = 50

We observed that the inversely proportional equation is, x y = k, where k is a real number called the proportionality constant.

Let us draw the graph of the equation for the given value of k, say k = 20
Therefore we have the equation x y = 20.

Let us prepare the table of values satisfying the above condition.

x	1	2	4	20
y	20	10	5	1
co-ordinates	(1, 20)	(2, 10)	(4, 5)	(20, 1)

By plotting these points, we get the following graph.

From the above graph, we observe that as x increases, y decreases and the curve does not pass through the origin.

Solved Examples

Question 1: Rosy cycles to her school at an average speed of 12 miles per hour. It takes 25 minutes for her to reach the school. If she wants to reach her school in 15 minutes, what will be here average speed.
Solution:

From the above data we observe that Rosy takes 25 minutes if she travels at a speed of 12 miles per hour.
Let us convert minutes to hour by dividing it by 60

                                25 minutes = $\frac{25}{60}$

                                                 = $\frac{5}{12}$ hour

                                15 minutes = $\frac{15}{60}$

                                                 = $\frac{1}{4}$
Let us prepare the table of data.

Average Speed ( x )	12 mph	x mph
Time Taken ( y )	$\frac{5}{12}$ hr	$\frac{1}{4}$ hr

According to the definition of inverse proportionality, x y = k

                                     therefore we have 12 $\times$ $\frac{5}{12}$ = k

                                                          =>       $\frac{60}{12}$ = k

                                                          =>            5 = k

               The constant of proportionality, k = 5

Assuming the average speed to reach the school in 15 minutes to be x, we get,

              x $\times$ $\frac{1}{4}$ = 5

                                                                       => x = 5 $\times$ 4

                                                                           = 20 miles per hour

Therefore, Rosy has to travel at an average speed of 20 miles per hour to reach the school in 15 minutes.

Question 2: 28 pumps can empty a reservoir in 18 hours. In how many hours 48 such pumps can empty the reservoir?
Solution:

Let us fill in the data in the table.

Number of Pumps ( x )	28	48
Time Taken ( y hours )	18	y

since the number of pumps and the time taken are inversely proportional ,let us use the equation xy = k, where k is the proportionality constant.
From the above table, we have x. y = k
                               =>      28 x 18 = k
                               =>            504 = k
Assuming the time taken to empty the reservoir by 48 pumps to be, y, we get
   48 . y = 504

                                        =>     y = $\frac{504}{48}$

                                                    = 10.5 hours
Therefore, it will take 10 hours and 50 minutes for the 48 pumps to empty the reservoir.