Inversely Proportional

Before learning what is inversely proportional, wouldn’t you want to know what a proportion means? A proportion is a statement that shows how two quantities or variables are related to each other. These variables can be of two types, directly proportional and inversely proportional. When one quantity increases along with the other then it is called directly proportion whereas if one increases and the other decreases then it is inversely proportional. For example, if there are more workers on a job then it won't take much time to complete a work. So what is actually happening here? As the number of workers increases, the time taken to complete work decreases. Now, this is what we call is inversely proportional. This is how inversely proportional meaning can be represented. We will say that the workers and the time are inversely proportional to each other. Inversely proportional is basically a relationship between two variables when their product equals to a constant value. There is a symbol to represent inversely proportional. Therefore, an inversely proportional symbol can be represented as ∝. The inversely proportional symbol is very much like the symbol of infinity. 

Inversely Proportional Example

1. The example of speed and time is a basic and appropriate example of inversely proportional. As the speed increases, the time to complete your trip will decrease. 

2. Another inversely proportional example could be the volume and the pressure of an ideal gas. They both are proportional to each other (Boyle’s law). 

3. Electric current is inversely proportional to resistance.

4. More the distance between you and the source of light, lesser would be the brightness. 

Methods To Solve Inverse Proportional

There are two ways to solve a problem having inversely proportional variables. 

Method 1) In an inverse proportion, x1 y1 = x2 y2 = x2 y2 = x2 y2

Therefore, to solve this problem we can use the equation to find the unknown terms as one pair would always be given. 

Method 2) We also know that in an inverse proportion, equation x + y = k becomes x = k/y. So in order to find the value of k, we can use the known values to find the unknown ones with the help of the above. 

You can also use an inversely proportional formula to solve the sums. 

Setting up an Equation

You can set up an equation using these four simple steps: 

Step 1) First write down the proportional relationship.

Step 2) The second step is to convert to an equation by using a constant of proportionality.

Step 3) In the third step, use the given information to find the constant of proportionality.

Step 4) The last and final step is to substitute the constant of proportionality into an equation. 

Solved Examples of Inverse Proportion

Question 1) The time taken(t seconds), for the water heater to boil the water is inversely proportional to the power(p watt), consumed by the water heater. If p = 2000W, t = 252 s. Find the time taken to boil the water when p = 800W.

Solution 1) t ∝\[\frac{1}{p}\] 

              t = \[\frac{k}{p}\]

252 = \[\frac{k}{2000}\]

k = 504000

t = \[\frac{504000}{p}\]

t = \[\frac{504000}{800}\]

t = 630 seconds.

Question 2) The force(F newtons) exerted by a magnet on a metal object tends to be inversely proportional to the square of the distance(d cm). If d = 2 cm and F = 50 N, then express:

1. F in terms of d.

2. Find the force when the distance between the metal object and the magnet is 10cm

3. Find the distance between the two when the force is 8N.

4. Also, explain what happens to F when d is halved.

Solution 2) Expressing F in terms of d: 

F  ∝ \[\frac{1}{d^{2}}\]

F = \[\frac{k}{d^{2}}\]

50 = \[\frac{k}{2^{2}}\]

50 = \[\frac{k}{4}\]

k = 200

1. Therefore, \[\frac{200}{d^{2}}\]

Finding force:

F = \[\frac{200}{10^{2}}\]

\[\frac{200}{100}\] = 2

2. Therefore, F = 2 N

Finding distance:

8 = \[\frac{200}{d^{2}}\]

\[8d^{2}\]=200

\[d^{2}\]=25

3. Therefore, d = 5

F when d is halved:

d = 10     F = 2

d = 5       F = 8

4. Therefore, F becomes four times larger when d is halved. 

Question 3) A and B are positive numbers and are inversely proportional to each other. If A = 4 and B = 36, find the value of A when B = A.

Solution 3) A  ∝ \[\frac{1}{b}\]

A =  \[\frac{k}{b}\]

4 = \[\frac{k}{36}\]

k = 144

A = \[\frac{144}{B}\]

\[A^{2}\] = 144

Therefore, A = 12

Question 4) The time (t) taken for the passengers to check-in for a flight is inversely proportional to the square of the number of staff (s) working. If it takes 30 minutes for the passengers to check-in when 10 staffs are working, the find,

1. An equation connecting t and s

2. The minimum number of staff must be working if the time taken is to be 60 minutes. 

Solution 4) Connecting t and s

t ∝ \[\frac{1}{s^{2}}\]

t = \[\frac{k}{s^{2}}\]

30 = \[\frac{k}{10^{2}}\]

30 = \[\frac{k}{100}\]

k = 3000

a) Therefore, t = \[\frac{3000}{s^{2}}\]

b) 60 = \[\frac{3000}{s^{2}}\]

\[60s^{2}\] = 3000

\[s^{2}\] = 50

s = 7.071

Therefore, 8 staff is needed. 

Question 5) The number of days (d) to complete research is inversely proportional to the number of researchers who are working. If the researchers take 125 days to complete the work of 16 people, then how many people are needed to complete the research in 40 days? 

Solution 5) D ∝ \[\frac{1}{R}\]

D = \[\frac{F}{R}\]

125 = \[\frac{F}{16}\]

2000 = F

D = \[\frac{2000}{R}\]

40 = \[\frac{2000}{R}\]

40R = 2000

R = 50

Therefore, 50 people will be needed to complete the research in 40 days.