Inverse Proportion

An inverse proportion is a relationship between two quantities in which one quantity increases as the other decreases. In other words, the two values change in opposite directions, with the product of their values remaining constant.

When two variables are inversely proportional, we can use the proportionality symbol to represent their relationship. If the variable x is inversely proportional to the variable y, we write x \propto \frac{1}{y}. To convert this relationship into an equation, we introduce a constant of proportionality, denoted by k. The resulting equation for an inverse proportion is xy = k.

To identify an inverse proportion from the information you are given, look for a relationship in which the product of the two variables remains constant as one variable increases and the other decreases. In other words, if the product of the two variables is always the same value, then they are inversely proportional.

Solving Inverse Proportion Problems

Finding the constant of proportionality

To find the constant of proportionality, use the provided information about the relationship between the two variables. Plug these values into the inverse proportion equation, xy = k, and solve for the constant, k.

For example, if x = 4 when y = 3, then the constant of proportionality, k, equals 4 \times 3 = 12.

Using the inverse proportion equation

Once you have determined the constant of proportionality, you can use the inverse proportion equation to address problems involving the two variables. Insert the value of k and the known variable value into the equation, and solve for the unknown variable.

For example, consider a situation where x = 6 and the constant of proportionality, k, is 12. To find the value of y, follow these steps:

xy = k

6y = 12

y = \frac{12}{6} y = 2

In this case, when x = 6, the corresponding value of y is 2.

Examples

Example 1:

Suppose the time taken to complete a task is inversely proportional to the number of people working on it. If 3 people can complete the task in 8 hours, how long will it take for 6 people to complete the same task?

Given that the time taken (t) is inversely proportional to the number of people (p) working on the task, we can write the relationship as:

pt = kpt = k

We are given that 3 people can complete the task in 8 hours, so we can find the constant k as follows:

p = 3, t = 8

3 \times 8 = k3 \times 8 = k

k = 24k = 24

Now we have the constant of proportionality, k = 24k = 24. Let’s find out how long it would take for 6 people to complete the task:

p = 6p = 6

6t = 246t = 24

Now, solve for t:

t = \frac{24}{6}t = \frac{24}{6}

t = 4t = 4

It would take 6 people 4 hours to complete the same task.

Example 2:

The brightness of a light source is inversely proportional to the square of the distance from the source. If the brightness is 100 units at a distance of 2 metres, what will be the brightness at a distance of 5 metres?

Write the inverse proportion equation: y = \frac{k}{x^2}y = \frac{k}{x^2}

Using the given information, x = 2x = 2 and y = 100y = 100:

100 = \frac{k}{2^2}100 = \frac{k}{2^2}

k = 100 \times 4 = 400k = 100 \times 4 = 400

Now, find the brightness at a distance of 5 metres, with x = 5x = 5:

y = \frac{400}{5^2} = \frac{400}{25} = 16y = \frac{400}{5^2} = \frac{400}{25} = 16

The brightness at a distance of 5 metres will be 16 units.

Example 3:

The speed of a car is inversely proportional to the time taken to travel a fixed distance. If a car takes 6 hours to travel the distance at a speed of 30 km/h, how long will it take to travel the same distance at a speed of 60 km/h?

Given that the speed (s) of a car is inversely proportional to the time taken (t) to travel a fixed distance, we can write the relationship as:

st = kst = k

We are given that the car takes 6 hours to travel the distance at a speed of 30 km/h, so we can find the constant k as follows:

s = 30 km/h, t = 6 hours

30 \times 6 = k30 \times 6 = k

k = 180k = 180

Now we have the constant of proportionality, k = 180. Let’s find out how long it will take to travel the same distance at a speed of 60 km/h:

s = 60 km/h

60 \times t = 18060 \times t = 180

Now, solve for t:

t = \frac{180}{60}t = \frac{180}{60}

t = 3t = 3

It would take 3 hours to travel the same distance at a speed of 60 km/h.

Example 4:

Suppose y is inversely proportional to \sqrt{x} and y=10, when x=25. Find the equation connecting x and y, then use it to find the value of y when x=100.

y \propto \dfrac{1}{\sqrt{x}}y \propto \dfrac{1}{\sqrt{x}}

To convert this into an equation, we introduce a constant of proportionality, k:

y = \dfrac{k}{\sqrt{x}}y = \dfrac{k}{\sqrt{x}}

We are given that y = 10y = 10 when x = 25x = 25. Let’s use this information to find the constant k:

10 = \dfrac{k}{\sqrt{25}}10 = \dfrac{k}{\sqrt{25}}

10 = \dfrac{k}{5}10 = \dfrac{k}{5}

k = 10 \times 5k = 10 \times 5

k = 50k = 50

Now we have the constant of proportionality, k = 50. The equation connecting x and y is:

y = \dfrac{50}{\sqrt{x}}y = \dfrac{50}{\sqrt{x}}

To find the value of y when x = 100, substitute x into the equation:

y = \dfrac{50}{\sqrt{100}}y = \dfrac{50}{\sqrt{100}}

y = \dfrac{50}{10}y = \dfrac{50}{10}

y = 5y = 5

So, when x = 100x = 100, y = 5y = 5.