# Inverse Proportion

🏆Practice reason

Inverse proportionality describes a situation in which, when one term is multiplied by a certain number of times, the second term is decreased by the same number of times. This also occurs in reverse, if one term decreases by a certain number of times, the second term increases by the same number of times.

#### Let's see an example to illustrate this concept.

Given the following table:

We see two values, $X$ and $Y$. It can be very clearly seen that, when the value of $X$ increases by $2$, the value of $Y$ also decreases $2$ times. Therefore, it can be said that there is inverse proportionality here.

## Test yourself on reason!

There are 18 balls in a box, $$\frac{2}{3}$$ of which are white.

How many white balls are there in the box?

## Let's look at an example from everyday life

Imagine traveling in some vehicle while the roads are quite empty, without any traffic jams.

As you travel more kilometers and more time passes, the amount of gasoline decreases.

We can say that, as the distance increases the amount of gasoline decreases.

Let's see a graphical representation of inverse proportionality:

The function: $Y=\frac{a}{X}$

represents inverse proportionality.

As the $X$ increases the $Y$ decreases.

## How can we check if there is inverse proportionality?

To find out if there is inverse proportionality, we will examine if, when one term is multiplied by a certain amount of times, the second term is decreased by the same amount of times.

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### Let's look at an example

Given the following table:

Let's check if every time $X$ increases by a specific amount, $Y$ also decreases in the same proportion.

If this occurs, it means there is inverse proportionality. If not, then there isn't.

By how much does $X$ increase from $5$ to $10$?

The answer is it multiplies by $2$.

And by how much does $Y$ decrease from $20$ to $10$?

The answer is it divides by $2$.

Let's continue,

By how much does $X$ increase from $5$ to $20$? The answer is it multiplies by $4$.

And by how much does $Y$ decrease from $20$ to $5$?

The answer is it divides by $4$.

We will continue examining and discover that indeed every time $X$ multiplies by a certain number, $Y$ also decreases divided by the same number.

We will see it in the following way:

## Examples and exercises with solutions on inverse proportionality

### Exercise #1

There are 18 balls in a box, $\frac{2}{3}$ of which are white.

How many white balls are there in the box?

12

### Exercise #2

In a box there are 28 balls, $\frac{1}{4}$ of which are orange.

How many orange balls are there in the box?

7

### Exercise #3

There are two circles.

The length of the diameter of circle 1 is 4 cm.

The length of the diameter of circle 2 is 10 cm.

How many times larger is the area of circle 2 than the area of circle 1?

### Video Solution

$6\frac{1}{4}$

### Exercise #4

There are two circles.

The length of the radius of circle 1 is 6 cm.

The length of the diameter of circle 2 is 12 cm.

How many times greater is the area of circle 2 than the area of circle 1?

They are equal.

### Exercise #5

There are two circles.

One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.

How many times greater is the area of the second circle than the area of the first circle?

### Video Solution

$6\frac{1}{4}$