# Direct and Inverse Proportion

## Direct Proportion

If two variables are in direct proportion it means they both increase at the same rate.

**Example**:

If the amount of money I get paid is directly proportional to the hours I work then both values will increase at the same rate.

Therefore if I get paid £10 per hour, then if I work 4 hours I will get £40.

**Writing an equation for two variables in direct proportion**

· Variables in direct proportion will be linked by a constant, also known as k.

· We use ∝ to signify that two values are in direct proportion to each other

If we are given two variables we can find k and then create an equation to find further variables.

**Example and basic steps:**

A and B are directly proportional when A=8 B=2

Step 1: A ∝ B

Step 2: A = kB 8 = k x 2

Step 3: (8÷2 = k) k = 4

Step 4: A = 4

**Example**

I know that my pay is directly proportional to the hours I work. I get £20 when I work for 2 hours.

Step 1: Pay ∝ Hours

Step 2: P = kH 20= k x 2

Step 3: (20÷2 = k) k = 10

Step 4: P= 10H

This graph shows direct proportion. It goes through the origin (0,0) and both variables increase at the same rate.

The graph is P=10.

**Other types of direct proportion.**

Be aware that you may have to square or root aspects of your equations.

When t is directly proportional to the square of u then t∝ u^{2}this means t=ku^{2}

When t is directly proportional to the cube of u then t∝ u^{3 }this means t=ku^{3}

When t is directly proportional to the square root then t∝ u this means t=ku

You always multiply by the constant.

**Example:**

If E is directly proportional to the square of L and E is 16 when L is 2, find E when L is 4.

Firstly create an equation like before;

E∝ L^{2} so…

E=kL^{2}

16= k(2)^{2}

16= 4k

4=k

Therefore the equation will be E=4L^{2}

So if L =4 E= 4(4)^{2}

```
E=64
```

## Inverse Proportion

If two variables are inversely proportional it means that as one increases the other decreases at the same rate.

**Example**

The time it takes to paint a fence is inversely proportional to the number of people there to paint it. The more people there are, the quicker we will get the job done!

For example, if it takes 1 person 6 days to paint the fence it’ll take 3 people 2 days.

**Writing equations for inverse proportion**

When two variables are inversely proportional they are linked by a 1/x relationship.

We can write this as y∝1/x or y= k/x

**Example**

Here’s an example of how we can create an equation about people painting fences.

We know that 1 person will take 6 days.

Step 1: People ∝ Days

Step 2: 1 = k/6 Rearrange to find k

Step 3: (6 x1 = k) k = 6 Now we have k put it into the format y=k/x

Step 4: P= 6/Days

Now that we have the formula we can draw the graph and show it graphically.

This is the graph, as you can see it is a reciprocal graph. If you think about it even if you have an infinite amount of people painting the fence it will take some time, even if it’s a nano-second!

**More complicated inverse proportion.**

As with direct proportion, sometimes things that are inversely proportional may be squared, cubed or rooted.

In these cases remember:

- When t is inversely proportional to the square of u then t∝ 1/u
^{2}this means t=k/u^{2} - When t is inversely proportional to the cube of u then t∝ 1/u
^{3}this means t=k/u^{3} - When t is inversely proportional to the square root then t∝ 1/u
^{1/2}this means t=k/u^{1/2}

**Example**

W is inversely proportional to the square root of T. When W = 16 T =4, what is W when t= 20

W∝1/T

16=k/4

64=k

Therefore our formula is W=64/T

W=64/20

W= 64/25

= 32/5

- R is inversely proportional to the cube of E . When R=10 E=4 what is E when R is 50? Leave your answer to 2 decimal places.
- 2.34