# Directly Proportional and Inversely Proportional Directly proportional: as one amount increases, another amount increases at the same rate.

 ∝ The symbol for "directly proportional" is ∝ (Don't confuse it with the symbol for infinity ∞)

### Example: you are paid \$20 an hour

How much you earn is directly proportional to how many hours you work

Work more hours, get more pay; in direct proportion.

This could be written:

Earnings Hours worked

• If you work 2 hours you get paid \$40
• If you work 3 hours you get paid \$60
• etc ...

## Constant of Proportionality

The "constant of proportionality" is the value that relates the two amounts

### Example: you are paid \$20 an hour (continued)

The constant of proportionality is 20 because:

Earnings = 20 × Hours worked

This can be written:

y = kx

Where k is the constant of proportionality

### Example: y is directly proportional to x, and when x=3 then y=15. What is the constant of proportionality?

They are directly proportional, so:

y = kx

Put in what we know (y=15 and x=3):

15 = k × 3

Solve (by dividing both sides by 3):

15/3 = k × 3/3
5 = k × 1
k = 5

The constant of proportionality is 5:

y = 5x

When we know the constant of proportionality we can then answer other questions

### Example: (continued)

What is the value of y when x = 9?

y = 5 × 9 = 45

What is the value of x when y = 2?

2 = 5x
x = 2/5 = 0.4

## Inversely Proportional

 Inversely Proportional: when one value decreases at the same rate that the other increases.

### Example: speed and travel time

Speed and travel time are Inversely Proportional because the faster we go the shorter the time.

• As speed goes up, travel time goes down
• And as speed goes down, travel time goes up
This:y is inversely proportional to x
Is the same thing as:y is directly proportional to 1/x
Which can be written:

y = kx ### Example: 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? (Assume everyone works at the same rate)

It is an Inverse Proportion:

• As the number of people goes up, the painting time goes down.
• As the number of people goes down, the painting time goes up.

We can use:

t = k/n

Where:

• t = number of hours
• k = constant of proportionality
• n = number of people

"4 people can paint a fence in 3 hours" means that t = 3 when n = 4

3 = k/4
3 × 4 = k × 4 / 4
12 = k
k = 12

So now we know:

t = 12/n

And when n = 6:

t = 12/6 = 2 hours

So 6 people will take 2 hours to paint the fence.

### How many people are needed to complete the job in half an hour?

½ = 12/n
n = 12 / ½ = 24

So it needs 24 people to complete the job in half an hour.
(Assuming they don't all get in each other's way!)

## Proportional to ...

It is also possible to be proportional to a square, a cube, an exponential, or other function!

### Example: Proportional to x2 A stone is dropped from the top of a high tower.

The distance it falls is proportional to the square of the time of fall.

The stone falls 19.6 m after 2 seconds, how far does it fall after 3 seconds?

We can use:

d = kt2

Where:

• d is the distance fallen and
• t is the time of fall

When d = 19.6 then t = 2

19.6 = k × 22
19.6 = 4k
k = 4.9

So now we know:

d = 4.9t2

And when t = 3:

d = 4.9 × 32
d = 44.1

So it has fallen 44.1 m after 3 seconds.

## Inverse Square Inverse Square: when one value decreases as the square of the other value.

### Example: light and distance

The further away we are from a light, the less bright it is. In fact the brightness decreases as the square of the distance. Because the light is spreading out in all directions.

So a brightness of "1" at 1 meter is only "0.25" at 2 meters (double the distance leads to a quarter of the brightness), and so on.