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 x^{2}

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 = kt^{2}

Where:

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

When d = 19.6 then t = 2

19.6 = k × 2^{2}

19.6 = 4k

k = 4.9

So now we know:

d = 4.9t^{2}

And when t = 3:

d = 4.9 × 3^{2}

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 2meters (double the distance leads to a quarter of the brightness), and so on.

8943, 8945, 8947, 8948, 8951, 8952, 7005, 8064, 8065, 8067, 8072

Proportions Ratios Algebra Index