Directly Proportional
and Inversely Proportional

proportional dogs

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

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.

inverse square law

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.

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Directly Proportional and Inversely Proportional