This revision bite gives you the information you need to be able to solve problems involving proportion, and to show proportional relationships on graphs.
Direct proportion
If two quantities are in direct proportion, as one increases, the other increases by the same percentage.
If y is directly proportional to x, this can be written as y ∝ x
A simple example of two things that are in the same proportion is the amount of apples you might buy and the amount you pay for them. If you buy twice as many apples as your friend, you pay twice as much.
We can write the connection between the cost and the amount as an equation:
- Cost of apples = price per apple × number of apples bought.
This can also be written as y = kx, where k is the cost (the price per apple).
This means that, for some constant k, y = kx for all values of x and k is called the constant of proportionality.
Example
- If y is directly proportional to x.
- When x = 12 then y = 3
- Find the constant of proportionality and the value of x when y = 8.
We know that y is proportional to x so y = kx
We also know that when x = 12 then y = 3
To find the value of k substitute the values y = 3 and x = 12 into y = kx
3 = k × 12
So k = 3/12 = 1/4
To find the value of x , when y = 8 substitute y = 8 and k= 3 into y = kx
8 = (1/4) x
So x = 32 when y = 8
Direct proportion to powers
y can be directly proportional to x2 , x3 and other powers of x.
They can always form an equation with k, a constant multiplier (the constant of proportionality), at the start.
eg y = kx2
- Question
y ∝ x 3
If y = 1 when x = 2, find the value of y when x = 4
- Answer
- y ∝ x 3
- So y = kx3
Substitute the value y = 1 and x = 2 into y = kx3 to find the value of k.
- 1 = k × 23
- So k = 1/8
Now use the values k = 1/8 and x = 4. y = 1/8 × 64
Gives the answer y = 8
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