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What Does Direct Proportion Mean?
Two quantities are directly proportional to each other if they are always in the same ratio.
For example, if x and y are directly proportional, then x doubles as y doubles, x halves as y halves and so on.
A real world example of directly proportional quantities would be the ingredients for some cupcakes. If you need 2 eggs to make 12 cakes, then you would need 3 eggs to make 18 cakes, 4 eggs to make 24 cakes and so on. The ratio of eggs to cakes is always 2:12 (or 1:6 if you want to simplify it) hence the amounts of eggs and cakes are directly proportional to each other.
Notation for Proportionality
If two quantities, let's call them x and y, are directly proportional we write this as follows:
y α x
Solving Proportionality Questions in Tables
Let's look at a typical table-based proportionality question.
x and y are Directly Proportional, Find the Missing Values
As x and y are directly proportional to each other, then as we multiply one of them by a number, we must also multiply the other by the same number.
In the example above, we can see that x goes from 5 to 20 in the second and third columns. This is a multiplication by 4, hence we also multiply the y value of 10 by 4 to get 40.
Another way we can solve these problems is by looking at how the x and y in each column compare to each other. As they are proportional, the ratio between them must remain constant. The ratio x:y in the second column is 5:10 = 1:2. This tells us that the y value must always be double the x value. Using this information we can see that our remaining blank in the top right corner must be 50 ÷ 2 = 25.
We can double check our calculation here by seeing that both values in column 4 are 5 times the values in column 2.
Solving Worded Proportionality Questions
y is directly proportional to x.
When y = 30, x = 5.
What is the value of y when x = 35?
To solve this we remember that direct proportion mean that y and x are in a constant ratio. We can therefore rewrite y α x as y = kx where k is a constant to be found, known as the constant of proportionality. By substituting the known values of 30 and 5 into this we get:
30 = k × 5
k = 6
This means that y and x are linked by the equation y = 6x. We can now substitute our value of x into this to find the answer.
y = 6 × 35 = 210.
Worded Example 2
y is directly proportional to x.
When y = 23, x = 184.
Find y when x = 16.
y α x
y = kx
23 = k × 184
k = 23 ÷ 184
Therefore y = 1/8 × x = x/8
When x = 16, y = 16/8 = 2
Find More Examples in This Quick Video Lesson
Worded Example 3: Proportion With Squared Values
The same rules apply when we look at direct proportion involving indices such as in the following question.
y is directly proportional to x2.
When y = 24, x = 2.
Find y when x = 5.
This time we have a situation where y doubles as x2 doubles, and so on, not as x doubles like in our previous examples. The solution, however, works in the same way by using k, our constant of proportionality.
First we set up the relationship using our proportionality symbol.
y α x2
As before, we now replace the proportionality symbol with = k × and substitute in our given pairing of y = 24 and x = 2 to work out k.
y = kx2
24 = k × 22 = 4k
k = 6
Therefore y = 6x2
We now have our equation showing the relationship between x and y, and so can substitute in x = 5 to find our answer.
When x = 5, y = 6 × 52 = 150.
Worded Example 4: Proportion With Square Roots
The exact same method works with square roots as well.
y is directly proportional to √x.
When y = 5, x = 9.
Find y when x = 25.
As always we start with our proportionality relationship and then convert into an equation with k.
y α √x
y = k√x
5 = k√9 = 3k
k = 5/3
Therefore y = 5√x / 3
When x = 25, y = 5√25 / 3 = 5 × 5 / 3 = 25/3
Direct Proportion in Physics
A cannonball is dropped from the top of a tall tower. The distance it falls is directly proportional to the square of the time taken to fall. If the cannonball drops 4.9 metres in 1 second, how far will it have fallen after 3 seconds.
If we let d = distance fallen and t = time taken, then
d α t2
d = kt2
4.9 = k × 12
k = 4.9
Therefore d = 4.9t2
After 3 seconds, d = 4.9 × 32 = 44.1 metres.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2022 David