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3.1 F) Inverse Proportion
3.1 F) Inverse Proportion
Variables that are inversely proportional to each other move in opposite directions. If one variable increases, the other variable decreases. If one variable decreases, the other variable increases.
An example would be the number of road workers and the time taken to resurface a road. The more road workers there are, the less time it takes to resurface the road. The fewer road workers there are, the more time that it takes to resurface the road.
Another example would be the average speed of a runner and the time taken to finish a race. The higher the average speed of the runner, the less time it takes to finish the race. The lower the average speed of the runner, the greater the time it takes to finish the race.
If y was inversely proportional to x, the notation would be:
An example would be the number of road workers and the time taken to resurface a road. The more road workers there are, the less time it takes to resurface the road. The fewer road workers there are, the more time that it takes to resurface the road.
Another example would be the average speed of a runner and the time taken to finish a race. The higher the average speed of the runner, the less time it takes to finish the race. The lower the average speed of the runner, the greater the time it takes to finish the race.
If y was inversely proportional to x, the notation would be:
With the general equation:
The value of the k in the general equation is a number.
Let’s have a few examples.
Let’s have a few examples.
Example 1
y is inversely proportional to x. When the value of y is 4, the value of x is 9.
a) Form an equation for y in terms of x.
b) Find the value of y when x is 12.
c) Find the value of x when y is 18.
Part a
Similar to direct proportion questions, the first step in answering inverse proportion questions is to form an equation for the variables involved. This is the case even if we are not asked to form an equation for the variables.
The variables in the above question are x and y. This means that general equation will be:
y is inversely proportional to x. When the value of y is 4, the value of x is 9.
a) Form an equation for y in terms of x.
b) Find the value of y when x is 12.
c) Find the value of x when y is 18.
Part a
Similar to direct proportion questions, the first step in answering inverse proportion questions is to form an equation for the variables involved. This is the case even if we are not asked to form an equation for the variables.
The variables in the above question are x and y. This means that general equation will be:
When we are forming an equation for the variables, we are essentially finding the value of k. We find the value of k by subbing in the pair of values for the variables that we are given in the question. We were told in the question, that when y is 4, x is 9. We therefore sub y in as 4 and x in as 9 into the general equation.
We want to find the value of k. At the moment, k is being divided by 9. We can get rid of the divided by 9 by doing the opposite; the opposite of dividing by 9 is multiplying by 9. Therefore, we multiply both sides of the equation by 9.
This tells us that k is 36. We sub k as 36 into the general equation. The general equation becomes:
We now have the general equation for y in terms of x. This general equation is useful because we can sub the value of one of the unknowns in to find the value of the other unknown; we can sub in a value for x to find y, or sub in a value for y to find x. We will be doing this in part b and part c.
Part b
Part b asks us to find the value of y when x is 12. We are able to do this by subbing in x as 12 into the general equation with a known value for k.
Part b
Part b asks us to find the value of y when x is 12. We are able to do this by subbing in x as 12 into the general equation with a known value for k.
Therefore, when x is 12, y is 3.
Part c
Part c asks us to find the value of x when y is 18. We are able to do this by subbing in y as 18 into the general equation with a known value of k.
Part c
Part c asks us to find the value of x when y is 18. We are able to do this by subbing in y as 18 into the general equation with a known value of k.
We want to find the value of x and currently x is on the denominator of a fraction. We are able to bring the x up from the denominator of the fraction by multiplying both sides of the equation by x (the denominator of the fraction). For the right side of the equation, the divide by x (which comes from the denominator of the fraction) and the multiply by x (from multiplying both sides by x) cancel each other out, which just leaves the numerator of the fraction; the right sides of the equation becomes 36.
We want to find the value of x and not 18x. Therefore, we divide both sides of the equation by 18.
Therefore, when y is 18, x is 2.
Example 2
Sometimes we will be given an inverse proportion questions that use variables that are not x and y. The method used to answer these types of questions is pretty much exactly the same as before; the only difference is that we change x and y in the notation and general equation for the variables that we are given in the question.
w is inversely proportional to z. When w is 48, z is 9.
a) Find the value of w when z is 12.
b) Find the value of z when w is 144.
The first step in answering any inverse proportion question is to find the general equation. We always need to do this, even if the question does not ask us to. The general equation for y being inversely proportional to x is:
Sometimes we will be given an inverse proportion questions that use variables that are not x and y. The method used to answer these types of questions is pretty much exactly the same as before; the only difference is that we change x and y in the notation and general equation for the variables that we are given in the question.
w is inversely proportional to z. When w is 48, z is 9.
a) Find the value of w when z is 12.
b) Find the value of z when w is 144.
The first step in answering any inverse proportion question is to find the general equation. We always need to do this, even if the question does not ask us to. The general equation for y being inversely proportional to x is:
However, in this question, we are using different variables to x and y. We are told that w is inversely proportional to z. This means that the general equation with the variables w and z instead of x and y is:
In order for this equation to be useful, we need to find the value of k. We can find the value of k by subbing in the values for w and z that we are given in the question. The question told us that when w is 48, z is 9. Therefore, we sub w in as 48 and z in as 9 into the general equation.
We want to find the value of k and not k divided by 9. Therefore, we need to multiply both sides of the equation by 9.
This tells us that k is 432. We now replace the k in the general equation with 432. The general equation becomes:
We now have the general equation, which means that we can sub in any value for w or z to find the value of the other unknown.
Part a
Part a asked us to find the value of w when z is 12. We can do this by subbing in x as 12 into the general equation with a known value of k.
Part a
Part a asked us to find the value of w when z is 12. We can do this by subbing in x as 12 into the general equation with a known value of k.
This tells us that when z is 12, w is 36.
Part b
Part b asks us to find the value of z when w is 144. We can do this by subbing w in as 144 into the general equation with a known value of k.
Part b
Part b asks us to find the value of z when w is 144. We can do this by subbing w in as 144 into the general equation with a known value of k.
We are looking for z and currently z is on the denominator of the fraction. We can get rid of the fraction by multiplying both sides of the equation by the denominator of the fraction. The denominator of the fraction is z, so we multiply both sides of the equation by z.
We want to find the value of z and not 144z. Therefore, we divide both sides of the equation by 144.
This tells us that when w is 144, z is 3.
End Note
Inverse proportion questions are not too tricky providing that you are carful with your working and that you remember the notation and the general equation. The notation is:
Inverse proportion questions are not too tricky providing that you are carful with your working and that you remember the notation and the general equation. The notation is:
With the general equation:
We can find the value of k by subbing in the pair of values that we are given in the question. After we have found the value of k, we replace k in the general equation with the value that we have found. We are then able to sub any value in for x or y to obtain the value of the other unknown.
Inverse proportion questions aren’t too tricky providing that you remember the notation and general equation. Therefore, it is worth getting the notation, the general equation and (maybe) a worked example down on a revision card.
Inverse proportion questions aren’t too tricky providing that you remember the notation and general equation. Therefore, it is worth getting the notation, the general equation and (maybe) a worked example down on a revision card.