Back to Edexcel Direct & Inverse Proportion (H) Home
3.1 E) Direct Proportion – Part 3
3.1 E) Direct Proportion – Part 3
In this section, we are going to look at some more typical direct proportion questions. From the previous section, we learnt that the notation for direct proportion is:
With the general equation:
Let’s now have a few examples.
Example 1
y is directly proportional to x. When the value of y is 28, the value of x is 7.
a) Form an equation for y in terms of x.
b) Find the value of y when x is 11.
c) Find the value of x when y is 64.
Part a
We are told in the question that the variables x and y are in direct proportion. This means that they will have the notation as:
y is directly proportional to x. When the value of y is 28, the value of x is 7.
a) Form an equation for y in terms of x.
b) Find the value of y when x is 11.
c) Find the value of x when y is 64.
Part a
We are told in the question that the variables x and y are in direct proportion. This means that they will have the notation as:
With the general equation:
Part a asks us to form an equation for y in terms of x. Essentially, this is asking us to find the value of k in the above general equation. We are able to do this by subbing in the pair of values for x and y that we are given in the question into the general equation. The question told us that when y is 28, the value of x is 7. Therefore, we sub in y as 28 and x as 7 into the general equation for direct proportion.
We want to find the value of k and not 7k. Therefore, we divide both sides of the equation by 7.
This tells us that k is 4.
We can now replace the value of k in the general equation for direct proportion with 4. This gives us the general equation:
We can now replace the value of k in the general equation for direct proportion with 4. This gives us the general equation:
We now have an equation for y in terms of x, thus answering part a.
Part b
Part b asks us to find the value of y when x is 11. We find the value of y when x is 11 by subbing in x as 11 into the general equation with the value of k known. The working is shown below.
Part b
Part b asks us to find the value of y when x is 11. We find the value of y when x is 11 by subbing in x as 11 into the general equation with the value of k known. The working is shown below.
This tells us that when x is 11, y is 44.
Part c
Part c asks us to find the value of x when y is 64. We find the value of x when y is 64 by subbing in y as 64 into the general equation with the value of k known. The working is shown below:
Part c
Part c asks us to find the value of x when y is 64. We find the value of x when y is 64 by subbing in y as 64 into the general equation with the value of k known. The working is shown below:
We want to find the value of x and not 4x. Therefore, we divide both sides of the equation by 4.
This tells us that when y is 64, x is 16.
A bit of Advice
The first part of this question asks us to form an equation for y in terms of x. Sometimes it will be the case that we are not asked to do this in the question. However, we must always form an equation for y in terms of x to be able to answer a direct proportion question. Therefore, even if you are not asked to form an equation for y in terms of x, the first step in answering direct proportion questions will be to form an equation for y in terms of x.
Also, it may be the case that the variables that you are given in the question are not x and y. This does not matter as we just change the variables that are involved in the notation and the general equation. This will be the case in the next example.
The first part of this question asks us to form an equation for y in terms of x. Sometimes it will be the case that we are not asked to do this in the question. However, we must always form an equation for y in terms of x to be able to answer a direct proportion question. Therefore, even if you are not asked to form an equation for y in terms of x, the first step in answering direct proportion questions will be to form an equation for y in terms of x.
Also, it may be the case that the variables that you are given in the question are not x and y. This does not matter as we just change the variables that are involved in the notation and the general equation. This will be the case in the next example.
Example 2
a is directly proportional to b. When a is 45, b is 9.
a) Find the value of a when b is 16.
b) Find the value of b when a is 120.
The question tells us that a is directly proportional to b. This means that the notation for this will be:
a is directly proportional to b. When a is 45, b is 9.
a) Find the value of a when b is 16.
b) Find the value of b when a is 120.
The question tells us that a is directly proportional to b. This means that the notation for this will be:
With the general equation:
The first step in answering any direct proportion question is to write an equation for the variables; we need to find the value of k in the general equation (a = kb). We can find the value of k by subbing in the pair of values that we are given in the question. We are told in the question that when a is 45, b is 9. Therefore, we can find the value of k by subbing in a as 45 and b as 9.
We want to find the value of k and not 9k. Therefore, we divide both sides of the equation by 9.
This tells us that k is 5.
We now replace the value of k in the general equation for direct proportion with 5. This gives us the general equation:
We now replace the value of k in the general equation for direct proportion with 5. This gives us the general equation:
We are now in a position to answer parts a and b.
Part a
Part a asks us to find the value of a when b is 16. We can do this by subbing in b as 16 into the general equation with the value of k known.
Part a
Part a asks us to find the value of a when b is 16. We can do this by subbing in b as 16 into the general equation with the value of k known.
This tells us that when b is 16, a is 80.
Part b
Part b asks us to find the value of b when a is 120. We can do this by subbing in a as 120 into the general equation with the value of k known.
Part b
Part b asks us to find the value of b when a is 120. We can do this by subbing in a as 120 into the general equation with the value of k known.
We want to find the value of b and not 5b. Therefore, we need to divide both sides of the equation by 5.
This tells us that when a is 120, b is 24.
Final Note
I am just going to recap the key steps involved in answering direct proportion questions. The first step in answering direct proportion questions is to place the values that you are given into the generic equation (y = kx) to find the value of k. When we have found the value of k, we then have our generic equation. We can use this equation to find the value of either x or y if we are given the value for the other variable (if we are given the value of y, we can find the value of x by subbing y into the general equation with a known value of k. If we are given the value of x, we can find the value of y by subbing x into the equation with a known value of k).
Also, if we are given variables that are in direct proportion that are not x and y, we just modify the notation and the general equation for the variables that we are given (like we did with example 2).
Finally, the value of k doesn’t always have to be an integer.
I am just going to recap the key steps involved in answering direct proportion questions. The first step in answering direct proportion questions is to place the values that you are given into the generic equation (y = kx) to find the value of k. When we have found the value of k, we then have our generic equation. We can use this equation to find the value of either x or y if we are given the value for the other variable (if we are given the value of y, we can find the value of x by subbing y into the general equation with a known value of k. If we are given the value of x, we can find the value of y by subbing x into the equation with a known value of k).
Also, if we are given variables that are in direct proportion that are not x and y, we just modify the notation and the general equation for the variables that we are given (like we did with example 2).
Finally, the value of k doesn’t always have to be an integer.