YOU ARE LEARNING:
Values that change relative to each other in a regular way can be described as proportional. Algebraic values can also be proportional, and we can use the constant of proportionality to find unknown values.
If two quantities are directly proportional, the rate at which one increases is proportional to the other. This means that the ratio between the two quantities is constant.
If two quantities are proportional, this means that...
If y is proportional to x:
where k will be a number. k is the constant of proportionality, and represents the ratio between the two proportional values.
For example, if y is proportional to x and y=10 when x=5, what is the value of k?
Rearrange the equation to make k the subject
Since y=kx, we can divide both sides by x to find that k=xy
Put the values we know into the equation
510=2. Therefore, we know that k=2.
Form an equation
We can form an equation with our value of k=2: y=2x. We can use this to find other values of y.
If y=2x, what is y when x=20?
When x=20, y=40
We can find this by putting x=20 into our equation: y=2×20.
y=4 when x=8. Find k
Ify=12 , what is x?
Now we know how to find k, we can use it to find the value of an unknown x or y in algebraic problems.
y is proportional to x3. When x=2, y=40. What is y when x=3?
The two quantities are proportional
Therefore, we can express them as y=kx3.The first step is to find the value of k, so we can apply this to find y when x=3.
To find k, put the values in the equation
We are given the values x=2 and y=40. Therefore, using our equation, 40=k(23), so 40=k(8).
Rearrange to make k the subject
To isolate k, we need to divide both sides by 8. 40=k(8), so k=840. Therefore, k=5.
Use k to find y when x=3
Now we know k, we can substitute this into our equation when x=3. Therefore, the equation is y=5×33, so y=135.