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Direct Proportion

Direct Proportion

Direct Proportion

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 yy is proportional to xx:

y=kxy = kx

where kk will be a number. kk is the constant of proportionality, and represents the ratio between the two proportional values.

For example, if yy is proportional to xx and y=10y = 10 when x=5x = 5, what is the value of kk?



Rearrange the equation to make kk the subject

Since y=kxy=kx, we can divide both sides by xx to find that k=yxk=\dfrac{y}{x}


Put the values we know into the equation

105=2\dfrac{10}{5} = 2. Therefore, we know that k=2k=2.


Form an equation

We can form an equation with our value of k=2k=2: y=2xy = 2x. We can use this to find other values of yy.


If y=2xy=2x, what is yy when x=20x = 20?


When x=20x=20, y=40y=40

We can find this by putting x=20x=20 into our equation: y=2×20y=2\times 20.


y=4y = 4 when x=8x = 8. Find kk


Ify=12y = 12 , what is xx?

Now we know how to find kk, we can use it to find the value of an unknown xx or yy in algebraic problems.

yy is proportional to x3x^3. When x=2x = 2, y=40y = 40. What is yy when x=3x = 3?


The two quantities are proportional

Therefore, we can express them as y=kx3y=kx^3.The first step is to find the value of kk, so we can apply this to find yy when x=3x=3.


To find kk, put the values in the equation

We are given the values x=2x=2 and y=40y=40. Therefore, using our equation, 40=k(23)40=k(2^3), so 40=k(8)40=k(8).


Rearrange to make kk the subject

To isolate kk, we need to divide both sides by 8. 40=k(8)40=k(8), so k=408k=\dfrac{40}{8}. Therefore, k=5k=5.


Use kk to find yy when x=3x=3

Now we know kk, we can substitute this into our equation when x=3x=3. Therefore, the equation is y=5×33y=5 \times 3^3, so y=135y=135.