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Inverse Proportion

Inverse Proportion

Inverse Proportion

Inversely proportion is where one value increases, and the other decreases, at the same rate.

If two quantities are inversely proportional, then the rate at which one increases is the same as the rate at which the other decreases. For example, we would expect that as the temperature increases, sales of winter coats decrease.

We call a relationship between two quantities inversely proportional if the rate at which they change is the same. For example, if the temperature doubles, to be inversely proportional, the sales of coats has to halve.

Likewise, if the sales of coats double, the temperature has to:

If we call the number of coats xx and the temperature yy, we can describe the relationship using algebra.

Algebraically, if yy is inversely proportional to xx:

y=kxy = \dfrac{k}{x}

where kk is the constant of proportionality, representing the ratio between the proportional values.

Let's let y=10y=10 and x=100x =100



What would be the value of kk?


As k=1000k=1000, what is the value of yy when we double xx to be 200200?

Graphically, when y is proportional to xx, it is a straight line relationship, but inverse proportion is a non-linear relationship.

If two quantities are inversely proportional, then...

We can also use these ideas to solve algebraic problems.

For example, if yy is inversely proportional to xx and y=1y = 1 when x=5x = 5, what is kk? y=kxy=\dfrac{k}{x}


Rearrange the equation to make kk the subject

Since y=kxy=\dfrac{k}{x}, we can multiply both sides by xx to find that k=xyk=xy.


Put the values we know into k=xyk=xy

5×1=55 \times 1 = 5. Therefore, we know that k=5k=5.


Form an equation

Use the value we just calculated in place of kk --> y=5xy = \dfrac{5}{x}


We can apply this to all values of xx and yy

yy is always equal to 5÷x5 \div x


If y=5xy=\dfrac{5}{x} what is xx when y=5y= 5?


If y=20y=20 when x=4x = 4, what is kk?

yy is inversely proportional tox3x^3. When x=3x = 3, y=10y = 10. What is yy when x=10x = 10?


We know they are inversely proportional

Therefore, we know that y=kx3y=\dfrac{k}{x^3}


Input the values we know into the equation

In the question, we can see that when x=3x=3, y=10y=10. We can put these values into the equation: 10=k3310=\dfrac{k}{3^3}, so 10=k2710=\dfrac{k}{27}.


To find kk, rearrange the equation

To find yy when x=10x=10, we need the constant, kk. To find this, we can rearrange the equation we derived by multiplying both sides by 2727. Therefore, since 10=k2710=\dfrac{k}{27}, k=10×27k=10 \times 27, so k=270k=270.


We can apply this to find yy when x=10x=10

Since they are inversely proportional, y=kxy=\dfrac{k}{x}. Now we know kk, we can fill in the equation as y=27010y=\dfrac{270}{10}, so y=27y=27


If y=16y=16 when x=10x = 10, what is kk?