Inverse proportion - Intermediate and Higher tier
If two values are inversely proportional, this means that as one value increases the other decreases.
Speed and time can be inversely proportional; as the speed increases, time taken to complete the journey will decrease.
C is inversely proportional to D.
We can write this as \({C}∝\frac{1}{D}\)
This can be converted into a formula:
\({C}={k}\times\frac{1}{D}\)
where \(\text{k}\) is the constant of proportionality. This can also be written as:
\({C}=\frac{k}{D}\)
The number of plumbers is inversely proportional to the number of days of work needed. 32 plumbers can complete a job in 15 days.
This can be written as an equation:
15 days = \(\frac{k}{32}\) plumbers.
To find the value of \(\text{k}\) multiply both sides of the equation by 32:
15 × 32 = \(\text{k}\) = 480
We can rewrite the equation as \(days~=\frac{480}{plumbers}\) and use this to calculate one value given the other.
Example
If there are 20 plumbers, how many days will it take to complete the job?
Days = \(\frac{480}{plumbers}\)
Days = \(\frac{480}{20}\)
Days = 24
Question
The time it takes to be served at a café is inversely proportional to the number of baristas working.
It takes 9 minutes to be served when there are 2 baristas working.
a) Find an equation connecting time (T) and number of baristas (B).
b) How many baristas need to be working to be served in at least 4 minutes?
a) \({T}=\frac{k}{B}\)
\({9}=\frac{k}{2}\)
\({k}={9}\times{2}={18}\)
Equation connecting time and number of baristas: \({T}=\frac{18}{B}\)
b) Complete the information you have for the equation:
\({4}=\frac{18}{B}\)
Multiply both sides by B:
\({4}\times{B}={18}\)
Then divide both sides by 4:
\({B}={18}\div{4}={4.5}\)
If rounded down it will take longer than 4 minutes, so we must round up.
The number of baristas that need to be working to be served in at least 4 minutes is 5 baristas.
If E is inversely proportional to the square of F we write \({E}~=~\frac{k}{F^2}\)
Manipulating the equation to find the constant k works in the same way.
Question
Given that G is inversely proportional to the square of H, and that G = 3 and H = 5.
a) Find an expression for G in terms of H
b) Calculate the value of G when H = 2
c) Calculate the value of H when G = 30
a) \({G}~=~\frac{k}{H^2}\)
\({3}~=~\frac{k}{5^2}\)
\({3}\times{25}~=~{k}\)
\({k}~=~{75}\)
\({G}~=~\frac{75}{H^2}\)
b) \({G}~=~{75}\div{2^2}\)
\({G}~=~{75}\div{4}\)
\({G}~=~{18.75}\)
c) \({30}~=~\frac{75}{H^2}\)
\({H^2}~=~{75}\div{30}~=~{2.5}\)
\({H}~=~\sqrt{2.5}\)
\({H}~=~{1.58}\) (to two decimal places)
