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 indirectly 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 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 nine minutes to be served when there are two 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 four 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 four minutes, so we must round up.
The number of baristas that need to be working to be served in at least four minutes is five 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)
