Worked example · Easy
Identify direct proportion
One ticket costs £4. Is total ticket price directly proportional to the number bought?
The cost of \(n\) tickets is \(C=4n\).
Therefore \(C\propto n\); the constant is 4.
Chapter 26
Proportion
Turn direct and inverse relationships into equations, find constants of proportionality, and use square, cube and square-root variation confidently.
By the end of this chapter, you should be able to
- distinguish direct and inverse proportion from statements and tables
- write a proportion statement using the correct symbol
- form an equation by finding the constant of proportionality
- solve direct and inverse proportion word problems
- work with square, cubic and square-root variations
1
Subtopic 1
Proportion
Two quantities are in direct proportion when one is a constant multiple of the other. If \(y\propto x\), doubling \(x\) doubles \(y\). The ratio \(y/x\) stays constant.
Two quantities are in inverse proportion when their product stays constant. If \(y\propto1/x\), doubling \(x\) halves \(y\). Do not conclude direct proportion merely because both quantities increase; their ratio must be constant.
Essential rule\[\text{Direct: }y\propto x,\quad\frac yx=k\qquad\text{Inverse: }y\propto\frac1x,\quad xy=k\]
Worked example · Medium
Identify inverse proportion
A journey takes 6 hours at 40 km/h. At 80 km/h, how long is it if the distance is unchanged?
The distance is \(40\times6=240\) km.
Time equals distance ÷ speed: \(t=\frac{240}{80}=3\) hours. Speed and time are inversely proportional.
Worked example · Exam-style
Test a table
Measurements are \((x,y)=(3,12),(5,20),(8,32)\). Does \(y\) vary directly as \(x\)?
Compute ratios: \(12/3=4,\ 20/5=4,\ 32/8=4\).
All are equal, so \(y=4x\) and direct proportion holds.
Exam method
Write down the appropriate rule, substitute carefully, show each rearrangement and check that the final result answers the question.
Practise: Proportion
The questions progress from straightforward recall to multi-step exam-style reasoning. Answers are hidden until selected.
Question 1
State whether \(y=9x\) shows direct or inverse proportion.
Show Answer
Direct: \(y/x=9\) is constant.
Question 2
State whether \(y=30/x\) shows direct or inverse proportion.
Show Answer
Inverse: \(xy=30\) is constant.
Question 3
A worker earns $75 for 5 hours. At the same rate, what is earned in 12 hours?
Show Answer
Hourly pay is \(75/5=15\).
For 12 hours: \(12\times15=\$180\).
Question 4
8 workers take 15 days to complete a job. How many days would 12 equally efficient workers take?
Show Answer
Work requires \(8\times15=120\) worker-days.
Time for 12 workers \(120/12=10\) days.
Question 5
Classify the table \(x:2,4,8;\ y:12,6,3\).
Show Answer
Products \(2(12)=4(6)=8(3)=24\).
It is inverse proportion: \(y=24/x\).
Question 6
Classify the pairs \((x,y)=(2,3),(4,8),(6,11)\). Are they directly proportional?
Show Answer
Ratios \(3/2,\ 8/4,\ 11/6\) are not the same.
So not directly proportional.
Question 7
A truck travels 180 km on 15 litres of fuel at a steady rate. How many litres for 276 km?
Show Answer
Fuel per km \(15/180=1/12\).
Required fuel \(276/12=23\) litres.
Question 8
A fixed sum is shared equally among \(n\) people. Each gets $18 when \(n=10\). Find each share when \(n=12\).
Show Answer
Total sum \(18\times10=180\).
New share \(180/12=\$15\).
2
Subtopic 2
Proportion as a formula
Convert “\(y\) is directly proportional to \(x\)” into \(y=kx\). Convert “\(y\) is inversely proportional to \(x\)” into \(y=k/x\). Find \(k\) using a given pair of corresponding values, then substitute the new value.
Always record which variable is proportional to which. If the relationship is inverse, it may be helpful to rearrange it as \(xy=k\). Write the constant with appropriate units where needed.
Essential rule\[y=kx\quad\text{or}\quad y=\frac{k}{x}\quad(\text{determine }k\text{ from known data})\]
Worked example · Easy
Find the direct formula
\(y\propto x\) and \(y=21\) when \(x=7\). Find the equation.
Write \(y=kx\).
Substitute: \(21=7k\Rightarrow k=3\).
Therefore \(y=3x\).
Worked example · Medium
Find the inverse formula
\(y\propto1/x\) and \(y=6\) when \(x=5\). Find \(y\) when \(x=12\).
\(xy=k=6(5)=30\).
\(y=\frac{30}{12}=\boxed{2.5}\).
Worked example · Exam-style
Use both unknown values
The quantity \(a\) is directly proportional to \(b\). When \(a=18,b=12\). Find \(b\) when \(a=45\).
\(a=kb,\quad k=18/12=3/2\).
\(45=\frac32 b\Rightarrow b=45\times\frac23=\boxed{30}\).
Exam method
Write down the appropriate rule, substitute carefully, show each rearrangement and check that the final result answers the question.
Practise: Proportion as a formula
The questions progress from straightforward recall to multi-step exam-style reasoning. Answers are hidden until selected.
Question 1
\(p\propto q\) and \(p=36\) when \(q=9\). Find the formula.
Show Answer
\(p=kq;\ 36=9k\Rightarrow k=4;\ p=4q\).
Question 2
\(v\propto1/t\) and \(v=9\) when \(t=8\). Find the formula.
Show Answer
\(vt=k=9\times8=72\); \(v=72/t\).
Question 3
Using \(y=3.5x\), find \(x\) when \(y=70\).
Show Answer
\(70=3.5x\Rightarrow x=20\).
Question 4
\(y\propto1/x\) and \(y=4\) when \(x=15\). Find \(y\) for \(x=24\).
Show Answer
\(k=xy=4\times15=60\).
\(y=60/24=2.5\).
Question 5
The mass of uniform wire is proportional to its length. A 2.4 m piece has mass 108 g. Find the mass of 7 m.
Show Answer
\(k=108/2.4=45\text{ g/m}\).
\(m=45(7)=315\text{ g}\).
Question 6
The frequency \(f\) is inversely proportional to wavelength \(w\). If \(f=300\) at \(w=1.2\), find \(f\) at \(w=0.8\).
Show Answer
\(fw=300(1.2)=360\).
\(f=360/0.8=450\).
Question 7
A machine makes 240 items in 8 minutes at constant output. Write a formula for items \(N\) after \(t\) minutes; find items in 17 minutes.
Show Answer
\(k=240/8=30\); \(N=30t\).
\(N(17)=510\).
Question 8
The variable \(a\) is inversely proportional to \(b\). When \(a=5\), \(b=12\). Find \(b\) when \(a=8\).
Show Answer
\(ab=60\); \(8b=60\Rightarrow b=7.5\).
3
Subtopic 3
Other types of proportion
A quantity may vary directly or inversely as a power or root, not merely as the variable itself. If \(y\propto x^2\), doubling \(x\) multiplies \(y\) by \(4\). If \(y\propto x^3\), doubling \(x\) multiplies \(y\) by \(8\).
Write the correct power or root first, insert the constant \(k\), use the known pair to find \(k\), and then substitute. For inverse square-root variation, doubling \(x\) divides \(y\) by \(\sqrt2\), not by 2.
Essential rule\[y\propto x^n\Rightarrow y=kx^n\qquad y\propto\frac1{x^n}\Rightarrow y=\frac{k}{x^n}\]
Worked example · Easy
Square variation
\(y\propto x^2\); \(y=45\) when \(x=3\). Find \(y\) at \(x=5\).
\(y=kx^2,\ 45=9k\Rightarrow k=5\).
\(y=5(25)=125\).
Worked example · Medium
Inverse square-root variation
\(y\propto1/\sqrt x\); \(y=9\) when \(x=4\). Find \(y\) when \(x=36\).
\(y=k/\sqrt x;\ k=9\sqrt4=18\).
\(y=18/\sqrt{36}=\boxed{3}\).
Worked example · Exam-style
Braking distance
Braking distance is proportional to the square of speed. At 20 m/s it is 16 m. Find the distance at 35 m/s.
\(d=kv^2,\quad k=16/20^2=0.04\).
\(d=0.04(35^2)=0.04(1225)=\boxed{49\text{ m}}\).
Exam method
Write down the appropriate rule, substitute carefully, show each rearrangement and check that the final result answers the question.
Practise: Other types of proportion
The questions progress from straightforward recall to multi-step exam-style reasoning. Answers are hidden until selected.
Question 1
\(y\propto x^2\) and \(y=12\) at \(x=2\). Find \(y\) at \(x=5\).
Show Answer
\(k=12/2^2=3\); \(y=3(25)=75\).
Question 2
\(y\propto x^3\) and \(y=16\) at \(x=2\). Find \(y\) at \(x=4\).
Show Answer
\(k=16/8=2\); \(y=2(64)=128\).
Question 3
\(y\propto1/x^2\) and \(y=5\) at \(x=3\). Find \(y\) at \(x=5\).
Show Answer
\(k=yx^2=5(9)=45\); \(y=45/25=1.8\).
Question 4
\(y\propto\sqrt x\) and \(y=15\) at \(x=25\). Find \(y\) at \(x=81\).
Show Answer
\(k=15/\sqrt{25}=3\); \(y=3\sqrt{81}=27\).
Question 5
\(y\propto1/\sqrt x\) and \(y=20\) at \(x=9\). Find \(y\) at \(x=100\).
Show Answer
\(k=20\sqrt9=60\); \(y=60/10=6\).
Question 6
The area of a circle is directly proportional to the square of its radius. If the radius triples, by what factor does area change?
Show Answer
\(A\propto r^2\); factor \(3^2=9\).
Question 7
The volume of a cube is proportional to the cube of its side. The side grows by 20%. What is the percentage increase in volume?
Show Answer
New side factor \(1.2\); volume factor \(1.2^3=1.728\).
Increase \(0.728\times100\%=72.8\%\).
Question 8
A force varies inversely as the square of a distance. It is 80 N at 2 m. How far away gives 20 N?
Show Answer
\(F=k/r^2,\quad k=80(2^2)=320\).
\(20=320/r^2\Rightarrow r^2=16\Rightarrow r=4\text{ m}\).