Learning Objectives
  • Express direct and inverse proportion algebraically.
  • Find the constant of proportionality from known values.
  • Use linear, square, square-root, cube and cube-root proportion models.
  • Solve combined proportion problems.
  • Interpret graphs and changes associated with proportional relationships.
Key Terms
Direct proportion
Two quantities related so that their ratio is constant.
Inverse proportion
Two quantities related so that their product, or an appropriate powered product, is constant.
Constant of proportionality
The fixed number k linking the variables.
Proportional symbol
\propto, meaning is proportional to.
Combined proportion
A relationship involving direct and inverse factors together.
8th Edition Chapter Map
  • Direct proportion and its graph
  • Inverse proportion and reciprocal curves
  • Power proportions
  • Combined proportion
  • Interpreting scaling effects
Direct Proportion

If y is directly proportional to x, write y\propto x, then replace the symbol with an equation y=kx. The ratio y/x is constant and the graph is a straight line through the origin.

y\propto x\quad\Longrightarrow\quad y=kx
Worked Example: Direct Proportion

Question: y\propto x and y=18 when x=6. Find y when x=14.

  1. 18=6k, so k=3.
  2. y=3x.
  3. When x=14, y=42.

Answer: 42.

Direct Proportion To A Power

The wording determines the model. If y\propto x^2, then y=kx^2. If y\propto\sqrt{x}, then y=k\sqrt{x}. The Cambridge syllabus includes linear, square, square-root, cube and cube-root proportion.

Worked Example: Square Proportion

Question: A\propto r^2 and A=72 when r=3. Find A when r=5.

  1. 72=k(3^2)=9k, so k=8.
  2. A=8r^2.
  3. A=8(25)=200.

Answer: 200.

Inverse Proportion

If y is inversely proportional to x, write y=k/x, so xy=k. As one variable increases, the other decreases. The graph is a reciprocal curve approaching but not touching the axes.

y\propto\frac1x\quad\Longrightarrow\quad y=\frac{k}{x}
Worked Example: Inverse Proportion

Question: p\propto1/q and p=12 when q=5. Find p when q=8.

  1. 12=k/5, so k=60.
  2. p=60/q.
  3. p=60/8=7.5.

Answer: 7.5.

Inverse Square And Other Powers

If y\propto1/x^2, doubling x makes y one quarter as large. If y\propto1/\sqrt{x}, multiplying x by 9 divides y by 3.

Combined Proportion

A statement may combine variables. If F is directly proportional to m and inversely proportional to d^2, then F=km/d^2. Keep the complete numerator and denominator structure visible.

Worked Example: Combined Proportion

Question: F\propto m/d^2. When m=8,d=2,F=30, find F when m=15,d=5.

  1. 30=k(8)/4=2k, so k=15.
  2. F=15m/d^2.
  3. F=15(15)/25=9.

Answer: 9.

Scaling Without Finding K

Sometimes the effect can be reasoned directly. If y\propto x^3 and x doubles, y is multiplied by 2^3=8. If y\propto1/x^2 and x triples, y is divided by 9.

Relationship Invariant or scaling fact
y\propto x y/x is constant.
y\propto x^2 y/x^2 is constant.
y\propto1/x xy is constant.
y\propto1/x^2 yx^2 is constant.
Examination Guidance
  • Always replace \propto by an equation containing k.
  • Copy powers and roots exactly from the wording.
  • Use known values to find k before substituting new values.
  • Check whether the final change agrees with the direction of the relationship.
Common Mistakes
  • Writing y=x/k for inverse proportion.
  • Assuming a straight line not through the origin represents direct proportion.
  • Forgetting the square, cube or root in the model.
  • Rounding k unnecessarily before the final step.
Knowledge Check And Practice

1. If y\propto x and y=24 when x=8, find y when x=15.

Answer: k=3, so y=45.

2. If y\propto x^2, what happens to y when x triples?

Answer: It is multiplied by 9.

3. If p\propto1/q and p=9,q=4, find p when q=12.

Answer: k=36, so p=3.

4. If z\propto\sqrt{x} and z=10 when x=25, find z when x=81.

Answer: k=2, so z=18.

5. If T\propto m^2/r, write the equation.

Answer: T=km^2/r.
Extended Worked Practice
Direct Proportion To A Cube

Question: y is directly proportional to x^3. When x=2, y=20. Find y when x=5.

  1. y=kx^3.
  2. 20=8k, so k=2.5.
  3. y=2.5(125)=312.5.

Answer: 312.5.

Inverse Proportion To A Square Root

Question: p is inversely proportional to \sqrt q. When q=16, p=9. Find p when q=81.

  1. p=\frac{k}{\sqrt q}.
  2. 9=k/4, so k=36.
  3. p=36/9=4.

Answer: 4.

Combined Proportion

Question: F is directly proportional to m and inversely proportional to r^2. What happens to F if m is tripled and r is doubled?

F\propto\frac{m}{r^2}. The scale factor is \frac{3}{2^2}=\frac34.

Answer: F becomes three quarters of its original value.

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