Ratio and Proportion • Topic 3 of 3

Inverse Variation

What is inverse variation? Two quantities are said to vary inversely if one increases while the other decreases proportionally. If x and y vary inversely, then their product remains constant:

xy = k or y = k/x

where k is the constant of variation.

Key properties:

  • As x increases, y decreases proportionally
  • As x decreases, y increases proportionally
  • The graph is a rectangular hyperbola (never touches the axes)
  • If x doubles, y halves; if x is halved, y doubles

Real-life examples of inverse variation:

  • Speed and time for a fixed distance — faster speed = less time
  • Number of workers and time to complete a job — more workers = less time
  • Number of people sharing a bill and each person's share — more people = smaller share
  • Pressure and volume of a gas (Boyle's Law) — \(P_{1}V_{1}\) = \(P_{2}V_{2}\)

Inverse variation equation: If y ∝ 1/x (reads "y is inversely proportional to x"), then y = k/x or xy = k.

Graph of inverse variation y equals k over x, a rectangular hyperbola in the first quadrant that never touches the axesInverse variation: y = k/x (k > 0)xyOy = k/xcurve never meets the axes: x doubles → y halves (xy = k)Table of speed against time for a fixed 120 km journey showing the product speed times time is constantSpeed ∝ 1/time for a fixed 120 km tripSpeedTime (h)Speed×Time3041206021201201120
Solved Examples
1
Worked Example
If y varies inversely as x, and y = 12 when x = 5, find y when x = 15.
Solution
  1. Step 1: y ∝ 1/x → xy = k (constant)
  2. Step 2: Find k: 5 × 12 = 60 → k = 60
  3. Step 3: Equation: xy = 60
  4. Step 4: When x = 15: 15 × y = 60 → y = 60/15 = 4

Answer: y = 4

2
Worked Example
8 workers can complete a job in 15 days. How many days will 12 workers take to complete the same job?
Solution
  1. Step 1: Workers and days vary inversely (more workers = fewer days)
  2. Step 2: \(W_{1} \times D_{1}\) = \(W_{2} \times D_{2}\) (constant = total work)
  3. Step 3: 8 × 15 = 12 × \(D_{2}
  4. Step\) 4: 120 = 12 × \(D_{2}\) → \(D_{2}\) = 120/12 = 10

Answer: 10 days

3
Worked Example
y varies inversely as the square of x. If y = 8 when x = 3, find y when x = 6.
Solution
  1. Step 1: y ∝ 1/\(x^{2}\) → y = k/\(x^{2}
  2. Step\) 2: Find k: 8 = k/9 → k = 8 × 9 = 72
  3. Step 3: Equation: y = 72/\(x^{2}
  4. Step\) 4: When x = 6: y = 72/36 = 2

Answer: y = 2

Key Points

  • Inverse variation: y ∝ 1/x → xy = k (constant)
  • Graph is a hyperbola (curved, never touches axes)
  • Product xy remains constant for all pairs
  • If x doubles, y halves; if x halves, y doubles
  • To solve: find k using given pair, then use equation
  • Can extend to squares: y ∝ 1/\(x^{2}\), cubes, etc.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In inverse variation, $xy=$
Explanation: Product is constant.
Q2.If $x\propto\tfrac1y$ and $x$ doubles, then $y$:
Explanation: Inverse relationship.
Q3.If $xy=12$ and $x=3$, then $y=$
Explanation: $y=12/3=4$.
Q4."More workers, less time" is an example of:
Explanation: Inverse variation.
Practice more MCQs → 📝 Assignment · 35 marks