Ratio and Proportion • Topic 2 of 3

Direct Variation

What is direct variation? Two quantities are said to vary directly if they increase or decrease together in such a way that their ratio remains constant. If x and y vary directly, then:

y = kx or y/x = k (constant)

where k is called the constant of variation (or constant of proportionality).

Key properties:

  • As x increases, y increases proportionally
  • As x decreases, y decreases proportionally
  • The graph of a direct variation is a straight line passing through the origin
  • If x doubles, y doubles; if x is halved, y is halved

Real-life examples of direct variation:

  • Distance travelled and time (at constant speed) — more time = more distance
  • Cost of apples and number of apples — more apples = more cost
  • Work done and number of workers (same time) — more workers = more work
  • Circumference of a circle and its diameter (C = πd)

Direct variation equation: If y ∝ x (reads "y is proportional to x"), then y = kx.

Graph of direct variation y equals k x, a straight line passing through the originDirect variation: y = kx (k > 0)xyOpasses through origin (0,0)slope = kratio y/x stays constant: x doubles → y doublesTable for cost of apples at fifty rupees per kilogram showing constant ratioCost ∝ quantity (Rs 50 per kg)Qty (kg)Cost (Rs)Cost/Qty15050210050420050
Solved Examples
1
Worked Example
If y varies directly as x, and y = 15 when x = 5, find y when x = 12.
Solution
  1. Step 1: y ∝ x → y = kx
  2. Step 2: Find k: 15 = k × 5 → k = 15/5 = 3
  3. Step 3: Equation: y = 3x
  4. Step 4: When x = 12: y = 3 × 12 = 36

Answer: y = 36

2
Worked Example
The cost of 6 kg of sugar is ₹210. Find the cost of 10 kg of sugar.
Solution
  1. Step 1: Cost ∝ Quantity → C = k × Q
  2. Step 2: Find k: 210 = k × 6 → k = 210/6 = 35
  3. Step 3: C = 35Q
  4. Step 4: When Q = 10, C = 35 × 10 = 350

Answer: ₹350

3
Worked Example
y varies directly as the square of x. If y = 48 when x = 4, find y when x = 6.
Solution
  1. Step 1: y ∝ \(x^{2}\) → y = \(kx^{2}
  2. Step\) 2: Find k: 48 = k × (4)\(^{2}\) = k × 16 → k = 48/16 = 3
  3. Step 3: Equation: y = \(3x^{2}
  4. Step\) 4: When x = 6: y = 3 × 36 = 108

Answer: y = 108

Key Points

  • Direct variation: y ∝ x → y = kx (k = constant)
  • Graph is a straight line through the origin
  • Ratio y/x remains constant (k)
  • If x doubles, y doubles; if x halves, y halves
  • To solve: find k using given pair, then use equation
  • Direct variation can involve powers: y ∝ \(x^{2}\), y ∝ \(\sqrt{x}\), etc.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In direct variation, $y=$
Explanation: $y=kx$.
Q2.If $y\propto x$ and $x$ doubles, then $y$:
Explanation: Direct proportion.
Q3.The graph of direct variation is a:
Explanation: Line through origin.
Q4.If $y=3x$, the constant $k$ is:
Explanation: $k=3$.
Practice more MCQs → 📝 Assignment · 35 marks