Variation describes how one variable changes in relation to another; understanding direct, inverse, and joint variation is essential for solving real-world business problems.
## Core concept
Variation is the relationship between two or more quantities where one changes predictably as the other changes. It is distinct from ratio and proportion—while proportion involves equality of ratios, variation describes the *nature* of the relationship itself.
Three types are tested at CA Foundation level:
- Direct variation (y ∝ x): As one quantity increases, the other increases proportionally. If y varies directly with x, then y = kx, where k is the constant of variation.
- Inverse variation (y ∝ 1/x): As one quantity increases, the other decreases proportionally. If y varies inversely with x, then y = k/x or xy = k.
- Joint variation: One quantity varies with the product or quotient of two or more other quantities (e.g., y ∝ xz means y = kxz).
The constant of variation (k) is found by substituting known values into the variation equation.
## Formula / rule
| Type | Equation | Finding k | |------|----------|-----------| | Direct | y = kx | k = y/x | | Inverse | y = k/x | k = xy | | Joint (direct) | y = kxz | k = y/(xz) | | Joint (mixed) | y = kx/z | k = yz/x |
Steps to solve variation problems: 1. Identify the type of variation from the problem statement ("varies directly," "is inversely proportional," etc.). 2. Write the variation equation with constant k. 3. Substitute given values to find k. 4. Use the equation with k to find unknown values.
## Common exam applications
Example 1 (Direct variation): If the cost of producing x units is directly proportional to x, and 50 units cost ₹2,500, find the cost of 120 units.
- Given: C ∝ x, so C = kx
- When x = 50, C = 2,500: k = 2,500/50 = 50
- For x = 120: C = 50 × 120 = ₹6,000
Example 2 (Inverse variation): The time taken to complete a job is inversely proportional to the number of workers. If 10 workers complete it in 8 days, how long will 16 workers take?
- Given: T ∝ 1/W, so T = k/W
- When W = 10, T = 8: k = 10 × 8 = 80
- For W = 16: T = 80/16 = 5 days
Example 3 (Joint variation): The profit P of a business varies jointly with investment I and time t. If P = ₹5,000 when I = ₹50,000 and t = 2 years, find P when I = ₹75,000 and t = 3 years.
- Given: P = kIt
- k = 5,000/(50,000 × 2) = 0.05
- P = 0.05 × 75,000 × 3 = ₹11,250
## Common mistakes
- Confusing direct and inverse: "y increases as x increases" = direct; "y increases as x decreases" = inverse.
- Forgetting to find k first: Always substitute one known pair of values to find k before solving for unknowns.
- Wrong equation for joint variation: Carefully check whether quantities are in numerator (direct) or denominator (inverse) of the variation equation.
- Ignoring units: Ensure consistency of units when calculating k and final answers (₹, days, units, etc.).
- Assuming all two-variable relationships are variations: A variation must show a *constant ratio* (direct) or *constant product* (inverse). Check using given data.
Exam tip: Always state the variation equation explicitly before substituting numbers—examiners expect to see your method, not just the answer.