How to Solve Direct and Inverse Variation Problems
Solve direct and inverse variation aptitude problems using the constant-ratio and constant-product tests, with practice questions.
Expected Interview Answer
Direct variation means y = kx, so both quantities rise and fall together proportionally, while inverse variation means y = k/x, so one quantity rises as the other falls, and the deciding test is whether the product xy or the ratio y/x stays constant.
In direct variation, the ratio y/x is a fixed constant k, so doubling x doubles y and halving x halves y. In inverse variation, the product xy is the fixed constant k, so doubling x halves y and vice versa. Many real problems combine both: work problems have time varying inversely with workers (more workers, less time) while total work varies directly with time for a fixed workforce. The universal solving method is to find k from one given pair of values, then substitute the new value of the known variable and solve for the unknown.
- One test, constant ratio vs constant product, classifies any variation problem
- Finding k from a given pair makes every subsequent value a one-step calculation
- Extends cleanly to joint and combined variation with multiple variables
AI Mentor Explanation
A batter’s total runs vary directly with balls faced at a fixed strike rate: runs = k × balls, so doubling the balls faced doubles the runs, keeping runs/balls constant. Meanwhile, for a fixed target score, the required run rate varies inversely with overs remaining: fewer overs left means a higher required rate, keeping rate × overs roughly constant. Spotting whether the ratio or the product is the fixed quantity is exactly how variation problems are classified.
Worked example (inverse variation, work problem)
Given
- 8 workers, 15 days
Constant product
- k = 8×15 = 120
New value
- 12×d = 120
- d = 10 days
Step-by-Step Explanation
Step 1
Classify the relationship
Check if the ratio y/x (direct) or the product xy (inverse) stays constant.
Step 2
Find the constant k
Use the given pair of values to compute k = y/x or k = xy.
Step 3
Write the variation equation
y = kx for direct variation, y = k/x for inverse variation.
Step 4
Substitute and solve
Plug in the new known value and solve for the unknown quantity.
What Interviewer Expects
- Correct classification of direct vs inverse variation from the problem wording
- Accurate computation of the constant k from given values
- Correct substitution to solve for the unknown quantity
- Recognition of combined variation (e.g. work problems with workers and days)
Common Mistakes
- Setting up y = kx when the relationship is actually inverse (y = k/x)
- Forgetting to recompute k when a third variable changes the relationship
- Cross-multiplying inverse variation incorrectly
- Confusing “varies as the square” with simple linear direct variation
Best Answer (HR Friendly)
“The quickest test is whether the ratio between the two quantities stays fixed, which is direct variation, or whether the product stays fixed, which is inverse variation. Once I know which one applies, I use the given pair of values to find the constant, then plug in the new value to solve for whatever is missing. Work and rate problems are the classic combined case, since total work varies directly with time but time varies inversely with the number of workers.”
Follow-up Questions
- How do you handle joint variation with three or more variables?
- How does a work-rate problem combine both direct and inverse variation?
- How would you solve a variation problem where y varies as the square of x?
- What is the difference between variation and simple proportion?
MCQ Practice
1. If y varies directly with x, and y = 20 when x = 4, what is y when x = 9?
k = 20/4 = 5, so y = 5×9 = 45.
2. 6 machines complete a job in 20 days. How many days would 8 machines take?
Machines×days = 6×20 = 120 (inverse variation). 8×d = 120, d = 15.
3. Which equation represents inverse variation between x and y?
Inverse variation means the product xy is constant, so y = k/x.
Flash Cards
Direct variation equation? — y = kx, where the ratio y/x is constant.
Inverse variation equation? — y = k/x, where the product xy is constant.
How to find k? — Use one given pair of values: k = y/x (direct) or k = xy (inverse).
Classic combined variation example? — Work problems: work varies directly with time, but time varies inversely with workers.