Fundamentals of Business Mathematics and Statistics · Ratios, Variations and Proportions
Direct, Inverse and Joint Variation Explained with Examples
Updated 10 October 2026 · Fact-checked
Variation shows how one quantity changes when others change. In direct variation, y = kx, so both rise or fall together. In inverse variation, y = k/x, so xy stays constant. In joint variation, y = kxz. Find k from the given data, then use it to find the unknown value.
Understand Direct, Inverse and Joint Variation
A variation links quantities so that a change in one forces a change in another. The link is fixed by a number called the constant of variation (also called the constant of proportionality), written k.
In direct variation, two quantities move in the same direction at the same rate. If you buy more pens at a fixed price, the cost rises in step. We write y ∝ x, which means y = kx. The ratio y ÷ x never changes.
In inverse variation, one quantity rises when the other falls, and their product stays fixed. If more workers do a fixed job, the days needed fall. We write y ∝ 1/x, which means y = k/x, or xy = k.
In joint variation, one quantity depends on two or more others together. Simple interest depends on principal and time at a fixed rate, so I ∝ P × T. We write y = kxz. You can also mix types, for example y ∝ x/z, where y varies directly with x and inversely with z.
Every problem has the same shape. Use one set of data to find k, then use k for the new set. In many questions you can skip k and compare the two sets directly.
Key formulas to remember
- Direct variation
- y ∝ x ⇒ y = kx ⇒ y₁ ÷ x₁ = y₂ ÷ x₂
- The ratio y ÷ x is constant. Both quantities increase or decrease together.
- Inverse variation
- y ∝ 1/x ⇒ y = k ÷ x ⇒ x₁y₁ = x₂y₂
- The product xy is constant. When one rises, the other falls.
- Joint variation
- y ∝ xz ⇒ y = kxz ⇒ y₁ ÷ (x₁z₁) = y₂ ÷ (x₂z₂)
- y varies directly with each of x and z when the other is held fixed.
- Mixed variation
- y ∝ x/z ⇒ y = kx ÷ z
- Put directly varying quantities on top and inversely varying ones at the bottom.
- Constant of variation
- k = y ÷ x (direct), k = xy (inverse), k = y ÷ (xz) (joint)
- Find k from the first set of data. It stays the same for the second set.
How to solve Direct, Inverse and Joint Variation questions
Use this method for any variation question, whether direct, inverse, joint or mixed.
- 1Read the statement and name the variables, for example y varies with x and z.
- 2Decide the type of each link. Same direction means direct. Opposite direction means inverse.
- 3Write the equation with k, for example y = kxz or y = kx ÷ z.
- 4Put in the first set of values and solve for k.
- 5Write the equation again with the value of k.
- 6Put in the second set of values and solve for the unknown.
- 7Check the direction. If a direct quantity rose, the answer should rise. If an inverse quantity rose, the answer should fall.
Quickest way: Compare the two sets without finding k
When to use it: Use it when the question gives two sets of data and asks for one missing value. It saves time in an MCQ.
- For direct links, put the quantity on the top and multiply by the ratio new ÷ old.
- For inverse links, multiply by the ratio old ÷ new.
- Multiply all these factors with the original value of y.
- Cancel numbers before multiplying to keep the sums small.
- Match the answer with the options and check the direction.
Common mistakes in Direct, Inverse and Joint Variation
Treating an inverse link as direct, for example more men means more days.
Students match the words 'more' and 'more' without thinking about the real situation.
Fix: Ask: if one quantity doubles, does the other double or halve? Halve means inverse.
Using y = kx when the question says y varies as the square of x.
Students forget that the power also goes into the equation.
Fix: Write exactly what is stated. 'y varies as x²' means y = kx².
Multiplying the inverse ratio the wrong way, such as new ÷ old.
The direct-ratio habit carries over.
Fix: For inverse links, flip the ratio. Or use xy = k and check that the product stays equal.
Forgetting to find k and then mixing values from the two sets.
Students rush and substitute into the wrong equation.
Fix: Find k from the first set only. Then use it in the second set.
In joint variation, changing only one variable's effect.
Students apply the change in x but ignore the change in z.
Fix: Multiply the factors for every variable that changes. Check each one.
Worked examples
Example 1
y varies directly with x and inversely with z. When x = 6 and z = 4, y = 9. Find y when x = 8 and z = 12.
Show the solution
- Write y = kx ÷ z.
- Put x = 6, z = 4, y = 9: 9 = k × 6 ÷ 4, so 9 = 1.5k.
- So k = 6.
- Now y = 6 × 8 ÷ 12 = 48 ÷ 12 = 4.
Answer: y = 4
Example 2
The simple interest on a sum varies jointly with the principal and the time at a fixed rate. The interest is ₹1,200 on ₹5,000 for 2 years. Find the interest on ₹7,500 for 4 years.
Show the solution
- Write I = kPT.
- Put in the first data: 1,200 = k × 5,000 × 2, so 1,200 = 10,000k.
- So k = 0.12.
- New interest = 0.12 × 7,500 × 4 = 0.12 × 30,000 = 3,600.
Answer: ₹3,600
Exam tips
- Decide direct or inverse from the real situation before you write any equation.
- If the question gives two sets of data, use the ratio method and skip k to save time.
- Watch for powers and roots such as x², √x or 1/x². They change the equation.
- In MCQs, check the direction of your answer. It removes at least one or two wrong options quickly.
- There is no negative marking, so attempt every variation question even if you must guess.
Practice questions from Ratios, Variations and Proportions
- A sum of ₹7,200 is to be divided among Arun, Bala and Chitra so that Arun's share : Bala's share = 2 : 3 and Bala's share : Chitra's share =…
- The ratio 5:8 is to be made equal to a ratio in proportion with 35:k. Find k.
- The monthly electricity cost C of a factory varies jointly with the number of machines M and the hours H run per day. When 8 machines run 6 …
- The monthly incomes of Sunil and Tarun are in the ratio 5:4 and their expenditures are in the ratio 3:2. If Sunil saves ₹4,000 and Tarun sav…
- The quantity y varies directly as x and inversely as the square of z. When x = 6 and z = 2, y = 9. What is y when x = 8 and z = 4?
Direct, Inverse and Joint Variation in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Direct, Inverse and Joint Variation: frequently asked questions
What is the difference between direct and inverse variation?
In direct variation, both quantities rise or fall together, and y ÷ x is constant. In inverse variation, one rises when the other falls, and x × y is constant.
What is the constant of variation?
It is the fixed number k that links the quantities. You find it by putting the first set of values in the equation. It then stays the same for the other sets.
How do I solve joint variation problems?
Write y = kxz and find k from the given values. Then use that k with the new values of x and z. You can also multiply the original y by the ratio of change for each variable.
Can a quantity vary directly with one variable and inversely with another?
Yes. This is mixed variation, such as y = kx ÷ z. Directly varying quantities go on top and inversely varying ones go at the bottom.