Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Variation and Problems on Ratio for CA Foundation

Updated 1 October 2026 · Fact-checked

Variation describes how one quantity changes when another changes. In direct variation, x/y stays constant. In inverse variation, x × y stays constant. In joint variation, a quantity varies with the product of two others. To solve ratio problems, write quantities as multiples of a common unit k, form an equation, and find k.

Understand Variation and Problems on Ratio

A ratio compares two quantities of the same kind. Variation goes one step further. It tells you how one quantity moves when another quantity moves.

In direct variation, both quantities rise or fall together at the same rate. If you buy more pens at a fixed price, the cost goes up in the same ratio. We write x ∝ y, which means x = k·y, where k is a constant. So x/y never changes.

In inverse variation, when one quantity rises, the other falls. More workers means fewer days for the same job. We write x ∝ 1/y, which means x·y = k. So the product never changes.

In joint variation, a quantity depends on two or more others together. If x ∝ y when z is fixed, and x ∝ z when y is fixed, then x ∝ yz, so x = k·y·z. You can also mix the two types: x ∝ y/z means x = k·y/z.

Ratio word problems on ages, mixtures and sharing use one idea. Write each quantity as a multiple of a common unit, such as 3k and 5k. Then use the given fact, such as a total or a difference, to find k.

Key formulas to remember

Direct variation: x ∝ y ⇒ x = ky, so x₁/y₁ = x₂/y₂; Use when both quantities increase or decrease together.
Inverse variation: x ∝ 1/y ⇒ xy = k, so x₁y₁ = x₂y₂; Use when one goes up as the other goes down, such as men and days.
Joint variation: x ∝ yz ⇒ x = kyz, so x₁/(y₁z₁) = x₂/(y₂z₂); Quantity depends on the product of two others.
Sharing a total in a ratio: Share of A = a/(a + b) × Total, for ratio a : b; For three parts a : b : c, divide by a + b + c.
Mixture ratio after adding: New ratio = (a·k + added A) : (b·k + added B); Write the original quantities as ak and bk first.
Age problems: Present ages ak and bk; after n years: (ak + n) and (bk + n); The difference of ages stays the same over time.
Replacement in a mixture: Final quantity of original liquid = Initial × (1 − x/T)ⁿ; Here T is the total volume of the mixture. Each time, x of the mixture is drawn out and replaced by the other liquid, and this is done n times. Initial is the original liquid's initial quantity.

How to solve Variation and Problems on Ratio questions

Use this method for any variation or ratio word problem.

1Read the question and identify the quantities involved. Note which are given and which are asked.
2Decide the type of relation. Ask: if one quantity doubles, does the other double (direct), halve (inverse), or depend on two others (joint)?
3Write the relation as an equation with a constant k, such as x = ky, xy = k or x = kyz.
4Find k using the first set of values, or compare the two sets directly using x₁/y₁ = x₂/y₂ or x₁y₁ = x₂y₂.
5For ratio problems, write each quantity as a multiple of k, such as 3k and 5k.
6Use the given fact (a total, a difference, a future age or an added amount) to form one equation in k.
7Solve for k, then compute what the question asks. Do not stop at k.
8Check that your answer fits the options and the conditions in the question.

Quickest way: Compare, don't find k

When to use it: Use this in MCQs where two sets of values are given and you need one unknown.

Skip finding k. Directly set up x₁/y₁ = x₂/y₂ for direct variation, or x₁y₁ = x₂y₂ for inverse variation.
For joint variation, compare the products: x₁/(y₁z₁) = x₂/(y₂z₂).
For ratio sharing, add the ratio parts and divide the total by that sum to get one unit.
In age problems, test the options. Add the given years to each option's ages and check the ratio.
Eliminate options that are not divisible by the ratio sum or break the direction of variation, such as a larger answer when it must be smaller.
Skip a question if setup takes more than about a minute, since wrong answers cost 0.25 marks.

Common mistakes in Variation and Problems on Ratio

Using a direct relation when the relation is inverse, for example in men and days problems.
Students match numbers by habit instead of thinking about the direction of change.
Fix: Ask: if one quantity increases, should the other increase or decrease? If it decreases, use xy = constant.
Adding the same number to ratio parts when time passes, but forgetting to add to the actual ages (ak + n), not to the ratio numbers.
Students treat 3 : 5 as the actual ages.
Fix: Always write ages as 3k and 5k, then add years to 3k and 5k.
Stopping after finding k.
Students feel the work is done once the equation is solved.
Fix: Re-read the question and compute the exact quantity asked, such as the larger share or the present age.
Mixing up 'x varies as y' with 'x varies inversely as y'.
The wording is similar and students read quickly.
Fix: Translate the words into symbols first: 'varies as' means x = ky, 'varies inversely' means x = k/y.
In mixture problems, adding ratios instead of actual quantities when something is added or removed.
Students forget that ratios are not quantities.
Fix: Convert to actual quantities using k, apply the addition or removal, then form the new ratio.
Squaring or cubing wrongly when the relation involves powers, such as x ∝ y².
Students ignore the power when comparing two sets of values.
Fix: Keep the power in the comparison: x₁/y₁² = x₂/y₂².

Worked examples

Example 1

If x varies directly as y, and x = 12 when y = 4, what is x when y = 10? (a) 25 (b) 30 (c) 40 (d) 48

Show the solution

x ∝ y, so x = ky.
Find k: 12 = k × 4, so k = 3.
Now x = 3 × 10 = 30.
Check by ratio: 12/4 = 3 and 30/10 = 3. It matches.

Answer: (b) 30

Example 2

The ages of A and B are in the ratio 3 : 5. After 6 years, the ratio becomes 2 : 3. What is the present age of B? (a) 15 (b) 18 (c) 25 (d) 30

Show the solution

Let present ages be 3k and 5k.
After 6 years: (3k + 6)/(5k + 6) = 2/3.
Cross-multiply: 3(3k + 6) = 2(5k + 6).
9k + 18 = 10k + 12, so k = 6.
B's present age = 5 × 6 = 30.
Check: A = 18, B = 30. After 6 years: 24 and 36, ratio 2 : 3. Correct.

Answer: (d) 30

Example 3

₹7,200 is to be shared among A, B and C in the ratio 2 : 3 : 4. How much does C get more than A? (a) ₹1,600 (b) ₹1,800 (c) ₹2,400 (d) ₹3,200

Show the solution

Sum of ratio parts = 2 + 3 + 4 = 9.
One part = 7,200 ÷ 9 = ₹800.
A gets 2 × 800 = ₹1,600. C gets 4 × 800 = ₹3,200.
Difference = 3,200 − 1,600 = ₹1,600.
Shortcut: difference is 2 parts, so 2 × 800 = ₹1,600.

Answer: (a) ₹1,600

Exam tips

Read the wording carefully. 'Varies as', 'varies inversely' and 'varies jointly' each give a different equation.
In age problems, the difference between ages never changes. Use this to check your answer.
Always test the options when the equation looks long. It is often faster than solving.
Check that the answer is the quantity asked for, not k or an intermediate value.
Do not spend more than a minute on a question. Negative marking makes a wrong guess costly.

Practice questions from Ratio and Proportion, Indices and Logarithms

Variation and Problems on Ratio: frequently asked questions

What is the difference between direct and inverse variation?

In direct variation, the ratio of the two quantities stays constant, so both rise or fall together. In inverse variation, their product stays constant, so when one rises the other falls. Direct: x/y = k. Inverse: xy = k.

What is joint variation with an example?

Joint variation means a quantity depends on the product of two or more others. For example, simple interest varies jointly with principal and time when the rate is fixed, so I = k × P × T.

How do I solve ratio word problems quickly?

Write each quantity as a multiple of k, such as 3k and 5k. Use the given total, difference or changed condition to form one equation. Solve for k, then compute the asked value. Testing options can be even faster.

Does the age difference change with time?

No. The difference between two people's ages stays the same every year. This is a useful check for any age problem.