Quantitative Aptitude · Equations

Applications of Equations in Business for CA Foundation

Updated 1 October 2026 · Fact-checked

Business equation problems turn words into linear or quadratic equations. Write cost as fixed cost plus variable cost, and revenue as price times quantity. Set revenue equal to cost for break-even. For equilibrium, set demand equal to supply. Solve for quantity or price, then check the answer is positive and fits the question.

Understand Applications of Equations in Business

A business problem is a normal equation problem with a story around it. Your job is to name the unknown (usually quantity x or price p), write each business quantity as an expression, and then join two of them with an equals sign.

Cost has two parts. Fixed cost does not change with output (rent, salaries). Variable cost grows with each unit made. So total cost = fixed cost + variable cost per unit × units. Revenue is what you earn: price × units sold. Profit is revenue minus cost.

The break-even point is the output where revenue equals cost, so profit is zero. Below it you make a loss. Above it you make a profit (when each unit sells for more than its variable cost). If cost or revenue contains x², the break-even equation becomes quadratic and can have two roots.

Market equilibrium is where the quantity buyers demand equals the quantity sellers supply. Demand usually falls as price rises. Supply usually rises as price rises. Set the two expressions equal and solve for price. Put that price back into either expression to get the equilibrium quantity.

In the exam, the maths is easy. Marks are lost in setting up the equation and in reading what the question asks for: units, rupees, price or quantity.

Key formulas to remember

Total cost (linear): C = F + v·x; F = fixed cost, v = variable cost per unit, x = units produced.
Revenue: R = p·x; p = selling price per unit, x = units sold.
Profit: P = R − C; Profit is zero at break-even. A negative value means loss.
Break-even quantity (linear): x = F ÷ (p − v); Valid when p > v. The term (p − v) is the contribution per unit.
Break-even sales value: Sales value = p × F ÷ (p − v); Break-even units multiplied by the price.
Average cost: AC = C ÷ x; Total cost divided by units. Do not confuse it with variable cost per unit.
Market equilibrium: Quantity demanded = Quantity supplied; Solve for price first. Then substitute to find quantity.
Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a; Use when the break-even equation is ax² + bx + c = 0 and does not factorise easily.

How to solve Applications of Equations in Business questions

Use the same routine for every cost, revenue, break-even or equilibrium question. It stops you from mixing up quantities.

1Read the last line first. Note exactly what is asked: units, price, rupees, or profit.
2Define the unknown, such as x = units or p = price.
3Write cost, revenue, demand or supply as expressions in that unknown. Keep fixed and variable parts separate.
4Join the right pair with an equals sign: revenue = cost for break-even, demand = supply for equilibrium.
5Simplify to the form ax + b = 0 (linear) or ax² + bx + c = 0 (quadratic) and solve. Factorise first, use the formula if it does not factorise.
6Reject any root that is negative or does not make sense for the story.
7Substitute back if the question asks for a second quantity such as equilibrium quantity or profit.
8Check by putting your answer into both sides of the original equation.

Quickest way: Option-substitution and direct formula

When to use it: Use it in MCQs when the setup is simple. Direct formulas save time on linear break-even. Substitution saves time on quadratics.

For linear break-even, skip the equation and use x = F ÷ (p − v) directly.
For equilibrium, subtract the two sides in one line: collect price terms on one side and numbers on the other.
For a quadratic, test the four options in the equation. Start with the smallest easy number. Two checks often settle it.
Check the last digit. If F ÷ (p − v) must be a whole number of units, discard options that do not divide evenly.
Eliminate options that give a negative price or quantity.
If the setup is confusing and takes more than about 90 seconds, skip it. Wrong answers cost 0.25 marks.

Common mistakes in Applications of Equations in Business

Treating fixed cost as per-unit cost, or multiplying it by x.
Students see a rupee figure and attach it to every unit.
Fix: Only the variable cost is multiplied by x. Fixed cost is added once: C = F + vx.
Using selling price instead of (price − variable cost) in the break-even formula.
Students divide fixed cost by price because it looks simpler.
Fix: Divide by the contribution per unit, p − v, because each unit first covers its own variable cost.
Stopping after finding the equilibrium price when the question asks for quantity.
Students feel the work is done once one variable is found.
Fix: Re-read the last line. Substitute the price into demand or supply to get quantity.
Giving both roots of a quadratic break-even equation without checking them.
Students copy the algebra result without thinking about context.
Fix: Reject negative roots. If both are positive, both are break-even points. With a convex cost curve (such as C = x² + 10x + 500) and a linear revenue line, profit is positive between the two break-even points and negative outside them. To be sure, test a value between the roots.
Sign errors when moving terms, for example in 120 − 3p = 20 + 2p.
Students rush and move terms without changing sign.
Fix: Collect the p terms on one side and numbers on the other in a fixed order. Then verify by substituting the answer.
Mixing up units, such as thousands of units with rupees.
Some problems state costs in thousands while others give units directly.
Fix: Write the unit next to each number as you set up. Convert everything to one unit before forming the equation.

Worked examples

Example 1

A firm has fixed cost of ₹60,000. Variable cost is ₹40 per unit and the selling price is ₹100 per unit. The break-even output is: (a) 600 units (b) 1,000 units (c) 1,500 units (d) 2,400 units

Show the solution

Cost: C = 60,000 + 40x.
Revenue: R = 100x.
Set R = C: 100x = 60,000 + 40x.
Subtract 40x: 60x = 60,000.
Divide: x = 1,000.
Check: R = ₹1,00,000. C = 60,000 + 40,000 = ₹1,00,000. They are equal.

Answer: (b) 1,000 units

Example 2

The demand function is Qd = 120 − 3p and the supply function is Qs = 20 + 2p, where p is price in rupees. The equilibrium quantity is: (a) 20 (b) 40 (c) 60 (d) 80

Show the solution

At equilibrium Qd = Qs.
120 − 3p = 20 + 2p.
Collect terms: 120 − 20 = 2p + 3p, so 100 = 5p.
p = 20.
Quantity from demand: Qd = 120 − 3(20) = 60.
Check with supply: Qs = 20 + 2(20) = 60. Both match.

Answer: (c) 60. The equilibrium price is ₹20 and the quantity is 60 units.

Example 3

A firm's revenue is R = 70x and its cost is C = x² + 10x + 500, where x is output in units. The highest output at which the firm just breaks even is: (a) 10 (b) 25 (c) 50 (d) 500

Show the solution

Set R = C: 70x = x² + 10x + 500.
Bring all terms to one side: x² + 10x − 70x + 500 = 0, so x² − 60x + 500 = 0.
Factorise: find two numbers with product 500 and sum −60. These are −10 and −50.
(x − 10)(x − 50) = 0, so x = 10 or x = 50.
Both roots are positive, so both are break-even points. The higher one is 50.
Check x = 50: R = 3,500. C = 2,500 + 500 + 500 = 3,500. They are equal.

Answer: (c) 50 units. The firm also breaks even at 10 units.

Exam tips

Look for the words 'just covers cost', 'no profit no loss' and 'neither gain nor loss'. They all mean revenue = cost.
Questions often give two or three linked parts. Read the options first to see whether they are quantities, prices or rupee amounts.
For quadratics, try factorising or substituting options before using the formula. It is usually faster in MCQs.
Always run a quick check of your answer in the original equation. It takes 15 seconds and catches sign errors.
Do not spend more than 90 seconds on a wordy question. Mark it, move on and return if time remains.

Practice questions from Equations

Applications of Equations in Business: frequently asked questions

What is the break-even point in CA Foundation Quantitative Aptitude?

It is the output at which total revenue equals total cost, so profit is zero. For a linear model it is fixed cost divided by (price − variable cost per unit). Below this output the firm makes a loss.

How do I find market equilibrium from demand and supply equations?

Set quantity demanded equal to quantity supplied and solve for price. Then substitute that price into either equation to get the equilibrium quantity. Both equations must give the same quantity, so use that as a check.

Why does a break-even equation sometimes have two answers?

If cost or revenue contains an x² term, the equation is quadratic and can have two positive roots. Each root is an output where revenue equals cost. Reject any negative root because output cannot be negative.

Is the formula x = F ÷ (p − v) always valid?

It applies to linear cost and revenue where price per unit and variable cost per unit are constant, and p is greater than v. If p is not greater than v, the firm never breaks even. If cost is quadratic, form and solve the equation instead.