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CA Foundation · Quantitative Aptitude

Equations for CA Foundation Quantitative Aptitude: Chapter Guide

Equations are statements that two expressions are equal, and you solve them by finding the values of the unknowns that make them true. In the CA Foundation MCQ paper, you solve linear, simultaneous, quadratic and simple cubic equations quickly, often by substituting the options instead of doing full algebra.

What this chapter covers

This chapter is about finding unknown values. You start with one unknown in a linear equation, move to two unknowns in simultaneous equations, and then to quadratic equations where the unknown is squared and can have two values. The chapter ends with cubic equations and business applications.

The core tools are few. You rearrange terms, eliminate or substitute a variable, factorise, or use the quadratic formula. For a quadratic ax² + bx + c = 0 with a ≠ 0, the roots are x = (-b ± √(b² - 4ac)) ÷ 2a. The expression b² - 4ac is the discriminant. It tells you the nature of the roots before you solve anything.

This chapter connects to the rest of Quantitative Aptitude. Ratio, percentages, simple and compound interest, profit and loss, and time and work all turn into equations once you read the word problem. Linear programming-style and cost-revenue questions also use it. If your equation skills are weak, other chapters become slower too.

Equations is a chapter where practice converts directly into marks. The questions are short, the methods are fixed, and many MCQs can be solved in under a minute by substituting options or using the sum and product of roots. Since Paper 3 has negative marking of 0.25 per wrong answer, a chapter where you can be both fast and accurate lets you attempt more questions with confidence. The skill also supports word problems in other chapters, so the effort pays back more than once.

Equations: topics in the order to study them

  1. 1Linear Equations in One VariableIt builds the habit of balancing both sides, which every later topic depends on.
  2. 2Simultaneous Linear EquationsIt extends the same skill to two unknowns using elimination and substitution.
  3. 3Quadratic Equations and Their RootsYou need factorising, the formula and the discriminant before using any root shortcuts.
  4. 4Sum and Product of RootsIt gives fast checks and lets you build equations from roots, so it follows the roots topic.
  5. 5Word Problems on Quadratic EquationsYou can only set up these problems well once you can solve a quadratic without effort.
  6. 6Cubic and Higher-Degree EquationsThese are solved by trial of simple values and factorising, which builds on quadratic methods.
  7. 7Applications of Equations in BusinessIt is the last step: you turn cost, revenue and break-even situations into equations using all earlier skills.

How to prepare Equations

Aim for method first, then speed, then option-based shortcuts. Do each step before moving to the next.

  1. Revise basic algebra: transposing terms, removing brackets and clearing fractions. Solve 15 simple linear equations without a calculator until errors stop.
  2. For simultaneous equations, practise both elimination and substitution. Then learn to pick the quicker one by looking at the coefficients.
  3. For quadratics, practise factorising by splitting the middle term. Keep the formula for cases where factors are not obvious. Check the discriminant first when a question only asks about the nature of roots.
  4. Learn the sum of roots = -b ÷ a and product of roots = c ÷ a for ax² + bx + c = 0. Use them to verify your answers and to answer questions without solving.
  5. For word problems, write what x stands for, form the equation, solve, then reject any value that does not make sense, such as a negative length or age.
  6. Practise MCQs with a timer. Try substituting the four options when the equation is messy. Skip a question if setting it up takes more than about a minute and a half, and return later.
  7. Attempt chapter-wise tests and keep an error log noting whether each mistake was a concept, a setup or a calculation slip.

Common mistakes in Equations

  • Dropping or flipping a sign while moving terms across the equals sign

    Fix: Expand brackets first, write each step on a new line, and substitute your answer back into the original equation to check.

  • Forgetting that a quadratic has two roots and giving only one

    Fix: Always write both x values, then decide which one the question accepts. In MCQs, check whether the options include both values or a sum or product.

  • Using the sum and product of roots with the wrong signs or without writing the equation in standard form

    Fix: Move every term to one side so the other side is 0 first. Then sum = -b ÷ a and product = c ÷ a.

  • Accepting a root that does not fit the word problem

    Fix: After solving, ask whether the value can be a length, age, number of items or price. Discard values that cannot.

  • Defining the variable vaguely in word problems

    Fix: Write one line such as 'Let x = the number of units' before forming the equation, and use the same unit throughout.

  • Doing full algebra on every MCQ

    Fix: In an objective paper, only the answer counts. Substitute the options, starting with the middle values, when the equation is long or the numbers are small.

Last-day revision: Equations

  • A linear equation in one variable has one solution: ax + b = 0 gives x = -b ÷ a, for a ≠ 0.
  • In simultaneous equations, multiply to match one coefficient, then add or subtract to eliminate a variable.
  • A quadratic ax² + bx + c = 0 needs a ≠ 0 and has at most two real roots.
  • Roots by formula: x = (-b ± √(b² - 4ac)) ÷ 2a.
  • Discriminant D = b² - 4ac. If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, no real roots.
  • Sum of roots = -b ÷ a. Product of roots = c ÷ a.
  • A quadratic with roots α and β is x² - (α + β)x + αβ = 0.
  • If D is a perfect square and a, b, c are rational, the roots are rational.
  • In word problems, reject roots that are negative or fractional when the quantity must be a positive whole number.
  • For cubics and higher degrees, test small values like 1, -1, 2, -2 to find one root, then factorise.
  • If x = k is a root, then (x - k) is a factor.
  • Break-even occurs where total revenue equals total cost.

Equations practice questions

Equations: frequently asked questions

Which topics in Equations should I do first for CA Foundation?

Start with linear equations in one variable, then simultaneous equations, then quadratics. Do sum and product of roots right after quadratics. Word problems and business applications come last because they need all earlier skills.

Can I solve Equations MCQs by putting in the options?

Yes, in many questions. It works best when the equation is complex but the numbers are small. Start with an option that is easy to compute, and eliminate wrong options as you go.

Do I need to memorise the quadratic formula?

Yes. Factorising is faster when it works, but the formula always works for a quadratic. Also memorise the discriminant, because it answers questions on the nature of roots without full solving.

How do I avoid losing marks to negative marking in this chapter?

Attempt only when you can either solve the question or eliminate at least some options with certainty. Each wrong answer costs 0.25 marks. Skip long set-up questions on the first pass and return if time remains.