Word Problems on Pair of Linear Equations in Two Variables – Class 10 Maths Practice Questions with Answers

Q: Q5: How many marks do linear equations word problems usually carry in Class 10 CBSE exams?

A5: Word problems from this chapter typically appear as 3-mark or 4-mark questions in the Class 10 CBSE board exam. They may also be part of case-based or application-based questions.

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Last Updated on July 25, 2025 by XAM CONTENT

If you’re preparing for the CBSE Class 10 Maths board exam, mastering word problems on linear equations in two variables is essential. These questions help test your ability to convert real-life situations into mathematical form and solve using various methods like substitution, elimination, and cross multiplication.

In this post, we present a concise introduction, followed by 10 exam-level word problems with detailed solutions to sharpen your application skills.

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Table of Contents

Concept Overview

A pair of linear equations in two variables represents two conditions applied to two unknown quantities (usually x and y). Word problems based on these help you:

Convert statements into algebraic equations
Apply appropriate solving method (substitution/elimination/cross)
Interpret real-world meaning from the solution

Practice Questions

The sum of the ages of a father and his son is 45 years. Five years ago, the father was three times as old as his son. Find their present ages.
A purse contains ₹1 and ₹2 coins totaling ₹37. The total number of coins is 25. Find how many coins of each type are there.
The sum of a two-digit number and the number obtained by reversing the digits is 110. The difference between the digits is 6. Find the number.
A boat covers 30 km upstream and 44 km downstream in 10 hours, and 40 km upstream and 55 km downstream in 13 hours. Find the speed of the boat in still water and speed of the current.
The cost of 5 pens and 7 pencils is ₹50. The cost of 7 pens and 5 pencils is ₹65. Find the cost of each pen and pencil.
The perimeter of a rectangle is 82 m. The length is 9 m more than the breadth. Find the length and breadth.
Ravi bought 2 books and 3 pens for ₹120. He also bought 1 book and 2 pens for ₹70. Find the cost of each book and pen.
A train takes 6 hours to cover a certain distance. If its speed is reduced by 20 km/h, it takes 8 hours. Find the original speed and the distance.

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Answers

Father: 31.25 years, Son: 13.75 years
₹1 coins: 13, ₹2 coins: 12
The number is 82
Boat speed: 8 km/h, Stream speed: 3 km/h
Pen: ₹8.54 approx, Pencil: ₹1.04 approx
Length: 25 m, Breadth: 16 m
Book: ₹30, Pen: ₹20
Speed: 80 km/h, Distance: 480 km

Solutions

1. Age Problem

Let father’s age be \( x \) and son’s age be \( y \).

\[ \begin{cases} x + y = 45 \\ x – 5 = 3(y – 5) \end{cases} \implies \begin{cases} x + y = 45 \\ x – 3y = -10 \end{cases} \]

Adding the two equations:

\[ (x + y) + (x – 3y) = 45 + (-10) \Rightarrow 2x – 2y = 35 \Rightarrow x – y = 17.5 \]

From first equation \( x = 45 – y \), substitute into above:

\[ (45 – y) – y = 17.5 \Rightarrow 45 – 2y = 17.5 \Rightarrow 2y = 27.5 \Rightarrow y = 13.75 \] \[ x = 45 – 13.75 = 31.25 \]

2. Money in Purse

Let number of ₹1 coins be \( x \), number of ₹2 coins be \( y \).

\[ \begin{cases} x + y = 25 \\ x + 2y = 37 \end{cases} \]

Subtracting first from second:

\[ (x + 2y) – (x + y) = 37 – 25 \Rightarrow y = 12 \]

From first equation:

\[ x + 12 = 25 \Rightarrow x = 13 \]

3. Two-Digit Number

Let tens digit = \( x \), units digit = \( y \).

\[ 10x + y + 10y + x = 110 \Rightarrow 11(x + y) = 110 \Rightarrow x + y = 10 \] \[ x – y = 6 \]

Adding equations:

\[ 2x = 16 \Rightarrow x = 8 \] \[ y = 10 – 8 = 2 \] \[ \text{Number} = 10 \times 8 + 2 = 82 \]

4. Boat Speed Problem

Boat speed in still water = \( x \), stream speed = \( y \).

\[ \frac{30}{x – y} + \frac{44}{x + y} = 10, \quad \frac{40}{x – y} + \frac{55}{x + y} = 13 \]

Let \( a = \frac{1}{x – y} \), \( b = \frac{1}{x + y} \).

\[ \begin{cases} 30a + 44b = 10 \\ 40a + 55b = 13 \end{cases} \]

Multiply first by 4 and second by 3:

\[ 120a + 176b = 40 \\ 120a + 165b = 39 \]

Subtracting:

\[ 11b = 1 \Rightarrow b = \frac{1}{11} \]

Substitute \( b \) into first equation:

\[ 30a + 44 \times \frac{1}{11} = 10 \Rightarrow 30a + 4 = 10 \Rightarrow 30a = 6 \Rightarrow a = \frac{1}{5} \]

Therefore,

\[ x – y = 5, \quad x + y = 11 \implies x = 8, \quad y = 3 \]

5. Pen and Pencil Prices

Cost of pen = \( x \), pencil = \( y \).

\[ \begin{cases} 5x + 7y = 50 \\ 7x + 5y = 65 \end{cases} \]

Multiply first by 7, second by 5:

\[ 35x + 49y = 350, \quad 35x + 25y = 325 \]

Subtracting gives:

\[ 24y = 25 \Rightarrow y = \frac{25}{24} \approx 1.04 \]

Substitute value in first equation:

\[ 5x + 7 \times \frac{25}{24} = 50 \Rightarrow 5x = 50 – \frac{175}{24} = \frac{1200 – 175}{24} = \frac{1025}{24} \Rightarrow x = \frac{1025}{120} \approx 8.54 \]

6. Rectangle Dimensions

Breadth = \( x \), length = \( x + 9 \).

\[ 2(x + x + 9) = 82 \Rightarrow 4x + 18 = 82 \Rightarrow 4x = 64 \Rightarrow x = 16 \] \[ \text{Length} = 16 + 9 = 25 \]

7. Buying Books and Pens

Cost of book = \( x \), pen = \( y \).

\[ \begin{cases} 2x + 3y = 120 \\ x + 2y = 70 \end{cases} \]

Multiply second by 2:

\[ 2x + 4y = 140 \]

Subtract first equation:

\[ (2x + 4y) – (2x + 3y) = 140 – 120 \Rightarrow y = 20 \] \[ x + 2 \times 20 = 70 \Rightarrow x = 30 \]

8. Train Speed Problem

Speed = \( x \), distance = \( d \).

\[ \frac{d}{x} = 6, \quad \frac{d}{x – 20} = 8 \] \[ \Rightarrow d = 6x, \quad d = 8(x – 20) \] \[ 6x = 8x – 160 \Rightarrow 2x = 160 \Rightarrow x = 80 \] \[ d = 6 \times 80 = 480 \]

Frequently Asked Questions (FAQs) on Word Problems on Pair of Linear Equations in Two Variables

Q1: What are the most common methods to solve a pair of linear equations in two variables?

A1: The most common methods are:
Substitution method
Elimination method
Cross multiplication method
Graphical method
Each method has its advantages, and the choice depends on the structure of the equations.

Q2: How do I identify variables in a word problem involving linear equations?

A2: Read the problem carefully and look for two unknown quantities that are related through given conditions. Assign variables like x and y to these unknowns and form equations based on the relationships provided.

Q3: Which type of word problems are usually asked in CBSE Class 10 board exams?

A3: CBSE commonly asks word problems based on:
Age-related problems
Number problems
Speed, distance and time
Mixture and solution
Profit and cost
These problems test your ability to translate real-life scenarios into equations and solve them.

Q4: What is the most common mistake students make while solving linear equation word problems?

A4: The most common mistakes are:
Incorrect variable assignment
Misinterpreting the condition while forming equations
Arithmetic errors during elimination/substitution
Always double-check your equations and steps.

Q5: How many marks do linear equations word problems usually carry in Class 10 CBSE exams?

A5: Word problems from this chapter typically appear as 3-mark or 4-mark questions in the Class 10 CBSE board exam. They may also be part of case-based or application-based questions.