0
0

Word Problems on Linear Equations

Introduction

Word problems involving linear equations test your ability to translate real-world situations into mathematical expressions. These problems typically involve relationships such as ages, speed-distance-time, work, or money where the conditions can be modeled as linear equations.

This pattern is important because it strengthens your logical thinking and helps you apply algebraic reasoning to everyday situations.

Pattern: Word Problems on Linear Equations

Pattern

The key concept: Convert the given word statement into an equation and solve for the unknown variable.

Common forms include:

  • “One number is twice another” → x = 2y
  • “Sum of two numbers is 20” → x + y = 20
  • “Difference between a number and 5 is 7” → x - 5 = 7

Step-by-Step Example

Question

The sum of two numbers is 45. One number is twice the other. Find the numbers.

Solution

  1. Step 1: Define the smaller number

    Let the smaller number be x. Then the other number = 2x.

  2. Step 2: Form the equation from the sum

    According to the question, their sum is 45.
    ⇒ x + 2x = 45

  3. Step 3: Solve for x

    Simplify → 3x = 45 ⇒ x = 45 ÷ 3 = 15.

  4. Step 4: Find the other number

    The other number = 2x = 2 × 15 = 30.

  5. Final Answer:

    The two numbers are 15 and 30

  6. Quick Check:

    15 + 30 = 45 ✅ and 30 is twice 15 ✅

Quick Variations

1. Age-related problems (e.g., father’s age is double the son’s age).

2. Speed-distance problems (e.g., distance = speed × time).

3. Money or mixture problems (e.g., sum of coins, total value, etc.).

4. Consecutive numbers problems (e.g., n and n+1, or even numbers n and n+2).

Trick to Always Use

  • Step 1: Define variables clearly - usually start with “Let the unknown be x.”
  • Step 2: Translate words to equations using relationships (“sum,” “difference,” “twice,” etc.).
  • Step 3: Solve and verify by substituting the answer back into the question.

Summary

Summary

In the Word Problems on Linear Equations pattern:

  • Translate the problem into an algebraic equation carefully.
  • Keep track of relationships like “sum,” “difference,” or “multiple.”
  • Always verify the answer with a quick substitution check.

Practice

(1/5)
1. The sum of two numbers is 36. One number is twice the other. Find the numbers.
easy
A. 12 and 24
B. 10 and 26
C. 9 and 27
D. 14 and 22

Solution

  1. Step 1: Represent the numbers

    Let the smaller number be x. Then the other number = 2x.

  2. Step 2: Form the equation

    Using the sum: x + 2x = 36 → 3x = 36.

  3. Step 3: Solve for x

    x = 36 ÷ 3 = 12. The other number = 2x = 24.

  4. Final Answer:

    12 and 24 → Option A.

  5. Quick Check:

    12 + 24 = 36 and 24 = 2 × 12 ✅
Hint: Translate 'one is twice the other' as x and 2x, then apply the sum.
Common Mistakes: Taking the larger number as x, or arithmetic slips when dividing.
2. A number exceeds its half by 18. Find the number.
easy
A. 24
B. 36
C. 45
D. 40

Solution

  1. Step 1: Define the number

    Let the number be x. Its half is x/2.

  2. Step 2: Form the equation

    Given: x - x/2 = 18.

  3. Step 3: Solve the equation

    x/2 = 18 → x = 18 × 2 = 36.

  4. Final Answer:

    36 → Option B.

  5. Quick Check:

    Half of 36 is 18, and 36 - 18 = 18 ✅
Hint: Write 'exceeds its half' as x - x/2 and solve.
Common Mistakes: Incorrectly writing x + x/2 instead of x - x/2.
3. The sum of two consecutive even numbers is 54. Find the numbers.
easy
A. 22 and 24
B. 24 and 26
C. 26 and 28
D. 28 and 30

Solution

  1. Step 1: Represent the numbers

    Let the smaller even number be x. The next even number = x + 2.

  2. Step 2: Form the equation

    x + (x + 2) = 54 → 2x + 2 = 54.

  3. Step 3: Solve

    2x = 52 → x = 26. The numbers are 26 and 28.

  4. Final Answer:

    26 and 28 → Option C.

  5. Quick Check:

    26 + 28 = 54 ✅
Hint: For consecutive even numbers use x and x+2.
Common Mistakes: Using x and x+1 (odd consecutive), giving wrong results.
4. The sum of two numbers is 45. One number is 9 more than the other. Find the numbers.
medium
A. 18 and 27
B. 17 and 28
C. 20 and 25
D. 16 and 29

Solution

  1. Step 1: Represent the numbers

    Let the smaller number be x. Then the other = x + 9.

  2. Step 2: Form the equation

    x + (x + 9) = 45 → 2x + 9 = 45.

  3. Step 3: Solve

    2x = 36 → x = 18. Other number = 27.

  4. Final Answer:

    18 and 27 → Option A.

  5. Quick Check:

    18 + 27 = 45 and 27 = 18 + 9 ✅
Hint: Translate 'x is 9 more' as (x + 9) and apply the sum.
Common Mistakes: Forgetting which number is larger or misplacing the +9.
5. A man's age is three times his son's age. After 12 years, the father's age will be twice the son's age. Find their present ages.
medium
A. Father 45, Son 15
B. Father 42, Son 14
C. Father 48, Son 16
D. Father 36, Son 12

Solution

  1. Step 1: Represent current ages

    Let son's present age be x. Then father's present age = 3x.

  2. Step 2: Form the future age equation

    After 12 years: son = x + 12, father = 3x + 12. Given: 3x + 12 = 2(x + 12).

  3. Step 3: Solve

    3x + 12 = 2x + 24 → x = 12. Son = 12, Father = 36.

  4. Final Answer:

    Father = 36, Son = 12 → Option D.

  5. Quick Check:

    After 12 years → Father = 48, Son = 24 and 48 = 2 × 24 ✅
Hint: Use ratios (3x) and add 'after years' equally before forming the relation.
Common Mistakes: Not adding 12 to both ages or sign mistakes when expanding.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes