0
0

Word Problems on Linear Equations

Introduction

Word problems involving linear equations आपकी ability को test करते हैं कि आप real-world situations को math equations में कैसे बदलते हैं। ऐसे problems में आमतौर पर ages, speed-distance-time, work, या money जैसे relations होते हैं जिन्हें linear equations से model किया जा सकता है।

यह pattern महत्वपूर्ण है क्योंकि यह आपकी logical सोचना-समझना और algebraic reasoning को रोज़मर्रा की situations में apply करना सिखाता है।

Pattern: Word Problems on Linear Equations

Pattern

Key concept: दिए गए word statement को equation में convert करें और unknown variable का value निकालें।

Common forms शामिल हैं:

  • “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

दो numbers का sum 45 है। एक number दूसरे का double है। Numbers निकालें।

Solution

  1. Step 1: Smaller number define करें

    Smaller number = x. तो दूसरा number = 2x.

  2. Step 2: Sum से equation बनाएं

    According to question, sum = 45
    ⇒ x + 2x = 45

  3. Step 3: x के लिए solve करें

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

  4. Step 4: दूसरा number निकालें

    दूसरे number = 2x = 2 × 15 = 30.

  5. Final Answer:

    Numbers हैं 15 और 30

  6. Quick Check:

    15 + 30 = 45 ✅ और 30, 15 का double है ✅

Quick Variations

1. Age-related problems (जैसे पिता की age बेटे की age का double).

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

3. Money या mixture problems (coins, total value, etc.).

4. Consecutive numbers problems (जैसे n और n+1, या even numbers n और n+2).

Trick to Always Use

  • Step 1: Variables clearly define करें - usually “Let the unknown be x.”
  • Step 2: Words को equations में translate करें (“sum,” “difference,” “twice,” आदि).
  • Step 3: Solve करके answer को वापस substitute करके verify करें।

Summary

Summary

Word Problems on Linear Equations pattern में:

  • Problem को ध्यान से पढ़कर algebraic equation में बदलें।
  • “Sum,” “difference,” “multiple” जैसी relationships का सही use करें।
  • अपने answer को quick substitution से verify करना न भूलें।

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