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Fractions & Rational Equations

Introduction

Fractions और rational equations algebraic problem-solving का एक important हिस्सा हैं, जहाँ variable denominator में आता है। ऐसे problems में आमतौर पर simplifying, cross-multiplying और ऐसे values निकालना शामिल होता है जो equation को true बनाते हों-और denominator को zero न बनाते हों।

यह pattern इसलिए ज़रूरी है क्योंकि यह ratios, rates और inverse relationships वाली equations समझने की foundation बनाता है, जो aptitude tests में अक्सर पूछी जाती हैं।

Pattern: Fractions & Rational Equations

Pattern

Key idea: Denominators हटाने के लिए पूरी equation को Least Common Denominator (LCD) से multiply करें, फिर बनी linear या quadratic equation को solve करें।

Simple form में, अगर आपके पास 1/x + 1/y = 1/6 हो, तो denominators हटाने के लिए दोनों sides को xy से multiply करें।

Step-by-Step Example

Question

Solve: 1/x + 1/4 = 1/2

Solution

  1. Step 1: Denominators पहचानें

    Denominators हैं x, 4 और 2. इनका LCM = 4x.

  2. Step 2: Fractions clear करने के लिए multiply करें

    पूरी equation को 4x से multiply करें:
    4x(1/x) + 4x(1/4) = 4x(1/2)

  3. Step 3: Equation simplify करें

    Simplify → 4 + x = 2x.

  4. Step 4: Rearrange करके solve करें

    2x - x = 4 → x = 4.

  5. Final Answer:

    4

  6. Quick Check:

    1/4 + 1/4 = 1/2 ✅

Quick Variations

1. Reciprocals वाले problems (जैसे 1/x + 1/y = 1/6).

2. Time-Work या Speed-Distance problems जो fractional rates में लिखे जाते हैं।

3. “Together” या “inversely proportional” वाली word problems।

4. Fractions जिनमें numerator और denominator दोनों में variables हों।

Trick to Always Use

  • Step 1: Least Common Denominator (LCD) निकालें।
  • Step 2: LCD से multiply करके denominators को clear करें।
  • Step 3: बनी हुई equation को normally simplify और solve करें।
  • Step 4: हमेशा check करें कि आपका answer denominator को zero न बनाए - ऐसे values reject करें।

Summary

Summary

Fractions & Rational Equations pattern में:

  • LCD से multiply करके denominators हटाएँ।
  • Step-by-step simplify करके linear या quadratic form में solve करें।
  • Zero-denominator values को हमेशा reject करना याद रखें।
  • Final answer को original equation में substitute करके verify करें।

Practice

(1/5)
1. Solve: 1/x + 1/3 = 1/2
easy
A. x = 6
B. x = 3
C. x = 5
D. x = 4

Solution

  1. Step 1: Identify the equation

    1/x + 1/3 = 1/2.

  2. Step 2: Isolate 1/x

    1/x = 1/2 - 1/3 = (3 - 2)/6 = 1/6.

  3. Step 3: Invert to find x

    Invert both sides → x = 6.

  4. Final Answer:

    6 → Option A.

  5. Quick Check:

    1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 ✅
Hint: Compute 1/x as RHS - the other fraction, then invert.
Common Mistakes: Subtracting fractions without using common denominators.
2. Solve: 1/(x + 2) = 1/4
easy
A. x = 1
B. x = 2
C. x = 3
D. x = 4

Solution

  1. Step 1: Identify the equation

    1/(x + 2) = 1/4.

  2. Step 2: Clear the denominator

    Cross-multiply → x + 2 = 4.

  3. Step 3: Solve for x

    x = 4 - 2 = 2.

  4. Final Answer:

    2 → Option B.

  5. Quick Check:

    1/(2 + 2) = 1/4 ✅
Hint: Cross-multiply directly when a simple fraction equals another fraction.
Common Mistakes: Performing unnecessary operations instead of simple cross-multiplication.
3. Solve: 1/(x + 1) + 1/2 = 1
easy
A. x = 0
B. x = 2
C. x = 1
D. x = 3

Solution

  1. Step 1: Identify the equation

    1/(x + 1) + 1/2 = 1.

  2. Step 2: Isolate the fractional term

    1/(x + 1) = 1 - 1/2 = 1/2.

  3. Step 3: Solve for x

    x + 1 = 2 → x = 1.

  4. Final Answer:

    1 → Option C.

  5. Quick Check:

    1/(1 + 1) + 1/2 = 1/2 + 1/2 = 1 ✅
Hint: Move the known fraction to the RHS before inverting the remaining fraction.
Common Mistakes: Adding instead of subtracting the known fraction from 1.
4. Solve: 2/x + 3/4 = 1
medium
A. x = 8
B. x = 6
C. x = 4
D. x = 3

Solution

  1. Step 1: Identify the equation

    2/x + 3/4 = 1.

  2. Step 2: Isolate 2/x

    2/x = 1 - 3/4 = 1/4.

  3. Step 3: Invert/cross-multiply to solve

    2 = x × 1/4 → x = 2 × 4 = 8.

  4. Final Answer:

    8 → Option A.

  5. Quick Check:

    2/8 + 3/4 = 1/4 + 3/4 = 1 ✅
Hint: Isolate the term with x, then invert or cross-multiply to solve.
Common Mistakes: Forgetting to include all terms when clearing denominators or inverting incorrectly.
5. Solve: 1/(x + 2) + 1/3 = 1/6
medium
A. x = -5
B. x = -6
C. x = -7
D. x = -8

Solution

  1. Step 1: Identify the equation

    1/(x + 2) + 1/3 = 1/6.

  2. Step 2: Isolate 1/(x + 2)

    1/(x + 2) = 1/6 - 1/3 = 1/6 - 2/6 = -1/6.

  3. Step 3: Invert to find x

    x + 2 = -6 → x = -8.

  4. Final Answer:

    -8 → Option D.

  5. Quick Check:

    1/(-8 + 2) + 1/3 = 1/(-6) + 1/3 = -1/6 + 1/3 = -1/6 + 2/6 = 1/6 ✅
Hint: Compute RHS - known fraction carefully (watch signs), then invert.
Common Mistakes: Sign errors when subtracting fractions or when inverting a negative fraction.

Mock Test

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