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Fractions & Rational Equations

Introduction

Fractions and rational equations are an essential part of algebraic problem-solving where the variable appears in the denominator. These problems often involve simplifying, cross-multiplying, and finding values that make the equation true without making the denominator zero.

This pattern is important because it builds your foundation for understanding equations involving ratios, rates, and inverse relationships, which frequently appear in aptitude tests.

Pattern: Fractions & Rational Equations

Pattern

The key idea: Multiply through by the Least Common Denominator (LCD) to eliminate denominators, then solve the resulting linear or quadratic equation.

In simple form, if you have 1/x + 1/y = 1/6 multiply both sides by xy to get rid of the denominators.

Step-by-Step Example

Question

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

Solution

  1. Step 1: Identify denominators

    Denominators are x, 4, and 2. LCM = 4x.

  2. Step 2: Multiply to clear fractions

    Multiply the entire equation by 4x:
    4x(1/x) + 4x(1/4) = 4x(1/2)

  3. Step 3: Simplify the equation

    Simplify → 4 + x = 2x.

  4. Step 4: Rearrange and solve

    2x - x = 4 → x = 4.

  5. Final Answer:

    4

  6. Quick Check:

    1/4 + 1/4 = 1/2 ✅

Quick Variations

1. Problems involving reciprocals (e.g., 1/x + 1/y = 1/6).

2. Time-Work or Speed-Distance problems represented as fractional rates.

3. Word problems with “together” or “inversely proportional” relationships.

4. Fractions with variables in both numerator and denominator.

Trick to Always Use

  • Step 1: Find the Least Common Denominator (LCD).
  • Step 2: Multiply through by the LCD to eliminate fractions.
  • Step 3: Simplify and solve the resulting equation normally.
  • Step 4: Always check for values that make the denominator zero (reject them).

Summary

Summary

In the Fractions & Rational Equations pattern:

  • Clear denominators by multiplying with the LCD.
  • Simplify step by step to get a linear or quadratic equation.
  • Always check for invalid (zero-denominator) values.
  • Verification is key - substitute back into the original equation to confirm.

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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