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Fractions Simplification

Introduction

Fraction simplification problems test your ability to handle ratios quickly. These are common in exams and become easy once you know how to use reciprocals and cancel numbers step by step.

Pattern: Fractions Simplification

Pattern

Key idea: For division, multiply by the reciprocal. Then cancel common factors and simplify step by step.

Step-by-Step Example

Question

Simplify: (3/4 ÷ 2/5) × (5/6)

Options:

  • A) 25/16
  • B) 16/25
  • C) 15/18
  • D) 75/48

Solution

  1. Step 1: Convert division into multiplication

    Change division into multiplication by reciprocal. (3/4 ÷ 2/5) = (3/4 × 5/2).

  2. Step 2: Include the third fraction

    Now include the third fraction: (3/4 × 5/2 × 5/6).

  3. Step 3: Multiply numerators and denominators

    Numerator = 3 × 5 × 5 = 75, Denominator = 4 × 2 × 6 = 48.

  4. Step 4: Simplify the resulting fraction

    Simplify 75/48 by dividing both by 3. 75 ÷ 3 = 25, 48 ÷ 3 = 16 → 25/16.

  5. Final Answer:

    25/16 → Option A.

  6. Quick Check:

    As decimals → (3/4 ÷ 0.4) × 0.833 ≈ 1.5625, and 25/16 = 1.5625 ✅

Quick Variations

1. Addition/Subtraction → use common denominator.

2. Mixed numbers → convert to improper fractions.

3. Division with whole numbers → write whole numbers as n/1.

Trick to Always Use

  • Step 1 → Turn division into multiplication by reciprocal.
  • Step 2 → Cancel common numbers before multiplying.
  • Step 3 → Simplify your fraction at the end.

Summary

Summary

  • Convert division to multiplication using reciprocals.
  • Cancel common factors early to reduce calculation effort.
  • Use common denominators only for addition or subtraction.
  • Simplify the final fraction to the lowest terms.

Example to remember:
(3/4 ÷ 2/5) × (5/6) = 25/16

Practice

(1/5)
1. Simplify: (2/3 ÷ 4/9)
easy
A. 3/2
B. 2/9
C. 9/8
D. 1/6

Solution

  1. Step 1: Change division into multiplication by reciprocal

    Change division to multiplication by reciprocal: (2/3 ÷ 4/9) = (2/3 × 9/4).

  2. Step 2: Multiply numerators and denominators

    Multiply numerators and denominators: numerator = 2 × 9 = 18, denominator = 3 × 4 = 12.

  3. Step 3: Simplify the fraction

    Simplify 18/12 = 3/2.

  4. Final Answer:

    3/2 → Option A.

  5. Quick Check:

    As decimals: 0.6667 ÷ 0.4444 ≈ 1.5 ✅
Hint: Flip the second fraction and multiply.
Common Mistakes: Multiplying denominators without flipping the second fraction.
2. Simplify: (5/8 × 16/15)
easy
A. 2/3
B. 3/4
C. 4/5
D. 8/15

Solution

  1. Step 1: Cancel common factors before multiplying

    Cancel before multiplying: 16/8 = 2, so expression becomes (5 × 2) / 15.

  2. Step 2: Compute numerator and denominator

    Compute numerator and denominator: 10/15.

  3. Step 3: Simplify the fraction

    Simplify 10/15 = 2/3.

  4. Final Answer:

    2/3 → Option A.

  5. Quick Check:

    0.625 × 1.0667 ≈ 0.6667 ✅
Hint: Always cancel common factors first to keep numbers small.
Common Mistakes: Multiplying fully first and forgetting to reduce early.
3. Simplify: (7/9 ÷ 14/27)
easy
A. 2/3
B. 3/2
C. 9/14
D. 27/126

Solution

  1. Step 1: Flip the divisor and multiply

    Flip and multiply: (7/9 ÷ 14/27) = (7/9 × 27/14).

  2. Step 2: Cancel common factors before multiplying

    Cancel 27/9 = 3 → expression becomes (7 × 3) / 14 = 21/14.

  3. Step 3: Simplify the fraction

    Simplify 21/14 = 3/2.

  4. Final Answer:

    3/2 → Option B.

  5. Quick Check:

    0.7778 ÷ 0.5185 ≈ 1.5 ✅
Hint: Look for easy cancellations like 27/9 before multiplying.
Common Mistakes: Multiplying denominators without cancelling common factors first.
4. Simplify: (3/4 ÷ 9/8)
medium
A. 3/4
B. 4/3
C. 2/3
D. 3/2

Solution

  1. Step 1: Flip the divisor and convert to multiplication

    Flip and multiply: (3/4 ÷ 9/8) = (3/4 × 8/9).

  2. Step 2: Cancel factors before multiplying

    Cancel 8/4 = 2 → expression becomes (3 × 2) / 9 = 6/9.

  3. Step 3: Simplify the fraction

    Simplify 6/9 = 2/3.

  4. Final Answer:

    2/3 → Option C.

  5. Quick Check:

    0.75 ÷ 1.125 = 0.6667 ✅
Hint: Flip the divisor and cancel before multiplying.
Common Mistakes: Forgetting to flip the second fraction or not cancelling 8/4 early.
5. Simplify: (7/8 ÷ 7/16) × (3/5)
medium
A. 3/8
B. 5/6
C. 7/16
D. 6/5

Solution

  1. Step 1: Divide by multiplying with reciprocal

    First divide: (7/8 ÷ 7/16) = (7/8 × 16/7).

  2. Step 2: Cancel identical factors

    Cancel the 7s → becomes 16/8 = 2.

  3. Step 3: Multiply by the third fraction

    Now multiply by 3/5 → 2 × 3/5 = 6/5.

  4. Final Answer:

    6/5 → Option D.

  5. Quick Check:

    (7/8 ÷ 7/16) = 2, and 2 × 0.6 = 1.2 = 6/5 ✅
Hint: Cancel identical factors (7) early to simplify the division.
Common Mistakes: Multiplying incorrectly before cancelling or forgetting to flip the divisor.

Mock Test

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