Multiplying Mixed Fractions A Step-by-Step Guide

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In the realm of mathematics, mixed fractions often pose a challenge for students. However, with a clear understanding of the underlying principles and a systematic approach, multiplying mixed fractions can become a straightforward task. This comprehensive guide will delve into the intricacies of multiplying mixed fractions, providing step-by-step explanations, illustrative examples, and valuable insights to enhance your understanding and problem-solving skills.

Understanding Mixed Fractions

Before we embark on the journey of multiplying mixed fractions, it is crucial to grasp the fundamental concept of what mixed fractions are. A mixed fraction is a combination of a whole number and a proper fraction. For instance, 2562 \frac{5}{6} is a mixed fraction, where 2 is the whole number and 56\frac{5}{6} is the proper fraction. The whole number represents the number of complete units, while the proper fraction represents a part of a unit.

Converting Mixed Fractions to Improper Fractions

The cornerstone of multiplying mixed fractions lies in converting them into improper fractions. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert a mixed fraction to an improper fraction, we employ a simple two-step process:

  1. Multiply the whole number by the denominator of the fractional part.
  2. Add the product obtained in step 1 to the numerator of the fractional part. Keep the same denominator.

Let's illustrate this with an example. Consider the mixed fraction 2562 \frac{5}{6}.

  1. Multiply the whole number (2) by the denominator (6): 2 × 6 = 12
  2. Add the product (12) to the numerator (5): 12 + 5 = 17

Therefore, the improper fraction equivalent of 2562 \frac{5}{6} is 176\frac{17}{6}. The denominator remains the same.

Multiplying Mixed Fractions: A Step-by-Step Approach

Now that we have mastered the art of converting mixed fractions to improper fractions, we can delve into the process of multiplying them. The procedure involves the following steps:

  1. Convert mixed fractions to improper fractions: As discussed earlier, the first step is to convert all mixed fractions involved in the multiplication into their corresponding improper fractions. This transforms the problem into a straightforward multiplication of fractions.
  2. Multiply the numerators: Multiply the numerators of the improper fractions. The product obtained will be the numerator of the resulting fraction.
  3. Multiply the denominators: Multiply the denominators of the improper fractions. The product obtained will be the denominator of the resulting fraction.
  4. Simplify the resulting fraction: If the resulting fraction is an improper fraction, convert it back to a mixed fraction. Additionally, simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF) if possible. This ensures the fraction is expressed in its simplest form.

Let's illustrate this with a practical example. We will solve the problem 256×23102 \frac{5}{6} \times 2 \frac{3}{10}.

  1. Convert to improper fractions:
    • 256=(2×6)+56=1762 \frac{5}{6} = \frac{(2 \times 6) + 5}{6} = \frac{17}{6}
    • 2310=(2×10)+310=23102 \frac{3}{10} = \frac{(2 \times 10) + 3}{10} = \frac{23}{10}
  2. Multiply the numerators: 17 × 23 = 391
  3. Multiply the denominators: 6 × 10 = 60
  4. Simplify the resulting fraction: 39160\frac{391}{60}. To convert this improper fraction to a mixed fraction, we divide 391 by 60. The quotient is 6, and the remainder is 31. Therefore, the mixed fraction equivalent is 631606 \frac{31}{60}. This fraction is already in its simplest form, as 31 and 60 have no common factors other than 1.

Therefore, 256×2310=631602 \frac{5}{6} \times 2 \frac{3}{10} = 6 \frac{31}{60}.

Illustrative Examples

To solidify your understanding, let's work through a few more examples:

Example 1

Solve: 237×2482 \frac{3}{7} \times 2 \frac{4}{8}

  1. Convert to improper fractions:
    • 237=(2×7)+37=1772 \frac{3}{7} = \frac{(2 \times 7) + 3}{7} = \frac{17}{7}
    • 248=(2×8)+48=2082 \frac{4}{8} = \frac{(2 \times 8) + 4}{8} = \frac{20}{8}
  2. Multiply the numerators: 17 × 20 = 340
  3. Multiply the denominators: 7 × 8 = 56
  4. Simplify the resulting fraction: 34056\frac{340}{56}. First, we can simplify this fraction by dividing both the numerator and denominator by their GCF, which is 4. This gives us 8514\frac{85}{14}. Now, we convert this improper fraction to a mixed fraction by dividing 85 by 14. The quotient is 6, and the remainder is 1. Therefore, the mixed fraction equivalent is 61146 \frac{1}{14}.

Therefore, 237×248=61142 \frac{3}{7} \times 2 \frac{4}{8} = 6 \frac{1}{14}.

Example 2

Solve: 238×2692 \frac{3}{8} \times 2 \frac{6}{9}

  1. Convert to improper fractions:
    • 238=(2×8)+38=1982 \frac{3}{8} = \frac{(2 \times 8) + 3}{8} = \frac{19}{8}
    • 269=(2×9)+69=2492 \frac{6}{9} = \frac{(2 \times 9) + 6}{9} = \frac{24}{9}
  2. Multiply the numerators: 19 × 24 = 456
  3. Multiply the denominators: 8 × 9 = 72
  4. Simplify the resulting fraction: 45672\frac{456}{72}. First, we can simplify this fraction by dividing both the numerator and denominator by their GCF, which is 24. This gives us 193\frac{19}{3}. Now, we convert this improper fraction to a mixed fraction by dividing 19 by 3. The quotient is 6, and the remainder is 1. Therefore, the mixed fraction equivalent is 6136 \frac{1}{3}.

Therefore, 238×269=6132 \frac{3}{8} \times 2 \frac{6}{9} = 6 \frac{1}{3}.

Example 3

Solve: 525×2365 \frac{2}{5} \times 2 \frac{3}{6}

  1. Convert to improper fractions:
    • 525=(5×5)+25=2755 \frac{2}{5} = \frac{(5 \times 5) + 2}{5} = \frac{27}{5}
    • 236=(2×6)+36=1562 \frac{3}{6} = \frac{(2 \times 6) + 3}{6} = \frac{15}{6}
  2. Multiply the numerators: 27 × 15 = 405
  3. Multiply the denominators: 5 × 6 = 30
  4. Simplify the resulting fraction: 40530\frac{405}{30}. First, we can simplify this fraction by dividing both the numerator and denominator by their GCF, which is 15. This gives us 272\frac{27}{2}. Now, we convert this improper fraction to a mixed fraction by dividing 27 by 2. The quotient is 13, and the remainder is 1. Therefore, the mixed fraction equivalent is 131213 \frac{1}{2}.

Therefore, 525×236=13125 \frac{2}{5} \times 2 \frac{3}{6} = 13 \frac{1}{2}.

Key Takeaways

  • Mastering the conversion of mixed fractions to improper fractions is paramount. This is the foundational step in multiplying mixed fractions.
  • Multiply numerators and denominators separately. This simplifies the multiplication process.
  • Always simplify the resulting fraction. Express the answer in its simplest form, either as a proper fraction or a mixed fraction.
  • Practice regularly to solidify your understanding. The more you practice, the more confident you will become in multiplying mixed fractions.

Conclusion

Multiplying mixed fractions is a fundamental skill in mathematics. By understanding the underlying concepts and following the systematic approach outlined in this guide, you can confidently tackle any multiplication problem involving mixed fractions. Remember to convert mixed fractions to improper fractions, multiply numerators and denominators, and simplify the resulting fraction. With consistent practice, you will master this skill and unlock new mathematical horizons.