Answered

  1. $\frac{13}{14} \div \frac{1}{7}$
  2. $\frac{3}{8} \div \frac{3}{8}$
  3. $\frac{3}{4} \div \frac{3}{8}$
  4. $16 \div \frac{4}{15}$
  5. $-\frac{8}{9} \div 3$
  6. $\frac{7}{9} \div(-2)$
  7. $\frac{7}{8} \div \frac{3}{4}$
  8. $\frac{9}{10} \div \frac{3}{5}$
  9. $-2 \frac{1}{2} \div \frac{5}{8}$

Verified Answer :

  • Convert all mixed numbers to improper fractions and express integers as fractions with a denominator of 1.
  • Divide by a fraction by multiplying the dividend by the reciprocal of the divisor.
  • Simplify each resulting fraction or mixed number.
  • 8: \(\frac{13}{2}\); 9: \(1\); 10: \(2\); 11: \(60\); 12: \(-\frac{8}{27}\); 13: \(-\frac{7}{18}\); 14: \(\frac{7}{6}\); 15: \(\frac{3}{2}\); 16: \(-4\).

Explanation

  1. General Approach to Division of Fractions

To divide fractions or mixed numbers, we follow a systematic process:

  1. Convert any mixed numbers into improper fractions. For example, \(-2 \frac{1}{2} = -\frac{2 \times 2 + 1}{2} = -\frac{5}{2}\).
  2. Rewrite the division problem as multiplication by the reciprocal of the divisor. The rule is \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\).
  3. Simplify the resulting product by canceling common factors in the numerator and denominator to ensure the final result is in its simplest form.
  4. Apply the sign convention: the quotient of two numbers with the same sign is positive, and the quotient of two numbers with different signs is negative.
  1. Calculations for Expressions 8-16

We evaluate each expression individually:

  1. \(\frac{13}{14} \div \frac{1}{7} = \frac{13}{14} \times \frac{7}{1} = \frac{13 \times 7}{14 \times 1} = \frac{13}{2} = 6.5\)
  1. \(\frac{3}{8} \div \frac{3}{8} = \frac{3}{8} \times \frac{8}{3} = \frac{3 \times 8}{8 \times 3} = 1\)
  1. \(\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \times \frac{8}{3} = \frac{3 \times 8}{4 \times 3} = \frac{8}{4} = 2\)
  1. \(16 \div \frac{4}{15} = \frac{16}{1} \times \frac{15}{4} = \frac{16 \times 15}{4} = 4 \times 15 = 60\)
  1. \(-\frac{8}{9} \div 3 = -\frac{8}{9} \times \frac{1}{3} = -\frac{8 \times 1}{9 \times 3} = -\frac{8}{27}\)
  1. \(\frac{7}{9} \div (-2) = \frac{7}{9} \times (-\frac{1}{2}) = -\frac{7 \times 1}{9 \times 2} = -\frac{7}{18}\)
  1. \(\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3} = \frac{7 \times 4}{8 \times 3} = \frac{7}{2 \times 3} = \frac{7}{6} = 1 \frac{1}{6}\)
  1. \(\frac{9}{10} \div \frac{3}{5} = \frac{9}{10} \times \frac{5}{3} = \frac{9 \times 5}{10 \times 3} = \frac{3 \times 1}{2 \times 1} = \frac{3}{2} = 1.5\)
  1. \(-2 \frac{1}{2} \div \frac{5}{8} = -\frac{5}{2} \div \frac{5}{8} = -\frac{5}{2} \times \frac{8}{5} = -\frac{8}{2} = -4\)
  1. Final Summary of Results

We have successfully computed the results for each of the nine expressions provided. Each calculation relied on multiplying by the reciprocal and simplifying the resulting fractions.
The final answers are:

  1. \(\frac{13}{2} = 6.5\)
  2. \(1\)
  3. \(2\)
  4. \(60\)
  5. \(-\frac{8}{27}\)
  6. \(-\frac{7}{18}\)
  7. \(\frac{7}{6} = 1\frac{1}{6}\)
  8. \(\frac{3}{2} = 1.5\)
  9. \(-4\)

Examples

In everyday life, dividing fractions is essential when adjusting recipes. For instance, if a recipe calls for \(3/4\) of a cup of flour and you only have enough ingredients to make \(3/8\) of the recipe, you might need to perform division to scale your portions correctly. Understanding how to divide by a fraction—which is the same as multiplying by its reciprocal—ensures that you keep the proportions of your ingredients balanced, resulting in a perfectly baked treat!

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