Dividing fractions by fractions Knowledge-Based Study

Study notes

1. What Does It Mean to Divide by a Fraction?

   Division fundamentally answers the question: How many groups of a certain size can be made from a total amount?

   For instance, in whole-number division, $6 \div 2 = 3$ asks how many groups of 2 are in 6. The same logic applies directly to fractions:

   Consider the problem: $$\frac{3}{4} \div \frac{1}{8}$$

   This asks: How many $\frac{1}{8}$ pieces fit into $\frac{3}{4}$?

   • One whole contains eight eighths ($\frac{8}{8}$).
   • Three-quarters of a whole is equivalent to six eighths ($\frac{6}{8}$).
   • Therefore, exactly 6 pieces of size $\frac{1}{8}$ fit into $\frac{3}{4}$, meaning $\frac{3}{4} \div \frac{1}{8} = 6$.

   The Multiplicative Inverse (Reciprocal)

   Two numbers are reciprocals of each other if their product equals 1. To find the reciprocal of a non-zero fraction, invert the numerator and the denominator:

   • The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$, because $\frac{2}{5} \times \frac{5}{2} = 1$.
   • The reciprocal of $\frac{1}{7}$ is $\frac{7}{1} = 7$.

   The Standard Rule

   Dividing by any non-zero number produces the exact same result as multiplying by its reciprocal. For any fractions where $b \neq 0$, $c \neq 0$, and $d \neq 0$:

   $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

   This method is commonly remembered through three simple steps:

   1. Keep the first fraction (dividend) exactly as it is.
   2. Change the division symbol to multiplication.
   3. Flip the second fraction (divisor) to its reciprocal.

   Calculating and Simplifying

   Let us evaluate: $$\frac{5}{6} \div \frac{10}{9}$$

   Step 1: Rewrite as multiplication

   Keep the first fraction, change to multiplication, and take the reciprocal of the second fraction:

   $$\frac{5}{6} \times \frac{9}{10}$$

   Step 2: Simplify common factors before multiplying

   Identify common factors between numerators and denominators to keep calculations simple:

   • 5 in the numerator and 10 in the denominator share a common factor of 5: $5 \div 5 = 1$ and $10 \div 5 = 2$.
   • 9 in the numerator and 6 in the denominator share a common factor of 3: $9 \div 3 = 3$ and $6 \div 3 = 2$.

   Rewrite the product with simplified terms:

   $$\frac{1}{2} \times \frac{3}{2}$$

   Step 3: Multiply across

   $$\frac{1 \times 3}{2 \times 2} = \frac{3}{4}$$

   Thus, $\frac{5}{6} \div \frac{10}{9} = \frac{3}{4}$.

   Handling Mixed Numbers

   Before applying the reciprocal rule, any mixed numbers must be converted into improper fractions.

   Example Problem

   Evaluate: $$2\frac{1}{4} \div 1\frac{1}{2}$$

   1. Convert each mixed number to an improper fraction:
      $2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$
      $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$
   2. Rewrite the division problem:
      $$\frac{9}{4} \div \frac{3}{2}$$
   3. Multiply by the reciprocal of the divisor:
      $$\frac{9}{4} \times \frac{2}{3}$$
   4. Simplify across diagonals:
      Divide 9 and 3 by 3 (yielding 3 and 1).
      Divide 2 and 4 by 2 (yielding 1 and 2).
      $$\frac{3}{2} \times \frac{1}{1} = \frac{3}{2}$$
   5. Write the final answer:
      $\frac{3}{2}$ or $1\frac{1}{2}$.