Dividing Fractions for 5th Grade: Keep, Change, Flip

Dividing fractions is used when splitting amounts into equal parts of a given size—for example, “a rope \(\frac{3}{4}\) meter long is cut into pieces of \(\frac{1}{8}\) meter each—how many pieces?” In Grade 5, students divide fractions by multiplying by the reciprocal of the divisor. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\) (swap numerator and denominator). So \(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}\).

The rule “multiply by the reciprocal” works because division is the inverse of multiplication. Dividing by \(\frac{4}{5}\) is the same as multiplying by \(\frac{5}{4}\), since \(\frac{4}{5} \times \frac{5}{4} = 1\). So \(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\).

DETAILED EXPLANATION

Rule: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\) (multiply by the reciprocal of the divisor).

Steps:

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1. Keep the first fraction.

2. Change the division sign to multiplication.

3. Flip the second fraction (write its reciprocal).

4. Multiply the fractions.

5. Simplify if needed.

To divide by a whole number: write the whole number as a fraction with denominator 1, then apply the rule. Example: \(\frac{3}{4} \div 2 = \frac{3}{4} \div \frac{2}{1} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}\).

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WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

Divide \(\frac{2}{3} \div \frac{4}{5}\)

Solutions:

Step 1: To divide by \(\frac{4}{5}\), multiply by its reciprocal \(\frac{5}{4}\).

Step 2: \(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}\).

Step 3: Multiply: \(2 \times 5 = 10\), \(3 \times 4 = 12\). Product: \(\frac{10}{12}\).

Step 4: Simplify: \(\frac{10}{12} = \frac{5}{6}\).

Answer: \(\frac{5}{6}\)

Example 2

A rope \(\frac{3}{4}\) meter long is cut into pieces of \(\frac{1}{8}\) meter each. How many pieces?

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Solutions:

Step 1: We need to find how many \(\frac{1}{8}\)s fit into \(\frac{3}{4}\). That is \(\frac{3}{4} \div \frac{1}{8}\).

Step 2: Multiply by the reciprocal: \(\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1}\).

Step 3: Multiply: \(\frac{3 \times 8}{4 \times 1} = \frac{24}{4} = 6\).

Step 4: There are 6 pieces.

Answer: 6 pieces

Example 3

\(\frac{5}{6} \div \frac{2}{3}\) = ?

Solutions:

Step 1: Multiply by the reciprocal: \(\frac{5}{6} \times \frac{3}{2}\).

Step 2: Multiply: \(\frac{5 \times 3}{6 \times 2} = \frac{15}{12}\).

Step 3: Simplify: \(\frac{15}{12} = \frac{5}{4}\). Convert to mixed: \(\frac{5}{4} = 1 \frac{1}{4}\).

Answer: \(1 \frac{1}{4}\)

Example 4

Divide \(\frac{3}{8} \div \frac{1}{4}\)

Solutions:

Step 1: \(\frac{3}{8} \div \frac{1}{4} = \frac{3}{8} \times \frac{4}{1}\).

Step 2: \(\frac{3 \times 4}{8 \times 1} = \frac{12}{8} = \frac{3}{2} = 1 \frac{1}{2}\).

Answer: \(1 \frac{1}{2}\)

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