Multiplying Mixed Numbers for 5th Grade: Convert and Multiply

Multiplying mixed numbers is used when finding areas of rectangles with fractional dimensions, scaling recipes that use mixed numbers, or solving real-world problems like “a garden is \(2 \frac{1}{4}\) m by \(1 \frac{1}{3}\) m—what is the area?” In Grade 5, students multiply mixed numbers by first converting them to improper fractions, then multiplying fractions as usual, and finally converting the result back to a mixed number if needed.

The key step is converting mixed numbers to improper fractions: \(a \frac{b}{c} = \frac{a \times c + b}{c}\). For example, \(2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\). Once both mixed numbers are improper fractions, we multiply numerators and denominators, simplify, and convert back if the product is improper.

DETAILED EXPLANATION

Steps to multiply mixed numbers:

1. Convert each mixed number to an improper fraction.

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2. Multiply the fractions (numerators × numerators, denominators × denominators).

3. Simplify if possible.

4. Convert the product to a mixed number if it is improper.

Conversion: \(a \frac{b}{c} = \frac{a \times c + b}{c}\). Example: \(3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}\).

Example: \(2 \frac{1}{2} \times 3 \frac{1}{3}\). Convert: \(\frac{5}{2}\) and \(\frac{10}{3}\). Multiply: \(\frac{5}{2} \times \frac{10}{3} = \frac{50}{6} = \frac{25}{3} = 8 \frac{1}{3}\).

WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

Multiply \(2 \frac{1}{2} \times 3 \frac{1}{3}\)

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Solutions:

Step 1: Convert to improper fractions. \(2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\). \(3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}\).

Step 2: Multiply: \(\frac{5}{2} \times \frac{10}{3} = \frac{5 \times 10}{2 \times 3} = \frac{50}{6}\).

Step 3: Simplify: \(\frac{50}{6} = \frac{25}{3}\).

Step 4: Convert to mixed number: \(25 \div 3 = 8\) remainder 1, so \(\frac{25}{3} = 8 \frac{1}{3}\).

Answer: \(8 \frac{1}{3}\)

Example 2

A garden is \(2 \frac{1}{4}\) m by \(1 \frac{1}{3}\) m. Find the area.

Solutions:

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Step 1: Area = length × width = \(2 \frac{1}{4} \times 1 \frac{1}{3}\).

Step 2: Convert: \(2 \frac{1}{4} = \frac{9}{4}\); \(1 \frac{1}{3} = \frac{4}{3}\).

Step 3: Multiply: \(\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3\).

Step 4: The area is 3 square meters.

Answer: 3 square meters

Example 3

\(1 \frac{2}{5} \times 2 \frac{1}{2}\) = ?

Solutions:

Step 1: Convert: \(1 \frac{2}{5} = \frac{7}{5}\); \(2 \frac{1}{2} = \frac{5}{2}\).

Step 2: Multiply: \(\frac{7}{5} \times \frac{5}{2} = \frac{35}{10}\).

Step 3: Simplify: \(\frac{35}{10} = \frac{7}{2}\). Convert: \(7 \div 2 = 3\) remainder 1, so \(\frac{7}{2} = 3 \frac{1}{2}\).

Answer: \(3 \frac{1}{2}\)

Example 4

Multiply \(1 \frac{1}{3} \times 2 \frac{1}{4}\)

Solutions:

Step 1: Convert: \(1 \frac{1}{3} = \frac{4}{3}\); \(2 \frac{1}{4} = \frac{9}{4}\).

Step 2: Multiply: \(\frac{4}{3} \times \frac{9}{4} = \frac{36}{12} = 3\).

Answer: 3

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