Multiplication with Mixed Numbers

Grade 5 Math – Notes & Formulae

Multiply Mixed Number × Whole Number (Model)

• Break the mixed number into whole and fraction parts.
• Model: Use repeated groups for whole, then for fraction.
• Formula:
  \( w\frac{a}{b} \times n = (w \times n) + \left(n \times \frac{a}{b}\right) \)
• Example: \( 2\frac{1}{4} \times 3 = (2 \times 3) + \left(3 \times \frac{1}{4}\right) = 6 + \frac{3}{4} = 6\frac{3}{4} \)

Multiply Mixed Number × Whole Number (Convert to Improper)

• First, convert the mixed number to improper fraction:
• Formula: \( w\frac{a}{b} = \frac{w \times b + a}{b} \)
• Then multiply by the whole number:
• Formula: \( n \times \frac{c}{d} = \frac{n \times c}{d} \)
• Example: \( 1\frac{3}{5} \times 4:\)
  Convert: \( 1\frac{3}{5} = \frac{8}{5} \)
  Calculate: \( 4 \times \frac{8}{5} = \frac{32}{5} = 6\frac{2}{5} \)

Multiply Mixed Numbers (Area Model)

• Draw a rectangle: partition sides into whole and fraction lengths.
• Calculate the area for each part:
• • Whole × Whole
  • Whole × Fraction
  • Fraction × Whole
  • Fraction × Fraction
• Add all parts for the total product.
• Example: \(1\frac{1}{2} \times 2\frac{1}{3}\)
  Convert to improper: \( \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = 3\frac{1}{2} \)

Quick Reference

• Multiply mixed × whole: multiply separately, add results, or convert to improper fraction.
• Convert mixed to improper: \( w\frac{a}{b} = \frac{w \times b + a}{b} \)
• For mixed × mixed, convert both to improper, multiply, then simplify back to mixed.
• Area model: draw for visual support, especially with tasks involving rectangles or arrays.
• Simplify and always check the context for proper labeling.