Scaling by Fractions

Grade 5 Math – Notes & Formulae

Scaling Whole Numbers by Fractions

• Multiplying a whole number by a fraction scales or resizes the number.
• If the fraction is less than 1: Product is smaller.
• If the fraction is more than 1: Product is larger.
• Formula: \( n \times \frac{a}{b} = \frac{n \times a}{b} \)
• Example: \( 6 \times \frac{1}{3} = \frac{6}{3} = 2 \) (makes 6 smaller)
• This shows part of a number, used in scaling recipes, maps, models, etc.

Scaling Fractions by Fractions

• Multiply two fractions to scale one by the other.
• Formula: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
• Example: \( \frac{1}{2} \) scaled by \( \frac{3}{4} \):
  \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \) (makes \(\frac{1}{2}\) smaller)
• If the scaling fraction is <1, result shrinks; if >1, result grows.

Scaling Mixed Numbers by Fractions

• Convert mixed number to improper fraction.
• Multiply as with fractions: numerator × numerator, denominator × denominator.
• Simplify and convert back to mixed number if needed.
• Example: \( 2\frac{1}{2} \) scaled by \( \frac{2}{3} \):
  Convert: \( 2\frac{1}{2} = \frac{5}{2} \); \( \frac{5}{2} \times \frac{2}{3} = \frac{10}{6} = \frac{5}{3} = 1\frac{2}{3} \)

Scaling by Fractions & Mixed Numbers

• For any scaling scenario, rewrite numbers as improper fractions where needed.
• Multiply all numerators together, all denominators together.
• Interpret result: Does it shrink or grow the original?
• Use models/number lines to visually check your work.
• Example: \( 4 \times \frac{2}{3} \times \frac{9}{5} = \frac{72}{15} = 4\frac{12}{15} \)

Quick Reference

• Scaling = Multiplying by a fraction; shrink if fraction <1, expand if >1.
• For whole numbers: numerator × whole, denominator stays.
• For fractions/mixed numbers: Convert all to improper fractions then multiply.
• Models (area diagrams, number lines) make scaling clear.
• Word problem? Always label answer with correct units!