Multiplying fractions and mixed numbers Knowledge-Based Study

Study notes

1. The Basic Rule

   To multiply two fractions, multiply their numerators to find the new numerator, and multiply their denominators to find the new denominator:

   Rule: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ (where b and d are not zero).

   Simplifying Before Multiplying

   You can make calculations simpler by dividing out common factors between any numerator and any denominator before multiplying. This technique is often called cross-canceling.

   Example: Calculate $\frac{3}{8} \times \frac{4}{9}$.

   1. Look for common factors between numerators and denominators: 3 and 9 share a factor of 3; 4 and 8 share a factor of 4.
   2. Divide 3 and 9 by 3: $3 \div 3 = 1$ and $9 \div 3 = 3$.
   3. Divide 4 and 8 by 4: $4 \div 4 = 1$ and $8 \div 4 = 2$.
   4. Multiply the simplified numbers: $\frac{1 \times 1}{2 \times 3} = \frac{1}{6}$.

   Working with Mixed Numbers

   Mixed numbers cannot be multiplied simply by multiplying their whole number parts and fraction parts separately. You must first convert every mixed number into an improper fraction.

   Step-by-Step Procedure

   1. Convert: Change each mixed number to an improper fraction using $A\frac{b}{c} = \frac{(A \times c) + b}{c}$.
   2. Simplify: Divide out any common factors between numerators and denominators.
   3. Multiply: Multiply straight across (numerator by numerator, denominator by denominator).
   4. Convert back: If the result is an improper fraction, write it in simplest form or as a mixed number as required.

   Example: Multiply $1\frac{3}{5} \times 2\frac{1}{2}$.

   • Convert to improper fractions: $1\frac{3}{5} = \frac{8}{5}$ and $2\frac{1}{2} = \frac{5}{2}$.
   • Set up the product: $\frac{8}{5} \times \frac{5}{2}$.
   • Divide out common factors: 5 and 5 divide to 1; 8 and 2 divide by 2 to give 4 and 1.
   • Compute the final product: $\frac{4 \times 1}{1 \times 1} = 4$.

   Multiplying by a Whole Number

   Any whole number n can be written as a fraction with a denominator of 1: $n = \frac{n}{1}$.

   For example, to evaluate $6 \times \frac{2}{9}$:

   $\frac{6}{1} \times \frac{2}{9} = \frac{2 \times 2}{1 \times 3} = \frac{4}{3} = 1\frac{1}{3}$

   The Meaning of "Of"

   In mathematics, the word of usually signals multiplication when dealing with fractions.

   Example: A recipe requires $2\frac{1}{4}$ cups of flour. If you make $\frac{2}{3}$ of the recipe, how much flour do you need?

   • Calculation: $\frac{2}{3} \times 2\frac{1}{4} = \frac{2}{3} \times \frac{9}{4}$
   • Divide out common factors: $2$ and $4$ reduce to $1$ and $2$; $9$ and $3$ reduce to $3$ and $1$.
   • Result: $\frac{1 \times 3}{1 \times 2} = \frac{3}{2} = 1\frac{1}{2}$ cups of flour.