Class Notes: Multiplying Fractions by Whole Numbers

Objective

Learn how to multiply fractions by whole numbers and how to convert the results to mixed numbers.

---

How to Multiply a Fraction by a Whole Number

Step 1: Convert the Whole Number to a Fraction

• Express the whole number as a fraction by putting it over 1.
  • Example: ( 32 = \frac{32}{1} )

Step 2: Multiply Across

• Multiply the numerators together.
• Multiply the denominators together.

Step 3: Simplify Before Multiplying (if possible)

• Factor the whole number to see if any numerator and denominator can be cancelled.
  • Example: ( 32 = 8 \times 4 )
  • If multiplying ( \frac{5}{8} \times 32 ):
    • ( 32 = 8 \times 4 ), so ( \frac{5}{8} \times \frac{8 \times 4}{1} )
    • Cancel the 8s: ( \frac{5}{\cancel{8}} \times \frac{\cancel{8} \times 4}{1} = \frac{5 \times 4}{1} = 20 )

---

Examples

Example 1: ( \frac{5}{8} \times 32 )

1. Convert ( 32 ) to ( \frac{32}{1} )
2. ( \frac{5}{8} \times \frac{32}{1} )
3. Factor: ( 32 = 8 \times 4 )
4. Cancel 8s: ( \frac{5}{8} \times \frac{8 \times 4}{1} = \frac{5 \times 4}{1} = 20 )

Final Answer: 20

---

Example 2: ( \frac{4}{7} \times 35 )

1. Convert ( 35 ) to ( \frac{35}{1} )
2. ( 35 = 7 \times 5 )
3. Cancel 7s: ( \frac{4}{\cancel{7}} \times \frac{\cancel{7} \times 5}{1} = \frac{4 \times 5}{1} = 20 )

Final Answer: 20

---

Example 3: ( \frac{4}{5} \times 3 )

1. Convert ( 3 ) to ( \frac{3}{1} )
2. ( \frac{4}{5} \times \frac{3}{1} )
3. No cancelling possible; multiply across.
   • Numerator: ( 4 \times 3 = 12 )
   • Denominator: ( 5 \times 1 = 5 )
4. Improper fraction: ( \frac{12}{5} )

Converting to a Mixed Number:

• ( \frac{12}{5} = \frac{10}{5} + \frac{2}{5} )
• ( \frac{10}{5} = 2 )
• Mixed number: ( 2 \frac{2}{5} )

---

Example 4: ( \frac{5}{9} \times 8 )

1. Convert ( 8 ) to ( \frac{8}{1} )
2. ( \frac{5}{9} \times \frac{8}{1} )
3. Multiply across:
   • Numerator: ( 5 \times 8 = 40 )
   • Denominator: ( 9 \times 1 = 9 )
4. Improper fraction: ( \frac{40}{9} )

Converting to a Mixed Number:

• ( 9 \times 4 = 36 ), ( 40 - 36 = 4 )
• ( \frac{40}{9} = \frac{36}{9} + \frac{4}{9} = 4 + \frac{4}{9} )
• Mixed number: ( 4 \frac{4}{9} )

---

Key Points

• Always convert the whole number to a fraction by putting it over 1.
• Multiply numerators and denominators across.
• Simplify by cancelling common factors if possible before multiplying.
• Convert improper fractions to mixed numbers by dividing the numerator by the denominator and expressing the remainder as a fraction.