Multiplying a fraction by a whole number Knowledge-Based Study

Study notes

1. Multiplication as Repeated Addition

   Multiplying a fraction by a whole number can be understood as repeatedly adding the fraction to itself, exactly like multiplying two whole numbers.

   For example, consider the expression:

   $3 \times \frac{2}{7}$

   This means adding $\frac{2}{7}$ three times:

   $\frac{2}{7} + \frac{2}{7} + \frac{2}{7} = \frac{2 + 2 + 2}{7} = \frac{6}{7}$

   Because the denominator represents the size of each part, it remains unchanged during addition. Only the count of the parts—the numerator—is multiplied.

   The Multiplication Procedure

   To multiply any whole number by a fraction, multiply the whole number by the numerator and keep the denominator the same.

   In algebraic terms:

   $a \times \frac{b}{c} = \frac{a \times b}{c}$

   You can also think of the whole number $a$ as a fraction with a denominator of 1:

   $\frac{a}{1} \times \frac{b}{c} = \frac{a \times b}{1 \times c} = \frac{a \times b}{c}$

   Worked Example

   Calculate $5 \times \frac{3}{16}$:

   1. Multiply the whole number by the numerator: $5 \times 3 = 15$.
   2. Place the product over the original denominator: $\frac{15}{16}$.

   Simplifying Your Answer

   When multiplying a whole number by a fraction, the resulting fraction should always be written in simplest form.

   Method 1: Simplifying at the End

   Compute $4 \times \frac{3}{8}$:

   $4 \times \frac{3}{8} = \frac{4 \times 3}{8} = \frac{12}{8}$

   Divide the numerator and denominator by their greatest common factor, which is 4:

   $\frac{12 \div 4}{8 \div 4} = \frac{3}{2}$

   Method 2: Simplifying Before Multiplying

   You can divide the whole number and the denominator by their common factor before carrying out the multiplication:

   $4 \times \frac{3}{8} = \frac{\cancel{4}^{1} \times 3}{\cancel{8}_{2}} = \frac{1 \times 3}{2} = \frac{3}{2}$

   Simplifying early keeps calculations smaller and reduces errors.

   Improper Fractions to Mixed Numbers

   When the resulting product has a numerator larger than or equal to its denominator, it is an improper fraction. In many practical situations, it is helpful to convert it to a mixed number.

   Step-by-Step Conversion

   Calculate $7 \times \frac{3}{5}$:

   • Step 1: Multiply the whole number by the numerator.
     $7 \times 3 = 21$, so the result is $\frac{21}{5}$.
   • Step 2: Divide the numerator by the denominator to find the whole number part.
     $21 \div 5 = 4$ with a remainder of $1$.
   • Step 3: Write the remainder over the original denominator.
     $\frac{21}{5} = 4\frac{1}{5}$.

   Interpreting 'Fraction of' as Multiplication

   In word problems, the word of often represents multiplication. Finding a fraction of a whole number uses the exact same calculation.

   Example Scenario

   A recipe calls for $\frac{3}{4}$ cup of flour per serving. How many cups of flour are needed for 6 servings?

   We need to find 6 groups of $\frac{3}{4}$ cup:

   $6 \times \frac{3}{4} = \frac{6 \times 3}{4} = \frac{18}{4}$

   Simplify by dividing by 2:

   $\frac{18 \div 2}{4 \div 2} = \frac{9}{2} = 4\frac{1}{2}$

   Therefore, $4\frac{1}{2}$ cups of flour are needed.