3-5 Divide with Whole and Mixed Numbers

Property

To divide a fraction by a whole number, you apply the same rule: multiply the fraction by the reciprocal of the whole number.
Since the reciprocal of a whole number $c$ is $\frac{1}{c}$, this operation makes the fractional parts smaller.

Examples

Property

To handle multiplication or division, first convert mixed numbers into improper fractions. To do this, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Examples

• $$3\frac{1}{2} = \frac{2 \times 3 + 1}{2} = \frac{7}{2}$$
• $$5\frac{2}{3} = \frac{3 \times 5 + 2}{3} = \frac{17}{3}$$
• $$4\frac{3}{5} = \frac{5 \times 4 + 3}{5} = \frac{23}{5}$$

Explanation

Think of it as counting slices! A mixed number like $3\frac{1}{2}$ means three whole pizzas and one extra slice. To make it one big fraction, you need to count all the slices. The shortcut is to multiply the whole number by the bottom number (denominator) and then add the top number (numerator) to get your new total.

Property

To divide a mixed number by a whole number, convert the mixed number to an improper fraction and write the whole number as a fraction with a denominator of 1. Then, multiply by the reciprocal of the whole number.

Examples

• $2\frac{1}{2} \div 5 = \frac{5}{2} \div \frac{5}{1} = \frac{5}{2} \times \frac{1}{5} = \frac{5}{10} = \frac{1}{2}$
• $3\frac{1}{3} \div 2 = \frac{10}{3} \div \frac{2}{1} = \frac{10}{3} \times \frac{1}{2} = \frac{10}{6} = \frac{5}{3} = 1\frac{2}{3}$

Explanation

To divide a mixed number by a whole number, you must first convert both numbers into fractional form. Change the mixed number into an improper fraction. Write the whole number as a fraction by placing it over a denominator of 1. Finally, multiply the first fraction by the reciprocal of the second fraction and simplify the result.

Property

To divide mixed numbers, you must complete two steps. First, rewrite all mixed numbers or whole numbers as improper fractions. Then, to perform the division, multiply the first fraction by the reciprocal of the second fraction.

Examples

$3\frac{1}{3} \div 1\frac{1}{2} = \frac{10}{3} \div \frac{3}{2} = \frac{10}{3} \times \frac{2}{3} = \frac{20}{9} = 2\frac{2}{9}$
$2\frac{1}{2} \div 1\frac{2}{3} = \frac{5}{2} \div \frac{5}{3} = \frac{5}{2} \times \frac{3}{5} = \frac{15}{10} = 1\frac{1}{2}$
$8 \div 1\frac{1}{3} = \frac{8}{1} \div \frac{4}{3} = \frac{8}{1} \times \frac{3}{4} = \frac{24}{4} = 6$

Explanation

Think of this as a two-part mission! First, you have to get your mixed numbers into their 'improper fraction' disguises. Then, instead of a direct fight (division), you bring in the second number's secret agent twin (its reciprocal) and multiply. It's a clever switcheroo that makes solving the problem much simpler!