4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Multiplication and division of mixed numbers

Multiplication and division of mixed numbers

Before you can multiply or divide mixed numbers, you need to convert them to improper fractions. Here's how that conversion works:

Converting a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.
2. Add the numerator to that result.
3. Place that total over the original denominator.

For example, $2\frac{1}{3}$: multiply $2 \times 3 = 6$, then add $6 + 1 = 7$, giving you $\frac{7}{3}$.

Multiplying mixed numbers:

1. Convert each mixed number to an improper fraction.
2. Multiply the numerators together and the denominators together.
3. Simplify the result and convert back to a mixed number if needed.

Dividing mixed numbers:

1. Convert each mixed number to an improper fraction.
2. Flip the second fraction (find its reciprocal).
3. Multiply the first fraction by that reciprocal.
4. Simplify and convert back to a mixed number if needed.

Verbal to fractional expression conversion

Word problems often describe fractions in plain English, so you need to translate them into math.

• Identify the parts and the whole in the description. In "two-thirds of a cup," the parts are two-thirds and the whole is a cup.
• Write the fraction using the numbers given: $\frac{2}{3}$ cup.
• Simplify if possible by dividing the numerator and denominator by their greatest common factor. For example, "six out of eight students" becomes $\frac{6}{8}$, which simplifies to $\frac{3}{4}$ of the students.

Reduction of complex fractions

Reduction of complex fractions

A complex fraction is a fraction that has a fraction in its numerator, its denominator, or both. Something like $\frac{\frac{2}{3}}{\frac{1}{4}}$ looks intimidating, but there's a clean method for simplifying it.

The LCD method:

1. Find the least common denominator (LCD) of all the smaller fractions inside the complex fraction. For $\frac{\frac{2}{3}}{\frac{1}{4}}$, the denominators are 3 and 4, so the LCD is 12.
2. Multiply both the top and bottom of the complex fraction by that LCD. This clears out the smaller fractions: $\frac{\frac{2}{3} \times 12}{\frac{1}{4} \times 12} = \frac{8}{3}$
3. Simplify the result: $\frac{8}{3} = 2\frac{2}{3}$

You can also think of any complex fraction as a division problem. $\frac{\frac{2}{3}}{\frac{1}{4}}$ is the same as $\frac{2}{3} \div \frac{1}{4}$, so you can flip and multiply. Both approaches give the same answer.

Simplification of fraction bar expressions

A fraction bar acts as a grouping symbol, just like parentheses. That means you need to simplify everything above the bar and everything below the bar before you divide.

1. Simplify the numerator expression: for $\frac{2 + \frac{1}{3}}{4 - \frac{1}{2}}$, the numerator is $2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}$

2. Simplify the denominator expression: $4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2}$

3. Divide the simplified numerator by the simplified denominator: $\frac{\frac{7}{3}}{\frac{7}{2}} = \frac{7}{3} \times \frac{2}{7} = \frac{2}{3}$

Notice in step 3 that the 7s cancel. Always look for common factors you can cancel before multiplying.

Additional concepts for fraction operations

• Simplification: Reduce fractions to their simplest form by dividing both the numerator and denominator by their greatest common factor.
• Cancellation: Before multiplying fractions, look for common factors shared between any numerator and any denominator and divide them out first. This keeps your numbers smaller and saves you from simplifying at the end.
• Cross multiplication: A shortcut for comparing two fractions or solving equations with fractions. You multiply the numerator of each fraction by the denominator of the other. For instance, to compare $\frac{3}{5}$ and $\frac{2}{3}$, compute $3 \times 3 = 9$ and $5 \times 2 = 10$. Since $9 < 10$, you know $\frac{3}{5} < \frac{2}{3}$.
• Common denominator: When adding or subtracting fractions, you need the denominators to match. Find the least common denominator, rewrite each fraction with that denominator, then combine the numerators.