# Reduce, add, subtract, multiply, and divide with fractions

Working with fractions is a very important foundation to algebra. Here we will briefly review reducing, multiplying, dividing, adding, and subtracting fractions. As this is a review, concepts will not be explained in detail as other lessons are.

World View Note: The earliest known use of fraction comes from the Middle Kingdom of Egypt around 2000 BC! We always like our final answers when working with fractions to be reduced. Reducing fractions is simply done by dividing both the numerator and denominator

by the same number. This is shown in the following example.

Example 1:

Reduce $\frac{36}{84}$

Solution:

Both numerator and denominator are divisible by 4

$\frac{36:4}{84:4}=\frac{9}{21}$

Both numerator and denominator are still divisible by 3

$\frac{9:3}{21:3}=\frac{3}{7}$

The previous example could have been done in one step by dividing both numerator and denominator by 12. We also could have divided by 2 twice and then divided by 3 once (in any order). It is not important which method we use as long as we continue reducing our fraction until it cannot be reduced any further.

The easiest operation with fractions is multiplication. We can multiply fractions by multiplying straight across, multiplying numerators together and denominators together.

Example 2:

Multiply $\frac{6}{7}$ with $\frac{3}{5}$

Solution:

Multiply numerators across and denominators across

$\frac{6}{7}\text{ x }\frac{3}{5}=\frac{6.3}{7.5}=\frac{18}{35}$

When multiplying we can reduce our fractions before we multiply. We can either reduce vertically with a single fraction, or diagonally with several fractions, as long as we use one number from the numerator and one number from the denominator.

Example 3:

Multiply $\frac{25}{24}$ with $\frac{32}{55}$.

Solution:

Reduce 25 and 55 by dividing by 5. Reduce 32 and 24 by dividing by 8

$\frac{5}{3}.\frac{4}{11}$

Multiply numerators across and denominators across

$\frac{5}{3}.\frac{4}{11}=\frac{20}{33}$.

Dividing fractions is very similar to multiplying with one extra step. Dividing fractions requires us to first take the reciprocal of the second fraction and multiply. Once we do this, the multiplication problem solves just as the previous problem.

Example 4:

Divide $\frac{21}{16}$ with $\frac{28}{6}$.

Solution:

$\frac{21}{16}:\frac{28}{6}$

Multiply by the reciprocal

$\frac{21}{16}.\frac{6}{28}$

Reduce 21 and 28 by dividing by 7. Reduce 6 and 16 by dividing by 2

$\frac{3}{8}.\frac{3}{4}=\frac{9}{32}$.