# Fractions

## Arithmetic Problems: Fractions

## Introduction

A fraction is a numerical quantity that is not whole. They are often encountered in weights and measurements. Examples include one-half ($\frac{1}{2}$), a third ($\frac{1}{3}$), and one-quarter ($\frac{1}{4}$).

A fraction is made up of two values: a numerator and a denominator. The numerator is the top value of the fraction and indicates how many pieces of a whole you have. The denominator is the bottom value of the fraction and indicates how many pieces make up the whole.

In the case of one-half ($\frac{1}{2}$), 2 pieces makes a whole (the denominator), and you have 1 piece (the numerator). In other words, you have 1 of 2 pieces, commonly referred to as one-half.

## Simplifying Fractions

A fraction is simplified (or reduced) when there is no number that can be divided into both the numerator and denominator.

For example, in the fraction $\frac{2}{4}$ (two-quarters) the numerator and the denominator can both be divided by 2, making the simplified fraction $\frac{1}{2}$ (one-half).

In the example $\frac{10}{35}$ the top and the bottom values can both be divided by 5, making the simplified fraction $\frac{2}{7}$.

## Adding & Subtracting Fractions

To add or subtract fractions:

- First, find their least common denominator.
- Second, add (or subtract) the the numerator.
- Finally, simplify the answer.

In the example of $\frac{1}{6}$ + $\frac{2}{6}$ = $\frac{1}{2}$, because the fractions $\frac{1}{6}$ and $\frac{2}{6}$ already have a common denominator (6), we simply add the numerators 1 and 2 together (making 3) and place the result over the common denominator (6), giving us $\frac{3}{6}$. Finally, we simplify our answer of $\frac{3}{6}$, giving us a final answer of $\frac{1}{2}$.

In the example of $\frac{2}{5}$ + $\frac{1}{2}$ = $\frac{9}{10}$, the denominators are different. So we must first find the least common denominator. We find the number 10 can be divided by both 5 and 2, so 10 is our least common denominator. Now we adjust the numerator in both fractions to their new common denominator. In the case of $\frac{2}{5}$ we know 5 goes into 10 2 times, so we multiply 2 by 2, giving us $\frac{4}{10}$. In the case of $\frac{1}{2}$ we know 2 goes into 10 5 times, so we multiply 1 by 5, giving us $\frac{5}{10}$. Now our problem becomes $\frac{4}{10}$ + $\frac{5}{10}$. Both of these fractions have a common denominator of 10, so now we add the numerators 4 and 5 together (giving us 9) and place the answer over our common denominator of 10, giving us an answer of $\frac{9}{10}$. Since there is no number that factors into both 9 and 10, we don't need to simplify our answer.

## Multiplying Fractions

To multiply fractions:

- First, multiply the numerators together.
- Second, multiply the denominators together.
- Finally, simplify the answer.

For example: $\frac{2}{5}$ ⨯ $\frac{1}{2}$ = $\frac{2}{10}$ which simplifies to $\frac{1}{5}$.

## Dividing Fractions

To divide fractions:

- First, get the reciprocal of one of the fraction, by flipping it.
- Second, multiply the numerators together.
- Third, multiply the denominators together.
- Finally, simplify the answer.

For example: $\frac{2}{5}$ ÷ $\frac{1}{2}$ = $\frac{2}{5}$ ⨯ $\frac{2}{1}$ = $\frac{4}{5}$