

# Adding fractions with a common denominator

To add fractions with a common denominator, you can simply add the numerators and keep the common denominator. For example:

$$\frac{1}{2} + \frac{2}{2} = \frac{1 + 2}{2} = \frac{3}{2}$$

Another example:

$$\frac{3}{4} + \frac{1}{4} = \frac{3 + 1}{4} = \frac{4}{4} = 1$$

## Exercise
Add the following fractions:

$$\frac{2}{5} + \frac{3}{5}$$

### Solution
$$\frac{2 + 3}{5} = \frac{5}{5} = 1$$

# Adding fractions with unlike denominators

When adding fractions with unlike denominators, you need to find a common denominator that can be used to represent both fractions. This can be done by finding the least common multiple (LCM) of the two denominators. The LCM is the smallest number that is divisible by both numbers.

For example, to add $\frac{1}{2}$ and $\frac{1}{3}$, you would find the LCM of 2 and 3, which is 6. Then, you can rewrite the fractions as:

$$\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$$

and

$$\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}$$

Now you can add the fractions:

$$\frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}$$

So, $\frac{1}{2}$ and $\frac{1}{3}$ can be added together as $\frac{5}{6}$.

## Exercise
Add the following fractions:

$$\frac{1}{4} + \frac{2}{5}$$

### Solution
$$\frac{1}{4} = \frac{1 \cdot 5}{4 \cdot 5} = \frac{5}{20}$$

and

$$\frac{2}{5} = \frac{2 \cdot 4}{5 \cdot 4} = \frac{8}{20}$$

Now, add the fractions:

$$\frac{5}{20} + \frac{8}{20} = \frac{5 + 8}{20} = \frac{13}{20}$$

# Subtracting fractions

To subtract fractions, you can treat them as negative fractions and add them together. For example, to subtract $\frac{1}{2}$ from $\frac{1}{3}$, you would rewrite the fractions as:

$$\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$$

and

$$\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}$$

Now, you can subtract the fractions by adding the negative of the second fraction to the first fraction:

$$\frac{3}{6} - \frac{2}{6} = \frac{3}{6} + \frac{-2}{6} = \frac{3 - 2}{6} = \frac{1}{6}$$

So, $\frac{1}{2}$ and $\frac{1}{3}$ can be subtracted as $\frac{1}{6}$.

## Exercise
Subtract the following fractions:

$$\frac{3}{4} - \frac{1}{2}$$

### Solution
$$\frac{3}{4} = \frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}$$

and

$$\frac{1}{2} = \frac{1 \cdot 4}{2 \cdot 4} = \frac{4}{8}$$

Now, subtract the fractions:

$$\frac{6}{8} - \frac{4}{8} = \frac{6 - 4}{8} = \frac{2}{8}$$

# Multiplying fractions

To multiply fractions, you can use the product of the numerators and the product of the denominators. For example, to multiply $\frac{1}{2}$ and $\frac{2}{3}$, you would multiply the numerators (1 and 2) and the denominators (2 and 3):

$$\frac{1}{2} \times \frac{2}{3} = \frac{1 \cdot 2}{2 \cdot 3} = \frac{2}{6} = \frac{1}{3}$$

So, $\frac{1}{2}$ and $\frac{2}{3}$ can be multiplied together as $\frac{1}{3}$.

## Exercise
Multiply the following fractions:

$$\frac{3}{4} \times \frac{1}{2}$$

### Solution
$$\frac{3}{4} \times \frac{1}{2} = \frac{3 \cdot 1}{4 \cdot 2} = \frac{3}{8}$$

# Dividing fractions

To divide fractions, you can use the quotient of the numerators and the quotient of the denominators. For example, to divide $\frac{1}{2}$ by $\frac{2}{3}$, you would divide the numerator of the first fraction (1) by the numerator of the second fraction (2) and the denominator of the first fraction (2) by the denominator of the second fraction (3):

$$\frac{1}{2} \div \frac{2}{3} = \frac{1}{1} \div \frac{3}{2} = \frac{1 \cdot 2}{1 \cdot 3} = \frac{2}{3}$$

So, $\frac{1}{2}$ divided by $\frac{2}{3}$ is $\frac{2}{3}$.

## Exercise
Divide the following fractions:

$$\frac{3}{4} \div \frac{1}{2}$$

### Solution
$$\frac{3}{4} \div \frac{1}{2} = \frac{3 \cdot 2}{4 \cdot 1} = \frac{6}{4} = \frac{3}{2}$$

# Simplifying fractions

To simplify a fraction, you can divide the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify $\frac{6}{12}$, you would find the GCD of 6 and 12, which is 6, and divide both the numerator and the denominator by 6:

$$\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}$$

So, $\frac{6}{12}$ can be simplified as $\frac{1}{2}$.

## Exercise
Simplify the following fraction:

$$\frac{10}{20}$$

### Solution
$$\frac{10}{20} = \frac{10 \div 10}{20 \div 10} = \frac{1}{2}$$

# Comparing and ordering fractions

To compare fractions, you can compare their values. For example, to compare $\frac{1}{2}$ and $\frac{1}{3}$, you would compare the values of both fractions:

$$\frac{1}{2} > \frac{1}{3}$$

So, $\frac{1}{2}$ is greater than $\frac{1}{3}$.

## Exercise
Compare the following fractions:

$$\frac{3}{4} \text{ and } \frac{1}{2}$$

### Solution
$$\frac{3}{4} < \frac{1}{2}$$

# Applications of fractions in real-world situations

Fractions are used in many real-world situations, such as:

- Measuring the proportion of a whole, such as the percentage of students who passed an exam.
- Calculating the cost of items, such as the price per unit of a product.
- Expressing the amount of time, such as the number of hours worked in a day.

For example, if a product costs $10 and you want to know the cost per item, you can use the fraction $\frac{10}{1}$ to represent the cost per item.

## Exercise
Find the cost per item of a product that costs $20 and is sold in packs of 4.

### Solution
$$\frac{20}{4} = \frac{5}{1}$$

So, the cost per item is $\frac{5}{1}$.

# Converting fractions to decimals and percents

To convert a fraction to a decimal, you can divide the numerator by the denominator. For example, to convert $\frac{1}{2}$ to a decimal, you would divide 1 by 2:

$$\frac{1}{2} = \frac{1}{2} = 0.5$$

To convert a fraction to a percent, you can multiply the numerator by 100 and divide the result by the denominator. For example, to convert $\frac{1}{2}$ to a percent, you would multiply 1 by 100 and divide the result by 2:

$$\frac{1}{2} = \frac{1 \cdot 100}{2} = 50\%$$

## Exercise
Convert the following fraction to a decimal and a percent:

$$\frac{3}{4}$$

### Solution
To convert $\frac{3}{4}$ to a decimal, divide 3 by 4:

$$\frac{3}{4} = \frac{3}{4} = 0.75$$

To convert $\frac{3}{4}$ to a percent, multiply 3 by 100 and divide the result by 4:

$$\frac{3}{4} = \frac{3 \cdot 100}{4} = 75\%$$

# Introduction to mixed numbers

A mixed number is a whole number added to a fraction. For example, $\frac{5}{2}$ can be written as a mixed number as 2 1/2.

To convert a mixed number to a fraction, you can multiply the whole number by the denominator of the fraction and add the numerator of the fraction. For example, to convert 2 1/2 to a fraction, you would multiply 2 by 2 and add 1:

$$2 \frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{5}{2}$$

## Exercise
Convert the following mixed number to a fraction:

$$3 \frac{1}{4}$$

### Solution
$$3 \frac{1}{4} = \frac{3 \cdot 4 + 1}{4} = \frac{13}{4}$$

# Remainders when dividing fractions

When dividing fractions, you may encounter a remainder. For example, when dividing $\frac{7}{12}$ by $\frac{2}{3}$, you would find the remainder:

$$\frac{7}{12} \div \frac{2}{3} = \frac{7 \cdot 3 - 2 \cdot 12}{12 \cdot 3} = \frac{21 - 24}{36} = \frac{-3}{36}$$

So, the remainder when dividing $\frac{7}{12}$ by $\frac{2}{3}$ is $\frac{-3}{36}$.

## Exercise
Find the remainder when dividing $\frac{5}{8}$ by $\frac{3}{4}$.

### Solution
$$\frac{5}{8} \div \frac{3}{4} = \frac{5 \cdot 4 - 3 \cdot 8}{8 \cdot 4} = \frac{20 - 24}{32} = \frac{-4}{32}$$

So, the remainder when dividing $\frac{5}{8}$ by $\frac{3}{4}$ is $\frac{-4}{32}$.