In this lesson, we’ll learn how to add 1/2 + 1/4 + 1/8. We’ll see the steps involved in this process, and in adding fractions in general, using an example from everyday life.

## Adding Fractions

In everyday life, we encounter scenarios in which we need to add three fractions. Suppose you want to ride your bike to your friend’s house. To get there, you must first ride 1/2 of a mile down Elm Street, then 1/4 of a mile down Maple Street, and finally 1/8 of a mile down Oak Street. Before you head out, you want to know how far you have to ride.

You can add the distances on each street, or 1/2 + 1/4 + 1/8. If we’re adding fractions that have the same denominator such as 1/4 and 2/4, all we have to do is add together the numerators to find the result. Since 1 + 2 = 3, our numerator is 3. The denominator, 4, stays the same. Thus, 1/4 + 2/4 = 3/4.

So, how do we add fractions with different denominators? To do this, we have to add a step to our procedure. Before we add the numerators, we have to convert our fractions so that they have the same denominator.

## Finding a Common Denominator

A **common denominator** is the same denominator in two or more fractions. The common denominator will be the least common multiple of the denominators of the fractions. The **least common multiple** (abbreviated **LCM**) of a group of numbers is the smallest positive number that all of the numbers in the group divide into evenly. To find this, it’s helpful to list multiples of each of the numbers.

Let’s say we want to find a common denominator for the fractions 1/3 and 2/5. We need to find the least common multiple of 3 and 5. We can list the multiples of each number:

- Multiples of 3 are: 3, 6, 9, 12, 15, 18, and so on
- Multiples of 5 are: 5, 10, 15, 20, and so on

The first number in common in these two lists is 15, so the least common multiple of 3 and 5 is 15.

If we wanted to add 1/3 and 2/5, we would first need to convert both fractions to denominators of 15. How do we change the denominator of a fraction without changing the value of that fraction? To do this, we have to multiple both the numerator and denominator by the same number. To change 1/3 into 15ths, we look at what we would need to multiple 3 by to get 15. 3 times 5 is 15, so if we multiple the numerator of the fraction (1) by the same number (5) we get 5. 1/3 is equal to 5/15.

## One Example of Adding 3 Fractions

Let’s practice these steps to find out how far you have to ride your bike to get to your friend’s house. We want to add 1/2 + 1/4 + 1/8. There are a couple of ways to go about adding three fractions:

The **associative property** states (*a* + *b*) + *c* = *a* + (*b* + *c*). This means that it doesn’t matter which fractions we add together first. We can add any two of the fractions together and then add the result to the third fraction. Let’s start by adding 1/2 and 1/4. First we need a common denominator. This will be the least common multiple of the denominators in the fractions. In our case, we want to find the least common multiple of 2 and 4:

- Mutliples of 2 are: 2, 4, 6, 8, 10, and so on
- Multiples of 4 are: 4, 8, 12, 16, and so on

Comparing the list, we see that 4 is the least common multiple of 2 and 4. Next, we need to make 4 the denominator of both fractions. We don’t have to do anything to 1/4 since it already has this denominator. To make 4 the denominator of 1/2, we need to multiply both the numerator and denominator by 2, since 2 times 2 equals 4. When we do this, we find that 1/2 is equal to 2/4. Now, we’re ready to add our fractions. Remember, to add two fractions with the same denominator, simply add the numerators: the denominator stays the same. When we add 2/4 and 1/4, we add 2 + 1 = 3 to find the numerator. Since the denominator is 4, we find that 2/4 + 1/4 = 3/4.

We’re almost there! Now, we need to add our third fraction, 1/8, to the result. We want to find 3/4 + 1/8. Again, we have to first find a common denominator by determining the least common multiple of 4 and 8:

- Multiples of 4 are: 4, 8, 12, 16, and so on
- Multiples of 8 are: 8, 16, 24, and so on

We see that 8 is the least common multiple of 4 and 8. Since 8 is already the denominator for 1/8, we just need to manipulate 3/4 so that it also has the denominator 8. To do this, we multiply both the numerator and denominator by 2. This gives us 6/8. Now, we add the numerators of 6/8 and 1/8: 6 + 1 = 7, which means that 3/4 + 1/8 = 7/8. Thus, by first adding 1/2 plus 1/4 and then adding 1/8 to the result, we find that 1/2 + 1/4 + 1/8 = 7/8.

## Another Way to Solve

Another way to solve this equation is to add all three fractions at once. To do this, we must find a common denominator for all three fractions. In this case, we would start by find the least common multiple of 2, 4, and 8:

- Multiples of 2 are: 2, 4, 6, 8, 10, and so on
- Multiples of 4 are: 4, 8, 12, 16, and so on
- Multiples of 8 are: 8, 16, 24, and so on

8 is the smallest number each of these numbers divides into evenly. Now, we just need to make 8 our common denominator. The denominator of 1/8 is already 8, so we leave it alone. To get a denominator of 8 for 1/2, we multiple the numerator and denominator by 4, giving us 4/8. To get a denominator of 8 for 1/4, we multiple the numerator and denominator by 2, giving us 2/8. Now we have three fractions with the denominator of 8. We need to add 4/8 + 2/8 + 1/8. We simply add the numerators: 4 + 2 + 1 = 7. Once again, we find that 1/2 + 1/4 + 1/8 = 7/8.

Both methods yielded the same result, whether we add two fractions first and then add the last one to the result, or add all three together at once, the answer is the same. You’d have to ride your bike 7/8 of a mile to get from your house to your friend’s house.

## Lesson Summary

When adding fractions, the fractions must have a **common denominator**. To change fractions so that they have a common denominator, we first find the **least common multiple**, or the smallest positive number that all of the numbers in the group divide into evenly. We find this by making a list of the multiples of each denominator. Once we’ve found a common denominator, we convert our fractions by multiplying both the numerator and denominator by the same number, so that the fraction does not change its value. Then, we simply add together the numerators and keep the denominator the same to give us our result. Because of the **associative property**, which states that numbers can be added in any order, it does not matter which fractions we add together first. We can also add all three at once. Knowing how to add fractions is very useful for real-world applications.