# How to Find 1/2 + 1/4 + 1/8

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• 1:01 Finding a Common Denominator
• 2:26 One Example of Adding…
• 4:59 Another Way to Solve
• 6:23 Lesson Summary

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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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:

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