Jerry is building a small birdhouse. He looks through the plans and determines the wood block lengths that he’ll need to complete his project. Unfortunately, not every measurement was equal to a full inch. Therefore, Jerry needs to use fractions to list the measurements he needs. Then, he needs to add fractions to determine how much wood to purchase.
Fractions are used to denote parts of a whole. If you have less than a whole pie, you can use fractions to show how much pie you really have. Some fractions include:
When Jerry is ready to add the fractions together, there are certain things he needs to know to be sure that he ends up with enough wood.
How to Add Fractions with Like Denominators
Adding fractions is pretty simple as long as you start with like denominators. The denominator is the bottom number, and the numerator is the number on top. Let’s say you have the following problem:
Since the denominators are the same (6), all you have to do is add the numerators together:
Maybe you bought two pies: a cherry pie and an apple pie. After a few of your guests have helped themselves to dessert, you find that one pie has 2 slices left out of the 6 slices you cut and the other has 3 slices out of 6 left. By adding them together, you know that you have 5 slices left that you can share with others.
After adding the fractions together, it’s important to give your answer in the simplest form. So, you need to make sure that the solution is reduced to the lowest terms. For example,
Before moving on to the next equation, you need to check your answer and make sure that it can’t be simplified any further. To do this,
- 1. Consider the factors of the numerator and denominator.
- 2. Determine the GCF (Greatest Common Factor). This is the highest number that divides evenly into both the numerator and denominator.
6 − 1, 2, 3, 6
12 − 1, 2, 3, 4, 6, 12
For the above example, the GCF is 6.
- 3. If we divide the numerator and denominator by this number, we end up with ½. This is the lowest term possible for this equation.
It’s important to make sure that the fraction cannot be reduced lower. This is especially true if you don’t find the GCF, so you should continue to divide the top and bottom by the same number until you reduce the fraction as much as possible.
Mixed Numbers and Improper Fractions
A mixed number is one that contains a fraction as well as a whole number, such as:
Adding these numbers can be difficult. To simplify the process, first convert the number to an improper fraction. An improper fraction has a numerator that is larger than the denominator. Because of this, it’s important to simplify the fraction after finding the solution since you’ll probably end up with another mixed number. Check out this equation:
In this equation, you know that 1 = 4/4. Therefore, 2 = 8/4. Add the fraction to the converted whole number:
After finding the solution, it’s time to simplify the fraction. In this case, the final solution is simply the whole number 3.
Adding Fractions with Different Denominators
Knowing the basic principles of adding fractions is important, but oftentimes equations have fractions with different denominators. When this happens, what should you do? Before adding fractions together, you need to have like denominators.
One of the easiest ways to find common denominators is by multiplying the first denominator by the second denominator and vice versa. When doing this, it’s important to multiply the numerator and denominator by the same number so the value doesn’t change. If you’re unsure what this means, take a look at this example:
Say we start with the following equation:
Since the fractions have different denominators, the first thing that we need to do is convert the fractions to get common denominators. To do this, look at the denominators and multiply each by the other denominator. And, multiply the numerator by the same number as its denominator to keep the value the same:
Once the fractions have the same denominator, add the numerators together:
The final solution is an improper fraction, so you need to convert it to a mixed number and simplify the solution:
By dividing the numerator and denominator by 2, you can simplify the solution to:
Take a look at the following 5 practice examples. If you need to convert the denominators, make sure that you complete this step first. Then, solve and simplify!
Practice Example 1 Solution
Since the fractions don’t have like denominators, the first thing to do is convert the fractions. The simplest way to do this would be to multiply the fractions by the opposite denominator. Remember to multiply the numerator and denominator by the same number. You may also notice that both numbers are factors of 30, so you could multiply them by the appropriate factor to get a denominator of 30 (6 x 5 and 10 x 3).
After the fractions have like denominators, add the numerators together. Since the sum is an improper fraction, change it to a mixed number before simplifying the fraction.
Practice Example 2 Solution
In this practice problem, you need to change the fractions to have like denominators. Of course, you could multiply the denominators by each other. However, since 3 is a factor of 12, you can multiply just one of the fractions to get like denominators before solving the equation and simplifying.
Since the sum is an improper fraction, you need to reduce the equation to a mixed number and simplify the final fraction.
Practice Example 3 Solution
For practice problem 3, the equation has a mixed number, so start by making an improper fraction. Since 1 = 4/4, we can surmise that 2 = 8/4. Add this number to the numerator (3). Then, it’s time to ensure that there are like denominators for the equation. Similar to the previous example, 4 is a factor of 12. Therefore, you can multiply the first fraction’s numerator and denominator by 3 to convert the fraction.
The sum is an improper fraction. Thirty-eight can be divided by 12 three times, (12 × 3 = 36). This gives you a remainder of 2. Then, check to see if this is the simplest form.
Practice Example 4 Solution
Once again, you can start solving the equation by converting the fractions into like denominators. Multiply each fraction by the other fraction’s denominator to get a new denominator of 56.
After converting the fraction, add the numerators together and see if you can simplify the fraction.
The solution is an improper fraction. Therefore, first make the number into a mixed number. Then, simplify the solution. The factors of 56 include 1, 2, 4, 7, 8, 14, 28, and 56. The factors of 25 include 1, 5, and 25. Since they don’t have common factors, the final solution is:
Practice Example 5 Solution
For the final practice problem, start by converting the mixed number into an improper fraction. For this equation, 1 = 8/8 so 3 = 24/8.
Next, make sure that the fractions have like denominators. You can multiply each fraction by the other fraction’s denominator. Then, add the products together.
With the solution as an improper fraction, simplify it into a mixed number:
Then, simplify! The factors of 51 include 1, 3, 17, 51. And, the factors of 72 include 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Both numbers can be divided by 3. Divide the numerator and denominator by 3 giving you:
Since 17 is prime, the fraction is as simplified as it can be.
How did you do with the practice questions? What questions do you have about adding fractions? Tell us in the comments below.