How do you add two or more fractions together?
Understanding the basics of fractions is essential to adding them successfully. In a fraction, the top number is called the numerator, and the bottom number is called the denominator.
\frac{numerator}{denominator}
When adding fractions, it’s important to ensure that the denominators are the same.
If they are not, you will need to find a common denominator by changing the fractions into their equivalent form.
Once you have the same denominator, you can add the numerators together and place the sum over the common denominator. Finally, simplify the fraction if possible.
With a little practice, adding fractions will become second nature to you.
Understanding Fractions
What are Fractions?
Fractions are a way of representing a part of a whole. They are used to describe a part of something that has been divided into equal parts. A fraction consists of two parts: the numerator and the denominator. The numerator represents the number of parts you have, while the denominator represents the total number of parts in the whole.
For example, in the fraction \frac{3}{4} , the numerator is 3 and the denominator is 4 .
Parts of a Fraction
The numerator and denominator are important parts of a fraction.
The numerator is the top number in a fraction and represents the number of parts you have. The denominator is the bottom number in a fraction and represents the total number of parts in the whole.
For example, in the fraction \frac{2}{5} , the numerator is 2 and the denominator is 5 .
The denominator cannot be zero, as the denominator represents the total number of parts in the whole, and if the whole is zero this means there is no whole!
Additionally, fractions can be simplified by dividing both the numerator and denominator by their greatest common factor.
For example, the fraction \frac{4}{8} can be simplified to \frac{1}{2} by dividing both the numerator and denominator by 4 .
\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}
Fractions can also be converted to decimals by dividing the numerator by the denominator. For example, the fraction \frac{3}{5} can be converted to the decimal 0.6 by dividing 3 by 5 .
\frac{3}{5} = 3 \div 5 = 0.6
In summary, fractions represent a part of a whole and consist of a numerator and denominator. The numerator represents the number of parts you have, while the denominator represents the total number of parts in the whole. Fractions can be simplified by dividing both the numerator and denominator by their greatest common factor, and can be converted to decimals by dividing the numerator by the denominator.
Adding and Subtracting Fractions
When working with fractions, it’s important to know how to add and subtract them. Adding and subtracting fractions involves working with both the numerator and denominator of the fractions.
Adding Fractions With Same Denominators
When adding fractions with like denominators, you simply add the numerators together and keep the same denominator.
For example, if you want to add \frac{1}{7} and \frac{3}{7} , you would add the numerators together to get \frac{4}{7} .
\frac{1}{7} + \frac{3}{7} = \frac{1 + 3}{7} = \frac{4}{7}
Adding Fractions With Different Denominators
When adding fractions with different denominators, you need to find the lowest common denominator (LCD) before you can add the fractions together. The LCD is the smallest multiple that both denominators have in common.
To find the LCD, write down all the multiples of each denominator and then identify the smallest number shared between both sets.
For example, if you want to add \frac{1}{3} and \frac{1}{4} , you need to find the LCD first between 3 and 4 . The multiples of 3 are \{ 3, 6, 9, 12, 15, 18, ... \ \} and so on. The multiples of 4 are \{ 4, 8, 12, 16, 20, ... \ \} and so on.
\{ 3, 6, 9, 12, 15, 18, ... \} \\ \{4, 8, 12, 16, 20, ... \}
The smallest multiple that both 3 and 4 share is 12 .
\{ 3, 6, 9, \fcolorbox{red}{yellow}{12}, 15, 18, ... \} \\ \{4, 8, \fcolorbox{red}{yellow}{12}, 16, 20, ... \}
To convert \frac{1}{3} to have a denominator of 12 , you need to multiply both the numerator and denominator by 4 .
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}
To convert \frac{1}{4} to have a denominator of 12 , you need to multiply both the numerator and denominator by 3 .
\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
Now that both fractions have the same denominator, you can add the numerators together, like so:
\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}
When subtracting fractions, you follow the same process as adding fractions, but instead of adding the numerators, you subtract them.
Adding Mixed Numbers
When dealing with fractions, you may come across mixed numbers. A mixed number is a combination of a whole number and a proper fraction.
Mixed \ Number \ \frac{numerator}{denominator}
For example, 2 \frac{1}{3} is a mixed number.
Adding mixed numbers involves adding the whole numbers and the fractions separately, or converting the mixed number into an improper fraction and adding the fractions as you normally would with any type of fraction.
Here’s how each approach can work:
Adding Mixed Numbers Then Fractions
Here is a step by step guide on how to add mixed fractions together.
- Add the fraction parts together first.
- If the addition of the fraction creates an improper fraction convert it to a mixed fraction.
- Add the mixed numbers together last.
Here’s how this works by way of a couple of examples.
Example 1 – No Improper Fraction Result
1 \frac{1}{2} + 2 \frac{1}{3}
Separate out the numbers like so:
1 + \frac{1}{2} + 2 + \frac{1}{3}
Then focus on the addition of the fractions together first:
\frac{1}{2} + \frac{1}{3}
Both numbers have different denominators therefore to add these two fractions together you need to find the LCD, which is 6 .
Converting these fractions into their equivalent types would result in the following:
\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \\ \ \\ \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \\ \ \\ \frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}
Now that the fraction part of this operation has been done and it did not result in an improper fraction you would add the mixed numbers together:
1 + 2 = 3
Then to conclude it would be a simple case of adding the whole number with the fraction:
3 + \frac{5}{6} = 3 \frac{5}{6}
Example 2 – Improper Fraction Result
Suppose the following fraction addition problem:
3 \frac{3}{4} + 2 \frac{4}{5}
Separate out each of the mixed numbers and fractions out:
3 + \frac{3}{4} + 2 + \frac{4}{5}
Then focus on adding the fractions together, as they have different denominators you would need to find their equivalent where both fractions have the same denominator:
\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \\ \ \\ \frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20} \\ \ \\ \frac{15}{20} + \frac{16}{20} = \frac{15 + 16}{20} = \frac{31}{20} = 1 \frac{11}{20}
Separate the result of the fraction addition, which produced an improper fraction and was converted to a mixed fraction and add the mixed numbers up and add that to the fraction, like so:
3 + 2 + 1 + \frac{11}{20} = 6 + \frac{11}{20} = 6 \frac{11}{20}
Adding Fractions Summary
To add fractions together, each fraction needs to have a common denominator.
If the denominators are different, you’ll need to find the least common multiple (LCM) of the denominators. To do this, either list the multiples of each denominator until you find a common one. Once you have the LCM, you can convert each fraction to an equivalent fraction with the same denominator.
To learn more about equivalent fractions, read this article.
Then once the denominators are the same, add their numerators and write the sum over the common denominator. With the resulting fraction see if both the numerator and the denominator can be divided by a common factor to reduce the result to it’s simplest term.