# Adding and Subtracting Fractions

## Adding and Subtracting Fractions

Hi, and welcome to this video on adding and subtracting fractions!

Before we get into it, let’s review some terminology needed to understand the concepts.

A fraction is a **ratio** of values that reflect a “part” to a “whole.” The “part” is called the **numerator** and is written above the division line. The “whole” is referred to as the **denominator** and is written below the division line:

When combining fractions by addition or subtraction, the work is done only with the numerators. The denominator does not change. For example, suppose there were seven rolls in a basket on the dinner table. You ate one and your brother ate two. What is the fraction that represents the number of rolls that were eaten by you and your brother?

You can probably conceptualize this example of adding fractions pretty quickly. You simply add up the number of rolls eaten by you and your brother and divide by the total number of rolls that were on the table at the start of dinner. Remember, the denominator remains the same. One seventh plus two sevenths, which can be seen as one plus two over seven, equals three sevenths. The fraction \(\frac{3}{7}\) represents the number of rolls eaten by you and your brother.

\(\frac{1}{7}+\frac{2}{7}=\frac{1+2}{7}=\frac{3}{7}\)Subtracting fractions can be thought of in the same way.

Let’s say you and your friends are playing a game of cards. You are holding three of the four Kings that are in a deck of cards. You throw down the King of Hearts on the next play. What fraction represents the number of Kings in your hand now?

Because there are a total of four Kings in a deck of cards, the fraction, 34, represents the three Kings that you had in your hand to start with. If you give away the King of Hearts, then the numerator of this fraction decreases by one. The fraction of the Kings that you have in your hand changes as follows: three fourths minus one fourth, which can be seen as three minus one over four, equals two fourths, which simplifies to one half.

\(\frac{3}{4}-\frac{1}{4}=\frac{3-1}{4}=\frac{2}{4}\text{ or }\frac{1}{2}\)These examples are pretty straightforward because the denominators of the fractions being added or subtracted are the same. If denominators are not the same, there is a bit more work involved. Specifically, one or both of the fractions must be algebraically adjusted to create **common denominators**.

Let’s work on a few examples:

\(\frac{2}{5}+\frac{3}{10}\)Like I said before, these fractions cannot be added until they share a common denominator. In fact, the denominator must be the smallest value that both denominators can divide into evenly. This value is known as the **Least Common Denominator (LCD)**.

When considering the denominators 5 and 10, it becomes clear that 10 is the smallest number that both 5 and 10 can divide into evenly. This means that we will have to algebraically adjust the first fraction so that the denominator becomes 10. We do this by multiplying both the numerator and denominator. The rules for multiplying fractions require multiplying the numerator times the numerator and the denominator times the denominator: two times two is four and two times five is ten.

\(\frac{2}{2}\times \frac{2}{5}=\frac{4}{10}\)This work creates the fraction four tenths, which is equivalent to the original fraction, two fifths. At this point, the fractions with common denominators can be added: four tenths plus three tenths, which can be seen as four plus three over ten, equals seven tenths.

\(\frac{4}{10}+\frac{3}{10}=\frac{4+3}{10}=\frac{7}{10}\)This last example requires you to adjust both fractions to achieve a common denominator. Let’s work through this one together, then I’ll give you one to try on your own.

\(\frac{5}{8}-\frac{1}{6}\)So, first things first: What is the least common multiple of 6 and 8? 24. Because 24 divided by 8 is 3, we will have to adjust the first fraction by multiplying both the numerator and denominator by 3. Twenty-four divided by six equals four, so we will have to multiply the numerator and denominator of the second fraction by four.

\(\frac{3}{3} \times \frac{5}{8}-\frac{4}{4} \times \frac{1}{6}\)These adjustments create equivalent fractions that have a common denominator of 24. Once that is taken care of, the numerators can be subtracted as follows:

\(\frac{15}{24}-\frac{4}{24}=\frac{11}{24}\)Alright, now here’s one for you to try. Pause the video and see if you can solve it.

\(\frac{3}{5}+\frac{3}{7}\)How did you do? Let’s walk through it. The least common multiple of 5 and 7 is 35. Adjust the first fraction by multiplying the numerator and denominator by 7, and adjust the second fraction by multiplying the numerator and denominator by 5. Once the denominators match, add the numerators. This gives us thirty-six thirty-fifths!

\(\frac{7}{7} \times \frac{3}{5}+\frac{5}{5} \times \frac{3}{7}\) \(\frac{21}{35}+\frac{15}{35}=\frac{36}{35}\)Let’s recap before we go. Whether you are adding or subtracting fractions, remember that the denominator will always stay the same, and sometimes, you will have to create a common denominator and find the least common multiple before you can proceed with the problem.

I hope this review was helpful! Thanks for watching and happy studying!