Hi, 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 7 rolls in a basket on the dinner table.
You ate 1 and your brother ate 2.
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 1 plus 2 over 7, equals three-sevenths.
The fraction, three-sevenths, represents the number of rolls eaten by you and your brother.
Subtracting fractions can be thought of in the same way.
Let’s say you and your friends are playing a game of cards.
You're holding 3 of the 4 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 4 Kings in a deck of cards, the fraction, three-fourths, represents the 3 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 1.
The fraction of the Kings that you have in your hand changes as follows:
three-fourths minus one-fourth, which can be seen as 3 minus 1 over 4, equals two-fourths, which simplifies to one-half.
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:
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, or 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 the denominator.
The rules for multiplying fractions require multiplying the numerator times the numerator and the denominator times the denominator:
2 times 2 is 4 and 2 times 5 is 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 4 plus 3 over 10, equals seven-tenths.
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.
So, the 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.
24 divided by 6 equals 4, so we'll have to multiply the numerator and denominator of the second fraction by 4.
These adjustments create equivalent fractions that have a common denominator of 24.
Once that's taken care of, the numerators can be subtracted as follows:
Alright, now here’s one for you to try.
Pause the video and see if you can solve it.
How'd you do?
Let’s walk through it together.
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!
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!