Addition and Subtraction of Fractions with Common Denominators

[One speaker] 

Narrator: Hi, welcome to the video on the addition and subtraction of fractions with common denominators. Common denominator just means that all of the pieces are the same size.

[Two circles are shown, one is divided into eighths with half of the pieces shaded in red. The other circle is divided into eighths with three pieces shaded red]

For example, here we have two pizzas. Each pizza has been divided into eight pieces, so they have a common denominator of eight. But, both pizzas have had some slices eaten, so the remaining amount in the pizza on the left is four-eighths [Writes 4/8 under the pizza with 4 shaded red.] and the remaining amount in the pizza on the right is three-eighths. [Writes 3/8 under the other pizza.] Or in other words, three out of the eight pieces still remain.

If we wanted to combine these two pizzas just into one, how many slices would we have? Well, let’s go ahead and combine these two pizzas.

[Clicks on the pizza of 3/8 and drags it onto the pizza of 4/8]

Here we see if we put all of the slices together, we have seven out of the eight pieces. So somehow four-eighths plus [Writes +] three-eighths equals seven-eighths. [Writes = 7/8] Well, 4 plus 3 equals 7 and the denominator stayed the same because we didn’t change the size of any of the slices.

So from this we see that as long as the shapes, or our slices, are all the same size, or in other words, as long as the denominator is the same, we can add the numbers in the numerator. Four plus three is seven and all of the pizza slices are the size of one-eighth of a pizza. So we have seven-eighths left over.

[Shows 7/8 pizza with “7/8 3/8 =” underneath.]

Now let’s suppose we have seven-eighths of the pizza and we want to give three-eighths of the pizza away. How much would be left? This is the same as seven-eighths minus [Writes -] three-eighths.

Now once again, since we’re talking about each slice being the same size, so all of these slices are only one-eighth of a pizza, it’s easy for us to calculate, and we can just take away our three slices [Drags the 3/8 pizza away from the 7/8 pizza leaving 4/8 of the pizza left.] and we’re left with four slices. So the answer is four-eighths. [Writes 4/8]

So seven-eighths minus three-eighths equals four-eighths. Here we have our representation of four-eighths. Notice, this is the same as one half. Equals one half. [Writes = 1/2 next to 4/8.] This makes sense because four is two times two and eight is two times two times two. [Writes a fraction with 2 · 2 in the numerator and 2 · 2 · 2 in the denominator.] If we break it down into its prime factorization. Two divided by two is one [Crosses out a 2 in the numerator and denominator.] and two divided by two is one. [Crosses out a 2 in the numerator and denominator.] And this is the same as times one. [Writes · 1 in numerator.] So one over two is what is left, which equals one half. [Writes = 1/2 next to the fraction.] 

[Two pies divided into 5 equal sections. One has 2 sections shaded in the other has 1.]

Here’s another example of two pies. These pies have been divided into five equal pieces. The number of slices in the pie on the left is two out of five, [Writes 2/5 under the pie.] or two-fifths of the pizza remain, and the pie on the right has one out of five pieces, [Writes 1/5 under pie.] or one-fifth of the pizza.

Now, if we want to combine the two, we can do that because all of their slices are the same size. We bring this one over, [Drags one pie on top of the other.] the two combined equal three-fifths or three out of the five possible slices. [Writes 3/5.] 

So now we have three-fifths, [Writes 3/5.] or three out of five, and let’s say we want to take away two out of the five slices and give those to two friends. [Writes - 2/5 =] How many would we have left? How many slices would we have left?

Once again, since all of the pizza slices are the same size, and that’s a really important feature in that the denominators have to be the same when doing addition or subtraction.

As long as all the pizza slices are the same, if we take away two out of the five pieces, let’s grab it and take away two and move it over here, [Drage pie with 2 sections shaded away from the combined pie leaving a pie with just one section shaded.] we are only left with one slice, so the answer is one-fifth. [Writes 1/5 ] 

I’m referring to this one-fifth over here [Draws arrow to pie with one section shaded.] because we took these two away. So again, as long as the denominators are the same, we can subtract the numerators. So three minus two equals one. We’re left with one-fifth.

[End of video.]