To add two fractions, they must have the same denominator. When they do have the same denominator,
we add fractions by simply adding their numerators and putting the result over that denominator that they both have.
If we add 2/7 and 3/7, the result is going to be 5/7 because the denominators are the same; so to add, we add the 2 and the 3 to get 5.
Now, the reason why this is true is because of the distributive property. And I've written out the steps right here, just so that you can see that—
that it is the distributive property that requires a common denominator for fractions;
and it's the distributive property that tells us just to add the numerators. 2/7 can be written as 2 times 1/7. 3/7 can be written as 3 times 1/7.
When the expression is written in this form, I see I have a 1/7 common to this term and this term.
I can write it outside using the distributive property and group the 2 and the 3 together. Now, (1/7)(3) is this; (1/7)(2) is this.
So, I know this expression is equivalent to this one. Now, when I add 2 + 3, I end up with 5; (5)(1/7) = 5/7. So,
following this line of reasoning right here, you can see that 2/7 + 3/7 = 5/7, and the reason why is because of the distributive property.
It also shows you that I must have the same denominator right here, so that this number and this number are exactly the same.
So to summarize, then: we add fractions by first making sure that they have the same denominator;
then we add by adding numerators and putting that result over that common denominator.
Let's look at our first example: 1/4 + 2/4 + 3/4. I notice that the denominators are all the same,
so to add these fractions, I'm simply going to add numerators, 1 + 2 + 3, and put that result over that common denominator.
Now, 1 + 2 = 3, + 3 = 6. I have 6/4; and of course, I want to reduce to lowest terms whenever I can.
I notice there's a factor of 2 common to the numerator and denominator, I divide it out, and I end up with the reduced form of that sum of 3/2.
Now, what we want to do next is add some fractions that don't have the same denominator; and in order to do that,
we're going to have to change to equivalent fractions that contain a common denominator.
Now, we have a definition right here for the least common denominator:
the least common denominator for a set of fractions is the smallest number that we can find that's divisible by all the numbers in the problem.
So, the least common denominator for a set of fractions is the smallest number that's divisible by all the numbers in the fraction.
Let's go to the board now and work a problem that involves adding fractions with different denominators: 1/8 + 3/4.
I want to add these two fractions. I notice the denominators aren't the same.
I want to look for the least common denominator, and it will be the smallest number that's divisible by 4 and also by 8—
that happens to be the number 8. So, I want to write each one of these fractions as an equivalent fraction with that denominator 8.
This fraction already has the 8; this one doesn't. In order to get an 8, I need to multiply the denominator by 2;
I do the same thing for the numerator. Let's see, I'm going to rewrite this a little bit. Let me do it this way. I'll just extend that here.
I'll multiply the numerator and denominator by 2. That's my property one for fractions, and I know that doing that produces an equivalent fraction.
Now, what I have is 1/8 + 6/8. So, 3/4 is equivalent to the fraction 6/8. But now, I have two fractions with the same denominator,
so to add them, I simply add numerators and put the result over that common denominator. I end up with a sum of 7/8.
So, to add 1/8 and 3/4, I see first of all that the least common denominator is 8.
It's 8 because 8 is the smallest number that's divisible by 4 and also by 8. So, I see I have to change this fraction, 3/4, to an equivalent fraction:
that is, a fraction with the same value that has that denominator 8. I multiply its numerator and denominator by 2—
that's property one for fractions; that produces the equivalent fraction 6/8. Now I'm ready to add, and the rest is very easy.
Simply add numerators; put that result over the common denominator. Let's try another one: 2 + 1/3. I'm going to rewrite this then as,
let's see, 2 over 1, plus 1/3. So, remember the number 2 is the same as the fraction 2 over 1.
Now, I see that the smallest number that's divisible by 1—that's divisible by 1 and by 3 is 3.
So, I need to multiply this fraction by—numerator and denominator by 3; that gives me 6/3 + 1/3.
Now the rest is easy; 6 + 1; I add numerators, put the result over the common denominator, and I end up with the improper fraction 7/3.
Now, notice here that I multiplied the numerator and denominator by 3. That gives me the fraction 6/3. Isn't 6/3 exactly the same as 2?
So, I mean, you know intuitively that I haven't changed the value of this number 2, right here. But when I write it in this form, 6 over 3,
I know now that I have two fractions with the same denominator, and that's the requirement for being able to add fractions.
Let's look now at our next example: I want to add 17/30 + 11/42.
Now, I can't see just by looking at these what's the smallest number that is divisible by 30 and also by 42.
So, what I'm going to have to do is write these denominators in factored form and build a least common denominator from them.
So, let's start by writing this as 17 over, let's see, 30 = 6(5); 6 = 2(3), and then times 5.
The other fraction will be 11/42, which is 6(7); so, 6 again is 2(3), times 7.
Now, I want to build my least common denominator from these factors.
So, my least common denominator, in order for 2(3)(5) to divide it, it must have factors of 2(3)(5).
So, now I know that this denominator will divide my least common denominator. For this denominator to divide it, it needs to factor a 2;
it has one, so I don't put another in. It needs a factor of 3; it has one, so I don't put another one in.
It needs a factor of 7. It doesn't have one, so I put in a 7.
So, this least common denominator, 2(3)(5)(7), is divisible by both of these denominators.
If I divide (2)(3)(5) into this, what I get is 7. If I divide (2)(3)(7) into this, what I get is 5.
Next—the next thing we want to do is write each of these fractions as an equivalent fraction with this denominator.
So, I look at my first fraction, and I see factors of 2, 3, and 5 in the denominator. This is the denominator I want it to have;
it needs a factor of 7. So, I'm going to multiply the numerator and denominator of this fraction by the number 7.
I look at my next fraction; it has a denominator with factors of 2, 3, and 7. It needs a factor of 5; it doesn't have one.
So, I'm going to put a factor of 5 in by multiplying the numerator and denominator by 5. This is the step at which I change to equivalent fractions.
Now, let's actually go do this multiplication. So, here I have 7(17) = 119;
over (2)(3)(5)(7); plus 11(5) = 55, over 2(3)(5)(7). So, I'll add these two together,
and I'll end up with 174 over (2)(3)(5)(7). Now, notice that I haven't multiplied this denominator back out to see what it is
because at this step, right here, I may be able to reduce to lowest terms. If I can reduce to lowest terms,
it will be because the numerator and denominator have some factors in common. So, why not leave my denominator in factored form
until the very end of the problem. And I see that 174 is divisible by 2, so let's see what happens when we factor it.
Well, actually, I've already done this problem, and I know that 174 = 2(3)(29).
So, I'm going to just save myself a little time here and write that numerator in factored form; you can check it out:
2(3) = 6, times 29 is going to be 174; the denominator, 2(3)(5)(7); and as you can see, these common factors divide out,
and I'm left with 29 over 5(7), which is 35. Now, you might want to check this on a calculator.
Just take 29, divide by 35; you'll get some decimal number; I know we haven't covered decimals yet, but you can see
that one decimal number will be the same as another one. Do that division, do this on calculator too,
and you'll see that you get the exact same number: 29/35 is the sum of these two fractions.
So, what we do here is to first factor each denominator, build the least common denominator from that;
multiply the fractions, numerator and denominator, by the factors that the least common denominator has
that their denominators don't have. That produces these equivalent fractions.
The rest is simply a matter of adding the numerators and reducing to lowest terms. It's a little more complicated problem
because the numerator and denominator are large numbers. I suggest leaving that least common denominator in factored form
until you're right at the very end of the problem, after you've reduced to lowest terms. Then, you can multiply things back out
and see what you get. Let's try one more problem. I want to add these four fractions.
Now, I can build this least common denominator up by factoring these, or I can just simply look at these numbers and ask myself,
"What is the smallest number that's divisible by 2, by 3, by 4, and by 6?" I think, if you look at that for a minute,
you'll see that the least common denominator in this case must be the number 12.
So, to write 1/2 as an equivalent fraction with a denominator of 12, I multiply numerator and denominator by 6.
When I do that, I get 6/12. And isn't 6/12 the same as 1/2? Now, 1/3 with a denominator of 12: I would have to multiply this denominator by 4;
I do the same thing to the numerator; I end up with 4/12. 1/4, to write it as a fraction with a denominator of 12,
I'd have to multiply 4 by 3 to get 12; I do the same thing to that numerator; I have 3/12. And then 1/6, of course, is going to be 2/12.
So, whatever way you go about working this problem, these are the equivalent fractions you'll get
when you take these fractions and rewrite them as fractions with a denominator of 12.
Now, the rest is simply a matter of adding the numerators and then reducing to lowest terms, if we can. Let's see what that looks like:
6 + 4 + 3 + 2, all divided by 12; that's going to be 6 + 4 = 10, + 3 = 13, + 2 = 15. And then, of course, 15 over 12:
3 divides the numerator and 3 divides the denominator. I'll do that division, and I end up with 5/4. So, kind of a surprising result, I guess.
I get an improper fraction, 5/4 when I add these four fractions right here. If you can't see that that least common denominator is 12,
factor each of denominators: this will be 2(2); this will be 2(3); and build that least common denominator
from the factors of each of these denominators, just like we did in the previous example.