Common denominators: 1/2 and 1/3 | Math | 4th grade | Khan Academy

You have two fractions: 1/4 and 5/6, and you want to rewrite them so they have the same denominator and have whole number numerators. What numbers could you use for the denominator?

So here's our fractions: 1/4 and 5/6, and we want to rewrite these fractions to have new denominators. We currently have a 4 and a 6 as our denominators. Can we just put any new thing, like maybe 5? Could we say, "Let's change them both to have 5 as the denominator"? The answer is no. We have to pick a multiple of 4 and 6.

A multiple is some number that we can multiply 4 by to get this number as an answer. For example, for four, some multiples of four would be: four times one is four, four times two is eight, four times three is twelve, and so on. Those are multiples of 4.

And let's just pause here and look at why we have to pick a multiple of 4 and 6. Well, we can't just pick any number; we have to pick a multiple of our denominators. So the fraction we were just talking about was one-fourth. We could look at either one, but let's look at one-fourth here. We have a picture showing fourths, and to show one-fourth, we shade one of these four equal-sized pieces.

Well, maybe I want to change this, and I want to say, "I want two." I want two as my numerator. So to have a numerator of two, I'm gonna need to split this fourth up here into two pieces. Now I have two shaded pieces. So can I say this is two? One, two out of one, two, three, four, five pieces?

It's not two-fifths because these are not equal-sized pieces. So if I split this fourth right here in half, I need to split all of them in half, and what I'm doing is doubling the amount of pieces. So now this is two pieces, this is two, and this is two because we need equal size pieces.

So now this is one, two pieces out of one, two, three, four, five, six, seven, eight equal-sized pieces. So 2/8. And you can see 8 is a multiple of 4 because we multiplied by 2. That's what we did. We multiplied each of our pieces by 2, and we also multiplied our numerator by two because that was also doubled.

The amount of shaded pieces doubled when the entire amount of pieces doubled. And we don't have to do this just with two; we could do any multiple of four. For example, we can do one more here. If we again, let's shade one-fourth, one of the four pieces, and maybe this time we want to split it into three equal-sized pieces.

So to have a new numerator of three, and these should be equal-sized, here's a numerator of three. But we can't do our denominator yet because we don't have equal-sized pieces. So to get equal-sized pieces, we'll need to split each of these fourths into three. So we are tripling the amount of pieces.

So now we have three shaded pieces out of a total of one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. Out of twelve. And I could have figured that out without even counting because I knew that we tripled this time. We multiplied our original denominator times three; we tripled, and we also multiplied our numerator times three.

So these are the multiples: 8, 12, and so on. Those are the denominators we can pick. Something that we can multiply our denominator by so we can multiply the entire amount of pieces.

And again, so this is super clear: 1/4 and 2/8 and 3/12, they all represent the same amount. Whether we had one-fourth: the original. Here's two-eighths, three-twelfths; they're all equivalent. They all represent the same amount. They're just different ways of writing the same number.

Back to our original question: What denominators can we use for fourths and for six? Well, we know we need to use multiples. So let's look at the multiples for four. We've already gone through some of these. The first multiple of four is four times one, which is four.

The second multiple of four is eight: four times two is eight. So we could split our fourths in half and get eighths, or we could say four times three is twelve, which we showed again where we split our fourth, see each fourth into three equal pieces. Or we could do four times four, which is sixteen, four times five is twenty, four times six is twenty-four, and so on.

The reason I'm stopping at 24 is I've looked at my answer choices, and I can see the largest possible answer is 24. So I don't need to write any larger multiples. There are many, many, many more multiples of 4, but we don't need to list them all because the largest number we're going to have to consider is 24.

Let's do the same for six. We could leave our six alone: six times one and keep six pieces. Or we could double our six: six times two would be twelve. If we doubled the pieces, we would have twelve pieces.

We could say six times three, which is eighteen, or six times four. We could divide each of our six into four pieces, and we'd have six times four, which is 24, or 20 fourths, and so on. Again, I'll stop at 24 since it's the largest number we need to consider.

So down to our answer choices: What numbers could we use for the denominator? Could we use eight? Let's look at these lists: eight is a multiple of four, so we could definitely split fourths into eighths, but eight is not a multiple of six, so we cannot split six into eighths. So eight will not work as a denominator for both fractions.

How about twelve? Twelve, we can see, is a multiple of four, and we showed that; we drew that already. And twelve is a multiple of six. We could split our six into two equal pieces each, and we would have twelfths. So twelve does work. Twelve is a denominator, a common denominator for fourths and sixths.

Eighteen: eighteen is here on the sixth. We could split six into eighteen because eighteen is a multiple of six, but it is not a multiple of four. So we can rule out eighteen. Eighteen is not a common denominator.

And 24, you may remember, was the last number we wrote on both of them. So yes, 24 could be a denominator for fourths and sixths. So we could use either 12 or 24, and there's a lot more numbers we could use as common denominators, but from these choices, we could use 12 or 24 as a common denominator for fourths and six.

And just a note: Lots of times people like to use the smallest one, the least common denominator, which in this case is 12, and it makes a lot of sense because it's easier to do computation with smaller numbers. But you don't absolutely have to use the smallest one.

You could use twelfths or twenty-fourths or lots of other options. But again, twelfths is probably the simplest one to work with just because generally, it's easier to work with smaller numbers. But for this question, the common denominators we can use from these choices for 4th and 6th are 12 and 24.