Common denominators: 1/4 and 5/6 | Math | 4th grade | Khan Academy

You have two fractions: 1/4 and 56, 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 56, and we want to rewrite these fractions to have new denominators. We currently have a four and a six as our denominator. Can we just put any new thing, like maybe five? Could we say, "Let's change them both to have five as the denominator?" The answer is no. We have to pick a multiple of four and six. A multiple is some number that we can multiply four to get this number as an answer.

For example, for four, some multiples of four would be:

• 4 * 1 = 4
• 4 * 2 = 8
• 4 * 3 = 12
• And so on.

Those are multiples of four. Now, let’s pause here and look at why we have to pick a multiple of four and six, and why 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 1/4. We could look at either one, but let's look at 1/4 here. We have a picture showing fourths, and to show 1/4, we shade one of these four equal size 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 going to need to split this fourth up here into two pieces. Now I have two shaded pieces. So, can I say this is 2 out of 1, 2, 3, 4, five pieces? It's not two-fifths because these are not equal size 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. Now this is one, two pieces out of 1, 2, 3, 4, 5, 6, 7, eight equal size pieces. So, 28.

You can see 8 is a multiple of four because we multiplied by two. That's what we did: we multiplied each of our pieces by two, 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.

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 shade 1/4, one of the four pieces, and maybe this time we want to split it into three equal size pieces.

So, to have a new numerator of three, 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 size pieces. So, to get equal size pieces, we'll need to split each of these fourths into three. We are tripling the amount of pieces, so now we have three shaded pieces out of a total of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12.

I could have figured that out without even counting because I knew that we tripled. This time we multiplied our original denominator by three. We tripled, and we also multiplied our numerator by 3.

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. We can multiply the entire amount of pieces.

And again, so this is super clear: 1/4 and 28 and 31/12. They all represent the same amount. Whether we had 1/4, the original here, 2/8, or 3/12, 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 sixes? Well, we know we need to use multiples.

Let's look at the multiples for four. We've already gone through some of these. The first multiple of four is 4 * 1, which is 4. The second multiple of four is 8, 4 * 2 = 8. So we could split our fourths into halves and get eighths. Or we could say 4 * 3 = 12, which we showed again where we split our fourths, each fourth, into three equal pieces.

Or we could do 4 * 4 = 16, 4 * 5 = 20, and 4 * 6 = 24, and so on. The reason I'm stopping at 24 is I've looked at my answer choices, and I can see the large possible answer is 24, so I don't need to write any larger multiples. There are many, many more multiples of four, 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, 6 * 1, and keep six pieces, or we could double our six. 6 * 2 would be 12. If we doubled the pieces, we would have 12 pieces. We could say 6 * 3, which is 18, or 6 * 4.

We could divide each of our six into four pieces, and we'd have 6 * 4, which is 24. 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 eights, but eight is not a multiple of six, so we cannot split six into eights. So 8 will not work as a denominator for both fractions.

How about 12? 12, we can see, is a multiple of four, and we showed that; we drew that already. And 12 is a multiple of six. We could split our six into two equal pieces each, and we would have twelfths. So, 12 does work; 12 is a common denominator for fourths and six.

18? 18 is here on the six. We could split six into 18 because 18 is a multiple of six, but it is not a multiple of four, so we can rule out 18. 18 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 six. So we could use either 12 or 24.

There's a lot more numbers we could use too as common denominators, but from these choices, we could use 12 or 24 as a common denominator for fourths and six.

Just a note: lots of times, people like to use the smallest one, the least common denominator, which in this case is 12. 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 12 or 24 or lots of other options.

But again, 12 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 fourths and sixths are 12 and 24.