Common denominators: 1/4 and 5/6

- [Voiceover] 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. So we currently have a four and a six as our denominator, and could we just pick 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, some number that we can multiply four and get this number as an answer. So for example, for four, some multiples of four would be four times one is four, four times two is eight, four times three is 12, and so on. Those are multiples of four. And let's just pause here and look at why we have to pick a multiple of four and six, why we can't just pick any number but 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. Maybe I wanna change this and I wanna 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 2/5 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. So now, this is one, two pieces out of one, two, three, four, five, six, seven, eight equal-size pieces. So 2/8. And you can see, eight is a multiple of four because we multiplied by two, and that's what we did. We multiplied each of our pieces by two. 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. Now, 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 1/4, one of the four pieces. And maybe this time we wanna split it into three equal-size 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-size pieces. So to get equal-size 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, 10, 11, 12, out of 12. 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, eight, 12, and so on, and so 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 1/4, the original, here's 2/8, 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 sixths? 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. 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 12, which we showed again where we split our fourths, each fourth into three equal pieces. Or we could do four times four which is 16. Four times five is 20, four times six is 24, 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 four, but we don't need to list them all 'cause the largest number we're gonna have to consider is 24. Let's do the same for sixths. We could leave our sixths alone, six times one, and keep six pieces. Or we could double our sixths. Six times two would be 12, if we doubled the pieces we would have 12 pieces. We could say six times three, which is 18, or six times four, we could divide each of our sixths into four pieces, and we'd have six times four which is 24 or 24ths. 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 can not split sixths into eighths. So... Eight 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 sixths into two equal pieces each and we would have 12ths. So 12 does work, 12 is a denominator, a common denominator for fourths and sixths. 18, 18 is here on the sixths. We could split sixths 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 sixths. So we could use either 12 or 24, and 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 sixths. 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 12ths or 24ths or lots of other options. But again, 12ths is probably the simplest one to work with just 'cause 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.