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# Nested fractions

CCSS Math: HSA.APR.D.7

## Video transcript

- Let's deal with some
algebraic expressions that involve multiplying fractions. So let's say that I had a over b, a over b times c over d. What is this going to be? I encourage you to pause the video and try to figure it out on your own. Well, when you multiply fractions, you just multiply the numerators and multiply the denominators. So the numerators here, a, c, you're just going to multiply those out. It's going to a times c, which we can just write as ac, that just a times c, all of that over the denominator, b times d, b times d. Instead of multiplying,
what would of happened if we were dividing? So if we had a over b, a over b divided by, divided by c over d, what would this be? And once again, encourage you, encourage you to pause the video and figure it out on your own. Well, when you divide by a fraction, it is equivalent to multiplying by it's, by it's reciprocal. So this is going to be the same thing as a over b, a over b times, times the reciprocal of this. So times d over, I'm going to use the same color just so
I don't confuse you, that d was purple, times d over c, times d over c and then it
reduces to a problem like this. You know, and I shouldn't even use this multiplication symbol
now that we're in algebra because you might confuse that with an x, so let me write that as times, times d, d over c, times d over c, and what are you left with? Well the numerator you're
going to have a times d, so it's ad, a, d, over, over bc, over b times c. Now let's do one that's maybe
a little bit more involved and see if you can pull it off. So let's say that I had,
let's say that I had, I don't know, let me write it as 1 over a, minus 1 over b, all of that over, all of that over c, and let's say, let's also divide that by 1 over d. So this is a more involved expression then what we've seen so
far but I think we have all the tools to tackle
it so I encourage you to pause the video and see
if you can simplify this, if you can actually carry
out these operations and come up with a one
fraction that represents this. All right so let's work
through it step by step, so 1 over a minus 1 over b, let me work through just
that part by itself. So 1 over a minus 1 over b, we know how to tackle that, we can find a common denominator, let me write it up here. So 1 over a minus 1 over b, is going to be equal to, we can multiply 1 over a times b over b, so it's going to be b over ba, notice I haven't changed it's value, I just multiplied it times 1, b over b, minus, well I'm going to multiply the numerator and denominator here by a, - a over ab, or I could write that as ba. And the whole reason why I did that is to have the same denominator. So this is going to be equal to b-a, over, I could write is a ba or ab. So this is going to be equal to, this is going to be equal to
this numerator right over here, b-a, over ab, and then, if I'm dividing it by c, that's the same thing as multiplying by the reciprocal of c. So if I'm dividing it by c, that's the same thing as multiplying, that's the same thing as
multiplying times 1 over c. And if I am, and I'll
just keep going here, if I'm dividing by 1 over d, if I'm dividing, notice
this is the same thing as division right over here. If I'm dividing by c,
that's the same thing as multiplying by the reciprocal of c. And then finally, I'm
dividing by 1 over d, that's the same thing as multiplying by the reciprocal of 1 over d. So the reciprocal of 1 over d is d over, d over 1. And so what does this result with? Well in the numerator I
have b-a times 1 times d. So we can write this as d times (b-a), times (b-a) and then in the denominator I have abc, ab and c. And then finally, we can use
the distributive property here, we can distribute this d, and we're going to be left with, we deserve a minor drum
roll at this point, we can write this as d times b, d times b minus d,
woops, I want to do that in the same green color so you really see how it got distributed, minus d times a, all of that over ab, abc. And we are done.