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# Nested fractions

## 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 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 numerator is here a C you're just going to multiply those out so it's going to be a times C which we could just write as a C that's just a times C all of that over the denominator b times d ve times b times d instead of multiplying what would have 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 its by its 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 let me do that 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 what this multiplication symbol now that we're in algebra because that you might confuse that with an X let me write that as x times d D over C times D over C and what are you left with well the numerator are going to have a times D so it's a d a.a d over Oh over B C 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 and 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 just let's also divide that by 1 over D so this is a more involved expression than 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 could 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 B a notice I haven't changed its value I just multiplied it times 1 B over B minus well I'm going to multiply the numerator and denominator here by a minus a over a B or I can write that as B a and the whole reason why I did that is to have to come the same denominator so this is going to be equal to B minus a over I could write it as B A or a B so this is going to be equal to this is going to be equal to this numerator right over here B minus a over a B 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 have B minus a times 1 times D so we could write this as D times B minus a times B minus a and then in the denominator I have a b c a b and c and then finally we can use the distributive property here we can distribute we can distribute this d and we're going to be left with we're going to be left with we deserve a a minor drumroll at this point we could write this as d times b B times B minus D whoops I'm go into that same green color so you really see how it got distributed minus D times a all of that over all of that over a be a B C and we are done