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How To easily Multiply & Simplify Rational Numbers
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Feb 19, 2025
When it comes to math, there are a lot of rules and procedures that can seem confusing at first. But once you understand the basics, everything else falls into place. Today we're going to be discussing rational numbers, and more specifically, how to multiply and simplify them. This is a fundamental skill that will come in handy in a variety of situations, so pay close attention and don't hesitate to ask your teacher for additional help if you need it.
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so we're going to talk about
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multiplying rational numbers now but the
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first thing that we need to understand
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is what is a rational number we need to
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know the definition okay so basically if
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you look on my board any number that can
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be written in the form of A over B of a
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quotient of two integers is called a
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rational number as long as B which is
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the denominator is not equal to zero so
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basically to recap it a rational number
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is a number that will be written as a
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ratio of two integers as long as the
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denominator is not equal to zero and
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I'll give you some example
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for example if you have five
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five is a rational number y because 5
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can be written as five over one A over B
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so this is a rational number
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one half is a rational number because
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this is one over two again the quotient
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is also called a division or a ratio so
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anytime you can write a number as a
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division of two numbers as long as the
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denominator is not equal to zero is
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called a rational number an example
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negative three is a rational number
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because negative three is a quotient of
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what negative three and one these are
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both integers okay integers and Compass
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positive and negative numbers so an
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integer can also be called a rational
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number okay and once you you also have
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0.5 so 0.5 is also a rational number
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because 0.5 is basically 5 over 10 okay
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so if you have a DOT here this is the
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tenth space so every time you move the
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decimal from one place
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you always divide the number by ten so
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if I have zero when zero five this will
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be 5 over 100 because I'm moving one
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place two place this is the tenth place
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this is a hundredth place so every time
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you move the decimal one place to the
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right you add a zero and that becomes a
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quotient okay so 0.05 is also a rational
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number okay now here's the thing when
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you have rational number you always want
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to make sure you simplify them okay you
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don't leave them you don't leave 0.5 as
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five okay
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here's what you do you can break down 5
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over 10. right
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so 0.5
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is basically 5 over 10. now we don't
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leave this like this we have to simplify
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simplify
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meaning the meaning of simplifying is
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you want to put these terms in its
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lowest form okay so now what how do you
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do that you need to find a number that
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both divides 10 and 5 at the same time
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we call that number the GCF
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g c f are the greatest common factor
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greatest
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common
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Factor okay
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so basically when you have five over ten
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what goes into five that goes into ten
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five so this is how you break it down so
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let's break that up okay
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so I'm going to invite this
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five would just be five times one and
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ten
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will be
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five times two and now you have the same
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time on on the numerator and the
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denominator you could just cancel it out
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so this becomes one
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over two so five over ten is one over
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two now the next thing we're going to
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learn is how do you multiply rational
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numbers okay this is pretty easy you
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should pay attention
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so let's get to that
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so first I'm going to give you the
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general formula and then we're going to
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work on some examples
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so A over B
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times C over D equals a times C over B
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over D so basically when you multiply
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rational numbers you just multiply
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across okay the numerators are going to
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be multiplied together and the
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denominators as well so this becomes a
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times C over B times D let me give you
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an example
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if I have one over two
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times three over five this becomes one
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times three over two times five which is
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basically three over ten okay so one
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half over times three over five is three
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tenths now there's one thing when you
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multiply rational numbers you always
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have to make sure
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you you heard them is this in their
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simplest form basically if I'm here okay
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I wanna see if
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um three and ten have a GCF a greatest
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common factor they don't because there's
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no number that divides three and ten and
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the only number that goes into ten
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thousand that goes into three is one so
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I'm gonna stop here okay but then we
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have another case
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let's assume that I have
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three over four right
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times seven over nine all right so to
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multiply these numbers
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we're gonna do three times seven right
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which is 21 and then four times nine and
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that is 36. so when you get here okay
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most people who multiply and then trying
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to find the GCF which is the greatest
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common factor so y goes into 21 that
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goes into 36 if you know your
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multiplication table you know that 3
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goes into 21 and 3 also goes into 36. so
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basically if you divide this by three
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you get a seven and if you divide it by
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three you get a 12 and that gives you 7
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over 12. but here's the problem I don't
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like doing this this way because it gets
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harder because basically you multiply
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these numbers and then you're going to
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try to simplify them so is there an
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easier way to do this yes there is so
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we're gonna get to that
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so if I have 3 over 4
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times 7 over 9. so what I'm trying to do
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here is this I don't know that there can
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be a connection between 3 and 9. so I'm
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going to try to break down nine in its
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prime factors okay the prime factors so
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basically you're going to find the
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lowest terms that goes into nine so you
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stay the same I have three over four and
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then that becomes seven
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over three times three because I know
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that three times three is nine now
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I can send your values okay I don't have
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to multiply them first I can put all of
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them on the same line so this is going
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to be turned into three times seven over
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four times three times three now I can
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cancel this out and I'm left with 7 over
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4 times 3 which is 12. so this is a
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simpler way of simplifying fractions
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once you multiply them okay so Multiply
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rational numbers you always want to
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multiply them and then trying to find a
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way to simplify them okay so this is the
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best way to do it I'll give you one more
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example and then we're gonna end this
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lecture
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so I assume that I have
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4 over 10
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times let's see 25
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over 16. now the tendency for people is
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to bought Grinders too and then simplify
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them that would be hard that'll be too
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much work so what I'm going to try to do
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is this I'm going to see if there's any
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connection between 4 and 16 and then 10
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and 25 or if you don't want to do that
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just break this down into their prime
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factors okay but I know that there's a
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connection between 4 and 16 so I'm going
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to leave for us four but I'm going to
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try to break down 10 into what five
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times two okay
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I'm gonna do the same thing here 25
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would be five
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times five and sixteen will be four
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times four now this can all go on the
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same line because we are multiplying
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across so this is four times five
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times five over five times two times
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four times four you can cancel the force
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you can guess one five and you are left
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with five over two times four which is
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basically five over eight okay
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and this is basically how to multiply
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rational numbers and then simplifying
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them all right thank you and see you
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next time
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