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# Expressions with rational numbers

CCSS.Math:

## Video transcript

we have four different expressions here what I want you to do is think about which of these expressions are equal to negative 2/3 which of these are equal to negative 2/3 and I encourage you now to pause this video and try this on your own so let's go to this first expression right over here I have 1/9 and I'm going to add to that five ninths so how many ninths am I going to have well I had 1/9 now I'm adding five ninths so I'm going to have six ninths if I have one of something and I have five more of that same something so in this case that something is 1/9 1/9 plus five nines is six ninths now can we simplify this in any way well both 6 & 9 are divisible by 3 so let's divide them both by 3 to try to get this fraction in a simpler form 6 divided by 3 is 2 9 divided by 3 is 3 so this is 2/3 while what we're trying to get to is negative 2/3 so these are not equal this expression does not equal not equal negative 2/3 so I'll write no for that one now let's go to this this green expression right over here give myself a little bit more real estate to work in now we have negative 1/6 plus negative 1/2 now we can view this as being the same thing as just to clarify right now the negative is in front of the entire 1/6 the negatives in front of the entire 1/2 but this is the same thing as negative 1 over 6 plus negative 1 over negative 1 over 2 negative 1/2 is the same thing as negative 1/2 is one way to think about it and the whole reason why I did this is so we can simplify what the- the negatives are right now only in our numerator so whenever we had two fractions we want to have this that we want to have the same denominator and we see that 6 is already a multiple of 2 so we could leave this first fraction the way it is we can rewrite it as negative 1 over 6 and then the second fraction we can write it as something over 6 well to go from 2 to 6 we have to multiply by 3 so let's also multiply the numerator by 3 negative 1 times 3 is negative three so if I have negative one sixths and I add to that negative three six this is going to be negative 1 plus negative 3 6 which is equal to negative 4 over 6 now let's see if we can simplify it both negative 4 or I guess we can say both 4 & 6 are divisible by 2 so let's divide them both by 2 let's divide them both by 2 and in the numerator we're left with negative 4 divided by 2 is negative 2 6 divided by 2 is 3 negative 2 divided by 3 well that's the same thing as negative 2/3 which is exactly what we're going to get to so yes this thing in green is equal to negative 2/3 now let's go over here so we have negative 1/3 times negative 2 well if you multiply negative times a negative we're going to get a positive and we're going to get a positive 1/3 times 2 so one way to think about it this is going to be the same thing as 1/3 times 2 which is the same thing and there's a couple of ways to think about it if you have 1/3 and now you're going to have you're gonna multiply it by 2 you now have 2/3 you now have 2/3 another way to think about this is that this is the same thing as 1/3 times 2 over 1 and you know that we can just multiply that when we multiply two fractions so this time we've expressed the 2 as a fraction we can multiply their numerators so it's 1 times 2 over the product of their denominators 3 times 1 which is 2 over 3 so either way you look at it this goes to positive 2/3 a negative times a negative is a positive so it gets this to positive 2/3 not negative 2/3 so like this first one no it does not it does not equal it does not equal negative 2/3 now let's look at this one negative 1/3 divided by 1/2 so when you divide by a fraction so when you divide when you take negative 1/3 dividing let me write it this way so negative 1/3 divided by 1/2 this is the same thing as negative 1/3 and let me let me color just so you see what I'm doing so make that that green color let me make this a blue color so negative 1/3 divided by 1/2 is the same thing as negative 1/3 times the reciprocal of 1/2 so x times 2 over 2 over 1 2 over 1 and what is this going to be equal to well we look we can assume instead of just going this is negative 1/3 we could view this as negative 1/3 that might help us keep track of the signs a little bit more and so if we do this is let me actually let me write it that way just to make it a little bit clearer let me write this as let me write this as negative 1/3 so our numerator is now going to be our numerator it's now going to be negative 1 times 2 when you multiply two fractions you just multiply the numerator to get the new the new numerator multiply the two numerators to get the new numerator and it's over 3 times 1/3 times 1 and you know you normally wouldn't have to do all of these steps but I'm just doing them to make sure you understand what's going on and so this is going to be equal to negative 1 times 2 is negative 2 negative 2 and 3 times 1 is positive 3 negative 2 over 3 well that's the same thing that's the same thing as negative 2/3 so this one works out it is equal to negative 2/3