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let's get some practice solving some equations and we're going to set up some equations that a little bit hairier than normal they're going to have some decimals and fractions in them so let's say I had the equation 1.2 times C is equal to 0.6 so what do I have to multiply times 1.2 to get 0.6 and it might not jump out immediately in your brain but lucky for us we can think about this a little bit methodically so one thing I like to do is say okay I already have to see on the left hand side and I'm just multiplying it by 1.2 it'd be great if this just said see if this just said C instead of one point two C so what could I do there well I could just divide by one point two but as we've seen multiple times you can't just do that to the left hand side that would change that would you no longer could say that this is equal to that if you only do if you only operate on one side so you have to divide by one point two on both sides so on your left hand side one point to C divided by one point two well that's just going to be C you're going to be left with C and you're going to have C is equal to zero point six over one point two now what is that equal to and there's a bunch of ways you can approach it the way I like to do it is well let's just let's just get rid of the decimals let's multiply the numerator and the denominator by a large enough number so that the decimals go away so what happens if we multiply the numerator and the denominator by let's see if we multiply them by 10 you're going to have a you're going to have six in the numerator and a 12 in the number actually let's do that so let's multiply the numerator and the denominator by 10 so once again this is the same thing as multiplying by 10 over 10 it's not changing the value of the fraction so 0.6 times 10 is 6 and 1.2 times 10 is 12 is 12 so it's equal to 6 12 and if we want we can write that in a little bit of a simpler way we could rewrite that as to divide the numerator and denominator by 6 you get 1 over 2 so this is equal to 1/2 and if you look back at the original at the original equation 1.2 times 1/2 well you could view this is 12 10 twelve tenths times one half is going to be equal to six tenths so we can feel pretty good that C is equal to one half let's do another one let's say that we have one say we have 1 over 4 is equal to Y over 12 so how do we solve for y here so we have a y on the right hand side and it's being divided by 12 well the best way I could think of of getting rid of this 12 and just having Y on the right hand side is by multiplying both sides by 12 let me do that in yellow so if I multiply the right-hand side by 12 I have to multiply the left-hand side by 12 and once again why did I pick 12 well I wanted to multiply by some number that when I multiply it by Y over 12 I'm just left with Y and so Y times 12 divided by 12 well that's just going to be 1 and then 12 and then you're on the left hand side you can have 12 times 1/4 which is 12 fourths so you get 12 over 4 12 over 4 is equal to Y or you could say Y is equal to 12 over 4 Y is equal to let me do that in that just so you see what I'm doing I'm just swapping the sides doesn't change what's being said Y is equal to 12 over 4 now what is 12 fourths well you can view this as 12 divided by 4 which is 3 or you could view this as 12 fourths which would be literally three wholes so you could say this would be equal to 3y is equal to 3 and you could check that 1/4 is equal to 3 over 12 so it all works out that's the neat thing about equations you can always check to see if you got the right answer let's do another one can't stop four point five is equal to zero point five and so like always I have my n already on the right-hand side so but but it's been multiplied by 0.5 would be great if it was just an N so what can I do well I can divide both sides I can divide both sides by 0.5 once again if I do to the right-hand side I have to do to the left-hand side and why am i dividing by 0.5 so I'm just with an N on the right-hand side so this is going to be so on the left-hand side I have four point five over 0.5 now let me just don't wanna skip too many steps four point five over zero point five over zero point five is equal to n because you have zero point 5 divided by zero point five you're just left with an N over here so what is that equal to well four point five divided by 0.5 there's a couple of ways to view that you could view this as 45 tenths divided by five tenths which would tell you okay so that's just going to be nine or if that seems a little bit confusing or a little bit daunting you can do what we did over here we could multiply the numerator the denominator by the same number so we get rid of the decimals and in this case you multiply by ten you can move the decimal one to the right so once again it has to be multiplying the numerator denominator by the same thing we're multiplying by ten over ten which is equivalent to one which tells us that we're not changing the value of this fraction so let's see this is going to be 45 45 over 5 over 5 is equal to N and some of you might say wait wait wait wait hold on a second Sal you just told us that whatever we do to one side of an equation we have to do to the other side of the equation and here you are you're just multiplying the left-hand side of this equation by 10 over 10 now remember what is 10 over 10 10 over 10 is just 1 yes if I wanted to I could multiply the left-hand side by 10 over 10 and I could multiply the right-hand side by 10 over 10 but that's not going to change the value of the right-hand side I'm not actually changing the values of the two sides I'm just trying to rewrite the left-hand side by multiplying it by 1 in kind of a creative way but notice n times 10 over 10 well that's still going to just be n so I'm not violating this principle of whatever I do to the left-hand side and do the right-hand side you can always multiply one side by one and you can do that as many times as you want like the same way you could add zero or subtract zero from one side without necessarily having to show you're doing it to the other side because it doesn't change the value but anyway you have n is equal to 45 over 5 well what's 45 over 5 well that's going to be 9 so we have 9 is equal to why did I switch to green we have we have nine is equal to n or we could say n is equal to nine and you could check that four point five is equal to zero point five times nine yep half of 9 is four point five let's do one more because once again I can't stop alright now let me let me get some space here so we can keep keep the different problems apart that we had so let's do let's do let's change let's have a different variable now let's say we have G over four is equal to three point two well I want to get rid of this dividing by four so the easiest way I can think of doing that is multiplying both sides by four so I'm multiplying both sides by four and the whole reason is so because I have four divided by four gives me one so I'm going to have G is equal to what's three what's three point two times four let's see three times four is twelve and two tenths times four is is eight tenths so it's going to be twelve and eight tenths it's go G is going to be twelve point eight and you can verify this is right twelve point eight divided by four is three point two