One-step multiplication & division equations: fractions & decimals

Let's get some practice solving some equations, and we're gonna set up some equations that are a little bit hairier than normal, they're gonna 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 have the c on the left hand side, and I'm just multiplying it by 1.2, it would be great if this just said c. If this just said c instead of 1.2c. So what can I do there? Well I could just divide by 1.2 but as we've seen multiple times, you can't just do that to the left hand side, that would change, you no longer could say that this is equal to that if you only operate on one side. So you have to divide by 1.2 on both sides. So on your left hand side, 1.2c divided by 1.2, well that's just going to be c. You're just going to be left with c, and you're going to have c is equal to 0.6 over 1.2 Now what is that equal to? There's a bunch of ways you could approach it. The way I like to do it is, well let's just, let's just get rid of the decimals. Let's just multiply the numerator and 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 gonna have a 6 in the numerator and 12 in the denominator, actually let's do that. Let's multiply the numerator and 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. So it's equal to six twelfths, and if we want we can write that in a little bit of a simpler way. We could rewrite that as, divide the numerator and denominator by 6, you get 1 over 2, so this is equal to one half. And if you look back at the original equation, 1.2 times one half, you could view this as twelve tenths. 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 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 can think of of getting rid of this 12 and just having a y on the right hand side is multiplying both sides by 12. We 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 on the left hand side you're going to have 12 times one fourth, which is twelve fourths. So you get 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 just so you can see what I'm doing, just flopping the sides, doesn't change what's being said, y is equal to 12 over 4. Now what is twelve fourths? Well, you can view this as 12 divided by 4, which is 3, or you could view this as twelve fourths which would be literally, 3 wholes. So you could say this would be equal to 3. Y is equal to 3, and you can check that. One fourth 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. 4.5 is equal to 0.5n So like always, I have my n already on the right hand side. But it's being multiplied by 0.5, it would be great if it just said 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 it to the right hand side I have to do it to the left hand side. And why am I dividing by 0.5? So I'm just left with an n on the right hand side. So this is going to be, so on the left hand side, I have 4.5 over 0.5, let me just, I don't want to skip too many steps. 4.5 over 0.5, is equal to n, because you have 0.5 divided by 0.5, you're just left with an n over here. So what does that equal to? Well 4.5 divided by 0.5, there's a couple ways to view this. You could view this as forty-five tenths divided by five tenths, which would tell you okay, this is going to be 9. Or if that seems a little bit confusing or a little bit daunting, you can do what we did over here. You could multiply the numerator and the denominator by the same number, so that we get rid of the decimals. And in this case, if you multiply by 10 you can move the decimal one to the right. So once again, it has to be multiplying the numerator and the denominator by the same thing. We're multiplying by 10 over 10, which is equivalent to 1, which tells us that we're not changing the value of this fraction. So let's see, this is going to be 45 over 5, is equal to n. And some of you might say wait wait wait, hold on a second, you just told us whatever we do to one side of the 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 I do to the right hand side. You can always multiply one side by 1 and you can do that as many times as you want. Like the same way you can add 0 or subtract 0 from one side, without necessarily having to show you're doing it to the other side, because it doesn't change the value. But anyways, 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 9 is equal to n, or we could say n is equal to 9. And you could check that: 4.5 is equal to 0.5 times 9, yup half of 9 is 4.5 Let's do one more, because once again I can't stop. Alright, let me get some space here, so we can keep the different problems apart that we had. So let's do, let's have a different variable now. Let's say we have g over 4 is equal to 3.2. Well I wanna get rid of this dividing by 4, so the easiest way I can think of doing that is multiplying both sides by 4. So I'm multiplying both sides by 4, and the whole reason is 4 divided by 4 gives me 1, so I'm gonna have g is equal to, what's 3.2 times 4? Let's see 3 times 4 is 12, and two tenths times 4 is eight tenths, so it's gonna be 12 and eight tenths. G is going to be 12.8, and you can verify this is right. 12.8 divided by 4 is 3.2.