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## 6th grade (Illustrative Mathematics)

### Unit 6: Lesson 5

Lesson 5: A new way to interpret a over b# One-step multiplication & division equations: fractions & decimals

CCSS.Math:

Learn how to solve equations in one step by multiplying or dividing a number from both sides. These problems involve decimals and fractions.

## Want to join the conversation?

- What if it is something like 3/4=5x?(21 votes)
- still divide by 5 or multiply by 1/5, answer would be x = 3/20(11 votes)

- Are there subtitles for the videos and translations for the articles and exercises for Non-Native english speakers in Khan Academy?(3 votes)
- If you go under your profile (name on top right corner of screen), click on settings, there is a language tab to choose a variety of languages, see if one matches what your most comfortable language is.(9 votes)

- how do you solve a problem like this:"3e = 3/4" how do you solve this?(3 votes)
- You need to get "e" by itself. So, you need to move the 3 from "3e".

"3e" means "3 times e", do you do the opposite of multiplication: divide by 3

3e/3 = 3/4 / 3/1

e = 3/4 * 1/3

e = 1/4

Hope this helps.(8 votes)

- For 4.5/0.5, can't you just multiply the numerator and denominator by 2 to get 9/1 ?(4 votes)
- Yes, that works because dividing by 0.5 is the same as multiplying by 2.(5 votes)

- im confused why do we have to use x and y not like a and e(3 votes)
- Technically you could use whatever you wanted but x and y don't look like any numbers and that's what people choose to use.(5 votes)

- Why we swap the inequality sign when we divide or multiply by a negative?(2 votes)
- Several ways to think about it:

If -x ≥ 3, look at values, if x=3, you have -3≥3 which is false. 0 gives 0 ≥ 3 which is false, even -2 gives -(-2)≥3 which is 2≥3 which is false. When we reach -3, we get -(-3)≥3 which gives 3≥3 which is finally true. The more negative we get, we will always stay true -(-10)≥3 gives 10≥3. So we found that x ≤ -3 as the solution.

Second way to look at it. If - x ≥ 3, we could add x to both sides and subtract 3 to both sides to get -3 ≥ x. Then flipping the equation around, we would have x ≤ -3.(6 votes)

- then how would you do 4=t/2.5 ?(2 votes)
- You have "t divided by 2.5"

Use the opposite operation - multiply both sides by 2.5 to get your answer.

Hope this helps.(5 votes)

- What if we multiply 6.4 to 3.2?(2 votes)
- What problem are you asking about? The last problem is the only one with 3.2, but there is no 6.4 in the problem. Clarify you question if you still need help.(4 votes)

- What if it is something like 4/5=6x(2 votes)
- You divide both sides by 6, and use rules for working with fractions to do the division.

x = 4/5 divided by 6/1

x = 4/5 * 1/6

x = (4*1)/(5*6)

x = 4/30

Then reduce the fraction by 2

x = 2/15

Hope this helps.(2 votes)

- Um, so I have issues with problems like 3/4=5x. How do you do that? Easily so I can understand it?(0 votes)
- Another way to do it would be to convert the fraction into an integer by multiplying both sides by four making it 3=20x. To find 1x (normally just expressed as x) we just divide both sides of the equation by 20 (because 20x/20 = 1x) resulting in your answer being x=3/20.(1 vote)

## Video transcript

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.