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## 7th grade (Eureka Math/EngageNY)

### Course: 7th grade (Eureka Math/EngageNY) > Unit 2

Lesson 2: Topic B: Multiplication and division of integers and rational numbers- Why a negative times a negative is a positive
- Why a negative times a negative makes sense
- Signs of expressions
- Multiplying positive & negative numbers
- Dividing positive and negative numbers
- Multiplying negative numbers
- Dividing negative numbers
- One-step equations with negatives (multiply & divide)
- Multiplying negative numbers review
- Dividing negative numbers review
- Rewriting decimals as fractions: 2.75
- Write decimals as fractions
- Rewriting decimals as fractions challenge
- Fraction to decimal: 11/25
- Worked example: Converting a fraction (7/8) to a decimal
- Fraction to decimal with rounding
- Converting fractions to decimals
- Multiplying positive and negative fractions
- Multiplying positive and negative fractions
- Dividing negative fractions
- Dividing positive and negative fractions
- Negative signs in fractions
- Negative signs in fractions
- Negative signs in fractions (with variables)
- Dividing mixed numbers
- Dividing mixed numbers with negatives
- Simplifying complex fractions
- Expressions with rational numbers
- Simplify complex fractions
- Equivalent expressions with negative numbers (multiplication and division)
- Equivalent expressions with negative numbers (multiplication and division)
- Why dividing by zero is undefined
- Dividing by zero

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# Expressions with rational numbers

Learn to compare expressions with positive and negative fractions. Created by Sal Khan.

## Want to join the conversation?

- I know Sal is working through solving each answer to see if it equals -2/3, however with the initial question being that, shouldn't we eliminate examples one and three right away as being positive regardless of their values?(80 votes)
- Yes, if this were an exam question, then that would be a very sensible shortcut. Here, I think Sal was more interested in showing examples of working with rational numbers rather than simply getting the answer.(103 votes)

- is a fraction rational or irrational?(4 votes)
- Rational. A rational number is a number that can be represented by the fraction of two integers. So, fractions are naturally rational. Hope this is helpful! :-)(4 votes)

- at2:18Sal said that -1+ -3= -4, but we learned that a negative + a negative = positive, not negative.(4 votes)
- he said that negative times negative = positive but a negative added to another negative is still a negative(3 votes)

- Why is the new Khan Academy learner so depressed and discouraged?

I feel that way too sometimes (not depressed, but ofc discouraged). Math is actually really hard and not really needed in life. The probably only thing you might need of math is 1 + 1. I wonder why math is a required unit in class. You don't learn much from it, anyways. So?(5 votes) - what is a rational number?(2 votes)
- A rational number is a number that can be written as a fraction, for example, 2/1, or 4/5. It doesn't matter if the fraction can be turned into a whole number or not. In comparison, an irrational number is one that is a recurring decimal with no repetition, eg pie, or 5.624319678.(6 votes)

- If we have one fraction with a negative numerator plus a fraction with a negative denominator, do we simply pretend that both fractions have negative numerators? For example, if our equation is -5/3 + 2/-3, should our answer be -7/3?(2 votes)
- I personally tend to think about the negative sign being before the fraction, like -(5/3) -(2/3). I think having it in the numerator is also acceptable, but it probably shouldn't be in the denominator. And yes, the answer to the example equation would be -7/3.(6 votes)

- how can you solve this equation (7j+1/8q+3)-(5/8-11+2j)? Then also how can I showed my work fro this,do I just rigth the frist equation or slove the seconed equation.(1 vote)
- First you should probably learn how to spell. Personally, I do not know how to solve this equation, but you should start there.(5 votes)

- How do you match a rational decimal?(3 votes)
- If you look at the factors of 4 and 8, you will see that the greatest common factor (GCF) is 4. So, if you divide the numerator and denominator by the GCF, which in this case is 4, you will get 1/2. Also, if this is a division problem, try dividing the denominator by the numerator, so in this case it would be 8 divided by 4. Hope this helps! (P.S.- If you ever get mixed up on what the numerator and denominator are, denominator and down both start with "d" and the denominator is on the bottom if that makes any sense. Idk if that works with how your brain thinks or not but just something to try if you struggle with remembering the difference.) :-)(2 votes)

- the third problem that you solved didnt have the negative sign.(2 votes)

## Video transcript

We have four different
expressions here, and what I want
you to do is think about which of these expressions
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 5/9. So how many ninths
am I going to have? Well, I had 1/9, now I'm adding
5/9, so I'm going to have 6/9. If I have one of something
and I have five more of that same something--
so in this case, that something is a ninth--
1/9 plus 5/9 is 6/9. Now, can we simplify
this in any way? Well, both six and nine 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 negative 2/3, so I'll write "no" for that one. Now let's go to 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 negative's in front of the entire 1/2. But this is the same
thing as negative 1/6, plus negative 1/2. Negative 1/2 is the same thing
as negative 1 divided by 2 is one way to think about it. And the whole reason
why I did this is so we can simplify what the
negatives are right now only in our numerator. So whenever we
add two fractions, 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 1/6. 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 3. So if I have negative 1/6 sixth
and I add to that negative 3/6, this is going to be negative 1
plus negative 3 sixths, which is equal to negative 4/6. Now, let's see if
we can simplify it. Both negative 4-- I guess
we can say both four and six are divisible by 2, so
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 our goal
value we're trying 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 a
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 this is going to be the same thing
as one third 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 multiply it by 2, we now have 2/3. You now have 2/3. Another way to
think about this is that this is the same thing
as 2/3 times 2 over 1. And you know that when we
multiply two fractions, so this time we've expressed
the 2 as a fraction, we can multiply the 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 us to positive
2/3 not negative 2/3, so like this first one, no, it
does not equal negative 2/3. Now let's look at this one. Negative 2/3
divided by one half. So when you divide
by a fraction, so 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 color code it just
so you see what I'm doing. So let me make that green color. Let me make this a blue color. So negative 1/3
divided by 1/2 half is the same thing
as negative 1/3 times the reciprocal of
1/2, so times 2 over 1. And what is this
going to be equal to? Well, we could assume
instead of just doing this as negative 1/3, we could do
this as negative 1 divided by 3, that might help us keep
track of the signs a little bit more. And let me actually
write it that way, just to make it a little bit clearer. Let me write this as
negative 1 divided by 3. So our numerator is now going
to be negative 1 times 2. When you multiply two
fractions, you just multiply the two numerators
to get the new numerator, and it's over 3 times 1. And you normally wouldn't
have to do all 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 1. And 3 times 1 is positive 3. Negative 2 over 3? Well, that's the same
thing as negative 2/3. So this one works out. It is equal to negative 2/3.