Multiply & divide negative fractions
Let's do some examples dividing fractions. Let's say that I have negative 5/6 divided by positive 3/4. Well, we've already talked about when you divide by something, it's the exact same thing as multiplying by its reciprocal. So this is going to be the exact same thing as negative 5/6 times the reciprocal of 3/4, which is 4/3. I'm just swapping the numerator and the denominator. So this is going to be 4/3. And we've already seen lots of examples multiplying fractions. This is going to be the numerators times each other. So we're going to multiply negative 5 times 4. I'll give the negative sign to the 5 there, so negative 5 times 4. Let me do 4 in that yellow color. And then the denominator is 6 times 3. Now, in the numerator here, you see we have a negative number. You might already know that 5 times 4 is 20, and you just have to remember that we're multiplying a negative times a positive. We're essentially going to have negative 5 four times. So negative 5 plus negative 5 plus negative 5 plus negative 5 is negative 20. So the numerator here is negative 20. And the denominator here is 18. So we get 20/18, but we can simplify this. Both the numerator and the denominator, they're both divisible by 2. So let's divide them both by 2. Let me give myself a little more space. So if we divide both the numerator and the denominator by 2, just to simplify this-- and I picked 2 because that's the largest number that goes into both of these. It's the greatest common divisor of 20 and 18. 20 divided by 2 is 10, and 18 divided by 2 is 9. So negative 5/6 divided by 3/4 is-- oh, I have to be very careful here. It's negative 10/9, just how we always learned. If you have a negative divided by a positive, if the signs are different, then you're going to get a negative value. Let's do another example. Let's say that I have negative 4 divided by negative 1/2. So using the exact logic that we just said, we say, hey look, dividing by something is equivalent to multiplying by its reciprocal. So this is going to be equal to negative 4. And instead of writing it as negative 4, let me just write it as a fraction so that we are clear what its numerator is and what its denominator is. So negative 4 is the exact same thing as negative 4/1. And we're going to multiply that times the reciprocal of negative 1/2. The reciprocal of negative 1/2 is negative 2/1. You could view it as negative 2/1, or you could view it as positive 2 over negative 1, or you could view it as negative 2. Either way, these are all the same value. And now we're ready to multiply. Notice, all I did here, I rewrote the negative 4 just as negative 4/1. Negative 4 divided by 1 is negative 4. And here, for the negative 1/2, since I'm multiplying now, I'm multiplying by its reciprocal. I've swapped the denominator and the numerator. Or I swapped the denominator and the numerator. What was the denominator is now the numerator. What was the numerator is now the denominator. And I'm ready to multiply. This is going to be equal to-- I gave both the negative signs to the numerator so it's going to be negative 4 times negative 2 in the numerator. And then in the denominator, it's going to be 1 times 1. Let me write that down. 1 times 1. And so this gives us, so we have a negative 4 times a negative 2. So it's a negative times a negative, so we're going to get a positive value here. And 4 times 2 is 8. So this is a positive 8 over 1. And 8 divided by 1 is just equal to 8.