Multiply & divide negative fractions
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Let's do a few examples multiplying fractions. So let's multiply negative 7 times 3/49. So you might say, I don't see a fraction here. This looks like an integer. But you just to remind yourself that the negative 7 can be rewritten as negative 7/1 times 3/49. Now we can multiply the numerators. So the numerator is going to be negative 7 times 3. And the denominator is going to be 1 times 49. 1 times 49. And this is going to be equal to-- 7 times 3 is 21. And one of their signs is negative, so a negative times a positive is going to be a negative. So this is going to be negative 21. You could view this as negative 7 plus negative 7 plus negative 7. And that's going to be over 49. And this is the correct value, but we can simplify it more because 21 and 49 both share 7 as a factor. That's their greatest common factor. So let's divide both the numerator and the denominator by 7. Divide the numerator and the denominator by 7. And so this gets us negative 3 in the numerator. And in the denominator, we have 7. So we could view it as negative 3 over 7. Or, you could even do it as negative 3/7. Let's do another one. Let's take 5/9 times-- I'll switch colors more in this one. That one's a little monotonous going all red there. 5/9 times 3/15. So this is going to be equal to-- we multiply the numerators. So it's going to be 5 times 3. 5 times 3 in the numerator. And the denominator is going to be 9 times 15. 9 times 15. We could multiply them out, but just leaving it like this you see that there is already common factors in the numerator and the denominator. Both the numerator and the denominator, they're both divisible by 5 and they're both divisible by 3, which essentially tells us that they're divisible by 15. So we can divide the numerator and denominator by 15. So divide the numerator by 15, which is just like dividing by 5 and then dividing by 3. So we'll just divide by 15. Divide by 15. And this is going to be equal to-- well, 5 times 3 is 15. Divided by 15 you get 1 in the numerator. And in the denominator, 9 times 15 divided by 15. Well, that's just going to be 9. So it's equal to 1/9. Let's do another one. What would negative 5/9 times negative 3/15 be? Well, we've already figured out what positive 5/9 times positive 3/15 would be. So now we just have to care about the sign. If we were just multiplying the two positives, it would be 1/9. But now we have to think about the fact that we're multiplying by a negative times a negative. Now, we remember when you multiply a negative times a negative, it's a positive. The only way that you get a negative is if one of those two numbers that you're taking the product of is negative, not two. If both are positive, it's positive. If both are negative, it's positive. Let's do one more example. Let's take 5-- I'm using the number 5 a lot. So let's do 3/2, just to show that this would work with improper fractions. 3/2 times negative 7/10. I'm arbitrarily picking colors. And so our numerator is going to be 3 times negative 7. 3 times negative 7. And our denominator is going to be 2 times 10. 2 times 10. So this is going to be the numerator. Positive times a negative is a negative. 3 times negative 7 is negative 21. Negative 21. And the denominator, 2 times 10. Well, that is just 20. So this is negative 21/20. And you really can't simplify this any further.