Division of Rational Numbers

Video transcript

Let's do a little bit of practice dividing rational numbers, or another way to think about it, dividing fractions. Same thing. So we have part A. Well, number 1. Find the multiplicative inverse of each of the following. Now, all that means is if I have some number a, its inverse is going to be 1/a. When I take a number and I multiply it by its multiplicative inverse, I'm going to get a/a, which is equal to 1. So a times its multiplicative inverse is going to be 1. So let's do these problems. So part A. We have 100. So the multiplicative inverse of that is just going to be 1/100. And if you multiplied that times that, you'd get 1. Part B. I'll do all of these. We have 2/8. The inverse of 2/8 is 1 over 2/8, which is the same thing-- this is equal to 1 times the inverse of this. Times 8/2, which is the same thing as 4. Now immediately, you might see something. I guess, you see a pattern here. If I take 1 over a fraction, the result is the fraction swapped. So the inverse of 2/8 is 8/2. You just swap the numerator and the denominator. So let's apply that to part C. If I have negative 19/21, its inverse is just going to be you swap the numerator and the denominator. Negative 21/19. And if you were to multiply these two numbers, you would get 1. The negatives would cancel out. The 21 and the 21 would cancel out, the 19 and the 19 would cancel out. Part D. The inverse of 7, multiplicative inverse, is 1/7. And then finally, let's do E. So the inverse of-- so they give us z to the third over-- so it's negative z to the third over 2xy to the third. So the multiplicative inverse of that-- one of the harder words for me to say-- is just going to be equal to-- so that's the inverse-- is going to be 1 over this, which is just going to be negative. The denominator becomes the numerator. 2y to the third over z cubed. Now let's go to this next section. Divide the following rational numbers. Be sure that your answer is in simplest form. So I'm just going to do every other one of these problems, just so that I don't use up all of your time. So let's do this first one: part A. Well, I'll do it over here. Part A. 5/2 divided by 1/4. This is the same thing as 5/2 times 4/1. You divide by something is the same thing as multiplying by its inverse. So I'm multiplying by the inverse of 1/4. So this is going to be equal to-- well, we could divide the numerator and denominator by 2, so the 4 becomes a 2, the 2 becomes a 1. 5 times 2 is 10/1, So it's just 10. Part B. Or let me do every other one. C. 5/11 divided by 6/7. Once again, this is the same thing as 5/11 times the inverse of 6/7, so times 7/6. And so we get, this is 5 times 7 is 35 over 66. And that's in lowest common form or simplified form. So E. Let's do E. E, we have negative x over 2 divided by 5 over 7. Once again, this is the same thing as negative x over 2 times 7/5, which is equal to negative 7x over 10. Think you're getting the hang of it. Let's do part G here. We have a negative 1/3 divided by negative 3/5. Well, this is going to be the same thing as negative 1/3 times the inverse of this, times negative 5/3. I just swapped the numerator and the denominator. So this is going to be equal to-- the negatives cancel out. A negative times a negative is a positive. 1 times 5 is 5. 3 times 3 is 9. Let's do one more. Let's do I. Part I. 11 divided by negative x/4. Once again, this is the same thing as 11 times the inverse of this. The multiplicative inverse of it, times negative 4/x. And if this confuses you-- actually, I shouldn't write multiplying. That looks just like an x. I should say this the same thing as 11 times negative 4/x. Or you could view this as 11 over 1 times-- you could even view it as negative 4/x, which is equal to minus 44/x, or negative 44/x. Let's do this last problem right here. The world's largest trench digger, Bagger 288, moves at 3/8 miles per hour. So its speed, or its velocity, or its rate is equal to 3/8 miles per hour. We could write it like this. How long will it take to dig a trench 2/3 miles long? So we want to go a distance that's equal to 2/3 miles. So we just remember that distance is always going to be equal to rate times time. So our distance is 2/3 miles. That's going to be equal to our rate, 3/8 miles per hour, times our time. Or we could divide both sides by 3/8 miles per hour, so if we divide both sides by 3/8 miles per hour, we'll get our time. So divide by 3/8 miles per hour. Divide by 3/8 miles per hour. This cancels out. So time, we're going to get time is equal to this. So we have 2/3 miles. Let me write it this way. 2/3 miles divided by 3/8 miles per hour. Or another way to think of it, that's the same thing as times the inverse, times 8/3 hours per mile. I just took the inverse of this. We divided by this. That's the same thing as multiplying by its inverse. And we inverted the units as well. And so we'll see that the units cancel out. Mile in the numerator, mile in the denominator. We're left with hours. And so this is equal to 2 times 8 is 16 over 3 times 3, 9, 16/9 hours. So like an hour and 7/9, or 1 and 7/9 hours to dig the 2/3 of a mile.