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# Division of Rational Numbers

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- 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.

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