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# 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.