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# Multiple rates word problem

Video transcript

Starting at home,
Umaima traveled uphill to the gift
store for 45 minutes at just 8 miles per hour. She then traveled back
home along the same path downhill at a speed
of 24 miles per hour. What is her average speed
for the entire trip from home to the gift store and back? So we're trying to figure
out her average speed for the entire trip. That's going to be equal
to the total distance that she traveled
over the total time. Well, what's the total
distance going to be? Well, the total
distance is going to be the distance
to the gift store and then the distance
back from the gift store, which are the same distances. So it's really you could say two
times the distance to the gift store. And then what is going
to be her total time? Well, it's time to gift store
plus the time coming back from the gift store. Now, we know that the
distance to the gift store and the distance back from
the gift store is the same. So that's why I just said that
the total distance is just going to be two times the
distance to the gift store. We don't know-- in
fact we know we're going to have different times
in terms of times to the gift store and times coming back. How do I know that? Well, she went at
different speeds. So it's going to
take her-- actually, she went there much
slower than she came back. So it would take her
longer to get there than it took her to get back. So let's see which of these we
can actually-- we already know. So how do we figure out the
distance to the gift store? At not point here do they
say, hey, the gift store is this far away. But they do tell us, this
first sentence right over here, "Umaima traveled uphill to
the gift store for 45 minutes at just 8 miles per hour." So we're given a time. And we are given a speed. We should be able to
figure out a distance. So let's just do a
little bit of aside here. We should be able to figure out
the distance to the-- actually let me write it this way. The distance to
the store will be equal to-- now we've got to make
sure we have our units right. Here they gave it in minutes. Here they have 8 miles per hour. So let's convert
this into hours. So 45 minutes in hours,
so it's 45 minutes out of 60 minutes per hour. So that's going
to give us 45/60. Divide both by 15. That's the same thing as 3/4. So it's going to be 3/4 hours is
the time times an average speed of 8 miles per hour. So what is the
distance to the store? Well, 3/4 times 8. Or you could view it
as 3/4 times 8 times 1, is going to be-- well,
it's going to be 24 over 4. Let me just write that. That's going to
be 24 over 4 which is equal to-- did I get it? Yeah. 24 over 4, which is equal to 6. And units-wise, we're
just left with miles. So the distance to
the store is 6 miles. 2 times the distance
to the gift store, well, this whole thing
is going to be 12 miles. 12 miles is the total
distance she traveled. Now, what is the time
to the gift store? Well, they already
told that to us. They already told us
that it's 45 minutes. Now, I want to put
everything in hours. I'm assuming that they want
our average speed in hours. So I'm going to put
everything in hours. So the time to the gift
store was 3/4 of an hour. And what's the time coming
back from the gift store? Well, we know her speed. We know her speed coming back. We already know the distance
from the gift store. It's the same as the
distance to the gift store. So we can take this distance,
we can take 6 miles, that's the distance to the gift
store, 6 miles divided by her speed coming back,
which is 24 miles per hour, so divided by 24 miles per hour. It gives us-- well, let's see. We're going to have 6 over
24 is the same thing as 1/4. It's going to be 1/4. And then miles divided
by miles per hour is the same thing as miles
times hours per mile. The miles cancel out. And you'll have 1/4 of an hour. So it takes her 1/4 of
an hour to get back. And that fits our intuition. Actually, let me write that
in the same green color since I'm writing all
the times in green color. So 1/4 of an hour. So she went there. Going to the gift
store was slow. It took her 3/4 of an hour. Coming back only took
her 1/4 of an hour. So now we're ready to
calculate her average speed for the entire trip. Average speed for
the entire trip is going to be equal to
the total distance, which is 12 miles, divided
by her total time. 3/4 hours plus 1/4
hour is exactly 1 hour. So her average
speed is 12 over 1, which is just 12 miles per hour. Her average speed is
just 12 miles per hour. And you might have been
tempted to say hey, wait. Why don't I just
average 24 and 8. But that wouldn't
have been right, because she's traveling those
for different amounts of time. So what you really
have to do is just think in terms of go back to
your basics-- total distance, total time. Figure out the total distance. This first sentence
right over here gives us half of
the total distance, the time to the store. We just doubled that
to get the time back. And then our total
time we can figure out. They tell us the
time to the store. And then we can figure out
the distance from the store and using that and the speed
to figure out her time back. And then we get total
distance divided by total time--
12 miles per hour.