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CCSS Math: 5.MD.A.1, HSA.CED.A.1, HSN.Q.A.1

Jamir is training for a
race and is running laps around a field. If the distance around the field
is 300 yards, how many complete laps would he need to
do to run at least 2 miles? So they tell us how far one lap
is, it's 300 yards, but we need to figure out how many
laps to go 2 miles. So a good starting point would
be to get everything into the same units. We have distance here in terms
of miles, we have it here in terms of yards. So let's just get everything
into yards. So he needs to run 2 miles. How do we convert
that to yards? Well, I don't have it memorized
how many yards there are per mile, but I do have it
memorized how many feet there are per mile. And it's a good thing to have
in the back of your brain someplace, that in general you
have 5,280 feet per mile. It's a good number to know. 5,280 feet per mile. So if we want to convert, we can
first convert the miles to feet, and then we know that
there are 3 feet per yard, and then we'll have 2 miles
in terms of yards. So 2 miles, if we want it
converted to feet, we want miles in the denominator and we
want feet in the numerator. And the reason why I say that
is so that this miles will cancel out with that
miles, and we'll just have feet there. And I just wrote down, there's
5,280 feet per mile, or you say 5,280 feet for
every 1 mile. You can write it either
way, but let's just write it like that. And then we can multiply. So this is going to
give us what? If we just multiply the
numbers 2 times 5,280. So what is that going to be? Maybe I should get
a calculator out. Or we could do that
in our head. Let's think of it this way:
2 times 80 is 160. 2 times 200 is 400. So it's going to be 400 plus
160 is going to be 560. And then 2 times 5,000
thousand is 10,000. So it's 10,560. And then the miles cancel
out, and we are just left with feet. And let me actually
multiply it out. I did it in my head that time,
but that's not always useful. Let me verify for
you that 5,280 times 2 is indeed 10,560. So 2 times 0 is 0. 2 times 8 is 16. Carry the 1. 2 times 2 is 4, plus 1 is 5. 2 times 5 is 10. 10,560. So he needs to run
10,560 feet. Now, we want this in
terms of yards. So 10,560 feet. Let's convert this to yards. Well, we want it in yards. So we want yards in the
numerator, and we want feet in the denominator, so that the
feet cancel out with that feet right there. And we know that there are
3 feet for every 1 yard. Or another way to read this is
that you have 1/3 of a yard for every foot. And now we can multiply. And it makes sense. If you're going from feet to
yards, the number should get smaller because yards
is a bigger unit. You need fewer yards to go the
same distance as a certain number of feet. So it makes sense that
we're dividing. Same thing: 2 miles is a ton of
feet, so it made sense that we were multiplying
by a large number. Here it makes sense that
we're dividing. So let's do this. So this becomes 10,560
times 1 divided by 3. So it's 10,560/3. That's that and that part. And then the feet cancel
out, and we are just left with yards. So 2 miles is 10,560
divided by 3. And let's figure out
what that is. So 3 goes into 10,560. It doesn't go into 1. It goes into 10 three times. 3 times 3 is 9. And we subtract. We get 1. Bring down this 5. It becomes a 15. 3 goes into 15 five times. 5 times 3 is 15. We have no remainder, or 0. You bring down the 6. 3 goes into 6 two times. Let me scroll down
a little bit. 2 times 3 is 6. Subtract. No remainder. Bring down this last 0. 3 goes into 0 zero times. 0 times 3 is 0. And we have no remainder. So 2 miles is the equivalent
to 3,520 yards. That's the total distance
he has to travel. That's the equivalent
of 2 miles. Now we want to figure out
how many laps there are. We want this in terms of laps,
not in terms of yards. So we want the yards
to cancel out. And we want laps in the
numerator, right? Because when you multiply, the
yards will cancel out, and we'll just be left with laps. Now, how many laps are there
per yard or yards per lap? Well, they say the distance
around the field is 300 yards. So we have 300 yards
for every 1 lap. So now, multiply this
right here. The yards will cancel out,
and we will get 3,520. Let me do that in a
different color. We will get 3,520, that right
there, times 1/300. When you multiply it times
1, it just becomes 3,520 divided by 300. And in terms of the units,
the yards canceled out. We're just left with the laps. So this is how many laps
he needs to run. So 3,520 divided by 300. Well, we can eyeball
this right here. What is 11 times 300? Let's just approximate
this right here. So if we did 11 times
300, what is that going to be equal to? Well, 11 times 3 is 33, and then
we have two zeroes here. So this will be 3,300. So it's a little bit
smaller than that. If we have 12 times 300, what
is that going to be? 12 times 3 is 36, and then we
have these two zeroes, so it's equal to 3,600. So this is going to be
11 point something. It's larger than 11, right? 3,520 is larger than 3,300. So when you divide by 300 you're
going to get something larger than 11. But this number right here is
smaller than 3,600 so when you divide it by 300, you're going
to get something a little bit smaller than 12. So the exact number of laps is
going to be a little bit lower than 12 laps. So 2 miles is a little bit
lower than 12 laps. But let's make sure we're
answering their question. How many complete laps would
he need to do to run at least 2 miles? So they're telling us that,
look, this might be, 11 point something, something,
something laps. That would be the exact number
of laps to run 2 miles. But they say how many complete
laps does he have to run? 11 complete laps would
not be enough. He would have to run 12. So our answer here is
12 complete laps. That complete tells us
that they want a whole number of laps. We can't just divide this. If we divide this, we're going
to get some 11 point something, something. You can do with the calculator
or do it by hand if you're interested. But we have to do at least 12
because that's the smallest whole number of laps that will
get us to at least this distance right here, or this
number of laps, or the equivalent of 2 miles.