重量單位的應用題 (英)

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重量單位的應用題 (英): U06_L1_T2_we3 Application problems involving units of weight

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A standard elevator in a mid-rise building can hold a
maximum weight of 1 and 1/2 tons.
Assuming an average adult weight of 160 pounds, what is
the maximum number of adults who could
safely ride the elevator?
So what we need to do is we have to get the maximum weight
that the elevator could hold in terms of pounds.
And then say, OK, well, how many adults is that?
So they gave us the maximum weight in terms of tons.
So they say it is 1 and 1/2 tons.
And it's always easier to deal with improper fractions than
mixed numbers, so let's write this as an improper fraction.
1 and 1/2 tons is the same thing as-- well, 1 ton is 2
halves, and then you add another 1
half, that's 3 halves.
So you get 3/2 tons.
Or another way to think about it is 2 times 1 is
2, plus 1 is 3.
So the maximum capacity is 3/2 tons.
Let's think about how many pounds that is.
And to do that, we have to know that there are 2,000
pounds per ton.
Let me write this up here.
We know that there are 2,000 pounds per ton.
This wasn't given in the problem anywhere.
This is something I knew from past experience, and it's a
good thing to know, in general, that a
ton is 2,000 pounds.
So I'm going to write it right over here.
Now, how do we convert these 3/2 tons into pounds?
Well, we're going to multiply it by something, and the units
that we're going to multiply by, we want the
tons to cancel out.
So we're going to want to have tons in the denominator, so it
cancels out with this tons up here.
And then we want pounds in the numerator.
And that's exactly what we wrote up here.
There are 2,000 pounds for every 1 ton, or you could just
say 2,000 pounds per ton.
You could put the 1 there, but it doesn't really change the
expression.
Now, if we multiply 3/2 tons times 2,000 pounds per ton,
what'll happen is that the tons cancel out.
That was the whole point of multiplying it by this.
And we would be left with 3/2 times 2,000, and the only unit
left is pounds.
And if you do it this way, you'll never get confused.
You'll know that the units cancel out so you're getting
the right units.
But if you just think about it in your mind, it should also
make sense.
If there are 2,000 pounds per ton and there are 1 and 1/2
tons, I should multiply 1 and 1/2 times 2,000 to get the
number of pounds.
That makes sense.
And we know 1 and 1/2 times 2,000 is 3,000, but we'll
figure it out right here.
So what is 3/2 times 2,000?
Well, I just told you answer.
We can actually simplify it right over here.
This is going to be 3 times 2,000/2 pounds.
We can divide the numerator and the denominator by 2.
This will become 1,000, and this will become 1, so it's 3
times 1,000 pounds, or this is equal to 3,000 pounds.
So what we've done so far, we've just figured out the
maximum capacity of the elevator.
It can hold 1 and 1/2 tons, which is the exact same thing
as 3,000 pounds.
Now, what we need to figure out is 3,000 pounds is the
equivalent to how many average adults of 160 pounds?
Or how many 160 pound people would it take to weigh a total
of 3,000 pounds?
Well, we can just divide by 160.
And if you want to make sure that the units work out,
remember, we want our answer to be in terms of people, and
we want the pounds to cancel out.
So we have pounds in the numerator here, so if we
divide by pounds, the pounds will cancel out.
And then we want our leftover to be people, or maybe person.
Person, people, same thing.
Let me do people.
The grammar of it, singular, plural, might make it a little
confusing, but I think you get the general idea.
Now, if we were to write this out, what does it tell us?
1 person weighs 160 pounds, so there's one person for every
160 pounds.
So notice, if we multiply these two expressions, the
pounds will cancel out.
We'll be just left with people.
We're multiplying 3,000 times 1/160, but it's really just
taking 3,000 and dividing by 160, which makes sense.
We have a capacity of 3,000.
Each of our people weighs 160 pounds.
Divide by 160.
It tells you how many people.
But this way, you know that the units are working out.
So this is going to be equal to 3,000/160 people.
That is the maximum capacity of the elevator in terms of
average people.
Now, what is this?
Well, we can divide the numerator and the
denominator by 10.
If we divide the numerator and the denominator by 10, this
becomes 300/16.
If we divide 300 by 2, this becomes 150.
If we divide 16 by 2, this becomes 8.
Now, let's see, what can we do more here?
We could divide by 2 again.
Let me rewrite it.
So this is the same thing as 150/8 people.
150, we can divide by 2.
It gives us 75.
And if you take 8 divided by 2, that is 4.
So we have 75 divided by 4 people.
Let me just do that, work it out.
So you have 75 divided by 4.
4 goes into 7 one time.
1 times 4 is 4.
You subtract.
7 minus 4 is 3.
Bring down this 5.
4 goes into 35 eight times.
8 times 4 is 32.
Subtract.
5 minutes 2 is 3.
And then you have the decimal.
We're going to the right of the ones place.
We're going to the tenths place now.
So we can bring down a zero over here.
4 goes into 30 seven times.
7 times 4 is 28.
You subtract.
You get 2.
Bring down another zero.
4 goes into 20 exactly five times.
5 times 4 is 20, and then we are done.
So this expression is exactly-- the 3,000 divided by
160, or the 150 divided by 8, the 75 divided by 4, it all
turns out to be 18.75 people is the
capacity of the elevator.
18.75 average 160-pound people would weigh 3,000 pounds.
Now, do they want a decimal?
What is the maximum number of adults who could safely ride
the elevator?
Well, if they're all going to be average, then the maximum
number of adults, since you can't have 3/4 of a person, or
75/100 of a person, the maximum number
is going to be 18.
If you have 19 average-weight adults, you are going to be
too heavy, and the elevator will fall or something.

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