Overtaking Word Problem

3 people overtaking each other at different rates

Here's another problem that
Kortaggio sent me and,

like the other two, this
is quite interesting.

Although, I think you'll start
to see a pattern forming

in how these are solved.

And that's the whole point, not
just to show you neat problems,

but also to give you an
intuition of how to solve it.

So what does it say?

It says Alice is 100 meters
from Bill and Bill is 300

meters away from Chelsea.

Maybe this is a bit of a
Clintonian reference, but who

knows, or maybe they just
wanted people with starting

names a, b, and c.

So let's draw that out.

So Alice is 100 meters
from Bill-- all right.

Alice is 100 meters from Bill--
and actually he drew a little

diagram, so we know that also
Chelsea is to the

right of Bill.

So let's see, the diagram he
sent me looks like this.

So you have Alice.

And then 100 meters
away you have Bill.

And then 300 meters to the
right of that you have Chelsea.

This distance right
here is 100 meters.

And then this distance
right there is 300 meters.

And the problem tells us they
are all facing east and are

standing on the same line.

Fair enough.

We'll say that's east.

So they're all facing in
that direction standing

in the same line.

They all travel to the
east at constant speeds.

OK, but they don't say that
they're all traveling

at the same speed.

Each individual
speed is constant.

So this could be velocity of
Alice, lowercase a, velocity

of Bill, and then this is
the velocity of Chelsea.

Alright, they all travel
east at constant speeds.

In 6 minutes, Alice
overtakes Bill.

So what has to happen?

Let me do this in another color
and write down what that

equation would look like.

So I'll do it in green.

In 6 minutes, Alice
overtakes Bill.

This tells the distance that
Alice travels in 6 minutes.

So what's the distance that
Alice travels in 6 minutes?

It would be the velocity
of Alice times 6.

And she overtakes Bill.

So that means she goes 100
meters more than Bill went.

So that means she went 100
meters plus the distance that

Bill went in that 6 minutes.

And the distance that Bill
went in 6 minutes is

velocity of Bill times 6.

Right?

This tell us in 6 minutes,
Alice went 100 meters more

than however far Bill
went in 6 minutes.

That's that equation there.

We got it from
this information.

And then the next thing,
this problem tells us, in

another 6 minutes Alice
overtakes Chelsea.

So from the beginning of
our time, it essentially

takes 12 minutes.

Right?

6 minutes to overtake Bill
and then another 6 minutes

to overtake Chelsea.

So 12 minutes to
overtake Chelsea.

So we could write that as the
distance that Alice travels

in 12 minutes is equal to--

Well, she has to
overtake Chelsea.

That means she made up 400
meters of distance, so that

she traveled 400 more meters.

So the distance that Alice
travels is going to be 400

meters more than the distance
that Chelsea travels

in 12 minutes.

And remember, distance is just
equal to rate times time.

Right?

So in 12 minutes this is how
far-- oh sorry --this is how

far Chelsea travels,
that's a c.

And this is how far Alice
travels, and she is going

to travel 400 meters more.

OK, that's what those
two equations tell us.

And what is their question?

They're asking us, how many
minutes did it take Bill

to overtake Chelsea?

How many minutes did it take
Bill to overtake Chelsea?

All right, so let's say we
want to know time in minutes.

How long does it take Bill
to overtake Chelsea?

Let's say we know
that it's time, t.

So that tells us that the
velocity of Bill times this

time is going to be equal to
--for him to overtake Chelsea,

he has to travel 300 meters
more than her in that

same amount of time.

So he has to go 300 meters
more than how far Chelsea

travels in that time.

So that's what this is, and
then we essentially are just

trying to solve for this time.

We want to be able
to solve for this.

Let's solve this equation.

Remember, all these equations
really have the same form.

In order for this person
to overtake this

person in 6 minutes.

That means in 6 minutes she
would have to travel 100

meters more than him.

In 12 minutes she would
have to travel 400 meters

more than Chelsea.

And then in t minutes-- and
that's what we're going to have

to solve the problem for --b is
going to have to travel

300 meters more than
Chelsea right there.

We just solve for t, so
let's just solve for t.

So what do you get?

You get vbt minus vct
is equal to 300.

I just subtracted Chelsea's
velocity times time from

both sides of the equation.

We can factor out the
time, so let's do that.

So you get vb minus vc times
time-- I'm just factoring out

the time --is equal to 300.

Or time is equal to
300/vb minus vc.

So if we could somehow figure
out what this is equal

to, then we would have
solved our problem.

Let's see if we can use
the other information.

Let's see, if we solve for vb
here and solved for vc here,

hopefully the va's
will cancel out.

And I suspect they will,
otherwise this problem

would not be solvable.

So let's do that.

Let's rewrite this
equation up here.

Let's solve for vb,
for Bill's velocity.

Let's see, if we divide both
sides by 6-- well, let's

subtract 100 from both sides.

So you have Bill's velocity
times 6-- that's just that

--is equal to 6 times
Alice's velocity minus 100.

All I did is I subtracted
100 from both sides of

this equation and I
swapped the sides.

Divide both sides by 6.

You got the velocity of Bill
is equal to the velocity

of Alice minus 100/6.

100/6 or 50/3-- I'll just
keep it as a 100/6.

Fair enough.

Let's see if we can do the same
thing with this equation.

If we subtract 400 from both
sides, we get 12 times

Chelsea's velocity is equal
to 12 times Alice's

velocity minus 400.

Divide both sides of
this equation by 12.

You get Chelsea's velocity
is equal to Alice's

velocity minus 400/12.

All right.

So let's if we can substitute
this back in for vb, and

substitute this back in for vc.

And hopefully the
va's cancel out.

It already looks
like they should.

So our time, the time that it
takes for Bill to overtake

Chelsea-- remember that's what
t was, it's always good to

remind yourself what this whole
problem was about to begin

with --is equal to 300/vb.

Well instead of vb
let's write this.

Va, the velocity of Alice,
minus 100/6 minus vc.

So minus this right here.

Minus va minus 400/12.

This should simplify to
t-- I'll just arbitrarily

switch colors.

t is equal to 300/va
minus-- what's 100/6?

100/6 is the same
thing as 50/3.

Distribute the minus sign.

Minus va-- we can already see
these are going to cancel out

--minus times a minus plus
400/12 is the same thing

is 200/6 or 100/3.

Right?

Plus 100/3.

These two cancel
out, va minus va.

So then we're left with
t is equal to 300 over

100/3 minus 50/3.

Right?

Or I could say minus 50/3 plus
100/3, but either way, minus

50/3 plus 100/3 is just 50/3.

Did I do that right?

If I divide the top and the
bottom by 4, I get this.

And then sure, this is
a 100/3 and than 50/3.

And so when you divide by a
fraction that's the same thing

is multiplying by its inverse.

300 times 3/50 over 1.

And then we can cancel out
some terms, so we don't have

to multiply big numbers.

So if we divide the top and the
bottom by 50 that becomes 1,

this becomes 6, and then we're
left with t is equal to 6 times

3/1, which is 18, and all the
time units we were working with

the whole time were in minutes.

And we're done.

That was a little hairier than
the other two, but notice we

were able to solve it by just
algebraically writing down the

information that they actually
gave us in the problem and just

seeing what we have to solve
for and then substituting back

in and then miraculously
things canceled out.

And in general, if you're given
a nice problem like this you

can normally just, you know,
march forward knowing that if

you do things correctly
probably nice things will

happen and terms
will cancel out.

But I thought you would find
that interesting and once

again, thanks to Kortaggio
for that problem.