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Here's another problem that Kortaggio sent me and,

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like the other two, this is quite interesting.

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

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in how these are solved.

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

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

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So what does it say?

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

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meters away from Chelsea.

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

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knows, or maybe they just wanted people with starting

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names a, b, and c.

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So let's draw that out.

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

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

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diagram, so we know that also Chelsea is to the

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right of Bill.

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

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So you have Alice.

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And then 100 meters away you have Bill.

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

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This distance right here is 100 meters.

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And then this distance right there is 300 meters.

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

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standing on the same line.

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Fair enough.

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We'll say that's east.

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So they're all facing in that direction standing

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in the same line.

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They all travel to the east at constant speeds.

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

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at the same speed.

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Each individual speed is constant.

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

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

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Alright, they all travel east at constant speeds.

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In 6 minutes, Alice overtakes Bill.

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So what has to happen?

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

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equation would look like.

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So I'll do it in green.

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In 6 minutes, Alice overtakes Bill.

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This tells the distance that Alice travels in 6 minutes.

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

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It would be the velocity of Alice times 6.

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And she overtakes Bill.

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

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So that means she went 100 meters plus the distance that

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Bill went in that 6 minutes.

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And the distance that Bill went in 6 minutes is

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velocity of Bill times 6.

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Right?

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

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than however far Bill went in 6 minutes.

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That's that equation there.

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We got it from this information.

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

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another 6 minutes Alice overtakes Chelsea.

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So from the beginning of our time, it essentially

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takes 12 minutes.

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Right?

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6 minutes to overtake Bill and then another 6 minutes

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to overtake Chelsea.

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So 12 minutes to overtake Chelsea.

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So we could write that as the distance that Alice travels

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in 12 minutes is equal to--

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Well, she has to overtake Chelsea.

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

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she traveled 400 more meters.

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So the distance that Alice travels is going to be 400

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meters more than the distance that Chelsea travels

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in 12 minutes.

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

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Right?

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

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far Chelsea travels, that's a c.

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

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to travel 400 meters more.

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

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And what is their question?

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

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to overtake Chelsea?

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How many minutes did it take Bill to overtake Chelsea?

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

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How long does it take Bill to overtake Chelsea?

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

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So that tells us that the velocity of Bill times this

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

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he has to travel 300 meters more than her in that

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same amount of time.

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

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travels in that time.

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

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trying to solve for this time.

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We want to be able to solve for this.

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Let's solve this equation.

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Remember, all these equations really have the same form.

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In order for this person to overtake this

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person in 6 minutes.

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That means in 6 minutes she would have to travel 100

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meters more than him.

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In 12 minutes she would have to travel 400 meters

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more than Chelsea.

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

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

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300 meters more than Chelsea right there.

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

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So what do you get?

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You get vbt minus vct is equal to 300.

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I just subtracted Chelsea's velocity times time from

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both sides of the equation.

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

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

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the time --is equal to 300.

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Or time is equal to 300/vb minus vc.

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So if we could somehow figure out what this is equal

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to, then we would have solved our problem.

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

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

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hopefully the va's will cancel out.

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And I suspect they will, otherwise this problem

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would not be solvable.

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So let's do that.

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Let's rewrite this equation up here.

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

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

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subtract 100 from both sides.

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

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

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All I did is I subtracted 100 from both sides of

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this equation and I swapped the sides.

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Divide both sides by 6.

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You got the velocity of Bill is equal to the velocity

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of Alice minus 100/6.

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

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Fair enough.

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

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

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

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velocity minus 400.

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Divide both sides of this equation by 12.

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

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velocity minus 400/12.

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All right.

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

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substitute this back in for vc.

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And hopefully the va's cancel out.

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It already looks like they should.

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

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

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remind yourself what this whole problem was about to begin

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with --is equal to 300/vb.

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Well instead of vb let's write this.

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

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So minus this right here.

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Minus va minus 400/12.

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

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switch colors.

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

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

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Distribute the minus sign.

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

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

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is 200/6 or 100/3.

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Right?

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Plus 100/3.

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These two cancel out, va minus va.

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

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100/3 minus 50/3.

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Right?

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

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

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Did I do that right?

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

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

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

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is multiplying by its inverse.

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300 times 3/50 over 1.

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

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to multiply big numbers.

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

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

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

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the whole time were in minutes.

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And we're done.

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

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were able to solve it by just algebraically writing down the

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information that they actually gave us in the problem and just

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seeing what we have to solve for and then substituting back

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in and then miraculously things canceled out.

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

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

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you do things correctly probably nice things will

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happen and terms will cancel out.

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But I thought you would find that interesting and once

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again, thanks to Kortaggio for that problem.

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