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This is the last of the word problems that Kortaggio had

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sent, and they're all interesting because they're

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all involving distances and rates and times.

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They all involve you having to do a little bit of implicit

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algebra without having to explicitly solve for values of

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variables, and that's why they're fun.

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So this one says Ann and Betty, two cyclists,

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start simultaneously.

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Ann at point A and Betty at point B.

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They travel to each other at constant speeds.

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So let's say that Ann is traveling with velocity A

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and Betty is traveling with velocity B.

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Constant speeds.

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They pass each other in 30 minutes.

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OK, so they're going to pass each other someplace

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in between A and B.

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I'll just arbitrarily draw that.

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They're going to pass each other there.

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And what are these distances?

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It took them 30 minutes to both to get this point, right?

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They pass each other in 30 minutes.

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And Ann, headed from point B, arrives there in

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another 20 minutes.

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OK, so this distance takes Ann 30 minutes.

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So what is the actual distance?

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Well, it's always useful to write distance is equal

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to rate times time.

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So what's the actual distance she traveled?

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Well, the rate is va.

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She traveled a rate of va and her time -- they tell us -- it

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took her 30 minutes to get there.

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So va times 30.

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Let's say this velocity is expressed in minutes.

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Fair enough.

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And then Ann, headed for point B, arrives there

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in another 20 minutes.

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So this is going to take her another 20 minutes.

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So this distance right here is what?

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She's going at the same velocity, but it

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takes her 20 minutes.

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So the distance is velocity times time.

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So it's va times 20, 20va.

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Fair enough.

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How many minutes after Ann arrives at point B will

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Betty arrive at point A?

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So Betty was going in this direction, they meet right

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here, and then she keeps going and hits point A.

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So the key realization here is that this took Ann 30 minutes

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to get here, but it also took Betty 30 minutes to get here.

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So what's this distance in terms of Betty's velocity?

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What's this distance?

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Well, distance is equal to rate times time.

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It took Betty 30 minutes to go this distance

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at a velocity of vb.

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So it took her 30vb meters, or whatever, actually.

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We don't even have to specify the units.

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Whatever the units the velocity is in.

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If it's in meters per minute, then this is

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the distance in meters.

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This distance is 30vb.

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Right?

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So essentially, we just have to figure out how long it takes

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her to travel this remaining distance and we would

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have solved the problem.

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So how can we do that?

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Well, we can already set up a relationship between

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vb and va, because this distance is the same.

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It doesn't matter how fast you travel it, the

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distance doesn't change.

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So we know that 30 times Betty's velocity is equal to 20

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times Ann's velocity, 20va.

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And so if we wanted to express Ann's velocity in terms of

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Betty's velocity -- let's just do it that way -- if we divide

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both sides by 20 we get 3/2 of Betty's velocity is equal

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to Ann's velocity.

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Fair enough.

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So what is this distance in terms of Betty's velocity?

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We know it in terms of Ann's velocity.

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In terms of Ann's velocity, it's 30 times Ann's velocity.

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Right?

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So what is 30 times Ann's velocity in terms of

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Betty's velocity?

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Well, if we multiply this times 30, that's 30 times Ann's

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velocity, so we can multiply both sides of this times 30.

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So if you multiply 30 times here and times thirty,

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this is the distance.

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This distance right here.

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Ann's velocity times 30 minutes.

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It took her 30 minutes to travel that far.

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So this is the same distance expressed in Betty's velocity.

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So 30 times 3 is 90 divided by 2 is 45.

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So it's 45vb.

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So this distance up here is 45vb.

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So distance is equal to rate times -- so this

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is rate -- times time.

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So this is how many minutes it took Betty to

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travel this distance.

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So what is the question asking us?

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How many minutes after Ann reaches point B will

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Betty arrive at point A.

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So they both reach right here.

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In another 20 minutes, Ann gets here.

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And in another 45 minutes, Betty will get over there.

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Another 45 minutes.

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So it takes Ann 20 minutes to get here.

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It takes Betty 45 minutes to get here, after their meeting

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right at the center.

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So it's going to take Betty another 25 minutes.

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Right?

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After 20 minutes Ann is here and then we have to wait

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another 25 minutes for Betty to get here, 45 minutes

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after this meeting.

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So the answer is 25 minutes.

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How many minutes after Ann arrives at point B will

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Betty arrive at point A.

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Yes, 25 minutes.

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They could have just asked how many minutes after the meeting

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will it take Betty to reach point A.

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It would have been 45.

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But they want to know how many more minutes

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after Ann gets there.

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Anyway, I thought you would find that useful and, a final

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time, thanks to Kortaggio for sending that problem.