Friday Quiz, Number Four

Happy Friday. Our word problem today was submitted by Tommy Kirkman, who describes himself as "someone who loves mathematics and wants to expose its beauty and utility and FUN to others." Kirkman lives in Maryland (somewhere between Washington and Baltimore!)

Here's his problem for you:

If I drive from Washington to Baltimore at 30 mph, how fast do I have to drive back in order to average 60mph for the entire round trip?


I'm off duty for the next week, though I may pop up now and then on the blog. Happy holidays!!

By Michael Alison Chandler  |  December 19, 2008; 9:00 AM ET  | Category:  Friday Quiz
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Hmm, so maybe *that's* what going on on I-95 when there's not a whole bunch of traffic...NOW I get it!

Posted by: | December 19, 2008 10:20 AM | Report abuse

How many people thought, "90 mph!"? It's kind of funny how incredibly far off that is, but it's a very common mistake. Speed doesn't average by distance, it averages by time...

Say the distance between Washington and Baltimore is d, and the speed on the way back is r2. The travel time on the first leg is t1, and on the return trip is t2. Distance = rate x time, so

d = 30 mph * t1..........(1)
d = r2 * t2..............(2)

Now, to average 60 mph for the whole trip means that

d + d = 60 mph*(t1 + t2)


2d = 60 mph*(t1 + t2).....(3)

Now, we've got 3 equations and 4 unknowns. It's pretty easy to get d. Google maps says it's 40 miles, so we can plug that in.

40 mi = 30 mph * t1..........(1)
40 mi = r2 * t2..............(2)
2*40 mi = 60 mph*(t1 + t2)...(3)

From (1), we get

t1 = 40 mi/30 mph = 4/3 hr

Plugging that into (3) gives

80 mi = 60 mph*(4/3 hr + t2)
80 mi/60 mph = 4/3 hr + t2
4/3 hr = 4/3 hr + t2
t2 = 0 hr

Then, from (2), we have

40 mi = r2 * 0 hr
r2 = 40 mi/0 hr
r2 = inf mph

Of course, at infinity mph, you'll have some relativistic time dilation which might affect your answer. But the point to take home is, if you go from point A to point B at speed x, you're never going to average 2x on the round trip, because you have zero time left.

The one thing I'm getting hung up on is, this result is independent of the distance - it's equally true for driving from Washington state to Baltimore. So d should drop out of the equations, and I shouldn't need to go to Google Maps to get it. I'll have to think about that one - something to do with r2 = 2*r1...

Posted by: tomsing | December 19, 2008 10:21 AM | Report abuse

My conclusion was "impossible", not "infinity," and I worked in a different direction than tomsing did.

I immediately looked up the distance, 40 miles. Then I found the time it would take to average 60 miles per hour for the round trip by calculating:
(80 miles)/(60 miles/hour)=1.3 hours

Then I checked to see how long the first leg of the trip would take:
(40 miles)/(30 miles/hour)=1.3 hours

Lo and behold, there's no way to get back to DC in 0 hours!!

Posted by: carysbrookgirl | December 19, 2008 11:01 AM | Report abuse

Nice explanation, but this one is more general (you won't need Google maps for anything).

We know distance = velocity * time. So D1 = V1 * T1 on the first leg. For the total trip (that's both legs considered together), Dt = Vt * Tt.

We also know that Dt = 2*D1, and Vt = 2*V1. Substitute into Dt = Vt * Tt and we get:

2*D1 = 2*V1 * Tt

Divide both sides by 2*V1 and this becomes:

D1/V1 = Tt

From the first equation, we know D1/V1 = T1. Therefore T1 = Tt.

Finally, we know that Tt = T1 + T2, so T2 = 0. In English, we must travel instantaneously on the return leg, i.e. an infinite velocity.

Posted by: JeffRandom | December 19, 2008 11:15 AM | Report abuse

I solved this one using the harmonic mean, which is used to find the average between rates.

The formula is n / [(1/x) + (1/y)] where n is the number of rates, and x,y variables are the given rates.

So, I set up the equation: 2 / [(1/30) + (1/x)] = 60

We need to find an x that satisfies this equation for an average rate of 60. Here's the algebra:

2 / [(x+30)/30x] = 60

60x / (x+30) = 60

60x = 60x + 1800

Obviously this is a contradiction! Thus, we have no solution.

Posted by: DCMathTutor | December 19, 2008 11:23 AM | Report abuse

You can ignore "t" as an unknown, if you look at the equation like this: for the whole trip, d/60 = t; for each leg, there's .5d/30 + .5d/x = t. Since both are equal to "t," combine and solve:

d/60 = .5d/30 + .5d/x

Divide both sides by "d," and you don't have to worry about the distance, either.

So what you end up with is something like 1/30 + 1/x = 2/60 ... and at that point, yes, x goes to infinity.

There's just no real speed you can travel for the exact reason "tomsing" pointed out.

Posted by: exerda | December 19, 2008 11:28 AM | Report abuse

Dear Michael Alison Chandler
We need to challenge brilliant young minds at X=why? I would ask them to not only think about X=why? but also,
X=who ()
X=where $
X=when 1983
X=what *
We need people working on this and soon it will have to go mainstream, from
Wall Street to Main Street. Here are some
more clues;
X=why? why privlaged imfromation
X=who? ()= powerful
X=where? where is the $ (billions) going
X=when? Dec. 30, 1983 Washington Post
X=what? "brown drawf"
"We" need to get to work.

Posted by: 454everafter1 | December 23, 2008 2:39 PM | Report abuse

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