kitchen table math, the sequel: help desk - a girl rides her bike

## Wednesday, July 21, 2010

### help desk - a girl rides her bike

A girl rides her bicycle to school at an average speed of 8 mph. She returns to her house using the same route at an average speed of 12 mph. If the round trip took 1 hour, how many miles is the round trip.

A. 8
B. 9 3/5
C. 10
D. 11 1/5
E. 12

Xiggi says:
Use a simple formula for average rates ....

(2 x speed1 x speed2) ÷ (speed1 + speed2) or in this case: (2 x 8 x 12) ÷ (8 + 12).

Most everyone will notice that the answer is 2*96/20 or simply 96/10. This yields 9.6 or 9 3/5. The total time to do this, probably 20-45 seconds. Not a bad method to know!
How did he derive this formula?

Meet Xiggi.

Catherine Johnson said...

wow - I just checked the thermometer.

95 degrees

It's been amazing, this summer, seeing what I can and can't do in 95 degrees.

Catherine Johnson said...

Which is not to say I would figure this out inside the house -- BUT my speed, accuracy, and insight shoot up as soon as I'm back inside an air-conditioned room.

Daniel Ethier said...

Average speed is not the average of the speeds. You need to use the harmonic mean.

But it's probably just as easy to pick a distance that works nicely with both speeds then fix it at the end:

Say 24 miles each way for a round trip of 48 miles. To school at 8 mph is 3 hours. Home is 2 hours at 12 mph. Total time is 5 hours for 48 miles, but we need 1 hour, so divide by 5 to get 9.6 miles in one hour.

Catherine Johnson said...

Hey Daniel - thanks!

I know how to solve this problem, but I don't know what a harmonic mean is or why it works --- sad to say.

Actually, I've looked up 'harmonic mean,' so in theory I know the definition.

But I don't know why I need a harmonic mean...

Catherine Johnson said...

I must say, I'm impressed that Xiggi has his own entry in urbandictionary.

I had no idea urbandictionary and college confidential intersected.

Daniel Ethier said...

The derivation will be hard to read in text like this, but here goes.

You are making a trip of D each way, going rate R1 in time T1 out and rate R2 in time T2 back.

Average speed is total distance/total time. Total distance for the round trip is 2D. Total time is T1+T2. So 2D/(T1+T2)

Remember that D=RT. And solving for T, we get T=D/R. Doing this for both T1 and T2 and substituting in the denominator, we get the very messy:

2D/(D/R1 + D/R2)

To eliminate the fractions inside fractions business, multiple top and bottom by R1*R2. This cancels the denominators in the fractions on the bottom:

2D*R1*R2/(D*R2 + D*R1)

Notice we have a D we can factor out in the denominator and a D on the top, so all those cancel and we get:

2*R1*R2/(R1 + R2)

I reversed the R1 and R2 on the bottom for appearance. But that is the harmonic mean.

I think my suggestion for solving is easier -- the basic principle being that if they don't tell you something, you can assume a convenient value and fix it in the end if necessary.

David said...

The reason for using the harmonic is that the two legs of the trip cover equal distances instead of equal times.

It might be clearer if we use hours per mile. The girl rides to school at 1/8 hours per mile, and returns at 1/12 hours per mile. Since the miles are equal, the average 'rate' is (1/8 + 1/12)/2 = 1/9.6 hours per mile.

The calculation (8 mph + 12 mph)/2 = 10 mph would be valid if the times were equal. But of course she spent more time riding to school than returning home.

Anonymous said...

1. distance = rate x time

2. If the trip to school takes time t, the trip home takes 1-t since the round trip takes time 1.

3. If r1 is the rate inbound and r2 is the rate outbound, since the distance from home to school is the same as the distance from school to home,

(r1)(t) = (r2)(1-t)

t = r2/(r1+r2)

4. t is the inbound time, so the inbound distance is

(r1)t = (r1)(r2)/(r1+r2)

and, of course, the round trip distance is twice this.

Allison said...

If you read the rest of Chapter 2 of Wu's book, you should see how to do this problem.

Amanda said...

Keep it simple - no need to memorise formulas, just use the simplest algebra and addiing fractions

if the distance to school is x miles

time to school = x/8 hours
time back = x/12 hours

Round trip = 1 hour

so x/8 + x/12 = 1

Multiply through by 24 (lowest common multiple)

3x + 2x = 24

Rearrange and divide

X = 24/5 = 9 3/5

Amanda said...

sorry, missed out a bit

x = 24/5

so round trip = 2x = 48/5 = 9 3/5

LynnG said...

I also didn't use a formula but got the same answer starting with the Singapore bar models and ratios. Our bicyclist covered the second part of her journey in 2/3 the time it took to go to school -- 3:2. Here's my bar graph:

___ ____ _____ time to go
___ ____ time to return

You have five blocks of equal time, so each block is worth 12 minutes (total time is 60 minutes).

She took 36 minutes to pedal to school and 24 minutes to return home. If she pedaled 8 mph for 36 minutes (36 minutes is 3/5 of an hour) then multiply 8 times 3/5 (the distance she covers at 8 mph for 36 minutes). You get 4.8 miles. Double that for the round trip and you get 9.6 or 9 3/5

SteveH said...

Amanda is right. Don't try to remember formulas.

I used the units to figure it out.

mi/(mi/hr) = hr

So, x/8 + x/12 = 1

Let the units lead you to the answer and remember that d=rt has to be applied to each leg.

Of course, it took me a while to figure out (reread the problem) that what they wanted was 2X! If one of the multiple choice answers was just X, then I would have picked that. This is after I drill into my son's head to look carefully at what they want.

Catherine Johnson said...

The reason for using the harmonic is that the two legs of the trip cover equal distances instead of equal times.

Thank you!