Distance rate time word problem

[Ara Anderson]

Hi,

I'm trying to solve this problem:

Phyllis is navigating her boat upstream in the Orinoco River. She plans to do a roundtrip voyage, heading back to where she began after circling around a buoy upstream. The speed of the current is 3 miles per hour. If Phyllis’ boat can travel 16 miles per hour in still water, how long does it take her to make her trip if the buoy is 2 miles from her starting point?

If I use my common sense, I would say that it takes Phyllis 15 minutes, because any time added going upstream would be made up for going downstream, so I could just solve this problem using 10mph as my rate:

t=d/r, t=4/16=1/4hr

But, if I break it down, calculating for the time on each individual leg of the trip, I get:

Upstream
r=16-3=13
Therefore,
t=d/r, t=2/13=approx 0.154

Downstream
t=d/r, r=16+3=19
Therefore,
t=d/r, t=2/19=approx 0.105

Total time = 0.154+0.105 = approx 0.259, which is a tad over 15 minutes.

Please help! Why do I get different answers?

Ara

[nona.m.nona]

Phyllis is navigating her boat upstream in the Orinoco River. She plans to do a roundtrip voyage, heading back to where she began after circling around a buoy upstream. The speed of the current is 3 miles per hour. If Phyllis’ boat can travel 16 miles per hour in still water, how long does it take her to make her trip if the buoy is 2 miles from her starting point?

If I use my common sense, I would say that it takes Phyllis 15 minutes, because any time added going upstream would be made up for going downstream, so I could just solve this problem using 10mph as my rate:

t=d/r, t=4/16=1/4hr

This assumption is in error. One may not "average" rates in the manner you propose. For instance, in the extreme case, suppose the current was 16 miles per hour. Then the net rate with the current would be 32 miles per hour, while the net rate against the current would be zero. The time then would be "infinite", in that the boat would never return to the starting point.

[jg.allinsymbols]

Make a Rate, Time, Distance table for all numbers in the description.


________________speed_________________time________________distance
upstream________16-3=13__________________(___)_________________2
downstream______16+3=19_________________(___)_________________2
RoundTrip______________________________(___)__________________4


We use r*t=d to get an expression for the missing times. in r*t=d, divide left and right by r and we have t=d/r. This means, we complete our speed,time,distance table like this:

________________speed_________________time(in hours)________________distance
upstream________16-3=13__________________(2/13)________________________2
downstream______16+3=19_________________(2/19)________________________2
RoundTrip______________________________(2/13+2/19)_______________________4

You simply now need to compute the sum of those two fractions as indicated; LCD will be 13*19. Remember the units for the sum will be hours, and in this case, less than 1 hour, so you might wish to convert the result to minutes.
(The total of sum to 4 miles is not important.)