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# Algebra

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Help how do I use rates here

Carissa the chessplayer is a very, very slow walker. In fact, she walks at 100 meters per hour when walking uphill, 120 meters per hour when walking across flat ground, and 180 meters per hour when walking downhill. One day, Carissa walks across Boston from a café to a boba shop, and then takes the same route in reverse to return to the café. What was Carissa's average speed during the entire round trip?

Feb 3, 2023

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Average speed = Total distance / Total time

2  x  [100 +  120  +  180] ==800 m - total distance walked back and forth

[1  +  1  +  1  +  180/100 + 1 + 100/180 ==4  +  1.8  +  5/9 ==6.35555....hours - total time spent walking

800 / 6.35555.... ==125.874 m/h - her average speed

Note: When the distance back and forth is the same, then:

Average speed ==Harmonic mean

Harmonic mean: 3 / (100^-1 + 120^-1 + 180^-1)==125.581 m/h - which is very close to the actual average speed. The reason they are not exactly the same, is because the time back and forth is not exactly the same because of the uphill and downhill walking makes a difference of 0.3555 of an hour due to the 180 m downhill + 100 m uphill.

Feb 3, 2023