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# Algebra

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Can someone help with rate problems

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?

Jan 27, 2023

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We can use the formula for average speed, which is the total distance traveled divided by the total time taken. Let's call the total distance traveled "d".

Since Carissa took the same route both ways, the distance traveled uphill is the same in both directions, and the same is true for the distance traveled downhill and across flat ground. So, the total distance traveled uphill is 2d/3, the total distance traveled across flat ground is d/3, and the total distance traveled downhill is 2d/3.

The time taken to travel uphill at 100 meters per hour is 2d/3 * (1 hour/100 meters) = 2d/(300 meters), the time taken to travel across flat ground at 120 meters per hour is d/3 * (1 hour/120 meters) = d/(360 meters), and the time taken to travel downhill at 180 meters per hour is 2d/3 * (1 hour/180 meters) = 2d/540 meters.

The total time taken is 2d/(300 meters) + d/(360 meters) + 2d/540 meters = (12d + 12d + 10d)/(180 meters) = 24d/180 meters = 4d/30 meters.

The average speed during the entire round trip is d/(4d/30 meters) = 30/4 = 7.5 meters per hour.

Feb 21, 2023