Solution
Content
Solution
One student solved this problem by the Guess and Check method. Another student solved it using algebra (see below).
Level
Level Time 40 min
Uphill
Uphill Time 240 min
Downhill Downhill Time 80 min 6 km 6 km 80 min
Total Time 360 min
Total Distance
2 km
2 km
40 min
6 km 6 km
240 min
360 min
18 km
I decided to make a guess and see what happens. First I made a table for all the information I would need. I guessed 2 kilometers for level walking and entered 2 km. in the Level column. The rate for level walking is 3 km/h. That means the Level Time would be 40 minutes. Because this time is in minutes, I changed the total time from 6 hours to 360 minutes. The Uphill and Downhill Times must be 360-40 minutes, or 320 minutes. The rate for uphill walking is 2 km/h and the rate for downhill walking is 6 km/h. The ratio of the Uphill rate to the Downhill rate is 1 to 3, so I divided 320 minutes according to this ratio. Walking Uphill would take three times as long as walking Downhill. So the Uphill Time would be 240 minutes and the Downhill Time would be 80 minutes. Then I wondered how far she could walk Uphill in 240 minutes at 2 km/h and how far she could walk Downhill in 80 minutes at 6 km/h. The answer is 6 km. for Uphill and 6 km. for Downhill. I wondered why the distance Uphill is the same as the distance Downhill and I realized that the distance she goes Uphill going to Andrew's house is the same distance she goes Downhill coming back, and the distance she goes Downhill going to Andrew's is the same distance she goes Uphill coming back. So the Uphill distance for the round trip must be the same as the Downhill distance for the round trip. Then I could fill in all the columns in my chart and get the Total Distance of 18 km. My first guess of 2 kilometers for Level walking resulted in 18 kilometers Total Distance. Then I tried some other guesses: 4 kilometers Level and 6 kilometers level. The result was still 18 kilometers total distance. So then I tried making the whole 18 kilometers Level, and I got 6 hours (360 minutes) for the Total Time. So 18 kilometers must be the right answer, and it doesn't matter how much Level, how much Uphill, and how much Downhill there is.
Level
Level Time 80 min. 120 min. 360 min
Uphill
Uphill Time 210 min 180 min.
Downhill Downhill Time 70 min 60 min.
Total Time 360 min. 360 min. 360 min.
Total Distance 18 k 18 k 18 k.
4 6 18
7 6
7 6
Solution using algebra:
One student solved this problem by the Guess and Check method. Another student solved it using algebra (see below).
Level
Level Time 40 min
Uphill
Uphill Time 240 min
Downhill Downhill Time 80 min 6 km 6 km 80 min
Total Time 360 min
Total Distance
2 km
2 km
40 min
6 km 6 km
240 min
360 min
18 km
I decided to make a guess and see what happens. First I made a table for all the information I would need. I guessed 2 kilometers for level walking and entered 2 km. in the Level column. The rate for level walking is 3 km/h. That means the Level Time would be 40 minutes. Because this time is in minutes, I changed the total time from 6 hours to 360 minutes. The Uphill and Downhill Times must be 360-40 minutes, or 320 minutes. The rate for uphill walking is 2 km/h and the rate for downhill walking is 6 km/h. The ratio of the Uphill rate to the Downhill rate is 1 to 3, so I divided 320 minutes according to this ratio. Walking Uphill would take three times as long as walking Downhill. So the Uphill Time would be 240 minutes and the Downhill Time would be 80 minutes. Then I wondered how far she could walk Uphill in 240 minutes at 2 km/h and how far she could walk Downhill in 80 minutes at 6 km/h. The answer is 6 km. for Uphill and 6 km. for Downhill. I wondered why the distance Uphill is the same as the distance Downhill and I realized that the distance she goes Uphill going to Andrew's house is the same distance she goes Downhill coming back, and the distance she goes Downhill going to Andrew's is the same distance she goes Uphill coming back. So the Uphill distance for the round trip must be the same as the Downhill distance for the round trip. Then I could fill in all the columns in my chart and get the Total Distance of 18 km. My first guess of 2 kilometers for Level walking resulted in 18 kilometers Total Distance. Then I tried some other guesses: 4 kilometers Level and 6 kilometers level. The result was still 18 kilometers total distance. So then I tried making the whole 18 kilometers Level, and I got 6 hours (360 minutes) for the Total Time. So 18 kilometers must be the right answer, and it doesn't matter how much Level, how much Uphill, and how much Downhill there is.
Level
Level Time 80 min. 120 min. 360 min
Uphill
Uphill Time 210 min 180 min.
Downhill Downhill Time 70 min 60 min.
Total Time 360 min. 360 min. 360 min.
Total Distance 18 k 18 k 18 k.
4 6 18
7 6
7 6
Solution using algebra:
