Depth · one concept from first principles · step 2 of 4
Two people, A and B, start apart and walk toward each other, A at per minute and B at per minute. At the same moment, a dog starts out with A and runs toward B at per minute; when it reaches B it turns back toward A, and it keeps running back and forth this way. How many kilometers has the dog run by the time A and B meet?
The speeds are in meters per minute, so write the gap in meters.
Mixing kilometers and meters would throw the answer off by a factor of 1000.
6.RP.A.3Analyze The UnitsLet them meet after x minutes. Walking toward each other, their distances add up to the whole gap.
Together they close 50 m every minute.
7.EE.B.4Convert To AlgebraThe dog runs the whole 40 minutes at 200 m per minute, no matter how many times it turns.
Turning around changes the dog's direction, not its speed or running time.
6.RP.A.3Change Focus Count The ComplementThe dog runs meters, however many times it turns.
The dog runs for exactly as long as A and B take to meet, which is minutes.
Walking toward each other, A and B each eat into the meter gap, so it shrinks by meters every minute.
At a steady meters a minute, closing meters takes minutes with , so .
Turning around changes only the dog's direction, so its whole run is just minutes at meters a minute.
The back-and-forth legs are pieces of one run with no gaps or overlaps, so they add up to the whole run.
At a steady speed, distance is speed times time.
When the path is messy but the clock is simple, find the time first.
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