How it works
distance = rate × time work = (workers × rate) × time
Work problems: the total "worker-days" (or machine-minutes) for a job stays the same. If 4 workers need 12 days, the job is 48 worker-days, so 8 workers need 6 days.
Average speed is total distance ÷ total time, not the average of the speeds. For two equal distances at speeds a and b, the average speed is 2ab ÷ (a + b), which is always less than (a + b) ÷ 2 because you spend more time going slowly.
Trains passing a pole travel their own length, so speed = length ÷ time.
Rate Problems practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
If 4 workers finish a job in 12 days, how many days do 8 workers need?
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Answer: 6 days
The job takes 4 × 12 = 48 worker-days. With 8 workers: 48 ÷ 8 = 6 days.
A car drives equal distances at 30 mph and 50 mph. Average speed?
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Answer: 37.5 mph
Pick a distance of 150 miles each way. Time: 150/30 + 150/50 = 5 + 3 = 8 hours for 300 miles, so 300 ÷ 8 = 37.5 mph. (Formula: 2 × 30 × 50 ÷ 80 = 37.5.)
A train 140 meters long passes a pole in 7 seconds. Speed in m/s?
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Answer: 20 m/s
To pass a pole, the train moves its own length: 140 ÷ 7 = 20 m/s.
If 10 machines make 20 parts in 4 minutes, how long do 5 machines need for 40 parts?
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Answer: 16 minutes
One machine makes 20 ÷ (10 × 4) = ½ part per minute. Five machines make 2.5 parts per minute, so 40 parts take 40 ÷ 2.5 = 16 minutes.
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