it takes 7 hours for Isabella to rake leaves by herself but her brother Matthew can work three times as fast. if they work together how long will it take them both to rake leaves  

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embizze eNotes educator| Certified Educator

If we let r be the rate Isabella rakes, then 3r is the rate at which Matthew rakes. For a unit job, their combined rate is r+3r=4r. Since a unit job took 7 hours, the time it takes for the combined effort is 7/4 hours.

It will take them 7/4 hours or 1 hour and 45minutes to finish the raking.

** To make this concrete, suppose the area to be raked is 28 square meters. Since Isabella can finish the job in 7 hours, she rakes 4 square meters per hour. Now Matthew is three times as fast, so he rakes at 12 square meters per hour. Together they rake 4+12=16 square meters per hour. Since the job is 28 square meters, the time required is 28/16=7/4 hours as above.