"Work" Word Problems


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"Work" Word Problems (page 3 of 4)

Working alone, Maria can complete a task in 100 minutes. Shaniqua can complete the same task in two hours. They work together for 30 minutes when Liu, the new employee, joins and begins helping. They finish the task 20 minutes later. How long would it take Liu to complete the task alone?

Convert their times to unit rates, making sure to convert "hours" to "minutes", so the units match:

minutes to complete job:
Maria: 100
Shaniqua: 120
Liu: t

completed per minute:
Maria: ¹/₁₀₀
Shaniqua: ¹/₁₂₀
Liu: ¹/t

They weren't all working together for the first half hour; only Maria and Shaniqua were working. To figure how much they've completed, remember that (fraction of work) = (rate per unit) × (number of units)

( ¹/₁₀₀ + ¹/₁₂₀ completed / minute) × (30 minutes)

= ( ¹¹/₆₀₀ completed / minute) × (30 minutes)

= 0.55 completed

That means that 1 – 0.55 = 0.45 (or 45%) of the job remained when Liu joined them. Use the "(fraction of work) = (rate per unit) × (number of units)" construction to set up your equation, remembering to add Liu's labor for 20 minutes to the mix:

finishing the job:

( ¹/₁₀₀ + ¹/₁₂₀ + ¹/t completed / minute) × (20 minutes)

= 0.45 completed

( ¹¹/₆₀₀ + ¹/t) × (20) = 0.45

¹¹/₆₀₀ + ¹/t = 0.0225

¹/t = ¹/₂₄₀

t = 240

That is, Liu's time for the task is 240 minutes, or four hours.

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