This GMAT Math practice question is a word problem in rates - work and time. The crux of solving work-time questions is your ability to frame equations from the information given. A medium difficulty, GMAT 650 level, problem solving sample question.

Question 7: Working together, Jose and Jane can complete an assigned task in 20 days. However, if Jose worked alone and completed half the work and then Jane takes over and completes the second half, the task will be completed in 45 days. How long will Jose take to complete the task if he worked alone? Assume that Jane is more efficient than Jose.

- 25 days
- 30 days
- 60 days
- 65 days
- 36 days

@ INR

Let Jose take 'x' days to complete the task if he worked alone.

Let Jane take 'y' days to complete the task if she worked alone.

**Statement 1 of the question**: They will complete the task in 20 days if they worked together.

In 1 day, Jose will complete \\frac{1}{x}) of the task.

In 1 day, Jane will complete \\frac{1}{y}) of the task.

Together, in 1 day they will complete \\frac{1}{20}) of the task.

Therefore,\\frac{1}{x}) + \\frac{1}{y}) = \\frac{1}{20}) .... (1)

**Statement 2 of the question**: If Jose worked alone and completed half the work and then Jane takes over and completes the second half, the task will be completed in 45 days.

Jose will complete half the task in \\frac{x}{2}) days.

Jane will complete half the task in \\frac{y}{2}) days.

∴ \\frac{x}{2} + \frac{y}{2}) = 45

Or, x + y = 90 or x = 90 - y .... (2)

Substitute the value of x as (90 - y) in the first equation.

\\frac{1}{90 - y} + \frac{1}{y} = \frac{1}{20})

\\frac{y + 90 - y}{y({90 - y)}} = \frac{1}{20})

Cross multiply and simplify: 1800 = 90y - y^{2}

Or y^{2} - 90 + 1800 = 0.

The quadratic equation factorizes as (y - 60)(y - 30) = 0

So, y = 60 or y = 30.

If y = 60, then x = 90 - y = 90 - 60 = 30 and

If y = 30, then x = 90 - y = 90 - 30 = 60.

The question states that Jane is more efficient than Jose. Therefore, Jane will take lesser time than Jose.

Hence, Jose will take 60 days to complete the task if he worked alone and Jane will take 30 days to complete the same task.

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