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A task can be completed in $20$ days, if Rose and Ryan work together. However, if Rose worked alone and completed half the task and then Ryan takes over and completes the second half, the task will be completed in $45$ days. How long will Rose take to complete the task if she worked alone ? Assume that Ryan is more efficient than Rose.

1. $25$ days
2. $30$ days
3. $60$ days
4. $65$ days
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Option C. 60 days

Let Rose take 'x' days to complete the task if she worked alone.

Let Ryan take 'y' days to complete the task if he worked alone.

In 1 day, Rose will complete $\frac{1}{x}$ of the task.

In 1 day, Ryan 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)

Rose will complete half the task in $\frac{x}{2}$ days.

Ryan will complete half the task in $\frac{y}{2}$ days.

∴ $\frac{x}{2}$ + $\frac{y}{2}$ = 45

Or, x + y = 90 …. (2)

By solving both the equation you will get, 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 clearly states that Ryan is more efficient than Rose. Therefore, Ryan will take lesser time than Rose.

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

Shortcut method:

Rose + Ryan = 90 then definitely Rose will take more then 45 days (option C. and D.)

and option C only satisfies the $eq^{n}$ (1) i.e. $\frac{1}{60}$ + $\frac{1}{30}$ = $\frac{1}{20}$

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