X can finish a job in 133 days. He worked for 49 days alone and the remaining work is completed by Y in 84 days. How many days would both together take to complete the entire job?

Given:

X can finish a job in = 133 days

X worked alone for = 49 days

Remaining work completed by Y in = 84 days

Formula used:

Efficiency = Total Work / Time

Work Done = Efficiency × Time

Time to complete job together = Total Work / Combined Efficiency

Calculation:

Let Total Work = 133 units.

X's Efficiency = Total Work / Time taken by X

⇒ X's Efficiency = \(\dfrac{133 \text{ units}}{133 \text{ days}}\) = 1 unit/day.

X worked for 49 days.

Work done by X = X's Efficiency × Days worked by X

⇒ Work done by X = 1 unit/day × 49 days = 49 units.

Remaining Work = Total Work - Work done by X

⇒ Remaining Work = 133 units - 49 units = 84 units.

Y completed the remaining 84 units of work in 84 days.

Y's Efficiency = Remaining Work / Time taken by Y

⇒ Y's Efficiency = \(\dfrac{84 \text{ units}}{84 \text{ days}}\) = 1 unit/day.

Combined Efficiency = X's Efficiency + Y's Efficiency

⇒ Combined Efficiency = 1 unit/day + 1 unit/day = 2 units/day.

Time (X + Y) = Total Work / Combined Efficiency

⇒ Time (X + Y) = \(\dfrac{133 \text{ units}}{2 \text{ units/day}}\)

⇒ Time (X + Y) = 66.5 days

∴ Both X and Y together would take 66.5 days to complete the entire job.