X can finish a job in 40 days. He works for 8 days, and Y completes the remaining work in 16 days. How long will X and Y together take to complete the whole work from start?

Problem restatement
Determine individual rates from the partial schedule, then compute their combined time for the entire work.

Given data

• X alone: 40 days ⇒ rate = 1÷40 per day.
• X works 8 days; Y finishes in 16 days.

Concept/Approach
Compute the remaining fraction after X works, deduce Y’s rate, then add rates to find combined time.

Step-by-step calculation
Work done by X in 8 days = 8 ÷ 40 = 1÷5Remaining work = 1 − 1÷5 = 4÷5Y does 4÷5 in 16 days ⇒ Y’s rate = (4÷5) ÷ 16 = 1÷20 per dayCombined rate = 1÷40 + 1÷20 = 3÷40 per dayTime together = 1 ÷ (3÷40) = 40÷3 = 13⅓ days

Verification/Alternative
Rates are consistent with the given completion periods for X and Y individually.

Common pitfalls
Do not average days; always add rates.

Final Answer
13⅓ days