Work Calculation: A's Effort
A completes the task working 5 hours a day for 8 days. The total hours A works is:
Total Hours for A = $5 \text{ h/day} \times 8 \text{ days} = 40 \text{ hours}$
Work Calculation: B's Effort
B completes the same task working 6 hours a day for 10 days. The total hours B works is:
Total Hours for B = $6 \text{ h/day} \times 10 \text{ days} = 60 \text{ hours}$
Combined Work Rate Calculation
Let the total amount of work required to complete the task be $W$ units.
- A's work rate is $\frac{W}{40}$ units per hour.
- B's work rate is $\frac{W}{60}$ units per hour.
When working together, their combined work rate is the sum of their individual rates:
Combined Rate = A's Rate + B's Rate = $\frac{W}{40} + \frac{W}{60}$
To add these fractions, find a common denominator, which is 120:
Combined Rate = $\frac{3W}{120} + \frac{2W}{120} = \frac{5W}{120} = \frac{W}{24}$ units per hour.
Joint Task Completion Time
A and B now work together, 8 hours a day. Let $D$ be the number of days they take to complete the task jointly.
The total hours they work together is $8 \times D$ hours.
The total work done is their combined rate multiplied by the total hours worked:
Total Work = Combined Rate $\times$ Total Hours Worked
$W = \frac{W}{24} \times (8 \times D)$
Simplify the equation:
$W = \frac{8DW}{24}$
Divide both sides by $W$ (assuming $W \neq 0$):
$1 = \frac{8D}{24}$
$1 = \frac{D}{3}$
Solving for $D$:
$D = 3$
Therefore, they can jointly complete the task in 3 days.
Solution Summary
The task requires 3 days to be completed when A and B work together for 8 hours daily.
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Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
