Work and Time Calculation: Finding B's Time
This problem involves calculating the time taken by an individual (B) to complete a task, given information about their work rate relative to another individual (A) and their combined working time.
Understanding Work Rates
Let $R_A$ be the rate at which A works and $R_B$ be the rate at which B works. Let $T_A$ be the time A takes and $T_B$ be the time B takes.
- We are given that A does half as much work as B in three-fourth the time. Let's express this using rates.
- Work done = Rate × Time.
- Work done by A in time $T_A$ is $W_A = R_A \times T_A$.
- Work done by B in time $T_B$ is $W_B = R_B \times T_B$.
- The condition states: $R_A \times (\frac{3}{4} T_B) = \frac{1}{2} (R_B \times T_B)$.
- Simplifying this equation: $R_A \times \frac{3}{4} = \frac{1}{2} R_B$.
- Solving for $R_A$: $R_A = \frac{1}{2} \times \frac{4}{3} R_B = \frac{2}{3} R_B$.
- This means A's work rate is two-thirds of B's work rate.
Calculating Combined Work Rate
When A and B work together, their combined rate is the sum of their individual rates:
Combined Rate $= R_A + R_B = \frac{2}{3} R_B + R_B = (\frac{2}{3} + 1) R_B = \frac{5}{3} R_B$.
Determining B's Individual Time
They complete the work together in 12 days. Let the total work be $W$. The relationship is:
Total Work $W = (\text{Combined Rate}) \times (\text{Time taken together})$.
$W = (\frac{5}{3} R_B) \times 12 \text{ days}$.
$W = 5 \times 4 \times R_B = 20 R_B$.
To find the time B takes to do the work alone ($T_B$), we use the formula:
$T_B = \frac{\text{Total Work}}{\text{B's Rate}} = \frac{W}{R_B}$.
$T_B = \frac{20 R_B}{R_B} = 20$ days.
Therefore, B shall take 20 days to do the work alone.
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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?
