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?
90 days
Solving Work and Time Problems: Finding B's Days
This problem involves calculating the time taken by individuals to complete a piece of work, given their combined work rates and the relative efficiency of two individuals. We are given information about how long A and B take together, how long B and C take together, and how the work efficiency of A compares to C. We need to find out how many days B alone would take to complete the work.
Understanding the Given Information
Let's denote the daily work rate of A, B, and C as $r_A$, $r_B$, and $r_C$ respectively. The work rate is the fraction of the total work completed in one day. If a person completes the work in $D$ days, their daily work rate is $\frac{1}{D}$. The total work is considered as 1 unit.
- A and B together can do the work in 18 days. This means their combined daily work rate is $\frac{1}{18}$.
- B and C together can do the work in 30 days. This means their combined daily work rate is $\frac{1}{30}$.
- A is twice as good a workman as C. This means A's work rate is double that of C.
Setting Up Equations based on Work Rates
From the information above, we can form the following equations:
- Combined daily work rate of A and B: $r_A + r_B = \frac{1}{18}$
- Combined daily work rate of B and C: $r_B + r_C = \frac{1}{30}$
- Relationship between A and C's work rates: $r_A = 2r_C$
Solving the System of Equations to Find Individual Work Rates
We have a system of three linear equations with three variables ($r_A, r_B, r_C$). We can solve this system to find the value of each individual's daily work rate. Our goal is to find $r_B$ so we can calculate the number of days B takes alone.
We can use the third equation ($r_A = 2r_C$) to substitute $r_A$ in the first equation:
Substituting $r_A = 2r_C$ into equation (1):
$(2r_C) + r_B = \frac{1}{18}$
So we have a new equation:
4. $r_B + 2r_C = \frac{1}{18}$
Now we have a system of two equations with two variables ($r_B$ and $r_C$):
- From equation (2): $r_B + r_C = \frac{1}{30}$
- From equation (4): $r_B + 2r_C = \frac{1}{18}$
Let's subtract equation (2) from equation (4):
$(r_B + 2r_C) - (r_B + r_C) = \frac{1}{18} - \frac{1}{30}$
$r_C = \frac{5}{90} - \frac{3}{90}$ (Finding a common denominator, 90)
$r_C = \frac{5 - 3}{90} = \frac{2}{90} = \frac{1}{45}$
So, C's daily work rate is $\frac{1}{45}$.
Now we can find $r_B$ by substituting the value of $r_C$ into equation (2):
$r_B + r_C = \frac{1}{30}$
$r_B + \frac{1}{45} = \frac{1}{30}$
$r_B = \frac{1}{30} - \frac{1}{45}$
$r_B = \frac{3}{90} - \frac{2}{90}$ (Finding a common denominator, 90)
$r_B = \frac{3 - 2}{90} = \frac{1}{90}$
Thus, B's daily work rate is $\frac{1}{90}$.
Calculating the Days B Takes Alone
If B's daily work rate is $\frac{1}{90}$, it means B completes $\frac{1}{90}$ of the total work each day. To complete the entire work (1 unit), B will take the reciprocal of the daily work rate.
Number of days B takes alone = $\frac{1}{r_B} = \frac{1}{1/90} = 90$ days.
Let's summarize the daily work rates we found:
| Person | Daily Work Rate |
|---|---|
| C | $\frac{1}{45}$ |
| A ($2 \times r_C$) | $2 \times \frac{1}{45} = \frac{2}{45}$ |
| B | $\frac{1}{90}$ |
We can verify our rates using the initial equations:
- $r_A + r_B = \frac{2}{45} + \frac{1}{90} = \frac{4}{90} + \frac{1}{90} = \frac{5}{90} = \frac{1}{18}$ (Matches A and B taking 18 days)
- $r_B + r_C = \frac{1}{90} + \frac{1}{45} = \frac{1}{90} + \frac{2}{90} = \frac{3}{90} = \frac{1}{30}$ (Matches B and C taking 30 days)
The daily work rate of B is $\frac{1}{90}$. Therefore, B alone can complete the work in 90 days.
Revision Table: Work and Time Concepts
| Concept | Explanation |
|---|---|
| Work Rate | The amount of work done per unit of time (e.g., per day). If work is completed in D days, rate is $\frac{1}{D}$. |
| Total Work | Usually considered as 1 unit. |
| Combined Work Rate | If individuals work together, their daily work rates are added up. For example, $(A+B)$'s rate is $r_A + r_B$. |
| Days to Complete | Reciprocal of the daily work rate. If rate is $r$, days = $\frac{1}{r}$. |
Additional Information: Efficiency and Work Rate
The term "efficiency" in work and time problems is directly proportional to the work rate. If person A is twice as efficient as person C, it means A does twice the amount of work C does in the same amount of time, or A's work rate is twice C's work rate ($r_A = 2r_C$). Conversely, if A is more efficient, A will take less time to complete the same amount of work compared to C, assuming they work alone. Specifically, if work rates are in ratio $r_A : r_C$, the times taken to complete the same work alone will be in the ratio $\frac{1}{r_A} : \frac{1}{r_C}$. So, if $r_A = 2r_C$, then $\frac{1}{r_A} = \frac{1}{2r_C}$, meaning A takes half the time C takes.
Understanding the relationship between efficiency, work rate, and time taken is crucial for solving work and time problems effectively.
Important Questions from Time and 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?
40 men can complete a work in 15 days. Three days after they started working, 20 more men joined them. In how many days the total work will be completed?
