Time And Work All Competence Aptitude

Let's first find the rate of work for each person.

1. P can do \( \dfrac{1}{4} \) of the work in 10 days, so the rate of work for P is:\[ P_{\text{rate}} = \dfrac{\dfrac{1}{4}}{10} = \dfrac{1}{40} \] of the work per day.

2. Q can do 40% of the work in 40 days, so the rate of work for Q is:\[ Q_{\text{rate}} = \dfrac{40\%}{40} = \dfrac{2}{5} \times \dfrac{1}{40} = \dfrac{1}{100} \] of the work per day.

3. R can do \( \dfrac{1}{3} \) of the work in 13 days, so the rate of work for R is:\[ R_{\text{rate}} = \dfrac{\dfrac{1}{3}}{13} = \dfrac{1}{39} \] of the work per day.

Now, let's compare the rates of work to determine who will complete the work first.

Since \( P_{\text{rate}} < Q_{\text{rate}} < R_{\text{rate}} \), R has the highest rate of work among them. Therefore, R will complete the work first. 

So, the answer is \boxed{R}.

Let's denote the time taken by B to complete the work as \( x \) days.

Given:
- A takes 50% more time than B. This means A takes \( x + \dfrac{50}{100}x = 1.5x \) days to complete the work.
- Together, A and B take 18 days to complete the work.

So, we can set up the equation based on their combined rates of work:

\[ \dfrac{1}{x} + \dfrac{1}{1.5x} = \dfrac{1}{18} \]

Now, let's solve this equation for \( x \):

\[ \dfrac{1}{x} + \dfrac{2}{3x} = \dfrac{1}{18} \]

\[ \dfrac{3 + 2}{3x} = \dfrac{1}{18} \]

\[ \dfrac{5}{3x} = \dfrac{1}{18} \]

\[ 5 \times 18 = 3x \]

\[ 90 = 3x \]

\[ x = \dfrac{90}{3} \]

\[ x = 30 \]

So, B takes \( \boxed{30} \) days to complete the work.

If Kamal can do a work in 15 days, then his rate of work is \( \dfrac{1}{15} \) of the work per day.

Bimal is 50% more efficient than Kamal, which means his rate of work is \( \dfrac{3}{2} \) times Kamal's rate of work.

So, Bimal's rate of work is \( \dfrac{3}{2} \times \dfrac{1}{15} = \dfrac{1}{10} \) of the work per day.

To find out how many days Bimal will take to complete the work, we divide the total work by Bimal's rate of work:

\[ \text{Number of days} = \dfrac{\text{Total work}}{\text{Rate of work}} = \dfrac{1}{\dfrac{1}{10}} = 10 \] days.

So, Bimal will take \(\boxed{10}\) days to do the work.

Let's denote the time taken by B to complete the work as \( x \) hours.

Given that A does 20% less work than B, we can write the equation:\[ \text{Work done by A} = \text{Work done by B} - 20\% \times \text{Work done by B} \]\[ \text{Work done by A} = \text{Work done by B} \times (1 - 0.20) \]\[ \text{Work done by A} = \text{Work done by B} \times 0.80 \]

Given that A can complete the work in \( 7\dfrac{1}{2} \) hours, we can find A's rate of work per hour:\[ \text{Rate of work for A} = \dfrac{1}{7\dfrac{1}{2}} \]

Given that A does 80% of the work compared to B:\[ \text{Rate of work for A} = 0.80 \times \text{Rate of work for B} \]\[ \dfrac{1}{7\dfrac{1}{2}} = 0.80 \times \dfrac{1}{x} \]

Now, let's solve for \( x \):\[ 7\dfrac{1}{2} \times 0.80 = \dfrac{1}{x} \]\[ 7.5 \times 0.80 = \dfrac{1}{x} \]\[ 6 = \dfrac{1}{x} \]

So, B can complete the work in \( \boxed{6} \) hours.

Let's denote the time taken by B to finish the job alone as \( B \) days. Therefore, A can finish the job alone in \( \dfrac{B}{3} \) days, as A is thrice as good a workman as B.

According to the given information, A is able to finish the job in 40 days less than B. So, we have:

\[ \dfrac{B}{3} = B - 40 \]

Now, let's solve this equation for \( B \):

\[ \dfrac{B}{3} = B - 40 \]\[ \Rightarrow B = 3B - 120 \]\[ \Rightarrow 2B = 120 \]\[ \Rightarrow B = 60 \]

So, B alone can finish the job in 60 days. And since A is three times as efficient as B, A alone can finish the job in \( \dfrac{60}{3} = 20 \) days.

Now, let's find the time taken when they work together:

The formula for the time taken when they work together is given by:

\[ \text{Time taken together} = \dfrac{(x \times y)}{(x + y)} \]

Where \( x \) and \( y \) are the individual times taken by A and B, respectively.

Substituting the values:

\[ \text{Time taken together} = \dfrac{(20 \times 60)}{(20 + 60)} \]\[ = \dfrac{(1200)}{(80)} \]\[ = 15 \text{ days} \]

So, working together, they can finish the job in 15 days. Therefore, the answer is 15 days.

Let's denote the time taken by A to finish the job alone as \( A \) days. Since A is twice as efficient as B, B would take \( 2A \) days to complete the job alone.

Given that A and B can do the job together in 12 days, we can use the formula for their combined work:

\[ \text{Time taken together} = \dfrac{(x \times y)}{(x + y)} \]

Where \( x \) and \( y \) are the individual times taken by A and B, respectively.

Substituting the values:

\[ 12 = \dfrac{(A \times 2A)}{(A + 2A)} \]\[ 12 = \dfrac{(2A^2)}{(3A)} \]\[ 12 = \dfrac{2A}{3} \]\[ A = 18 \]

So, A alone can complete the job in 18 days. Since B takes half the time of A, B alone would take \( 2 \times 18 = 36 \) days to complete the job.

Therefore, the answer is 36 days.

Let's denote the time taken by A to finish the job alone as \( A \) days. Since A is twice as good as B, B would take \( 2A \) days to complete the job alone.

Given that A and B can do the job together in 16 days, we can use the formula for their combined work:

\[ \text{Time taken together} = \dfrac{(x \times y)}{(x + y)} \]

Where \( x \) and \( y \) are the individual times taken by A and B, respectively.

Substituting the values:

\[ 16 = \dfrac{(A \times 2A)}{(A + 2A)} \]\[ 16 = \dfrac{(2A^2)}{(3A)} \]\[ 16 = \dfrac{2A}{3} \]\[ A = 24 \]

So, A alone can complete the job in 24 days.

Therefore, the answer is 24 days.

If 10 people can do a job in 20 days, we can find the rate at which one person can do the job by multiplying the number of people by the number of days:

Rate of 1 person = \(\dfrac{1}{20 \times 10} = \dfrac{1}{200}\) jobs per day.

Now, if 20 people with twice the efficiency work on the same job, their combined rate will be:

Rate of 1 person = \(\dfrac{1}{200} \times 2 = \dfrac{1}{100}\) jobs per day.

Now, we can find how many days it would take for 20 people to complete the job at this rate:

\[ \text{Number of days} = \dfrac{\text{Total work}}{\text{Rate of 20 people}} \]\[ = \dfrac{1}{\dfrac{1}{100} \times 20} \]\[ = \dfrac{1}{\dfrac{20}{100}} \]\[ = \dfrac{1}{\dfrac{1}{5}} \]\[ = 5 \text{ days} \]

So, 20 people with twice the efficiency can do the same job in 5 days.

Therefore, the answer is 5 days.

First, let's find the individual rates at which Sunil, Dinesh, and Ramesh work.

Sunil completes the work in 4 days, so his rate is \( \dfrac{1}{4} \) of the work per day.
Dinesh completes the work in 6 days, so his rate is \( \dfrac{1}{6} \) of the work per day.
Ramesh works 1 and 1/2 times as fast as Sunil, which means his rate is \( \dfrac{3}{2} \times \dfrac{1}{4} = \dfrac{3}{8} \) of the work per day.

Now, let's find the combined rate when they work together:

\[ \text{Combined rate} = \text{Rate of Sunil} + \text{Rate of Dinesh} + \text{Rate of Ramesh} \]\[ = \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{3}{8} \]\[ = \dfrac{2}{8} + \dfrac{1}{6} + \dfrac{3}{8} \]\[ = \dfrac{5}{8} + \dfrac{1}{6} \]\[ = \dfrac{15}{24} + \dfrac{4}{24} \]\[ = \dfrac{19}{24} \]

So, the combined rate at which they work together is \( \dfrac{19}{24} \) of the work per day.

Now, let's find how many days it would take for them to complete the work together:

\[ \text{Number of days} = \dfrac{\text{Total work}}{\text{Combined rate}} \]\[ = \dfrac{1}{\dfrac{19}{24}} \]\[ = 1 \times \dfrac{24}{19} \]\[ = \dfrac{24}{19} \]

So, the three together can complete the work in \( \dfrac{24}{19} \) days.

To convert this into a mixed number:

\[ \dfrac{24}{19} = 1 \dfrac{5}{19} \]

Therefore, the answer is 1 and 5/19 days.

We find the combined efficiency of B and C together is 7 units (3 for B and 4 for C). Since the total combined efficiency of all three is 12 units, A's efficiency alone must be 5 units (12 - 3 - 4).

Given that the total work is \(10 \text{ days} \times 12 \text{ units} = 120 \text{ units}\), and A's efficiency is 5 units, we can calculate the time taken by A alone:

\[ \text{Time taken by A alone} = \dfrac{\text{Total work}}{\text{Efficiency of A}} = \dfrac{120}{5} = 24 \text{ days}\]

So, the time taken by A to complete the work separately is 24 days, which confirms the answer.