Time And Work All Competence Aptitude

Let's first find the total man-hours required to make 460 chairs with 12 carpenters working 6 hours a day for 240 days.

Total man-hours \(= 12 \text{ carpenters} \times 6 \text{ hours/day} \times 240 \text{ days} = 12 \times 6 \times 240 = 17280\) man-hours.

Now, let's find the number of chairs made by 18 carpenters working 8 hours a day for 36 days.

Total man-hours available \(= 18 \text{ carpenters} \times 8 \text{ hours/day} \times 36 \text{ days} = 18 \times 8 \times 36 = 5184\) man-hours.

Let's assume \(x\) as the number of chairs made in this time.

Now, the ratio of the total man-hours to the number of chairs made will be the same in both scenarios:

\[\dfrac{\text{Total man-hours}}{\text{Number of chairs made}} = \dfrac{17280}{460} = \dfrac{5184}{x}\]

Let's solve for \(x\):

\[x = \dfrac{5184 \times 460}{17280}\]\[x = \dfrac{2381440}{17280}\]\[x = 138\]

So, the number of chairs made by 18 carpenters in 36 days, each working 8 hours a day, is 138.

Therefore, the correct answer is 138.

First, let's calculate the volume of the first wall:\[ \text{Volume of the first wall} = \text{Length} \times \text{Breadth} \times \text{Height} \]\[ \text{Volume of the first wall} = 18 \times 2 \times 12 \]\[ \text{Volume of the first wall} = 432 \text{ cubic meters} \]

Now, let's find the work done by 8 workers in 10 days, working 9 hours a day:\[ \text{Work done by 8 workers in 1 day} = \dfrac{\text{Volume of the first wall}}{\text{Total man-hours}} \]\[ \text{Total man-hours} = \text{Number of workers} \times \text{Number of days} \times \text{Number of hours per day} \]\[ \text{Total man-hours} = 8 \times 10 \times 9 \]

Substituting the values:\[ \text{Work done by 8 workers in 1 day} = \dfrac{432}{8 \times 10 \times 9} \]\[ \text{Work done by 8 workers in 1 day} = \dfrac{432}{720} \]

\[ \text{Work done by 8 workers in 1 day} = \dfrac{3}{5} \text{ cubic meters} \]

Now, let's find the number of workers needed to build the second wall in 8 days, working 6 hours a day:\[ \text{Number of workers} = \dfrac{\text{Volume of the second wall}}{\text{Work done by 8 workers in 1 day} \times \text{Total number of days} \times \text{Number of hours per day}} \]

Substituting the values:\[ \text{Number of workers} = \dfrac{32 \times 3 \times 9}{\dfrac{3}{5} \times 8 \times 6} \]\[ \text{Number of workers} = \dfrac{864}{\dfrac{144}{5}} \]\[ \text{Number of workers} = \dfrac{864 \times 5}{144} \]\[ \text{Number of workers} = \dfrac{4320}{144} \]\[ \text{Number of workers} = 30 \]

So, 30 workers will be able to build the second wall in 8 days, working 6 hours a day.

Therefore, the correct answer is 30.

Let's denote the number of hours per day the 48 men need to work to complete the job in 7 days as \(h\).

Using the concept of man-days, the total work done by 20 men working 8 hours per day in 21 days is equal to the total work done by 48 men working \(h\) hours per day in 7 days.

So, we can set up the equation:

\[ 20 \times 8 \times 21 = 48 \times h \times 7 \]

Let's solve for \(h\):

\[ 20 \times 8 \times 21 = 48 \times h \times 7 \]

\[ 3360 = 336h \]

\[ h = \dfrac{3360}{336} \]

\[ h = 10 \]

So, 48 men must work 10 hours per day to complete the same job in 7 days.

Therefore, the correct answer is 10.

Let's assume the amount of work required to lay a road of length 100m is 1 unit.

So, the rate at which 20 women can lay the road is:\[ \text{Rate of 20 women} = \dfrac{1 \text{ road}}{10 \text{ days}} \]

Now, we need to find the time it takes for 10 women to lay a road of length 50m.

If the amount of work required is 1 unit, and the rate of 20 women is \( \dfrac{1}{10} \) of the work per day, then the amount of work done by 10 women will be half of that, since they are half the number of workers. 

So, the rate of 10 women is \( \dfrac{1}{20} \) of the work per day.

Now, let's find the time it takes for 10 women to lay the road:\[ \text{Rate of 10 women} = \dfrac{1 \text{ road}}{x \text{ days}} \]

Since the amount of work is halved and the rate of work is also halved, the time taken by 10 women will be the same as that of 20 women, which is 10 days.

Therefore, the correct answer is 10 days.

Let's denote the rate of work for the father as \(F\), the rate of work for one son as \(S_1\), and the rate of work for the other son as \(S_2\).

Given:
1. The father can do a job as fast as his two sons working together, so his rate of work is equal to the sum of the rates of work of his two sons:\[ F = S_1 + S_2 \]

2. One son can do the job in 3 hours, so his rate of work is:\[ S_1 = \dfrac{1}{3} \]

3. The other son can do the job in 6 hours, so his rate of work is:\[ S_2 = \dfrac{1}{6} \]

Now, let's substitute the values of \(S_1\) and \(S_2\) into the equation for the rate of work of the father:\[ F = \dfrac{1}{3} + \dfrac{1}{6} \]\[ F = \dfrac{2}{6} + \dfrac{1}{6} \]\[ F = \dfrac{3}{6} \]\[ F = \dfrac{1}{2} \]

So, the father's rate of work is \( \dfrac{1}{2} \) of the job per hour.

Now, let's find the number of hours taken by the father to do the job alone:\[ \text{Number of hours taken by the father} = \dfrac{1}{F} \]\[ \text{Number of hours taken by the father} = \dfrac{1}{\dfrac{1}{2}} \]\[ \text{Number of hours taken by the father} = 2 \]

So, the number of hours taken by the father to do the job alone is 2.

Therefore, the correct answer is 2.

Let's denote the rate of work of 1 man as \(M\), the rate of work of 1 woman as \(W\), and the rate of work of 1 boy as \(B\).

Given:
1. 1 man, 2 women, or 3 boys can do the work in 44 days, so their combined rate of work is the same:\[ 1 \times M = 2 \times W = 3 \times B \]

Now, let's set up the equation based on their combined rates of work:\[ 1 \times M = 2 \times W = 3 \times B \]

We know that the ratio of men's rate to women's rate to boys' rate is 1:2:3. 

Now, let's assign a common multiple to make the calculation easier. Let's say the common multiple is 6. So, 1 man's rate is \( \dfrac{6}{6} \), 1 woman's rate is \( \dfrac{3}{6} \), and 1 boy's rate is \( \dfrac{2}{6} \). 

Now, let's calculate the combined rate of work:\[ 1 \times \dfrac{6}{6} = 2 \times \dfrac{3}{6} = 3 \times \dfrac{2}{6} \]\[ M = 2W = 3B \]

Now, let's solve for the rate of work of 1 man, 1 woman, and 1 boy combined:\[ M + W + B = \dfrac{6}{6} + \dfrac{3}{6} + \dfrac{2}{6} \]\[ M + W + B = \dfrac{11}{6} \]

So, the combined rate of work of 1 man, 1 woman, and 1 boy is \( \dfrac{11}{6} \) of the work per day.

Now, let's find out how many days it will take for 1 man, 1 woman, and 1 boy combined to do the work:\[ \text{Number of days} = \dfrac{1}{\text{Combined rate of work}} \]\[ \text{Number of days} = \dfrac{1}{\dfrac{11}{6}} \]\[ \text{Number of days} = \dfrac{6}{11} \]

Now, let's calculate:\[ \text{Number of days} = \dfrac{6}{11} \times 44 \]\[ \text{Number of days} = \dfrac{264}{11} \]\[ \text{Number of days} = 24 \]

So, the same piece of work will be done by 1 man, 1 woman, and 1 boy in 24 days.

Therefore, the correct answer is 24 days.

Let's find the fraction of work completed by A and B individually in 5 days.

A can complete the work in 10 days, so in 5 days, A completes \( \dfrac{1}{10} \) of the work.
Similarly, B can complete the work in 20 days, so in 5 days, B completes \( \dfrac{1}{20} \) of the work.

Now, let's find the fraction of work completed by both A and B together in 5 days:

\[ \text{Fraction of work completed by A in 5 days} = \dfrac{1}{10} \times 5 = \dfrac{1}{2} \]\[ \text{Fraction of work completed by B in 5 days} = \dfrac{1}{20} \times 5 = \dfrac{1}{4} \]

Now, let's find the fraction of work completed by both A and B together in 5 days:

\[ \text{Fraction of work completed by A and B together in 5 days} = \dfrac{1}{2} + \dfrac{1}{4} = \dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4} \]

So, the fraction of the work completed by both A and B together in 5 days is \( \dfrac{3}{4} \).

Now, let's find the fraction of the work that is left:

\[ \text{Fraction of work left} = 1 - \text{Fraction of work completed by A and B together in 5 days} \]\[ \text{Fraction of work left} = 1 - \dfrac{3}{4} \]\[ \text{Fraction of work left} = \dfrac{4}{4} - \dfrac{3}{4} \]\[ \text{Fraction of work left} = \dfrac{1}{4} \]

So, the fraction of the work that is left is \( \dfrac{1}{4} \).

Therefore, the correct answer is \( \dfrac{1}{4} \).

if Bimal is 50% more efficient, then his efficiency would be 150% of Kamal's efficiency, resulting in him completing the work in \( \dfrac{15}{1.5} = 10 \) days.

So, Bimal will take 10 days to complete the work.

Therefore, the correct answer is 10 days.

Let's denote the number of days required for 12 men to complete the work as \(d\).

According to the given information, the number of men and the number of days taken to complete the work are inversely proportional. This means that if the number of men increases, the number of days required to complete the work decreases, and vice versa.

So, we can set up the following proportion:

\[\text{Men} \times \text{Days} = \text{Constant}\]

For the initial scenario with 10 men completing the work in 12 days:\[10 \times 12 = 12 \times d\]\[120 = 12d\]

Now, we can solve for \(d\):\[d = \dfrac{120}{12}\]\[d = 10\]

So, the time taken by 12 men to do the same piece of work will be 10 days.

Therefore, the correct answer is 10 days.

Let's denote the number of hours per day \(h\) that 64 persons need to work to complete the job in 15 days.

We know that the work done is the same, regardless of the number of persons or the number of hours they work per day. So, we can set up the equation:

\[80 \text{ persons} \times 6 \text{ hours/day} \times 16 \text{ days} = 64 \text{ persons} \times h \text{ hours/day} \times 15 \text{ days}\]

Let's solve for \(h\):

\[80 \times 6 \times 16 = 64 \times h \times 15\]\[7680 = 960h\]\[h = \dfrac{7680}{960}\]\[h = 8\]

So, 64 persons should work 8 hours per day to finish the job within 15 days.

Therefore, the correct answer is 8 hours.