Time And Work All Competence Aptitude

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

According to the given information, P is able to finish the job in 48 days less than Q. So, we have:

\[ \dfrac{Q}{3} = Q - 48 \]

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

\[ \dfrac{Q}{3} = Q - 48 \]\[ \Rightarrow Q = 3Q - 144 \]\[ \Rightarrow 2Q = 144 \]\[ \Rightarrow Q = 72 \]

So, Q alone can finish the job in 72 days. And since P is three times as efficient as Q, P alone would take \( \dfrac{72}{3} = 24 \) days to finish the job.

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 P and Q, respectively.

Substituting the values:

\[ \text{Time taken together} = \dfrac{(24 \times 72)}{(24 + 72)} \]\[ = \dfrac{(1728)}{(96)} \]\[ = 18 \text{ days} \]

So, working together, they can finish the job in 18 days. 

Therefore, the answer is 18 days.

Given:
- Efficiency of A relative to B is \( k = 30\% = 1.3 \).
- A can finish the work alone in 23 days.

We want to find out how long it takes for A and B, working together, to finish the job.

We can use the formula for the time taken when both A and B work together, which is derived from the concept of relative efficiency:

\[ \text{Time taken together} = \dfrac{k \times \text{Time taken by A alone}}{1 + k} \]

Plugging in the given values:

\[ \text{Time taken together} = \dfrac{1.3 \times 23}{1 + 1.3} \]

\[ \text{Time taken together} = \dfrac{29.9}{2.3} = 13 \]

So, A and B, working together, can finish the job in \( \mathbf{13 \text{ days}} \).

This is the comprehensive solution utilizing the concept of relative efficiency and the formula for the time taken when working together.

To solve this problem, we can use the concept of "man-days," which represents the total amount of work done by a certain number of men in a day. 

Given that 10 men can complete the work in 12 days, we can find the total man-days required for the work:

\[ \text{Total man-days} = \text{Number of men} \times \text{Number of days} = 10 \times 12 = 120 \text{ man-days} \]

Now, if we increase the number of men to 12, we can find out how many days it would take for them to complete the same work. Let \( x \) be the number of days required for 12 men to complete the work.

Since the amount of work remains the same, we have:

\[ 10 \times 12 = 12 \times x \]

\[ 120 = 12x \]

\[ x = \dfrac{120}{12} = 10 \]

So, it would take 12 men \( \mathbf{10} \) days to complete the same piece of work.

Therefore, the correct answer is:

\[ \text{The time taken by 12 men to do the same piece of work will be } \mathbf{10 \text{ days}} \]

Let's denote the original number of carpenters as \( x \). 

The job was supposed to be completed in 9 days with \( x \) carpenters, but due to the absence of 5 carpenters, the remaining \( x - 5 \) carpenters completed the job in 12 days.

We can set up equations based on the work done:

1. With \( x \) carpenters, the job was supposed to be completed in 9 days. Therefore, the total work done by the carpenters in 9 days is \( 9x \).

2. With \( x - 5 \) carpenters, the job was completed in 12 days. Therefore, the total work done by the remaining carpenters in 12 days is \( 12(x - 5) \).

Since the amount of work done is the same in both cases, we can equate these expressions:

\[ 9x = 12(x - 5) \]

Let's solve for \( x \):

\[ 9x = 12x - 60 \]

\[ 60 = 12x - 9x \]

\[ 60 = 3x \]

\[ x = \dfrac{60}{3} \]

\[ x = 20 \]

So, the original number of carpenters was \( \mathbf{20} \).

To solve this problem, we can use the concept of "man-hours," which represents the total amount of work done by a certain number of people in an hour. 

Given that 80 persons can finish the work in 16 days by working 6 hours a day, we can find the total man-hours required for the work:

\[ \text{Total man-hours} = \text{Number of persons} \times \text{Number of days} \times \text{Number of hours per day} \]

\[ = 80 \times 16 \times 6 \]

\[ = 7680 \text{ man-hours} \]

Now, we need to determine how many hours per day 64 persons should work to finish the same job within 15 days. Let \( x \) be the number of hours per day.

Since the amount of work remains the same, we have:

\[ 80 \times 16 \times 6 = 64 \times 15 \times x \]

\[ 7680 = 960x \]

\[ x = \dfrac{7680}{960} \]

\[ x = 8 \]

So, 64 persons should work \( \mathbf{8} \) hours a day to finish the job within 15 days.

Therefore, the correct answer is:

\[ \text{The number of hours a day 64 persons should work to finish the job within 15 days is } \mathbf{8 \text{ hours}} \]

To solve this problem, we can use the concept of "man-days," which represents the total amount of work done by a certain number of workers in a day. 

Given that 30 men can repair the road in 18 days, we can find the total man-days required for the work:

\[ \text{Total man-days} = \text{Number of men} \times \text{Number of days} = 30 \times 18 = 540 \text{ man-days} \]

Now, if 6 more workers join them, making a total of \( 30 + 6 = 36 \) workers, we can find out how many days it would take for them to complete the same work.

Let \( x \) be the number of days required for 36 workers to complete the work.

Since the amount of work remains the same, we have:

\[ 30 \times 18 = 36 \times x \]

\[ 540 = 36x \]

\[ x = \dfrac{540}{36} = 15 \]

So, with 36 workers, the road can be repaired in \( \mathbf{15} \) days.

Therefore, the correct answer is:

\[ \text{The road can be repaired in } \mathbf{15 \text{ days}} \]

To solve this problem, we can use the concept of man-days, which represents the total amount of work done by a certain number of workers in a day.

Given that the contractor employed 120 men to finish the work in 124 days and after 64 days, \( \dfrac{2}{3} \) of the work is already done, let's denote the total work by \( W \). Therefore, \( \dfrac{2}{3}W \) work is already done.

The total work can be represented as \( \text{Total Work} = \text{Number of men} \times \text{Number of days} \). Therefore, the total work done in 64 days with 120 men is \( 64 \times 120 = 7680 \) man-days.

Since \( \dfrac{2}{3} \) of the work is done, the remaining work is \( \dfrac{1}{3}W \). We know that this remaining work will be done in the next \( 124 - 64 = 60 \) days. Let's denote the number of men required to complete this remaining work in time as \( x \).

So, we have the equation:

\[ \text{Remaining Work} = \text{Number of men} \times \text{Number of days} \]

\[ \dfrac{1}{3}W = 60x \]

We already know the total work done in 64 days with 120 men, which is 7680 man-days:

\[ 7680 = \dfrac{2}{3}W \]

Now, let's solve for \( W \):

\[ W = \dfrac{7680 \times 3}{2} = 11520 \]

Now, substitute the value of \( W \) in the equation of remaining work:

\[ \dfrac{11520}{3} = 60x \]

\[ x = \dfrac{11520}{3 \times 60} = 64 \]

So, 64 men are required to complete the remaining work in time.

Since initially, there were 120 men, and now only 64 men are needed, the number of men to be discharged now is:

\[ \text{Number of men to be discharged} = 120 - 64 = 56 \]

Therefore, \( \mathbf{56} \) men can be discharged now so that the work may finish in time.

Given that \( 2 \times 14 \) men + 14 women = 16 men + 32 women,

which simplifies to \( 28 \) men - \( 16 \) men = \( 32 \) women - \( 14 \) women,

which further simplifies to \( 12 \) men = \( 18 \) women.

This implies \( 2 \) men = \( 3 \) women.

Therefore, for each woman to receive the same amount as \( 2 \) men, we have:

Amount received by \( 1 \) woman per day \( = \dfrac{2}{3} \) × 600 = ₹ 400.

So, the correct answer is ₹ 400.

Let's denote the initial number of workers as \(15x\) and their initial wages as \(22y\). After the reduction in the number of workers in the ratio \(15 : 11\), the number of workers becomes \(11x\), and after the increment in their wages in the ratio \(22 : 25\), the wages become \(25y\). 

The total wages initially would be \(15x \times 22y\) and after the changes, it becomes \(11x \times 25y\).

We want to find the ratio by which the total wages are decreased, so we'll compare the initial total wages to the final total wages.

Initial total wages = \(15x \times 22y\)

Final total wages = \(11x \times 25y\)

Now, let's find the ratio:

\[ \text{Ratio} = \dfrac{\text{Initial total wages}}{\text{Final total wages}} = \dfrac{15x \times 22y}{11x \times 25y}\]

Simplify:

\[ \text{Ratio} = \dfrac{15 \times 22}{11 \times 25} = \dfrac{330}{275}\]

Now, let's simplify this ratio:

\[ \text{Ratio} = \dfrac{330}{275} = \dfrac{6 \times 55}{5 \times 55} = \dfrac{6}{5}\]

So, the ratio by which the total wages of the workers should be decreased is \(6 : 5\). 

Therefore, the correct answer is \(6 : 5\).

Let's first find out the work rates of A, B, and C individually.

A can complete the job in 15 days, so his work rate is \( \dfrac{1}{15} \) of the job per day.

Similarly, B can complete the job in 12 days, so his work rate is \( \dfrac{1}{12} \) of the job per day.

Let's denote the work rate of C as \( c \). Together, A, B, and C can complete the job in 5 days, so their combined work rate is \( \dfrac{1}{5} \) of the job per day.

Now, we can set up the equation:

\[ \dfrac{1}{15} + \dfrac{1}{12} + c = \dfrac{1}{5} \]

To solve for \( c \), let's first find a common denominator:

\[ \dfrac{4}{60} + \dfrac{5}{60} + c = \dfrac{12}{60} \]

\[ \dfrac{9}{60} + c = \dfrac{12}{60} \]

\[ c = \dfrac{12}{60} - \dfrac{9}{60} \]

\[ c = \dfrac{3}{60} \]

\[ c = \dfrac{1}{20} \]

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

Now, let's find the share of the amount that C earns. Since C works for 5 days and his work rate is \( \dfrac{1}{20} \) of the job per day, his total contribution to the job is \( 5 \times \dfrac{1}{20} = \dfrac{1}{4} \) of the job.

The total amount for the job is 1200.

So, C's share of the amount is \( \dfrac{1}{4} \times 1200 = 300 \).

Therefore, the share of the amount that C earns is 300.