Time And Work All Competence Aptitude

Let's denote the amount of work done by B in one day as \( x \), then A does half as much work as B in one-third of the time taken by B, which means A does \( \dfrac{1}{2}x \) work in \( \dfrac{1}{3} \) of the time B takes to do \( x \) work.

So, A does \( \dfrac{1}{2}x \) work in \( \dfrac{1}{3} \) of the time B takes to do \( x \) work, which implies A does \( \dfrac{1}{2}x \times \dfrac{3}{1} = \dfrac{3}{2}x \) work in the same amount of time as B.

Now, let's denote the time taken by B to complete the work alone as \( T \) days.

So, in one day, A does \( \dfrac{3}{2}x \) work and B does \( x \) work.

Together, in one day, they do \( \dfrac{3}{2}x + x = \dfrac{5}{2}x \) work.

Given that together they take 10 days to complete the work, we can write:

\[ \dfrac{5}{2}x \times 10 = 1 \]

This equation represents the total work done by A and B together. Solving for \( x \):

\[ x = \dfrac{2}{5 \times 10} = \dfrac{1}{25} \]

So, B does \( \dfrac{1}{25} \) work in one day.

If B does \( \dfrac{1}{25} \) work in one day, to complete the whole work, which is \( 1 \), B would take \( \dfrac{1}{\dfrac{1}{25}} = 25 \) days.

So, the correct answer is:

\[ \boxed{\text{25 days}} \]

Let's denote the rates of work for the man, woman, and boy as \( M \), \( W \), and \( B \) respectively. Then we have:

1. The combined rate of the man, woman, and boy together is \( M + W + B \).
2. The rate at which a man and a woman work together is \( M + W \).

Given that the man, woman, and boy together finish a piece of work in 6 days, we can set up the equation:

\[ (M + W + B) \times 6 = 1 \]

Given that a man can do the work in 10 days, his rate is \( M = \dfrac{1}{10} \) per day.

Given that a woman can do the work in 24 days, her rate is \( W = \dfrac{1}{24} \) per day.

We're asked to find the rate of the boy, denoted as \( B \), and ultimately the days taken by the boy to finish the work. 

Now, we can solve for the rate of the boy:

\[ \dfrac{1}{10} + \dfrac{1}{24} + B = \dfrac{1}{6} \]

\[ B = \dfrac{1}{6} - \left(\dfrac{1}{10} + \dfrac{1}{24}\right) \]

\[ B = \dfrac{1}{6} - \left(\dfrac{12 + 5}{120}\right) \]

\[ B = \dfrac{1}{6} - \dfrac{17}{120} \]

\[ B = \dfrac{20}{120} - \dfrac{17}{120} \]

\[ B = \dfrac{3}{120} \]

\[ B = \dfrac{1}{40} \]

This means the boy completes \(\dfrac{1}{40}\) of the work per day.

To find the number of days it takes for the boy to finish the work alone, we set up:

\[ \dfrac{1}{40} \times D = 1 \]

Where \( D \) represents the number of days the boy takes to complete the work alone.

Solving for \( D \):

\[ D = \dfrac{1}{\dfrac{1}{40}} \]

\[ D = 40 \]

So, the boy alone takes 40 days to finish the work.

Therefore, the correct answer is \(\boxed{40}\) days.

Let's first find the work rate of a man and a woman separately.

If 3 men can do a work in 12 days, then the work rate of one man is:\[ \text{Rate of one man} = \frac{1}{3 \times 12} = \frac{1}{36} \text{ per day} \]

Similarly, if 5 women can do the same work in 12 days, then the work rate of one woman is:\[ \text{Rate of one woman} = \frac{1}{5 \times 12} = \frac{1}{60} \text{ per day} \]

Now, we need to find how much work 6 men and 5 women can do in one day together.

The combined work rate of 6 men and 5 women is:\[ \text{Combined rate} = 6 \times \text{Rate of one man} + 5 \times \text{Rate of one woman} = 6 \times \frac{1}{36} + 5 \times \frac{1}{60} \]

So, the total work they can do in one day is:\[ \text{Combined rate} = \frac{1}{6} + \frac{1}{12} = \frac{1}{4} \text{ of the work per day} \]

Therefore, the number of days required to complete the work together is:\[ \text{Days} = \frac{1}{\text{Combined rate}} = \frac{1}{1/4} = 4 \text{ days} \]

Let's denote the total work as \( W \) and the rate at which one man works as \( M \).

Given that 15 men can finish a piece of work in 40 days, the total work can be calculated as:\[ W = 15 \times 40 = 600 \]

So, the rate at which one man works (\( M \)) is:\[ M = \dfrac{W}{\text{total days}} = \dfrac{600}{40} = 15 \]

Now, let's say after \( x \) days, 5 men leave the work. So, for the remaining \( 45 - x \) days, we'll have 10 men working.

In the first \( x \) days, the work done by 15 men is:\[ \text{Work done in } x \text{ days} = 15 \times x \]

In the remaining \( 45 - x \) days, the work done by 10 men is:\[ \text{Work done in } 45 - x \text{ days} = 10 \times (45 - x) \]

Given that the total work remains the same, we can set up the equation:

\[ 15x + 10(45 - x) = 600 \]

Let's solve for \( x \):

\[ 15x + 450 - 10x = 600 \]

\[ 5x + 450 = 600 \]

\[ 5x = 150 \]

\[ x = 30 \]

Therefore, 5 men should leave the work after \( \boxed{30} \) days so that the work is finished in 45 days altogether.

Let's denote the total work as \( W \) and the rate at which one man works as \( M \). 

Given that a certain number of men can do the work in 40 days, we have:

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

\[ W = M \times 40 \]

Now, if there were 8 more men, the work would be completed in 10 days less, so:

\[ W = (M + 8) \times (40 - 10) \]

\[ W = (M + 8) \times 30 \]

Since both expressions represent the same work, we can equate them:

\[ M \times 40 = (M + 8) \times 30 \]

Let's solve this equation for \( M \), the number of men initially:

\[ 40M = 30(M + 8) \]

\[ 40M = 30M + 240 \]

\[ 10M = 240 \]

\[ M = 24 \]

So, initially there were \( \boxed{24} \) men.

Let's denote the total work as \( W \) and the rate at which one man works as \( M \) and the rate at which one boy works as \( B \).

Given that 12 men or 24 boys can do the work in 66 days, we have:

\[ W = \text{Men} \times \text{Days} = 12M \times 66 = 24B \times 66 \]

From this, we can find the ratio of the work done by one man to the work done by one boy:

\[ 12M \times 66 = 24B \times 66 \]

\[ 12M = 24B \]

\[ M = 2B \]

This tells us that one man does the same amount of work as two boys.

Now, we need to find how much work 15 men and 6 boys can do in one day together.

The work done by 15 men and 6 boys in one day is:

\[ \text{Work done in one day} = 15M + 6B \]

\[ \text{Work done in one day} = 15(2B) + 6B \]

\[ \text{Work done in one day} = 30B + 6B \]

\[ \text{Work done in one day} = 36B \]

So, 15 men and 6 boys together can do \( 36B \) work in one day.

Now, we need to find the number of days it takes for them to complete the entire work, which we know is equal to \( W \).

\[ W = \text{Total work} = 24B \times 66 \]

\[ \text{Number of days} = \dfrac{\text{Total work}}{\text{Work done in one day}} \]

\[ \text{Number of days} = \dfrac{24B \times 66}{36B} \]

\[ \text{Number of days} = \dfrac{24 \times 66}{36} \]

\[ \text{Number of days} = \dfrac{8 \times 66}{12} \]

\[ \text{Number of days} = \dfrac{528}{12} \]

\[ \text{Number of days} = 44 \]

Therefore, \( 15 \) men and \( 6 \) boys together can finish the work in \( \boxed{44} \) days.

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

If A can do a piece of work in 20 days, his rate of work (\(A_{\text{rate}}\)) is:\[ A_{\text{rate}} = \dfrac{1}{20} \]

Similarly, if B can do the same work in 12 days, his rate of work (\(B_{\text{rate}}\)) is:\[ B_{\text{rate}} = \dfrac{1}{12} \]

Now, let's calculate how much work B has done in 9 days. Since we know his rate is \( \dfrac{1}{12} \) per day, in 9 days he will do:\[ \text{Work done by B in 9 days} = 9 \times B_{\text{rate}} = 9 \times \dfrac{1}{12} = \dfrac{3}{4} \]

So, after 9 days, B has completed \( \dfrac{3}{4} \) of the work.

Now, we need to find how much work is left to be done.

Total work is \( 1 \), and B has completed \( \dfrac{3}{4} \), so the remaining work is:\[ \text{Remaining work} = 1 - \dfrac{3}{4} = \dfrac{1}{4} \]

Now, let's find how long it takes for A to finish this remaining \( \dfrac{1}{4} \) of the work.

A's rate of work is \( \dfrac{1}{20} \) per day, so the number of days A needs to complete \( \dfrac{1}{4} \) of the work is:\[ \text{Days taken by A to finish } \dfrac{1}{4} \text{ of the work} = \dfrac{\dfrac{1}{4}}{\dfrac{1}{20}} = \dfrac{1}{4} \times 20 = 5 \]

Therefore, A can finish the remaining work in \( \boxed{5} \) days.

Let's denote the number of men who can finish the work in 10 days as \(x\).

We can use the concept of man-days to solve this problem. Man-days represent the total amount of work done by a certain number of men in a certain number of days.

Given that 35 men can finish the work in 8 days, the total man-days for this scenario would be \(35 \times 8\).

So, the man-days for this scenario is \(35 \times 8 = 280\).

Now, let's denote the total man-days when \(x\) men finish the work in 10 days. It would be \(x \times 10\).

Since the amount of work remains the same, we can set up the equation:

\[35 \times 8 = x \times 10\]

Solving for \(x\):

\[280 = 10x\]

\[x = \dfrac{280}{10}\]

\[x = 28\]

Therefore, the number of men who can finish the same work in 10 days is \(\boxed{28}\).

Let's use the concept of man-days to solve this problem. Man-days represent the total amount of work done by a certain number of men in a certain number of days.

Given that 24 villagers can clean the canal in 12 days, the total man-days for this scenario would be \(24 \times 12\).

So, the man-days for this scenario is \(24 \times 12 = 288\).

Now, let's denote the total man-days when 36 villagers clean the canal. Let's call the number of days needed for this scenario \(x\). The total man-days for this scenario would be \(36 \times x\).

Since the amount of work remains the same, we can set up the equation:

\[24 \times 12 = 36 \times x\]

Solving for \(x\):

\[288 = 36x\]

\[x = \dfrac{288}{36}\]

\[x = 8\]

Therefore, 36 villagers can clean the canal in \(\boxed{8}\) days.

To find the time it takes for A and B to finish the work together, we need to calculate their combined work rate.

Let's denote the work rate of A as \( A_{\text{rate}} \) and the work rate of B as \( B_{\text{rate}} \).

A can complete the work in 12 days, so his work rate is:\[ A_{\text{rate}} = \dfrac{1}{12} \text{ per day} \]

Similarly, B can complete the work in 24 days, so his work rate is:\[ B_{\text{rate}} = \dfrac{1}{24} \text{ per day} \]

When they work together, their combined work rate is the sum of their individual work rates:

\[ \text{Combined rate} = A_{\text{rate}} + B_{\text{rate}} \]\[ \text{Combined rate} = \dfrac{1}{12} + \dfrac{1}{24} \]

\[ \text{Combined rate} = \dfrac{2}{24} + \dfrac{1}{24} \]\[ \text{Combined rate} = \dfrac{3}{24} \text{ per day} \]

Now, to find how many days it takes for them to finish the work together, we use the formula:

\[ \text{Time} = \dfrac{\text{Total work}}{\text{Combined rate}} \]

Since the total work is 1, as they complete the entire piece of work:

\[ \text{Time} = \dfrac{1}{\dfrac{3}{24}} \]

\[ \text{Time} = \dfrac{24}{3} \]

\[ \text{Time} = 8 \]

So, A and B will finish the work together in \(\boxed{8}\) days.