Time And Work All Competence Aptitude

To find out how long it takes for x and y to finish the job together, we need to calculate their combined work rate.

Let's denote the work rate of x as \(X_{\text{rate}}\) and the work rate of y as \(Y_{\text{rate}}\).

If x can finish a job in 4 hours, his work rate is:\[X_{\text{rate}} = \dfrac{1}{4}\text{ job per hour}\]

Similarly, if y can finish the job in 8 hours, his work rate is:\[Y_{\text{rate}} = \dfrac{1}{8}\text{ job per hour}\]

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

\[\text{Combined rate} = X_{\text{rate}} + Y_{\text{rate}}\]\[\text{Combined rate} = \dfrac{1}{4} + \dfrac{1}{8}\]

\[\text{Combined rate} = \dfrac{2}{8} + \dfrac{1}{8}\]\[\text{Combined rate} = \dfrac{3}{8}\text{ job per hour}\]

Now, to find out how many hours it takes for them to finish the job together, we use the formula:

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

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

\[\text{Time} = \dfrac{1}{\dfrac{3}{8}}\]

\[\text{Time} = \dfrac{8}{3}\]

\[\text{Time} = 2 \dfrac{2}{3}\text{ hours}\]

Since we're asked for the answer in minutes:

\[2 \dfrac{2}{3} \text{ hours} = 2 \times 60 + \dfrac{2}{3} \times 60 \text{ minutes}\]\[= 120 + 40\]\[= 160\text{ minutes}\]

So, x and y will finish the job together in \(\boxed{160}\) minutes.

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

If 3 men can plow a field in 43 days, then the work rate of one man is:\[ \text{Rate of one man} = \dfrac{1}{3 \times 43} = \dfrac{1}{129} \text{ per day} \]

Similarly, if 4 women can plow the same field in 43 days, then the work rate of one woman is:\[ \text{Rate of one woman} = \dfrac{1}{4 \times 43} = \dfrac{1}{172} \text{ per day} \]

Now, we need to find how much work 7 men and 5 women can do in one day together.

The combined work rate of 7 men and 5 women is:\[ \text{Combined rate} = 7 \times \text{Rate of one man} + 5 \times \text{Rate of one woman} \]\[ \text{Combined rate} = 7 \times \dfrac{1}{129} + 5 \times \dfrac{1}{172} \]\[ \text{Combined rate} = \dfrac{7}{129} + \dfrac{5}{172} \]

Now, let's find a common denominator for the fractions:

\[ \text{Combined rate} = \dfrac{7 \times 172}{129 \times 172} + \dfrac{5 \times 129}{129 \times 172} \]\[ \text{Combined rate} = \dfrac{1204}{22188} + \dfrac{645}{22188} \]\[ \text{Combined rate} = \dfrac{1849}{22188} \]

Now, we can calculate how many days it will take for them to plow the field:

\[ \text{Days required} = \dfrac{1}{\text{Combined rate}} = \dfrac{22188}{1849} \]

\[ \text{Days required} = 12 \]

Therefore, 7 men and 5 women together will take \(\boxed{12}\) days to plow the field.

Let's first find out the work rate of x and y individually.

If x completes \( \dfrac{1}{4} \) of the job in 6 days, his work rate is:\[ x_{\text{rate}} = \dfrac{\dfrac{1}{4}}{6} = \dfrac{1}{24} \]

Now, since y completes the rest of the job in 12 days, and \( \dfrac{3}{4} \) of the job is left for y, his work rate is:\[ y_{\text{rate}} = \dfrac{\dfrac{3}{4}}{12} = \dfrac{1}{16} \]

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

\[ \text{Combined rate} = x_{\text{rate}} + y_{\text{rate}} = \dfrac{1}{24} + \dfrac{1}{16} \]

\[ \text{Combined rate} = \dfrac{1}{24} + \dfrac{3}{48} \]

\[ \text{Combined rate} = \dfrac{2 + 3}{48} \]

\[ \text{Combined rate} = \dfrac{5}{48} \]

Now, let's find how long it will take for them to complete the job together:

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

Since the total work is 1 (the entire job):

\[ \text{Time} = \dfrac{1}{\dfrac{5}{48}} \]

\[ \text{Time} = \dfrac{48}{5} \]

\[ \text{Time} = 9 \dfrac{3}{5} \text{ days} \]

Therefore, x and y could complete the job together in \(\boxed{9 \dfrac{3}{5}}\) days.

Let's denote the individual rates of work for A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively.

From the given information, we can form the following equations:

1. The combined work rate of A and B is \(A_r + B_r = \dfrac{1}{10}\).
2. The combined work rate of B and C is \(B_r + C_r = \dfrac{1}{12}\).
3. The combined work rate of C and A is \(C_r + A_r = \dfrac{1}{15}\).

We're asked to find the rate at which C works alone, denoted as \(C_r\).

Now, let's add equations 1, 2, and 3:

\[ (A_r + B_r) + (B_r + C_r) + (C_r + A_r) = \dfrac{1}{10} + \dfrac{1}{12} + \dfrac{1}{15} \]

\[ 2(A_r + B_r + C_r) = \dfrac{6 + 5 + 4}{60} \]

\[ 2(A_r + B_r + C_r) = \dfrac{15}{60} \]

\[ A_r + B_r + C_r = \dfrac{15}{120} \]

\[ A_r + B_r + C_r = \dfrac{1}{8} \]

Now, since A and B can do the work in 10 days together, their combined work rate is \( \dfrac{1}{10} \). Similarly, we find \(A_r + B_r + C_r = \dfrac{1}{8}\). This means \(C_r = \dfrac{1}{8} - \dfrac{1}{10} = \dfrac{5}{40} - \dfrac{4}{40} = \dfrac{1}{40}\).

This implies that C can finish the work alone in 40 days.

Therefore, C will finish the work alone in \(\boxed{40}\) days.

Let's denote the individual rates of work for A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively.

From the given information, we can form the following equations:

1. The combined work rate of A and B is \(A_r + B_r = \dfrac{1}{8}\).
2. The combined work rate of B and C is \(B_r + C_r = \dfrac{1}{12}\).
3. The combined work rate of C and A is \(C_r + A_r = \dfrac{1}{8}\).

We're asked to find the combined work rate of A, B, and C together, denoted as \(A_r + B_r + C_r\).

Now, let's add equations 1, 2, and 3:

\[ (A_r + B_r) + (B_r + C_r) + (C_r + A_r) = \dfrac{1}{8} + \dfrac{1}{12} + \dfrac{1}{8} \]

\[ 2(A_r + B_r + C_r) = \dfrac{3 + 2 + 3}{24} \]

\[ 2(A_r + B_r + C_r) = \dfrac{8}{24} \]

\[ A_r + B_r + C_r = \dfrac{8}{48} \]

\[ A_r + B_r + C_r = \dfrac{1}{6} \]

Therefore, A, B, and C together can complete the work in \(\boxed{6}\) days.

Let's denote the individual rates of work for A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively.

From the given information, we can form the following equations:

1. The combined work rate of A, B, and C is \(A_r + B_r + C_r = \dfrac{1}{30}\).
2. The combined work rate of A and B is \(A_r + B_r = \dfrac{1}{50}\).

We're asked to find the work rate of C alone, denoted as \(C_r\).

Now, let's subtract equation 2 from equation 1 to find \(C_r\):

\[ (A_r + B_r + C_r) - (A_r + B_r) = \dfrac{1}{30} - \dfrac{1}{50} \]

\[ C_r = \dfrac{1}{30} - \dfrac{1}{50} \]

\[ C_r = \dfrac{5}{150} - \dfrac{3}{150} \]

\[ C_r = \dfrac{2}{150} \]

\[ C_r = \dfrac{1}{75} \]

This means that C can complete \( \dfrac{1}{75} \) of the work in one minute.

Therefore, C alone can complete the work in \(\boxed{75}\) minutes.

Let's denote the rates of work for A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively. 

From the given information, we can form the following equations:

1. The combined work rate of A and B is \(A_r + B_r = \dfrac{1}{36}\).
2. The combined work rate of B and C is \(B_r + C_r = \dfrac{1}{60}\).
3. The combined work rate of A and C is \(A_r + C_r = \dfrac{1}{45}\).

We're asked to find the rate of work of C alone, denoted as \(C_r\).

Now, let's add equations 1, 2, and 3:

\[ (A_r + B_r) + (B_r + C_r) + (A_r + C_r) = \dfrac{1}{36} + \dfrac{1}{60} + \dfrac{1}{45} \]

\[ 2(A_r + B_r + C_r) = \dfrac{5 + 3 + 4}{180} \]

\[ 2(A_r + B_r + C_r) = \dfrac{12}{180} \]

\[ A_r + B_r + C_r = \dfrac{12}{360} \]

\[ A_r + B_r + C_r = \dfrac{1}{30} \]

We found that \(A_r + B_r + C_r = \dfrac{1}{30}\). 

We know that the combined work rate of A, B, and C is \(A_r + B_r + C_r = \dfrac{1}{30}\). We also know that the combined work rate of A and B is \(A_r + B_r = \dfrac{1}{36}\). 

Therefore, the work rate of C alone is \(C_r = \dfrac{1}{30} - \dfrac{1}{36}\):

\[ C_r = \dfrac{6}{180} - \dfrac{5}{180} \]

\[ C_r = \dfrac{1}{180} \]

This means that C can complete \( \dfrac{1}{180} \) of the work in one day.

Therefore, C alone can complete the work in \( \boxed{180} \) days.

To find out how many days it will take for A and B to complete the work together, we need to calculate their combined work rate.

Let's denote the work rate of A as \(A_r\) and the work rate of B as \(B_r\).

If A can do the work in 6 days, his work rate is:\[ A_r = \dfrac{1}{6} \text{ job per day} \]

Similarly, if B can do the work in 9 days, his work rate is:\[ B_r = \dfrac{1}{9} \text{ job per day} \]

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

\[ \text{Combined rate} = A_r + B_r = \dfrac{1}{6} + \dfrac{1}{9} \]

\[ \text{Combined rate} = \dfrac{3}{18} + \dfrac{2}{18} \]

\[ \text{Combined rate} = \dfrac{5}{18} \text{ job per day} \]

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

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

Since the total work is 1 (the entire job):

\[ \text{Time} = \dfrac{1}{\dfrac{5}{18}} \]

\[ \text{Time} = \dfrac{18}{5} \]

\[ \text{Time} = 3.6 \text{ days} \]

Therefore, A and B together will take \(\boxed{3.6}\) days to complete the work.

Let's denote the total work as \( W \).

If a team of 10 men can complete the job in 12 days, then the rate of work for one man is:\[ \text{Rate of one man} = \dfrac{W}{\text{Total days} \times \text{Number of men}} = \dfrac{W}{12 \times 10} = \dfrac{W}{120} \]

Similarly, if a team of 10 women can complete the same job in 6 days, then the rate of work for one woman is:\[ \text{Rate of one woman} = \dfrac{W}{\text{Total days} \times \text{Number of women}} = \dfrac{W}{6 \times 10} = \dfrac{W}{60} \]

Now, if the two teams work together, their combined rate of work will be the sum of their individual rates:

\[ \text{Combined rate} = \text{Rate of one man} + \text{Rate of one woman} \]\[ \text{Combined rate} = \dfrac{W}{120} + \dfrac{W}{60} \]

\[ \text{Combined rate} = \dfrac{W}{120} + \dfrac{2W}{120} \]\[ \text{Combined rate} = \dfrac{3W}{120} \]\[ \text{Combined rate} = \dfrac{W}{40} \]

Now, to find out how many days it takes for both teams to complete the work together, we use the formula:

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

Since the total work is \( W \) (the entire job):

\[ \text{Time} = \dfrac{W}{\dfrac{W}{40}} = 40 \]

Therefore, if the two teams work together, they will complete the job in \(\boxed{4}\) days.

Let's denote the rates of work for A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively.

From the given information, we can form the following equations:

1. The combined work rate of A and B is \(A_r + B_r = \dfrac{1}{72}\).
2. The combined work rate of B and C is \(B_r + C_r = \dfrac{1}{120}\).
3. The combined work rate of A and C is \(A_r + C_r = \dfrac{1}{90}\).

We're asked to find the work rate of A alone, denoted as \(A_r\).

Now, let's add equations 1, 2, and 3:

\[ (A_r + B_r) + (B_r + C_r) + (A_r + C_r) = \dfrac{1}{72} + \dfrac{1}{120} + \dfrac{1}{90} \]

\[ 2(A_r + B_r + C_r) = \dfrac{5 + 3 + 4}{360} \]

\[ 2(A_r + B_r + C_r) = \dfrac{12}{360} \]

\[ A_r + B_r + C_r = \dfrac{12}{720} \]

\[ A_r + B_r + C_r = \dfrac{1}{60} \]

Therefore, \(A_r + B_r + C_r = \dfrac{1}{60}\). 

We know that the combined work rate of A, B, and C is \(A_r + B_r + C_r = \dfrac{1}{60}\). We also know that the combined work rate of B and C is \(B_r + C_r = \dfrac{1}{120}\). 

Therefore, the work rate of A alone is \(A_r = \dfrac{1}{60} - \dfrac{1}{120}\):

\[ A_r = \dfrac{2}{120} - \dfrac{1}{120} \]

\[ A_r = \dfrac{1}{120} \]

This means that A can complete \( \dfrac{1}{120} \) of the work in one day.

Therefore, A alone can complete the work in \( \boxed{120} \) days.