Time And Work All Competence Aptitude

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

If A can complete the work alone in 12 days, then the rate of work for A is:\[ A_r = \dfrac{1}{12} \text{ job per day} \]

Now, if A and B together can complete the work in 8 days, their combined rate of work is:\[ A_r + B_r = \dfrac{1}{8} \text{ job per day} \]

We're asked to find how long it will take for B alone to complete the work.

From the given information, we have the equations:

1. \( A_r + B_r = \dfrac{1}{8} \)
2. \( A_r = \dfrac{1}{12} \)

Let's substitute the value of \(A_r\) from equation 2 into equation 1:

\[ \dfrac{1}{12} + B_r = \dfrac{1}{8} \]

\[ B_r = \dfrac{1}{8} - \dfrac{1}{12} \]

\[ B_r = \dfrac{3}{24} - \dfrac{2}{24} \]

\[ B_r = \dfrac{1}{24} \]

This means that B can complete \( \dfrac{1}{24} \) of the work in one day.

Therefore, B alone can complete the work in \( \boxed{24} \) days.

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

Given:
1. \(A_r = B_r + C_r\)
2. \(A_r + B_r = \dfrac{1}{9.6}\) (since A and B together complete the work in 9 hours 36 minutes, which is \(9.6\) hours)
3. \(C_r = \dfrac{1}{48}\) (since C alone can complete the work in 48 hours)

From equation 2, we can express \(A_r\) in terms of \(B_r\):\[A_r = \dfrac{1}{9.6} - B_r\]

Now, we substitute this expression for \(A_r\) into equation 1:\[B_r + C_r = \dfrac{1}{9.6} - B_r + \dfrac{1}{48}\]

Now, we can solve for \(B_r\):\[2B_r + \dfrac{1}{48} = \dfrac{1}{9.6}\]\[2B_r + \dfrac{1}{48} = \dfrac{1}{\dfrac{48}{5}}\]\[2B_r + \dfrac{1}{48} = \dfrac{5}{48}\]

Now, we can solve for \(B_r\):\[2B_r = \dfrac{5}{48} - \dfrac{1}{48}\]\[2B_r = \dfrac{4}{48}\]\[B_r = \dfrac{2}{48}\]\[B_r = \dfrac{1}{24}\]

This means that B can complete \( \dfrac{1}{24} \) of the work in one hour.

Therefore, B alone can complete the work in \( \boxed{24} \) hours.

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

If A can complete the work alone in 4 days, then the rate of work for A is:\[ A_r = \dfrac{1}{4} \text{ job per day} \]

Similarly, if B can complete the work alone in 12 days, then the rate of work for B is:\[ B_r = \dfrac{1}{12} \text{ job per day} \]

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

\[ \text{Combined rate} = A_r + B_r = \dfrac{1}{4} + \dfrac{1}{12} \]

\[ \text{Combined rate} = \dfrac{3}{12} + \dfrac{1}{12} \]

\[ \text{Combined rate} = \dfrac{4}{12} \text{ job per day} \]

\[ \text{Combined rate} = \dfrac{1}{3} \text{ job per day} \]

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), as they complete the entire piece of work:

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

\[ \text{Time} = 3 \text{ days} \]

Therefore, A and B together will take \(\boxed{3}\) days to complete the work.

Let's denote the rate of work for x as \(X_r\) (pages per hour) and the rate of work for y as \(Y_r\) (pages per hour).

Given:
1. \(X_r = \dfrac{80}{20} = 4\) (since x can copy 80 pages in 20 hours)
2. \(X_r + Y_r = \dfrac{135}{27} = 5\) (since x and y together can copy 135 pages in 27 hours)

We have two equations:

1. \(X_r = 4\)
2. \(X_r + Y_r = 5\)

Now, let's solve for \(Y_r\):\[X_r + Y_r = 5\]\[4 + Y_r = 5\]\[Y_r = 5 - 4\]\[Y_r = 1\]

This means that y can copy \(1\) page per hour.

Now, to find out how many pages y can copy in 20 hours, we use the formula:\[ \text{Pages} = \text{Rate} \times \text{Time} \]\[ \text{Pages} = 1 \times 20 \]\[ \text{Pages} = 20 \]

Therefore, y can copy \(\boxed{20}\) pages in 20 hours.

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

Given:
1. A and B together can do the work in 6 days, so their combined work rate is \(\dfrac{1}{6}\) (the fraction of work they complete per day).
2. A alone can do the work in 18 days, so \(A_r = \dfrac{1}{18}\).

We have two equations:

1. \(A_r + B_r = \dfrac{1}{6}\)
2. \(A_r = \dfrac{1}{18}\)

Now, let's substitute \(A_r\) from equation 2 into equation 1:\[\dfrac{1}{18} + B_r = \dfrac{1}{6}\]

Now, let's solve for \(B_r\):\[B_r = \dfrac{1}{6} - \dfrac{1}{18}\]\[B_r = \dfrac{3}{18} - \dfrac{1}{18}\]\[B_r = \dfrac{2}{18}\]\[B_r = \dfrac{1}{9}\]

This means that B can complete \( \dfrac{1}{9} \) of the work in one day.

Now, to find out how many days it will take for B to complete the work alone, we use the formula:\[ \text{Time} = \dfrac{\text{Total work}}{\text{Rate}} \]

Since the total work is 1 (the entire job), as B completes the entire piece of work:\[ \text{Time} = \dfrac{1}{\dfrac{1}{9}} \]\[ \text{Time} = 9 \]

Therefore, B alone will take \(\boxed{9}\) days to complete the work.

Let's denote the rate of work for x as \(X_r\) (pages per hour) and the rate of work for y as \(Y_r\) (pages per hour).

Given:
1. \(X_r = \dfrac{80}{20} = 4\) (since x can copy 80 pages in 20 hours)
2. \(X_r + Y_r = \dfrac{135}{27} = 5\) (since x and y together can copy 135 pages in 27 hours)

We have two equations:

1. \(X_r = 4\)
2. \(X_r + Y_r = 5\)

Now, let's solve for \(Y_r\):\[X_r + Y_r = 5\]\[4 + Y_r = 5\]\[Y_r = 5 - 4\]\[Y_r = 1\]

This means that y can copy \(1\) page per hour.

Now, to find out how many hours it will take for y to copy 20 pages, we use the formula:\[ \text{Time} = \dfrac{\text{Pages}}{\text{Rate}} \]

\[ \text{Time} = \dfrac{20}{1} \]\[ \text{Time} = 20 \]

Therefore, y can copy \(\boxed{20}\) pages in 20 hours.

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

Given:
1. A and B together can complete the work in 6 days, so their combined rate is \(\dfrac{1}{6}\) (the fraction of work they complete per day).
2. A alone can complete the work in 18 days, so \(A_r = \dfrac{1}{18}\).

We have two equations:

1. \(A_r + B_r = \dfrac{1}{6}\)
2. \(A_r = \dfrac{1}{18}\)

Now, let's substitute \(A_r\) from equation 2 into equation 1:\[\dfrac{1}{18} + B_r = \dfrac{1}{6}\]

Now, let's solve for \(B_r\):\[B_r = \dfrac{1}{6} - \dfrac{1}{18}\]\[B_r = \dfrac{3}{18} - \dfrac{1}{18}\]\[B_r = \dfrac{2}{18}\]\[B_r = \dfrac{1}{9}\]

This means that B can complete \( \dfrac{1}{9} \) of the work in one day.

Now, to find out how many days it will take for B to complete the work alone, we use the formula:\[ \text{Time} = \dfrac{\text{Total work}}{\text{Rate}} \]

Since the total work is 1 (the entire job), as B completes the entire piece of work:\[ \text{Time} = \dfrac{1}{\dfrac{1}{9}} \]\[ \text{Time} = 9 \]

Therefore, B alone will take \(\boxed{9}\) days to complete the work.

You're correct; I made an error in the calculation. Let's approach the problem again.

Let's denote the rate 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. \(A + B\) together can complete the work in 8 days, so their combined rate is \(\dfrac{1}{8}\) (the fraction of work they complete per day).
2. \(B + C\) together can complete the work in 12 days, so their combined rate is \(\dfrac{1}{12}\).
3. \(A + B + C\) together can complete the work in 6 days, so their combined rate is \(\dfrac{1}{6}\).

We have three equations:

1. \(A_r + B_r = \dfrac{1}{8}\)
2. \(B_r + C_r = \dfrac{1}{12}\)
3. \(A_r + B_r + C_r = \dfrac{1}{6}\)

Now, we can subtract equation 2 from equation 3 to find \(A_r\):\[A_r = \dfrac{1}{6} - \dfrac{1}{12}\]\[A_r = \dfrac{2}{12} - \dfrac{1}{12}\]\[A_r = \dfrac{1}{12}\]

Now, we can find \(B_r\) using equation 1:\[A_r + B_r = \dfrac{1}{8}\]\[\dfrac{1}{12} + B_r = \dfrac{1}{8}\]\[B_r = \dfrac{1}{8} - \dfrac{1}{12}\]\[B_r = \dfrac{3}{24} - \dfrac{2}{24}\]\[B_r = \dfrac{1}{24}\]

Now that we have \(A_r\) and \(B_r\), we can find \(C_r\) using equation 2:\[B_r + C_r = \dfrac{1}{12}\]\[\dfrac{1}{24} + C_r = \dfrac{1}{12}\]\[C_r = \dfrac{1}{12} - \dfrac{1}{24}\]\[C_r = \dfrac{2}{24} - \dfrac{1}{24}\]\[C_r = \dfrac{1}{24}\]

This means that each of them can complete \(\dfrac{1}{24}\) of the work per day.

Now, let's find out the time taken by A and C together to complete the work:\[A_r + C_r = \dfrac{1}{12} + \dfrac{1}{24}\]\[A_r + C_r = \dfrac{2}{24} + \dfrac{1}{24}\]\[A_r + C_r = \dfrac{3}{24}\]

So, the time required for A and C together to finish the work is:\[ \text{Time} = \dfrac{1}{A_r + C_r} \]\[ \text{Time} = \dfrac{1}{\dfrac{3}{24}} \]\[ \text{Time} = 8 \text{ days} \]

Therefore, A and C together will take \(\boxed{8}\) days to complete the work.

Let's denote the rate of work for each woman as \(W_r\) (meters of road laid per day).

From the given information, we can use the formula: 

\[ \text{Total work} = \text{Rate} \times \text{Time} \]

Given:
1. 20 women can lay a road of length 100m in 10 days.

So, we can find the rate of work for each woman:\[ W_r = \dfrac{100 \text{ meters}}{20 \text{ women} \times 10 \text{ days}} \]\[ W_r = \dfrac{100}{200} \text{ meters/day/woman} \]\[ W_r = \dfrac{1}{2} \text{ meter/day/woman} \]

Now, we need to find the time required for 10 women to lay a road of length 50 meters. Let's use the formula:\[ \text{Time} = \dfrac{\text{Total work}}{\text{Rate}} \]

\[ \text{Time} = \dfrac{50 \text{ meters}}{\dfrac{1}{2} \text{ meter/day/woman} \times 10 \text{ women}} \]\[ \text{Time} = \dfrac{50}{5} \text{ days} \]\[ \text{Time} = 10 \text{ days} \]

Therefore, 10 women can lay the same road of length 50 meters in \(\boxed{10}\) 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 set up the following equations:

1. The combined rate of A and B is \(\dfrac{1}{24}\) job per day.
2. The combined rate of A, B, and C is \(\dfrac{1}{8}\) job per day.

We can represent the combined rates as follows:

\[ A_r + B_r = \dfrac{1}{24} \]\[ A_r + B_r + C_r = \dfrac{1}{8} \]

Now, let's subtract the first equation from the second to isolate \(C_r\):

\[ (A_r + B_r + C_r) - (A_r + B_r) = \dfrac{1}{8} - \dfrac{1}{24} \]\[ C_r = \dfrac{1}{8} - \dfrac{1}{24} \]\[ C_r = \dfrac{3}{24} - \dfrac{1}{24} \]\[ C_r = \dfrac{2}{24} \]\[ C_r = \dfrac{1}{12} \]

Now that we have the rate of work for C, let's find out how many days C alone will take to finish the job:

\[ \text{Time taken by C alone} = \dfrac{\text{Total work}}{\text{Rate of C}} \]\[ \text{Time taken by C alone} = \dfrac{1}{\dfrac{1}{12}} \]\[ \text{Time taken by C alone} = 12 \text{ days} \]

Therefore, C alone will take \(\boxed{12}\) days to finish the job.