Time And Work All Competence Aptitude

Let's first find out how much work A completes in 12 days.

Since A can do the work in 18 days, his daily rate of work is \( \dfrac{1}{18} \) of the work per day.

So, in 12 days, A completes \( 12 \times \dfrac{1}{18} = \dfrac{12}{18} = \dfrac{2}{3} \) of the work.

Now, the remaining work that needs to be done is \( 1 - \dfrac{2}{3} = \dfrac{1}{3} \) of the total work.

Since B finishes the remaining work in 8 days, we can find B's daily rate of work. Let's denote B's rate of work as \( B_r \).

B's rate of work is \( \dfrac{1}{8} \) of the work per day.

Now, let's denote the total work as \( W \). Since B finishes the remaining \( \dfrac{1}{3} \) of the work in 8 days, we can set up the equation:

\[ B_r \times 8 = \dfrac{1}{3} \times W \]

Substituting the value of \( B_r \):

\[ \dfrac{1}{8} \times 8 = \dfrac{1}{3} \times W \]

\[ W = 3 \]

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

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 complete the work in 9 days, so their combined work rate is \(\dfrac{1}{9}\) (the fraction of work they complete per day).
2. A does thrice the work of B in a given time.

Let's denote the time it takes for B to complete the work alone as \(t\) days. Therefore, the time it takes for A to complete the work alone would be \(3t\) days.

We have two equations:

1. \(A_r + B_r = \dfrac{1}{9}\) (from the fact that A and B together can do the work in 9 days)
2. \(A_r = 3B_r\) (from the fact that A does thrice the work of B)

Now, let's substitute \(A_r\) from equation 2 into equation 1:\[3B_r + B_r = \dfrac{1}{9}\]

Now, let's solve for \(B_r\):\[4B_r = \dfrac{1}{9}\]\[B_r = \dfrac{1}{36}\]

So, B's work rate is \(\dfrac{1}{36}\) of the work per day.

Now, let's find A's work rate using equation 2:\[A_r = 3B_r\]\[A_r = 3 \times \dfrac{1}{36}\]\[A_r = \dfrac{1}{12}\]

So, A's work rate is \(\dfrac{1}{12}\) of the work per day.

Now, let's find out how many days it will take for A to complete the work alone:\[A_r = \dfrac{1}{12}\]\[\text{Time} = \dfrac{\text{Total work}}{\text{Rate}}\]\[\text{Time} = \dfrac{1}{\dfrac{1}{12}}\]\[\text{Time} = 12\]

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

Let's assume that the rate at which cats kill mice remains constant over time.

Given:
- 100 cats kill 100 mice in 100 days.

If we consider one cat, it would kill one mouse in 100 days.

Now, if we have 4 cats, the total number of mice they kill in 100 days remains the same (as each cat kills one mouse in 100 days). Therefore, 4 cats would still kill 100 mice in 100 days.

So, the number of days required for 4 cats to kill 4 mice would be the same as the number of days required for 100 cats to kill 100 mice, which is \(\boxed{100}\) days.

Let's denote the time taken by the father to complete the job alone as \(F\) hours.

Given that the father can do the job as fast as his two sons working together, we can set up the following equation based on their rates of work:

The father's rate of work + the combined rate of his two sons = 1 job per unit time (as they complete the job together)

The father's rate of work is \(\dfrac{1}{F}\) job per hour.

The first son's rate of work is \(\dfrac{1}{3}\) job per hour.

The second son's rate of work is \(\dfrac{1}{6}\) job per hour.

So, the equation becomes:

\[\dfrac{1}{F} + \dfrac{1}{3} + \dfrac{1}{6} = 1\]

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

\[\dfrac{1}{F} + \dfrac{1}{3} + \dfrac{1}{6} = 1\]\[\dfrac{1}{F} + \dfrac{2}{6} + \dfrac{1}{6} = 1\]\[\dfrac{1}{F} + \dfrac{3}{6} = 1\]\[\dfrac{1}{F} + \dfrac{1}{2} = 1\]\[\dfrac{1}{F} = \dfrac{1}{2}\]

So, \(F = 2\) hours.

Therefore, the number of hours taken by the father to do the job alone is \(\boxed{2}\) hours.

Let's find the rates of work for each person first:

A's rate = \( \dfrac{1}{24} \) of the work per day
B's rate = \( \dfrac{1}{9} \) of the work per day
C's rate = \( \dfrac{1}{12} \) of the work per day

Now, let's calculate the work done by B and C in 3 days:

Work done by B in 3 days = B's rate × 3 = \( \dfrac{1}{9} \) × 3 = \( \dfrac{1}{3} \) of the work
Work done by C in 3 days = C's rate × 3 = \( \dfrac{1}{12} \) × 3 = \( \dfrac{1}{4} \) of the work

Together, B and C complete \( \dfrac{1}{3} + \dfrac{1}{4} = \dfrac{7}{12} \) of the work in 3 days.

Now, let's find out how much work is left:

Remaining work = 1 - \( \dfrac{7}{12} \) = \( \dfrac{5}{12} \) of the work

Now, A completes the remaining \( \dfrac{5}{12} \) of the work:

Time taken by A to complete \( \dfrac{5}{12} \) of the work = \( \dfrac{\text{Remaining work}}{\text{A's rate}} \)
= \( \dfrac{\dfrac{5}{12}}{\dfrac{1}{24}} \) = \( \dfrac{5}{12} \times 24 \) = 10 days

Therefore, the remaining work was completed by A in \(\boxed{10}\) days.

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 8 days, so their combined rate is \(\dfrac{1}{8}\) (the fraction of work they complete per day).
2. B alone can complete the work in 12 days, so \(B_r = \dfrac{1}{12}\).

Let's find the individual rates of A and B when working together. Since the combined rate of A and B is \(\dfrac{1}{8}\), and we know \(B_r = \dfrac{1}{12}\), we can subtract \(B_r\) from the combined rate to find \(A_r\):

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

Now, let's find out how much work B completes in 4 days:\[ \text{Work done by B in 4 days} = B_r \times 4 \]\[ \text{Work done by B in 4 days} = \dfrac{1}{12} \times 4 \]\[ \text{Work done by B in 4 days} = \dfrac{1}{3} \]

Now, the remaining work for A to complete is \(1 - \dfrac{1}{3} = \dfrac{2}{3}\) of the total work.

Now, let's find out how many days it will take for A to complete the remaining work alone:\[ \text{Time} = \dfrac{\text{Remaining work}}{\text{Rate of A}} \]\[ \text{Time} = \dfrac{\dfrac{2}{3}}{\dfrac{1}{24}} \]\[ \text{Time} = \dfrac{2}{3} \times 24 \]\[ \text{Time} = 16 \text{ days} \]

Therefore, A alone will take \(\boxed{16}\) days to complete the remaining work.

Let's first find the rates of work for each person:

A's rate = \( \dfrac{1}{10} \) of the work per day
B's rate = \( \dfrac{1}{15} \) of the work per day
C's rate = \( \dfrac{1}{20} \) of the work per day

Now, let's calculate the work done by A and C working together for 2 days:

Work done by A and C together in 2 days:\[ \text{Work by A} = A's \ rate \times \text{Time} = \dfrac{1}{10} \times 2 = \dfrac{1}{5} \]\[ \text{Work by C} = C's \ rate \times \text{Time} = \dfrac{1}{20} \times 2 = \dfrac{1}{10} \]

Now, let's find the total work done by A and C together in 2 days:\[ \text{Total work in 2 days} = \dfrac{1}{5} + \dfrac{1}{10} = \dfrac{2}{10} + \dfrac{1}{10} = \dfrac{3}{10} \]

Now, the remaining work that needs to be done is \( 1 - \dfrac{3}{10} = \dfrac{7}{10} \) of the total work.

Now, B replaces A, and B and C work together to complete the remaining work.

The combined rate of B and C is:\[ \text{B's rate} + \text{C's rate} = \dfrac{1}{15} + \dfrac{1}{20} = \dfrac{4}{60} + \dfrac{3}{60} = \dfrac{7}{60} \]

Now, let's find out how long it takes for B and C to complete the remaining work:\[ \text{Time taken} = \dfrac{\text{Remaining work}}{\text{Combined rate of B and C}} \]\[ \text{Time taken} = \dfrac{\dfrac{7}{10}}{\dfrac{7}{60}} \]\[ \text{Time taken} = \dfrac{7}{10} \times \dfrac{60}{7} \]\[ \text{Time taken} = 6 \text{ days} \]

Therefore, the work was completed in \( 2 \text{ days (A and C)} + 6 \text{ days (B and C)} = \boxed{8} \text{ days} \).

Let's first find the rates of work for each person:

A's rate = \( \dfrac{1}{18} \) of the work per day
B's rate = \( \dfrac{1}{20} \) of the work per day
C's rate = \( \dfrac{1}{30} \) of the work per day

Now, let's calculate the work done by B and C together in 2 days:

Work done by B and C together in 2 days:\[ \text{Work by B} = B's \ rate \times \text{Time} = \dfrac{1}{20} \times 2 = \dfrac{1}{10} \]\[ \text{Work by C} = C's \ rate \times \text{Time} = \dfrac{1}{30} \times 2 = \dfrac{1}{15} \]

Now, let's find the total work done by B and C together in 2 days:\[ \text{Total work in 2 days} = \dfrac{1}{10} + \dfrac{1}{15} = \dfrac{3}{30} + \dfrac{2}{30} = \dfrac{5}{30} = \dfrac{1}{6} \]

Now, the remaining work that needs to be done is \( 1 - \dfrac{1}{6} = \dfrac{5}{6} \) of the total work.

Now, A completes the remaining \( \dfrac{5}{6} \) of the work alone.

Now, let's find out how long it takes for A to complete the remaining work alone:\[ \text{Time taken} = \dfrac{\text{Remaining work}}{\text{A's rate}} \]\[ \text{Time taken} = \dfrac{\dfrac{5}{6}}{\dfrac{1}{18}} \]\[ \text{Time taken} = \dfrac{5}{6} \times \dfrac{18}{1} \]\[ \text{Time taken} = \dfrac{5 \times 18}{6} \]\[ \text{Time taken} = \dfrac{90}{6} \]\[ \text{Time taken} = 15 \text{ days} \]

Therefore, the time taken by A alone to complete the remaining work is \(\boxed{15}\) days.

Let's first find the combined rate of work when A is assisted by B and C on every third day.

A's rate = \( \dfrac{1}{20} \) of the work per day
B's rate = \( \dfrac{1}{30} \) of the work per day
C's rate = \( \dfrac{1}{60} \) of the work per day

When A is assisted by B and C on every third day, the combined rate of work on those days is:

\[ \text{Combined rate} = A's \ rate + B's \ rate + C's \ rate \]\[ \text{Combined rate} = \dfrac{1}{20} + \dfrac{1}{30} + \dfrac{1}{60} \]\[ \text{Combined rate} = \dfrac{3}{60} + \dfrac{2}{60} + \dfrac{1}{60} \]\[ \text{Combined rate} = \dfrac{6}{60} \]\[ \text{Combined rate} = \dfrac{1}{10} \]

This means that A, B, and C together can complete \( \dfrac{1}{10} \) of the work per day when A is assisted by B and C on every third day.

Now, on the days when A is not assisted by B and C, A alone will complete the work.

So, every third day, A completes \( \dfrac{1}{10} \) of the work, and for the other two days, A completes \( \dfrac{1}{20} \) of the work.

Let's find out how many days it takes for A to complete the work alone with this alternating pattern:

\[ \text{Work done in 3 days} = \dfrac{1}{10} + \dfrac{1}{20} + \dfrac{1}{20} \]\[ \text{Work done in 3 days} = \dfrac{1}{10} + \dfrac{1}{10} \]\[ \text{Work done in 3 days} = \dfrac{2}{10} \]\[ \text{Work done in 3 days} = \dfrac{1}{5} \]

Therefore, A completes \( \dfrac{1}{5} \) of the work every 3 days.

To find out how many 3-day cycles it takes to complete the work, we'll divide the total work (\(1\) job) by the work done in one 3-day cycle (\( \dfrac{1}{5} \) job):

\[ \text{Number of 3-day cycles} = \dfrac{1}{ \dfrac{1}{5}} = 5 \]

So, it takes 5 cycles of 3 days each to complete the work.

Therefore, the total number of days required for A to complete the work is \(5 \times 3 = \boxed{15} \) days.

Let's first find the rate of work of each man.

If 40 men can complete the work in 18 days, then the rate of each man is:

\[ \text{Rate of each man} = \dfrac{1}{(40 \times 18)} = \dfrac{1}{720} \]

Now, let's find out how much work the initial group of 40 men completes in 8 days.

\[ \text{Work done by 40 men in 8 days} = \text{Rate of each man} \times \text{Number of men} \times \text{Number of days} \]\[ \text{Work done by 40 men in 8 days} = \dfrac{1}{720} \times 40 \times 8 = \dfrac{320}{720} = \dfrac{4}{9} \]

So, after 8 days, the initial group of 40 men completes \( \dfrac{4}{9} \) of the work. This means \( \dfrac{5}{9} \) of the work is left.

Now, 10 more men join them. So, the total number of men working together becomes 50.

Now, let's find out how many days it will take for 50 men to complete \( \dfrac{5}{9} \) of the work.

\[ \text{Number of days} = \dfrac{\text{Total work remaining}}{\text{Work done per day}} \]\[ \text{Number of days} = \dfrac{\dfrac{5}{9}}{\dfrac{1}{720} \times 50} \]\[ \text{Number of days} = \dfrac{\dfrac{5}{9}}{\dfrac{50}{720}} \]\[ \text{Number of days} = \dfrac{5}{9} \times \dfrac{720}{50} \]\[ \text{Number of days} = \dfrac{5 \times 720}{9 \times 50} \]\[ \text{Number of days} = \dfrac{3600}{450} = 8 \]

Therefore, it will take \( \boxed{8} \) days for the remaining men to complete the work.