Time And Work All Competence Aptitude

Let's denote the rate of work for each person. The rate is the amount of work they can complete per hour.

- Let \( R_A \), \( R_B \), and \( R_C \) be the rates of work for A, B, and C, respectively.

Given information:
1. A can do the work in 4 hours, so \( R_A = \dfrac{1}{4} \) (A's rate is one-fourth of the work per hour).
2. B and C can do the work in 3 hours, so \( R_B + R_C = \dfrac{1}{3} \) (the combined rate of B and C is one-third of the work per hour).
3. A and C can do the work in 2 hours, so \( R_A + R_C = \dfrac{1}{2} \) (the combined rate of A and C is one-half of the work per hour).

Now, let's set up and solve the equations:

Equation 1: \( R_A = \dfrac{1}{4} \)

Equation 2: \( R_B + R_C = \dfrac{1}{3} \)

Equation 3: \( R_A + R_C = \dfrac{1}{2} \)

Now, substitute Equation 1 into Equation 3:\[ \dfrac{1}{4} + R_C = \dfrac{1}{2} \]\[ R_C = \dfrac{1}{2} - \dfrac{1}{4} \]\[ R_C = \dfrac{1}{4} \]

Now, substitute \( R_C \) into Equation 2:\[ R_B + \dfrac{1}{4} = \dfrac{1}{3} \]\[ R_B = \dfrac{1}{3} - \dfrac{1}{4} \]\[ R_B = \dfrac{1}{12} \]

Now, we know \( R_B \), which is the rate at which B works. To find the time \( T \) B alone will take to do the work, we use the formula:\[ T = \dfrac{\text{Total work}}{\text{Rate of B}} \]\[ T = \dfrac{1}{R_B} \]

Substitute the value of \( R_B \):\[ T = \dfrac{1}{\dfrac{1}{12}} \]\[ T = 12 \]

Therefore, B alone will take 12 hours to do the work. Hence, the correct answer is 12 hours.

Let's denote the rates of work for A and B as \( R_A \) and \( R_B \) respectively.

Given information:
1. A and B together can complete the work in 15 days, so their combined rate is \( R_A + R_B = \dfrac{1}{15} \) (the work completed per day by A and B combined is one-fifteenth of the total work).
2. B alone can complete the work in 20 days, so \( R_B = \dfrac{1}{20} \) (the work completed per day by B alone is one-twentieth of the total work).

Now, let's set up and solve the equations:

Equation 1: \( R_A + R_B = \dfrac{1}{15} \)

Equation 2: \( R_B = \dfrac{1}{20} \)

Now, substitute Equation 2 into Equation 1:\[ R_A + \dfrac{1}{20} = \dfrac{1}{15} \]\[ R_A = \dfrac{1}{15} - \dfrac{1}{20} \]\[ R_A = \dfrac{1}{60} \]

Now, we know \( R_A \), which is the rate at which A works. To find the time \( T \) A alone will take to do the work, we use the formula:\[ T = \dfrac{\text{Total work}}{\text{Rate of A}} \]\[ T = \dfrac{1}{R_A} \]

Substitute the value of \( R_A \):\[ T = \dfrac{1}{\dfrac{1}{60}} \]\[ T = 60 \]

Therefore, A alone will take 60 days to complete the work. Hence, the correct answer is 60 days.

Let \( R_A \) be the rate of work for A, and \( R_B \) be the rate of work for B.

Given information:
1. A and B together can complete the work in 12 days, so their combined rate is \( R_A + R_B = \dfrac{1}{12} \).
2. B alone can complete the work in 30 days, so \( R_B = \dfrac{1}{30} \).

Now, let's set up and solve the equations:

Equation 1: \( R_A + R_B = \dfrac{1}{12} \)

Equation 2: \( R_B = \dfrac{1}{30} \)

Now, substitute Equation 2 into Equation 1:\[ R_A + \dfrac{1}{30} = \dfrac{1}{12} \]\[ R_A = \dfrac{1}{12} - \dfrac{1}{30} \]\[ R_A = \dfrac{1}{20} \]

Now, we know \( R_A \), which is the rate at which A works. To find the time \( T \) A alone will take to do the work, we use the formula:\[ T = \dfrac{\text{Total work}}{\text{Rate of A}} \]\[ T = \dfrac{1}{R_A} \]

Substitute the value of \( R_A \):\[ T = \dfrac{1}{\dfrac{1}{20}} \]\[ T = 20 \]

Therefore, A alone will take 20 days to complete the work. Hence, the correct answer is 20 days.

Consider the given information about the time it takes for different pairs of workers to complete a task:

1. (A + B) together can complete the work in 18 days.
2. (B + C) together can complete the work in 9 days.
3. (C + A) together can complete the work in 12 days.

Let's calculate the one-day work rates for each pair:

- The one-day work rate of (A + B) is \( \dfrac{1}{18} \) (equation 1).
- The one-day work rate of (B + C) is \( \dfrac{1}{9} \) (equation 2).
- The one-day work rate of (C + A) is \( \dfrac{1}{12} \) (equation 3).

Now, by adding equations (1), (2), and (3), we find the combined one-day work rate of 2(A + B + C):

\[ \text{One-day work rate of 2(A + B + C)} = \left(\dfrac{1}{18}\right) + \left(\dfrac{1}{9}\right) + \left(\dfrac{1}{12}\right) \]

Simplifying the expression, we get:

\[ \text{One-day work rate of 2(A + B + C)} = \dfrac{2 + 4 + 3}{36} \]

Further simplification yields:

\[ \text{One-day work rate of 2(A + B + C)} = \dfrac{9}{36} \]

Now, we find the one-day work rate of (A + B + C):

\[ \text{One-day work rate of (A + B + C)} = \dfrac{1}{8} \] (equation 4).

By subtracting equation (3) from equation (4), we determine the one-day work rate of B:

\[ \text{One-day work rate of B} = \dfrac{1}{8} - \dfrac{1}{12} \]

This simplifies to:

\[ \text{One-day work rate of B} = \dfrac{3 - 2}{24} = \dfrac{1}{24} \]

In conclusion, the one-day work rate of B is \( \dfrac{1}{24} \), and to find the time B alone will take to complete the work, we use the formula:

\[ \text{Time (B alone)} = \dfrac{1}{\text{One-day work rate of B}} = \dfrac{1}{\dfrac{1}{24}} = 24 \]

Hence, B alone can complete the work in 24 days.

Let's denote the work rates of A, B, and C as \(A_r\), \(B_r\), and \(C_r\) respectively. Also, let \(T\) be the total amount of work.

The given information can be translated into equations:

1. A and B together can complete the work in 10 days:
   \[\dfrac{T}{A_r + B_r} = 10\]

2. B and C together can complete the work in 15 days:
   \[\dfrac{T}{B_r + C_r} = 15\]

3. C and A together can complete the work in 20 days:
   \[\dfrac{T}{C_r + A_r} = 20\]

Now, let's solve these equations. Combining equations 1, 2, and 3, we get:

\[A_r + B_r = \dfrac{T}{10}\]\[B_r + C_r = \dfrac{T}{15}\]\[C_r + A_r = \dfrac{T}{20}\]

Adding all three equations gives:\[2(A_r + B_r + C_r) = \dfrac{T}{10} + \dfrac{T}{15} + \dfrac{T}{20}\]

Simplifying the right side:\[2(A_r + B_r + C_r) = \dfrac{3T + 2T + T}{60}\]\[2(A_r + B_r + C_r) = \dfrac{6T}{60}\]\[A_r + B_r + C_r = \dfrac{T}{10}\]

Now, substitute this back into any of the original equations, let's use equation 1:\[\dfrac{T}{\dfrac{T}{10}} = 10\]\[T = 10\]

Now, we can find the rate at which C alone can do the work:\[C_r = \dfrac{T}{20} = \dfrac{10}{20} = \dfrac{1}{2}\]

So, C alone can complete the work in \(2 \times \text{days} = 2 \times 60 = 120\) days.

Therefore, the correct answer is 120 days.

Let's denote the efficiency of A, B, and C as \(a\), \(b\), and \(c\) respectively. The given information can be expressed as the following three equations:

1. \(A + B = \dfrac{1}{72}\)  (A and B can do the work in 72 days)
2. \(B + C = \dfrac{1}{120}\)  (B and C can do the work in 120 days)
3. \(A + C = \dfrac{1}{90}\)  (A and C can do the work in 90 days)

Now, we need to find \(A + B + C\) which represents the combined efficiency of all three working together.

Adding the three equations above:

\[A + B + B + C + A + C = \dfrac{1}{72} + \dfrac{1}{120} + \dfrac{1}{90}\]

Combining like terms:

\[2A + 2B + 2C = \dfrac{5 + 3 + 4}{360}\]

\[2(A + B + C) = \dfrac{12}{360}\]

\[A + B + C = \dfrac{6}{360}\]

\[A + B + C = \dfrac{1}{60}\]

So, all three together can do the work in 60 days. Therefore, the correct answer is:

\[\text{60 days}\]

Let's denote the rate of work of one man as \( M \) (i.e., the amount of work one man can do in one day). 

The total work \( W \) is given by the product of the number of men and the number of days:

\[ W = \text{Number of men} \times \text{Number of days} \times \text{Rate of work per man} \]

Given that 8 men can do the work in 12 days, we can write the equation as:

\[ W = 8 \times 12 \times M \]

Now, after 6 days of work, the work completed by 8 men is \( 8 \times 6 \times M \). The remaining work to be done is \( W - 8 \times 6 \times M \).

Then, 4 more men join, making the total number of men \( (8 + 4) = 12 \). Let \( D \) be the number of days needed to complete the remaining work with 12 men.

So, the equation becomes:

\[ W - 8 \times 6 \times M = 12 \times D \times M \]

Now, we can substitute the value of \( W \) from the first equation:

\[ 8 \times 12 \times M - 8 \times 6 \times M = 12 \times D \times M \]

Solving for \( D \):

\[ 96 - 48 = 12D \]

\[ 48 = 12D \]

\[ D = 4 \]

Therefore, the remaining work would be completed in 4 days. Hence, the correct answer is:

\[ \text{4 days} \]

Let's find the rate at which A and B work individually. 

The work rate of A is \( \dfrac{1}{20} \) (because A can complete the work in 20 days).

The work rate of B is \( \dfrac{1}{12} \) (because B can complete the work in 12 days).

When A started working alone for 4 days, the amount of work completed by A alone is \( 4 \times \dfrac{1}{20} = \dfrac{1}{5} \).

Now, the remaining work to be done when B joins is \( 1 - \dfrac{1}{5} = \dfrac{4}{5} \).

Now, A and B work together. The combined rate of A and B is \( \dfrac{1}{20} + \dfrac{1}{12} \).

Let \( t \) be the time (in days) B works with A. So, the equation representing the remaining work is:

\[ t \times \left(\dfrac{1}{20} + \dfrac{1}{12}\right) = \dfrac{4}{5} \]

Combine the fractions and solve for \( t \):

\[ t \times \left(\dfrac{6 + 10}{120}\right) = \dfrac{4}{5} \]

\[ t \times \left(\dfrac{16}{120}\right) = \dfrac{4}{5} \]

\[ t \times \left(\dfrac{4}{30}\right) = \dfrac{4}{5} \]

\[ t \times \left(\dfrac{2}{15}\right) = \dfrac{4}{5} \]

\[ t = \dfrac{4}{5} \div \dfrac{2}{15} \]

\[ t = \dfrac{4}{5} \times \dfrac{15}{2} \]

\[ t = 6 \]

So, B works with A for 6 days. A worked alone for 4 days before that. Therefore, the total time the work lasted is \( 4 + 6 = 10 \) days.

Hence, the correct answer is:

\[ \text{10 days} \]

Let's find the rates of work for both men and women first.

The work rate of 5 men is \( \dfrac{1}{6} \) (because 5 men can complete the work in 6 days).

The work rate of 10 women is \( \dfrac{1}{5} \) (because 10 women can complete the work in 5 days).

Now, we can find the rates for men and women individually:

1 man's rate = \( \dfrac{1}{6 \times 5} = \dfrac{1}{30} \)
1 woman's rate = \( \dfrac{1}{10 \times 5} = \dfrac{1}{50} \)

Now, let \( t \) be the number of days it takes for 5 women and 3 men to complete the work together. The combined rate of 5 women and 3 men is \( 5 \times \dfrac{1}{50} + 3 \times \dfrac{1}{30} \).

The equation representing the work completed is:

\[ t \times \left(5 \times \dfrac{1}{50} + 3 \times \dfrac{1}{30}\right) = 1 \]

Combine the fractions and solve for \( t \):

\[ t \times \left(\dfrac{1}{10} + \dfrac{1}{10}\right) = 1 \]

\[ t \times \left(\dfrac{2}{10}\right) = 1 \]

\[ t \times \left(\dfrac{1}{5}\right) = 1 \]

\[ t = 5 \]

So, 5 women and 3 men can complete the work together in 5 days.

Hence, the correct answer is:

\[ \text{5 days} \]

Let's denote the rate of work for one man as \(M\) and for one woman as \(W\).

The work rate of 4 men and 6 women is \(4M + 6W\), and the work rate of 3 men and 7 women is \(3M + 7W\).

The information can be expressed in terms of rates and time:

1. \( (4M + 6W) \times 8 = 1 \) (because they complete the work in 8 days)
2. \( (3M + 7W) \times 10 = 1 \) (because they complete the work in 10 days)

Now, let's solve these two equations to find the values of \(M\) and \(W\).

From equation (1):

\[ 32M + 48W = 1 \]

From equation (2):

\[ 30M + 70W = 1 \]

Now, we can solve these two equations simultaneously. One way to do this is by subtracting equation (2) from equation (1):

\[ (32M + 48W) - (30M + 70W) = 1 - 1 \]

\[ 2M - 22W = 0 \]

\[ 2M = 22W \]

\[ M = 11W \]

Now, let's substitute this back into equation (1):

\[ 32(11W) + 48W = 1 \]

\[ 352W + 48W = 1 \]

\[ 400W = 1 \]

\[ W = \dfrac{1}{400} \]

Now, we want to find how many days 10 women will take to complete the work. The rate of 10 women is \(10 \times \dfrac{1}{400} = \dfrac{1}{40}\).

Let \(D\) be the number of days:

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

\[ D = 40 \]

So, 10 women will take 40 days to complete the work.

Hence, the correct answer is:

\[ \text{40 days} \]