Time And Work All Competence Aptitude

Let's denote the rate of work for one man as \(M_r\), for one woman as \(W_r\), and for one boy as \(B_r\).

Given that:
1. If 1 man can do the work in 44 days, then the rate of work for one man is \(M_r = \dfrac{1}{44}\) (work per day for one man).
2. If 2 women can do the work in 44 days, then the rate of work for one woman is \(W_r = \dfrac{1}{(2 \times 44)} = \dfrac{1}{88}\) (work per day for one woman).
3. If 3 boys can do the work in 44 days, then the rate of work for one boy is \(B_r = \dfrac{1}{(3 \times 44)} = \dfrac{1}{132}\) (work per day for one boy).

Now, let's find the combined rate of work for 1 man, 1 woman, and 1 boy:

\[ \text{Combined rate of work} = M_r + W_r + B_r \]\[ = \dfrac{1}{44} + \dfrac{1}{88} + \dfrac{1}{132} \]

Let's find a common denominator for these fractions, which is \( 44 \times 2 \times 3 = 264 \).

\[ \text{Combined rate of work} = \dfrac{6}{264} + \dfrac{3}{264} + \dfrac{2}{264} \]\[ = \dfrac{11}{264} \]

Now, let's find out how many days it will take for 1 man, 1 woman, and 1 boy to complete the work:

\[ \text{Number of days} = \dfrac{\text{Total work}}{\text{Rate of work}} = \dfrac{1}{\dfrac{11}{264}} \]\[ = \dfrac{264}{11} \]\[ = 24 \]

So, 1 man, 1 woman, and 1 boy together will complete the work in \(\boxed{24}\) days.

Let's denote the rate of work for one man as \(M_r\) and for one boy as \(B_r\).

Given that:
1. 12 men and 16 boys can do a piece of work in 5 days, which means the combined rate of work for 12 men and 16 boys is \( \dfrac{1}{5} \).
2. 13 men and 24 boys can do the same work in 4 days, which means the combined rate of work for 13 men and 24 boys is \( \dfrac{1}{4} \).

Now, let's set up equations based on these rates of work:

For the first scenario:\[12M_r + 16B_r = \dfrac{1}{5}\] (Combined rate of work for 12 men and 16 boys)

For the second scenario:\[13M_r + 24B_r = \dfrac{1}{4}\] (Combined rate of work for 13 men and 24 boys)

Now, let's solve these equations to find the values of \(M_r\) and \(B_r\).

From the first equation:\[12M_r + 16B_r = \dfrac{1}{5}\]\[60M_r + 80B_r = 1\] (Multiplying both sides by 5)

From the second equation:\[13M_r + 24B_r = \dfrac{1}{4}\]\[52M_r + 96B_r = 1\] (Multiplying both sides by 4)

Now, let's subtract the first equation from the second:\[(52M_r + 96B_r) - (60M_r + 80B_r) = 1 - 1\]\[52M_r - 60M_r + 96B_r - 80B_r = 0\]\[-8M_r + 16B_r = 0\]

Now, let's solve for \(B_r\):\[16B_r = 8M_r\]\[B_r = \dfrac{8}{16}M_r\]\[B_r = \dfrac{1}{2}M_r\]

This means the rate of work for one boy is half the rate of work for one man.

So, the ratio of the daily work done by a man to that of a boy is \(\boxed{2:1}\).

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

Given that:
1. 3 men and 5 women can do a piece of work in 14 days, which means the combined rate of work for 3 men and 5 women is \(\dfrac{1}{14}\).
2. 5 men can do the same work in 14 days, which means the combined rate of work for 5 men is also \(\dfrac{1}{14}\).

Now, let's set up equations based on these rates of work:

For the first scenario:\[3M_r + 5W_r = \dfrac{1}{14}\] (Combined rate of work for 3 men and 5 women)

For the second scenario:\[5M_r = \dfrac{1}{14}\] (Combined rate of work for 5 men)

Now, let's solve these equations to find the values of \(M_r\) and \(W_r\).

From the second equation:\[5M_r = \dfrac{1}{14}\]\[M_r = \dfrac{1}{14 \times 5} = \dfrac{1}{70}\] (Rate of work for one man)

Now, let's substitute the value of \(M_r\) into the first equation:\[3 \times \dfrac{1}{70} + 5W_r = \dfrac{1}{14}\]\[\dfrac{3}{70} + 5W_r = \dfrac{1}{14}\]

Now, let's solve for \(W_r\):\[5W_r = \dfrac{1}{14} - \dfrac{3}{70}\]\[5W_r = \dfrac{5}{70} - \dfrac{3}{70}\]\[5W_r = \dfrac{2}{70}\]\[W_r = \dfrac{2}{350} = \dfrac{1}{175}\] (Rate of work for one woman)

Now, let's find out how many days it will take for 5 men and 5 women to complete the work:

\[5M_r + 5W_r = 5 \times \dfrac{1}{70} + 5 \times \dfrac{1}{175}\]\[= \dfrac{5}{70} + \dfrac{5}{175}\]\[= \dfrac{25 + 10}{350}\]\[= \dfrac{35}{350} = \dfrac{1}{10}\]

So, 5 men and 5 women together can complete the work in \(\boxed{10}\) days.

Let's first find the individual rates of work for A and B.

If A alone can do a piece of work in 20 days, then the rate of work for A is \( \dfrac{1}{20} \) of the work per day.

Similarly, if B alone can do the same work in 30 days, then the rate of work for B is \( \dfrac{1}{30} \) of the work per day.

When A and B work together, their combined rate of work is the sum of their individual rates.

So, the combined rate of work for A and B is:

\[ \text{Combined rate} = \dfrac{1}{20} + \dfrac{1}{30} \]

Now, let's find a common denominator for these fractions, which is 60:

\[ \text{Combined rate} = \dfrac{3}{60} + \dfrac{2}{60} \]\[ = \dfrac{5}{60} \]

Now, let's find the time it takes for A and B to complete half of the work.

Since they are working together at a combined rate of \( \dfrac{5}{60} \) of the work per day, the time it takes to complete half of the work is:

\[ \text{Time} = \dfrac{\text{Work}}{\text{Rate}} = \dfrac{\dfrac{1}{2}}{\dfrac{5}{60}} \]\[ = \dfrac{1}{2} \times \dfrac{60}{5} \]\[ = \dfrac{60}{10} \]\[ = 6 \]

So, they will finish half of the work in \(\boxed{6}\) days.

Let's denote the time it takes for A to complete the work alone as \(A_t\) and the time it takes for B to complete the work alone as \(B_t\). 

Given:
1. A working alone would take 8 hours more to complete the work than when they work together.
2. B working alone would take 4 and 1/2 hours more to complete the work than when they work together.

Let's denote the time it takes for them to complete the work together as \(T\).

So, we have:

\[ A_t = T + 8 \]\[ B_t = T + 4.5 \]

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

So, the equation representing their combined rate is:

\[ \dfrac{1}{A_t} + \dfrac{1}{B_t} = \dfrac{1}{T} \]

Now, substituting the values of \(A_t\) and \(B_t\) from the given information, we get:

\[ \dfrac{1}{T + 8} + \dfrac{1}{T + 4.5} = \dfrac{1}{T} \]

Now, let's solve this equation for \(T\).

Multiplying both sides by \(T(T + 8)(T + 4.5)\) to clear the denominators:

\[ T(T + 8)(T + 4.5) \left(\dfrac{1}{T + 8} + \dfrac{1}{T + 4.5}\right) = T(T + 8)(T + 4.5) \left(\dfrac{1}{T}\right) \]

\[ T(T + 4.5) + T(T + 8) = (T + 8)(T + 4.5) \]

\[ T^2 + 4.5T + T^2 + 8T = T^2 + 12.5T + 36 \]

\[ 2T^2 + 12.5T = T^2 + 12.5T + 36 \]

\[ T^2 - 36 = 0 \]

\[ (T - 6)(T + 6) = 0 \]

\[ T = 6 \text{ or } -6 \]

Since time cannot be negative, \(T = 6\).

So, it takes them \(\boxed{6}\) hours to complete the work together.

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

Given:
1. A can do \(1/6\) of the work in 5 days. So, A's rate of work is:\[A_r = \dfrac{1/6}{5} = \dfrac{1}{30}\] of the work per day.
2. B can do \(2/5\) of the work in 8 days. So, B's rate of work is:\[B_r = \dfrac{2/5}{8} = \dfrac{1}{20}\] of the work per day.

When A and B work together, their combined rate is the sum of their individual rates.

So, the combined rate of work for A and B is:\[A_r + B_r = \dfrac{1}{30} + \dfrac{1}{20} = \dfrac{2 + 3}{60} = \dfrac{5}{60} = \dfrac{1}{12}\] of the work per day.

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

So, both A and B together can complete the work in \(\boxed{12}\) days.

Let's first find the fraction of the work done by A and B together in one day.

Given:
- A can do the work in 10 days, so A's rate of work is \( \dfrac{1}{10} \) of the work per day.
- B can do the work in 20 days, so B's rate of work is \( \dfrac{1}{20} \) of the work per day.

So, the combined rate of work for A and B together is:\[ \text{Combined rate} = \dfrac{1}{10} + \dfrac{1}{20} = \dfrac{2}{20} + \dfrac{1}{20} = \dfrac{3}{20} \] of the work per day.

Now, if they work together for 5 days, the fraction of the work done by them is:\[ \text{Work done in 5 days} = \text{Combined rate} \times \text{Number of days} = \dfrac{3}{20} \times 5 = \dfrac{15}{20} = \dfrac{3}{4} \]

So, the fraction of the work that is left is \( 1 - \dfrac{3}{4} = \dfrac{1}{4} \).

Therefore, the fraction of the work that is left is \(\boxed{\dfrac{1}{4}}\).

Let's first find the fraction of the work completed in 50 days. 

Given that one fourth of the work has been done in 50 days, the fraction of the work completed in 50 days is \( \dfrac{1}{4} \).

Now, let's find the rate at which the workers are completing the work. 

If 200 workers can complete the whole work in 150 days, then their combined rate is \( \dfrac{1}{150} \) of the work per day.

So, the work done by 200 workers in 50 days is:\[ \text{Work done} = \text{Rate} \times \text{Time} = \dfrac{1}{150} \times 50 = \dfrac{1}{3} \]

Now, let's find out how much work is left to be done:\[ \text{Work left} = 1 - \text{Work done} = 1 - \dfrac{1}{3} = \dfrac{2}{3} \]

Now, let's find out how many workers are needed to complete \( \dfrac{2}{3} \) of the work in the remaining 100 days.

Let's denote the number of additional workers needed as \( x \). The total number of workers after adding \( x \) additional workers will be \( 200 + x \).

Since the total work remains the same, the combined rate of work for the \( 200 + x \) workers is \( \dfrac{1}{150} \) of the work per day.

So, the work done by \( 200 + x \) workers in 100 days is:\[ \text{Work done} = \text{Rate} \times \text{Time} = \dfrac{1}{150} \times 100 = \dfrac{2}{3} \]

Now, let's solve for \( x \):\[ \dfrac{200}{150} + \dfrac{x}{150} = \dfrac{2}{3} \]\[ \dfrac{4}{3} + \dfrac{x}{150} = \dfrac{2}{3} \]\[ \dfrac{x}{150} = \dfrac{2}{3} - \dfrac{4}{3} \]\[ \dfrac{x}{150} = -\dfrac{2}{3} \]

Since the number of additional workers cannot be negative, we disregard the negative solution.

\[ x = \dfrac{-2}{3} \times 150 = -100 \]

Since we can't have negative workers, the additional workers needed should be \( 100 \).

So, the number of additional workers to be employed is \(\boxed{100}\).

Let's denote the total work Janardan needs to complete as \( W \).

Given that Janardan completes \( \dfrac{2}{3} \) of his work in 10 days, we can find his rate of work per day:

\[ \text{Rate} = \dfrac{\text{Work}}{\text{Time}} = \dfrac{\dfrac{2}{3}W}{10} = \dfrac{1}{15}W \]

Now, let's find out how much time Janardan will take to complete \( \dfrac{3}{5} \) of the same work.

Let \( T \) be the time he will take to complete \( \dfrac{3}{5} \) of the work. So, we can set up the equation:

\[ \text{Work} = \text{Rate} \times \text{Time} \]\[ \dfrac{3}{5}W = \dfrac{1}{15}W \times T \]

Now, let's solve for \( T \):

\[ T = \dfrac{\dfrac{3}{5}W}{\dfrac{1}{15}W} \]\[ T = \dfrac{3}{5} \times 15 \]\[ T = 9 \]

So, Janardan will take \( \boxed{9} \) days to complete \( \dfrac{3}{5} \) of the same work.

Let's denote the share of A, B, and C as \( A_s \), \( B_s \), and \( C_s \), respectively.

Given:
- A and B together are to do \( \dfrac{9}{11} \) of the work, which means their combined share is \( \dfrac{9}{11} \) of the total amount.
- The total amount for the work is Rs. 440.

So, the combined share of A and B is:\[ A_s + B_s = \dfrac{9}{11} \times 440 \]\[ A_s + B_s = \dfrac{9}{11} \times 440 \]\[ A_s + B_s = 360 \]

Now, let's find out the individual shares of A and B. Since their shares are equal, we can split the combined share equally between A and B:\[ A_s = B_s = \dfrac{360}{2} = 180 \]

Now, the remaining share goes to C. We can find C's share by subtracting the combined share of A and B from the total amount:\[ C_s = 440 - (A_s + B_s) \]\[ C_s = 440 - 360 \]\[ C_s = 80 \]

So, the share of C should be Rs. 80.