Time And Work All Competence Aptitude

Let's denote the daily earnings of A, B, and C as \(a\), \(b\), and \(c\) respectively.

Given:
1. A, B, and C together earn 150 per day: \(a + b + c = 150\)
2. A and C together earn 94 per day: \(a + c = 94\)
3. B and C together earn 76 per day: \(b + c = 76\)

We can solve these equations to find the daily earnings of A, B, and C.

From equation (2), we can express \(a\) in terms of \(c\):\[a = 94 - c\]

From equation (3), we can express \(b\) in terms of \(c\):\[b = 76 - c\]

Now, substitute these expressions into equation (1) to solve for \(c\):\[a + b + c = 150\]\[(94 - c) + (76 - c) + c = 150\]\[94 - c + 76 - c + c = 150\]\[170 - c = 150\]\[-c = 150 - 170\]\[-c = -20\]\[c = 20\]

So, the daily earning of C is 20. Therefore, the correct answer is 20.

Let's denote the amount paid to A, B, and C as \(A\), \(B\), and \(C\) respectively.

Given:
1. A, B, and C are employed to do a piece of work for 575: \(A + B + C = 575\)
2. A and C together are supposed to finish \( \dfrac{19}{23} \) of the work: \(A + C = \dfrac{19}{23} \times 575\)

We can use these equations to find the amounts paid to A, B, and C.

From equation (2), we can express \(A\) in terms of \(C\):\[A = \dfrac{19}{23} \times 575 - C\]

Now, substitute this expression into equation (1) to solve for \(C\):\[A + B + C = 575\]\[\left(\dfrac{19}{23} \times 575 - C\right) + B + C = 575\]\[\dfrac{19}{23} \times 575 - C + B + C = 575\]\[\dfrac{19}{23} \times 575 + B = 575\]\[B = 575 - \dfrac{19}{23} \times 575\]\[B = 575 \left(1 - \dfrac{19}{23}\right)\]\[B = 575 \times \dfrac{4}{23}\]\[B = \dfrac{2300}{23}\]\[B \approx 100\]

So, the amount paid to B is approximately 100. Therefore, the correct answer is 100.

Let's denote the daily work rate of A as \(a\) and the daily work rate of B as \(b\).

Given that A can do the work in 12 days and B can do it in 15 days, we can express their daily work rates as:

\[a = \dfrac{1}{12}\] (A's work rate)

\[b = \dfrac{1}{15}\] (B's work rate)

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

\[a + b = \dfrac{1}{12} + \dfrac{1}{15}\]

To add these fractions, we need a common denominator, which is \(12 \times 15 = 180\):

\[a + b = \dfrac{15}{180} + \dfrac{12}{180}\]\[a + b = \dfrac{27}{180}\]

This simplifies to:

\[a + b = \dfrac{3}{20}\]

So, together A and B can complete \(\dfrac{3}{20}\) of the work in one day.

Now, let's find out how many days they take to complete the entire work when working together. Since they complete \(\dfrac{3}{20}\) of the work in one day, to complete the entire work, they will take:

\[ \text{Number of days} = \dfrac{1}{\dfrac{3}{20}} \]\[ \text{Number of days} = \dfrac{20}{3} \]

Now, let's find out the total earning of A when they complete the work together:

\[ \text{A's share} = \text{A's daily earning} \times \text{Number of days} \]

\[ \text{A's share} = a \times \dfrac{20}{3} \]\[ \text{A's share} = \dfrac{1}{12} \times \dfrac{20}{3} \]\[ \text{A's share} = \dfrac{20}{36} \]\[ \text{A's share} = \dfrac{5}{9} \]

Now, let's find out the total amount of money A earns:

\[ \text{Total amount of money A earns} = \dfrac{5}{9} \times \text{Total amount of money earned together} \]

\[ \text{Total amount of money A earns} = \dfrac{5}{9} \times 450 \]\[ \text{Total amount of money A earns} = \dfrac{5 \times 450}{9} \]\[ \text{Total amount of money A earns} = \dfrac{2250}{9} \]\[ \text{Total amount of money A earns} = 250 \]

So, the share of A in the amount of money is $250.

Therefore, the correct answer is $250.

Let's denote the number of men initially as \(x\). 

The rate at which \(x\) men can complete the job in 30 days is:\[ \text{Rate}_1 = \dfrac{1}{30} \]

If there were 5 men more, then the total number of men would be \(x + 5\), and they could complete the job in 10 days less, i.e., in 20 days. 

So, the rate at which \(x + 5\) men can complete the job in 20 days is:\[ \text{Rate}_2 = \dfrac{1}{20} \]

According to the given condition, the work done by both groups of men is the same:

\[ \text{Rate}_1 \times \text{Number of men}_1 \times \text{Time}_1 = \text{Rate}_2 \times \text{Number of men}_2 \times \text{Time}_2 \]

Substituting the values:

\[ \dfrac{1}{30} \times x \times 30 = \dfrac{1}{20} \times (x + 5) \times 20 \]

Solving this equation:

\[ x = (x + 5) \times \dfrac{20}{30} \]

\[ x = \dfrac{2}{3} \times (x + 5) \]

\[ 3x = 2(x + 5) \]

\[ 3x = 2x + 10 \]

\[ x = 10 \]

So, the initial number of men was 10.

Therefore, the correct answer is 10.

Let's denote the time taken by the son to complete the work alone as $x$ days.

Given:
1. The father can do the work in 5 days, so his rate is $\frac{1}{5}$ of the work per day.
2. With the help of his son, the father can do the work in 3 days, so their combined rate is $\frac{1}{3}$ of the work per day.

Now, let's set up the equation based on the rates of work:

The father's rate + son's rate = combined rate:
$$\frac{1}{5}+\frac{1}{x}=\frac{1}{3}$$

Now, we can solve this equation for $x$:

$$\frac{1}{5}+\frac{1}{x}=\frac{1}{3}$$

Multiply all terms by $15x$ to clear the denominators:

$$3x+15=5x$$

Subtract $3x$ from both sides:

$$15=2x$$

Divide both sides by 2:

$$x=7\frac{1}{2}$$

So, the son can do the work alone in $7\frac{1}{2}$ days.

Therefore, the correct answer is $7\frac{1}{2}$ days.

Let's first find the rate of work for 12 persons:

12 persons can complete the work in 4 days, so the total work done by 12 persons in 1 day is:

\[ \text{Work done by 12 persons in 1 day} = \dfrac{1}{4} \]

This implies the work done by 1 person in 1 day is:

\[ \text{Work done by 1 person in 1 day} = \dfrac{1}{12 \times 4} = \dfrac{1}{48} \]

Now, let's find out how many persons are required to complete 8 times the work in half the time.

If 12 persons can do 1 work in 4 days, then they can do \(8\) works in \(8 \times 4 = 32\) days.

To complete 8 times the work in half the time, we need to find how many persons are required. Let's denote this number of persons as \(x\).

The total work is multiplied by 8, and the time is halved, so the new total work is \(8 \times 8 = 64\) work units, and the new time is \(\dfrac{32}{2} = 16\) days.

So, the work rate needed to complete 64 units of work in 16 days is:

\[ \text{Required work rate} = \dfrac{\text{Total work}}{\text{Time}} = \dfrac{64}{16} = 4 \]

Now, let's find out how many persons are required:

\[ \text{Number of persons} \times \text{Work done by 1 person in 1 day} = \text{Required work rate} \]

\[ x \times \dfrac{1}{48} = 4 \]

\[ x = 4 \times 48 \]

\[ x = 192 \]

So, 192 persons are required to complete 8 times the work in half the time.

Therefore, the correct answer is 192.

First, let's determine the rate at which the men work. If 12 men can complete the job in 12 days, then the rate of work for each man is:\[ \text{Rate of one man} = \dfrac{1 \text{ job}}{12 \times 12 \text{ days}} = \dfrac{1}{144} \text{ job/day} \]

Now, let's find out how much work is done by 12 men in 6 days. The total work done by 12 men in 6 days is:\[ \text{Work done by 12 men in 6 days} = 12 \times \dfrac{1}{144} \times 6 = \dfrac{1}{2} \text{ job} \]

So, after 6 days, half of the job is completed. This means the remaining work is also half of the total job.

Now, 6 men leave, leaving 6 men to complete the remaining work. The rate at which the remaining 6 men work is the same as before, as their working conditions haven't changed. So, the remaining work will be completed at the same rate as before.

Since half of the job is already done, the remaining work is also half of the total job. And, we have half the number of men to complete it.

Let \( x \) be the number of extra days needed to complete the job. With 6 men, it will take \( x \) days to complete half of the total job.

\[ \text{Work done by 6 men in } x \text{ days} = \dfrac{1}{2} \text{ job} \]

We know the rate of one man is \( \dfrac{1}{144} \) job/day, so we can write the equation as:

\[ 6 \times \dfrac{1}{144} \times x = \dfrac{1}{2} \]

Solving for \( x \):

\[ x = \dfrac{1}{2} \times \dfrac{144}{6} \]\[ x = 12 \]

So, 12 extra days will be needed to complete the job if 6 men leave after working for 6 days.

Therefore, the correct answer is 12 days.

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

Given:
1. A can complete the work in 12 days, so his rate of work is \( A_w = \dfrac{1}{12} \) of the work per day.
2. A worked alone for 3 days and then B joined him. Together, they completed the work in 3 more days. This means they completed the remaining work in 3 days.
3. When A and B worked together, their combined rate of work is \( A_w + B_w \).

Now, let's calculate the amount of work A completes in 3 days:\[ \text{Work done by A in 3 days} = A_w \times \text{Number of days} \]\[ \text{Work done by A in 3 days} = \dfrac{1}{12} \times 3 = \dfrac{1}{4} \]

This means \( \dfrac{1}{4} \) of the work was done by A alone in the first 3 days.

Since A and B completed the remaining work in 3 days, the total work done by both A and B is \( 1 - \dfrac{1}{4} = \dfrac{3}{4} \).

Now, let's find the combined rate of work for A and B:\[ A_w + B_w = \dfrac{\text{Work done by A and B}}{\text{Number of days}} \]\[ A_w + B_w = \dfrac{\dfrac{3}{4}}{3} \]\[ A_w + B_w = \dfrac{1}{4} \]

Now, we know A's rate of work, which is \( \dfrac{1}{12} \). So, we can substitute this into the equation and solve for B's rate of work:

\[ \dfrac{1}{12} + B_w = \dfrac{1}{4} \]\[ B_w = \dfrac{1}{4} - \dfrac{1}{12} \]\[ B_w = \dfrac{3}{12} - \dfrac{1}{12} \]\[ B_w = \dfrac{2}{12} \]\[ B_w = \dfrac{1}{6} \]

So, B's rate of work is \( \dfrac{1}{6} \) of the work per day.

Now, to find out how many days B alone needs to complete the work, we use his rate of work:

\[ \text{Number of days for B alone} = \dfrac{\text{Total work}}{\text{B's rate of work}} \]\[ \text{Number of days for B alone} = \dfrac{1}{\dfrac{1}{6}} \]\[ \text{Number of days for B alone} = 6 \]

So, B alone can finish the work in 6 days.

Therefore, the correct answer is 6 days.

If the working efficiencies of P and Q are in the ratio 3:4, then let's say P can complete 3 units of work in 1 day, and Q can complete 4 units of work in 1 day.

Let's denote the number of days taken by P to complete the work as \(x\) days, and the number of days taken by Q to complete the work as \(y\) days.

Since work done = rate × time, we have:

\[ \text{Work done by P in } x \text{ days} = 3x \]\[ \text{Work done by Q in } y \text{ days} = 4y \]

Now, as the amount of work done is the same, we can equate them:

\[ 3x = 4y \]

To find the ratio of the number of days taken by them to complete the work, we need to find a relationship between \(x\) and \(y\).

From the equation \(3x = 4y\), we can solve for \(y\) in terms of \(x\):

\[ y = \dfrac{3}{4}x \]

So, the ratio of the number of days taken by them to complete the work is \(3 : 4\).

However, this isn't one of the options provided. Let's check our calculation.

We know \(y = \dfrac{3}{4}x\). To find the ratio of the number of days taken by P to the number of days taken by Q, we can set up the ratio:

\[ \dfrac{x}{y} = \dfrac{x}{\dfrac{3}{4}x} = \dfrac{4}{3} \]

So, the correct ratio of the number of days taken by P to the number of days taken by Q is \(4 : 3\).

Therefore, the correct answer is \(4 : 3\).

Let's denote the work capacity of one woman per day as \(W\) and the work capacity of one man per day as \(M\).

Given:
1. Twenty women together can complete a work in 16 days.
2. Sixteen men together can complete the same work in 15 days.

From the first statement, we can find the work capacity of one woman per day:\[20 \text{ women} \times 16 \text{ days} = 1 \text{ work}\]\[320 \text{ woman-days} = 1 \text{ work}\]\[1 \text{ woman-day} = \dfrac{1}{320} \text{ work}\]

From the second statement, we can find the work capacity of one man per day:\[16 \text{ men} \times 15 \text{ days} = 1 \text{ work}\]\[240 \text{ man-days} = 1 \text{ work}\]\[1 \text{ man-day} = \dfrac{1}{240} \text{ work}\]

Now, let's find the ratio of the work capacity of a man to that of a woman:\[ \dfrac{M}{W} = \dfrac{\dfrac{1}{240}}{\dfrac{1}{320}} = \dfrac{320}{240} = \dfrac{4}{3}\]

So, the ratio of the working capacity of a man to that of a woman is \(4 : 3\).

Therefore, the correct answer is \(4 : 3\).