Time And Work All Competence Aptitude

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

A can do the work in 20 days, so in one day, A completes 1/20th of the work.
B can do the work in 40 days, so in one day, B completes 1/40th of the work.

Together, in one day, they complete:

1/20 (A's rate) + 1/40 (B's rate) = 2/40 + 1/40 = 3/40

So, together A and B complete 3/40th of the work in one day.

Now, they work together for 5 days, so the fraction of work completed in those 5 days is:

(3/40) * 5 = 15/40 = 3/8

Now, the fraction of work left after 5 days is:

1 - 3/8 = 5/8

So, the correct option is:

5/8

We have:

A does half as much work as B in three fourths of the time.

Together they take 18 days to complete the work.

Let's denote:

- A's time to complete the work as $a$ days.
- B's time to complete the work as $b$ days.
- The total work to be done as $W$.

From the given information:

A does half as much work as B in three fourths of the time:

$\frac{1}{2}=\frac{\mathit{B}}{\mathit{A}}\times \frac{3}{4}$

This equation gives us the ratio of the time taken by A and B. Let's solve it:

$\frac{1}{2}=\frac{\mathit{B}}{\mathit{A}}\times \frac{3}{4}$
$1=\frac{3\mathit{B}}{2\mathit{A}}$
$\mathit{A}=\frac{3}{2}\mathit{B}$ 

Together they take 18 days to complete the work:

Their combined work rate is:

$\frac{1}{\mathit{A}}+\frac{1}{\mathit{B}}=\frac{1}{18}$

Substituting the value of $A$ from equation (1):

$\frac{1}{\frac{3}{2}\mathit{B}}+\frac{1}{\mathit{B}}=\frac{1}{18}$
$\frac{2}{3\mathit{B}}+\frac{1}{\mathit{B}}=\frac{1}{18}$
$\frac{2+3}{3\mathit{B}}=\frac{1}{18}$
$\frac{5}{3\mathit{B}}=\frac{1}{18}$

Solving for $\mathbf{B}$:

$\mathit{B}=\frac{5\times 18}{3}=\frac{90}{3}=30$

So, B alone would take 30 days to complete the work. 

Thus, the correct option is 30 days.

Let's denote the time taken by Tapas to complete the work as \(T\) days and the time taken by Mihir to complete the work as \(M\) days.

Given that Tapas works twice as fast as Mihir, we can express this relationship as:

\[T = \dfrac{1}{2}M\]

And, we know that their combined work rate is:

\[\dfrac{1}{T} + \dfrac{1}{M} = \dfrac{1}{12}\]

Substituting \(T = \dfrac{1}{2}M\) into the combined work rate equation:

\[\dfrac{1}{\dfrac{1}{2}M} + \dfrac{1}{M} = \dfrac{1}{12}\]\[2 \times \dfrac{1}{M} + \dfrac{1}{M} = \dfrac{1}{12}\]\[\dfrac{3}{M} = \dfrac{1}{12}\]

Now, solving for \(M\):

\[M = \dfrac{3 \times 12}{1} = 36\]

So, Mihir alone would take 36 days to complete the work.

Now, we can find the time taken by Tapas alone:

\[T = \dfrac{1}{2}M = \dfrac{1}{2} \times 36 = 18\]

So, Tapas alone can complete the work in 18 days.

Therefore, the correct option is 18 days.

Given that 4 mat-weavers can weave 4 mats in 4 days, we can deduce that each mat-weaver weaves one mat in 4 days. So, each mat-weaver's rate is 1 mat in 4 days.

Now, if there are 8 mat-weavers, their combined rate would be 8 times that of a single mat-weaver.

So, the combined rate of 8 mat-weavers would be $8\times \frac{1}{4}$ mats per day.

To find out how many mats they would weave in 8 days, we multiply their combined rate by the number of days:

$$8\times \frac{1}{4}\times 8=2\times 8=16$$

Therefore, 8 mat-weavers would weave 16 mats in 8 days.

So, the correct answer is 16 mats.

Let's first find the rate at which one man works:

If 7 men can complete the work in 12 days, then one man's work rate is given by:

\[ 7 \text{ men} \times 12 \text{ days} = 84 \text{ man-days} \]

So, the work rate of one man is \( \dfrac{1}{84} \) of the work per day.

Now, to complete double the work in 8 days, the total work will be twice that of before, which is \(2 \times 84 = 168\) man-days.

Now, let's find out how many men are needed to complete 168 man-days of work in 8 days.

Let \(x\) be the number of additional men required.

So, the equation becomes:

\[ (7 + x) \times 8 = 168 \]

\[ 56 + 8x = 168 \]

\[ 8x = 168 - 56 \]

\[ 8x = 112 \]

\[ x = \dfrac{112}{8} \]

\[ x = 14 \]

So, an additional 14 men will be required to complete double the work in 8 days.

Therefore, the correct answer is 14.

Let's first find the daily earning of one man and one woman.

Given that 5 men or 7 women can earn $5,250 per day, we can set up a proportion to find out the daily earning of one man or one woman.

For men:\[5 \text{ men} = 5250 \text{ dollars/day}\]

So, 1 man would earn:\[1 \text{ man} = \dfrac{5250}{5} \text{ dollars/day} = 1050 \text{ dollars/day}\]

For women:\[7 \text{ women} = 5250 \text{ dollars/day}\]

So, 1 woman would earn:\[1 \text{ woman} = \dfrac{5250}{7} \text{ dollars/day} \approx 750 \text{ dollars/day}\]

Now, let's calculate the daily earnings of 7 men and 13 women:

\[7 \times 1050 \text{ dollars/day} + 13 \times 750 \text{ dollars/day}\]\[= 7350 \text{ dollars/day} + 9750 \text{ dollars/day}\]\[= 17,100 \text{ dollars/day}\]

So, 7 men and 13 women would earn $17,100 per day.

Therefore, the correct answer is 17,100. 

From the given information, we have:

\[ (2 \text{ men} + 1 \text{ woman}) \times 14 = (4 \text{ women} + 2 \text{ men}) \times 8 \]

Which simplifies to:

\[ 28 \text{ men} + 14 \text{ women} = 32 \text{ women} + 16 \text{ men} \]

Further simplifying gives:

\[ (28 - 16) \text{ men} = (32 - 14) \text{ women} \]\[ 12 \text{ men} = 18 \text{ women} \]

Then, dividing both sides by 2 gives:

\[ 2 \text{ men} = 3 \text{ women} \]

From this relation, we can infer that 1 woman is equivalent to \( \dfrac{2}{3} \) of a man. 

Given that the wages per day of 1 man is Rs. 180, the wages per day of 1 woman will be:

\[ \text{Wages per day of 1 woman} = \dfrac{2}{3} \times 180 = \dfrac{360}{3} = Rs. 120 \]

So, the correct answer is Rs. 120.

A finishes \( \dfrac{1}{4} \) of the work in 15 days, leaving \( \dfrac{3}{4} \) of the work.

B's rate is \( \dfrac{1}{40} \) of the work per day.

When A and B work together, their combined rate is \( \dfrac{1}{60} + \dfrac{1}{40} = \dfrac{1}{24} \) of the work per day.

So, together A and B can finish the work in 24 days.

If two persons, with equal abilities, can do two jobs in two days, it implies that each person can do one job in two days. So, the rate of work per person is 1 job per 2 days.

If 100 persons with equal abilities work together, their combined rate of work will be 100 times the rate of work of one person. Therefore, they can complete 100 similar jobs in:

\[ \text{Number of days} = \dfrac{\text{Total jobs}}{\text{Rate of work per person}} = \dfrac{100}{1/2} = 200 \]

So, 100 persons can complete 100 similar jobs in 2 days. 

Therefore, the answer is 2 days.

Given that A and B together can complete the work in 24 days, we can find their combined rate of work per day:

Let \( A_r \) be the rate of work for A per day and \( B_r \) be the rate of work for B per day.

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

Given that B alone completes 1/3rd of the work in 12 days, we can find \( B_r \) as:

\[ B_r = \dfrac{1/3}{12} = \dfrac{1}{36} \]

Now, let's find \( A_r \):

\[ A_r + \dfrac{1}{36} = \dfrac{1}{24} \]\[ A_r = \dfrac{1}{24} - \dfrac{1}{36} = \dfrac{3}{72} - \dfrac{2}{72} = \dfrac{1}{72} \]

So, the rate of work for A per day is \( \dfrac{1}{72} \).

Now, to find out how many days A alone will take to complete the remaining work:

\[ \text{Days taken by A alone} = \dfrac{\text{Remaining work}}{\text{Rate of work for A per day}} \]

The remaining work for A is \( \dfrac{2}{3} \), and A's rate of work per day is \( \dfrac{1}{72} \).

\[ \text{Days taken by A alone} = \dfrac{\dfrac{2}{3}}{\dfrac{1}{72}} = \dfrac{2}{3} \times 72 = 48 \]

So, A alone will take 48 days to complete the remaining work.

Therefore, the answer is 48 days.