Unit 1: Time and Work, Pipes and Cisterns - Subjective Questions

Work rate is the fraction of a job completed in one unit of time.

• If a worker completes a job in days, the worker's one-day work is:
  
• If the rate of work is , the work completed in days is:
  
• Therefore:
  

For a fixed amount of work, efficiency is proportional to the rate of work. Hence:

Since for fixed :

Thus, if the efficiencies of two workers and are in the ratio , their times are in the inverse ratio:

The one-day work of is:

The one-day work of is:

Their combined one-day work is:

Therefore, the time required to complete one job is:

Hence, and together complete the work in days.

Let the one-day work rates of , , and be , , and .

Given:

Adding the three equations:

Therefore, all three complete the work in days.

Individual rates are:

Thus:

• takes days.
• takes days.
• takes days.
• All three together take days.

Efficiency and time are inversely proportional. Therefore:

Given days:

Their combined rate is:

Hence, their combined time is:

Therefore:

• alone takes days.
• and together take days.

Given that two men are equivalent to three women:

Therefore, one woman is equivalent to:

The equivalent strength of women is:

Thus, men and women are equivalent to:

The total work is:

Time required by the equivalent of men is:

Hence, men and women complete the job in days.

The daily work rates of the given groups are equal:

Thus, the daily work of one person in each category is:

The combined one-day work of men, women, and children is:

Therefore, the required time is:

Hence, the group completes the job in days, approximately days.

Wages must be divided in proportion to the work done by each worker.

Their efficiency ratio is:

Therefore, the wage ratio is also:

The total number of ratio parts is:

Thus, receives:

And receives:

Hence, the wages should be divided as follows:

• receives .
• receives .

The combined one-day work of and is:

Work completed in days is:

Remaining work is:

After leaves, works at the rate per day. The time taken by to complete the remaining work is:

Therefore, the total time is:

Hence, the job is completed in days.

The one-day work rates are:

Work completed in one two-day cycle is:

In complete cycles, or days, the work completed is:

The remaining work is:

On the fifteenth day, it is 's turn. The fraction of a day required by is:

Therefore, total time required is:

Hence, the work is completed in days.

The total work is:

Work completed during the first days is:

Remaining work is:

To complete this work in days, the required number of men is:

Since men are already employed, the number of additional men required is:

Hence, additional men must be employed.

• An inlet pipe fills a tank, so its work rate is taken as positive.
• An outlet pipe empties a tank, so its work rate is taken as negative.

If an inlet fills a tank in hours, its rate is:

If an outlet empties a full tank in hours, its rate is:

For several pipes, the net rate is:

For one inlet and one outlet:

If , the tank fills, and the required time is:

If , the tank empties. If , the water level remains unchanged.

The rates of the two inlet pipes are:

The rate of the outlet pipe is:

The net filling rate is:

Therefore, the time required to fill the tank is:

Hence, the tank will be filled in hours.

The combined hourly rate of the two pipes is:

The fraction filled in hours is:

Thus, half of the tank is filled in hours.

The remaining fraction is:

The additional time required is:

Hence:

• Fraction filled after hours: .
• Additional time required: hours.
• Total filling time: hours.

The inlet's filling rate is:

The actual net filling rate with the leak is:

Let the leak empty the full tank in hours. Its rate is . Therefore:

Rearranging:

Thus:

Hence, the leak alone can empty the full tank in hours.

In the first hours, only the inlet operates. The fraction filled is:

The remaining fraction is:

After the outlet is opened, the net filling rate is:

The time needed to fill the remaining of the tank is:

Therefore, the total time is:

Hence, the tank becomes full hours after the inlet is first opened.

The combined rate of pipes and is:

The fraction filled in hours is:

The remaining fraction is:

After pipe is closed, pipe alone fills at the rate . The time required for the remaining part is:

Total time is:

Hence, the tank is filled in hours.

The magnitudes of their rates are:

Their efficiency ratio is:

Multiplying by gives:

Since is an outlet, its contribution is negative. The net rate is:

Therefore, the tank fills in:

Hence:

• Efficiency ratio by magnitude: .
• Net filling time: minutes.

The initial filled fraction is:

Therefore, the remaining fraction is:

The net rate when both pipes are opened is:

The time required to fill the remaining is:

Hence, the tank will become full in hours.

The rates of the two inlet pipes are:

The outlet rate is negative:

Therefore, the net filling rate is:

Taking the least common denominator :

Since time is the reciprocal of the rate, the required filling time is:

The tank can be filled only if the net rate is positive:

Equivalently:

If equality holds, the water level remains constant. If the outlet rate is greater, the tank cannot be filled while all three pipes remain open.

Given:

Therefore:

Convert the entire group into man-equivalents:

The total work is:

Time required by man-equivalents is:

The contribution ratio of men, women, and children is:

The total number of ratio parts is:

Wage distribution:

• Men's group:
  
• Women's group:
  
• Children's group:
  

Hence, the group completes the job in days, and the wages are divided as to the men, to the women, and to the children.