Unit 1: Time, Work and Cisterns - Subjective Questions

Definitions:

• Work is the total task to be completed.
• Rate of work is the fraction of the task completed per unit time.
• Efficiency is the capacity of a person or machine to perform work in a given time.

If a person completes work in days, then:

Therefore:

When the total work is treated as one unit:

Efficiency is proportional to the rate of work and inversely proportional to the time taken:

Thus, a more efficient worker takes less time to complete the same work.

Step 1: Find individual one-day work.

• A's one-day work is .
• B's one-day work is .

Step 2: Add their rates.

Together, they complete of the work per day.

Step 3: Calculate the required time.

Therefore, A and B together complete the work in days.

• A's one-day work is .
• The one-day work of A and B together is .

Therefore, B's one-day work is:

Hence, the time required by B alone is:

Therefore, B alone can complete the work in 10 days.

For the same amount of work, efficiency and time are inversely proportional:

Given:

Therefore:

Since A takes 10 days:

Their combined one-day work is:

Thus, their combined time is:

Result: B takes 15 days, and together they take 6 days.

Work completed in one two-day cycle:

In 7 complete cycles, or 14 days, the work completed is:

The remaining work is:

On the 15th day, A works at the rate per day. Time taken by A to complete the remainder is:

Therefore, the total time is:

Hence, the work is completed in days.

For a fixed amount of work, the number of workers and the number of days are inversely proportional.

Total work in man-days is:

If 20 men perform the work, the required number of days is:

Alternatively:

Therefore, 20 men can complete the work in 9 days.

Given:

Therefore:

The equivalent strength of 9 women is:

Thus, 8 men and 9 women are equivalent to:

Total work is:

Time required by 14 equivalent men is:

Therefore, 8 men and 9 women complete the work in 15 days.

Given:

Express every worker in terms of men:

• 8 women are equivalent to men.
• 12 children are equivalent to men.

Total equivalent workforce is:

Total work is:

Required time is:

Therefore, the combined group completes the work in 14 days.

When workers work for the same duration, wages are divided in the ratio of their efficiencies or rates of work.

A's and B's rates are:

Multiplying by 36 gives:

Total ratio units are:

A's share is:

B's share is:

Therefore:

• A receives ₹5,400.
• B receives ₹3,600.

Step 1: Calculate total work.

Step 2: Calculate work completed in the first 4 days.

Remaining work is:

After 5 men leave, 10 men remain. Additional time required is:

Thus, total time is:

Step 3: Divide wages according to days worked.

The wage per man-day is:

Each of the 10 men who stays works for 16 days and receives:

Each of the 5 men who leaves works for 4 days and receives:

Therefore, the work takes 16 days; each continuing man earns ₹4,800, and each departing man earns ₹1,200.

Pipes and cistern problems use the same rate principle as time-and-work problems. The tank represents the total work, usually treated as one unit.

• An inlet pipe fills the tank and performs positive work.
• An outlet pipe empties the tank and performs negative work.

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

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

The net rate is the algebraic sum of all pipe rates:

The required time is:

A positive net rate fills the tank, whereas a negative net rate empties it.

The rates of the two inlet pipes are:

Their combined rate is:

Thus, they fill of the tank in one hour.

The time required to fill the whole tank is:

Therefore, both pipes together fill the tank in hours, or 7 hours 12 minutes.

• Inlet rate is tank per hour.
• Outlet rate is tank per hour.

The net filling rate is:

Therefore, the required filling time is:

Hence, the tank will be filled in 30 hours when both pipes remain open.

The net rate is:

Taking 60 as the common denominator:

Thus, of the tank is filled per hour.

The time required is:

Therefore, all three pipes together fill the tank in 10 hours.

The pipe fills the whole tank in 8 hours. Therefore, its hourly rate is:

Let the time required to fill of the tank be hours. Then:

Therefore:

Since the rate is constant, the filling time is directly proportional to the fraction of the tank filled.

Thus, the required time is hours, or 4 hours 48 minutes.

Stage 1: Inlet operating alone.

In 2 hours, the inlet fills:

The unfilled portion is:

Stage 2: Both pipes operating.

Their net rate is:

Time to fill the remaining is:

Therefore, total time is:

Hence, the tank is filled in 18 hours from the time the inlet was first opened.

The inlet's filling rate without the leak is:

The net filling rate when the leak is present is:

Therefore, the leak's emptying rate is:

Hence, the leak alone would empty the tank in:

Therefore, the leak can empty the full tank in 30 hours.

Since the tank is initially full, the outlets remove water while the inlet adds water.

The net emptying rate is:

Using 72 as the common denominator:

Thus, of the tank is emptied per hour.

The required time is:

Therefore, the full tank becomes empty in hours.

Case 1: Two workers or two inlets

Their individual rates are and . The combined rate is:

Therefore, the combined time is:

Case 2: One inlet and one outlet

If the inlet fills in hours and the outlet empties in hours, the net filling rate is:

Provided , the inlet is faster than the outlet and the tank fills. The filling time is:

Thus:

• For two positive rates: .
• For an inlet and an outlet: , provided .

Step 1: Determine relative efficiency.

Since 8 men and 12 women complete the same work in equal time:

Therefore:

Assign efficiency units of 3 to each man and 2 to each woman.

Step 2: Calculate total work.

The initial team's daily rate is:

Work completed in 8 days is:

Remaining work is:

After 3 women join, the daily rate becomes:

Additional time required is:

Total time is:

Step 3: Divide the wages.

Contribution of the 4 original men is:

Contribution of the 6 original women is:

Contribution of the 3 additional women is:

Their contribution ratio is:

Therefore, the wage shares are:

• Original men:
• Original women:
• Additional women:

Hence, the work takes days, and the three groups receive ₹27,040, ₹27,040, and ₹8,320, respectively.