Calculating Work and Time for A, B, and C

This problem involves calculating the individual work rates of A, B, and C based on their combined work times, and then determining the time A takes to complete a specific percentage of the total work.

Step-by-Step Solution

Let's denote the total work as 1 unit. The work rate of a person or a group is the amount of work they can complete in one day. If a group completes the work in \(d\) days, their combined daily work rate is \(\frac{1}{d}\).

• A and B together can do the work in 36 days.
• So, their combined daily work rate is \(\frac{1}{36}\).

In mathematical terms:

$$ \text{Rate}(A+B) = \frac{1}{36} \quad (1) $$

• B and C together can do the work in 30 days.
• So, their combined daily work rate is \(\frac{1}{30}\).

In mathematical terms:

$$ \text{Rate}(B+C) = \frac{1}{30} \quad (2) $$

• A, B, and C together can complete the same work in 24 days.
• So, their combined daily work rate is \(\frac{1}{24}\).

In mathematical terms:

$$ \text{Rate}(A+B+C) = \frac{1}{24} \quad (3) $$

Calculating Individual Work Rates

We can find the individual work rate of each person by subtracting the combined rates.

• To find A's daily work rate, we subtract the combined rate of B and C from the combined rate of A, B, and C:

$$ \text{Rate}(A) = \text{Rate}(A+B+C) - \text{Rate}(B+C) $$

Substitute the values from equations (3) and (2):

$$ \text{Rate}(A) = \frac{1}{24} - \frac{1}{30} $$

Find a common denominator, which is 120:

$$ \text{Rate}(A) = \frac{5}{120} - \frac{4}{120} = \frac{5-4}{120} = \frac{1}{120} $$

So, A's daily work rate is \(\frac{1}{120}\). This means A alone can complete the entire work in 120 days.

• To find C's daily work rate, we subtract the combined rate of A and B from the combined rate of A, B, and C:

$$ \text{Rate}(C) = \text{Rate}(A+B+C) - \text{Rate}(A+B) $$

Substitute the values from equations (3) and (1):

$$ \text{Rate}(C) = \frac{1}{24} - \frac{1}{36} $$

Find a common denominator, which is 72:

$$ \text{Rate}(C) = \frac{3}{72} - \frac{2}{72} = \frac{3-2}{72} = \frac{1}{72} $$

So, C's daily work rate is \(\frac{1}{72}\). C alone can complete the entire work in 72 days.

• To find B's daily work rate, we can subtract A's rate from the combined rate of A and B:

$$ \text{Rate}(B) = \text{Rate}(A+B) - \text{Rate}(A) $$

Substitute the values from equation (1) and A's calculated rate:

$$ \text{Rate}(B) = \frac{1}{36} - \frac{1}{120} $$

Find a common denominator, which is 120 or 360 (LCM of 36 and 120 is 360):

$$ \text{Rate}(B) = \frac{10}{360} - \frac{3}{360} = \frac{10-3}{360} = \frac{7}{360} $$

So, B's daily work rate is \(\frac{7}{360}\). B alone can complete the entire work in \(\frac{360}{7}\) days.

Calculating Time for A to Complete 33 \(\frac{1}{3}\) % of Work

The question asks for the time A takes to complete 33 \(\frac{1}{3}\) % of the same work.

First, convert the percentage to a fraction:

$$ 33 \frac{1}{3} \% = \frac{100}{3} \% = \frac{100}{3} \times \frac{1}{100} = \frac{1}{3} $$

So, A needs to complete \(\frac{1}{3}\) of the total work.

We know that A takes 120 days to complete the full work (which is 1 unit of work).

The time taken to complete a fraction of the work is (Fraction of work) \(\times\) (Time to complete full work).

Time taken by A to complete \(\frac{1}{3}\) of the work = \(\frac{1}{3} \times 120\) days.

$$ \text{Time} = \frac{1}{3} \times 120 = \frac{120}{3} = 40 $$

So, A alone can complete 33 \(\frac{1}{3}\) % of the work in 40 days.

Final Answer Calculation

The time taken by A alone to complete 33 \(\frac{1}{3}\) % of the work is 40 days.

Revision Table: Work and Time Concepts

Additional Information: Understanding Work Efficiency

Work and time problems often involve the concept of efficiency. Efficiency is directly proportional to the work rate. A person with higher efficiency completes more work in the same amount of time or takes less time to complete the same amount of work.

• If A is twice as efficient as B, then A's work rate is twice B's work rate.
• If A's rate = \(2 \times\) B's rate, and B takes \(x\) days to complete the work, then A takes \(\frac{x}{2}\) days to complete the same work. Time taken is inversely proportional to efficiency (and work rate).

In this problem, we calculated the individual daily work rates:

• A's rate = \(\frac{1}{120}\)
• B's rate = \(\frac{7}{360}\)
• C's rate = \(\frac{1}{72}\)

Comparing the rates by finding a common denominator (360):

• A's rate = \(\frac{3}{360}\)
• B's rate = \(\frac{7}{360}\)
• C's rate = \(\frac{5}{360}\)

This shows that among A, B, and C, B is the most efficient worker, followed by C, and then A is the least efficient. This aligns with A taking the longest time (120 days) to complete the full work compared to C (72 days) and B (approximately 51.4 days).