Understanding the Work and Time Problem

The question describes a scenario where a worker completes a certain fraction of a task in a given number of days. We need to find out how many days it will take the same worker to complete a different fraction of the same task, assuming their work rate remains constant.

Problems involving work and time often rely on the concept of the rate of work, which is the amount of work done per unit of time.

Calculating the Worker's Rate of Work

We are told that the worker completes \(\frac{3}{5}\) of the total work in 12 days.

The rate of work can be calculated as:

Rate = \(\frac{\text{Amount of Work Done}}{\text{Time Taken}}\)

In this case, the amount of work done is \(\frac{3}{5}\) and the time taken is 12 days.

Rate = \(\frac{\frac{3}{5}}{12}\) work per day

To simplify this fraction, we can write it as multiplication:

Rate = \(\frac{3}{5} \times \frac{1}{12}\) work per day

Rate = \(\frac{3 \times 1}{5 \times 12}\) work per day

Rate = \(\frac{3}{60}\) work per day

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

Rate = \(\frac{3 \div 3}{60 \div 3}\) work per day

Rate = \(\frac{1}{20}\) work per day

So, the worker completes \(\frac{1}{20}\) of the total work every day.

Calculating Time to Complete \(\frac{3}{4}\) of the Work

Now we want to find out how many days it will take the worker to complete \(\frac{3}{4}\) of the work at the rate of \(\frac{1}{20}\) work per day.

The relationship between time, work, and rate is:

Time = \(\frac{\text{Amount of Work to be Done}}{\text{Rate of Work per Day}}\)

Here, the amount of work to be done is \(\frac{3}{4}\) and the rate is \(\frac{1}{20}\) work per day.

Time = \(\frac{\frac{3}{4}}{\frac{1}{20}}\) days

To divide by a fraction, we multiply by its reciprocal:

Time = \(\frac{3}{4} \times \frac{20}{1}\) days

Time = \(\frac{3 \times 20}{4 \times 1}\) days

Time = \(\frac{60}{4}\) days

Now, we perform the division:

Time = 15 days

Therefore, the worker will complete \(\frac{3}{4}\) of the work in 15 days.

Summary of the Solution Steps

1. Identify the given information: Worker completes \(\frac{3}{5}\) of work in 12 days.
2. Identify the target: Find the time to complete \(\frac{3}{4}\) of the work.
3. Calculate the worker's daily rate: Rate = \(\frac{\text{Work Done}}{\text{Time Taken}}\) = \(\frac{3/5}{12} = \frac{1}{20}\) work per day.
4. Calculate the time for the target work: Time = \(\frac{\text{Target Work}}{\text{Rate}}\) = \(\frac{3/4}{1/20}\) = \(\frac{3}{4} \times 20 = 15\) days.

Final Answer

The worker will complete \(\frac{3}{4}\) of the work in 15 days.

Revision Table: Key Work and Time Concepts

Additional Information: Proportionality in Work Problems

Work and Time problems with a constant rate can also be solved using proportionality. If a worker completes \(W_1\) amount of work in \(T_1\) time, and \(W_2\) amount of work in \(T_2\) time, assuming a constant rate, the ratio of work to time is constant:

\(\frac{W_1}{T_1} = \frac{W_2}{T_2}\)

In this problem:

• \(W_1 = \frac{3}{5}\)
• \(T_1 = 12\) days
• \(W_2 = \frac{3}{4}\)
• \(T_2 = ?\)

Setting up the proportion:

\(\frac{3/5}{12} = \frac{3/4}{T_2}\)

This simplifies to:

\(\frac{3}{5 \times 12} = \frac{3}{4 \times T_2}\)

\(\frac{3}{60} = \frac{3}{4 \times T_2}\)

Simplifying the left side:

\(\frac{1}{20} = \frac{3}{4 \times T_2}\)

Cross-multiply:

\(1 \times (4 \times T_2) = 3 \times 20\)

\(4 \times T_2 = 60\)

Divide by 4:

\(T_2 = \frac{60}{4}\)

\(T_2 = 15\)

This confirms that it takes 15 days to complete \(\frac{3}{4}\) of the work.