Solving Work and Time: Job Completion Calculation

This problem requires us to calculate the total duration needed to finish a job based on individual work speeds and a specific work schedule. We need to find the total days until the job is fully completed.

Understanding Work Rate

Key concepts for solving this type of problem:

• Work Rate: The amount of job a person can complete in one day. If someone takes 'x' days to finish a job, their work rate is $1/x$ of the job per day.
• Total Work: We assume the total job is equivalent to 1 unit.

Calculating Individual Work Rates

We first determine the rate at which A and B work individually:

• A completes the job in 12 days. A's work rate is $1/12$ of the job per day.
• B completes the job in 18 days. B's work rate is $1/18$ of the job per day.

Work Done by A Initially

A starts the job and works alone for 3 days. Let's calculate the fraction of the job A completes during this period:

Work done by A = (A's daily rate) $\times$ (Number of days A worked alone)

Work done by A = $\frac{1}{12} \times 3$

Work done by A = $\frac{3}{12} = \frac{1}{4}$

After 3 days, $1/4$ of the job is completed by A.

Calculating Remaining Work

The next step is to find out how much of the job is left after A's initial work:

Remaining Work = Total Work - Work done by A

Remaining Work = $1 - \frac{1}{4}$

Remaining Work = $\frac{3}{4}$

So, $3/4$ of the job still needs to be completed.

Calculating Combined Work Rate

From the 4th day, A and B work together. To find the time they take together, we first calculate their combined daily work rate:

Combined Rate = A's Rate + B's Rate

Combined Rate = $\frac{1}{12} + \frac{1}{18}$

Find the least common multiple (LCM) of 12 and 18, which is 36, to add the fractions:

Combined Rate = $\frac{3}{36} + \frac{2}{36}$

Combined Rate = $\frac{5}{36}$

When working together, A and B complete $5/36$ of the job each day.

Time Taken to Finish Remaining Work

We can now calculate the number of days A and B will take to complete the remaining $3/4$ of the job working together:

Time = Remaining Work / Combined Rate

Time = $\frac{3/4}{5/36}$

To divide fractions, we multiply by the reciprocal of the divisor:

Time = $\frac{3}{4} \times \frac{36}{5}

Time = $\frac{3 \times 36}{4 \times 5} = \frac{108}{20}

Simplifying the fraction:

Time = $\frac{27}{5}$ days

As a decimal, this is $5.4$ days.

Total Days to Complete the Job

The total time to finish the job is the sum of the initial days A worked alone and the days A and B worked together:

Total Days = Days A worked alone + Days A and B worked together

Total Days = $3 + 5.4$

Total Days = $8.4$ days.

The calculation indicates that the job will be completed in $8.4$ days.

Comparing this result to the options:

• 1. 8 days
• 2. 11 days
• 3. 12 days
• 4. 10 days

The calculated total time is $8.4$ days. Option 1 (8 days) is the closest provided choice.