Solving Work and Time Problems Together

This problem involves calculating the time required to complete a job when multiple individuals work together.

Individual Work Rates

First, determine the rate at which each person works. The rate is the reciprocal of the time taken to complete the job.

• Rate of A = $ \frac{1}{102} $ job per day
• Rate of B = $ \frac{1}{153} $ job per day
• Rate of C = $ \frac{1}{204} $ job per day

Combined Work Rate Calculation

To find the combined rate when they work together, add their individual rates.

Combined Rate = Rate of A + Rate of B + Rate of C

Combined Rate = $ \frac{1}{102} + \frac{1}{153} + \frac{1}{204} $

To add these fractions, find the Least Common Multiple (LCM) of the denominators (102, 153, 204).

• $ 102 = 2 \times 3 \times 17 $
• $ 153 = 3^2 \times 17 $
• $ 204 = 2^2 \times 3 \times 17 $
• LCM = $ 2^2 \times 3^2 \times 17 = 4 \times 9 \times 17 = 612 $

Now, express each fraction with the LCM as the denominator:

Combined Rate = $ \frac{612/102}{612} + \frac{612/153}{612} + \frac{612/204}{612} $

Combined Rate = $ \frac{6}{612} + \frac{4}{612} + \frac{3}{612} $

Combined Rate = $ \frac{6 + 4 + 3}{612} = \frac{13}{612} $ job per day.

Total Time for Combined Work

The total time required to complete the job together is the reciprocal of the combined rate.

Time = $ \frac{1}{\text{Combined Rate}} $

Time = $ \frac{1}{13/612} = \frac{612}{13} $ days.

Convert the improper fraction to a mixed number:

$ \frac{612}{13} = 47 \frac{1}{13} $ days.

Therefore, working together, A, B, and C will require $ 47 \frac{1}{13} $ days to complete the job.