To find the time taken when A, B, and C work together, we first determine their individual work rates and then calculate their combined rate.

Individual Work Rates

• A finishes the job in 146 days. Rate of A = $\frac{1}{146}$ job per day.
• B finishes the job in 219 days. Rate of B = $\frac{1}{219}$ job per day.
• C finishes the job in 292 days. Rate of C = $\frac{1}{292}$ job per day.

Combined Work Rate Calculation

The combined rate is the sum of individual rates:

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

Combined Rate = $\frac{1}{146} + \frac{1}{219} + \frac{1}{292}$

To add these fractions, we find the Least Common Multiple (LCM) of 146, 219, and 292.

• Prime factorization:
  • $146 = 2 \times 73$
  • $219 = 3 \times 73$
  • $292 = 2^2 \times 73$
• LCM(146, 219, 292) = $2^2 \times 3 \times 73 = 4 \times 3 \times 73 = 876$

Now, we express each rate with the common denominator 876:

• Rate of A = $\frac{1}{146} = \frac{1 \times 6}{146 \times 6} = \frac{6}{876}$
• Rate of B = $\frac{1}{219} = \frac{1 \times 4}{219 \times 4} = \frac{4}{876}$
• Rate of C = $\frac{1}{292} = \frac{1 \times 3}{292 \times 3} = \frac{3}{876}$

Combined Rate = $\frac{6}{876} + \frac{4}{876} + \frac{3}{876} = \frac{6 + 4 + 3}{876} = \frac{13}{876}$ job per day.

Total Time Calculation

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

Time = $\frac{1}{\text{Combined Rate}} = \frac{1}{\frac{13}{876}} = \frac{876}{13}$ days.

Convert to Mixed Number

To express $\frac{876}{13}$ as a mixed number:

$876 \div 13 = 67$ with a remainder of $5$.

Therefore, the time taken is $67 \frac{5}{13}$ days.