Calculating Combined Work Time for X, Y, and Z

This problem involves calculating the total time required for three individuals, X, Y, and Z, to complete a piece of work when working together, given their individual times.

Individual Work Rates

First, determine the rate at which each person works. The rate is the amount of work done per day, which is the reciprocal of the time taken to complete the entire work.

• X's rate: $\frac{1}{153}$ of the work per day.
• Y's rate: $\frac{1}{255}$ of the work per day.
• Z's rate: $\frac{1}{340}$ of the work per day.

Combined Work Rate Calculation

When working together, their rates add up. To find the combined rate, we sum their individual rates:

Combined Rate = X's rate + Y's rate + Z's rate

Combined Rate = $\frac{1}{153} + \frac{1}{255} + \frac{1}{340}$

To add these fractions, find the Least Common Multiple (LCM) of the denominators (153, 255, 340).

• Prime factorization:
  • $153 = 3^2 \times 17$
  • $255 = 3 \times 5 \times 17$
  • $340 = 2^2 \times 5 \times 17$
• LCM$(153, 255, 340) = 2^2 \times 3^2 \times 5 \times 17 = 4 \times 9 \times 5 \times 17 = 3060$.

Convert fractions to the common denominator:

• $\frac{1}{153} = \frac{1 \times 20}{153 \times 20} = \frac{20}{3060}$
• $\frac{1}{255} = \frac{1 \times 12}{255 \times 12} = \frac{12}{3060}$
• $\frac{1}{340} = \frac{1 \times 9}{340 \times 9} = \frac{9}{3060}$

Add the converted fractions:

Combined Rate = $\frac{20}{3060} + \frac{12}{3060} + \frac{9}{3060} = \frac{20 + 12 + 9}{3060} = \frac{41}{3060}$ of the work per day.

Total Time Calculation

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

Time Together = $\frac{1}{\text{Combined Rate}} = \frac{1}{\frac{41}{3060}} = \frac{3060}{41}$ days.

Final Answer Conversion

Convert the improper fraction $\frac{3060}{41}$ into a mixed number:

Divide 3060 by 41:

$3060 \div 41 = 74$ with a remainder of $26$.

So, the time taken is $74 \frac{26}{41}$ days.