This problem involves calculating the time taken by individuals and a group to complete a certain amount of work. We are given the time taken by X, Y, and Z to complete a fraction of the work and the time taken by Y to complete the remaining work after X and Z leave. We need to find the number of days (n) all three worked together.

Calculating Individual Work Rates

First, let's determine the rate at which each person completes the total work per day. The rate is the reciprocal of the time taken to complete the entire work.

• X's Work Rate: X completes $1/3$ of the work in 6 days. To complete the whole work (1 unit), X would take $6 \text{ days} \times 3 = 18$ days.

Therefore, X's daily work rate is $\frac{1}{18}$ of the total work.

• Y's Work Rate: Y completes $1/3$ of the work in 8 days. To complete the whole work (1 unit), Y would take $8 \text{ days} \times 3 = 24$ days.

Therefore, Y's daily work rate is $\frac{1}{24}$ of the total work.

• Z's Work Rate: Z completes $3/4$ of the work in 12 days. To complete the whole work (1 unit), Z would take $12 \text{ days} \times \frac{4}{3} = 4 \times 4 = 16$ days.

Therefore, Z's daily work rate is $\frac{1}{16}$ of the total work.

Combined Work Rate of X, Y, and Z

When X, Y, and Z work together, their daily work rates add up to find their combined daily work rate.

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

Combined rate = $\frac{1}{18} + \frac{1}{24} + \frac{1}{16}$

To add these fractions, we find the least common multiple (LCM) of 18, 24, and 16.

• 18 = $2 \times 3^2$
• 24 = $2^3 \times 3$
• 16 = $2^4$

LCM(18, 24, 16) = $2^4 \times 3^2 = 16 \times 9 = 144$.

Now, express each fraction with the denominator 144:

• $\frac{1}{18} = \frac{1 \times 8}{18 \times 8} = \frac{8}{144}$
• $\frac{1}{24} = \frac{1 \times 6}{24 \times 6} = \frac{6}{144}$
• $\frac{1}{16} = \frac{1 \times 9}{16 \times 9} = \frac{9}{144}$

Combined rate = $\frac{8}{144} + \frac{6}{144} + \frac{9}{144} = \frac{8 + 6 + 9}{144} = \frac{23}{144}$ of the total work per day.

Work Done in Different Phases

The work is completed in two phases:

1. X, Y, and Z work together for 'n' days.
2. Y alone finishes the remaining work in $82/3$ days.

Work done by X, Y, and Z together:

In 'n' days, the work done by all three is their combined rate multiplied by the number of days.

Work done together = Combined rate $\times$ n

Work done together = $\frac{23}{144} \times n = \frac{23n}{144}$

Work done by Y alone:

Y works alone for $82/3$ days. Let's convert $82/3$ to an improper fraction: $82/3 = \frac{(8 \times 3) + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}$ days.

Work done by Y alone = Y's rate $\times$ time worked by Y alone

Work done by Y alone = $\frac{1}{24} \times \frac{26}{3} = \frac{26}{24 \times 3} = \frac{26}{72}$

This fraction can be simplified by dividing the numerator and denominator by 2:

Work done by Y alone = $\frac{13}{36}$

Setting Up and Solving the Equation

The total work done is the sum of the work done in the two phases, which must equal the total work (1 unit).

Work done together + Work done by Y alone = Total work

$\frac{23n}{144} + \frac{13}{36} = 1$

To solve for 'n', isolate the term with 'n':

$\frac{23n}{144} = 1 - \frac{13}{36}$

To subtract the fractions on the right side, find a common denominator, which is 36.

$1 = \frac{36}{36}$

$\frac{23n}{144} = \frac{36}{36} - \frac{13}{36} = \frac{36 - 13}{36} = \frac{23}{36}$

So, we have:

$\frac{23n}{144} = \frac{23}{36}$

To solve for 'n', we can multiply both sides by 144:

$23n = \frac{23}{36} \times 144$

$23n = 23 \times \frac{144}{36}$

Since $144 / 36 = 4$, the equation becomes:

$23n = 23 \times 4$

Divide both sides by 23:

$n = \frac{23 \times 4}{23}$

$n = 4$

Thus, X, Y, and Z worked together for 4 days.

Summary of Calculations

Combined Daily Rate (X+Y+Z) = $\frac{1}{18} + \frac{1}{24} + \frac{1}{16} = \frac{8+6+9}{144} = \frac{23}{144}$

Let 'n' be the number of days X, Y, and Z worked together.

Work done by X, Y, Z in n days = $n \times \frac{23}{144} = \frac{23n}{144}$

Time Y worked alone = $82/3$ days = $26/3$ days

Work done by Y alone = $\frac{1}{24} \times \frac{26}{3} = \frac{13}{36}$

Total work = Work done together + Work done by Y alone = 1

$\frac{23n}{144} + \frac{13}{36} = 1$

$\frac{23n}{144} = 1 - \frac{13}{36} = \frac{36-13}{36} = \frac{23}{36}$

$23n = \frac{23}{36} \times 144$

$23n = 23 \times 4$

$n = 4$

Conclusion

Based on the calculations, the number of days 'n' that X, Y, and Z worked together is 4.

Revision Table: Work and Time Concepts

Additional Information on Work and Time Problems

Work and time problems often involve calculating how long it takes individuals or groups to complete tasks. The key is to convert the given information into daily (or hourly) work rates. If someone completes a fraction of work in a certain time, you can find the time taken for the whole work and thus their rate.

• If a person can do a piece of work in 'd' days, their rate is $1/d$ work per day.
• If multiple people work together, their rates are usually added.
• The total work done is often treated as '1' unit. If only a fraction of the work is completed, that fraction is used.
• Problems can involve phases where different people work or the group changes. Calculate the work done in each phase and sum them up to the total work.
• Ensure units are consistent (e.g., days and work per day).

These problems are fundamentally about setting up equations based on the fraction of work done and the time taken, using the relationship: Work Done = Rate $\times$ Time.