X and Y complete a task in 14 days, Y and Z in 18 days. X works 6 days, Y works 8 days, Z works 12 days to complete it. How many days for Z alone(approximate)?

The correct option is 22 days.

Let us solve this step-by-step to find the number of days required for Z alone to complete the task.

Step 1: Define the work rates.
Let the total work be represented as 1 unit.
Let the efficiency (work done per day) of X, Y, and Z be $W_x$, $W_y$, and $W_z$ respectively.

From the question, we are given:

X and Y together can complete the work in 14 days.
So, 1 day's work of (X + Y) is:

$W_x+W_y=\frac{1}{14}$

Y and Z together can complete the work in 18 days.
So, 1 day's work of (Y + Z) is:

$W_y+W_z=\frac{1}{18}$

Step 2: Express total work done by X, Y, and Z.
X works for 6 days, Y works for 8 days, and Z works for 12 days to finish the total work.
Total work equation:

$6W_x+8W_y+12W_z=1$

We can regroup the days worked to use the combined rates of (X + Y) and (Y + Z):

$6(W_x+W_y)+2W_y+12W_z=1$

$6(W_x+W_y)+2(W_y+W_z)+10W_z=1$

Step 3: Substitute the combined daily work rates.

$6\×\frac{1}{14}+2\×\frac{1}{18}+10W_z=1$

$\frac{3}{7}+\frac{1}{9}+10W_z=1$

Find a common denominator for $\frac{3}{7}$ and $\frac{1}{9}$, which is 63:

$\frac{27+7}{63}+10W_z=1$

$\frac{34}{63}+10W_z=1$

$10W_z=1-\frac{34}{63}$

$10W_z=\frac{29}{63}$

$W_z=\frac{29}{630}$

Step 4: Calculate the total time taken by Z alone.
The number of days required for Z alone to complete 1 unit of work is:

$\text{Days for Z}=\frac{1}{W_z}=\frac{630}{29}$

$\frac{630}{29}\≈21.72\text{days}$

Rounding off to the nearest whole number gives approximately 22 days.