X can complete a project in 14 days, Y in 10 days. When Z joins them, the work is finished in 3.5 days. How many days would Z need alone?

The correct answer is 8.75 days.

Let's solve the problem step-by-step to find out how long Z would take to complete the project alone.

Step 1: Determine the work rates of X, Y, and the combined group (X + Y + Z).

Let the total work of the project be represented as 1 unit.

Work done by X in 1 day = $\frac{1}{14}$

Work done by Y in 1 day = $\frac{1}{10}$

When X, Y, and Z work together, they complete the project in 3.5 days.

3.5 days can be written as the fraction $\frac{7}{2}$ days.

Combined work done by (X + Y + Z) in 1 day = $\frac{1}{7/2}=\frac{2}{7}$

Step 2: Calculate the work done by Z in 1 day.

The combined 1-day work rate is the sum of their individual 1-day work rates:

$\text{Work rate of Z}=\text{Work rate of (X + Y + Z)}-\text{Work rate of X}-\text{Work rate of Y}$

$\text{Work rate of Z}=\frac{2}{7}-\frac{1}{14}-\frac{1}{10}$

Find a common denominator for the fractions 7, 14, and 10. The Least Common Multiple (LCM) of 7, 14, and 10 is 70.

Convert each fraction to have a denominator of 70:

$\frac{2}{7}=\frac{2\times 10}{70}=\frac{20}{70}$

$\frac{1}{14}=\frac{1\times 5}{70}=\frac{5}{70}$

$\frac{1}{10}=\frac{1\times 7}{70}=\frac{7}{70}$

Now, subtract the work rates:

$\text{Work rate of Z}=\frac{20}{70}-\frac{5}{70}-\frac{7}{70}=\frac{20-5-7}{70}=\frac{8}{70}=\frac{4}{35}$

Step 3: Calculate the total time taken by Z alone.

The total time required for Z to finish the project alone is the reciprocal of Z's 1-day work rate:

$\text{Time taken by Z}=\frac{35}{4}\text{days}$

Dividing 35 by 4 gives:

$\frac{35}{4}=8.75\text{days}$

Thus, Z alone would need 8.75 days to complete the project.