Evaluating Data Sufficiency for Project Completion Time

The question asks for the number of days required for P, Q, and R to complete a project working together. We need to determine if the given statements provide enough information.

Let the total amount of work required to complete the project be represented by 1 unit. Let the daily work rates of P, Q, and R be $p$, $q$, and $r$ units per day, respectively. The time taken to complete the project is calculated as $\frac{\text{Total Work}}{\text{Combined Work Rate}}$. We need to find the value of $\frac{1}{p+q+r}$.

Statement I Analysis

Statement I says that Q and R together can complete the project in 10 days.

This translates to the equation:

$(q + r) \times 10 = 1$

$\implies q + r = \frac{1}{10}$

This statement gives us the combined work rate of Q and R. However, it does not provide any information about P's work rate ($p$). With only $q+r$, we cannot determine $p+q+r$. Therefore, Statement I alone is not sufficient.

Statement II Analysis

Statement II says that P and R together can complete the project in 15 days.

This translates to the equation:

$(p + r) \times 15 = 1$

$\implies p + r = \frac{1}{15}$

This statement gives us the combined work rate of P and R. It does not provide any information about Q's work rate ($q$). With only $p+r$, we cannot determine $p+q+r$. Therefore, Statement II alone is not sufficient.

Combined Statements Analysis

Combining both statements, we have the following two equations:

1. $q + r = \frac{1}{10}$
2. $p + r = \frac{1}{15}$

We want to find $p+q+r$. Let's see if we can determine this sum from the two equations.

If we add the two equations:

$(q + r) + (p + r) = \frac{1}{10} + \frac{1}{15}$

$p + q + 2r = \frac{3}{30} + \frac{2}{30}$

$p + q + 2r = \frac{5}{30} = \frac{1}{6}$

This equation can be written as $(p + q + r) + r = \frac{1}{6}$.

We have one equation relating $p$, $q$, and $r$, but we have three unknown variables ($p$, $q$, $r$). We cannot determine the value of $p+q+r$ uniquely. For example, we don't know the individual rates $p$, $q$, or $r$. We need at least one more independent piece of information (like the rate of P alone, Q alone, R alone, or the combined rate of P and Q) to find the combined rate $p+q+r$.

Therefore, even when combined, the information in both statements is not sufficient to answer the question.

Conclusion

Neither statement alone nor both statements combined provide sufficient information to determine the number of days P, Q, and R will take to complete the project working together.