Question

If 1,200 employees, males and females, requested a raise, how many requests were granted?
(1) \(\frac{5}{12}\) of the males and \(\frac{7}{12}\) of the females had their request granted.
(2) 200 of the requests made by males were granted.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

This is a classic weighted average or mixture problem. Statement (1) gives the proportions but not the weights (number of males/females). Statement (2) gives a piece of absolute data. Combining them often allows you to find the weights and solve the problem.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Let \(M\) be the number of male employees and \(F\) be the number of female employees who requested a raise. We are given that \(M + F = 1200\).
Let \(G_M\) be the number of males whose requests were granted, and \(G_F\) be the number of females whose requests were granted.
The question asks for the total number of requests granted, which is \(G_{Total} = G_M + G_F\).
Step 2: Detailed Explanation:
Analyzing Statement (1):
This statement provides the fractions of males and females whose requests were granted:
\[ G_M = \frac{5}{12} M \quad \text{and} \quad G_F = \frac{7}{12} F \]
The total number of granted requests is:
\[ G_{Total} = \frac{5}{12} M + \frac{7}{12} F \]
Since we know \(F = 1200 - M\), we can substitute this into the equation:
\[ G_{Total} = \frac{5}{12} M + \frac{7}{12} (1200 - M) = \frac{5M}{12} + 700 - \frac{7M}{12} = 700 - \frac{2M}{12} = 700 - \frac{M}{6} \]
The total number of granted requests depends on the number of males, \(M\). Since \(M\) is unknown, we cannot find a unique value for \(G_{Total}\). Therefore, statement (1) alone is not sufficient.
Analyzing Statement (2):
This statement tells us the number of males whose requests were granted:
\[ G_M = 200 \]
The total number of granted requests is \(G_{Total} = G_M + G_F = 200 + G_F\).
This statement provides no information about \(G_F\), the number of females whose requests were granted. Therefore, statement (2) alone is not sufficient.
Combining Statements (1) and (2):
From statement (2), we know \(G_M = 200\).
From statement (1), we have the relationship \(G_M = \frac{5}{12} M\).
We can combine these to find the number of males, \(M\):
\[ 200 = \frac{5}{12} M \implies M = \frac{200 \times 12}{5} = 40 \times 12 = 480 \]
So, there were 480 males who requested a raise.
Now we can find the number of females, \(F\):
\[ F = 1200 - M = 1200 - 480 = 720 \]
Using the information from statement (1) again, we can find the number of females whose requests were granted:
\[ G_F = \frac{7}{12} F = \frac{7}{12} (720) = 7 \times 60 = 420 \]
Finally, we can find the total number of granted requests:
\[ G_{Total} = G_M + G_F = 200 + 420 = 620 \]
Since we can find a unique value, the statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but when combined, they provide enough information to answer the question. Therefore, the correct option is (C).