Analyzing Wage Data Sufficiency

The question asks for the average daily wage of X, Y, and Z. To find the average daily wage, we need to calculate the sum of their wages and divide by 3. That is, we need to find the value of $\frac{X + Y + Z}{3}$. This means we need to determine the value of the sum $X + Y + Z$. We will examine each statement separately to see if it provides enough information to find this sum.

Evaluating Statement 1: Y’s salary is half of (X + Z)

Statement 1 gives us a relationship between the wages of X, Y, and Z. We can write this relationship as an equation:

\( Y = \frac{1}{2}(X + Z) \)

Multiplying both sides by 2, we get:

\( 2Y = X + Z \)

Now let's consider the sum of the wages, $X + Y + Z$. We can group the terms $X$ and $Z$ together:

\( X + Y + Z = (X + Z) + Y \)

From Statement 1, we know that $X + Z = 2Y$. Substituting this into the sum equation:

\( X + Y + Z = (2Y) + Y \)

\( X + Y + Z = 3Y \)

So, the sum of the wages is equal to $3Y$. The average daily wage is $\frac{X+Y+Z}{3} = \frac{3Y}{3} = Y$. This means the average wage is equal to the wage of Y.

However, Statement 1 only gives a relationship ($2Y = X + Z$). It does not provide the actual value of Y (or X or Z). Without knowing the value of Y, we cannot determine the average wage.

For example, if Y = Rs. 100, then X + Z = Rs. 200, and the average is Rs. 100. If Y = Rs. 500, then X + Z = Rs. 1000, and the average is Rs. 500. Since Y can be any value consistent with some X and Z, the average wage cannot be uniquely determined.

Therefore, Statement 1 alone is insufficient to answer the question about the average daily wage.

Evaluating Statement 2: X and Y together earn Rs. 40 more than Z and Z earns Rs. 500

Statement 2 provides two pieces of information:

1. X and Y together earn Rs. 40 more than Z: \( X + Y = Z + 40 \)
2. Z earns Rs. 500: \( Z = 500 \)

From the second piece of information, we know the exact value of Z is Rs. 500. We can substitute this value into the first equation:

\( X + Y = 500 + 40 \)

\( X + Y = 540 \)

Now we have the sum of X and Y ($X+Y$) and the value of Z. To find the average daily wage, we need the sum $X + Y + Z$. We can write this sum as:

\( X + Y + Z = (X + Y) + Z \)

Substitute the values we found from Statement 2:

\( X + Y + Z = (540) + 500 \)

\( X + Y + Z = 1040 \)

The sum of the wages is Rs. 1040. Now we can calculate the average daily wage:

\( \text{Average Daily Wage} = \frac{X + Y + Z}{3} = \frac{1040}{3} \)

Since we can calculate a unique value for the average daily wage using the information in Statement 2 alone, Statement 2 is sufficient to answer the question.

Conclusion on Data Sufficiency

Based on our analysis:

• Statement 1 alone is insufficient because it only provides a relationship between the variables and does not allow us to find a unique value for the average wage.
• Statement 2 alone is sufficient because it provides the value of Z and the sum of X and Y, allowing us to calculate the total sum of wages $X + Y + Z$ and subsequently the unique average wage.

Therefore, only Statement 2 is sufficient to answer the question.

Revision Table: Average Daily Wage Sufficiency

This table summarizes the sufficiency of each statement in determining the average daily wage of X, Y, and Z.

Additional Information on Data Sufficiency and Averages

Data Sufficiency Questions: These questions test your ability to determine whether the given information is sufficient to answer a specific question, not necessarily to find the answer itself. You analyze each statement individually, and then possibly together, to see if the question can be uniquely answered.

Average Calculation: The average (mean) of a set of numbers is found by summing the numbers and dividing by the count of numbers. For the average daily wage of X, Y, and Z, the formula is $\frac{X+Y+Z}{3}$. To find the average, you need to determine the value of the sum $X+Y+Z$.

Solving Systems of Equations: Sometimes, data sufficiency problems involve setting up and potentially solving linear equations based on the given statements. If the number of independent equations equals the number of variables, you can usually find a unique solution for the variables. However, for sufficiency, you only need to determine if the target value (in this case, $X+Y+Z$) can be uniquely found.

In this problem, Statement 1 gave one equation with three variables, which is not enough to find unique values for X, Y, or Z, or their sum/average. Statement 2 provided information that directly led to finding the sum $X+Y+Z$, making it sufficient.