Solving Data Sufficiency Problems: Finding Marks Using Averages

This question asks us to determine if the given statements are sufficient to find the marks of student X, given the total marks of X, Y, and Z.

We are given the total marks of X, Y, and Z:

$\qquad X + Y + Z = 180 \quad \text{(Equation 1)}$

We need to find the value of X using the information provided in the statements.

Analyzing Statement 1: Average of Y and Z together is 50 marks

Statement 1 provides the average of Y and Z's marks. The average is calculated as the sum of their marks divided by the number of students (which is 2).

From Statement 1:

$\qquad \frac{Y + Z}{2} = 50$

Multiplying both sides by 2 gives us the sum of Y and Z's marks:

$\qquad Y + Z = 50 \times 2$

$\qquad Y + Z = 100 \quad \text{(Equation 2)}$

Now, we can use Equation 1 ($X + Y + Z = 180$) and substitute the value of $(Y + Z)$ from Equation 2:

$\qquad X + (Y + Z) = 180$

$\qquad X + 100 = 180$

To find X, subtract 100 from both sides:

$\qquad X = 180 - 100$

$\qquad X = 80$

Using Statement 1 alone, we were able to find a unique value for X's marks (which is 80). Therefore, Statement 1 alone is sufficient to answer the question.

Analyzing Statement 2: Average of X and Z together is 60 marks

Statement 2 provides the average of X and Z's marks.

From Statement 2:

$\qquad \frac{X + Z}{2} = 60$

Multiplying both sides by 2 gives us the sum of X and Z's marks:

$\qquad X + Z = 60 \times 2$

$\qquad X + Z = 120 \quad \text{(Equation 3)}$

Now, let's use Equation 1 ($X + Y + Z = 180$) and substitute the value of $(X + Z)$ from Equation 3:

$\qquad (X + Z) + Y = 180$

$\qquad 120 + Y = 180$

To find Y, subtract 120 from both sides:

$\qquad Y = 180 - 120$

$\qquad Y = 60$

Using Statement 2 alone, we found the value of Y's marks (which is 60). However, we need to find X's marks. Equation 3 gives us $X + Z = 120$. This is one equation with two unknown variables (X and Z). There are multiple possible pairs of values for X and Z that would satisfy this equation while also keeping the total sum $X+Y+Z=180$ (since we found Y=60). For example, if X=50, then Z=70 ($50+70=120$) and $50+60+70=180$. If X=60, then Z=60 ($60+60=120$) and $60+60+60=180$. Since X can have multiple possible values based on Statement 2 alone, Statement 2 alone is not sufficient to find a unique value for X's marks.

Conclusion on Sufficiency

Based on our analysis:

• Statement 1 alone is sufficient to find X's marks.
• Statement 2 alone is not sufficient to find X's marks.

Therefore, Statement 1 alone is sufficient while Statement 2 alone is not sufficient.

Revision Table: Data Sufficiency Analysis

Additional Information: Understanding Data Sufficiency

Data sufficiency questions test your ability to determine whether the information given in the statements is enough to answer the question asked. You do not necessarily need to find the answer itself, but rather ascertain if the statements provide sufficient information to find a unique answer.

The standard options in data sufficiency questions are usually variations of the following:

• Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
• Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
• Both statements together are sufficient, but neither statement alone is sufficient.
• Each statement alone is sufficient.
• Statements 1 and 2 together are not sufficient.

Always analyze each statement independently first. If a statement alone is sufficient, you don't need to consider the other statement or both together for that specific case. If neither statement alone is sufficient, then you combine the information from both statements and check for sufficiency.