Each of these questions consists of a question followed by informations in three statements. You have to study the question and the statements and decide that information in which of the statement(s) is/are necessary to answer the question.
What is the difference between two numbers X and Y?
(A) X is 20 percent more than another number Z
(B) Y is 20 percent less than Z
(C) The sum of Y and Z is 72 .

Options

A. Only A and B
B. Only A and C
C. All A ,B and C
D. Any two

Answer

To solve the question "What is the difference between two numbers X and Y?" using the provided statements, we will analyze each statement step by step.

### Step 1: Analyze Statement A
Statement A states that "X is 20 percent more than another number Z."
- This can be expressed mathematically as:
\[ X = Z + 0.2Z = 1.2Z \]
- This gives us a relationship between X and Z, but does not provide any information about Y. Therefore, Statement A alone is **not sufficient** to determine the difference \(X - Y\).

### Step 2: Analyze Statement B
Statement B states that "Y is 20 percent less than Z."
- This can be expressed mathematically as:
\[ Y = Z - 0.2Z = 0.8Z \]
- This gives us a relationship between Y and Z, but still does not provide any information about X. Therefore, Statement B alone is **not sufficient** to determine the difference \(X - Y\).

### Step 3: Analyze Statement C
Statement C states that "The sum of Y and Z is 72."
- We can express this as:
\[ Y + Z = 72 \]
- However, without knowing either X or Y in terms of Z, we cannot find the difference \(X - Y\). Thus, Statement C alone is also **not sufficient**.

### Step 4: Combine Statements A and B
From Statement A, we know:
\[ X = 1.2Z \]
From Statement B, we know:
\[ Y = 0.8Z \]
Now, we can find the difference \(X - Y\):
\[ X - Y = 1.2Z - 0.8Z = 0.4Z \]
At this point, we have expressed the difference in terms of Z, but we still need the value of Z to compute \(X - Y\).

### Step 5: Use Statement C with A and B
Now, we can use Statement C:
\[ Y + Z = 72 \]
Substituting \(Y\) from Statement B:
\[ 0.8Z + Z = 72 \]
This simplifies to:
\[ 1.8Z = 72 \]
Now, solving for Z:
\[ Z = \frac{72}{1.8} = 40 \]

### Step 6: Calculate X and Y
Now that we have Z, we can find X and Y:
- From Statement A:
\[ X = 1.2Z = 1.2 \times 40 = 48 \]
- From Statement B:
\[ Y = 0.8Z = 0.8 \times 40 = 32 \]

### Step 7: Find the Difference
Now we can find the difference \(X - Y\):
\[ X - Y = 48 - 32 = 16 \]

### Conclusion
To answer the question "What is the difference between two numbers X and Y?", we find that all three statements (A, B, and C) are necessary to arrive at the solution.