The sum of three numbers is 396. If the ratio between the first and the second number is 7:11 and that between the second and the third number is 11:15, then the difference between the first and the third number is:

Options

A. 86
B. 94
C. 96
D. 85

Answer

To solve the problem step by step, we will first establish the relationships between the three numbers based on the given ratios and then find the values of each number.

### Step 1: Define the Variables
Let the first number be \( x \), the second number be \( y \), and the third number be \( z \).

### Step 2: Set Up the Ratios
According to the problem:
- The ratio between the first and second number is \( 7:11 \). This can be expressed as:
\[ \frac{x}{y} = \frac{7}{11} \implies x = \frac{7}{11}y \]

- The ratio between the second and third number is \( 11:15 \). This can be expressed as:
\[ \frac{y}{z} = \frac{11}{15} \implies z = \frac{15}{11}y \]

### Step 3: Write the Sum Equation
The sum of the three numbers is given as 396:
\[ x + y + z = 396 \]

### Step 4: Substitute the Values of \( x \) and \( z \)
Substituting the expressions for \( x \) and \( z \) in terms of \( y \):
\[ \frac{7}{11}y + y + \frac{15}{11}y = 396 \]

### Step 5: Combine Like Terms
To combine the terms, we can express \( y \) as \( \frac{11}{11}y \):
\[ \frac{7}{11}y + \frac{11}{11}y + \frac{15}{11}y = 396 \]
\[ \frac{7 + 11 + 15}{11}y = 396 \]
\[ \frac{33}{11}y = 396 \]
\[ 3y = 396 \]

### Step 6: Solve for \( y \)
Dividing both sides by 3:
\[ y = \frac{396}{3} = 132 \]

### Step 7: Find \( x \) and \( z \)
Now, we can find \( x \) and \( z \):
\[ x = \frac{7}{11}y = \frac{7}{11} \times 132 = 84 \]
\[ z = \frac{15}{11}y = \frac{15}{11} \times 132 = 180 \]

### Step 8: Calculate the Difference Between \( x \) and \( z \)
Now, we need to find the difference between the first and third numbers:
\[ \text{Difference} = z - x = 180 - 84 = 96 \]

### Final Answer
The difference between the first and the third number is **96**.

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