The mean and median of the data a, b and c are 50 and 35 , where `a lt b lt c` . If `c - a = 55` , then find (b - a) .

Options

A. 8
B. 7
C. 3
D. 5

Explanation

To solve the problem step by step, we will use the information given about the mean, median, and the relationship between the variables \(a\), \(b\), and \(c\).

### Step 1: Understand the given information
We know:
- Mean of \(a\), \(b\), and \(c\) is 50.
- Median of \(a\), \(b\), and \(c\) is 35.
- The relationship \(c - a = 55\).
- The order \(a < b < c\).

### Step 2: Set up the equations
1. From the mean, we have:
\[ \text{Mean} = \frac{a + b + c}{3} = 50 \]
Therefore,
\[ a + b + c = 150 \quad \text{(Equation 1)} \]

2. Since the median is \(b\) and it is given as 35:
\[ b = 35 \quad \text{(Equation 2)} \]

3. From the relationship \(c - a = 55\):
\[ c = a + 55 \quad \text{(Equation 3)} \]

### Step 3: Substitute \(b\) into Equation 1
Substituting \(b = 35\) into Equation 1:
\[ a + 35 + c = 150 \]
This simplifies to:
\[ a + c = 115 \quad \text{(Equation 4)} \]

### Step 4: Substitute \(c\) from Equation 3 into Equation 4
Now substitute \(c = a + 55\) from Equation 3 into Equation 4:
\[ a + (a + 55) = 115 \]
This simplifies to:
\[ 2a + 55 = 115 \]

### Step 5: Solve for \(a\)
Subtract 55 from both sides:
\[ 2a = 115 - 55 \]
\[ 2a = 60 \]
Now divide by 2:
\[ a = 30 \quad \text{(Equation 5)} \]

### Step 6: Find \(b\) and \(c\)
Using Equation 2, we already know:
\[ b = 35 \]

Now substitute \(a = 30\) into Equation 3 to find \(c\):
\[ c = 30 + 55 = 85 \quad \text{(Equation 6)} \]

### Step 7: Calculate \(b - a\)
Now we can find \(b - a\):
\[ b - a = 35 - 30 = 5 \]

### Final Answer
Thus, the value of \(b - a\) is:
\[ \boxed{5} \]