If three successive positive integers a,b,c are such that `a lt b lt c`, then which of the following options will be correct?

Options

A. `(a+b)/(a+b+c)gt (b+c)/(a+b+c) gt (c+a)/(a+b+c)`
B. `(c+a)/(a+b+c)gt (b+c)/(a+b+c) gt (a+b)/(a+b+c)`
C. `(b+c)/(a+b+c)gt (c+a)/(a+b+c) gt (a+b)/(a+b+c)`
D. `(b+c)/(a+b+c)gt (a+b)/(a+b+c) gt (c+a)/(a+b+c)`

Answer

To solve the problem, we need to analyze the three successive positive integers \( a, b, c \) where \( a < b < c \).

### Step 1: Define the integers
Let’s define the three successive positive integers:
- Let \( a = n \)
- Then \( b = n + 1 \)
- And \( c = n + 2 \)

### Step 2: Write the expressions
Now, we will write the expressions based on the problem statement:
1. \( \frac{a + b}{a + b + c} \)
2. \( \frac{b + c}{a + b + c} \)
3. \( \frac{c + a}{a + b + c} \)

### Step 3: Calculate each expression
Now we will calculate each expression step by step.

#### Expression 1: \( \frac{a + b}{a + b + c} \)
- \( a + b = n + (n + 1) = 2n + 1 \)
- \( a + b + c = n + (n + 1) + (n + 2) = 3n + 3 \)
- So, \( \frac{a + b}{a + b + c} = \frac{2n + 1}{3n + 3} \)

#### Expression 2: \( \frac{b + c}{a + b + c} \)
- \( b + c = (n + 1) + (n + 2) = 2n + 3 \)
- So, \( \frac{b + c}{a + b + c} = \frac{2n + 3}{3n + 3} \)

#### Expression 3: \( \frac{c + a}{a + b + c} \)
- \( c + a = (n + 2) + n = 2n + 2 \)
- So, \( \frac{c + a}{a + b + c} = \frac{2n + 2}{3n + 3} \)

### Step 4: Compare the expressions
Now we need to compare the three fractions:
1. \( \frac{2n + 1}{3n + 3} \)
2. \( \frac{2n + 3}{3n + 3} \)
3. \( \frac{2n + 2}{3n + 3} \)

Since the denominators are the same, we can compare the numerators directly:
- \( 2n + 1 \)
- \( 2n + 3 \)
- \( 2n + 2 \)

### Step 5: Determine the largest and smallest
From the numerators:
- \( 2n + 3 \) is the largest
- \( 2n + 1 \) is the smallest

Thus, the order from largest to smallest is:
1. \( \frac{2n + 3}{3n + 3} \) (largest)
2. \( \frac{2n + 2}{3n + 3} \)
3. \( \frac{2n + 1}{3n + 3} \) (smallest)

### Conclusion
The correct option based on the analysis is the one corresponding to \( \frac{b + c}{a + b + c} \) being the largest.