`a: (b +c) ::1:3 `
`c: (a+b) ::5:7`
find `b: (a+c) = ?`

Options

A. `1:2`
B. `2:1`
C. `3:2`
D. `5:3`

Answer

To solve the problem step by step, we will analyze the given ratios and derive the required ratio \( b : (a + c) \).

### Step 1: Understand the given ratios
We have two ratios:
1. \( a : (b + c) = 1 : 3 \)
2. \( c : (a + b) = 5 : 7 \)

From these ratios, we can express \( a \), \( b + c \), \( c \), and \( a + b \) in terms of a common variable.

### Step 2: Express variables in terms of a common variable
From the first ratio \( a : (b + c) = 1 : 3 \):
- Let \( a = 1k \) (where \( k \) is a common multiplier)
- Then \( b + c = 3k \)

From the second ratio \( c : (a + b) = 5 : 7 \):
- Let \( c = 5m \) (where \( m \) is another common multiplier)
- Then \( a + b = 7m \)

### Step 3: Set the equations equal
We have:
1. \( a = k \)
2. \( b + c = 3k \)
3. \( c = 5m \)
4. \( a + b = 7m \)

From \( b + c = 3k \):
- Substitute \( c = 5m \):
\[ b + 5m = 3k \quad \text{(1)} \]

From \( a + b = 7m \):
- Substitute \( a = k \):
\[ k + b = 7m \quad \text{(2)} \]

### Step 4: Solve the equations
From equation (2):
\[ b = 7m - k \quad \text{(3)} \]

Substituting equation (3) into equation (1):
\[ (7m - k) + 5m = 3k \]
\[ 12m - k = 3k \]
\[ 12m = 4k \]
\[ k = 3m \]

### Step 5: Substitute back to find \( a \), \( b \), and \( c \)
Now substituting \( k = 3m \) back into our expressions:
- \( a = k = 3m \)
- \( c = 5m \)
- Substitute \( k \) into equation (3) to find \( b \):
\[ b = 7m - k = 7m - 3m = 4m \]

### Step 6: Find \( b : (a + c) \)
Now we need to find \( b : (a + c) \):
\[ a + c = 3m + 5m = 8m \]
Thus,
\[ b : (a + c) = 4m : 8m = 4 : 8 = 1 : 2 \]

### Final Answer
The ratio \( b : (a + c) = 1 : 2 \).

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