If `a: (b+c) = 1:3` and `c: (a+ b) = 5:7`, then `b: (a+c)` is equal to:

Options

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

Answer

To solve the problem, we need to find the ratio \( b : (a + c) \) given the ratios \( a : (b + c) = 1 : 3 \) and \( c : (a + b) = 5 : 7 \).

### Step 1: Express the ratios in terms of variables
From the first ratio \( a : (b + c) = 1 : 3 \), we can express:
- Let \( a = 1k \) (where \( k \) is a constant)
- Then \( b + c = 3k \)

From the second ratio \( c : (a + b) = 5 : 7 \), we can express:
- Let \( c = 5m \) (where \( m \) is another constant)
- Then \( a + b = 7m \)

### Step 2: Express \( b \) in terms of \( a \) and \( c \)
From the first equation \( b + c = 3k \), we can express \( b \) as:
\[ b = 3k - c \]

Substituting \( c = 5m \) into this expression gives:
\[ b = 3k - 5m \]

### Step 3: Substitute \( a \) into the second equation
From the second equation \( a + b = 7m \), we substitute \( a = 1k \) and \( b = 3k - 5m \):
\[ 1k + (3k - 5m) = 7m \]
This simplifies to:
\[ 4k - 5m = 7m \]
Combining like terms gives:
\[ 4k = 12m \]
Thus, we have:
\[ k = 3m \]

### Step 4: Substitute back to find \( a \), \( b \), and \( c \)
Now substituting \( k = 3m \) back into our expressions for \( a \), \( b \), and \( c \):
- \( a = 1k = 1(3m) = 3m \)
- \( c = 5m \)
- Substitute \( k \) into \( b = 3k - 5m \):
\[ b = 3(3m) - 5m = 9m - 5m = 4m \]

### Step 5: Find \( b : (a + c) \)
Now we can find \( a + c \):
\[ a + c = 3m + 5m = 8m \]

Now we can find the ratio \( b : (a + c) \):
\[ b : (a + c) = 4m : 8m \]

### Step 6: Simplify the ratio
This simplifies to:
\[ b : (a + c) = 4 : 8 = 1 : 2 \]

Thus, the final answer is:
\[ \boxed{1 : 2} \]