If `B : A = 2 : 3` and `A : C = 5 : 7`, then what is `(A + B): (B + C)`?

Options

A. `25:26`
B. `25:36`
C. `25:31`
D. `27: 43`

Answer

To solve the problem, we need to find the ratio \((A + B) : (B + C)\) given the ratios \(B : A = 2 : 3\) and \(A : C = 5 : 7\).

### Step 1: Express A, B, and C in terms of a common variable
From the ratio \(B : A = 2 : 3\), we can express \(B\) and \(A\) as:
- Let \(A = 3x\)
- Then \(B = 2x\)

From the ratio \(A : C = 5 : 7\), we can express \(A\) and \(C\) as:
- Let \(A = 5y\)
- Then \(C = 7y\)

### Step 2: Equate the expressions for A
Since both expressions represent \(A\), we can set them equal to each other:
\[ 3x = 5y \]
From this, we can express \(y\) in terms of \(x\):
\[ y = \frac{3x}{5} \]

### Step 3: Substitute y back to find C
Now substitute \(y\) back into the expression for \(C\):
\[ C = 7y = 7 \left(\frac{3x}{5}\right) = \frac{21x}{5} \]

### Step 4: Write down A, B, and C
Now we have:
- \(A = 3x\)
- \(B = 2x\)
- \(C = \frac{21x}{5}\)

### Step 5: Calculate \(A + B\) and \(B + C\)
Now we can calculate \(A + B\) and \(B + C\):
\[ A + B = 3x + 2x = 5x \]
\[ B + C = 2x + \frac{21x}{5} = \frac{10x}{5} + \frac{21x}{5} = \frac{31x}{5} \]

### Step 6: Find the ratio \((A + B) : (B + C)\)
Now we can find the ratio:
\[ (A + B) : (B + C) = 5x : \frac{31x}{5} \]
To simplify this, we can multiply both sides by 5:
\[ = 25x : 31x \]
Cancelling \(x\) from both sides, we get:
\[ = 25 : 31 \]

### Final Answer
Thus, the ratio \((A + B) : (B + C)\) is:
\[ \boxed{25 : 31} \]