Ratio Calculation: A + B : B + C : A + C

The problem requires finding a combined ratio based on two given ratios involving common terms.

Step 1: Combine the Ratios

We are given:

• $A : B = 5 : 6$
• $B : C = 6 : 7$

Since the value corresponding to 'B' is the same (6) in both ratios, we can combine them directly to find the ratio $A : B : C$.

Combined Ratio: $A : B : C = 5 : 6 : 7$

Step 2: Express Terms using a Common Factor

Let $A = 5k$, $B = 6k$, and $C = 7k$, where $k$ is a common constant factor.

Step 3: Calculate the Required Sums

We need to find the ratio $(A + B) : (B + C) : (A + C)$.

• $A + B = 5k + 6k = 11k$
• $B + C = 6k + 7k = 13k$
• $A + C = 5k + 7k = 12k$

Step 4: Determine the Final Ratio

The ratio $(A + B) : (B + C) : (A + C)$ is:

$11k : 13k : 12k$

Dividing by the common factor $k$, we get:

$11 : 13 : 12$

Therefore, the required ratio is $11 : 13 : 12$.