If `A :B=2 : 3` and `B : C=3 :7`, then `A+B : B+C : C+A` is.

Options

A. `4 : 10 : 9`
B. `5 : 10 : 9`
C. `5 : 8 : 9`
D. `4 : 8 : 9`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given ratios
We are given two ratios:
1. \( A : B = 2 : 3 \)
2. \( B : C = 3 : 7 \)

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

For some variable \( k \).

From the second ratio \( B : C = 3 : 7 \), we can express:
- \( B = 3m \)
- \( C = 7m \)

For some variable \( m \).

### Step 3: Equate the values of B
Since \( B \) is expressed in two different ways, we can equate them:
\[ 3k = 3m \]
This implies:
\[ k = m \]

### Step 4: Substitute k back to find A and C
Now we can substitute \( k \) back into the expressions for \( A \) and \( C \):
- \( A = 2k = 2m \)
- \( C = 7m \)

### Step 5: Write A, B, and C in terms of m
Now we have:
- \( A = 2m \)
- \( B = 3m \)
- \( C = 7m \)

### Step 6: Calculate A + B, B + C, and C + A
Now we can calculate:
1. \( A + B = 2m + 3m = 5m \)
2. \( B + C = 3m + 7m = 10m \)
3. \( C + A = 7m + 2m = 9m \)

### Step 7: Form the ratio A + B : B + C : C + A
Now we can express the ratio:
\[ A + B : B + C : C + A = 5m : 10m : 9m \]

### Step 8: Simplify the ratio
Since \( m \) is common in all parts, we can simplify:
\[ 5 : 10 : 9 \]

### Final Answer
Thus, the final answer is:
\[ A + B : B + C : C + A = 5 : 10 : 9 \]

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