`a:a+b+c=2:9`
`b:a+c=4:7`
Find `a:b:c=?`

Answer

To solve the problem step by step, we will use the given ratios to find the values of \(a\), \(b\), and \(c\).

### Step 1: Write down the given ratios
We have two ratios given:
1. \( a : (a + b + c) = 2 : 9 \)
2. \( b : (a + c) = 4 : 7 \)

### Step 2: Express \(a + b + c\) and \(a + c\) in terms of \(a\) and \(b\)
From the first ratio, we can express \(a\) in terms of \(a + b + c\):
\[ \frac{a}{a + b + c} = \frac{2}{9} \]
This implies:
\[ 9a = 2(a + b + c) \]
Expanding this gives:
\[ 9a = 2a + 2b + 2c \]
Rearranging, we have:
\[ 7a = 2b + 2c \quad \text{(1)} \]

From the second ratio, we can express \(b\) in terms of \(a + c\):
\[ \frac{b}{a + c} = \frac{4}{7} \]
This implies:
\[ 7b = 4(a + c) \]
Expanding this gives:
\[ 7b = 4a + 4c \]
Rearranging, we have:
\[ 4a + 4c = 7b \quad \text{(2)} \]

### Step 3: Solve the equations (1) and (2)
From equation (1):
\[ 2b + 2c = 7a \implies b + c = \frac{7a}{2} \quad \text{(3)} \]

Substituting \(c\) from equation (3) into equation (2):
\[ 4a + 4c = 7b \]
Substituting \(c = \frac{7a}{2} - b\) into this gives:
\[ 4a + 4\left(\frac{7a}{2} - b\right) = 7b \]
Expanding this:
\[ 4a + 14a - 4b = 7b \]
Combining like terms:
\[ 18a = 11b \]
Thus, we can express \(b\) in terms of \(a\):
\[ b = \frac{18a}{11} \quad \text{(4)} \]

### Step 4: Substitute \(b\) back to find \(c\)
Now substitute \(b\) from equation (4) into equation (3):
\[ b + c = \frac{7a}{2} \]
Substituting \(b = \frac{18a}{11}\):
\[ \frac{18a}{11} + c = \frac{7a}{2} \]
To solve for \(c\), we first find a common denominator:
\[ c = \frac{7a}{2} - \frac{18a}{11} \]
Finding a common denominator (22):
\[ c = \frac{77a}{22} - \frac{36a}{22} = \frac{41a}{22} \]

### Step 5: Express \(a\), \(b\), and \(c\) in terms of a common variable
Now we have:
- \(a = a\)
- \(b = \frac{18a}{11}\)
- \(c = \frac{41a}{22}\)

### Step 6: Find the ratio \(a : b : c\)
To express \(a : b : c\) in the simplest form, we can use a common multiple for the denominators:
Let \(a = 22k\) for some \(k\):
- \(b = \frac{18 \times 22k}{11} = 36k\)
- \(c = \frac{41 \times 22k}{22} = 41k\)

Thus, the ratio \(a : b : c\) is:
\[ a : b : c = 22k : 36k : 41k = 22 : 36 : 41 \]

### Final Answer
The final ratio is:
\[ a : b : c = 22 : 36 : 41 \]