`a:b+c=1:3`
`c:a+b=5:7`
Find `a:b:c`

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) based on the given ratios. The ratios provided are:

1. \( a : (b + c) = 1 : 3 \)
2. \( c : (a + b) = 5 : 7 \)

### Step 1: Express the ratios in terms of a common variable
From the first ratio \( a : (b + c) = 1 : 3 \), we can express this as:
\[ \frac{a}{b+c} = \frac{1}{3} \]
This implies:
\[ a = \frac{1}{3}(b + c) \]

From the second ratio \( c : (a + b) = 5 : 7 \), we can express this as:
\[ \frac{c}{a+b} = \frac{5}{7} \]
This implies:
\[ c = \frac{5}{7}(a + b) \]

### Step 2: Substitute and simplify
Now we have two equations:
1. \( a = \frac{1}{3}(b + c) \)
2. \( c = \frac{5}{7}(a + b) \)

Let's substitute \( c \) from the second equation into the first equation.

Substituting \( c \) in the first equation:
\[ a = \frac{1}{3}(b + \frac{5}{7}(a + b)) \]

### Step 3: Clear the fractions
To eliminate the fractions, multiply through by 21 (the least common multiple of 3 and 7):
\[ 21a = 7(b + \frac{5}{7}(a + b)) \]
This simplifies to:
\[ 21a = 7b + 5(a + b) \]
Expanding the right side gives:
\[ 21a = 7b + 5a + 5b \]
Combining like terms:
\[ 21a = 12b + 5a \]
Now, isolate \( a \):
\[ 21a - 5a = 12b \]
\[ 16a = 12b \]
\[ \frac{a}{b} = \frac{12}{16} = \frac{3}{4} \]

### Step 4: Find \( c \) in terms of \( a \) and \( b \)
Now we have \( a = \frac{3}{4}b \). We can substitute this back into one of our earlier equations to find \( c \).

Using \( c = \frac{5}{7}(a + b) \):
Substituting \( a \):
\[ c = \frac{5}{7}\left(\frac{3}{4}b + b\right) \]
\[ c = \frac{5}{7}\left(\frac{3}{4}b + \frac{4}{4}b\right) \]
\[ c = \frac{5}{7}\left(\frac{7}{4}b\right) \]
\[ c = \frac{5}{4}b \]

### Step 5: Express \( a, b, c \) in terms of a common variable
Now we have:
- \( a = \frac{3}{4}b \)
- \( b = b \)
- \( c = \frac{5}{4}b \)

To express everything in terms of \( b \), let \( b = 4k \) (for some \( k \)):
- \( a = \frac{3}{4}(4k) = 3k \)
- \( b = 4k \)
- \( c = \frac{5}{4}(4k) = 5k \)

### Step 6: Write the final ratio
Now we can express the ratio \( a : b : c \):
\[ a : b : c = 3k : 4k : 5k \]
This simplifies to:
\[ a : b : c = 3 : 4 : 5 \]

### Final Answer
Thus, the ratio \( a : b : c \) is \( 3 : 4 : 5 \).