If (a + b) : (b + c) : (c + a) = 4 : 9 : 7 and 3a + 4b + 7c = 399, then the value of c will be.

We are given the ratio \((a + b) : (b + c) : (c + a) = 4 : 9 : 7\) and the equation \(3a + 4b + 7c = 399\). We need to find the value of \(c\).

### Step 1: Express the ratios
Since \((a + b) : (b + c) : (c + a) = 4 : 9 : 7\), we can write:
\[
\frac{a+b}{4} = \frac{b+c}{9} = \frac{c+a}{7} = k
\]
where \(k\) is a constant.

Thus, we have:
\[
a + b = 4k, \quad b + c = 9k, \quad c + a = 7k
\]

### Step 2: Solve the system of equations
We can now solve for \(a\), \(b\), and \(c\) by manipulating these equations.

From \(a + b = 4k\) and \(c + a = 7k\), subtracting the first from the second gives:
\[
(c + a) - (a + b) = 7k - 4k
\]
\[
c - b = 3k \quad \text{(Equation 1)}
\]

Similarly, from \(b + c = 9k\) and \(c + a = 7k\), subtracting the second from the first gives:
\[
(b + c) - (c + a) = 9k - 7k
\]
\[
b - a = 2k \quad \text{(Equation 2)}
\]

Now, we can solve for \(b\) and \(c\) in terms of \(a\) using Equations 1 and 2:
\[
b = a + 2k \quad \text{(substitute in Equation 2)}
\]
\[
c = b + 3k = (a + 2k) + 3k = a + 5k \quad \text{(substitute in Equation 1)}
\]

### Step 3: Substitute in the given equation
We substitute \(a\), \(b\), and \(c\) into the equation \(3a + 4b + 7c = 399\):
\[
3a + 4(a + 2k) + 7(a + 5k) = 399
\]
Expanding and combining like terms:
\[
3a + 4a + 8k + 7a + 35k = 399
\]
\[
14a + 43k = 399
\]

### Step 4: Solve for \(a\) and \(k\)
We can solve for \(a\) in terms of \(k\):
\[
14a = 399 - 43k
\]
\[
a = \frac{399 - 43k}{14}
\]

Now, substitute \(a\) into \(c = a + 5k\):
\[
c = \frac{399 - 43k}{14} + 5k
\]

### Step 5: Test values of \(k\)
Let's try different integer values of \(k\) that satisfy the ratio condition and make \(c\) an integer.

After testing, we find that \(k = 7\) satisfies the condition:
\[
a = \frac{399 - 43(7)}{14} = \frac{399 - 301}{14} = \frac{98}{14} = 7
\]
\[
c = 7 + 5(7) = 7 + 35 = 42
\]

### Final Answer
The value of \(c\) is \( \boxed{42} \).

Do you want details or have any questions?

---
1. How would you solve for \(b\) in this problem?
2. Can the ratio method be applied to any linear equation?
3. What happens if the values of \(k\) don't work in an equation like this?
4. What is the importance of the ratio method in solving such problems?
5. Could the same method be applied if the coefficients of the linear equation were different?

**Tip:** Always verify the solution by plugging the values back into the original equation.

Explore how to solve a problem involving ratios and linear equations where (a + b) : (b + c) : (c + a) = 4 : 9 : 7 and 3a + 4b + 7c = 399. Detailed step-by-step solution included.

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