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

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 using variables
Let:
\[
a + b = 4k, \quad b + c = 9k, \quad c + a = 7k
\]
for some constant \(k\).

### Step 2: Solve the system of equations
We have three equations:
1. \( a + b = 4k \)
2. \( b + c = 9k \)
3. \( c + a = 7k \)

Adding the first and third equations:
\[
(a + b) + (c + a) = 4k + 7k = 11k
\]
This gives:
\[
2a + b + c = 11k \quad \text{(Equation 4)}
\]

Subtracting the first equation from the second:
\[
(b + c) - (a + b) = 9k - 4k = 5k
\]
This gives:
\[
c - a = 5k \quad \text{(Equation 5)}
\]

### Step 3: Substitute into the equation
We are given \(3a + 4b + 7c = 399\).

Substituting \(a\), \(b\), and \(c\) in terms of \(k\) and solving for \(k\):

From the equations, solve for \(a\), \(b\), and \(c\) and substitute in the equation \(3a + 4b + 7c = 399\). After solving, you should get the value of \(c\).

Let's calculate that.The value of \( c \) is \( 42 \).

Would you like more details on the solution process, or do you have any other questions?

Here are 5 related questions for practice:
1. If \( a + b + c = 12k \), what would be the value of \( k \)?
2. What is the value of \( a \) in the above problem?
3. How does changing the coefficients in the equation \(3a + 4b + 7c\) affect the value of \( c \)?
4. If the ratio \( (a + b) : (b + c) : (c + a) \) were \( 5 : 8 : 6 \), how would you solve for \( c \)?
5. How can you express \( a + c \) in terms of \( k \)?

**Tip:** When dealing with ratios, consider expressing variables in terms of a common multiplier to simplify the problem.

Learn how to solve a ratio problem involving linear equations and proportions. Given the ratio (a + b) : (b + c) : (c + a) = 4 : 9 : 7 and the equation 3a + 4b + 7c = 399, find the value of c step-by-step. Suitable for Grades 9-12.

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