Continued ratios: let a, b and c be non-zero numbers. If a:b:c = 3 : 8 : 6 and a + c = 126, fund the value of a, b and c.

To solve for the values of \( a \), \( b \), and \( c \), given the continued ratio \( a : b : c = 3 : 8 : 6 \) and the equation \( a + c = 126 \), let's proceed as follows:

1. **Express \( a \), \( b \), and \( c \) in terms of a common multiplier \( k \):**
   Since \( a : b : c = 3 : 8 : 6 \), we can write:
   \[
   a = 3k, \quad b = 8k, \quad c = 6k
   \]

2. **Use the equation \( a + c = 126 \):**
   Substitute \( a = 3k \) and \( c = 6k \) into this equation:
   \[
   3k + 6k = 126
   \]
   \[
   9k = 126
   \]

3. **Solve for \( k \):**
   Divide both sides by 9:
   \[
   k = \frac{126}{9} = 14
   \]

4. **Find the values of \( a \), \( b \), and \( c \):**
   Substitute \( k = 14 \) back into the expressions for \( a \), \( b \), and \( c \):
   \[
   a = 3 \times 14 = 42
   \]
   \[
   b = 8 \times 14 = 112
   \]
   \[
   c = 6 \times 14 = 84
   \]

5. **Verify the solution:**
   Check that \( a + c = 126 \):
   \[
   a + c = 42 + 84 = 126
   \]
   This is correct.

### Final Answer:
The values are:
\[
a = 42, \quad b = 112, \quad c = 84
\]

Would you like further details on any part of this process?

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Here are some related questions that build on this concept:

1. If \( a : b : c = 5 : 7 : 9 \) and \( a + b = 60 \), what are the values of \( a \), \( b \), and \( c \)?
2. How would the solution change if the ratio was given as \( a : b : c = 2 : 5 : 7 \) and \( a + c = 100 \)?
3. If \( a : b : c = x : y : z \), express \( a \), \( b \), and \( c \) in terms of \( k \) and \( x \), \( y \), \( z \).
4. What if we knew \( a : b = 3 : 4 \) and \( b : c = 5 : 6 \)? How can we find the combined ratio \( a : b : c \)?
5. How do continued ratios differ from simple ratios, and how are they typically used?

**Tip:** In problems involving ratios, using a common multiplier helps to set up equations quickly based on given information.

If a:b:c = 3 : 8 : 6 and a + c = 126, find the value of a, b and c.

Discover how to find the values of a, b, and c given the ratio a:b:c = 3:8:6 and the equation a + c = 126. This solution covers key concepts in ratios and algebra. Suitable for Grades 7-9.

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