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.