We are given the following information:

1. $a + c = 4:6$ (which simplifies to $a + c = 2:3$).
2. $c : (a + b) = 3:7$.

We need to find the ratio $a : (b + c)$.

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Step 1: Represent the ratios mathematically

• From $a + c = 2:3$, let $a + c = 2x$ and $c = 3x$.
• From $c : (a + b) = 3:7$, let $c = 3y$ and $a + b = 7y$.

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Step 2: Relating $c$ from both ratios

From the first equation, $c = 3x$, and from the second equation, $c = 3y$. Therefore: $3x = 3y \quad \Rightarrow \quad x = y$

So we can substitute $x = y$ throughout.

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Step 3: Substitute and solve for $b$

From the first ratio, $a + c = 2x$, and from the second, $a + b = 7x$. Using these: $b = (a + b) - a = 7x - a$

From $a + c = 2x$, we know: $a = 2x - c = 2x - 3x = -x$

Thus: $b = 7x - (-x) = 7x + x = 8x$

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Step 4: Find $a : (b + c)$

We now calculate $b + c$: $b + c = 8x + 3x = 11x$

The ratio $a : (b + c)$ is: $a : (b + c) = -x : 11x = -1 : 11$

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Final Answer:

The ratio is: $a : (b + c) = -1 : 11$

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Would you like further clarification or a step-by-step breakdown? Here are some related questions:

1. How do you simplify and interpret given ratios in mathematical problems?
2. What are common strategies for solving ratio-based equations?
3. How do we handle negative ratios in such problems?
4. Could you elaborate on real-world applications of such ratio problems?
5. What are key concepts in proportional reasoning?

Tip: Always check if simplifying a ratio can help solve the problem more effectively!