To find the ratio $a : b : c$, we need to combine the given ratios $a : b = 5 : 9$ and $b : c = 4 : 7$.

Step 1: Express $b$ in terms of both ratios.

• From $a : b = 5 : 9$, let $a = 5k$ and $b = 9k$.
• From $b : c = 4 : 7$, let $b = 4m$ and $c = 7m$.

Step 2: Equate the values of $b$.

• Since $b = 9k$ and $b = 4m$, equate these: $9k = 4m$.

Step 3: Solve for $k$ and $m$.

• $k = \frac{4m}{9}$.

Step 4: Express $a$, $b$, and $c$ in terms of $m$.

• $a = 5k = 5 \times \frac{4m}{9} = \frac{20m}{9}$.
• $b = 9k = 9 \times \frac{4m}{9} = 4m$.
• $c = 7m$.

Step 5: Find the ratio $a : b : c$.

• $a : b : c = \frac{20m}{9} : 4m : 7m$.

To eliminate $m$ and simplify:

• Multiply each term by 9 to clear the denominator for $a$:
  $20 : 36 : 63$.

Final Answer:

The ratio $a : b : c = 20 : 36 : 63$.

Would you like a detailed explanation of any steps, or do you have any questions? Here are some related questions you might find interesting:

1. If the ratio $a : b$ changes, how would it affect $a : b : c$?
2. How do you solve for three variables when given multiple ratios?
3. Can you express $a : c$ directly from the given ratios?
4. How would the result change if the second ratio was $b : c = 3 : 7$ instead?
5. What is the importance of clearing denominators in ratios?

Tip: When dealing with ratios, always ensure to express the common terms in a consistent form to combine them effectively.