If $a : b = 2 : 3$ and $b : c = 5 : 7$, then find the ratio $c : a$.

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**Solution:**

1. Given ratios:
   - $a : b = 2 : 3$
   - $b : c = 5 : 7$

2. To find $c : a$, first express both ratios with a common term for $b$.

3. The least common multiple (LCM) of $b$ terms (3 and 5) is 15.

4. Adjust the ratios to have $b = 15$:
   - For $a : b = 2 : 3$, multiply both terms by 5:
     $$a : b = 2 	imes 5 : 3 	imes 5 = 10 : 15$$
   - For $b : c = 5 : 7$, multiply both terms by 3:
     $$b : c = 5 	imes 3 : 7 	imes 3 = 15 : 21$$

5. Now combine the ratios:
   $$a : b : c = 10 : 15 : 21$$

6. From this, the ratio $c : a$ is:
   $$21 : 10$$

**Answer:** $21 : 10$

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