If $a : b = 3 : 4$, $b : c = 2 : 3$, and $c : d = 5 : 6$, then find the ratio $a : b : c : d$.

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**Solution:**

1. Given ratios:
   - $a : b = 3 : 4$
   - $b : c = 2 : 3$
   - $c : d = 5 : 6$

2. To find $a : b : c : d$, we need to express all ratios with common terms.

3. From $a : b = 3 : 4$ and $b : c = 2 : 3$, the common term is $b$.

4. Make $b$ equal in both ratios:
   - In $a : b = 3 : 4$, $b = 4$
   - In $b : c = 2 : 3$, $b = 2$

5. Find LCM of $4$ and $2$ which is $4$.

6. Adjust ratios:
   - Multiply $a : b = 3 : 4$ by $1$ (no change): $3 : 4$
   - Multiply $b : c = 2 : 3$ by $2$: $4 : 6$

7. Now, $a : b : c = 3 : 4 : 6$

8. Next, consider $c : d = 5 : 6$.

9. Current $c$ is $6$, but in $c : d = 5 : 6$, $c = 5$.

10. Find LCM of $6$ and $5$ which is $30$.

11. Adjust $a : b : c$ and $c : d$ to have $c = 30$:
    - Multiply $a : b : c = 3 : 4 : 6$ by $5$: $15 : 20 : 30$
    - Multiply $c : d = 5 : 6$ by $6$: $30 : 36$

12. Combine all to get:
    $$a : b : c : d = 15 : 20 : 30 : 36$$

**Answer:** $15 : 20 : 30 : 36$

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