Divide ₹ 1488 among X, Y and Z. If the shares are in the ratio $X : Y = 4 : 5$ and $Y : Z = 7 : 6$, then find the share of Z.

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**Solution:**

1. Given ratios:
   - $X : Y = 4 : 5$
   - $Y : Z = 7 : 6$

2. To find the combined ratio $X : Y : Z$, express both ratios with the same value for $Y$.

3. From $X : Y = 4 : 5$, $Y = 5k$ for some $k$.

4. From $Y : Z = 7 : 6$, $Y = 7m$ for some $m$.

5. Equate $Y$ values:
5k = 7m

6. Substitute $k$ back to find $X$:
   $$X = 4k = 4 	imes \frac{7m}{5} = \frac{28m}{5}$$

7. $Y = 7m$, $Z = 6m$

8. So the ratio $X : Y : Z$ is:
   $$\frac{28m}{5} : 7m : 6m$$

9. Multiply all terms by 5 to clear denominator:
   $$28m : 35m : 30m$$

10. Simplify ratio:
    $$28 : 35 : 30$$

11. Total parts = $28 + 35 + 30 = 93$

12. Total amount = ₹ 1488

13. Value of one part:
    $$\frac{1488}{93} = 16$$

14. Share of Z = $30$ parts:
    $$30 	imes 16 = 480$$

**Answer:** ₹ 480

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