Ritu and Priya invested in a business in the ratio $7 : 8$. They earned an annual profit of Rs $34450$. If Ritu withdrew her entire investment after $9$ months, what is the difference between their shares of the profit?

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**Solution:**

1. Let the investments of Ritu and Priya be $7x$ and $8x$ respectively.

2. Since Ritu withdrew her amount after 9 months, her investment duration is 9 months, while Priya's investment is for the full 12 months.

3. Calculate the effective investment for each:
   - Ritu's effective investment = $7x 	imes 9 = 63x$
   - Priya's effective investment = $8x 	imes 12 = 96x$

4. The ratio of their effective investments (which determines profit sharing) is:
   $$\frac{63x}{96x} = \frac{63}{96} = \frac{21}{32}$$

5. Total profit = Rs $34450$.

6. Sum of ratio parts = $21 + 32 = 53$.

7. Calculate individual shares:
   - Ritu's share = $\frac{21}{53} 	imes 34450 = 13650$
   - Priya's share = $\frac{32}{53} 	imes 34450 = 20800$

8. Difference between their shares:
   $$20800 - 13650 = 7150$$

**Answer:** Rs $7150$

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