A vessel contains $360$ liters of a mixture of milk and water in the ratio of $5:3$, respectively. If $X$ liters of milk and $35$ liters of water are added to the mixture, the ratio of milk to water in the new mixture becomes $30:17$. Find the value of $X$.

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**Solution:**

1. Initially, the total mixture is $360$ liters with milk and water in the ratio $5:3$.

2. Calculate the quantity of milk and water initially:
   - Milk = $\frac{5}{5+3} 	imes 360 = \frac{5}{8} 	imes 360 = 225$ liters
   - Water = $\frac{3}{8} 	imes 360 = 135$ liters

3. After adding $X$ liters of milk and $35$ liters of water, the new quantities become:
   - Milk = $225 + X$
   - Water = $135 + 35 = 170$

4. The new ratio of milk to water is given as $30:17$, so:
   $$\frac{225 + X}{170} = \frac{30}{17}$$

5. Cross multiply to solve for $X$:
   $$17(225 + X) = 30 	imes 170$$
   $$3825 + 17X = 5100$$
   $$17X = 5100 - 3825 = 1275$$
   $$X = \frac{1275}{17} = 75$$

**Answer:** $75$

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