A container contains 90 liters of milk and 18 liters of water. If X liter of mixture is taken out and 6 liter of water is added the new ratio of milk to water became 5:2, then find the value of X?

The correct answer is 72 (which corresponds to Option 72).

Step-by-step Explanation:

Step 1: Calculate the initial amounts and ratio of milk and water.
Initial volume of milk = 90 liters
Initial volume of water = 18 liters
Total volume of mixture = 90 + 18 = 108 liters

Ratio of milk to water in the initial mixture:
Milk : Water = 90 : 18 = 5 : 1
Total ratio parts = 5 + 1 = 6 parts

Step 2: Express the removed quantities in terms of X.
When $X$ liters of the mixture is removed, milk and water are taken out in the same ratio of 5 : 1.
Amount of milk removed = $\frac{5}{6}\times X$
Amount of water removed = $\frac{1}{6}\times X$

Step 3: Determine the remaining quantities and form the equation.
After removing $X$ liters of mixture and adding 6 liters of water:
Remaining Milk = $90-\frac{5X}{6}$
Remaining Water = $18-\frac{X}{6}+6=24-\frac{X}{6}$

The new ratio of milk to water is given as 5 : 2. Therefore:

$\frac{90-\frac{5X}{6}}{24-\frac{X}{6}}=\frac{5}{2}$

Step 4: Solve for X.
Cross-multiplying the equation:

$2\times \left(90-\frac{5X}{6}\right)=5\times \left(24-\frac{X}{6}\right)$

$180-\frac{10X}{6}=120-\frac{5X}{6}$

Rearranging the terms:

$180-120=\frac{10X}{6}-\frac{5X}{6}$

$60=\frac{5X}{6}$

$5X=60\times 6=360$

$X=\frac{360}{5}=72$

Thus, the value of $X$ is 72.