A large tank holds a blend of mango juice and lime water in the ratio of 9 : x respectively. When 500 ml of this blend is drawn out from the tank and poured into a separate container already holding 100 ml of lime water, the ratio of mango juice to lime water in the new container becomes 3 : 7. Find the value of x.

Correct Answer: The correct answer is 16.

Step-by-Step Explanation:

Step 1: Determine the composition of the drawn blend
The tank holds a blend of mango juice and lime water in the ratio of $9:x$.
When 500 ml of this blend is drawn out, the ratio of mango juice to lime water in the drawn 500 ml portion remains $9:x$.

The total number of ratio parts is:
$9+x$

The quantity of mango juice present in the 500 ml drawn blend is:
$\text{Mango Juice}=500\times \frac{9}{9+x}=\frac{4500}{9+x}$ ml

The quantity of lime water present in the 500 ml drawn blend is:
$\text{Lime Water in blend}=500\times \frac{x}{9+x}=\frac{500x}{9+x}$ ml

Step 2: Calculate the total quantities in the new container
The 500 ml blend is poured into a separate container that already holds 100 ml of lime water.
Therefore, the new container contains:

Total Mango Juice:
$\text{Mango Juice}=\frac{4500}{9+x}$ ml

Total Lime Water:
$\text{Lime Water}=\frac{500x}{9+x}+100$ ml

Simplifying the expression for total lime water:
$\text{Total Lime Water}=\frac{500x+100(9+x)}{9+x}=\frac{500x+900+100x}{9+x}=\frac{600x+900}{9+x}$ ml

Step 3: Set up the ratio equation
The ratio of mango juice to lime water in the new container becomes $3:7$.

Writing the ratio as an equation:
$\frac{\text{Mango Juice}}{\text{Lime Water}}=\frac{3}{7}$

Substituting the respective expressions:
$\frac{\frac{4500}{9+x}}{\frac{600x+900}{9+x}}=\frac{3}{7}$

Canceling the common denominator $9+x$ from both the numerator and denominator:
$\frac{4500}{600x+900}=\frac{3}{7}$

Step 4: Solve for x
Simplify the left fraction by dividing both numerator and denominator by 300:
$\frac{15}{2x+3}=\frac{3}{7}$

Cross-multiply to solve for $x$:
$15\times 7=3\times (2x+3)$

$105=6x+9$

Subtract 9 from both sides:
$6x=96$

Divide by 6:
$x=16$

Thus, the value of $x$ is 16.