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 value of x is 16 (Option 16).

Step-by-Step Explanation:

1. Understanding the Ratio in the Original Tank:

The original blend of mango juice and lime water in the tank is in the ratio of 9 : x.

When a portion of a uniform mixture is removed, the ratio of its components in the removed portion remains the same as in the original tank. Therefore, in the 500 ml of blend drawn out, the ratio of mango juice to lime water is also 9 : x.

2. Expressing Quantities in the 500 ml Drawn Blend:

The total ratio parts equal (9 + x).

The quantity of mango juice in the 500 ml mixture is:

$\text{Mango juice}=500\times \frac{9}{9+x}=\frac{4500}{9+x}\text{ml}$

The quantity of lime water in the 500 ml mixture is:

$\text{Lime water}=500\times \frac{x}{9+x}=\frac{500x}{9+x}\text{ml}$

3. Setting Up the Equation for the New Container:

The drawn 500 ml blend is poured into a container that already contains 100 ml of pure lime water.

Therefore, in the new container:

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

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

The new ratio of mango juice to lime water is given as 3 : 7. We set up the proportion:

$\frac{\frac{4500}{9+x}}{\frac{500x}{9+x}+100}=\frac{3}{7}$

4. Solving for x:

Simplify the denominator of the left-hand side:

$\frac{500x}{9+x}+100=\frac{500x+100(9+x)}{9+x}=\frac{500x+900+100x}{9+x}=\frac{600x+900}{9+x}$

Canceling out the common denominator $(9+x)$ from both numerator and denominator:

$\frac{4500}{600x+900}=\frac{3}{7}$

Divide the numerator and denominator of the left-hand side by 100:

$\frac{45}{6x+9}=\frac{3}{7}$

Cross-multiply to solve for x:

$45\times 7=3\times (6x+9)$

$315=18x+27$

$18x=315-27$

$18x=288$

$x=\frac{288}{18}=16$

Thus, the value of x is 16.