A 50-litre mixture contains juice and water in the ratio 3:2. How much water (in litres) must be added to this mixture to change the ratio of juice to water to 3:4?

The correct option is 20 litres.

Let's break down the problem step-by-step to find out how much water needs to be added to the mixture.

Step 1: Calculate the initial quantity of juice and water in the mixture.
The total volume of the initial mixture is 50 litres.
The ratio of juice to water in the initial mixture is 3 : 2.

The total number of ratio parts is:

$3+2=5$

Now, calculate the quantity of juice:

$\text{Juice}=\left(\frac{3}{5}\right)\times 50=30\text{litres}$

Now, calculate the quantity of water:

$\text{Water}=\left(\frac{2}{5}\right)\times 50=20\text{litres}$

Step 2: Set up the equation for the new ratio.
Let $x$ litres be the amount of water added to the mixture.
The quantity of juice remains unchanged at 30 litres.
The new quantity of water becomes $(20+x)$ litres.

The new ratio of juice to water is given as 3 : 4. Therefore, we can write the equation as:

$\frac{30}{20+x}=\frac{3}{4}$

Step 3: Solve for $x$.
Cross-multiply to solve the equation:

$30\times 4=3\times (20+x)$

$120=60+3x$

Subtract 60 from both sides:

$3x=120-60$

$3x=60$

Divide by 3:

$x=20$

Thus, 20 litres of water must be added to change the ratio of juice to water to 3 : 4.