Question

A container is full of mango juice. One fifth of juice is taken out from this container and then an equal amount of water is poured into the bottle. This process is repeated 3 more times. The final ratio of juice and water in the container is:

• A. \(\frac{254}{269}\)
• B. \(\frac{10}{15}\)
• C. \(\frac{254}{369}\)
• D. \(\frac{2}{5}\)

Answer 1

The Correct Option is C

To solve this problem, we will follow the process of dilution and calculate the final concentration of mango juice after repeated removal and replacement with water.

Initially, let the volume of the container be \( V = 1 \) (we can assume any unit since the ratio is dimensionless).

Initially, the entire volume is mango juice.

Step 1: One-fifth of the juice is taken out and replaced by water. Hence, the juice left in the container is:

\( \text{Juice left after step 1} = V \times \left( 1 - \frac{1}{5} \right) = \frac{4}{5} \times V \)

Step 2: Repeat the process:

\( \text{Juice left after step 2} = \frac{4}{5} \times \left(\frac{4}{5} \times V\right) = \left(\frac{4}{5}\right)^2 \times V \)

Step 3: Repeat the process again:

\( \text{Juice left after step 3} = \frac{4}{5} \times \left(\frac{4}{5}\right)^2 \times V = \left(\frac{4}{5}\right)^3 \times V \)

Step 4: Repeat the process one more time:

\( \text{Juice left after step 4} = \frac{4}{5} \times \left(\frac{4}{5}\right)^3 \times V = \left(\frac{4}{5}\right)^4 \times V \)

Substitute the value of \( V = 1 \), thus:

\( \text{Juice left after step 4} = \left(\frac{4}{5}\right)^4 = \frac{256}{625} \)

Since the container is now filled with juice and water, the amount of water is:

\( \text{Amount of water} = 1 - \frac{256}{625} = \frac{625-256}{625} = \frac{369}{625} \)

Hence, the final ratio of juice to water is:

\( \frac{256}{625} : \frac{369}{625} = \frac{256}{369} \)

Therefore, the final ratio of juice and water in the container is: \(\frac{254}{369}\).