A juice mixture is made by combining orange concentrate and water in a ratio of 2 to 5 by … — SAT

A juice mixture is made by combining orange concentrate and water in a ratio of $2$ to $5$ by volume. To make a large batch of this mixture, a worker uses a total of $35$ liters of water. If the worker wants to dilute the mixture so that the new ratio of orange concentrate to water is $1$ to $4$ by volume, how many additional liters of water must be added?

A
5
B
7
21Answer
D
56

Answer

The correct answer is 21. First, we find the volume of orange concentrate in the initial mixture. The ratio of orange concentrate to water is 2 to 5, and the volume of water is 35 liters. Setting up the proportion

\frac{2}{5} = \frac{\text{concentrate}}{35}

gives a concentrate volume of 14 liters. When the mixture is diluted, the amount of orange concentrate remains 14 liters, but the ratio of concentrate to water becomes 1 to 4. Setting up the new proportion

\frac{1}{4} = \frac{14}{\text{new water}}

gives a new water volume of 56 liters. The additional water required is the difference between the final and initial water volumes, which is

56 - 35 = 21

liters.

Step-by-Step Solution

Determine the volume of orange concentrate in the initial mixture.

The initial ratio of orange concentrate to water is

2:5

. Since the mixture contains

35

liters of water, the volume of orange concentrate,

x

, can be found using the proportion

\frac{2}{5} = \frac{x}{35}

. Solving for

x

gives

x = \frac{2 \times 35}{5} = 14

liters.

The amount of orange concentrate must be determined because it remains constant during dilution.

Calculate the total volume of water needed for the diluted mixture.

In the diluted mixture, the ratio of orange concentrate to water is

1:4

. Since the amount of orange concentrate is still

14

liters, the new volume of water,

y

, can be found using the proportion

\frac{1}{4} = \frac{14}{y}

. Solving for

y

gives

y = 14 \times 4 = 56

liters.

The final volume of water is required to determine the additional amount of water that needs to be added.

Find the additional volume of water that must be added.

Subtract the initial volume of water from the final volume of water:

56 - 35 = 21

liters.

Subtracting the starting amount of water gives the volume of new water needed to achieve the target ratio.

Key Concept

Using proportions to solve multi-step mixture and dilution problems.

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