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Difficulty: MediumRatios, Rates, and Proportions

A beverage company formulates a fruit punch by blending apple juice, cranberry juice, and sparkling water in the ratio 5:3:25 : 3 : 2 by volume. In a specific batch, there are 1212 liters of sparkling water. A technician then adds 1010 liters of cranberry juice and an additional amount of apple juice so that the new ratio of apple juice to cranberry juice becomes 3:23 : 2. How many liters of apple juice were added to the batch?

  1. 1212Answer
  2. B
    1515
  3. C
    2727
  4. D
    4242
  5. E
    2222

Answer

The technician added 12 liters of apple juice.
In the initial ratio of 5:3:25 : 3 : 2, 22 parts represent 1212 liters of sparkling water, meaning each part is 66 liters. Thus, the original mixture contains 3030 liters of apple juice and 1818 liters of cranberry juice. Adding 1010 liters of cranberry juice brings its total to 2828 liters. For the new apple juice to cranberry juice ratio to be 3:23 : 2, the required total apple juice is 32×28=42\frac{3}{2} \times 28 = 42 liters. Subtracting the initial 3030 liters yields 1212 liters of added apple juice.

Step-by-Step Solution

1
Determine the unit value per part in the original ratio.
Since 22 parts correspond to 1212 liters of sparkling water, 11 part = 122=6\frac{12}{2} = 6 liters.
The original ratio 5:3:25 : 3 : 2 defines the relative proportions of apple juice, cranberry juice, and sparkling water.
2
Calculate the original volumes of apple juice and cranberry juice.
Apple juice = 5×6=305 \times 6 = 30 liters; Cranberry juice = 3×6=183 \times 6 = 18 liters.
Multiply each component's ratio value by the single-part volume of 66 liters.
3
Calculate the new volume of cranberry juice after adding 10 liters.
New cranberry juice volume = 18+10=2818 + 10 = 28 liters.
1010 liters of cranberry juice were added to the initial 1818 liters.
4
Set up a proportion for the new ratio of apple juice to cranberry juice and solve for the added amount xx.
\frac{30 + x}{28} = \frac{3}{2} \implies 2(30 + x) = 3(28) \implies 60 + 2x = 84 \implies 2x = 24 \implies x = 12.
The new ratio of total apple juice (30+x30 + x) to total cranberry juice (2828) must equal 3:23 : 2.

Key Concept

Multi-step proportional reasoning and scaling ratios when quantities change.
Estimated Time:1m 15s
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Topics

Ratios, Rates, and Proportions

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