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Difficulty: HardRatios, Rates, and Proportions

A scientist is preparing a mixture by combining liquid A, liquid B, and liquid C. Initially, the ratio of the volume of liquid A to the volume of liquid B is 2:32:3, and the ratio of the volume of liquid B to the volume of liquid C is 4:54:5. The scientist then adds 120120 milliliters of liquid A to the mixture, while the volumes of liquid B and liquid C remain unchanged. This addition changes the ratio of the volume of liquid A to the volume of liquid B to 5:65:6. What was the original total volume of the mixture, in milliliters?

  1. A
    900
  2. B
    990
  3. C
    1,200
  4. 2,100Answer
  5. E
    2,220

Answer

The correct total original volume of the mixture is 2,100 milliliters.
The correct answer shows the sum of the initial volumes of all three liquids. By aligning the two ratios, the original ratio is established as A:B:C=8:12:15A:B:C = 8:12:15, representing a total of 3535 parts. Setting up the equation for adding 120120 milliliters of liquid A relative to liquid B reveals that each part corresponds to 6060 milliliters. Multiplying the total of 3535 parts by 6060 milliliters yields 2,1002,100 milliliters.

Step-by-Step Solution

1
Align the two given ratios by finding a common multiple for the shared component, liquid B.
The ratio of liquid A to liquid B is 2:3=8:122:3 = 8:12, and the ratio of liquid B to liquid C is 4:5=12:154:5 = 12:15. This yields the combined ratio A:B:C=8:12:15A:B:C = 8:12:15.
To analyze changes in mixtures with multiple components, all ratio terms must be defined relative to a common unit value.
2
Define the initial volumes using a variable xx and set up a proportion representing the addition of liquid A.
Let the initial volumes be A=8xA = 8x, B=12xB = 12x, and C=15xC = 15x. Adding 120120 milliliters of liquid A gives the new volume 8x+1208x + 120. The new ratio equation is 8x+12012x=56\frac{8x + 120}{12x} = \frac{5}{6}.
The new ratio of liquid A to liquid B is given as 5:65:6 while the volume of liquid B remains unchanged.
3
Solve the proportion for xx by cross-multiplying.
6(8x+120)=5(12x)  ⟹  48x+720=60x  ⟹  12x=720  ⟹  x=606(8x + 120) = 5(12x) \implies 48x + 720 = 60x \implies 12x = 720 \implies x = 60.
Finding the value of xx allows us to calculate the actual volumes of all three components.
4
Calculate the original total volume of the mixture.
Original total volume =8x+12x+15x=35x=35(60)=2,100= 8x + 12x + 15x = 35x = 35(60) = 2,100 milliliters.
The question asks for the sum of the initial volumes of all three liquids.

Key Concept

Combining multiple ratios to solve multi-step mixture and rate problems.

Alternative Method

Observe that the volume of liquid B does not change. Initially, the ratio of B to A is 3:23:2, which can be scaled to 12:812:8. After adding liquid A, the ratio of B to A becomes 6:56:5, which is equivalent to 12:1012:10. The change in liquid A's ratio units is 10−8=210 - 8 = 2 units. Since the actual change is 120120 milliliters, each unit represents 120÷2=60120 \div 2 = 60 milliliters. The original mixture has 8+12+15=358 + 12 + 15 = 35 units of volume, giving a total of 35×60=2,10035 \times 60 = 2,100 milliliters.
Estimated Time:1m 30s
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Topics

Ratios, Rates, and Proportions

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