Examkin

Question

Difficulty: HardRatios, Rates, and Proportions

A pharmaceutical company produces two strengths of a certain medication, Strength P and Strength Q, by mixing an active ingredient with a liquid base. In Strength P, the ratio of the volume of the active ingredient to the volume of the liquid base is 22 to 55. In Strength Q, the ratio of the volume of the active ingredient to the volume of the liquid base is 33 to 88. A technician prepares a batch of Strength P and a batch of Strength Q such that the volume of the liquid base used in the batch of Strength P is equal to the volume of the liquid base used in the batch of Strength Q. If the total volume of the active ingredient used in both batches combined is 6262 liters, what is the total volume of the liquid base, in liters, used in both batches combined?

Answer: 160 liters

Answer

The total volume of the liquid base used in both batches combined is 160 liters.
The total volume of the liquid base used in both batches combined is 160 liters. To find this, let the volume of the liquid base in each batch be LL liters. The volume of the active ingredient in Strength P is 25L\frac{2}{5}L liters, and the volume of the active ingredient in Strength Q is 38L\frac{3}{8}L liters. The sum of the active ingredients is 25L+38L=62\frac{2}{5}L + \frac{3}{8}L = 62 liters. Finding a common denominator gives 1640L+1540L=62\frac{16}{40}L + \frac{15}{40}L = 62, which simplifies to 3140L=62\frac{31}{40}L = 62. Solving for LL yields L=62×4031=80L = 62 \times \frac{40}{31} = 80 liters. Since both batches contain equal volumes of liquid base, the combined total volume of the liquid base is 2L=2×80=1602L = 2 \times 80 = 160 liters.

Step-by-Step Solution

1
Represent the volumes of the ingredients in terms of a single variable.
Let the volume of the liquid base used in each batch be 40x40x liters, where 40 is chosen as the least common multiple of 5 and 8. The volume of the active ingredient in Strength P is 25×40x=16x\frac{2}{5} \times 40x = 16x liters, and the volume of the active ingredient in Strength Q is 38×40x=15x\frac{3}{8} \times 40x = 15x liters.
This establishes a common algebraic scale for both mixtures, utilizing the fact that the volume of the liquid base is identical in both batches.
2
Formulate an equation using the total volume of the active ingredient.
16x+15x=6216x + 15x = 62, which simplifies to 31x=6231x = 62.
The problem states that the combined volume of the active ingredient from both batches is 62 liters.
3
Solve for the variable xx.
x=2x = 2
Dividing both sides of the equation by 31 yields the scaling factor.
4
Calculate the total volume of the liquid base used in both batches combined.
Total liquid base = 40x+40x=80x=80(2)=16040x + 40x = 80x = 80(2) = 160 liters.
Since each batch uses 40x40x liters of liquid base, the total volume for both batches is the sum of their individual base volumes, which is 80x80x liters.

Key Concept

Solving ratio and proportion problems with a shared quantity
Estimated Time:2m 30s
Rate this question

Topics

Ratios, Rates, and Proportions

More questions from this topic