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Difficulty: MediumAlgebraic Word Problems and Modeling

A boutique perfume workshop creates a signature fragrance blend by combining two fragrance oils, Oil A and Oil B. Oil A costs 12perounceandcontains4012 per ounce and contains 40% pure essential oil by volume. Oil B costs 20 per ounce and contains 80% pure essential oil by volume. The perfumer creates a 30-ounce batch of the signature blend at a total cost that averages exactly $15 per ounce. What is the total volume, in ounces, of pure essential oil contained in this 30-ounce blend?

Answer: 16.5 ounces

Answer

The total volume of pure essential oil contained in the 30-ounce blend is 16.5 ounces.
To find the total essential oil content, first set up a system of equations for the volumes of Oil A (xx) and Oil B (yy): x+y=30x + y = 30 and 12x+20y=45012x + 20y = 450. Solving this system yields x=18.75x = 18.75 ounces and y=11.25y = 11.25 ounces. Multiplying each by its respective essential oil concentration gives 0.40×18.75=7.50.40 \times 18.75 = 7.5 ounces from Oil A and 0.80×11.25=90.80 \times 11.25 = 9 ounces from Oil B, totaling 16.5 ounces of essential oil.

Step-by-Step Solution

1
Set up linear equations representing the total volume and total cost of the mixture.
x+y=30x + y = 30 and 12x+20y=45012x + 20y = 450, where xx is ounces of Oil A and yy is ounces of Oil B.
The total cost of the 30-ounce blend at 15perounceis15 per ounce is 15 \times 30 = 450$ dollars.
2
Solve for the quantities of Oil A and Oil B used in the mixture.
x=18.75x = 18.75 ounces of Oil A and y=11.25y = 11.25 ounces of Oil B.
Substituting y=30−xy = 30 - x into 12x+20(30−x)=45012x + 20(30 - x) = 450 yields −8x=−150-8x = -150, giving x=18.75x = 18.75.
3
Compute the amount of pure essential oil contributed by each component and sum them.
0.40(18.75)+0.80(11.25)=7.5+9=16.50.40(18.75) + 0.80(11.25) = 7.5 + 9 = 16.5 ounces.
Oil A contains 40% essential oil by volume and Oil B contains 80% essential oil by volume.

Key Concept

Systems of Linear Equations and Mixture Modeling
Estimated Time:2m 0s
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Topics

Algebraic Word Problems and Modeling

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