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Difficulty: MediumSolving Linear Equations

A craft cider company produces two specialty blends. The production cost, in dollars per gallon, of the premium blend is represented by the expression 34(d−8)\frac{3}{4}(d - 8), where dd is the wholesale cost, in dollars, of a bushel of apples. The production cost, in dollars per gallon, of the dry blend is represented by the expression 0.2(2d+5)0.2(2d + 5). If the production cost per gallon is the same for both blends, what is this production cost, in dollars per gallon?

  1. A
    1.2
  2. B
    1.8
  3. 9Answer
  4. D
    20
  5. E
    45

Answer

The production cost is 9 dollars per gallon.
The correct answer is the value obtained by setting the two cost expressions equal, solving for d=20d = 20, and then substituting d=20d = 20 back into either expression to find the cost of 99 dollars per gallon.

Step-by-Step Solution

1
Set the two production cost expressions equal to each other.
34(d−8)=0.2(2d+5)\frac{3}{4}(d - 8) = 0.2(2d + 5)
The problem states that the production cost per gallon is the same for both blends, so their algebraic representations must be equal.
2
Express the decimal as a fraction and solve for dd.
15(d−8)=4(2d+5)  ⟹  15d−120=8d+20  ⟹  7d=140  ⟹  d=2015(d - 8) = 4(2d + 5) \implies 15d - 120 = 8d + 20 \implies 7d = 140 \implies d = 20
Converting 0.20.2 to 15\frac{1}{5} allows us to clear fractions by multiplying both sides of the equation by the least common multiple of the denominators, which is 2020. We then distribute, isolate the variable terms, and solve for dd.
3
Substitute the value of dd back into either of the original cost expressions to find the production cost.
Cost=34(20−8)=34(12)=9\text{Cost} = \frac{3}{4}(20 - 8) = \frac{3}{4}(12) = 9
The question asks for the production cost per gallon, not the value of the variable dd. Substituting d=20d = 20 into the premium blend cost expression gives the final cost.

Key Concept

Solving linear equations involving fractions and decimals, and evaluating expressions using the solved variable value.

Alternative Method

Convert all numbers to decimals. The premium blend cost is 0.75(d−8)=0.75d−60.75(d - 8) = 0.75d - 6. The dry blend cost is 0.2(2d+5)=0.4d+10.2(2d + 5) = 0.4d + 1. Equating them gives 0.75d−6=0.4d+1  ⟹  0.35d=7  ⟹  d=200.75d - 6 = 0.4d + 1 \implies 0.35d = 7 \implies d = 20. Then find the cost: 0.75(20)−6=15−6=90.75(20) - 6 = 15 - 6 = 9 dollars.
Estimated Time:1m 30s
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Topics

Solving Linear Equations

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