Examkin

Question

Difficulty: Very hardSolving Linear Equations

An electronics retailer sells tablet computers. The retailer's weekly revenue from tablets, in thousands of dollars, is modeled by the expression 25(3x−4)\frac{2}{5}(3x - 4), where xx represents the average number of tablets sold per day. The weekly operating costs, in thousands of dollars, are modeled by the expression 13(2x−12)\frac{1}{3}\left(2x - \frac{1}{2}\right). The retailer's weekly profit is equal to the profit of a competitor, which is modeled by 14(x+5)−116\frac{1}{4}(x + 5) - \frac{11}{6} thousand dollars. If the retailer and the competitor earn the same weekly profit, what is the value of 12x−512x - 5?

  1. 31Answer
  2. B
    3
  3. C
    76717\frac{767}{17}
  4. D
    51
  5. E
    36797\frac{367}{97}

Answer

31
The correct answer is 31. By equating the profit models, we get 25(3x−4)−13(2x−0.5)=14(x+5)−116\frac{2}{5}(3x - 4) - \frac{1}{3}(2x - 0.5) = \frac{1}{4}(x + 5) - \frac{11}{6}. Multiplying the entire equation by the least common multiple of the denominators (60) yields 24(3x−4)−20(2x−0.5)=15(x+5)−11024(3x - 4) - 20(2x - 0.5) = 15(x + 5) - 110. Expanding the terms gives 72x−96−40x+10=15x+75−11072x - 96 - 40x + 10 = 15x + 75 - 110, which simplifies to 32x−86=15x−3532x - 86 = 15x - 35. Solving for xx gives 17x=5117x = 51, or x=3x = 3. Evaluating the required expression 12x−512x - 5 for x=3x = 3 gives 12(3)−5=3112(3) - 5 = 31.

Step-by-Step Solution

1
Set up the equation equating the retailer's profit (revenue minus cost) to the competitor's profit.
25(3x−4)−13(2x−12)=14(x+5)−116\frac{2}{5}(3x - 4) - \frac{1}{3}\left(2x - \frac{1}{2}\right) = \frac{1}{4}(x + 5) - \frac{11}{6}
Profit is calculated as revenue minus operating costs. Since the retailer and the competitor earn the same profit, their profit models are equal.
2
Multiply the entire equation by 60 to eliminate all fractional denominators.
24(3x−4)−20(2x−12)=15(x+5)−10(11)24(3x - 4) - 20\left(2x - \frac{1}{2}\right) = 15(x + 5) - 10(11)
The least common multiple (LCM) of 5, 3, 4, and 6 is 60. Multiplying both sides by 60 simplifies the equation into integer terms.
3
Distribute the coefficients to remove parentheses, taking care with the negative signs.
72x−96−40x+10=15x+75−11072x - 96 - 40x + 10 = 15x + 75 - 110
Distributing −20-20 across (2x−12)\left(2x - \frac{1}{2}\right) yields −40x+10-40x + 10, and distributing 2424 across (3x−4)(3x - 4) yields 72x−9672x - 96.
4
Combine like terms on both sides of the equation.
32x−86=15x−3532x - 86 = 15x - 35
On the left side, 72x−40x=32x72x - 40x = 32x and −96+10=−86-96 + 10 = -86. On the right side, 75−110=−3575 - 110 = -35.
5
Isolate the variable term xx on one side and the constants on the other.
17x=51  ⟹  x=317x = 51 \implies x = 3
Subtracting 15x15x from both sides gives 17x−86=−3517x - 86 = -35. Adding 8686 to both sides gives 17x=5117x = 51. Dividing by 17 yields x=3x = 3.
6
Substitute x=3x = 3 into the requested expression 12x−512x - 5 to find the final value.
12(3)−5=3112(3) - 5 = 31
The question asks for the value of the expression 12x−512x - 5 rather than just the variable xx.

Key Concept

Solving multi-step linear equations containing fractions by clearing denominators and distributing terms correctly.
Rate this question

Topics

Solving Linear Equations

More questions from this topic