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Question

Difficulty: HardLinear Equations in One Variable

A corporate enterprise allocates a total annual budget of $52,000\$52,000 across three departments: Marketing, Operations, and Development. The amount allocated to Operations is $5,000\$5,000 less than twice the amount allocated to Marketing. The amount allocated to Development is $1,000\$1,000 more than half of the combined allocations of Marketing and Operations. What is the budget allocation for Development?

  1. A
    $13,000\$13,000
  2. B
    $17,000\$17,000
  3. $18,000\$18,000Answer
  4. D
    $21,000\$21,000
  5. E
    $24,000\$24,000

Answer

The budget allocation for Development is $18,000\$18,000.
By defining the Marketing budget as xx, Operations becomes 2x−5,0002x - 5,000 and Development becomes 12(3x−5,000)+1,000=1.5x−1,500\frac{1}{2}(3x - 5,000) + 1,000 = 1.5x - 1,500. Adding these three expressions equals the total budget of $52,000\$52,000, giving 4.5x−6,500=52,0004.5x - 6,500 = 52,000. Solving for xx yields x=13,000x = 13,000. Substituting x=13,000x = 13,000 into the expression for Development gives 1.5(13,000)−1,500=$18,0001.5(13,000) - 1,500 = \$18,000.

Step-by-Step Solution

1
Define the variable and express each department's budget in terms of that variable.
Let xx be the Marketing budget in dollars. Then Operations =2x−5,000= 2x - 5,000. The combined Marketing and Operations budget =x+(2x−5,000)=3x−5,000= x + (2x - 5,000) = 3x - 5,000. Therefore, Development =12(3x−5,000)+1,000=1.5x−1,500= \frac{1}{2}(3x - 5,000) + 1,000 = 1.5x - 1,500.
Establishing a single linear variable allows all three department budgets to be combined into one linear equation.
2
Set up the total budget equation and solve for xx.
x+(2x−5,000)+(1.5x−1,500)=52,000  ⟹  4.5x−6,500=52,000  ⟹  4.5x=58,500  ⟹  x=13,000x + (2x - 5,000) + (1.5x - 1,500) = 52,000 \implies 4.5x - 6,500 = 52,000 \implies 4.5x = 58,500 \implies x = 13,000.
Summing the allocations for Marketing, Operations, and Development yields the total corporate budget of $52,000\$52,000.
3
Calculate the specific budget allocation requested for Development.
Development allocation =1.5(13,000)−1,500=19,500−1,500=18,000= 1.5(13,000) - 1,500 = 19,500 - 1,500 = 18,000.
Substituting x=13,000x = 13,000 back into the algebraic expression for Development gives the final required dollar amount.

Key Concept

Formulating and solving multi-step linear equations in one variable from word problems.
Estimated Time:2m 30s
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Topics

Linear Equations in One Variable

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