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Difficulty: Very hardLinear Equations in One Variable

A company's annual budget of $84,000\$84,000 is split among three departments: Research, Marketing, and Operations. The Marketing department receives 23\frac{2}{3} as much funding as the Research department. The Operations department receives $6,000\$6,000 more than half of the combined funding of the Research and Marketing departments. What is the amount, in dollars, allocated to the Research department?

  1. A
    24,00024,000
  2. B
    28,80028,800
  3. 31,20031,200Answer
  4. D
    33,60033,600
  5. E
    36,00036,000

Answer

31,20031,200
Defining Research funding as xx, Marketing funding becomes 23x\frac{2}{3}x, and Operations funding becomes 12(x+23x)+6000=56x+6000\frac{1}{2}(x + \frac{2}{3}x) + 6000 = \frac{5}{6}x + 6000. Summing all three department allocations yields x+23x+56x+6000=84,000x + \frac{2}{3}x + \frac{5}{6}x + 6000 = 84,000. Combining the variable terms gives 52x+6000=84,000\frac{5}{2}x + 6000 = 84,000, which simplifies to 52x=78,000\frac{5}{2}x = 78,000 and yields x=31,200x = 31,200.

Step-by-Step Solution

1
Define the unknown variable for the target quantity.
Let xx represent the dollar amount allocated to the Research department.
The problem asks specifically for the Research department allocation.
2
Express the allocations of Marketing and Operations in terms of xx.
Marketing =23x= \frac{2}{3}x. Combined Research and Marketing =x+23x=53x= x + \frac{2}{3}x = \frac{5}{3}x. Operations =12(53x)+6,000=56x+6,000= \frac{1}{2}\left(\frac{5}{3}x\right) + 6,000 = \frac{5}{6}x + 6,000.
Translating word problem relationships into algebraic expressions.
3
Set up the single-variable linear equation for the total budget.
x+23x+(56x+6,000)=84,000x + \frac{2}{3}x + \left(\frac{5}{6}x + 6,000\right) = 84,000
The sum of allocations across all three departments must equal the total budget of $84,000\$84,000.
4
Combine like terms using a common denominator.
66x+46x+56x+6,000=84,000  ⟹  156x+6,000=84,000  ⟹  52x+6,000=84,000\frac{6}{6}x + \frac{4}{6}x + \frac{5}{6}x + 6,000 = 84,000 \implies \frac{15}{6}x + 6,000 = 84,000 \implies \frac{5}{2}x + 6,000 = 84,000
Simplifying fractional coefficients by finding the common denominator 6.
5
Isolate xx to solve the linear equation.
52x=78,000  ⟹  5x=156,000  ⟹  x=31,200\frac{5}{2}x = 78,000 \implies 5x = 156,000 \implies x = 31,200
Subtracting 6,0006,000 from both sides and multiplying by 25\frac{2}{5}.

Key Concept

Linear Equations in One Variable
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Topics

Linear Equations in One Variable

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