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Difficulty: HardSolving Linear Equations

A company allocates its monthly advertising budget between online advertisements and print media. Last month, the company spent 300300 more than half of its total budget on online advertisements, and 16\frac{1}{6} of its total budget on print media. The remaining 700700 of the budget was spent on administrative fees. If BB represents the total budget in dollars, what is the value of B10+150\frac{B}{10} + 150?

  1. A
    270
  2. B
    300
  3. 450Answer
  4. D
    405
  5. E
    315

Answer

The correct value is 450.
Evaluating the total budget equation yields B=3000B = 3000. Substituting this value into the expression B10+150\frac{B}{10} + 150 gives 450450. This is correct because the individual allocations (online ads, print media, and administrative fees) sum to the total budget, and the fraction arithmetic is performed correctly.

Step-by-Step Solution

1
Define the variables and write the expressions for each category of expenses.
Let BB be the total budget. Online advertisements budget is 12B+300\frac{1}{2}B + 300, print media budget is 16B\frac{1}{6}B, and administrative fees are 700700.
Translating verbal descriptions into algebraic expressions is necessary to build the equation.
2
Set up the linear equation representing the sum of all expenses equaling the total budget BB.
B=(12B+300)+16B+700B = \left(\frac{1}{2}B + 300\right) + \frac{1}{6}B + 700
The total budget is the sum of its individual components.
3
Group like terms and solve for the total budget BB.
Combine constant terms: 300+700=1000300 + 700 = 1000. Combine fraction terms: 12B+16B=36B+16B=46B=23B\frac{1}{2}B + \frac{1}{6}B = \frac{3}{6}B + \frac{1}{6}B = \frac{4}{6}B = \frac{2}{3}B. The equation becomes B=23B+1000B = \frac{2}{3}B + 1000. Subtracting 23B\frac{2}{3}B from both sides gives 13B=1000\frac{1}{3}B = 1000, which yields B=3000B = 3000.
This isolates the variable BB to find the total budget value.
4
Evaluate the required expression B10+150\frac{B}{10} + 150 using the solved value of BB.
300010+150=300+150=450\frac{3000}{10} + 150 = 300 + 150 = 450
The question asks for the value of this specific expression rather than BB itself.

Key Concept

Solving linear equations in one variable, including translating word problems with fractional terms and evaluating algebraic expressions.

Alternative Method

Instead of solving the equation algebraically, we could test the options to find the total budget BB. Since each option represents the value of V=B10+150V = \frac{B}{10} + 150, we can express BB as B=10(V−150)B = 10(V - 150). For the correct option of 450450, we get B=10(450−150)=3000B = 10(450 - 150) = 3000. Substituting B=3000B = 3000 back into the original word problem description: half the budget plus 300300 is 1500+300=18001500 + 300 = 1800; one-sixth of the budget is 500500; the remaining is 3000−1800−500=7003000 - 1800 - 500 = 700, which matches the given administrative fees.
Estimated Time:2m 0s
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Topics

Solving Linear Equations

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