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Question

Difficulty: HardRatios, Rates, and Proportions

A company allocates its annual marketing budget among digital, print, and television advertisements in the ratio 4:3:54:3:5, respectively. Due to a strategy change, the company increases its digital advertising budget by 25%25\%, decreases its print advertising budget by 40%40\%, and increases its television advertising budget by 15%15\%. If the company's total marketing budget after these changes is $376,500\$376,500, what was the company's original print advertising budget?

  1. A
    $30,000\$30,000
  2. B
    $54,000\$54,000
  3. $90,000\$90,000Answer
  4. D
    $94,125\$94,125
  5. E
    $120,000\$120,000

Answer

The original print advertising budget was $90,000\$90,000.
The correct answer is $90,000\$90,000. We define the original digital, print, and television budgets as 4x4x, 3x3x, and 5x5x, respectively. Applying the shifts yields a new digital budget of 4x×1.25=5x4x \times 1.25 = 5x, a new print budget of 3x×0.60=1.8x3x \times 0.60 = 1.8x, and a new television budget of 5x×1.15=5.75x5x \times 1.15 = 5.75x. The sum of these new budgets is 5x+1.8x+5.75x=12.55x5x + 1.8x + 5.75x = 12.55x. Setting 12.55x=376,50012.55x = 376,500 gives x=30,000x = 30,000. The original print budget is therefore 3x=3(30,000)=90,0003x = 3(30,000) = 90,000.

Step-by-Step Solution

1
Represent the original budget allocations using a variable xx based on the given ratio.
Digital budget = 4x4x, Print budget = 3x3x, and Television budget = 5x5x. The original total budget was 4x+3x+5x=12x4x + 3x + 5x = 12x.
Establishing algebraic terms for each part allows us to apply the percentage changes.
2
Calculate the new budget allocations after applying the percentage changes.
New Digital budget = 4x×1.25=5x4x \times 1.25 = 5x. New Print budget = 3x×(1−0.40)=1.8x3x \times (1 - 0.40) = 1.8x. New Television budget = 5x×1.15=5.75x5x \times 1.15 = 5.75x.
Applying the strategic increases and decreases determines the new individual allocations in terms of xx.
3
Sum the new allocations and set the equation equal to the new total budget.
New Total = 5x+1.8x+5.75x=12.55x5x + 1.8x + 5.75x = 12.55x. Since the new total is $376,500\$376,500, we write: 12.55x=376,50012.55x = 376,500.
This allows us to solve for the multiplier xx.
4
Solve for xx and calculate the original print advertising budget.
x=376,50012.55=30,000x = \frac{376,500}{12.55} = 30,000. The original print advertising budget is 3x=3×30,000=90,0003x = 3 \times 30,000 = 90,000.
Finding the value of xx allows us to calculate any of the original or new budget components.

Key Concept

Solving multi-step word problems involving ratios, percentage changes, and linear equations.

Alternative Method

Alternatively, you can write the equation directly for the fraction of the budget that print represents: the original print budget is 3x3x, and the new total budget is 12.55x12.55x. The ratio of the original print budget to the new total budget is 312.55\frac{3}{12.55}. Multiplying this ratio by the new total budget yields 312.55×376,500=90,000\frac{3}{12.55} \times 376,500 = 90,000.
Estimated Time:2m 0s
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Topics

Ratios, Rates, and Proportions

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