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Zorluk: OrtaRatio and Proportion Word Problems

A corporate marketing team initially allocates its quarterly budget among Digital, Print, and Event advertising in the ratio 5:3:25 : 3 : 2, respectively. Midway through the quarter, $12,000\$12,000 is reallocated from Digital advertising to Event advertising, while the Print advertising budget remains unchanged. As a result of this transfer, the ratio of the Digital advertising budget to the Event advertising budget becomes 4:34 : 3. What was the total initial quarterly budget, in dollars, allocated across all three advertising categories?

Cevap: 120000 dollars

Cevap

The total initial quarterly budget allocated across all three categories was 120,000 dollars.
By representing the initial amounts as 5x5x, 3x3x, and 2x2x, the total initial budget is 10x10x. Accounting for the $12,000\$12,000 shift gives 5x−12,0002x+12,000=43\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3}. Solving yields x=12,000x = 12,000, which gives a total initial budget of 10×12,000=120,00010 \times 12,000 = 120,000 dollars.

Adım Adım Çözüm

1
Define initial allocations using a multiplier variable
Digital = 5x5x, Print = 3x3x, Event = 2x2x, Total Initial Budget = 10x10x
Representing ratio components algebraically allows set up of linear equations.
2
Adjust allocations according to the budget transfer
New Digital = 5x−12,0005x - 12,000, New Event = 2x+12,0002x + 12,000
Reallocating $12,000\$12,000 reduces Digital by 12,000 and increases Event by 12,000.
3
Set up and solve the ratio equation for xx
5x−12,0002x+12,000=43⇒15x−36,000=8x+48,000⇒7x=84,000⇒x=12,000\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3} \Rightarrow 15x - 36,000 = 8x + 48,000 \Rightarrow 7x = 84,000 \Rightarrow x = 12,000
Equating the altered ratio of Digital to Event to 4/34/3 allows solving for multiplier xx.
4
Compute the total initial budget
Total = 10×12,000=120,00010 \times 12,000 = 120,000
Substituting x=12,000x = 12,000 back into the total initial expression 10x10x gives the final answer.

Anahtar Kavram

Altering non-isolated ratios through part-to-part modifications
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Ratio and Proportion Word Problems

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