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Zorluk: ZorTranslating and Solving Algebraic Word Problems

A digital marketing firm allocates a total monthly advertising budget of BB dollars between search engine ads and social media ads. The amount allocated to social media ads is $800\$800 less than twice the amount allocated to search engine ads. The firm then increases its total monthly budget by 20%20\% and allocates this entire increase to search engine ads. As a result, the new amount allocated to search engine ads is equal to the amount originally allocated to social media ads. What was the firm's original total monthly advertising budget, BB?

  1. A
    $1,600\$1,600
  2. B
    $2,200\$2,200
  3. C
    $2,400\$2,400
  4. $4,000\$4,000Cevap
  5. E
    $8,000\$8,000

Cevap

The original total monthly advertising budget was $4,000\$4,000.
By defining the original search engine advertising budget as SS, the original social media advertising budget as M=2S−800M = 2S - 800, and the original total budget as B=3S−800B = 3S - 800, we can set up the equation S+0.20B=MS + 0.20B = M using the condition that the new search engine budget equals the original social media budget. Solving this equation yields S=1600S = 1600. Substituting S=1600S = 1600 into the expression for the total budget gives B=3(1600)−800=4000B = 3(1600) - 800 = 4000 dollars.

Adım Adım Çözüm

1
Define the variables representing each budget allocation.
Let SS represent the original amount allocated to search engine ads, and let MM represent the original amount allocated to social media ads. The original total budget is B=S+MB = S + M.
Establishing clear variables is necessary to translate the verbal relationships into equations.
2
Translate the relationship between the two advertising categories into an equation.
M=2S−800M = 2S - 800
The problem states that the social media ads budget is $800\$800 less than twice the search engine ads budget.
3
Express the original total budget BB in terms of a single variable, SS.
B=S+(2S−800)=3S−800B = S + (2S - 800) = 3S - 800
Substituting the expression for MM into the total budget equation simplifies the system to a single variable.
4
Set up the equation for the new budget allocation after the 20%20\% increase.
S+0.20B=MS + 0.20B = M
An increase of 20%20\% of the total budget is 0.20B0.20B, which is added entirely to the search engine ads budget, making it equal to the original social media ads budget.
5
Substitute the expressions for BB and MM in terms of SS into the new budget equation.
S+0.20(3S−800)=2S−800S + 0.20(3S - 800) = 2S - 800
Substituting the relationships found in steps 2 and 3 allows us to solve for SS directly.
6
Solve the equation for SS.
S+0.60S−160=2S−800⇒1.60S−160=2S−800⇒640=0.40S⇒S=1,600S + 0.60S - 160 = 2S - 800 \Rightarrow 1.60S - 160 = 2S - 800 \Rightarrow 640 = 0.40S \Rightarrow S = 1,600
Distributing, combining like terms, and isolating the variable yields the original search engine ads allocation.
7
Calculate the original total budget, BB.
B=3(1,600)−800=4,800−800=4,000B = 3(1,600) - 800 = 4,800 - 800 = 4,000
The question asks for the original total budget BB, so we must evaluate the expression for BB using the value of SS.

Anahtar Kavram

Translating verbal descriptions of algebraic relationships into systems of linear equations and solving them.
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Konular

Translating and Solving Algebraic Word Problems

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