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Zorluk: ZorTranslating and Solving Algebraic Word Problems

An online retailer sells two types of subscription boxes: a Basic Box and a Premium Box. Last month, the retailer sold a total of 250250 boxes. The number of Basic Boxes sold was 1010 more than 33 times the number of Premium Boxes sold. If the retailer made a total profit of 4,3504,350 dollars, and the profit from each Premium Box is 55 dollars less than twice the profit from each Basic Box, what is the profit, in dollars, for a single Premium Box?

  1. A
    15
  2. B
    35
  3. C
    31
  4. 25Cevap
  5. E
    28

Cevap

The profit for a single Premium Box is 2525 dollars.
Solving the system of equations for the box quantities gives 190190 Basic Boxes and 6060 Premium Boxes. Setting up the profit relation p=2b−5p = 2b - 5 and the total profit equation 190b+60p=4350190b + 60p = 4350, we substitute b=p+52b = \frac{p+5}{2} to get 155p=3875155p = 3875, which simplifies to p=25p = 25 dollars.

Adım Adım Çözüm

1
Define variables for the quantities of boxes sold and set up a system of equations.
Let BB be the number of Basic Boxes sold and PP be the number of Premium Boxes sold. The given relationships are: B+P=250B + P = 250 and B=3P+10B = 3P + 10.
This translates the word problem statements about the number of boxes into solvable linear equations.
2
Solve for the quantities of each box sold.
Substitute B=3P+10B = 3P + 10 into the first equation: (3P+10)+P=250  ⟹  4P+10=250  ⟹  4P=240  ⟹  P=60(3P + 10) + P = 250 \implies 4P + 10 = 250 \implies 4P = 240 \implies P = 60. Then, B=3(60)+10=190B = 3(60) + 10 = 190.
Finding the exact number of each type of box sold is required to formulate the profit equation.
3
Define variables for the profit of each box and set up the profit equations.
Let bb be the profit of a Basic Box and pp be the profit of a Premium Box. The total profit equation is 190b+60p=4350190b + 60p = 4350. The relationship between the profits is p=2b−5p = 2b - 5, which can be rearranged to b=p+52b = \frac{p+5}{2}.
This sets up the system of equations representing the profit values.
4
Substitute and solve for the profit of a Premium Box (pp).
Substitute b=p+52b = \frac{p+5}{2} into the profit equation: 190(p+52)+60p=4350  ⟹  95(p+5)+60p=4350  ⟹  95p+475+60p=4350  ⟹  155p=3875  ⟹  p=25190\left(\frac{p+5}{2}\right) + 60p = 4350 \implies 95(p+5) + 60p = 4350 \implies 95p + 475 + 60p = 4350 \implies 155p = 3875 \implies p = 25.
Solving this single-variable equation gives the final required value for the profit of a single Premium Box.

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