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Modeling Proportional Relationships - Pre Algebra

  1. Two quantities xx and yy are in a proportional relationship. When x=12x = 12, y=18y = 18. If this relationship is graphed and then translated 3 units up, what would be the equation of the new line?
    Answer: y=1.5x+3y = 1.5x + 3 First, find the constant of proportionality: k = y/x = 18/12 = 1.5, so the original equation is y = 1.5x. Translating 3 units up means adding 3 to the y-value: y = 1.5x + 3. Choice B translates down. Choice C translates horizontally (left 3 units). Choice D incorrectly changes the slope.
  2. A phone plan charges $0.15 per minute for calls. Which representation does NOT correctly model this proportional relationship where mm represents minutes and CC represents total cost?
    Answer: A graph showing a line through (0,0) and (30,4) For $0.15 per minute: Choice A is correct equation. Choice B: 20 minutes costs 20 × $0.15 = $3.00 ✓. Choice C: All values check (10 × $0.15 = $1.50, etc.) ✓. Choice D: 30 minutes should cost 30 × $0.15 = $4.50, not $4.00 ✗. The point (30,4) represents $4.00 for 30 minutes, which gives an incorrect rate of $4.00/30 ≈ $0.133 per minute.
  3. A car rental company charges a flat fee plus a rate per mile. After driving 150 miles, the total cost is $85. After driving 300 miles, the total cost is $130. Which equation represents the proportional part of this relationship, where yy is the variable cost and xx is the number of miles?
    Answer: y=0.3xy = 0.3x First, find the rate per mile: (130 - 85) ÷ (300 - 150) = 45 ÷ 150 = 0.3. The proportional part represents only the variable cost, which is $0.30 per mile, so y = 0.3x. Choice B includes the flat fee, making it the total cost equation. Choice C reverses the variables. Choice D uses the incorrect rate 85/150.

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