SAT Math Multiple Choice Question 475: Answer and Explanation

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Question: 475

10. Many wholesale businesses charge customers less per item when they buy those items in bulk. Suppose a baseball cap distributor charges $6 per cap for the first 25 caps the customer purchases, $5 per cap for the next 75 purchased, and $4 per cap for all additional caps over 100. Which of the following piecewise functions represents this scenario, where C represents the total cost and n represents the number of caps purchased?

  • A.
  • B.
  • C.
  • D.

Correct Answer: D

Explanation:

D

Difficulty: Hard

Category: Passport to Advanced Math / Functions

Strategic Advice: Don't let the format of the answer choices intimidate you. Use the Kaplan Method for Translating English into Math to systematically eliminate wrong answers.

Getting to the Answer: Let n be the number of caps ordered. The supplier charges $6 per cap for up to and including 25 caps, so the first inequality should be the cost of the cap multiplied by the number ordered, or 6n, given that n ≤ 25. You can eliminate B because it does not include 25 in the "if" statement. The distributor then charges $5 per cap for the next 75 caps, which means n is greater than 25, but less than or equal to 25 + 75 = 100. The cost of the total order would be the cost of the first 25 caps (6 × 25 = 150) added to the amount of the next set of caps, which is $5 times the number ordered minus the first 25 ordered (because they are at the $6 price point), or 150 + 5(n – 25), so you can now eliminate A. For the final price point, the inequality is simply any order of caps greater than 100, or n > 100. The cost equation is the cost of the first 25 caps (150) plus the cost of the next 75 caps (75 × 5 = 375) plus the cost of the final set of caps, $4 multiplied by the number ordered, minus the first 100 ordered (because they are at either the $5 or $6 price points), or 150 + 375 + 4(n – 100) = 525 + 4(n – 100), making (D) correct.

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