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Question 26 of 30

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M:C:SPR:SET:B

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Problem Solving and Data Analysis

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Questions 25 and 26 refer to the following information.

An international bank issues its Traveler credit cards worldwide. When a customer makes a purchase using a Traveler card in a currency different from the customer’s home currency, the bank converts the purchase price at the daily foreign exchange rate and then charges a 4% fee on the converted cost.

Sara lives in the United States, but is on vacation in India. She used her Traveler card for a purchase that cost 602 rupees (Indian currency). The bank posted a charge of $9.88 to her account that included the 4% fee.

A bank in India sells a prepaid credit card worth 7,500 rupees. Sara can buy the prepaid card using dollars at the daily exchange rate with no fee, but she will lose any money left unspent on the prepaid card. What is the least number of the 7,500 rupees on the prepaid card Sara must spend for the prepaid card to be cheaper than charging all her purchases on the Traveler card? Round your answer to the nearest whole number of rupees.

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Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No

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7,212

Let d dollars be the cost of the 7,500-rupee prepaid card. This implies that the exchange rate on this particular day is fraction numerator d over denominator 7 comma 500 end fraction dollars per rupee. Suppose Sara’s total purchases on the prepaid card were r rupees. The value of the r rupees in dollars is open parentheses fraction numerator d over denominator 7 comma 500 end fraction close parentheses r dollars. If Sara spent the r rupees on the Traveler card instead, she would be charged open parentheses 1.04 close parentheses open parentheses fraction numerator d over denominator 7 comma 500 end fraction close parentheses r dollars. To answer the question about how many rupees Sara must spend in order to make the Traveler card a cheaper option (in dollars) for spending the r rupees, we set up the inequality 1.04 open parentheses fraction numerator d over denominator 7 comma 500 end fraction close parentheses r greater or equal than d. Rewriting both sides reveals 1.04 open parentheses fraction numerator r over denominator 7 comma 500 end fraction close parentheses d greater or equal than open parentheses 1 close parentheses d comma from which we can infer 1.04 open parentheses fraction numerator r over denominator 7 comma 500 end fraction close parentheses greater or equal than 1.

Dividing on both sides by 1.04 and multiplying on both sides by 7,500 finally yields r greater or equal than 7 comma 212. Hence the least number of rupees Sara must spend for the prepaid card to be cheaper than the Traveler card is 7,212.

Question Difficulty: 
hard
Objective: 

Students must use the currency conversion information and the constraints provided to determine a meaningful value in the context of the problem.