Hard Problem Solving Question with Solution

Hard Problem Solving Question

Last week I gave you a Level 5 Problem Solving question for the revised SAT to try. Today I will provide a solution for this problem. If you have not yet attempted the problem go back and take a look at it first so you can try it on your own. Here is the link: Hard Problem Solving Question for the Revised SAT

Level 5 – Problem Solving

Here is the problem once again followed by a solution:

John, a United States resident, is on vacation in Spain and is trying to decide if he should use his own credit card from the U.S., or to purchase a prepaid credit card for 500 euros in Spain.

The bank that issues John’s U.S. credit card converts all purchase prices at the foreign exchange rate for that day, and an additional fee of 6% of the converted cost is applied before the bank posts the charge.

If John decides to purchase the prepaid card, he can use this card spending dollars at the exchange rate for that day with no fee, but he loses any money left unspent on the card.

Suppose that John does decide to buy the prepaid card. What is the least number of the 500 euros John must spend for the prepaid card to have been the cheaper option? Round your answer to the nearest whole number of euros.

Solution: If we let d be the cost of the 500 euro card in dollars, and we let e be John’s total purchases on the prepaid card in euros, we need

d < 1.06(d/500)e.

We can divide each side of this inequality by d to get 1 < e(1.06/500), and then multiply each side of this last equation by 500/1.06 to get 500/1.06 < e, or equivalently e > 471.698 .

So the least number of the 500 euros John must spend is 472.

For an even more detailed solution to this problem check out New SAT Math Problems arranged by Topic and Difficulty Level.

Feel free to add your own solutions to the comments.