PSAT Math Grid-Ins Question 38: Answer and Explanation

Question: 38

Type of Metal	Weight (ounces)	Price ($)
Gold	2	2,500
Platinum	5	7,500

A jewelry supply store sells boxes of sheet metal for gold and platinum at the weights and prices listed in the table above. A jeweler wants to purchase at least 90 ounces of metal and plans to spend at least $170,000. If she has already ordered 10 boxes of gold, what is the minimum number of boxes of platinum she can order to satisfy the requirements?

Correct Answer: 20

Explanation:

20

The question asks for the least number of boxes of platinum that the jeweler can order to satisfy the requirements. Translate the English into math in Bite-Sized Pieces. Let the number of gold boxes purchased be g and the number of platinum boxes purchases be p. Since there are two ounces of gold per box, the total weight of the gold purchased is 2g. Since there are five ounces of platinum per box, the total weight of the platinum is 5p. Therefore, the total weight of the metal is 2g + 5p. The jeweler wants to purchase at least 90 ounces of metal, so 2g + 5p ≥ 90. Similarly, a box of gold costs $2,500, so the total cost, in dollars, of the gold is 2,500g. A box of platinum costs $7,500, so the total cost, in dollars, of the platinum is 7,500p. Therefore, the total cost of the metal is 2,500g + 7,500p. This must be at least $170,000, so 2,500g + 7,500p ≥ 170,000. According to the question, the jeweler already ordered 10 boxes of gold, so plug in g = 10 to get 2(10) + 5p ≥ 90 or 20 + 5p ≥ 90 for the first inequality and 2,500(10) + 7,500p ≥ 170,000 or 25,000 + 7,500p ≥ 170,000 for the second inequality. Solve each inequality. Subtract 20 from both sides of 20 + 5p ≥ 90 to get 5p ≥ 70. Divide both sides by 5 to get p ≥ 14. Now look at the other inequality. Subtract 25,000 from both sides of 25,000 + 7,500p ≥ 170,000 to get 7,500p ≥ 145,000. Divide both sides by 7,500 to get p ≥ 19.3. The least value of p that satisfies both inequalities is p = 20. The correct answer is 20.