PSAT Math Multiple-Choice Question 43: Answer and Explanation

Question: 43

An alloy needs to contain between 10 and 15% of titanium. Which of the following inequalities represents the amount in kilograms, x, of a 20% titanium alloy that should be mixed with a 5% titanium alloy to produce 10 kilograms of an alloy with the acceptable percentage of titanium?

• A. 1.33 ≤ x ≤ 1.67
• B. 2.25 ≤ x ≤ 3.25
• C. 3.33 ≤ x ≤ 6.67
• D. 7.25 ≤ x ≤ 9.75

Correct Answer: C

Explanation:

C

The question asks for the range of values of x that would satisfy the parameters in the question. Since the question asks for a specific range of values and the answers are increasing order, plug in the answers. Before plugging in, however, find the desired amount of titanium in the final alloy. The question says that the final alloy must contain between 10 and 15% titanium and must weigh 10 kilograms total. If the alloy has 10% titanium, it contains (0.10)(10 kg) = 1 kg of titanium. If the alloy has 15% titanium, it contains (0.15)(10 kg) = 1.5 kg of titanium. So, the final alloy must contain between 1 and 1.5 kg of titanium. Start with (B). Choose a value for x that is within the given range and is easy to work with. Choose x = 3. x represents the weight of the 20% alloy, so there are 3 kg of the 20% alloy. Since the two alloys must add up to 10 kg, there are 10 - 3 = 7 kg of the 5% alloy. Find the weight of titanium in each alloy. There are (0.20)(3 kg) = 0.6 kg in the 20% alloy and (0.05)(7 kg) = 0.35 kg in the 5% alloy. Therefore, there are 0.6 + 0.35 = 0.95 kg of titanium in the final alloy. Since this is not within the desired range of 1 to 1.5 kg, eliminate (B). Also eliminate (A) because it has even smaller values of x. Try (C). Choose x = 4. There are 4 kg of the 20% alloy and 10 - 4 = 6 kg of the 5% alloy. Find the weight of titanium in each alloy. There are (0.20)(4 kg) = 0.8 kg in the 20% alloy and (0.05)(6 kg) = 0.3 kg in the 5% alloy. Therefore, there are 0.8 + 0.3 = 1.1 kg of titanium in the final alloy. Since this is within the desired range, (C) works, so stop here. The correct answer is (C).