PSAT Math Grid-Ins Question 54: Answer and Explanation

Question: 54

Heating water accounts for a good portion of the average home's energy consumption. Tankless water heaters, which run on natural gas, are about 22 percent more energy efficient on average than electric hot water heaters. However, a tankless hot water heater typically costs significantly more. Suppose one tankless water heater costs $160 more than twice as much as a conventional hot water heater. If both water heaters cost $1,000 together, how many more dollars does the tankless water heater cost than the conventional one?

Correct Answer: 440

Explanation:

440

Difficulty: Medium

Category: Heart of Algebra/Systems of Linear Equations

Getting to the Answer: Translate from English into math to write a system of equations with t = the cost of the tankless heater in dollars, and c = the cost of the conventional heater in dollars. First, a tankless heater (t) costs $160 more (+ 160) than twice as much (2 c) as the conventional one, or t = 2 c + 160. Together, a tankless heater (t) and a conventional heater (c) cost $1,000, or t + c = 1,000. The system is:

The top equation is already solved for t, so substitute 2 c + 160 into the second equation for t and solve for c:

Be careful—that's not the answer! The conventional hot water heater costs $280, so the tankless heater costs 2(280) + 160 = $720. This means the tankless heater costs $720 - $280 = $440 more than the conventional heater.