Question
A homeowner has a budget of $135 for the installation of a light fixture. An electrician charges $60 per hour plus a one-time trip charge of $45. What is the greatest amount of time, in hours, the electrician can work on the installation while staying within the homeowner’s budget? Write your answer as a decimal.
Grid-in question: work out the answer and enter it as a number.
Show the answer
Correct answer: 1.5
Let x represent the number of hours the electrician can work.
The cost for the number of hours worked is $60 times the number of hours worked, x. The sum of the one-time trip charge ($45) and the cost of the number of hours worked () must be less than or equal to $135. Express this relationship as an inequality:
Solve the inequality for x:
Written as a decimal, .
Question analysis
A homeowner has $135 to spend, and an electrician charges $60 an hour plus a one-time $45 trip charge. This grid-in asks for the greatest number of hours the electrician can work within the budget, written as a decimal. Students write and solve a two-step inequality, and turning a length of time into decimal hours is the last trap.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Solving a word problem that leads to an inequality of the form px + q ≤ r is the grade 7 standard itself. The numbers are friendly; the only twists are treating the budget as a limit and writing the time in decimal hours.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Turning words into math moderate
- An answer in an unexpected form minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 8, 52 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
The arithmetic is light, $135 − $45 = $90 and then $90 ÷ $60, so the question is decided by the setup and by the form of the entry. The trip charge is paid once, not every hour, and has to come off the budget before dividing. The time works out to an hour and a half, and students who think in hours and minutes may write it the way a clock shows it rather than as a decimal.
Turning words into math moderate Understanding the question
The stem never says “at most”; the limit comes from “a budget of $135”, and the one-time $45 trip charge is added once while the $60 multiplies the hours, giving 45 + 60x ≤ 135. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
An answer in an unexpected form minor Choosing the answer
The grid-in demands a decimal, so an hour and a half must be entered as 1.5, not as 1:30 or 90 minutes, with no choices to confirm the form. (Enright, Morley & Sheehan, 2002)
Strategy: Take the one-time trip charge off the budget first, divide what is left by the hourly charge, and give the time in hours as a decimal rather than in hours and minutes.
Traps
- Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
- Stops one step early: A choice is a number from partway through the solution.
The wrong answers, classified
2.25 Misreads the question (Hegarty, Mayer & Monk, 1995)
Divides the whole $135 budget by $60 an hour, 135 ÷ 60 = 2.25, leaving out the one-time $45 trip charge that must come off first.
1.3 Units or scale (King et al., 2004)
Finds that the $90 left pays for 1 hour 30 minutes and writes that as 1.3, as if minutes were tenths of an hour rather than sixtieths.
90 Stops one step early (King et al., 2004; Gierl et al., 2017)
Subtracts the trip charge and enters the $90 left for labor, stopping before dividing by the $60 hourly charge to turn dollars into hours.
Worked solution, step by step
Formulas used
- Total cost with a one-time charge
- C is the total cost. f is the one-time charge, paid once no matter how long the job takes. r is the rate, the cost for each hour. h is the number of hours. The hourly part grows with time, but the one-time charge does not.
- Keeping an inequality balanced
- The sign ≤ says the left side is less than or equal to the right side. a and b are the two sides. c is a positive number. Subtracting c from both sides, or dividing both sides by c, keeps the statement true.
Sort out the question
The homeowner can spend $135 at most. That is the budget. The electrician has a one-time trip charge of $45. One-time means it is paid once, not every hour. On top of that, the electrician charges $60 for each hour. You need the greatest number of hours that keeps the total at $135 or less. The question asks for a decimal.
Write an inequality
Let h be the number of hours. Use the total cost formula with f = 45 and r = 60. The total must be less than or equal to $135. An inequality is like an equation, but it uses a sign such as ≤ instead of =.
Take off the trip charge
Subtract 45 from both sides. This keeps the inequality balanced.
So $90 is left to pay for the hours. Watch out: don’t divide the whole $135 by $60. That leaves out the trip charge, and you would grid 2.25.
Divide by the hourly rate
60 × h means 60 times h. Divide both sides by 60 to get h by itself.
The electrician can work at most 1.5 hours. Watch out: 1.5 hours is 1 hour 30 minutes, but don’t grid 1.3. Thirty minutes is half an hour, 0.5, not 0.3. Also don’t grid 90. That is dollars, not hours. So you would grid 1.5.
Check your answer
Put 1.5 hours back in: . That is exactly the budget. Now try 2 hours: . That is over $135. So 1.5 hours is the most.
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