Question
Johan leased a car for three years. He paid a one-time fee of $1,000, and an additional $300 per month for the full three years. At the end of the three years, what is the total amount Johan paid for leasing this car?
- A$1,900
- B$4,600
- C$10,800
- D$11,800
Show the answer
Correct answer: D
Three years is 36 months (12 × 3).
Set up an expression to find the total amount Johan paid:
Question analysis
This word problem about leasing a car combines a one-time fee with a monthly payment over three years and asks for the total amount paid. Students must convert years to months, multiply by the monthly payment and add the fee. Leaving out one piece of the cost, or mixing years with months, is the usual slip.
- NYS standards
- Grade level
- Grade 6 content, by NYS standard
- Compared with grade 7
- Below grade 7 This is a short total-cost calculation with whole numbers, easier than typical grade 7 word problems, which bring in percents, negative numbers or fractions. The only care needed is turning years into months.
- Estimated difficulty
- Routine: one familiar step, little to misread.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Much to keep track of minor
- A very close wrong answer minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 4, 41 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
This is the easiest kind of multi-step word problem: whole numbers, a familiar money situation and a direct method. The only places to slip are changing three years into 36 months and remembering the one-time fee after the long multiplication. Each slip has its own choice, so a quick check that both costs are in the total is worth the seconds.
Much to keep track of minor Working it out
The $1,000 fee is paid once while the $300 repeats monthly, so it must be held aside while 36 × 300 is computed. (Sweller, 1988; Carpenter, Just & Shell, 1990)
A very close wrong answer minor Choosing the answer
The choices $11,800 and $10,800 differ by only $1,000 out of about $11,000, so a rough total cannot separate them; only including the one-time fee does. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Strategy: Count the months in the lease first, multiply the monthly payment by that number, and then add the one-time fee.
Traps
- Stops one step early: A choice is a number from partway through the solution.
- Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
The wrong answers, classified
A Units or scale (King et al., 2004)
Multiplies $300 by 3 years instead of 36 months, getting $900, then adds the $1,000 fee to reach $1,900.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Charges the $300 for only 12 months, one year rather than the full three, so 3,600 + 1,000 gives $4,600.
C Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds the 36 monthly payments, 36 × $300 = $10,800, and stops before adding the one-time $1,000 fee.
Worked solution, step by step
Formulas used
- Total cost with a fee
- T is the total paid. f is the one-time fee, paid only once. m is the monthly payment. n is the number of months. You multiply first because the monthly payment repeats every month. Then you add the fee once.
Sort out what you know
Johan paid $1,000 one time. He also paid $300 every month for 3 years. You need the total he paid.
The payments are counted by the month, but the time is given in years. So first change the years to months.
Count the months
Each year has 12 months:
So Johan made 36 monthly payments. Watch out: multiplying $300 by 3 years instead of 36 months gives $900. Adding the fee then gives $1,900, which is choice A.
Add up the monthly payments
Multiply the monthly payment by the number of months:
The monthly payments add up to $10,800.
Add the one-time fee
Use the formula. The fee goes in once:
Watch out: $10,800 is choice C. It leaves out the $1,000 fee. So the answer is D, $11,800.
Check your answer
Round to check. $300 a month for 36 months is a bit more than $10,000. Add the $1,000 fee and you get a bit more than $11,000. $11,800 fits. Choices A and B are far too small for three years of payments.
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