Question
A person buys a used car with a down payment of $1,200 and makes monthly payments of $275 for 3 years. What is the total amount the person pays for the car?
- A$2,025
- B$9,900
- C$11,100
- D$13,500
Show the answer
Correct answer: C
Determine the total cost of the car by adding the down payment to the total monthly payments over 3 years.
There are 12 months in a year, so multiply (12 months/year × 3 years) to get a total of 36 months. Determine the total monthly payments by multiplying the monthly payment ($275) × the total number of months (36):
Determine the total cost of the car by adding the down payment ($1,200) to the total monthly payments ($9,900):
The total cost of the car is $11,100.
Question analysis
A used car is bought with a down payment of $1,200 followed by monthly payments of $275 for 3 years, and students find the total paid. The work is to count the payments, multiply, and add the down payment. Mixing up years and months, or forgetting the money paid up front, leads to the wrong totals among the choices.
- NYS standards
- Grade level
- Grade 6 content, by NYS standard
- Compared with grade 7
- Below grade 7 A total made of one payment up front plus equal monthly payments is a short whole-number calculation, simpler than grade 7 word problems with percents or negatives. The only conversion is from years to 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
- Error-prone arithmetic minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 5, 29 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 situation is familiar and the method direct, so this is one of the easier questions on the test. Points are lost in two places: turning 3 years into 36 payments, and remembering the money paid up front after the long multiplication. Each slip, like paying the down payment once a year, leads to a listed total, so checking that the answer holds exactly one $1,200 and 36 payments of $275 is worth the seconds.
Much to keep track of minor Working it out
The $1,200 down payment is paid once while the $275 repeats every month, so it has to be set aside while 36 payments are totaled and then added at the end. (Sweller, 1988; Carpenter, Just & Shell, 1990)
Error-prone arithmetic minor Working it out
The product 275 × 36 = 9,900 is a two-digit multiplication done by hand; splitting it as 275 × 4 = 1,100 and then 1,100 × 9 makes it quick and safe. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Strategy: Turn 3 years into a number of months, multiply the monthly payment by it, and add the down payment once at the end.
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.
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
The wrong answers, classified
A Units or scale (King et al., 2004)
Multiplies the $275 by 3, the number of years, instead of by 36 months, then adds the $1,200 down payment: 825 + 1,200 = 2,025.
B Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds the 36 monthly payments, 36 × $275 = $9,900, and stops without adding the $1,200 down payment made at the start.
D Misreads the question (Hegarty, Mayer & Monk, 1995)
Works out one year as $1,200 + 12 × $275 = $4,500 and multiplies by 3, so the down payment is paid three times instead of once.
Worked solution, step by step
Formulas used
- Changing years into months
- Each year has 12 months. Multiply the number of years by 12 to get the number of months. You need months here because a payment is made every month.
- Total cost with a down payment
- The down payment is money paid once, at the start. The monthly payment is paid every month. Multiply it by the number of months to get all the monthly payments. Then add the down payment only once, because it is paid only once.
Sort out the question
The person pays $1,200 once, at the start. This is the down payment. Then the person pays $275 every month for 3 years. You need the total amount paid for the car.
The payments come once a month, but the time is given in years. So change the years into months first.
Change 3 years into months
Use the months formula:
So the person makes 36 monthly payments.
Watch out: multiplying $275 by 3 counts only 3 payments. That leads to choice A, $2,025.
Add up the monthly payments
Multiply $275 by 36. A quick way is to split 36 into 4 × 9:
So the 36 payments add up to $9,900.
Watch out: stopping here gives choice B, $9,900. The down payment is still missing.
Add the down payment once
Use the total-cost formula:
The person pays $11,100 in all. Adding $1,200 for every year counts it three times. That gives choice D, $13,500.
So the answer is C, $11,100.
Check your answer
Estimate. $275 is a little less than $300, and . Add the $1,200: . The real total should be a little less than $12,000. $11,100 fits.
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