Question
Mr. Smith opened a retirement account with a deposit of $900. This account earns 5% simple interest annually. How many years will it take for his $900 deposit to earn $360 interest?
Grid-in question: work out the answer and enter it as a number.
Show the answer
Correct answer: 8
Calculate simple interest (y) by multiplying the initial deposit (p) by the interest rate (r) by the number of years (t):
Question analysis
Mr. Smith deposits $900 in a retirement account that earns 5% simple interest each year. This grid-in asks how many years it takes for the deposit to earn $360 in interest. Students find one year's interest and then how many years add up to the total, and the percent must become the right decimal.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Simple interest is grade 7 percent work, and a typical problem asks for the interest earned over a given time. This one runs the formula backward to find the time, one extra step, with friendly numbers.
- 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
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 7, 28 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 numbers are friendly once one year’s interest is found, so most of the difficulty is setting up the right relationship. Students who divide $360 by 5 treat the rate as a dollar amount, and students who compare $360 with $900 stop at a percent. Multiplying back, eight years of $45, confirms the $360.
Something you must already know minor Working it out
Simple interest is the deposit times the rate times the years; the stem does not give this rule, and it must be known to see that each year adds the same $45. (Anderson & Krathwohl, 2001)
Turning words into math minor Understanding the question
The question gives the total interest, $360, and asks for the time, so the usual interest calculation has to be run backward, as 45t = 360. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Error-prone arithmetic minor Working it out
Writing 5% as 0.05, not 0.5 or 0.005, is the one place-value decision, and a slip there changes the yearly interest by a power of ten. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Strategy: Work out the interest for one year, 5 percent of $900, then divide the $360 target by that yearly amount.
Traps
- Stops one step early: A choice is a number from partway through the solution.
- Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
- An arithmetic slip: The right method with a plausible arithmetic error: a digit, a place value or a times-table fact.
The wrong answers, classified
40 Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds that $360 is 40 percent of the $900 deposit and enters 40, stopping before dividing by the 5 percent earned each year.
72 Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reads 5% as $5 a year, a fixed amount instead of a share of the $900, and divides 360 by 5.
80 An arithmetic slip (Junker & Sijtsma, 2001)
Writes 5% as 0.005, so a year’s interest seems to be $4.50, and 360 ÷ 4.5 = 80 years.
Worked solution, step by step
Formulas used
- Simple interest
- I is the interest, the money the account earns. P is the deposit you start with. r is the yearly rate, written as a decimal. t is the number of years. With simple interest, every year earns the same amount, because it is always based on the first deposit.
Sort out the question
Mr. Smith puts $900 into an account. It earns 5% simple interest each year. Interest is extra money the bank pays you for keeping your money there. You need the number of years until the interest adds up to $360.
Write 5% as a decimal
Percent means “out of 100.” So 5% is 5 out of 100.
Watch out: 5% is not 0.5 or 0.005. Using 0.005 makes a year’s interest $4.50, and that leads to 80 years.
Find the interest for one year
Each year the account earns 5% of $900.
So it earns $45 every year. Watch out: 5% is not $5 a year. Dividing 360 by 5 gives 72, which is wrong.
Use the formula to find the years
Put the numbers into the formula. I is 360, P is 900 and r is 0.05.
To undo “times 45,” divide both sides by 45.
Give the answer
It takes 8 years. Watch out: $360 is 40% of $900. Gridding 40 stops too soon, because each year adds only 5%. So you would grid 8.
Check your answer
Add up the interest year by year. $45 a year for 8 years is dollars. Also, 8 years at 5% a year is 40%, and 40% of $900 is $360.
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