Question
A clothing store buys shirts for a cost of m dollars each. To determine the selling price of the shirts, the manager increases the purchase cost by 55%. Which expression represents the selling price of each shirt?
- A
- B
- C
- D
Show the answer
Correct answer: C
It is given that m represents the purchase cost.
represents 100%. The amount the purchase cost was increased is represented by 55%.
Therefore, the selling price will be .
The selling price can be represented by the expression .
Question analysis
A clothing store buys shirts for m dollars each, and the manager sets the selling price by increasing that purchase cost by 55%. Students choose the expression for the selling price of each shirt. It tests writing a percent increase as an algebraic expression, where a percent of an unknown cost has to be handled in letters.
- NYS standards
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Writing a percent markup as an expression is a named grade 7 idea and a common classroom task. This is a direct case with one percent and a letter for the cost, with no further steps such as tax.
- 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
- Variables instead of numbers minor
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 5, 40 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 markup is an increase by 55%, so the selling price is the whole cost plus 55% of it. The commonest mistake writes only the 55% part, which is the markup and not the price. Others subtract the percent as a discount would, or add 55 cents. Choosing a cost of $100 and finding a $155 price picks out the right expression at once.
Turning words into math moderate Understanding the question
“Increases the purchase cost by 55%” means the cost plus 55% of it, 100% + 55% = 155% of m, not 55% of m on its own. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Variables instead of numbers minor Understanding the question
The cost is the letter m, so the markup has to be written as 0.55m and added to m, with no dollar amount to test the expression against. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
Strategy: Pick an easy cost such as m = 100 dollars, work out the selling price after a 55% increase, and see which expression gives that price.
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.
- Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
The wrong answers, classified
A Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Subtracts the 55% from 100% instead of adding it, writing 0.45m, which is the price after a 55% discount rather than a 55% increase.
B Stops one step early (King et al., 2004; Gierl et al., 2017)
Writes only the amount of the markup, 55% of m, and stops before adding it to the cost the store paid for the shirt.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Adds 0.55 to m as if the 55% markup were 55 cents, a fixed amount, instead of 55% of whatever the shirt cost.
Worked solution, step by step
Formulas used
- Percent increase
- The old amount is what you start with, here the cost m. The 1 keeps the whole old amount. The percent over 100 adds the increase on top. Use it whenever an amount goes up by a percent.
Sort out the question
Each shirt costs the store m dollars. The manager raises that cost by 55% to get the selling price. You need an expression for the selling price. An expression is a math phrase with numbers, letters and signs, but no equal sign.
Find the amount of the increase
The increase is 55% of m. Write 55% as a decimal by dividing by 100.
So the increase is 0.55m dollars, which means 0.55 times m.
Watch out: 0.55m is only the increase. Stopping here gives choice B.
Add the increase to the cost
The selling price is the cost plus the increase. Since m is the same as 1m, add the numbers in front.
The percent increase formula gives the same thing:
Give the answer
Watch out: 0.45m is a 55% decrease, choice A. And m + 0.55 adds only 55 cents, choice D. The increase must grow when the cost grows. So the answer is C, .
Check your answer
Pick a cost of $100, so m = 100. A 55% increase adds $55, for a price of $155. Now try choice C: . It matches. Choice B gives only 55, and choice D gives 100.55.
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