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Take this question for an example:

A smartphone is now $\$500$ after a $20\%$ discount. What was its original price?

Now, this is an example of a type of math problem that students face usually in 7th or 8th grade1, and usually get the question incorrect due to not being careful.

The solution to this specific one would be as follows:

Since it is $500$ dollars after a $20$ percent discount, that means that $500$ dollars is going to be $80$ percent of the original price. Therefore, we can set up this equation$$0.8x=500$$which then we can multiply both sides by $10/8$ (or $5/4$ or $1.25$) to get that the original price of the smartphone would be$$x=\$625$$

While I am not a teacher, I do remember a few specific mistakes that me and my friends would make on these types of problems back in the day:

  1. Using the example question as an example, $\$500$ after a $20\%$ discount implies $0.2x=500$
  2. When tasked to find the discount, for example a price that was knocked down to $\$300$ when it was originally $\$700$, doing $300x=700\implies x=\dfrac73\approx233\%$ discount.

However that is all that I can think of, so my question is, what are other common mistakes that students will make when solving "What's the original price" percentage problems?

While I have found a Blackpenredpen video on this, any other references would be appreciated.

I did attempt a few Google searches, but nothing turned up.


1In America, this is students age 12-13, although it might be different in other places.

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    $\begingroup$ This doesn't just stump students, I think most adults will get it wrong. This type of question is often used in books like "Thinking Fast and Slow" and "Predictably Irrational" -- people think in shortcuts rather than working out the math. $\endgroup$
    – Barmar
    Jan 3 at 15:27
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    $\begingroup$ I'll note that this sort of question is a great way of confronting students with the need to check their work. $\endgroup$
    – Brian
    Jan 3 at 18:58
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    $\begingroup$ @Brian: Absolutely correct. This is a problem where checking your answer is much easier than finding the answer. $\endgroup$
    – James
    Jan 4 at 18:24

3 Answers 3

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The most common one I think is going from

"The product costs $x$ after a $y$ percent discount"

to trying to calculate the original price by

"original.price $ = x \times (100+y)/100$"

It possibly originates from a certain failure in realising that talking about discounts commonly involves a certain mixture of addition/subtraction and multiplication/division, and this subsequent mistaken application of 'inverse rules/operations'

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    $\begingroup$ The second almost equally common mistake is thinking that if the price decreased by 10% today and then by 20% tomorrow, the total drop is 30%. $\endgroup$
    – fedja
    Jan 3 at 0:25
  • $\begingroup$ @fedja this is also an interesting one, and I think it's related $\endgroup$
    – ac15
    Jan 3 at 1:21
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    $\begingroup$ I think a common mistake is that 20% of the original price is not 20% of the discounted price - so it's a bit hard to see why subtracting 20 % from the original price and then adding 20 % to the discounted price results in a different number for the original price, because you expect subtraction and addition of the same quantity (20 %) to cancel out. And the failure to realize that adding or subtracting percents is actually a multiplication. After writing this, you probably meant this, but I had difficulties understanding your condensed form. $\endgroup$
    – Arsenal
    Jan 3 at 15:40
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    $\begingroup$ @Arsenal I was amused that I have to wait for a while before I read someone wrote what you wrote in your first sentence. I thought that would be the most common mistake and the most common explanation of the mistake. $\endgroup$
    – justhalf
    Jan 4 at 11:27
  • $\begingroup$ @justhalf Yeah, I would also bet that this is the most common misconception, at least among adults facing this question outside of the context of a math course. $\endgroup$
    – xLeitix
    Jan 5 at 8:24
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I agree with ac15's answer (and fedja's comment) describing two specific instances of mistake categories, and I would like to propose a more general framing that explains why students make those mistakes and also suggests other mistakes.

The idea is that in general, many students who pattern-match (as opposed to reasoning from first principles) will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions are

  • "directionally correct" in the sense that they make a quantity increase or decrease as expected, and

  • "order-of-magnitude correct" in the sense that the final result is on a sensible order of magnitude compared to expectations.

It's essentially the result of applying Occam's razor in the case when common sense has granted you some significant but imperfect level of predictive ability. (As ac15 points out in the comments, this might be better characterized as the principle of least effort.)

Explaining Previously Mentioned Mistake Categories

If the price is $\$500$ after a $20\%$ discount, then what was the original price?

Well, the original price was obviously higher, and the simplest way to get a sensibly higher number using the quantities given in the problem is to just add $20\%$ of $\$500$ to the original price.

If a $\$500$ price is discounted by $10\%$ today and again by $20\%$ tomorrow, what is the price tomorrow?

Well, the discounted price is obviously lower, and a simple way to get a sensibly lower number using the quantities given in the problem is just to total up the discounts ($10\% + 20\% = 30\%$) and then apply that discount.

Suggesting Another Mistake Category

A third category of mistake suggested by this framing (that nobody has mentioned yet) is that when applying a discount, you might also see a student divide by the markup multiplier. For instance, if a $\$500$ price is discounted by $20\%,$ you might see a student divide $\$500 \div 1.2.$

I used to see this occasionally with students who were first learning about percents. If a student gets used to solving two-step problems about percent increase (write $1.\textrm{something}$ and then multiply or divide), then it's easy for them to mistakenly apply the same technique to get a directionally-correct, order-of-magnitude correct answer to a percent decrease problem. That said, this mistake tends to get hammered out quicker than the previous ones because the corresponding problems are more basic / foundational and therefore students get way more practice with them.

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    $\begingroup$ the "pattern-matching as opposed to reasoning from first principles" bit immediately reminded me of the '3 in the denominator' case Krantz mentions in (page 11 of) 'How to teach mathematics', and may well be a factor/cause in it too. But I think you were looking for/thinking of the principle of least effort instead of Occam, as the former seems to have more of an instinctual character, while the latter (besides demanding equal explanatory power/succesful solution, which the mentioned heuristics lack) seems to be more of a "I have arrived at a correct solution by more [...] $\endgroup$
    – ac15
    Jan 3 at 7:30
  • $\begingroup$ than one mean! Now I'll think about how is it that I did it, and which one of them is better" type $\endgroup$
    – ac15
    Jan 3 at 7:30
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    $\begingroup$ For the third category, dividing $500 by 1.2, it doesn't look like an "error" so much as a "shot in the dark". It looks like a student having no idea what to do and so randomly trying one of the rules they've learned without understanding it. In this case, it's "easy to hammer out" in the sense that the student already did not believe that what they did was correct, they just tried this formula because they felt it was better to use a random formula than to not give an answer at all. $\endgroup$
    – Stef
    Jan 3 at 16:16
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    $\begingroup$ @Stef: Dividing by 1.2 is not a "random formula" or a "shot in the dark". When students do this, they often believe it is correct. If a student gets used to solving two-step problems about percent increase (write 1.something and then multiply or divide), then it's easy for them to mistakenly apply the same technique to get a directionally-correct, order-of-magnitude correct answer to a percent decrease problem. Errors are not limited to carrying out a solution procedure; they also occur in matching a solution procedure to a problem. $\endgroup$ Jan 3 at 19:00
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    $\begingroup$ Your first sentence reminded me of this chestnut from a review: "In this paper are presented incorrect solutions to trivial problems. The basic error, however, is not new". This is from mathscinet.ams.org/mathscinet/relay-station?mr=39515. $\endgroup$ Jan 3 at 19:31
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Whatever their mistake, they should learn that working with percentages is always about performing multiplications:

  • Take $20\%$ of 500 => $\frac{20}{100} \cdot 500 = 0.2 \cdot 500 = 100$
  • Add $20\%$ to 500 => $(1+\frac{20}{200}) \cdot 500 = 1.2 \cdot 500 = 600$
  • Subtract $20\%$ from 500 => $(1 - \frac{20}{100}) \cdot 500 = 0.8 \cdot 500 = 400$

This will make it more or less "obvious" to not make the "$20\% + 30\%$" mistake, as: $500 + 20\% + 30\% = 500 \cdot 1.2 \cdot 1.3 = 500 \cdot 1.56 = 780$.

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  • $\begingroup$ So you think that if a student answers the question, "Increase $200$ by $50$%" with, $200+100=300.$ Then that should be seen as bad practise and pointed out to the student as such? $\endgroup$ Jan 4 at 19:30
  • $\begingroup$ +1 for the $1 \pm x\%$ formalism, -1 for missing the actual answer: OP asks given today's price of \$500 with a 20% discount what was the original price meaning the original price is found by solving $P_0 \cdot (1-.2) = \$500$. $\endgroup$
    – user121330
    Jan 4 at 21:12

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