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I've been looking at some discounted prices of goods.

They are listed with a $20\%$ discount, so to work this out I did:

$$ \$25.45 \cdot 0.8333 = \$21.21. $$

But their total was $20.34$, which I presume they got by doing $25.42 \cdot 0.8$.

To apply a $20\%$ discount or to subtract $20\%$, which of the above is correct?

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    $\begingroup$ $25.42\times (1 - 20/100)$ $\endgroup$ Jun 2, 2017 at 15:58
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    $\begingroup$ why multiply by $5/6$? $\endgroup$
    – Dando18
    Jun 2, 2017 at 16:00
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    $\begingroup$ You tell us neither where the $25.45$ comes from, nor the $0.8333$. $\endgroup$
    – Carsten S
    Jun 2, 2017 at 17:39
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    $\begingroup$ Care to check your figures. You quote 25.45 then 25.42. Can't both be accurate. $\endgroup$
    – Tim
    Jun 3, 2017 at 14:16
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    $\begingroup$ Umm, seriously? This is a school-level problem... $\endgroup$
    – nicael
    Jun 4, 2017 at 9:47

6 Answers 6

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A $20\%$ discount means that the price is $80\%$ of what it was originally, so you multiply by $1-0.2=0.8$.

$0.8\dot{3}$ is $1/1.2$, which is used to find out the original price when you have been given $120\%$ of it (if there's $20\%$ tax included in it, for example)

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    $\begingroup$ So $1-r$ is not the same as $\frac{1}{1+r}$. This is related to the fact that $(1-r)(1+r)$ is not one exactly (physicists would say it is $1$ only to the first order in $r$). Another well-known way of thinking of it is if you first remove $\frac{r}{100}$ percent, then add $\frac{r}{100}$ percent of the resulting amount, then you're not back where you started. $\endgroup$ Jun 4, 2017 at 7:17
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Note that multiplying by $0.83\bar3$ is really dividing by $1.2$. In order to know which one is right, you need to keep in mind which number represents $100\%$, or the origin. Calculating percentages from the origin is done by multiplying, while recovering the origin from some given percentage is done by dividing.

So finding $80\%$ of a price is done by multiplying with $0.8$. If you have the $80\%$ price and want to find the original price, you divide by $0.8$ (which becomes multiplying by $1.25$).

So multiplying with $0.83\bar3$, when we're talking about $20\%$ and not $16.6\bar6\%$, is, as mentioned above, really dividing by $1.2$. That means that what we find is the original price when we're given the new price after a $20\%$ price increase. That's not the same as finding $80\%$.

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An $x \%$ discount normally means you subtract $x \%$ of the original price.

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    $\begingroup$ @TheGreatDuck I disagree. $\endgroup$ Jun 2, 2017 at 22:46
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    $\begingroup$ @TheGreatDuck What don't you understand about my answer? $\endgroup$ Jun 3, 2017 at 2:13
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    $\begingroup$ @TheGreatDuck Not an ounce of multiplication is even mentioned here Well, "x% of the original price" counts as a multiplication in my books. You take the "original price", then multiply it by "x%" which is in this case "a 20% discount" according to the OP, meaning you multiply by $20\% = 0.2\,$. Once you subtract that from the original price $\,x\,$, what's left to pay is $\,x - 0.2 x = 0.8 x\,$. Not sure what your problem was with this answer. $\endgroup$
    – dxiv
    Jun 3, 2017 at 6:20
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    $\begingroup$ @TheGreatDuck I would expect the OP to very easily connect the x% discount from this answer to a 20% discount from the posted question, as to reckon that $x=20\,$. $\endgroup$
    – dxiv
    Jun 3, 2017 at 6:26
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    $\begingroup$ @TheGreatDuck Read the question again, what the OP wrote was to work this out I did ... x 0.8333. That is not equivalent to subtracting $20%$ or multiplying by $0.8$, which all other answers pointed out, and so did this one. That is the point I was trying to drive, but don't know how to make any more clear, so I'll leave it at this. $\endgroup$
    – dxiv
    Jun 3, 2017 at 6:41
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A $20\%$ increase in price can be computed by dividing by $0.83333\ldots$ (with $3$ repeating), but that does not mean a $20\%$ decrease results from multiplying by that number. The reason is that in the latter problem we're dealing with $20\%$ of a different quantity. A $20\%$ decrease is computed by multiplying by $1-0.2 = 0.8.$

For example, if you cut a $\$100$ price by $50\%$ and then increase it by $50\%,$ you don't get back $\$100,$ but rather $\$75.$

(Multiplying by $1.2$ is simpler than dividing by $0.83333\ldots$ and is more accurate unless you know how to take the infinite repetition of the $3$ into account (and calculators that I've seen do not know how to do that).

Postscript:

\begin{align} 80\% \text{ of } \$25.45 & = \$25.45 \times 0.8 = \$20.36 \\[10pt] 80\% \text{ of } \$25.45 & = \$25.45 \times \frac 4 5 = \$5.09\times 4 = \$20.36 \\[10pt] \$25.45 & = \frac 5 4 \times \$20.36 = 5\times\$5.09 \\[10pt] \$25.45 & = 1.25 \times \$20.36 = 125\% \text{ of } \$20.36 \end{align}

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  • $\begingroup$ Whatcha mean infinite? You'd only need enough precision to figure out the partial penny (e.g., a tenth of a penny). Even the partial penny only affects rounding, if you do that. If you know the tenth-of-a-penny, there's no reason to figure out the hundredth or thousandth (or infinitth) $\endgroup$
    – TOOGAM
    Jun 3, 2017 at 6:18
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    $\begingroup$ @TOOGAM Not necessarily, you might be buying a thousand units. Your suggestion could mean that you end up paying £833.00 for a thousand one pound items discounted by a sixth, where the correct amount would be £833.33. It's not a big difference, but it's a difference, and that kind of thing tends to make accountants unhappy. $\endgroup$
    – Mike Scott
    Jun 3, 2017 at 12:14
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    $\begingroup$ This is a simple explanation, and makes good sense. +1 Lots of folk (in real life) seem to not have grasped your middle para. $\endgroup$
    – Tim
    Jun 3, 2017 at 14:15
  • $\begingroup$ @TOOGAM : Rounding before the last step is far more hazardous than you seem to realize. $\endgroup$ Jun 4, 2017 at 23:38
  • $\begingroup$ @MikeScott and Michael Hardy: Obviously rounding too early can affect things. Why'd both of you think there's another step, such as multiplying the $25.45 by a thousand? Both of you seem to indicate there is at least one more step, but the question didn't say anything about a thousand units, nor did the question specify that the 20% discount is per final order (instead of per item). The question didn't say $25.45 "each unit". After discussing "the goods", it says "their total was 20.34", suggesting that we've already added (to handle quantities), and are looking at the more final totals. $\endgroup$
    – TOOGAM
    Jun 5, 2017 at 2:07
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The trouble with percentages (and a major source of math anxiety when people are doing mental arithmetic) is that they are inherently ambiguous. The question most people are not trained to ask is percent of what. Most of the problems occur because when percentages are use the percent of what is not explicitly specified.

In this question a 20% discount probably means deduct 20% from the stated price. But, unless this is stated very explicitly there is still room for confusion. If it really is 20% off the stated price then the correct result is price*0.8.

But it is a little ambiguous. Here is an example. In the UK we have a sales tax called VAT levied at 20%. But 20% of what? Legally it is 20% of the pre-tax price but shops have to quote the total price after tax has been applied (something the USA ought to mandate to avoid confusion for foreigners if not locals as well). So if a discount is described as "we pay your VAT" it sounds like a 20% discount to many but is actually a reduction of ~16.7% on the stated price (1/1.2 since the stated price is pre-tax price*1.2).

So, if you want to avoid confusion with percentages, always ask percent of what?

And, if you are going to be a good scientist always specify explicitly what your percentages mean. If your drug "reduces deaths by 20%" be very explicit in saying from what base (and tell us what percentage of people die without the drug so we can judge the absolute risk: 10 deaths reduced to 8 is a significant result when only 100 people were in the trial but not much when there are 10,000 people in the trial).

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You cannot multiply anything by 0.8333 to arrive at 80% of the original. All you end up with is 83.33% of the original. So to get a discount of 20%, you need to calculate 80% 0f that figure. Don't know where '.8333' came from! That only gives a discount of 16.7%. X times 0.8 does it right. To get that discounted figure back to its original, multiply by 1.25.

Please explain why you thought 0.8333 was a good idea, and also edit the question with one start point value - you wrote two different ones.

Basic answer, the 'less 20%' - aka times 0.8 can be the only way. Even if tax was applied to the item before and after discount, it would make no difference.

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  • $\begingroup$ We have phantom downvoters on this site too! I need to know what the reason is, please! $\endgroup$
    – Tim
    Jun 4, 2017 at 16:44
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    $\begingroup$ I didn't downvote, though my guess would be that downvoters didn't feel that this answer added anything beyond what's already been covered in prior answers. Repetition makes for clutter and consumes readers' time for no return. You'd probably have gotten a lot of upvotes if this were the first answer. $\endgroup$
    – Nat
    Jun 4, 2017 at 21:53

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