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?
On
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 $80\%$.
On
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}
On
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).
On
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.
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)