What is the difference between the 2 ways to remove percentages?

Question

Given the value $690$, I want to remove $10$% from that and then remove another $20$% from the resulting value, so as an example, I am doing:

$690 \over {(1 + 0.1 + 0.2})$ = $690 \over {1.3}$ = $530.76.$ Apparently, I was told this is wrong and it should be:

$690 * 0.9 =$ $621 \over {1.2}$ = $517.50$

In the above, the $10$% is non-compounded and the $20$% is compounded. Can someone explain what is the difference between the $2$ and is one way better than the other if I do not know certain values like the start or end value (690 in this case)

If I do $690 \over {1 + 0.111111111111 + 0.2}$, I get closer to $517.50.$

Answer 1

If you remove $10\%$ from $690$, your new value is $690-690\cdot \frac{10}{100}$. This is $690\left(1-\frac{10}{100}\right)$, or more simply $(690)(0.9)$.

To remove $20\%$ from the new value of $(690)(0.9)$, you go through essentially the same process, and arrive at $(690)(0.9)(0.8)$.

Note that we would have gotten to the same final answer if we removed $20\%$ from $690$, and then removed $10\%$ from the result.

I do not know what you mean by "if I do not know certain values." If we only know the percentages removed, we cannot know the dollar amounts.

But if the initial price is $A$, and you remove $10\%$ and then $20\%$, the same reasoning leads to the final price of $A(0.9)(0.8)$.

So if you know that discounts of $10\%$ and then $20\%$ were applied, and the final price was, say, $576$, then you can determine the original price as follows. Let $A$ be the original price. Then $$A(0.9)(0.8)=576,$$ and therefore $$A=\frac{576}{(0.9)(0.8)}.$$

Answer 2

If you remove 10 percent of 690 from 690, then you have 90 percent of 690 left; so removing 10 percent of 690 from 690 gives you $$ 690\cdot (.9)=621. $$

If you remove 20 percent of 621 from 621, you have 80 percent of 621 left; so so removing 20 percent of the new value 621 from 621 leaves you with $$ 621\cdot(.8) = 496.8 $$

It would be different to compute $690\cdot (.7)=483$; because here you are removing 10 percent of 690 and 20 percent of $690$.

I'm not sure what you were trying to do with your solution; but it seems you are doing the "backwards problem". An example

A dress is put on a 30 percent off sale. If the sale price was 690 dollars, what was the original price?

Solution: Let $O$ be the original price. We are told that $$ 690=O\cdot .7; $$ so $O=690/.7=985.71$.

Answer 3

Neither is correct - you should start with $690\times0.9=621$, to get the $10\%$ reduction, and then do $621\times0.8=496.8$ for the subsequent $20\%$ reduction. Note that this is not the same as just doing a $30\%$ reduction straight off, as the second percentage is of $621$, not $690$, so "each percent" is worth less (this is a common error).

To work backwards, if you're given the $496.8$ and told it was obtained in this way, then you should first undo the $20\%$ reduction. To do this, note that $496.8$ is $80\%$ of the value before the reduction, so you can find $1\%$ by dividing by $80$, and then multiply by $100$ to get the full amount; in one step this amounts to dividing by $0.8$. This gives you back $621$, and then by the same procedure you divide by $0.9$ to get the $690$ back.

As a general rule, you should first work out what $100\%$ is - in the original problem, it's the $690$ you start with, and if you're trying to "undo" a percentage reduction it's the amount you're trying to find (although if you're undoing more than one reduction it's probably easier to do them one at a time, so $100\%$ is the number you want to find in the next step). Then most problems can be reduced to dividing to find out what $1\%$ is, and then multiplying by the percentage you actually want. So when starting with the $690$ and reducing by $10\%$ you start with $100\%$ and want $90\%$, so divide by $100$ and multiply by $90$ (equivalently multiply by $0.9$), and when starting with $621$ and undoing the $10\%$ reduction you start with $90\%$ and want $100\%$, so divide by $90$ and multiply by $100$ (equivalently divide by $0.9$).

Answer 4

Rather than comment on the differences between the two approaches, here's the way I would do it. Think in terms of what you want to keep. In the first step, keep 90%, then in the second step keep 80% of that. So just multiply $690(0.9)(0.8) = 690(0.72) = 496.8$.