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Can someone help me understand percentages?

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You get a different result because they are different operations: dividing by .85 is like multiplying by 1/.85, which is 1.1765.

What makes you think dividing by .85 should be equal to multiplying by 1.15?

If you're trying to calculate 15% more or less of something:

\- multplying by 1 + percent will give you the increment (1.15 is 15% more)

\- **multiplying** by 1 - percent will give you the decrement (0.85 is 15% less)
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115 is 15% more than 100, but 100 is not 15% less than 115. 15% less than 115 would be 115 * 0.85 = 97.75. 100 is ~13.04% less than 115.
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Let's say something costs $1.00. The price rises by 15%. That's 15% of $1.00

Let's say something costs $1.15. The price drops by 15%. That's 15% of $1.15.

15% of $1.15 is more than 15% of $1.00. You shouldn't expect "15%" to always represent the same number, it depends on what you're taking 15% of.

>Ex: 100/.85 equals 117.6470588 100\*1.15 equals 115

Here you're asking the question, "what is 100 85% of? What would you take 15% off from to end up at $1.00?"

As I noted, 15% of $1.15 is more than 0.15. If you discount 15% from $1.15 you end up with less than $1.00.

You have to start with a larger number than $1.15 in order to take off 15% and end up with $1.00. Because 15% of $1.15 is more than 0.15. So that's why it's something like $1.18.
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Whenever you use portions, be clear about what is the whole from which these portions come. They’re not just portions, per se. They’re portions of something.

For example, let’s say we have 16 pencils and we wish to get rid of a fourth (or a quarter) of them. One fourth of 16 is 4. So we get rid of 4 pencils. Now we only have 12 pencils.

Here comes Eliana. Hi, Eliana!

We ask Eliana to add a fourth (or a quarter). She sees 12 pencils. She correctly calculates that a fourth of 12 is 3. So she adds 3 pencils. Now we have 15 pencils!

Maybe you can now see what’s happening here.

When we subtracted a fourth, we could have asked: “A fourth of what?” And the answer would have been: “A fourth of 16.” This means that, for us, 16 is the whole from which we were taking a fourth.

When Eliana added a fourth, we could have asked: “A fourth of what?” And Eliana would have answered: “A fourth of 12.” This means that, for Eliana, 12 is the whole to which we are adding a fourth.

A fourth of 16 is not the same as a fourth of 12.

Percentages are portions. And the same rule applies.

We can present the same example as percentages.

From our 100% of 16 pencils, we got rid of 25% of them, so we were left with 75% of what we had, that is: 12 pencils.

Then Eliana added 25% to that. 25% of 12 is 3. So now we have 15 pencils.

We didn’t end up with 16 pencils because 25% of 16 is not the same as 25% of 12.
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Well the percentage is always relative to something. In absokute units, you can tell for example, that 7 is greater than 4 by 3 units - vice versa 4 is less than 7 by the same 3 units. But if you use percentages, you take a percentage of one of the two numbers here - either 7 or 4.   



So if we take 4 as 100%, than 7 should be 7/4 = 1.75 = 175%, so 75% greater.   



On the other side, if we take 7 as 100%, then 4 is 4/7 ≈ 0.5714 = 57.14%, so 42.86% less.   



The percentage is never the same, it's a relative unit. By the way, here it's either 75% of 4 (which equals 3 by the way) or 42.86% of 7 (which is also 3, exactly 7 - 4).
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In my opinion, percentages are extremely confusing, but it can help to take it to an extreme: think about 100% more of something -- thats 2x the amount you started with. but 100% less of something is nothing -- so 100% more and 100% less are not opposites. Similarly 15% more and 15% less are not opposites, but the effect is smaller
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117.64... \* 0.85 = 100.  Ie take 15% of 117.64... and subtract it from 117.64... and you get 100.  In other words: 117.64... \* 0.15 - 17.64...

Let's rearrange your equation to see how it is similar to my operation above!  You said 100 / 0.85 = 117.64...  Multiply both sides by 0.85.  100 = 117.64... \* 0.85.  That's just what I did above!  I.e. 100/0.85 finds the number for which subtracting 15% gets you 100.  This is something I do all the time because I work in food production.  If my knowns are that *spice blend X* is 90% salt, and I have 50 kgs of salt, then I can figure out the number for which subtracting 10% gets me 50 kg, or more to the point the number for which 90% is 50 kg, or even more to the point how much spice blend I can make!  50 kgs / 0.90 = 55.55... kg.

100 \* 1.15 to be pedantic is equivalent to (100 \* 1) + (100 \* 0.15).  Ie you're adding 15% of what you have to what you have. We call that 15% more.  Whenever you're working with percentages, changes in terms of percentages, and changes **of** percentages there can be difficulty, misinterpretation, and miscommunication.  You just have to make yourself clear on the math and then be wary of interpreting the language!

If for some reason you just really wanted to divide by a decimal you would have to do something equivalent to 100 \* 1.15.  That would be 100 / (1/1.15) = 100 / .8696...  

That's probably not something you would want to do, but as you wrestle with converting language to math it's good to carry out these algebraic manipulations just so you can get comfortable with what's going on where and when and why to do it one way and not another!
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100÷(20/23)=115

(1/1.15=20/23)
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Dividing by 0.85 is equivalent to multiplying by  1 3/17

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