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I am having headaches whenever question requires choosing base appropriately when manipulating percentage related problems. I am sure I haven't made any sense so far, so lemme choose a example problem first :

A number is increased by 20% and then again increased by 20%. By what percent should the increased number be decreased so as to get back the original number?

My init solution was like :

let there be number $x$ which is increased sequentially twice by $ 20$%$ $ . So the difference between increased number and init number $x$ would be : $ 120$%$ 120$%$ x - x $

Now what to choose as base (increased number or init number $x$ ?) to make the ratio (part to whole) and then convert it in to percent ?

This was just an example of problem I often face , so I'd welcome any concepts/analogy which will make whole base selection procedure easy . Thanks

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It would be better to just consider the increased value of $x$, namely $1.2\cdot1.2\cdot x$ (not the difference). If this increased value is decreased by $y\%$, the final value is $(1-y/100)\cdot(1.2\cdot1.2\cdot x)$. But you know this later value is just $x$. So you have $(1-y/100)\cdot(1.2\cdot1.2\cdot x)=x$. Now solve this for $y$ (note the $x$'s cancel). – David Mitra Jan 31 '13 at 14:06

Always consider a "percent increase" of $n$ as multiplying the current number by $1 + n/100$, and a "percent decrease of $n$ as multiplying the current number by $1 - n/100$.

Always use the last calculated number as a "base". In this case, suppose your number is 100. Increasing it by 20% twice gives you 144. Then, calculate the percentage that 144 needs to be decreased to bring it back to 100, which is about 30.55%, rather than the 44% you might have gotten by using 100 as the "base".

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In short, use the "increased number". – Joe Z. Jan 31 '13 at 14:07
will you theory apply to this question : If A's salary is 20% less than B's salary. By how much percent is B's salary more than A's ? ? – Mr.Anubis Jan 31 '13 at 14:16
@Mr.Anubis Yes. B's salary is 25% more than A's. – Joe Z. Jan 31 '13 at 14:17
It seems working in this case too (which is hard to see where is last value calculated etc) , I mean your theory may work but it's not actual way to solve the problem/thinking of problem to solve. Also I am still not touching the concepts here I believe – Mr.Anubis Jan 31 '13 at 14:20
In order to determine that base, always determine which value the percentage operation is being performed on. (This gets into grammar as well.) For example, in the sentence "B is 20% more than A", A is the value being increased by 20%. – Joe Z. Jan 31 '13 at 14:30

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