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I came across the following shortcut formulae while going through a text book:

For questions of the type: If the price of a commodity increases by R%,  find the % decrease in the consumption given that the expenditure remains same 

One can directly find the value by applying: Decrease = 100R/(100 + R)

e.g. If the price of potato is increased by 20%,  by how much should the consumption be decreased  so as to have no change in the expenditure. Decrease = 100*20/(100 + 20) = 50/3 = 16.67% 

Can someone please help me know why the above shortcut formulae works for such type of problems?

Thanks in advance!

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    There is even more shortcut approximate formula that tells that the consumption should be decreased by 20%. :) See my answer http://math.stackexchange.com/questions/27202/what-does-x-faster-mean/27318#27318 .2011-06-12

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First of all, percentages are just another way of talking about hundredths. So whenever you see a percent sign, you might as well imagine it says "/100" instead.

In the given type of problem, the expenditure $x$ is given by the consumption $c$ times the price $p$ (per unit of consumption), i.e. $x=cp$. Now if $p$ increases by $R\%$ of $p$, that is, by $(R/100)p$, then $c$ needs to change to c' such that $x$ remains the same, i.e. c'(p+(R/100)p)=cp. Solving for c' yields

c'=\frac{p}{p+(R/100)p}c=\frac{1}{1+R/100}c=\frac{100}{100+R}c\;.

Then the relative change in consumption is

\frac{c'-c}{c}=\frac{\frac{100}{100+R}c-c}{c}=\frac{100}{100+R}-1=-\frac{R}{100+R}\;.

Since the relative change is the percentage change over $100$, the percentage change is $100$ times this, i.e. $-100R/(100+R)$.

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    Thanks a lot @joriki for explaining this so neatly!2011-06-12