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April 1st, 2018, 08:58 AM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  How to solve this percentage problem?
The price of rice is increased by 20%, By what percentage should the consumption be reduced so that the expenditure on rice remains the same? Ans : 16(2/3)% Mixed fraction Solution : My sir wrote: (P X C) = P + (20P/100) * C  XC/100? Why did my sir consider (P X C) in calculation? Shortcut method is 100 * X/ 100 + X I didn't understand shortcut formula also, please help. Last edited by Ganesh Ujwal; April 1st, 2018 at 09:20 AM. 
April 1st, 2018, 10:20 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 550 
$\text {physical units consumed } = c.$ $\text {price per physical unit } = p.$ $\text {total amount expended } = t = p * c.$ Label things. If the price goes up but the total amount expended stays the same, then the physical units consumed must go down. Do you see that? So $(p + \Delta p)(c + \Delta c) = t.$ But $\Delta p = 20\% \text { of } p \implies \Delta p = 0.2p \implies (p + \Delta p) = 1.2p \implies$ $1.2p(c + \Delta c) = t \implies (c + \Delta c) = \dfrac{t}{1.2p} = \dfrac{p * c}{1.2p} = \dfrac{c}{1.2} \implies$ $\Delta c = \dfrac{c}{1.2}  c = c \left (\dfrac{1}{1.2}  1 \right ) = c \left ( \ \dfrac{0.2}{1.2} \right) = \ c * \dfrac{1}{6} \implies$ $\dfrac{\Delta c}{c} = \ \dfrac{1}{6} \approx \ 16.67 \%.$ Do you follow that? 
April 1st, 2018, 04:05 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond 
$$p\cdot c_1=T$$ $$\frac65p\cdot c_2=T$$ Divide and rearrange... $$\frac65c_2=c_1\implies c_2=\frac56c_1$$ ,,, $c_1$ must be reduced by $\frac16$. 

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