Question 170205
Let x=amount of rice he starts with and y=amount he has leftover




<h4>After First Station</h4>


Since "At the first he gives up 1/3 of his rice", this tells us that after the first station, he has {{{x-(1/3)x}}} pounds of rice leftover


Note: I've subtracted 1/3 of the amount "x" from "x"




Combine like terms to get {{{(2/3)x}}}. So he has {{{(2/3)x}}} pounds of rice leftover after the first station (this makes sense because the guard took 1/3 so the man must have 2/3 leftover).




Note: remember, the "x" is just a number (a number in which we do NOT know). However, to help you get an idea of what is going on, let {{{x=12}}} (ie let's say that he started at 12 lbs of rice). After the first station, he would have {{{(2/3)12=24/3=8}}} lbs of rice.



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<h4>After Second Station</h4>



Keep in mind that the man still has {{{(2/3)x}}} lbs of rice. 




Since at "the second he is allowed to keep 1/3 of what is left", this means that they will take 2/3 of what he has left. 




So this means that you need to subtract like this


Amount he has left - (2/3)*(Amount he has left )



{{{(2/3)x - (2/3)*((2/3)x)}}}



{{{(2/3)x - (4/9)x}}} Multiply



{{{(2/9)x}}} Combine the fractions.




So after the second station, he has {{{(2/9)x}}} lbs of rice left.

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<h4>After Third Station</h4>



Finally, because "at the third he gives up 1/4 of what remains", this tells us that we subtract 1/4 of what he has left like this:


Amount he has left - (1/4)*(Amount he has left )



Which algebraically looks like:



{{{(2/9)x-(1/4)((2/9)x)}}}



{{{(2/9)x-(1/18)x}}} Multiply



{{{(1/6)x}}} Combine the fractions.




So after the third station, he has {{{(1/6)x}}} lbs of rice left.




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Since "he has 6 pounds less than he started with." this means that he ended up with {{{x-6}}} lbs. Also, we know that he ended up with {{{(1/6)x}}} lbs of rice (after passing through the third station)



So this means that the two are equal, so...


{{{x-6=(1/6)x}}}



{{{6x-36=x}}} Multiply EVERY term by the LCD 6 to clear the fraction.



{{{6x=x+36}}} Add {{{36}}} to both sides.



{{{6x-x=36}}} Subtract {{{x}}} from both sides.



{{{5x=36}}} Combine like terms on the left side.



{{{x=(36)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



{{{x=7.2}}} Divide.



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Answer:



So the man started with 7.2 lbs of rice and the amounts after each station are:



1st station: {{{(2/3)(7.2)=4.8}}} so he had 4.8 lbs after the 1st station



2nd station: {{{(2/9)(7.2)=1.6}}} so he had 1.6 lbs after the 2nd station



3rd station: {{{(1/6)(7.2)=1.2}}} so he had 1.2 lbs after the 3rd station




Since the difference between 7.2 and 1.2 is {{{7.2-1.2=6}}} this helps confirm our answer.