document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185956>Question 1185956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three brothers shared a box of biscuits.<BR>\n" );
document.write( "Roy ate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the box of biscuits and 1/3 of a biscuit.<BR>\n" );
document.write( "Sam ate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the remaining biscuits and 1/3 of a biscuit.<BR>\n" );
document.write( "Tom ate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the rest of the biscuits and 1/3 of a biscuit.<BR>\n" );
document.write( "There were no biscuits left after this. How many biscuits did Roy a eat? </SPAN>\n" );
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document.write( "When each brother eats 2/3 of the biscuits, 1/3 of them are left.  So when each brother eats some of the biscuits, the number of biscuits remaining gets multiplied by 1/3 and the reduced by 1/3.  So<br><BR>\n" );
document.write( "original number of biscuits: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "number after Roy eats his: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"%281%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "number after Sam eats his: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "number after Tom eats his: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3%29-1%2F3 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3%29-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The number of biscuits left at the end is 0, so<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3%29-1%2F3=0 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28%281%2F3%29%28%281%2F3%29x-1%2F3%29-1%2F3%29-1%2F3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That equation is actually not too hard to solve, although it is easy to get lost along the way....<br><BR>\n" );
document.write( "This kind of problem is quite often easier to work backwards, especially if there are a large number of steps.  (Imagine what the final equation above would look like if there had been five (or more) brothers.)<br><BR>\n" );
document.write( "At each step working the problem from beginning to end, the number of biscuits remaining is multiplied by 1/3 and then 1/3 is subtracted.  Working that backwards, each step consists of adding 1/3 to the number remaining (the opposite of subtracting 1/3) and multiplying that by 3 (the opposite of dividing by 3).<br><BR>\n" );
document.write( "Working the problem backwards is then simple:<br><BR>\n" );
document.write( "biscuits remaining at the end: 0<BR>\n" );
document.write( "biscuits remaining before Tom eats his: 3(0+1/3) = 3(1/3) = 1<BR>\n" );
document.write( "biscuits remaining before Sam eats his: 3(1+1/3) = 3(4/3) = 4<BR>\n" );
document.write( "biscuits remaining before Roy eats his: 3(4+1/3) = 3(13/3) = 13<br><BR>\n" );
document.write( "There were 13 biscuits originally; Roy ate 9.<br><BR>\n" );
document.write( "ANSWER: Roy ate 9 biscuits<br>\r<BR>\n" );
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