document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=35432>Question 35432</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box contains some apples. Andree takes 1/2 of them along with one extra apple. Beatrice takes 1/3 of the remaining apples along but put 2 apples back in thebox & finally, corrine takes 5/6 of the remaining apples along with one more apple. There are now 7 apples left in the box. How many apples were in the box before Andree took her share? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #21507 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1806)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=21507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 4><b>A box contains some apples. Andree takes 1/2 of them along\r\n" );
document.write( "with one extra apple. Beatrice takes 1/3 of the remaining\r\n" );
document.write( "apples along but put 2 apples back in thebox & finally,\r\n" );
document.write( "Corrine takes 5/6 of the remaining apples along with one more\r\n" );
document.write( "apple. There are now 7 apples left in the box. How many\r\n" );
document.write( "apples were in the box before Andree took her share?\r\n" );
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document.write( "Two ways to solve it.  The FUN way and the ALGEBRA way. ;-)\r\n" );
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document.write( "The FUN way:\r\n" );
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document.write( "Think of showing the movie of that backward and tell what you \r\n" );
document.write( "see at each step, then when you figure how many were in would\r\n" );
document.write( "the basket have at the first of that movie, that will be the \r\n" );
document.write( "answer.\r\n" );
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document.write( "Showing the movie backwards:\r\n" );
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document.write( " \"7 apples left in the box.\"\r\n" );
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document.write( "So in the backwards movie we start out seeing a box with 7\r\n" );
document.write( "apples in it.\r\n" );
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document.write( "\"Corrine takes 5/6 of the remaining apples along with one\r\n" );
document.write( "more apple\"\r\n" );
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document.write( "So in the backwards movie we see Corrine walking backwards\r\n" );
document.write( "carrying a bunch of apples up to the box which contains 7\r\n" );
document.write( "apples.  Then she puts one apple in the box.  Now in order\r\n" );
document.write( "for her to have been able (in the forward movie) to have taken\r\n" );
document.write( "away 5/6 and to have left those 8, then those 8 would have\r\n" );
document.write( "been the other 1/6 of what was there before she came along.\r\n" );
document.write( "Therefore there had to have been 6x8 or 48 when Corrine first\r\n" );
document.write( "walked up. Then she took 5/6 or 40 away, and left the other\r\n" );
document.write( "8.\r\n" );
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document.write( "So, back to the backwards movie, after seeing Corrine put one\r\n" );
document.write( "in the box, she then proceeds to put in 40 apples.  So we see\r\n" );
document.write( "her walk backwards to the box with an armload of 41 apples.\r\n" );
document.write( "She first puts one, pauses, and then dumps the other 40 in\r\n" );
document.write( "the box.  Now she walks backwards away appleless leaving the\r\n" );
document.write( "box containing 48 apples.  \r\n" );
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document.write( "So in the backwards movie we now see a box with 48 apples in\r\n" );
document.write( "it. Then we see Beatrice walking backwards carrying a bunch\r\n" );
document.write( "of apples up to the box which contains 48 apples. \r\n" );
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document.write( "  \"(Beatrice)...put 2 apples back in the box.\"\r\n" );
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document.write( "So in the backwards movie she takes 2 apples from the 48 in\r\n" );
document.write( "the box, and puts them in her arms along with her other\r\n" );
document.write( "apples, that leaves 46 in the box.  \r\n" );
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document.write( "  \"Beatrice takes 1/3 of the remaining apples...\"\r\n" );
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document.write( "Now in order for her to have been able (in the forward movie)\r\n" );
document.write( "to have taken away 1/3 and to have left those 46, then those\r\n" );
document.write( "46 would have been the other 2/3 of what was there before she\r\n" );
document.write( "came.  Half of 2/3 is 1/3, and half of 46 is 23. So Beatrice\r\n" );
document.write( "took away 23, before she put back those 2.      \r\n" );
document.write( " \r\n" );
document.write( "So, back to the backwards movie, after seeing Beatrice take\r\n" );
document.write( "two from the box, leaving 46 in the box, she then proceeds to\r\n" );
document.write( "put 23 more in the box.  So we see her walk backwards to the\r\n" );
document.write( "box with an armload of 21 apples. She takes 2 out of the box,\r\n" );
document.write( "pauses, and then dumps the 23 in the box.  Now she walks \r\n" );
document.write( "backwards away appleless, leaving the box containing 46+23\r\n" );
document.write( "or 69 apples.  \r\n" );
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document.write( "So in the backwards movie we now see a box with 69 apples in\r\n" );
document.write( "it. Then we see Andree walking backwards carrying a bunch of\r\n" );
document.write( "apples up to the box which contains 69 apples.\r\n" );
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document.write( " \"Andree takes 1/2 of them along with one extra apple.\"\r\n" );
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document.write( "Let's interpret that as \r\n" );
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document.write( "\"Andree takes 1/2 of what's there and then takes one more\r\n" );
document.write( "apple.  \r\n" );
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document.write( "So in the backwards movie, Andree first puts 1 apple in the\r\n" );
document.write( "box.  That makes 70 apples in the box.  In the forward movie,\r\n" );
document.write( "he took half and left half, so in the backwards movie he put\r\n" );
document.write( "another 70 in the box.  So the box contained 140 apples\r\n" );
document.write( "originally.\r\n" );
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document.write( "--------------------------------------------------------\r\n" );
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document.write( "Now for the ALGEBRA way:\r\n" );
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document.write( "A box contains N apples. Andree takes (1/2)N plus one extra\r\n" );
document.write( "apple. So he takes away (1/2)N + 1 from N.\r\n" );
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document.write( "Subtracting, N - [(1/2)N + 1] = N - (1/2)N - 1 = (1/2)N - 1\r\n" );
document.write( "in the box.\r\n" );
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document.write( "Beatrice takes 1/3 of the remaining (1/2)N - 1 \r\n" );
document.write( "So Beatrice takes away (1/3)[(1/2)N - 1] from the remaining\r\n" );
document.write( "(1/2)N-1.  Subtracting,\r\n" );
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document.write( "(1/2)N - 1 - (1/3)[(1/2)N - 1] = (1/2)N - 1 - (1/6)N + 1/3 =\r\n" );
document.write( "(1/3)N - 2/3 \r\n" );
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document.write( "That's how many are left in the box  \r\n" );
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document.write( "But puts 2 apples back in the box\r\n" );
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document.write( "Now there are (1/3)N - 2/3 + 2 = (1/3)N + 4/3 apples in the\r\n" );
document.write( "box.\r\n" );
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document.write( "& finally, Corrine takes 5/6 of the remaining (1/3)N + 4/3.\r\n" );
document.write( "So multiplying:\r\n" );
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document.write( "(5/6)[(1/3)N + 4/3] = (5/18)N + 10/9\r\n" );
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document.write( "Now we subtract these (5/18)N + 10/9 from the (1/3)N + 4/3\r\n" );
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document.write( "(1/3)N + 4/3 - [(5/18)N + 10/9] = (1/3)N + 4/3 - (5/18)N - 10/9 =\r\n" );
document.write( "(1/18)N + 2/9 left in the box \r\n" );
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document.write( "\" along with one more apple\"\r\n" );
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document.write( "So now we subtract one more and get \r\n" );
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document.write( "(1/18)N + 2/9 - 1 = (1/18)N - 7/9 left in the box.\r\n" );
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document.write( "Since there are now 7 apples left in the box, we have the\r\n" );
document.write( "equation:\r\n" );
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document.write( "(1/18)N - 7/9 = 7\r\n" );
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document.write( "Multiply through by 18\r\n" );
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document.write( "N - 14 = 126\r\n" );
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document.write( "N = 140\r\n" );
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document.write( "Edwin McCravy\r\n" );
document.write( "AnlytcPhil@aol.com</PRE><BR>\n" );
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