document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1092717>Question 1092717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If you have 23 heads and 72 total legs, How many cows and roosters do you have? I have already tried 12 cows to get 48 legs and then had 11 roosters to add 22 legs. (A cow has 4 legs and a rooster has 2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #707327 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52847)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=707327\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you have 23 heads and 72 total legs, How many cows and roosters do you have? <BR>\n" );
document.write( "(A cow has 4 legs and a rooster has 2)<BR>\n" );
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document.write( "You can solve this problem in three different ways. <BR>\n" );
document.write( "If you are familiar with systems of linear equations, &nbsp;you can reduce the problem to the system of two linear equations in two unknowns and solve it.<BR>\n" );
document.write( "This way is implemented in the &nbsp;<B>Solution 1</B>&nbsp; below.<BR>\n" );
document.write( "You can also reduce the problem to one equation with one unknown and solve it. &nbsp;This is done in the &nbsp;<B>Solution 2</B>&nbsp; below.<BR>\n" );
document.write( "Alternatively, &nbsp;you can solve the problem simply applying logical reasoning and not using equations at all. &nbsp;This is done in the &nbsp;<B>Solution 3</B>&nbsp; below.\r<BR>\n" );
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document.write( "<B>Solution 1</B>\r<BR>\n" );
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document.write( "Let &nbsp;<B>x</B>&nbsp; be the number of cows and &nbsp;<B>y</B>&nbsp; be the number of roosters at the farm.<BR>\n" );
document.write( "If you count the heads you get the equation<BR>\n" );
document.write( "x + y = 23.<BR>\n" );
document.write( "If you count the legs you get the equation <BR>\n" );
document.write( "4x + 2y = 72.\r<BR>\n" );
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document.write( "So you have the system of two equations with two unknowns<BR>\n" );
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document.write( "To solve this system of equations multiply the first equation by &nbsp;2&nbsp; and subtract the obtained equation from the second one. <BR>\n" );
document.write( "You will get, step by step,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system+%282x+%2B+2y+=+46%2C%0D%0A4x+%2B+2y+=+72%29%0D%0A ALIGN=MIDDLE  ALT=\"system+%282x+%2B+2y+=+46%2C%0D%0A4x+%2B+2y+=+72%29%0D%0A\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "4x - 2x = 72 - 46,<BR>\n" );
document.write( "2x = 26,<BR>\n" );
document.write( "x = 13.\r<BR>\n" );
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document.write( "So, &nbsp;there are &nbsp;13&nbsp; cows in the farm. <BR>\n" );
document.write( "Hence, &nbsp;the number of roosters is &nbsp;23 - 13 = 10.\r<BR>\n" );
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document.write( "Let us check the total number of legs. &nbsp;You have altogether<BR>\n" );
document.write( "4*13 + 2*10 = 52 + 20 = 72 legs.\r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;There are &nbsp;13&nbsp; cows and &nbsp;10&nbsp; roosters at the farm.\r<BR>\n" );
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document.write( "<B>Solution 2</B>\r<BR>\n" );
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document.write( "Let &nbsp;<B>x</B>&nbsp; be the number of cows at a farm.<BR>\n" );
document.write( "Then the number of roosters is &nbsp;23 - x&nbsp; in accordance with the condition.<BR>\n" );
document.write( "If you count the legs you get the equation <BR>\n" );
document.write( "4x + 2*(23-x) = 72.\r<BR>\n" );
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document.write( "To solve this equation open the brackets and combine like terms, &nbsp;step by step:<BR>\n" );
document.write( "4x + 2*23 - 2x = 72,<BR>\n" );
document.write( "2x + 46 = 72,<BR>\n" );
document.write( "2x = 72 - 46,<BR>\n" );
document.write( "2x = 26,<BR>\n" );
document.write( "x = 13.\r<BR>\n" );
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document.write( "So, &nbsp;there are &nbsp;13&nbsp; cows at the farm. <BR>\n" );
document.write( "Hence, &nbsp;the number of roosters is &nbsp;23 - 13 = 10.\r<BR>\n" );
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document.write( "Let us check the total number of legs. &nbsp;You have altogether<BR>\n" );
document.write( "4*13 + 2*10 = 52 + 20 = 72 legs.\r<BR>\n" );
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document.write( "You get the same answer as in the <B>Solution 1</B>.\r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;There are &nbsp;13&nbsp; cows and &nbsp;10&nbsp; roosters at the farm.\r<BR>\n" );
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document.write( "<B>Solution 3</B>\r<BR>\n" );
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document.write( "Let us suppose for a moment that all the animals at the farm have two legs each.<BR>\n" );
document.write( "Under this assumption, the total number of legs is &nbsp;23*2 = 46 legs.\r<BR>\n" );
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document.write( "This number is &nbsp;26 = 72 - 46&nbsp; less than &nbsp;72 &nbsp;legs given by condition. <BR>\n" );
document.write( "Certainly, &nbsp;these &nbsp;26&nbsp; legs belong to cows in the number of &nbsp;2&nbsp; legs to each cow &nbsp;(in addition to that two legs we just counted under the assumption). <BR>\n" );
document.write( "This means that the number of cows is &nbsp;26/2 = 13.<BR>\n" );
document.write( "Hence, &nbsp;the number of roosters is &nbsp;23 - 13 = 10.\r<BR>\n" );
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document.write( "You get the same answer as in the solutions &nbsp;<B>1</B>&nbsp; and &nbsp;<B>2</B>&nbsp; above.\r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;There are &nbsp;13&nbsp; cows and 10 roosters at the farm.\r<BR>\n" );
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document.write( "This kind of problems are traditionally considered as entertainment problems. <BR>\n" );
document.write( "And they traditionally are used to show the students all three approaches.\r<BR>\n" );
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document.write( "To see more similar problems of this kind and their solutions, look into the lessons \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Problem-on-animals-at-a-farm.lesson>Problem on animals at a farm</A>  (*)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/Problem-on-two-wheels-and-three-wheels-bicycles-in-a-sale.lesson>Problem on two-wheel and three-wheel bicycles</A>  \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/Problem-on-tablets-in-containers.lesson>Problem on pills in containers</A> &nbsp;&nbsp;&nbsp;&nbsp;and\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/What-type-of-problems-are-these.lesson>What type of problems are these?</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Notice that the lesson (*) has practically identical problem.<BR>\n" );
document.write( "It is because of tradition again.<BR>\n" );
document.write( "For this post, I simply made the global replacement \"rabbits ---> cows\" and \"turkey ---> roosters\" in that text file <BR>\n" );
document.write( "(and changed the numbers respectively).\r<BR>\n" );
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document.write( "Ha-ha-ha.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;H&nbsp;a&nbsp;p&nbsp;p&nbsp;y &nbsp;&nbsp;l&nbsp;e&nbsp;a&nbsp;r&nbsp;n&nbsp;i&nbsp;n&nbsp;g &nbsp;!&nbsp;!\r<BR>\n" );
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document.write( " </SPAN>\n" );
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