document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=949394>Question 949394</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A supply officer distributes 80 business envelops among 13 employees, giving half the number to the female employees and the other half to the male employees. Each female employee received 3 more than each male. How many male employees are there? </SPAN>\n" );
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document.write( "you are given that the total number of employees are equal to 13.<BR>\n" );
document.write( "you are also given that 80 envelopes are distributed among them, with 40 going to the males and 40 going to the females.<BR>\n" );
document.write( "you are also given that each female employee receives 3 more than each male.\r<BR>\n" );
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document.write( "let x = number of males.<BR>\n" );
document.write( "let y = number of females.<BR>\n" );
document.write( "let a = number of envelopes that each male received.<BR>\n" );
document.write( "let a+3 = number of envelopes that each female received.\r<BR>\n" );
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document.write( "since the males received 40 envelopes, and the number of envelopes each male received is equal to a, then a*x = 40\r<BR>\n" );
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document.write( "since the females received 40 envelopes, and the number of envelopes each female received is equal to a+3, then (a+3)*y = 40.\r<BR>\n" );
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document.write( "solve for x and solve for y in each of these equations to get both x and ys in terms of a.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "a*x = 40 results in x = 40/a\r<BR>\n" );
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document.write( "(a+3)*y = 40 results in y = 40/(a+3)\r<BR>\n" );
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document.write( "you know that x+y = 13\r<BR>\n" );
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document.write( "replace x with 40/a and replace y with 40/(a+3) to get:\r<BR>\n" );
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document.write( "40/a + 40/(a+3) = 13\r<BR>\n" );
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document.write( "all you have to do now is solve for a.\r<BR>\n" );
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document.write( "multiply both sides of that equation by a*(a+3) to get:\r<BR>\n" );
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document.write( "40*(a+3) + 40*a = 13*a*(a+3)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "40a + 120 + 40a = 13a^2 + 39a\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "80a + 120 = 13a^2 + 39a\r<BR>\n" );
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document.write( "subtract 80a from both sides of the equation and subtract 120 from both sides of the equation to get:\r<BR>\n" );
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document.write( "0 = 13a^2 + 39a - 80a - 120\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "0 = 13a^2 - 41a - 120\r<BR>\n" );
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document.write( "flip sides in the equation to get:\r<BR>\n" );
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document.write( "13a^2 - 41a - 120 = 0\r<BR>\n" );
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document.write( "this is a quadratic equation in standard form.\r<BR>\n" );
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document.write( "factor it and you will get:\r<BR>\n" );
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document.write( "a = 5 or a = -24/13.\r<BR>\n" );
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document.write( "a can't be negative, so the solution is a = 5\r<BR>\n" );
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document.write( "in the formula of ax = 40, replace a with 5 to get 5x = 40 which results in x = 8.\r<BR>\n" );
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document.write( "in the formula of (a+3)y = 40, replace a with 5 to get 8y = 40 which results in y = 5.\r<BR>\n" );
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document.write( "number of males is 8<BR>\n" );
document.write( "number of females is 5\r<BR>\n" );
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document.write( "8 males got 40 envelopes which results in 5 envelopes each.<BR>\n" );
document.write( "5 females got 40 envelopes which results i 8 envelopes each.<BR>\n" );
document.write( "each female got 3 more envelopes than each male.<BR>\n" );
document.write( "problem is solved.\r<BR>\n" );
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document.write( "answer to the question is there are 8 male employees.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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