document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=890570>Question 890570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mr Tong has some guavas and papayas in two crates. In crate A, the number of guavas and papayas are in the ratio 5:3. In crate B, the number of guavas is 4:9 the number of papayas. Mr Tong transfers half of the guavas from crate A to B. The number of fruits in crate A becomes 308 and the ratio of the number of guavas to the number of papayas in crate B becomes 5:6.\r<BR>\n" );
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document.write( "(a) How many guavas have been transferred from crate A to crate B?\r<BR>\n" );
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document.write( "(b) What is the number of fruits in crate B after the transfer? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #539097 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=539097\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "the ratio of guavas and papayas in crate A is equal to 5:3.\r<BR>\n" );
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document.write( "the ratio of guavas and papayas in crate B is equal to 4:9.\r<BR>\n" );
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document.write( "a ratio of 5:3 means that, for every 8 pieces of fruit in crate A, 5 of them are guavas and 3 of them are papayas.\r<BR>\n" );
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document.write( "the total number of fruit in crate A therefore has to be a multiple of 8.\r<BR>\n" );
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document.write( "we can let x represent that multiple for the number of fruit in crate A.\r<BR>\n" );
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document.write( "if x is 1, then there are 5 guavas and 3 papayas for a total of 8.\r<BR>\n" );
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document.write( "if x is 100, then there are 500 guavas and 300 papayas for a total of 800.\r<BR>\n" );
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document.write( "the ratio will hold no matter the value of x.\r<BR>\n" );
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document.write( "5/3 is the same ratio as 500/300.\r<BR>\n" );
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document.write( "so that's where we start for crate A.\r<BR>\n" );
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document.write( "now they say that, if you take away half of the guavas from crate A, the total number of fruit in crate A will be equal to 308.\r<BR>\n" );
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document.write( "since we started with 5x guavas and took away half, then we are left with 5x/2 guavas.\r<BR>\n" );
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document.write( "the formula for crate A now becomes:\r<BR>\n" );
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document.write( "5x/2 + 3x = 308\r<BR>\n" );
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document.write( "if we solve this equation for x, we will find that x = 56.\r<BR>\n" );
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document.write( "this means that the original number of fruit in crate A had to be 5(56) + 3(56) = 280 + 168 = 448\r<BR>\n" );
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document.write( "if we took away half the guavas, this means that we took away 140 guavas.\r<BR>\n" );
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document.write( "we took those guavas and put them in crate B.\r<BR>\n" );
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document.write( "so 140 guavas were transferred from crate A to crate B.\r<BR>\n" );
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document.write( "now we know the ratio of guavas to papayas in crate B is 4:9.\r<BR>\n" );
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document.write( "this means that, for every 13 pieces of fruit in crate B, 4 of them are guavas and 9 of them are papayas.\r<BR>\n" );
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document.write( "this means that the total amount of fruit in crate B has to be a multiple of 13.\r<BR>\n" );
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document.write( "if we let y equal that multiple, then the total number of fruit in crate B is equal to 13y.\r<BR>\n" );
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document.write( "this means that the number of guavas in crate B is equal to 4y and the number of papayas in crate B is equal to 9y.\r<BR>\n" );
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document.write( "the ratio of guavas to papayas is 4/9.\r<BR>\n" );
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document.write( "we get the equation:\r<BR>\n" );
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document.write( "4y/9y = 4/9\r<BR>\n" );
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document.write( "now when we add 140 guavas to crate B, the total number of guavas becomes 4y + 140.\r<BR>\n" );
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document.write( "the total number of papayas stays the same at 9y.\r<BR>\n" );
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document.write( "the total number of fruit becomes 13y + 140\r<BR>\n" );
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document.write( "now instead of the ratio being 4/9, the ratio becomes 5/6.\r<BR>\n" );
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document.write( "so we get the equation:\r<BR>\n" );
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document.write( "(4y+140) / 9y = 5/6\r<BR>\n" );
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document.write( "we can use this equation to solve for y.\r<BR>\n" );
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document.write( "cross multiply to get:\r<BR>\n" );
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document.write( "6*(4y+140) = 9y*5\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "24y + 840 = 45y\r<BR>\n" );
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document.write( "subtract 24y from both sides of the equation to get:\r<BR>\n" );
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document.write( "840 = 21y\r<BR>\n" );
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document.write( "divide both sides of this equation by 21 to get:\r<BR>\n" );
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document.write( "y = 40\r<BR>\n" );
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document.write( "now let's see what we have.\r<BR>\n" );
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document.write( "the original ratio is 4/9 and is composed of 4y + 9y = 13y\r<BR>\n" );
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document.write( "when y = 40, this becomes 4(40) + 9(40) = 13(40) which becomes 160 + 360 = 520 which becomes 520 = 520.\r<BR>\n" );
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document.write( "this means that we originally had 520 pieces of fruit in crate B and that 160 of them were guavas and 360 of them were papayas.\r<BR>\n" );
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document.write( "if we add 140 to that, then the pieces of fruit in crate B goes from 520 to 660.\r<BR>\n" );
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document.write( "but now we have a new ratio of 5/6.\r<BR>\n" );
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document.write( "this means that for every 11 pieces of fruit in crate B, 5 of them are guavas and 6 of them are papayas.\r<BR>\n" );
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document.write( "so the total number of fruit in crate B will now be a multiple of 11.\r<BR>\n" );
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document.write( "if we let z equal that multiple, then we get:\r<BR>\n" );
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document.write( "5z + 6z = 11z = 660\r<BR>\n" );
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document.write( "solve for z to get z = 60\r<BR>\n" );
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document.write( "5z + 6z becomes 5(60) + 6(60) = 11(60) which becomes 300 + 360 = 660 which becomes 660 = 660.\r<BR>\n" );
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document.write( "this confirms that the multiple of z = 60 is correct.\r<BR>\n" );
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document.write( "we originally had 160 guavas and 360 papayas in crate B.\r<BR>\n" );
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document.write( "we now have 300 guavas and 360 papayas in crate B.\r<BR>\n" );
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document.write( "the number of guavas in crate B grew by 140.\r<BR>\n" );
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document.write( "the numbers check out and we now have all answers to the problem.\r<BR>\n" );
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document.write( "the questions to the problem were:\r<BR>\n" );
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document.write( "a) How many guavas have been transferred from crate A to crate B?<BR>\n" );
document.write( "(b) What is the number of fruits in crate B after the transfer?\r<BR>\n" );
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document.write( "the answer to question A is that 140 guavas were transferred from crate A to crate B.\r<BR>\n" );
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document.write( "the answer to question B is that the number of fruit in crate B after the transfer is 660.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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