document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189654>Question 1189654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles, the ratio of Tony’s marbles to Peter’s marbles became 5:3. What was the ratio of Peter’s marbles to Tony’s marbles at first? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821076 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=821076\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "T = amount of marbles Tony has initially<BR>\n" );
document.write( "P = amount of marbles Peter has initially\r<BR>\n" );
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document.write( "Tony gives 4/14 = 2/7 of his marbles to Peter. <BR>\n" );
document.write( "That means Tony gives (2/7)T marbles to Peter.<BR>\n" );
document.write( "Tony's count T becomes T-(2/7)T = (5/7)T<BR>\n" );
document.write( "Peter's count P becomes P+(2/7)T\r<BR>\n" );
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document.write( "Let's say<BR>\n" );
document.write( "M = Tony's new count of marbles<BR>\n" );
document.write( "N = Peter's new count of marbles\r<BR>\n" );
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document.write( "To connect back to the previous variables, we would have:<BR>\n" );
document.write( "M = (5/7)T<BR>\n" );
document.write( "N = P+(2/7)T\r<BR>\n" );
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document.write( "After the gift is made, we have<BR>\n" );
document.write( "M:N = 5:3<BR>\n" );
document.write( "which is the same as writing <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%2FN+=+5%2F3 ALIGN=MIDDLE  ALT=\"M%2FN+=+5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's cross multiply to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%2FN+=+5%2F3 ALIGN=MIDDLE  ALT=\"M%2FN+=+5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3M+=+5N ALIGN=MIDDLE  ALT=\"3M+=+5N\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now apply substitution<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3M+=+5N ALIGN=MIDDLE  ALT=\"3M+=+5N\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%285%2F7%29T+=+5%28P%2B%282%2F7%29T%29 ALIGN=MIDDLE  ALT=\"3%285%2F7%29T+=+5%28P%2B%282%2F7%29T%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2F7%29T+=+5P%2B%2810%2F7%29T ALIGN=MIDDLE  ALT=\"%2815%2F7%29T+=+5P%2B%2810%2F7%29T\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Fractions are a bit of a pain to work with, but we can multiply both sides by 7 to clear them out. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2F7%29T+=+5P%2B%2810%2F7%29T ALIGN=MIDDLE  ALT=\"%2815%2F7%29T+=+5P%2B%2810%2F7%29T\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A%2815%2F7%29T+=+7%2A%285P%2B%2810%2F7%29T%29 ALIGN=MIDDLE  ALT=\"7%2A%2815%2F7%29T+=+7%2A%285P%2B%2810%2F7%29T%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15T+=+35P%2B10T ALIGN=MIDDLE  ALT=\"15T+=+35P%2B10T\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The ultimate goal is to find the ratio P:T where P and T will be replaced with actual numbers. That ratio is effectively the same as the fraction P/T\r<BR>\n" );
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document.write( "Let's see if we can somehow isolate P/T from that previous equation above<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15T+=+35P%2B10T ALIGN=MIDDLE  ALT=\"15T+=+35P%2B10T\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15T-10T+=+35P ALIGN=MIDDLE  ALT=\"15T-10T+=+35P\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5T+=+35P ALIGN=MIDDLE  ALT=\"5T+=+35P\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5T%2F5+=+35P%2F5 ALIGN=MIDDLE  ALT=\"5T%2F5+=+35P%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1T+=+7P ALIGN=MIDDLE  ALT=\"1T+=+7P\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%2FP+=+7%2F1 ALIGN=MIDDLE  ALT=\"T%2FP+=+7%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2FT+=+1%2F7 ALIGN=MIDDLE  ALT=\"P%2FT+=+1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This leads to the ratio P:T = <font color=red>1:7</font> telling us that Tony has 7 times more marbles compared to Peter when comparing the initial counts.\r<BR>\n" );
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document.write( "For example, let's say Peter starts with 10 marbles. That leads to Tony having 7*10 = 70 marbles.<BR>\n" );
document.write( "Next, Tony gives 4/14 of his count to Peter. He gives (4/14)*70 = 20 marbles.<BR>\n" );
document.write( "Tony's count becomes 70-20 = 50<BR>\n" );
document.write( "Peter's count becomes 10+20 = 30<BR>\n" );
document.write( "Then notice that the ratio Tony:Peter becomes 50:30 which reduces fully to 5:3<BR>\n" );
document.write( "This example helps confirm the answer. \r<BR>\n" );
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document.write( "As another example, if Peter starts with 5 marbles then Tony starts with 7*5 = 35 marbles.<BR>\n" );
document.write( "4/14 of 35 = 10 marbles are given<BR>\n" );
document.write( "Tony = 35-10 = 25<BR>\n" );
document.write( "Peter = 5+10 = 15<BR>\n" );
document.write( "The ratio Tony:Peter = 25:15 reduces to 5:3<BR>\n" );
document.write( "So as you can see, there are infinitely many possibilities for the values of P and T; however, their ratio P:T is fixed at <font color=red>1:7</font>\r<BR>\n" );
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document.write( "I'll let you try out other examples.\r<BR>\n" );
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document.write( "Answer is the ratio <font color=red>1:7</font><BR>\n" );
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