document.write( "Question 1006385: Can you help me answer this? Thanks
\n" ); document.write( "In a country then are 100 governors, every governor has 4 assistants. Every governor and assistant serve various committees. Each committee may be composed of 5 governors, 4 governors and 4 assistants, or 2 governors and 12 assistants. Each governor serves 5 committees, and each assistant serves 3 committees. How many committees are there?
\n" ); document.write( "a. 60
\n" ); document.write( "b. 80
\n" ); document.write( "c. 100
\n" ); document.write( "d. 120
\n" ); document.write( "e. 160
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Algebra.Com's Answer #622573 by KMST(5328)\"\" \"About 
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There is a total of \"100%2A4=400\" assistants,
\n" ); document.write( "and each will serve in exactly \"3\" committees,
\n" ); document.write( "so we will prepare \"3%2A400=1200\" assistant place cards.
\n" ); document.write( "\"x\"= number of committees made up of 2 governors and 12 assistants.
\n" ); document.write( "\"y\"= number of committees made up of 4 governors and 2 assistants.
\n" ); document.write( "\"12x%2B4y=1200\" number of assistant place cards prepared for the \"x%2By\" committees with assistants.
\n" ); document.write( "Since each of the \"100\" governors serve in exactly \"5\" committees,
\n" ); document.write( "we will prepare \"5%2A100=500\" governor place cards.
\n" ); document.write( "For those place cards to suffice for the committees with assistants, we need
\n" ); document.write( "\"2x%2B4y%3C=500\" .
\n" ); document.write( "We have the system \"system%2812x%2B4y%3C=1200%2C2x%2B4y=500%29\" ,which graphs as
\n" ); document.write( " We need to be in the green-dashed region and on the red line.
\n" ); document.write( "If we do not set up any 5-governor committees,
\n" ); document.write( "\"system%2812x%2B4y=1200%2C2x%2B4y=500%29\"-->\"system%283x%2By=300%2C2x%2B4y=500%29\"-->\"system%28x=70%2Cy=90%29\" .
\n" ); document.write( "So, forming \"70%2B90=160\" committees, including
\n" ); document.write( "\"70\" committees made up of 2 governors and 12 assistants, and
\n" ); document.write( "\"90\" committees made up of 4 governors and 2 assistants,
\n" ); document.write( "seems a possibility.
\n" ); document.write( "We could rearrange the committees differently, getting rid
\n" ); document.write( "Setting up 5-governor meetings sounds like it would lead to less meetings.
\n" ); document.write( "However, as shown by common sense , and by \"3x%2By=300\"<-->\"y=300-3x\" ,
\n" ); document.write( "for every extra 12-assistant committee, we must give up \"3\" 4-assistant committees,
\n" ); document.write( "for a total of \"3-1=2\" less committees with assistants.
\n" ); document.write( "using \"2\" more governor place cards for the extra 12-assistant committee,
\n" ); document.write( "and requiring \"3%2A4=12\" less governor place cards for the \"3\" eliminated 4-assistant committees.
\n" ); document.write( "That would leave us with \"12-2=10\" governor place cards,
\n" ); document.write( "which would be used in \"10%2F5=2\" 5-governor committee.
\n" ); document.write( "That way, every time we do that shift,
\n" ); document.write( "we eliminate a total of \"2\" committees with assistants,
\n" ); document.write( "but create \"2\" 5-governor committees,
\n" ); document.write( "leaving the total number of committees always at \"highlight%28160%29\" .
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