document.write( "Question 1164119: A survey of 539 adults aged 18-24 year olds was conducted in which they were asked what they did last Friday night. It found:\r
\n" ); document.write( "\n" ); document.write( "173 watched TV
\n" ); document.write( "176 hung out with friends
\n" ); document.write( "27 watched TV and ate pizza, but did not hang out with friends
\n" ); document.write( "41 watched TV and hung out with friends, but did not eat pizza
\n" ); document.write( "26 hung out with friends and ate pizza, but did not watch TV
\n" ); document.write( "39 watched TV, hung out with friends, and ate pizza
\n" ); document.write( "73 did not do any of these three activities
\n" ); document.write( "How may 18-24 year olds (of these three activities) only ate pizza last Friday night?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #788405 by ikleyn(52776)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "A survey of 539 adults aged 18-24 year olds was conducted in which they were asked what they did last Friday night. It found:\r
\n" ); document.write( "\n" ); document.write( "173 watched TV
\n" ); document.write( "176 hung out with friends
\n" ); document.write( "27 watched TV and ate pizza, but did not hang out with friends
\n" ); document.write( "41 watched TV and hung out with friends, but did not eat pizza
\n" ); document.write( "26 hung out with friends and ate pizza, but did not watch TV
\n" ); document.write( "39 watched TV, hung out with friends, and ate pizza
\n" ); document.write( "73 did not do any of these three activities
\n" ); document.write( "How may 18-24 year olds (of these three activities) only ate pizza last Friday night?
\n" ); document.write( "~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "We are given a universal set U of 539 persons and its subsets\r\n" );
document.write( "\r\n" );
document.write( "T  (watched TV)                                                          of 173\r\n" );
document.write( "\r\n" );
document.write( "F  (hung out with friends)                                               of 176\r\n" );
document.write( "\r\n" );
document.write( "TP \ TPF  (watched TV and ate pizza, but did not hang out with friends)  of  27\r\n" );
document.write( "\r\n" );
document.write( "TF \ TPF  (watched TV and hung out with friends, but did not eat pizza)  of  41\r\n" );
document.write( "\r\n" );
document.write( "FP \ TPF  (hung out with friends and ate pizza,  but did not watch TV)   of  26 \r\n" );
document.write( "\r\n" );
document.write( "TPF       (watched TV, hung out with friends, and ate pizza)             of  39\r\n" );
document.write( "\r\n" );
document.write( "The complement of (T U F U P) to U                                       of  73.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Couple comments regarding this table.\r\n" );
document.write( "\r\n" );
document.write( "    For brevity, I write AB for the intersection of subsets A and B (instead of traditional (A ∩ B) ).\r\n" );
document.write( "    For brevity, I write TPF for the triple intersection (T ∩ P ∩ F).\r\n" );
document.write( "    The sign  \" \ \"  means subtraction of subsets.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, in the elementary set theory, there is a REMARCABLE identity \r\n" );
document.write( "\r\n" );
document.write( "    n(A U B U C) = n(A) + n(B) + n(C) - n(AB) - n(AC) - nBC) + n(ABC)              (1)\r\n" );
document.write( "\r\n" );
document.write( "for any three subsets A, B, C of a universal set, their in-pair intersections AB, AC and BC, and the triple intersection ABC.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I will apply it for the given subsets T, F, P, their in-pairs and triple intersetions\r\n" );
document.write( "\r\n" );
document.write( "    n(T U F U P) = n(T) + n(F) + n(P) - n(TF)   - n(TP)   - n(FP)   + n(TPF).      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this equality, all the terms are given and are known, EXCEPT only one n(P).  It is my goal now to find n(P).\r\n" );
document.write( "For it, I substitute all the known values into equation (2)\r\n" );
document.write( "\r\n" );
document.write( "    539 - 73     = 173  + 176  + n(P) - (41+39) - (27+39) - (26+39) + 39.          (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From (3) I get\r\n" );
document.write( "\r\n" );
document.write( "    n(P) = (539-73) - 173 - 176 + (41+39) + (27+39) + (26+39) - 39 = one click in my Excel = 289.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now my last step is\r\n" );
document.write( "\r\n" );
document.write( "    the number of those who only eat Pizza = n(P) - n(TP) - n(FP) + n(TPF) = 289 - (27+39) - (26+39) + 39 = 197.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The number of those who only eat Pizza is 197.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "So, the problem is just solved, and the answer is 197 (by the modulus of my possible arithmetic errors).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "' ' ' ' ' ' ' ' \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I like to get  \"THANKS\"  from students for my work and for my teaching.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So,  I expect to get your  \"DEEPEST THANKS\"  soon.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );