document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=571178>Question 571178</A>: <I><SPAN STYLE=\"word-break: break-all;\"> After finding the average of 35 scores, a student carelessly included the average with the 35 scores and found the average of these 36 numbers. The ratio of the second average to the true average was\r<BR>\n" );
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document.write( "(A) 1 : 1 (B) 35 : 36 (C) 36 : 35 (D) 2 : 1 (E) None of these </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #368013 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=368013\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe the answer is 1:1 which would be selection A.<BR>\n" );
document.write( "there's a couple of ways you can prove this.<BR>\n" );
document.write( "the first way is to take any average number and any number of data points that make up that average.<BR>\n" );
document.write( "assume you have 556 data points whose average is equal to 375.<BR>\n" );
document.write( "these numbers were chosen at random.<BR>\n" );
document.write( "if we let x equal the total of the numbers, then we get:<BR>\n" );
document.write( "x / 556 = 375<BR>\n" );
document.write( "if we multiply both sides of this equation by 556, then we get:<BR>\n" );
document.write( "x = 375 * 556 = 208500<BR>\n" );
document.write( "we have a total of 208500 for 556 data points whose average is 375.<BR>\n" );
document.write( "we add the average to this set of data points to get a new total of 208500 + 375.<BR>\n" );
document.write( "this is equal to 208875/<BR>\n" );
document.write( "we divide this by 557 data points (556 of the original plus the additional data point of the average) to get:<BR>\n" );
document.write( "208875 / 557 = 375<BR>\n" );
document.write( "this is the exact same average that we original got which makes the ratio of the second average to the true average equal to 1:1.<BR>\n" );
document.write( "we can also prove this by formula as follows:<BR>\n" );
document.write( "the original average is equal to T/N<BR>\n" );
document.write( "T is the total of the data points and N is the number of data points.<BR>\n" );
document.write( "the second average is equal to T plus the original average divided by N+1<BR>\n" );
document.write( "we know that N is equal to 35 which makes N+1 equal to 37.<BR>\n" );
document.write( "our original average becomes T/35<BR>\n" );
document.write( "our second average becomes (T + the original average)/37<BR>\n" );
document.write( "since the original average is equal to T/35, the second average becomes:<BR>\n" );
document.write( "(T + T/35)/36<BR>\n" );
document.write( "we can simplify this expression to get:<BR>\n" );
document.write( "(36T/35)/36<BR>\n" );
document.write( "this is because T is equivalent to 35T/35 which, when added to T/35, becomes 36T/35.<BR>\n" );
document.write( "our second average is therefore equal to (36T/35)/36<BR>\n" );
document.write( "this is the same as 36T/35 * 1/36 which is the same as 36T/(35*36)<BR>\n" );
document.write( "since the 36 in the numerator cancels out the 36 in the denominator, we are left with:<BR>\n" );
document.write( "second average = T/35.<BR>\n" );
document.write( "this is the same as the original average, so the ratio is 1:1.<BR>\n" );
document.write( " </SPAN>\n" );
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