Question 1067722
Let {{{ n }}} = the number of integers in the original list
Let {{{ m }}} = the mean of the original list
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Originally, the total, {{{ T }}}, of the integers was:
{{{ m = T/n }}}
{{{ T = m*n }}}
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Add {{{ 15 }}} to the list
(1) {{{ m + 2 = ( m*n + 15 ) / ( n + 1 ) }}}
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Add {{{  1 }}} to the enlarged list
(2) {{{ m + 2 - 1 = ( m*n + 15 + 1 ) / ( n + 2 ) }}}
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(1) {{{ ( m + 2 )*( n + 1 ) = m*n + 15 }}}
(1) {{{ m*n + 2n + m + 2 = m*n + 15 }}}
(1) {{{ m + 2n = 13 }}}
and
(2) {{{ ( m + 1 )*( n + 2 ) = m*n + 16 }}}
(2) {{{ m*n + n + 2m + 2 = m*n + 16 }}}
(2) {{{ 2m + n = 14 }}}
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Multiply both sides of (2) by {{{ 2 }}} 
and subtract (1) from (2)
(2) {{{ 4m + 2n = 28 }}}
(1) {{{ -m - 2m = -13 }}}
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{{{ 3m = 15 }}}
{{{ m = 5 }}}
and
(1) {{{ m + 2n = 13 }}}
(1) {{{ 5 + 2n = 13 }}}
(1) {{{ 2n = 8 }}}
(1) {{{ n = 4 }}}
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There were 4 integers in the original list
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check:
(1) {{{ T = m*n }}}
(1) {{{ T = 5*4 }}}
(1) {{{ T = 20 }}}
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(1) {{{ m + 2 = ( m*n + 15 ) / ( n + 1 ) }}}
(1) {{{ 5 + 2 = ( 20 + 15 ) / ( 4 + 1 ) }}}
(1) {{{ 7 = 35/5 }}}
(1) {{{ 7 = 7 }}}
and
(2) {{{ m + 2 - 1 = ( m*n + 15 + 1 ) / ( n + 2 ) }}}
(2) {{{ 5 + 2 - 1 = ( 20 + 15 + 1 ) / ( 4 + 2 ) }}}
(2) {{{ 6 = 36/6 }}}
(2) {{{ 6 = 6 }}}
OK