Question 1152153

There is a mountain with {{{30}}} bat caves in a row that contain {{{340}}} bats in all. 

Any {{{7}}} caves in a row contain exactly {{{77 }}}bats.=>each cave contains {{{11}}} bats

how many times we have {{{7}}} caves in a row, divide {{{30}}} by {{{7}}}=>{{{4}}}

in these {{{4*7=28}}} caves we have {{{4*77=308}}} bats
means, we have {{{2}}} more caves: first and last

{{{first }}}cave , then {{{ 28}}} caves, and  {{{last }}}cave ={{{30}}}th cave

so,  the {{{29}}}th cave is last in the group of {{{ 28}}} caves that each contains {{{11}}} bats

=>the {{{29}}}th cave  contains {{{11}}} bats


 if the  first  cave {{{c[1]}}} has {{{7}}} times more bats than the  last  cave{{{c[30]}}} , then 

{{{c[1]=7c[30]}}} .....eq.1

since these {{{28}}} caves contain  {{{308}}} bats, the first and last cave together contain {{{340-308=32}}} bats

{{{c[1]+c[30]=32}}} .....eq.2....substitute {{{c[1]}}} from eq.1

{{{7c[30]+c[30]=32}}} .....eq.2

{{{8c[30]=32}}}

{{{c[30]=4}}}

go to {{{c[1]=7c[30]}}} .....eq.1..substitute {{{c[30]}}}

{{{c[1]=7*4}}} 
{{{c[1]=28}}} 


so,

{{{1st}}} cave:{{{28}}}
{{{2nd}}} cave: {{{11}}}
{{{3rd}}} cave: {{{11}}}
{{{4th}}} cave: {{{11}}}
:
:
:
:
{{{29th }}}cave: {{{11}}}
{{{30th}}} cave:{{{4}}}