# On her birthday seema decided to donate some money to children of an orphanage home. If there ware 8 children less, every one would have got Rs.10 more However, if there were 16 children more, every one would have got Rs.10 less. Using matrix method, find the number of children and the amount distributed by seema.

$Lettheamountdistributedbyseemabe\text{'}n\text{'}andnoofchildrenbex.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}So,amounteachchildgot=\frac{n}{x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Ifthere8childrenless,theywouldhavegot10rsmore.So,\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{n}{x-8}=\frac{n}{x}+10\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow nx=nx-8n+10{x}^{2}-80x\phantom{\rule{0ex}{0ex}}\Rightarrow 5{x}^{2}-40x=4n....\left(1\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Iftherewere16childrenmore,everyonewouldhavegotRs.10less.So,\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{n}{x+16}=\frac{n}{x}-10\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow nx=nx+16n-10{x}^{2}-160x\phantom{\rule{0ex}{0ex}}\Rightarrow 5{x}^{2}+80x=8n....\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Dividing\left(2\right)by\left(1\right),weget\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{5{x}^{2}+80x}{5{x}^{2}-40x}=2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 5{x}^{2}-160x=0\phantom{\rule{0ex}{0ex}}\Rightarrow x=32\phantom{\rule{0ex}{0ex}}Subsitutingin\left(1\right),weget\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}n=960\phantom{\rule{0ex}{0ex}}$

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