Question

Word Problems Workshee You have a jar of loose change which contains pen total value of the change in the jar is $104. In total, there are exactly 1000 coins in the jar There are four times as many pennies as there are nickels and there are twice as many pennies as there are dimes. How many of each type of coin are there? 4. nies, nickels, dimes, and quarters. The

Answer 1

$\dpi{80} \fn_jvn Suppose\,\,p,\,\,n,\,\,d\,\,and\,\,q\,\,are\,\,denoting\,\,number\,\,of\,\,pennies,\,\,nickels,\,\,dimes\,\,and\,\,quarters\,\,present\,\,in\,\,the\,\,jar.$

$\dpi{80} \fn_jvn Now\,\,there\,\,are\,\,1000\,\,coins\,\,in\,\,the\,\,jar.\\ \Rightarrow \,\,\,\,p+n+d+q=1000$$\dpi{80} \fn_jvn ......(1)$

$\dpi{80} \fn_jvn The\,\,total\,\,value\,\,of\,\,change\,\,in\,\,the\,\,jar\,\,is\,\, \$ 104 \\ \Rightarrow \,\,\,\,0.01p+0.05n+0.10d+0.25q=104$   $\dpi{80} \fn_jvn ......(2)$

$\dpi{80} \fn_jvn There\,\,are\,\,four\,\,times\,\,as\,\,many\,\,pennies\,\,as\,\,there\,\,are\,\,nickels \\ \Rightarrow \,\,\,\,p=4n$
$\dpi{80} \fn_jvn \Rightarrow \,\,\,\,n=0.25p$ $\dpi{80} \fn_jvn ......(3)$

$\dpi{80} \fn_jvn There\,\,are\,\,twice\,\,times\,\,as\,\,many\,\,pennies\,\,as\,\,there\,\,are\,\,dimes \\ \Rightarrow \,\,\,\,p=2d$
$\dpi{80} \fn_jvn \Rightarrow \,\,\,\,d=0.5p$ $\dpi{80} \fn_jvn ......(4)$

$\dpi{80} \fn_jvn Now\,\,values\,\,of\,\,n\,\,and\,\,d\,\,from\,\,the\,\,equations\,\,(3)\,\,and\,\,(4)\,\,intoequations\,\,(1)\,\,and\,\,(2)\\$
$\dpi{80} \fn_jvn p+0.25p+0.5p+q=1000\\ \Rightarrow \,\,1.75p+q=1000\\ \Rightarrow \,\,q=1000-1.75p$ $\dpi{80} \fn_jvn ......(5)$

$\dpi{80} \fn_jvn 0.01p+0.05\times 0.25p+0.10\times0.5p+0.25q=104\\ \Rightarrow \,\,0.0725p+0.25q=104\\ \Rightarrow \,\,0.0725p+0.25\times(1000-1.75p)=104$$\dpi{80} \fn_jvn [\because q=1000-1.75p\,\,from\,\, eq.(5)]$
$\dpi{80} \fn_jvn \\ \Rightarrow \,\,0.0725p+250-0.4375p=104\\ \Rightarrow \,\,0.4375p-.0725p=250-104\\ \Rightarrow \,\,0.365p=146\\\\ \Rightarrow \,\,p=\frac{146}{0.365}\\\\ \Rightarrow p=400$ $\dpi{80} \fn_jvn ......(6)$

$\dpi{80} \fn_jvn From\,\,eq.(6)\,\,and\,\,(3)\\ n=0.25p\\ \Rightarrow \,\,n=0.25\times400\\ \Rightarrow \,\,n=100$

$\dpi{80} \fn_jvn From\,\,eq.(6)\,\,and\,\,(4)\\ d=0.5p\\ \Rightarrow \,\,d=0.5\times400\\ \Rightarrow \,\,d=200$

$\dpi{80} \fn_jvn From\,\,eq.(6)\,\,and\,\,(5)\\ q=1000-1.75p\\ \Rightarrow \,\,q=1000-1.75\times400\\ \Rightarrow \,\,q=300$

$\dpi{80} \fn_jvn Therefore,\,\,$
  $\dpi{80} \fn_jvn Number\,\,of\,\,pennies\,\,p=400\\$
     $\dpi{80} \fn_jvn nickels\,\,n=100\\$
$\dpi{80} \fn_jvn dimes\,\,d=200\\$
$\dpi{80} \fn_jvn quarters\,\,q=300\\$