Mrs.Sharma sold a jewellery at a loss of 5%. If he had sold it at Rs.5,200 more, she would have gained 8%. Find the cost price of the jewellery ?

#color(blue)("Modelling the given conditions")#

Set the original selling price as #x#
Let the cost price be #y#

#color(brown)("Consider the 5% loss condition. ")#

To get get just the cost price back we have #(100)/(100)xxy#

But what she got back was: #(100-5)/100xxy #

And we know this is the original selling price so we have:

#95/100ycolor(white)("d")=color(white)("d")xcolor(white)("dddd") =>color(white)("dddd") ycolor(white)("d")=color(white)("d")100/95 x" "..Equation(1)#

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#color(brown)("Consider the 8% profit")#

To obtain this profit she sold it for Rs 5200 more than the original selling price of #x#. So we have:

#y+8%y=x+5200#

#y(1+8%)=x+5200#

#y(108/100)=x+5200#

#y=[(100/108)x]+[100/108xx5200]" "...Equation(2)#
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#color(blue)("Putting it all together")#

Substitute for #y# in #Equation(2)# using #Equation(1)#

#100/95 x=100/108x+[100/108xx5200] #

Subtract #100/108 x# from both sides

#(100x)/95-(100x)/108=(100xx5200)/108#

Lets make all the denominators the same so we need to change the 95 into 108.

Note that #95xx108/95 = 108# so by applying this we have:

#(100xx108/95xx x)/108-(100x)/108=(100xx5200)/108#

Multiply all of both sides by 108

#(100xx108/95xx x)-100x=100xx5200#

#10800/95 x-100x = 520000 larr" Using decimals will introduce"#
#color(white)("ddddddddddddddddddddddddd")"rounding errors so I am sticking"#
#color(white)("dddddddddddddddddddddddd")" with fractions"#

#260/19 x = 520000#

#x=520000xx19/260 = 38000#
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#color(blue)("Determine the cost price.")#

Using #Equation(1)#

#y=100/95 x color(white)("d")->color(white)("d")y=100/95xx38000#

#"cost price" =y=Rs40000#

color(white)("d")