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If simple interest and compound interest of a certain sum of money for two years are Rs. 8400 and Rs. 8652 respectively, then find the sum of money and the rate of interest.

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Let the principal be Rs. `x` and the rate of interet be r% per annum.
`:.` The simple interest of Rs. x in 2 years at the rate of r% annum
`=Rs. (x xxrxx2)/100=Rs. (2xr)/100`
Again the amount of Rs. `x` for 2 years at the rate of r% pr annum
`=Rs. {x xx(1+r/100)^(2)-x}`
`=Rs. x{(1+r/100)^(2)-(1)^(2)}`
`=Rs. x(1+r/100+1)(1+r/100-1)`
`=Rs x(2+r/100)xxr/100`
As per question `(2xr)/100=8400implies(xr)/100=4200`...............1
and `(xr)/100(2+r/100)=8652implies4200(2+r/100)=8652`[by(1)]
`implies2+r/100=8652/4200`
`impliesr/100=8652/4200-2`
`impliesr/100=(8652-8400)4200`
`impliesr=(252xx100)/4200`
`impliesr=6`
Then from 1 we get `(x xx6)/100=4200 [ :'r=6]`
`impliesx=(4200xx100)/6impliesx=70000`
Hence the required sum of money `=Rs. 70000` and the rate of interest `=6%` per annum.
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