The difference between ci and si on a certain sum at a certain rate of interest for 3years is 990.if si at the same sum at same rate of interest for 2years be rs 2000 then find the rate of interest?
Answers
Given :
Difference between Compound interest and Simple interest for 3 years = 990
Simple interest at same rate of interest and same sum = Rs. 2000
To find : Rate of interest
Solution :
Let the same sum i.e Principal be P and and same Rate of interest be R.
Simple interest for 2 years = Rs. 2000
=> PTR / 100 = 2000
=> 2PR / 100 = 2000
=> PR / 100 = 2000 / 2
=> PR × 1 / 100 = 1000
=> Simple interest for 1 years = 1000
Simple interest for 3 years = 3 × 1000 = Rs. 3000
=> PR = 10000
=> P = 100000 / R → EQ( 1 )
Difference between Compound interest and Simple interest for 3 years = 990
=> Compound interest - Simple interest = 990
=> Compound interest for 3 years = 3000 + 990 = Rs. 3990
=> P{ (1 + R/100)^T - 1} = 3990
=> P{ (1 + R/100)³ - 1 } = 3990
From EQ( 1 )
=> ( 100000 / R ) { ( 1 + R/100)³ - 1 } = 3990
=> ( 1 + R/100 )³ - 1 = 399R / 10000
Using identity a³ - b³ = ( a - b )( a² + b² + ab)
=> ( 1 + R/100 - 1 ){ ( 1 + R/100 )² + 1 + 1 + R/100) } = 399R / 10000
=> ( R/100 ) ( 3 + 2R/100 + R²/10000 + R/100 ) = 399R / 10000
=> 3 + R²/10000 + 3R/100 = 399 / 100
=> R²/10000 + 3 - 399 / 100 + 3R/100 = 0
=> R²/10000 + 3 - 399/100 + 3R/100 = 0
=> R² + 30000 - 39900 + 300R = 0
=> R² + 300R - 9900 = 0
=> R² + 300R - 9900 = 0
=> R² + 330R - 30R - 9900 = 0
=> R( R + 330 ) - 30( R + 330 ) = 0
=> ( R - 30 )( R + 330 ) = 0
=> R - 30 = 0 or R + 330 = 0
=> R = 30 or R = - 330
R ≠ 330 since R can't be negative
=> R = 30
Therefore the Rate of interest is 30 %.