A man lent a part of money at 10% p.a. and the rest at 15% p.a. his annual income is Rs.1900 . If he had interchanged the rate of interest on two sums , he would have earned Rs.200 more . Find the amount lent in each case.


Please answer with the Explaination

Answer :

Let  ,  Man lent a part of money at 10% p.a. =  x 
And
Man lent a part of money at 15% p.a. =  y  

So,

From first condition ,we get

$$\begin{gathered} \frac{10 x}{100} + \frac{15 y}{100} = 1900 \\ \Rightarrow \frac{x}{10} + \frac{3 y}{20} = 1900 \\ \Rightarrow \frac{2 x + 3 y }{20} = 1900 \\ \mathbf{\Rightarrow }\mathbf{2}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{ }\mathbf{3}\mathbf{ }\mathit{y}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{38000}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{)} \end{gathered}$$

And also given  :  If he had interchanged the rate of interest on two sums , he would have earned Rs.200 more .  So

$$\begin{gathered} \frac{10 y}{100} + \frac{15 x}{100} = 1900 \\ \Rightarrow \frac{y}{10} + \frac{3 x}{20} = 1900 \\ \Rightarrow \frac{2 y + 3 x }{20} = 1900 \\ \mathbf{\Rightarrow }\mathbf{2}\mathbf{ }\mathit{y}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{ }\mathbf{3}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{38000}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{)} \end{gathered}$$

Now we multiply by 3 in equation 1 and multiply by 2 in equation 2 , we get

6x  +  9y  =  114000                              ----- ( 4 )
And
4y  +  6x  =  84000                                ----- ( 5 ) 

Now we subtract equation 5 from equation 4  and get

5y =  30000 

y  =  6000  , Substitute that value in equation ( 1  )  and get

2x  +  3 (  6000 ) =  38000

2x  +  18000 =  38000

2x  =  20000

x  =  10000

So,

Man lent a part of money at 10% p.a. = Rs. 10, 000
And
Man lent a part of money at 15% p.a. = Rs. 6,000                                                                       ( Ans )