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Q.21. A person invested some money at 12 % simple interest and some other amount at 10 % simple interest. He received yearly interest of Rs. 1300. If he had interchanged the amounts, he would have received Rs. 40 more as yearly interest. How much did he invest at different rates?

Dear Student,

Please find below the solution to the asked query:

Let , Person invest total monet at 12% in first place  =  x

And

Person invest total monet at 10% in first place  =  y 

We know :  Simple interest  = $\frac{P \times R \times T}{100}$
From first condition we get :

$$\begin{gathered} \frac{x \times 12 \times 1}{100} + \frac{y \times 10 \times 1}{100} = 1300 \\ \Rightarrow \frac{12 x + 10 y}{100}= 1300 \\ \Rightarrow \frac{2\left( 6 x + 5 y\right)}{100}= 1300 \\ \Rightarrow \frac{ 6 x + 5 y}{50}= 1300 \\ \mathbf{\Rightarrow }\mathbf{6}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{5}\mathbf{ }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{65000}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{)} \end{gathered}$$
And from second condition we get :

$$\begin{gathered} \frac{y \times 12 \times 1}{100} + \frac{x\times 10 \times 1}{100} = 1300 +\hspace{0.17em}40 \\ \Rightarrow \frac{10 x + 12 y}{100}= 1340 \\ \Rightarrow \frac{2\left( 5 x + 6 y\right)}{100}= 1350 \\ \Rightarrow \frac{ 5 x + 6 y}{50}= 1340 \\ \mathbf{\Rightarrow }\mathbf{5}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{6}\mathbf{ }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{67000}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{(}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{)} \end{gathered}$$
Now we multiply by ' 6 ' in equation 1 and by ' 5 ' in equation 2 and get :

36 x  + 30 y  = 390000                           --- ( 3 )

And

25 x  + 30 y  = 335000                           --- ( 4 )

Now we subtract equation 4 from equation 3 and get :

11 x  = 55000 ,

x  = 5000 , Substitute that value in equation 1 and get :

6 ( 5000 ) + 5 y  = 65000 ,

30000 + 5 y  = 65000 ,

5 y  = 35000 ,

y  =  7000

Therefore ,

Person invest total monet at 12% in first place  = Rs. 5000

And

Person invest total monet at 10% in first place  =  Rs. 7000                                                 ( Ans )


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