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Vijay invested ${\text{Rs}}{\text{.50,000}}$ partly at $11\% $ per annum and partly $9\% $ per annum simple interest. If he annually receives ${\text{Rs}}.5400$ as interest , how much did he invest at each rate?

Answer
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Hint- In order to solve this question, we have to assume a value that he invested at a different rate of interests and then solve the equation. Thus, in this way we get our desired answer.

Complete step-by-step answer:
We have given that Vijay invested ${\text{Rs}}{\text{.50,000}}$ i.e.
Principal $ = 50,000$
Now, it is also given that he invested the amount partly at rate of $11\% $ per annum simple interest i.e.
R.O.I.=$11\% $
Also , it is given that he invested the amount partly at rate of $9\% $ per annum simple interest i.e.
R.O.I$ = 9\% $
Also , it is given that he receives ${\text{Rs}}{\text{.5400}}$ as interest annually.
So, we have to find how much he invests at each rate.
Let Vijay invest ${\text{Rs}}{\text{.}}x$ at $11\% $ per annum and ${\text{Rs}}{\text{.50,000 - }}x$ at $9\% $ per annum .
Now, the interest on ${\text{Rs}}{\text{.}}x$ at 11% per annum$ = {\text{Rs}}{\text{.}}\dfrac{{x \times 11}}{{100}}$

  And the interest on ${\text{Rs}}{\text{.50,000 - }}x$ at 9% per annum$ = {\text{Rs}}{\text{.}}\dfrac{{\left( {50,000 - x} \right) \times 9}}{{100}}$
Now he receives a total interest of Rs.5400
Or we can write the above thing as,
$\dfrac{{x \times 11}}{{100}} + \dfrac{{\left( {50000 - x} \right) \times 9}}{{100}} = 5400$
On solving the above question we get,
$
  11x + 450000 - 9x = 540000 \\
  2x = 540000 - 450000 \\
  2x = 90,000 \\
 $
Or $x = \dfrac{{90000}}{2}$
Or $x = 45000$
Thus, the value of $x$ is 45,000
And $50,000 - x = 5,000$
Thus, Vijay invested ${\text{Rs}}{\text{.45,000}}$ at 11% per annum and ${\text{Rs}}{\text{.5,000}}$ at 9% per annum.

Note- Whenever we face such types of questions the key concept is that we should read the question first and write down the things given to us and then apply the required formula. Here in this question we simply write what is given to us and we assumed a variable that he invested at 11% p.a. and 9% p.a. and then we equate them with the values provided in the question. Thus, we get our desired answer.