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A man invested an amount at 12% per annum simple interest and another amount at 10% per annum simple interest. He received an annual interest of Rs.1145. But, if he had interchanged the amounts invested, he would have received Rs.90 less. What amounts did he invest at the different rates?​

Sagot :

SalmonWorldTopper

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-}}[/tex]

Let we assume that

Sum invested at the rate of 12 % per annum be Rs x

Sum invested at the rate of 10 % per annum be Rs y

According to statement,

Annual interest earned = Rs 1145

We know,

Interest received on a sum of Rs p invested at the rate of r % per annum for n years is

[tex]\boxed{\tt{ \: I \: = \: \dfrac{p \times r \times n}{100} \: }}[/tex]

So, we get

[tex]\rm :\longmapsto\:\dfrac{x \times 12 \times 1}{100} + \dfrac{y \times 10 \times 1}{100} = 1145[/tex]

[tex]\rm :\longmapsto\:\dfrac{12x }{100} + \dfrac{10y}{100} = 1145[/tex]

[tex]\rm :\longmapsto\:\boxed{\tt{ 12x + 10y = 114500 }}- - - (1)[/tex]

According to second condition

Sum invested at the rate of 12 % per annum be Rs y

Sum invested at the rate of 10 % per annum be Rs x

According to statement,

Annual interest earned = Rs 1055

So,

[tex]\rm :\longmapsto\:\dfrac{x \times 10 \times 1}{100} + \dfrac{y \times 12 \times 1}{100} = 1055[/tex]

[tex]\rm :\longmapsto\:\dfrac{10x }{100} + \dfrac{12y}{100} = 1055[/tex]

[tex]\rm :\longmapsto\:\boxed{\tt{ 10x + 12y = 105500 }}- - - (2)[/tex]

Now, On adding equation (1) and (2), we get

[tex]\rm :\longmapsto\:22x + 22y = 220000[/tex]

[tex]\rm :\longmapsto\:22(x + y) = 220000[/tex]

[tex]\rm :\longmapsto\:\boxed{\tt{ x + y= 10000}} - - - - (3)[/tex]

On Subtracting equation (2) from equation (1), we get

[tex]\rm :\longmapsto\:2x - 2y = 9000[/tex]

[tex]\rm :\longmapsto\:2(x - y) = 9000[/tex]

[tex]\rm :\longmapsto \:\boxed{\tt{ x - y = 4500}} - - - - (4)[/tex]

On adding equation (3) and (4), we get

[tex]\rm :\longmapsto\:2x = 14500[/tex]

[tex]\bf\implies \:x = 7250[/tex]

On substituting the value of x in equation (3), we get

[tex]\rm :\longmapsto\:7250 + y = 10000[/tex]

[tex]\rm :\longmapsto\:y = 10000 - 7250[/tex]

[tex]\bf\implies \:y = 2750[/tex]

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So,

Sum invested at the rate of 12 % per annum be Rs 7250

Sum invested at the rate of 10 % per annum be Rs 2750

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