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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. $ 2600 . $ But, if he had interchanged the amounts invested, he would have received Rs. 140 less. What amounts did he invest at the different rates?

Given:

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.\( 2600 . \)

If he had interchanged the amounts invested, he would have received Rs. 140 less.

To do:

We have to find the amounts he invested.

Solution:

Let's say he has invested amounts in two different schemes A and B.

Let the amount invested in scheme A be $Rs.\ x$ and the amount invested in scheme B be $Rs.\ y$.

SI on $Rs.\ x$ at 12% per annum for 1 year $=Rs.\ \frac{x\times12\times1}{100}=Rs.\ \frac{12x}{100}$

SI on $Rs.\ y$ at 10% per annum for 1 year $=Rs.\ \frac{y\times10\times1}{100}=Rs.\ \frac{10y}{100}$

SI on $Rs.\ x$ at 10% per annum for 1 year $=Rs.\ \frac{x\times10\times1}{100}=Rs.\ \frac{10x}{100}$

SI on $Rs.\ y$ at 12% per annum for 1 year $=Rs.\ \frac{y\times12\times1}{100}=Rs.\ \frac{12y}{100}$

According to the question,

$\frac{12x}{100}+\frac{10y}{100}=2600$

$\frac{12x+10y}{100}=2600$

$12x+10y=2600(100)$......(i)

$\frac{10x}{100}+\frac{12y}{100}=2600-140$

$\frac{10x+12y}{100}=2460$

$10x+12y=2460(100)$......(ii)

Multiplying equation (i) by 12 on both sides, we get,

$12(12x+10y)=12(2600)(100)$

$144x+120y=31200(100)$.....(iii)

Multiplying equation (ii) by 10 on both sides, we get,

$10(10x+12y)=10(2460)(100)$

$100x+120y=24600(100)$.....(iv)

Subtracting equation (iv) from equation (iii), we get,

$(144x+120y)-(100x+120y)=31200(100)-24600(100)$

$144x-100x+120y-120y=6600(100)$

$44x=6600(100)$

$x=\frac{6600(100)}{44}$

$x=150(100)$

$x=15000$

Substituting $x=15000$ in equation (i), we get,

$12(15000)+10y=2600(100)$

$180000+10y=260000$

$10y=260000-180000$

$10y=80000$

$y=\frac{80000}{10}$

$y=8000$

The money he invested at the different rates is Rs. 15000 and Rs. 8000 respectively.