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A small software corporation borrowed 500,000 cash to expand its software line. The corporation borrowed some of the money at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest rate was $52,000 and the amount borrowed at 10% was 2.5 times the amount borrowed 9%.

I have no idea where to start with this problem. Can I get some help?

Thank you very much.

Here's what I have so far:

0.09x + 0.1y + 0.12z = ?

2.5x = y
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    Was all the 500,000 borrowed at each of the interest rates? I.e, some at 9%, some at 10% and the rest at 12%?2012-09-10
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    Yes, all was borrowed at each, respectively.2012-09-10

2 Answers 2

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Let $x$ be the amount borrowed at $10$% and let $y$ be the amount borrowed at $9$%. We know the amount borrowed at $12$% is $500,000 - (x+y)$ . Since the amount borrowed at $10$% is 2.5 times the amount borrowed at $9$% we have that $2.5y = x$. Now $$1.12(500,000 - 3.5y) + 1.1(2.5y) + 1.09y - 500,000 = 52,000.$$ The amount $1.12(500,000 - 3.5y) + 1.1(2.5y) + 1.09y$ gives us the money we borrowed plus interest. That is why we subtract the $500,000$.

Try solving for $y$. Once you have solved for $y$ you can find $x$ because $x = 2.5y$. Once you know those two you can easily find out how much is borrowed at $12$%. If you have further trouble let me know

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    From that equation, y= -23,529.4. That seems incorrect given the context?2012-09-10
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    I get $y = 100,000$2012-09-10
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A small software corporation borrowed 500,000 cash to expand its software line. The corporation borrowed some of the money at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest rate was $52,000 and the amount borrowed at 10% was 2.5 times the amount borrowed 9%.

Let $x$ = Amount borrowed at 9%
Let $y$ = Amount borrowed at 10%
Let $y$ = Amount borrowed at 12%

The equations are:

$0.09x + 0.10y + 0.12z = 52000$
$x+y+z = 500000$
$y = 2.5x$

  • 0
    Sorry for my ignorance, how would I set up the matrix from these equations? I'm not sure how to handle the third one in the context of the other two equations.2012-09-10