# Thread: Cramers rule and equations help

1. ## Cramers rule and equations help

hi guys i'm having some problem finding the correct 3 equations which go with the each of these problems. i know how to work it through cramer's rule already if you can help me out that would be great!

1. A company invests a total of $1 million in three different forms: short-term loans, treasury notes, and municipal bonds. The short-term loans pay annual interest at a rate of 10%, the treasury notes pay 8% interest, and the bonds pay 4% interest. The total interest for the year is$60,000 or $0.06 million. If the company has three times as much money invested in municipal bonds as in treasury notes, how much money is invested in each type of investment? 2. The annual report of a company contains a summary of personnel figures. It states that the company has a total of 900 employees in three classifications-clerical, factory, and trainee- and that it budgets a total of 2450 weeks for paid vacations for these employees during the year and 10, 450 days for paid sick leave. The report also states that a clerical employee receives 2 weeks of paid vacation and 8 days of sick leave each year, that a factory employee receives 3 weeks of vacation and 13 days of sick leave, and that a trainee gets 1 week of vacation and 3 days of sick leave. The report does not state how many employees the company has in each of the three categories. If possible, determine this from the information given. 2. Originally Posted by xsolutionx 1. A company invests a total of$1 million in three different forms: short-term loans, treasury notes, and municipal bonds. The short-term loans pay annual interest at a rate of 10%, the treasury notes pay 8% interest, and the bonds pay 4% interest. The total interest for the year is $60,000 or$0.06 million. If the company has three times as much money invested in municipal bonds as in treasury notes, how much money is invested in each type of investment?
Ok, let's take this step by step:

Let $\displaystyle x$ be the amount of money invested in loans
Let $\displaystyle n$ be the amount of money invested in notes
Let $\displaystyle b$ be the amount of money invested in bonds

Since the company invests a total of $1 million, we have:$\displaystyle x + n + b = 1$Since the short-term loans pay annual interest at a rate of 10%, the treasury notes pay 8% interest, and the bonds pay 4% interest, and the total interest for the year is$0.06 million, we have:

$\displaystyle 0.1x + 0.08n + 0.04b = 0.06$

Since, the company has three times as much money invested in municipal bonds as in treasury notes, we have:

$\displaystyle b = 3n$

or

$\displaystyle b - 3n = 0$

Thus, we need to solve the system:

$\displaystyle \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0.1 & 0.08 & 0.04 \\ 0 & -3 & 1 \end{array} \right) \left( \begin{array}{c} x \\ n \\ b \end{array} \right) = \left( \begin{array}{c} 1 \\ 0.06 \\ 0 \end{array} \right)$

Can you use this reasoning to set up the system for the second?

3. Originally Posted by Jhevon
Ok, let's take this step by step:

Let $\displaystyle x$ be the amount of money invested in loans
Let $\displaystyle n$ be the amount of money invested in notes
Let $\displaystyle b$ be the amount of money invested in bonds

Since the company invests a total of $1 million, we have:$\displaystyle x + n + b = 1$Since the short-term loans pay annual interest at a rate of 10%, the treasury notes pay 8% interest, and the bonds pay 4% interest, and the total interest for the year is$0.06 million, we have:

$\displaystyle 0.1x + 0.08n + 0.04b = 0.06$

Since, the company has three times as much money invested in municipal bonds as in treasury notes, we have:

$\displaystyle b = 3n$

or

$\displaystyle b - 3n = 0$

Thus, we need to solve the system:

$\displaystyle \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0.1 & 0.08 & 0.04 \\ 0 & -3 & 1 \end{array} \right) \left( \begin{array}{c} x \\ n \\ b \end{array} \right) = \left( \begin{array}{c} 1 \\ 0.06 \\ 0 \end{array} \right)$

Can you use this reasoning to set up the system for the second?
yes thank you very much! i appreciate all your guys' help tonight.