Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The following matrix gives the amounts of debt and the terms for the companies being purchased.
Due in 30 days  		Due in 60 days	
Ace  
Baker  
Champ  

$120,000
$75,000
$105,000
$180,000
$45,000
$174,000

(a) If ALCO pays 55% of the amount owed on each debt, write the matrix giving the remaining debts.
					

				
				


(b) Suppose ALCO decides to pay 95% of all debts due in 30 days and to increase the debts due in 60 days by 25%. Write a matrix that gives the debts after these transactions are made.
					

				
				


Let's tackle the problem step-by-step, beginning with the given matrix of debts. The matrix for the amounts due is as follows:

\[
\begin{matrix}
  & \text{Due in 30 days} & \text{Due in 60 days} \\
\text{Ace} & 120,000 & 180,000 \\
\text{Baker} & 75,000 & 45,000 \\
\text{Champ} & 105,000 & 174,000 \\
\end{matrix}
\]

### Part (a) - ALCO pays 55% of the amount owed on each debt

If ALCO pays 55% of the amount owed, then 45% of the debt remains unpaid. To find the remaining debts, we multiply each entry in the matrix by 0.45 (since $100\% - 55\% = 45\%$).

The calculations for each company will be:

\[
\begin{aligned}
\text{Ace (30 days)} & = 120,000 \times 0.45 = 54,000 \\
\text{Ace (60 days)} & = 180,000 \times 0.45 = 81,000 \\
\text{Baker (30 days)} & = 75,000 \times 0.45 = 33,750 \\
\text{Baker (60 days)} & = 45,000 \times 0.45 = 20,250 \\
\text{Champ (30 days)} & = 105,000 \times 0.45 = 47,250 \\
\text{Champ (60 days)} & = 174,000 \times 0.45 = 78,300 \\
\end{aligned}
\]

Thus, the matrix of the remaining debts is:

\[
\begin{matrix}
  & \text{Due in 30 days} & \text{Due in 60 days} \\
\text{Ace} & 54,000 & 81,000 \\
\text{Baker} & 33,750 & 20,250 \\
\text{Champ} & 47,250 & 78,300 \\
\end{matrix}
\]

### Part (b) - ALCO pays 95% of debts due in 30 days and increases debts due in 60 days by 25%

Now, let's adjust the debts for both time frames based on these new conditions:

1. **For debts due in 30 days**: ALCO pays 95%, so 5% of the debt remains. We multiply the amounts due in 30 days by 0.05.

2. **For debts due in 60 days**: The debts increase by 25%, so we multiply the amounts due in 60 days by 1.25.

The calculations are as follows:

\[
\begin{aligned}
\text{Ace (30 days)} & = 120,000 \times 0.05 = 6,000 \\
\text{Ace (60 days)} & = 180,000 \times 1.25 = 225,000 \\
\text{Baker (30 days)} & = 75,000 \times 0.05 = 3,750 \\
\text{Baker (60 days)} & = 45,000 \times 1.25 = 56,250 \\
\text{Champ (30 days)} & = 105,000 \times 0.05 = 5,250 \\
\text{Champ (60 days)} & = 174,000 \times 1.25 = 217,500 \\
\end{aligned}
\]

Thus, the new matrix of debts after these transactions is:

\[
\begin{matrix}
  & \text{Due in 30 days} & \text{Due in 60 days} \\
\text{Ace} & 6,000 & 225,000 \\
\text{Baker} & 3,750 & 56,250 \\
\text{Champ} & 5,250 & 217,500 \\
\end{matrix}
\]

---

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:
1. How would the matrix change if ALCO paid a different percentage of debts, like 75% instead of 55%?
2. What would be the total remaining debt after each transaction?
3. How can you represent these debt changes as a system of equations?
4. What would the debt matrix look like if only one of the companies (e.g., Ace) had its debts adjusted?
5. How could you model interest accumulation on the debts using matrices?

**Tip**: When working with percentage changes in matrix entries, remember that multiplying by $1 + \text{percentage increase}$ applies the change directly to each debt.

Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The following matrix gives the amounts of debt and the terms for the companies being purchased. (a) If ALCO pays 55% of the amount owed on each debt, write the matrix giving the remaining debts. (b) Suppose ALCO decides to pay 95% of all debts due in 30 days and to increase the debts due in 60 days by 25%. Write a matrix that gives the debts after these transactions are made.

Explore how to calculate the remaining debts for Ace, Baker, and Champ after being purchased by ALCO, Inc. This problem involves matrix operations and percentage calculations, suitable for Grades 10-12. Learn step-by-step how to adjust debts based on specific payment and increase percentages.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation