Loans To get the necessary funds for a planned expansion, a...

Loans To get the necessary funds for a planned expansion, a small company took out three loans totaling $\$ 25,000$ . Company owners were able to get interest rates of $8 \%, 9 \%,$ and 10$\% .$ They borrowed $\$ 1000$ more at 9$\%$ than they borrowed at 10$\% .$ The total annual interest on the loans was $\$ 2190$ . a. How much did they borrow at each rate? b. Suppose we drop the condition that they borrowed $\$ 1000$ more at 9$\%$ than at 10$\% .$ What can you say about the amount borrowed at 10$\%$ ? What is the solution if the amount borrowed at 10$\%$ is $\$ 5000 ?$ c. Suppose the bank sets a maximum of $\$ 10,000$ at the lowest interest rate of 8$\% .$ Is a solution possible that still meets all of the original conditions? d. Explain why $\$ 10,000$ at $8 \%, \$ 8000$ at $9 \%,$ and $\$ 7000$ at 10$\%$ is not a feasible solution for part $c .$

Loans To get the necessary funds for a planned expansion, a small company took out three loans totaling $\$ 25,000$ . Company owners were able to get interest rates of $8 \%, 9 \%,$ and 10$\% .$ They borrowed $\$ 1000$ more at 9$\%$ than they borrowed at 10$\% .$ The total annual interest on the loans was $\$ 2190$ .
a. How much did they borrow at each rate?
b. Suppose we drop the condition that they borrowed $\$ 1000$ more at 9$\%$ than at 10$\% .$ What can you say about the amount borrowed at 10$\%$ ? What is the solution if the amount borrowed at 10$\%$ is $\$ 5000 ?$
c. Suppose the bank sets a maximum of $\$ 10,000$ at the lowest interest rate of 8$\% .$ Is a solution possible that still meets all of the original conditions?
d. Explain why $\$ 10,000$ at $8 \%, \$ 8000$ at $9 \%,$ and $\$ 7000$ at 10$\%$ is not a feasible solution for part $c .$

Key Concepts

System of Linear Equations

This concept involves setting up and solving equations where multiple linear relationships must hold simultaneously. In the context of loan problems, different equations represent total loan amounts, interest rates, and specific relationships between loan parts. Solving these equations allows one to determine the unknown loan amounts that satisfy all given conditions.

Variable Assignment

Assigning variables is a fundamental approach in formulating a mathematical model from a word problem. By representing unknown quantities, such as the amount borrowed at different interest rates, as variables, one can translate the problem’s conditions into algebraic equations that are easier to manipulate and solve.

Interest Calculation

Interest calculation is the process of determining the cost of borrowing money, usually by applying a percentage rate to the principal amount. In problems involving multiple loans, the interest for each loan is calculated separately and then combined to form a total interest equation, which is key to linking the loan amounts to the overall interest paid.

Constraints and Conditions

Constraints in a problem are the specific conditions or limits that any acceptable solution must satisfy. These include equalities, such as the total sum borrowed, and additional relationships between different loan amounts. Ensuring that all constraints are met is critical in determining a correct and feasible solution.

Inequalities and Bound Analysis

Inequalities and bound analysis come into play when the problem includes conditions that set upper or lower limits on certain quantities. For example, a constraint that limits the amount borrowed at a particular interest rate requires the solution to be analyzed within those bounds, ensuring that any proposed solution is realistic and compliant with the restrictions.

Parametric Solutions

When some conditions are relaxed or when there is one degree of freedom in the system, the solution can be expressed in terms of a parameter. This means a range or family of solutions exists instead of a unique answer. Parametric solutions allow for exploring different scenarios within the constraints provided and then choosing a specific solution based on additional criteria.

Feasibility Analysis

Feasibility analysis involves checking whether a proposed solution meets all the problem’s constraints and conditions. In loan allocation problems, even if a numerical solution is found, it must be assessed against real-world constraints, such as maximum allowable amounts, to ensure that it is both valid and practical.

Transcript

00:01 Okay, so for this problem, i know, so i'm going to have three interest rates.

00:03 So i'm going to set x to be 8%, y to be 9%, and z to be 10%.

00:12 So for the, for part a, i'm going to have my three equations.

00:16 So it's asking us how much to, how much was borrowed at one time.

00:21 So they have three different loans totaling 25 ,000.

00:25 So which means x plus y plus z equals 25 ,000.

00:32 We are told the, of course, we then, we have the interest rates and we are told that we are paying a total of $2 ,090 in interest.

00:46 And we are told that we are going to take out the same amount for the 9%.

00:51 I'm sorry, we're going to take out a thousand more of the 9%.

00:56 So it's going to be y minus z is going to equal 1 ,000.

01:05 So i'm going to set these up in a matrix.

01:07 So it's going to be 1 .8.

01:11 1 .9 ,1, 1 .1, and negative 1, and then my solutions of 25 ,000, 2 ,190, and 1 ,000.

01:25 After utilizing a calculator, i find that i'm going to have $12 ,000 and the 8 % loan, $7 ,000 in the 9%, and then $6 ,000, and the $10.

01:44 So then for part b, we are, says it's supposed to we drop the condition, pretty much the last condition.

01:53 So what's going to happen? so if we look at this, we have our 0 .8.

02:06 Okay, so it's saying if, it's saying if we drop the one about the $1 ,000 more.

02:15 So we're going to have our first condition, which is saying that all of them together, we're going to have $25 ,000.

02:20 And then plus our interest.

02:22 And that's going to equal 2190.

02:28 So if i set up this matrix, it's going to be 1 .8, 1 .9, 1 .1, with 25 ,000, and 2190 being our solutions.

02:41 So our very last, so our last matrix that we should get for this one is going to be 1 -0 -9 -1, and then with a solution of 6 ,000.

03:02 And then, of course, we're going to have 0 -1 -2 -19 ,000.

03:09 So if we look at these two equations, what this ends up telling me is i'm going to have x minus z equals 6 ,000...

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