Solving Interest Applications
Solving Interest Applications
The formula to model interest applications is I = Prt. Interest, I, is the product of the principal, P, the rate, r, and the time, t. In our work here, we will calculate the interest earned in one year, so t will be 1.
We modify the column titles in the mixture table to show the formula for interest, as you’ll see in the example below.
Example
Translate to a system of equations and solve:
Adnan has $40,000 to invest and hopes to earn 7.1% interest per year. He will put some of the money into a stock fund that earns 8% per year and the rest into bonds that earns 3% per year. How much money should he put into each fund?
Solution
| Step 1. Read the problem. | A chart will help us organize the information. |
| Step 2. Identify what we are looking for. | We are looking for the amount to invest in each fund. |
| Step 3. Name what we are looking for. | Let \(s=\) the amount invested in stocks. \(\phantom{\rule{1.5em}{0ex}}b=\) the amount invested in bonds. |
| Write the interest rate as a decimal for each fund. Multiply: Principal · Rate · Time to get the Interest. |
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| Step 4. Translate into a system of equations. We get our system of equations from the Principal column and the Interest column. |
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| Step 5. Solve the system of equations Solve by elimination. Multiply the top equation by −0.03. |
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| Simplify and add to solve for s. | |
| To find b, substitute s = 32,800 into the first equation. | |
| Step 6. Check the answer in the problem. | We leave the check to you. |
| Step 7. Answer the question. | Adnan should invest $32,800 in stock and $7,200 in bonds. |
Did you notice that the Principal column represents the total amount of money invested while the Interest column represents only the interest earned? Likewise, the first equation in our system, s + b = 40,000, represents the total amount of money invested and the second equation, 0.08s + 0.03b = 0.071(40,000), represents the interest earned.
Example
Translate to a system of equations and solve:
Rosie owes $21,540 on her two student loans. The interest rate on her bank loan is 10.5% and the interest rate on the federal loan is 5.9%. The total amount of interest she paid last year was $1,669.68. What was the principal for each loan?
Solution
| Step 1. Read the problem. | A chart will help us organize the information. |
| Step 2. Identify what we are looking for. | We are looking for the principal of each loan. |
| Step 3. Name what we are looking for. | Let \(b=\) the principal for the bank loan. \(\phantom{\rule{1.5em}{0ex}}f=\) the principal on the federal loan |
| The total loans are $21,540. | |
| Record the interest rates as decimals in the chart. | |
| Multiply using the formula l = Pr t to get the Interest. | |
| Step 4. Translate into a system of equations.
The system of equations comes from the Principal column and the Interest column.
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| Step 5. Solve the system of equations We will use substitution to solve. Solve the first equation for b. |
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| Substitute b = −f + 21,540 into the second equation. | |
| Simplify and solve for f. | |
| To find b, substitute f = 12,870 into the first equation. | |
| Step 6. Check the answer in the problem. | We leave the check to you. |
| Step 7. Answer the question. | The principal of the bank loan is $12,870 and the principal for the federal loan is $8,670. |
Key Concepts
- Table for coin and mixture applications
- Table for concentration applications
- Table for interest applications
This lesson is part of:
Systems of Linear Equations I