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To model simple interest applications with systems of equations, the mixture table can be adapted to reflect the simple interest formula, $$I = Prt$$. Since the interest earned is typically calculated for one year, the time variable $$t$$ is $$1$$, making the formula $$I = Pr$$. The table organizes the principal ($$P$$), the rate ($$r$$), and the interest ($$I$$) for each fund or loan. This structure allows the problem to be translated into a system of two equations: the first equation comes from the Principal column, representing the total amount of money invested or borrowed, and the second equation comes from the Interest column, representing the total interest earned or paid.

Setting Up an Interest Application Using a System of Equations

Consider an investment of \$$$50{,}000$$ divided between a stock fund earning $$7\%$$ interest per year and a savings account earning $$2\%$$ interest per year. The goal is to earn a total of $$6.2\%$$ interest on the investment in one year. Let $$s$$ be the amount in stocks and $$v$$ be the amount in savings. The system of equations is:
$$s + v = 50{,}000$$
$$0.07s + 0.02v = 0.062(50{,}000)$$
The second equation simplifies to $$0.07s + 0.02v = 3{,}100$$. Solving this system yields $$s = 42{,}000$$ and $$v = 8{,}000$$. Thus, \$$$42{,}000$$ is invested in the stock fund and \$$$8{,}000$$ in the savings account.

Solving a Stock and Savings Investment Application Using a System of Equations

Consider an investment of \$$$7{,}000$$ distributed across two stocks, one paying $$11\%$$ interest and the other paying $$13\%$$ interest. The total investment earns $$12.5\%$$ interest in one year. Let $$x$$ be the amount in the first stock and $$y$$ be the amount in the second stock. The system of equations is:
$$x + y = 7{,}000$$
$$0.11x + 0.13y = 0.125(7{,}000)$$
The second equation simplifies to $$0.11x + 0.13y = 875$$. Solving this system yields $$x = 1{,}750$$ and $$y = 5{,}250$$. Therefore, \$$$1{,}750$$ is invested at $$11\%$$ and \$$$5{,}250$$ is invested at $$13\%$$.

Solving a Dual Stock Investment Application Using a System of Equations

Apply the seven-step problem-solving strategy and a mixture table to solve an interest application involving two investments. Problem: 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? 1. Read the problem. A chart will help organize the information. 2. Identify what to find: the amount to invest in each fund. 3. Name the unknowns. Let $$s$$ = the amount invested in stocks, and $$b$$ = the amount invested in bonds. Multiply Principal $$\cdot$$ Rate $$\cdot$$ Time ($$t=1$$) to get the Interest. Organize into a table: / Fund / Principal (\$) / Rate / Interest (\$) / /---/---/---/---/ / Stocks / $$s$$ / $$0.08$$ / $$0.08s$$ / / Bonds / $$b$$ / $$0.03$$ / $$0.03b$$ / / Total / $$40{,}000$$ / $$0.071$$ / $$0.071(40{,}000)$$ / 4. Translate into a system of equations from the Principal and Interest columns: $$\left\{\begin{array}{l} s + b = 40{,}000 \ 0.08s + 0.03b = 0.071(40{,}000) \end{array}ight.$$ 5. Solve the system by elimination. Multiply the top equation by $$-0.03$$: $$-0.03s - 0.03b = -1{,}200$$ $$0.08s + 0.03b = 2{,}840$$ Add the equations: $$0.05s = 1{,}640$$ $$s = 32{,}800$$ Substitute $$s = 32{,}800$$ into the first equation: $$32{,}800 + b = 40{,}000$$ $$b = 7{,}200$$ 6. Check the answer. $$32{,}800 + 7{,}200 = 40{,}000$$ $$\checkmark$$ $$0.08(32{,}800) + 0.03(7{,}200) = 2{,}624 + 216 = 2{,}840$$ $$0.071(40{,}000) = 2{,}840$$ $$\checkmark$$ 7. Answer the question. Adnan should invest $$\$32{,}800$$ in stock and $$\$7{,}200$$ in bonds.

Solving an Investment Fund Interest Application Using a System of Equations

Apply the seven-step problem-solving strategy and a mixture table to solve an interest application involving two investments. Problem: Leon had $$\$50{,}000$$ to invest and hopes to earn $$6.2\%$$ interest per year. He will put some of the money into a stock fund that earns $$7\%$$ per year and the rest in to a savings account that earns $$2\%$$ per year. How much money should he put into each fund? 1. Read the problem. 2. Identify what to find: the amount to invest in each fund. 3. Name the unknowns. Let $$s$$ = the amount invested in stocks, and $$a$$ = the amount invested in the savings account. Organize into a table: / Fund / Principal (\$) / Rate / Interest (\$) / /---/---/---/---/ / Stocks / $$s$$ / $$0.07$$ / $$0.07s$$ / / Savings / $$a$$ / $$0.02$$ / $$0.02a$$ / / Total / $$50{,}000$$ / $$0.062$$ / $$0.062(50{,}000)$$ / 4. Translate into a system of equations: $$\left\{\begin{array}{l} s + a = 50{,}000 \ 0.07s + 0.02a = 0.062(50{,}000) \end{array}ight.$$ 5. Solve the system by elimination. Multiply the top equation by $$-0.02$$: $$-0.02s - 0.02a = -1{,}000$$ $$0.07s + 0.02a = 3{,}100$$ Add the equations: $$0.05s = 2{,}100$$ $$s = 42{,}000$$ Substitute $$s = 42{,}000$$ into the first equation: $$42{,}000 + a = 50{,}000$$ $$a = 8{,}000$$ 6. Check the answer. $$42{,}000 + 8{,}000 = 50{,}000$$ $$\checkmark$$ $$0.07(42{,}000) + 0.02(8{,}000) = 2{,}940 + 160 = 3{,}100$$ $$0.062(50{,}000) = 3{,}100$$ $$\checkmark$$ 7. Answer the question. Leon should invest $$\$42{,}000$$ in the stock fund and $$\$8{,}000$$ in the savings account.

Solving a Savings and Stock Interest Application Using a System of Equations

Apply the seven-step problem-solving strategy and a mixture table to solve an interest application involving two investments. Problem: Julius invested $$\$7{,}000$$ into two stock investments. One stock paid $$11\%$$ interest and the other stock paid $$13\%$$ interest. He earned $$12.5\%$$ interest on the total investment. How much money did he put in each stock? 1. Read the problem. 2. Identify what to find: the amount invested in each stock. 3. Name the unknowns. Let $$x$$ = the amount invested at $$11\%$$, and $$y$$ = the amount invested at $$13\%$$. Organize into a table: / Fund / Principal (\$) / Rate / Interest (\$) / /---/---/---/---/ / 11% Stock / $$x$$ / $$0.11$$ / $$0.11x$$ / / 13% Stock / $$y$$ / $$0.13$$ / $$0.13y$$ / / Total / $$7{,}000$$ / $$0.125$$ / $$0.125(7{,}000)$$ / 4. Translate into a system of equations: $$\left\{\begin{array}{l} x + y = 7{,}000 \ 0.11x + 0.13y = 0.125(7{,}000) \end{array}ight.$$ 5. Solve the system by elimination. Multiply the top equation by $$-0.11$$: $$-0.11x - 0.11y = -770$$ $$0.11x + 0.13y = 875$$ Add the equations: $$0.02y = 105$$ $$y = 5{,}250$$ Substitute $$y = 5{,}250$$ into the first equation: $$x + 5{,}250 = 7{,}000$$ $$x = 1{,}750$$ 6. Check the answer. $$1{,}750 + 5{,}250 = 7{,}000$$ $$\checkmark$$ $$0.11(1{,}750) + 0.13(5{,}250) = 192.5 + 682.5 = 875$$ $$0.125(7{,}000) = 875$$ $$\checkmark$$ 7. Answer the question. Julius invested $$\$1{,}750$$ at $$11\%$$ and $$\$5{,}250$$ at $$13\%$$.

Solving a Dual Stock Interest Application Using a System of Equations

Apply the seven-step problem-solving strategy and a mixture table to find the principal given the amount of interest earned. Problem: 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? 1. Read the problem. A chart will help organize the information. 2. Identify what to find: the principal of each loan. 3. Name the unknowns. Let $$b$$ = the principal for the bank loan, and $$f$$ = the principal for the federal loan. Multiply Principal $$\cdot$$ Rate $$\cdot$$ Time ($$t=1$$) to get the Interest. Organize into a table: / Loan / Principal (\$) / Rate / Interest (\$) / /---/---/---/---/ / Bank / $$b$$ / $$0.105$$ / $$0.105b$$ / / Federal / $$f$$ / $$0.059$$ / $$0.059f$$ / / Total / $$21{,}540$$ / / $$1{,}669.68$$ / 4. Translate into a system of equations from the Principal and Interest columns: $$\left\{\begin{array}{l} b + f = 21{,}540 \ 0.105b + 0.059f = 1{,}669.68 \end{array}ight.$$ 5. Solve the system using substitution. Solve the first equation for $$b$$: $$b = -f + 21{,}540$$ Substitute into the second equation: $$0.105(-f + 21{,}540) + 0.059f = 1{,}669.68$$ $$-0.105f + 2{,}261.7 + 0.059f = 1{,}669.68$$ $$-0.046f + 2{,}261.7 = 1{,}669.68$$ $$-0.046f = -592.02$$ $$f = 12{,}870$$ To find $$b$$, substitute $$f = 12{,}870$$ into the first equation: $$b = -12{,}870 + 21{,}540$$ $$b = 8{,}670$$ 6. Check the answer. $$8{,}670 + 12{,}870 = 21{,}540$$ $$\checkmark$$ $$0.105(8{,}670) + 0.059(12{,}870) = 910.35 + 759.33 = 1{,}669.68$$ $$\checkmark$$ 7. Answer the question. The principal for the bank loan was $$\$8{,}670$$ and the principal for the federal loan was $$\$12{,}870$$.

Solving a Student Loan Interest Application Using a System of Equations

Apply the seven-step problem-solving strategy and a mixture table to find the principal given the amount of interest earned. Problem: Laura owes $$\$18{,}000$$ on her student loans. The interest rate on the bank loan is $$2.5\%$$ and the interest rate on the federal loan is $$6.9\%$$. The total amount of interest she paid last year was $$\$1{,}066$$. What was the principal for each loan? 1. Read the problem. 2. Identify what to find: the principal of each loan. 3. Name the unknowns. Let $$b$$ = the principal for the bank loan, and $$f$$ = the principal for the federal loan. Organize into a table: / Loan / Principal (\$) / Rate / Interest (\$) / /---/---/---/---/ / Bank / $$b$$ / $$0.025$$ / $$0.025b$$ / / Federal / $$f$$ / $$0.069$$ / $$0.069f$$ / / Total / $$18{,}000$$ / / $$1{,}066$$ / 4. Translate into a system of equations: $$\left\{\begin{array}{l} b + f = 18{,}000 \ 0.025b + 0.069f = 1{,}066 \end{array}ight.$$ 5. Solve the system using substitution. Solve the first equation for $$b$$: $$b = -f + 18{,}000$$ Substitute into the second equation: $$0.025(-f + 18{,}000) + 0.069f = 1{,}066$$ $$-0.025f + 450 + 0.069f = 1{,}066$$ $$0.044f + 450 = 1{,}066$$ $$0.044f = 616$$ $$f = 14{,}000$$ To find $$b$$, substitute $$f = 14{,}000$$ into the first equation: $$b + 14{,}000 = 18{,}000$$ $$b = 4{,}000$$ 6. Check the answer. $$4{,}000 + 14{,}000 = 18{,}000$$ $$\checkmark$$ $$0.025(4{,}000) + 0.069(14{,}000) = 100 + 966 = 1{,}066$$ $$\checkmark$$ 7. Answer the question. The principal for the bank loan was $$\$4{,}000$$ and the principal for the federal loan was $$\$14{,}000$$.

Solving a Dual Student Loan Interest Application Using a System of Equations

A financial analyst is setting up a system of equations to help a client distribute capital between two investment accounts. The analyst uses an organizational table to track the principal, the interest rates, and the annual interest for each account. Match each part of this table to its corresponding role in the resulting mathematical model for a one-year period.

A corporate treasurer is setting up a system of equations to manage two different short-term investments over a one-year period. If $x$ and $y$ represent the principal amounts invested at annual interest rates of $r_1$ and $r_2$ respectively, which equation correctly models the total interest ($I$) earned?

A small business owner is using a system of equations to analyze two different loans over a one-year period. When setting up the organizational table for this model, why is the interest ($I$) for each loan represented simply as the product of the principal ($P$) and the annual interest rate ($r$)? Briefly explain by referencing the standard simple interest formula $I = Prt$.

Simplifying the Interest Formula for Annual Models

A financial coordinator is managing two different investment accounts for a one-year period. To model the total principal and total interest using a system of equations, arrange the following setup steps in the correct chronological order.

A corporate accountant is setting up a system of equations to model two different investment funds for a one-year period. True or False: To build this system, the accountant typically derives one equation from the total principal and a second equation from the total annual interest using the formula $I = Pr$.

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