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Translate to a system of equations and solve:
Anatole needs to make 250 milliliters of a 25% solution of hydrochloric acid for a lab experiment. The lab only has a 10% solution and a 40% solution in the storeroom. How much of the 10% and how much of the 40% solutions should he mix to make the 25% solution?
Anatole should mix 125 ml of the 10% solution and 125 ml of the 40% solution.
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 [link] .
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?
| 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
the amount invested in stocks.
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 . |
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| To find b , substitute s = 32,800 into the first equation. |
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| Step 6. Check the answer in the problem. | We leave the check to you. |
| Step 7. Answer the question. | Adnan should invest $32,000 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, 0.08 s + 0.03 b = 0.071(40,000), represents the interest earned.
Translate to a system of equations and solve:
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?
Leon should put $42,000 in the stock fund and $8000 in the savings account.
Translate to a system of equations and solve:
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?
Julius invested $1,750 at 11% and $5,250 at 13%.
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?
| 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
the principal for the bank loan.
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. |
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|
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. |
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| Simplify and solve for f . |
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| To find
b , substitute
f = 12,870 into
the first equation. |
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|
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. |
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