Lesson 5: Solve Mixture Applications with Systems of Equations

New Concept

We'll tackle real-world mixture and interest problems by translating them into systems of two linear equations. This powerful method lets us find unknown quantities, like numbers of coins or investment amounts, using algebra.

What’s next

Next, we'll dive into worked examples. You'll use our interactive tables to set up and solve systems of equations for various mixture scenarios.

Property

Use a table to organize items by Number, Value, and Total Value. Two variables represent the number of each item (e.g., nickels and dimes). The system of equations is formed from the total number of items and the total monetary value. For a problem with two types of items, item 1 and item 2, the system is:

Examples

• A theater sold 250 tickets, earning 4500 dollars. Adult tickets cost 20 dollars and child tickets cost 10 dollars. Let $a$ be adult tickets and $c$ be child tickets. The system is $a + c = 250$ and $20a + 10c = 4500$. Solving gives $a=200$ and $c=50$.

• Dev has a collection of dimes and quarters with a total value of 10.55 dollars. The number of dimes is five more than twice the number of quarters. Let $d$ be dimes and $q$ be quarters. The system is $0.10d + 0.25q = 10.55$ and $d = 2q + 5$. Solving gives $d=47$ and $q=21$.

Property

To solve mixture problems, like trail mix, use a table to track the number of pounds (or other units), the value per unit, and the total value for each component and the final mixture. The system of equations comes from the total quantity and the total value.

Examples

• Carson wants to make 40 pounds of trail mix costing 6 dollars per pound by mixing nuts at 8 dollars per pound and chocolate chips at 2 dollars per pound. Let $n$ be pounds of nuts and $c$ be pounds of chips. The system is $n+c=40$ and $8n+2c=6(40)$. He should use 26.67 pounds of nuts and 13.33 pounds of chips.

• A coffee shop wants to make a 20-pound blend costing 15 dollars per pound using a coffee that costs 12 dollars per pound and another that costs 16 dollars per pound. Let $x$ and $y$ be the pounds of each coffee. The system is $x+y=20$ and $12x+16y=15(20)$. They should use 5 pounds of the first coffee and 15 pounds of the second.

Property

For mixture problems involving percent concentrations, use a table with columns for the number of units, concentration (as a decimal), and the total amount of the pure substance. The system of equations is derived from the total volume and the total amount of the dissolved substance.

Examples

• Sasheena needs to make 300 ml of a 50% solution of sulfuric acid. The lab has 20% and 60% solutions. Let $x$ be the ml of 20% solution and $y$ be the ml of 60% solution. The system is $x+y=300$ and $0.20x+0.60y=0.50(300)$. She should mix 75 ml of the 20% solution and 225 ml of the 60% solution.

• Jotham needs 50 liters of a 40% alcohol solution. He has a 25% and a 75% solution available. Let $x$ be liters of the 25% and $y$ be liters of the 75% solution. The system is $x+y=50$ and $0.25x+0.75y=0.40(50)$. He should mix 35 liters of the 25% and 15 liters of the 75% solution.

Property

The formula to model interest applications is $I = Prt$. Interest, $I$, is the product of the principal, $P$, the rate, $r$, and the time, $t$. For one year, $t=1$. The system of equations comes from the Principal column and the Interest column.

Examples

• Adnan has 60,000 dollars to invest and hopes to earn 6% interest per year. He puts some money into a stock fund that earns 8% per year and the rest into bonds that earn 3% per year. Let $s$ be the amount in stocks and $b$ be the amount in bonds. The system is $s+b=60000$ and $0.08s+0.03b=0.06(60000)$. He should put 36,000 dollars in stocks and 24,000 dollars in bonds.

• Rosie owes 50,000 dollars on two student loans. The interest rate on her bank loan is 8.5% and the interest rate on the federal loan is 4.5%. The total interest she paid last year was 3050 dollars. Let $b$ be the bank loan and $f$ be the federal loan. The system is $b+f=50000$ and $0.085b+0.045f=3050$. The bank loan is 20,000 dollars and the federal loan is 30,000 dollars.