Lesson 4.3: Solve Mixture Applications with Systems of Equations

New Concept

Learn to model real-world scenarios by creating systems of equations. We'll tackle mixture, interest, and cost/revenue problems, translating given information into two equations with two variables to find the specific amounts needed for each component.

What’s next

Now, let's apply this. You'll work through interactive examples and practice cards covering coin, mixture, and interest problems.

Property

Mixture applications involve combining two or more quantities. To solve, we create a table to organize the information. The table typically has columns for the type of item, the number of units, the value (or concentration) per unit, and the total value. This organization helps in translating the word problem into a system of two linear equations, which can then be solved to find the unknown quantities. The equations are derived from the 'Number' column and the 'Total Value' column.

Examples

• A movie theater sold 450 tickets. Adult tickets cost 13 dollars and child tickets cost 8 dollars. If receipts totaled 4650 dollars, how many of each were sold? Let $a$ be adult tickets and $c$ be child tickets. The system is $a + c = 450$ and $13a + 8c = 4650$. Solving gives $a=210$ and $c=240$.

• A blend of coffee is made from beans costing 8 dollars per pound and 14 dollars per pound. If the blend is 30 pounds and is valued at 10 dollars per pound, how many pounds of each bean are used? Let $x$ be pounds of the first bean and $y$ be pounds of the second. The system is $x + y = 30$ and $8x + 14y = 300$. Solving gives $x=20$ and $y=10$.

Property

The formula to model simple interest applications is $I = Prt$, where $I$ is interest, $P$ is principal, $r$ is the rate, and $t$ is the time. For calculations involving one year, $t=1$. A table is used to organize the information with columns for the account type, principal, rate, time, and interest. The system of equations is derived from the 'Principal' column (total amount invested) and the 'Interest' column (total interest earned).

Examples

• Jake invested 20,000 dollars, part in a fund earning 5% interest and the rest in a fund earning 9% interest. If his total interest for the year was 1,320 dollars, how much was in each fund? Let $x$ and $y$ be the amounts. The system is $x + y = 20000$ and $0.05x + 0.09y = 1320$. Solving gives $x=12000$ and $y=8000$.

• A retiree invests 70,000 dollars into two accounts. She wants to earn an overall interest rate of 6%. One account pays 4% and the other pays 7%. How much should she invest in each? The system is $x+y=70000$ and $0.04x + 0.07y = 0.06(70000)$. Solving gives $x=23333.33$ and $y=46666.67$.

Property

The cost function is the cost to manufacture each unit times $x$, the number of units manufactured, plus the fixed costs.

The revenue function is the selling price of each unit times $x$, the number of units sold.

The break-even point is when the revenue equals the costs.