Problem 62 - Precalculus Mathematics for Calculus

Question

A fruit stand sells two varieties of strawberries: standard and deluxe. A box of standard strawberries sells for \(\$ 7,\) and a box of deluxe strawberries sells for \(\$ 10 .\) In one day the stand sells 135 boxes of strawberries for a total of \(\$ 1110 .\) How many boxes of each type were sold?

Step-by-Step Solution

Answer

80 boxes of standard and 55 boxes of deluxe strawberries were sold.

Step 1: Define Variables

Let's define two variables: let \( x \) represent the number of boxes of standard strawberries sold, and \( y \) represent the number of boxes of deluxe strawberries sold.

Step 2: Set Up Equations

We know from the problem statement that the total number of boxes sold is 135, so we can write the equation \( x + y = 135 \). Additionally, the total revenue from selling these strawberries is \$1110, which gives us the second equation \( 7x + 10y = 1110 \) based on the individual prices.

Step 3: Solve the System of Equations - Substitution Method

First, solve the first equation for \( x \): \( x = 135 - y \). Substitute this expression into the second equation: \( 7(135 - y) + 10y = 1110 \).

Step 4: Simplify and Solve for \( y \)

Distribute the 7 in the equation: \( 945 - 7y + 10y = 1110 \). Simplify to \( 3y = 1110 - 945 \). Calculate \( 1110 - 945 = 165 \). Thus, \( 3y = 165 \). Solve for \( y \) by dividing by 3: \( y = \frac{165}{3} = 55 \).

Step 5: Solve for \( x \)

Substitute the value of \( y = 55 \) back into the equation \( x = 135 - y \). Thus, \( x = 135 - 55 = 80 \).

Key Concepts

Introduction to Algebra

Algebra is a fundamental part of mathematics that involves using symbols and letters to represent numbers and quantities in equations. This enables us to solve for unknowns, simplify expressions, and work with equations to uncover relationships between different variables. In the context of our fruit stand problem, algebra allows us to set up equations that model real-world scenarios. These equations help to determine how many boxes of strawberries were sold.

By defining variables, such as \(x\) for the standard strawberries and \(y\) for the deluxe strawberries, we can create a mathematical representation of the problem. Algebraic equations follow the rules of arithmetic, but they allow for much broader applications, especially when dealing with unknown quantities. This makes algebra a powerful tool for solving many types of mathematical problems.

Understanding the Substitution Method

The substitution method is a strategic approach in algebra to solve systems of equations. The key idea is to express one variable in terms of the other using one equation, then substitute this expression into the other equation. This technique simplifies the problem to a single equation with one variable.

In our strawberry problem, we started with the equations \(x + y = 135\) and \(7x + 10y = 1110\). By solving the first equation for \(x\), we found \(x = 135 - y\). We substituted \(135 - y\) in place of \(x\) in the second equation to create a simpler equation with only \(y\): \(7(135 - y) + 10y = 1110\).
This allowed us to solve for \(y\), and once we knew \(y\), it was easy to find \(x\) using the initial equation.

• The substitution method is particularly useful because it often reduces the complexity of the problem by reducing the number of variables involved in each equation.
• This approach is most effective when one of the equations can be easily manipulated to isolate one variable.

Solving Word Problems with Equations

Word problems are designed to test a student's ability to translate real-world situations into mathematical language. The key steps are understanding the problem, defining variables, setting up equations, and then solving them.

In this particular problem involving the fruit stand, we took stock of what we know: the number of boxes sold and the total revenue. Using these, we framed our questions into a system of equations.
Breaking down word problems into manageable parts takes practice, but here are some tips:

• Start by carefully reading the problem to ensure you understand what is being asked.
• Identify known quantities and relationships among them, then define variables to represent unknown quantities.
• Translate these relationships into mathematical equations using these variables.
• Use algebraic methods, like substitution or elimination, to solve the equations.

Practicing with word problems enhances your ability to apply algebra to real-life scenarios, making abstract concepts much more tangible.