Problem 52 - College Algebra

Question

Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

Step-by-Step Solution

Answer

1,250 vanilla and 3,750 chocolate cupcakes are sold each day.

Step 1: Define Variables

Let \( x \) be the number of vanilla cupcakes sold each day. Since chocolate cupcakes are 3 times as popular as vanilla ones, the number of chocolate cupcakes sold each day is \( 3x \).

Step 2: Set Up Equation

The total number of cupcakes sold every day is 5,000. Therefore, we can set up the equation: \( x + 3x = 5000 \).

Step 3: Simplify Equation

Simplify the equation to combine like terms: \( 4x = 5000 \).

Step 4: Solve for Vanilla Cupcakes

Divide both sides of the equation by 4 to find the value of \( x \): \( x = \frac{5000}{4} = 1250 \). So, 1,250 vanilla cupcakes are sold each day.

Step 5: Solve for Chocolate Cupcakes

Since chocolate cupcakes are 3 times the vanilla ones, calculate \( 3x \): \( 3 \times 1250 = 3750 \). Therefore, 3,750 chocolate cupcakes are sold each day.

Key Concepts

Linear Equations

Linear equations are equations that make a straight line when graphed. They typically have one or two variables. In the cupcake problem, the linear equation is used to represent the total number of cupcakes sold daily. The equation we set up is \( x + 3x = 5000 \).
This equation has one variable, \( x \), which stands for the vanilla cupcakes sold.A linear equation always aims to find values for the variables that make the equation true. This is done by isolating the variable on one side. Here, we simplified the equation to: \( 4x = 5000 \).By solving this equation, we discover how many vanilla cupcakes are sold. This is a perfect example of how linear equations can solve real-life problems by providing necessary insights through simple arithmetic operations.

Variables and Expressions

Variables and expressions are key elements in solving mathematical problems. A variable is a letter or symbol used to represent a number in equations. For instance, in our cupcake scenario, \( x \) is a variable that denotes how many vanilla cupcakes are sold each day. Expressions, on the other hand, are phrases that combine variables, numbers, and operations. In this case, \( 3x \) represents the number of chocolate cupcakes sold and is an expression indicating that chocolate cupcakes are three times more popular than vanilla cupcakes.Understanding variables and expressions helps us translate real-world situations—like the distribution of cupcake sales—into solvable mathematical problems. By assigning variables, we can build and manipulate expressions, leading us to accurate solutions efficiently.

Problem-Solving Strategies

Problem-solving strategies are essential tools when working through mathematical challenges. Here, we break down the cupcake problem into several manageable parts, a common strategy in dealing with systems of equations. - **Define the Variable**: This is the first step where we assign what \( x \) will represent. In this case, the vanilla cupcakes.- **Set Up the Equation**: We use the relationship between the two types of cupcakes and set the equation \( x + 3x = 5000 \).- **Combine Like Terms and Simplify**: Combining terms simplifies our equation to \( 4x = 5000 \). This helps make calculations more straightforward.- **Solve the Equation**: Divide both sides to isolate \( x \), determining that 1,250 vanilla cupcakes are sold.Through these steps, we ensure we tackle every part of the problem methodically, ensuring full comprehension and an accurate solution. Such strategies are valuable in many fields beyond just mathematics, aiding in systematic and logical approaches towards problem solving.