Problem 132 - College Algebra with Modeling and Visualization

Question

The total number of cigarettes sold overseas in 1990 and 2000 was 10.9 trillion cigarettes. The number sold in 2000 was 100 billion more than the number sold in \(1990 .\) How many cigarettes were sold each year?

Step-by-Step Solution

Answer

In 1990, 5.4 trillion cigarettes were sold, and in 2000, 5.5 trillion cigarettes were sold.

Step 1: Define Variables

Let's define two variables to represent the number of cigarettes sold in each year. Let \( x \) be the number of cigarettes sold in 1990, and \( y \) be the number of cigarettes sold in 2000.

Step 2: Set Up Equations

We have two pieces of information that allow us to set up two equations. First, from the problem, we know that the total number of cigarettes sold in both years is 10.9 trillion: \[ x + y = 10.9 \text{ trillion} \] Second, the problem states that the number sold in 2000 is 100 billion more than the number sold in 1990, which gives:\[ y = x + 0.1 \text{ trillion} \]

Step 3: Substitute Second Equation into First Equation

Substitute the expression for \( y \) from the second equation back into the first equation:\[ x + (x + 0.1) = 10.9 \]

Step 4: Solve the Simplified Equation

Combine like terms in the equation to simplify:\[ 2x + 0.1 = 10.9 \] Now, solve for \( x \) by subtracting 0.1 from both sides:\[ 2x = 10.8 \]Then, divide by 2:\[ x = 5.4 \text{ trillion} \] This represents the number of cigarettes sold in 1990.

Step 5: Use Original Equation to Find y

Now that we have \( x = 5.4 \text{ trillion} \), substitute this value into the equation \( y = x + 0.1 \):\[ y = 5.4 + 0.1 = 5.5 \text{ trillion} \] This represents the number of cigarettes sold in 2000.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and are equations involving variables raised only to the first power. In the context of this exercise, we used two linear equations to solve the problem of finding the number of cigarettes sold in two different years: 1990 and 2000. A linear equation can be written in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.
In our specific problem, we created two equations based on the information provided:

• The total sales in 1990 and 2000 added up to 10.9 trillion: \( x + y = 10.9 \text{ trillion} \).
• The sales in 2000 were 0.1 trillion (100 billion) more than in 1990: \( y = x + 0.1 \text{ trillion} \).

By understanding how to construct and manipulate these equations, we can find solutions for unknown variables by applying methods like substitution or combination.

Problem-solving Strategies

Problem-solving in algebra requires a strategic approach to identify and solve problems using a series of logical steps. When faced with a word problem, start by breaking down the problem's components. Here's how you can approach such problems:
1. **Understand the Problem:** Read the problem carefully to determine what is being asked. Identify key details, such as numbers, keywords, and relations among them.

• In our case, we noticed that the total sales and the difference in sales between the years are crucial details.

2. **Define Variables:** Assign variables to unknown quantities, as done with \( x \) for cigarettes in 1990 and \( y \) for 2000. This helps transform words into math expressions.3. **Set Up Equations:** Use the relationships defined in the problem to write one or more equations.4. **Solve the Equations:** Apply algebraic techniques, like substitution, as illustrated in the solution by entering \( y = x + 0.1 \) into the equation \( x + y = 10.9 \).
Following these strategies allows you to systematically approach and solve word problems in algebra.

Variables and Equations

Variables are symbols used to represent unknown values in equations and play a central role in algebraic problem-solving. In equations, variables can be manipulated to find solutions, giving us the ability to translate real-world problems into mathematical terms.
In this particular example, the variables \( x \) and \( y \) represent the unknown quantities: the number of cigarettes sold in 1990 and 2000, respectively.

• We set up two equations to describe their relationships.
• Using these equations, we were able to solve for the variables\( x \) and \( y \) by substitution method.

Equations are tools that include variables. They express relationships between quantities and allow us to solve for unknowns. By substituting known values and solving for the unknowns, you derive meaningful results, such as determining the sales figures in the years specified in the problem.
By mastering the concept of variables and how they function within equations, you're able to tackle various math problems more effectively.