Problem 42 - Algebra A Combined Function

Question

The number of registered vehicles on the road in the United States is constantly increasing. In \(2007,\) there were approximately 246 million registered vehicles. This represents a \(3 \%\) increase over \(2002 .\) How many registered vehicles were there in the United States in \(2002 ?\) Round to the nearest million. (Source: Federal Highway Administration)

Step-by-Step Solution

Answer

There were approximately 239 million registered vehicles in the United States in 2002.

Step 1: Understand the Problem

We are given that the number of registered vehicles in 2007 was approximately 246 million. This number represents a 3% increase over the number of registered vehicles in 2002. We need to find the number of registered vehicles in 2002.

Step 2: Setup the Equation

Let the number of registered vehicles in 2002 be \(x\). According to the problem, the number of vehicles in 2007 is 3% more than in 2002. This gives us the equation: \[ 1.03x = 246 \] million.

Step 3: Solve the Equation for x

We need to solve the equation \(1.03x = 246\). Divide both sides by 1.03 to isolate \(x\): \[ x = \frac{246}{1.03} \]

Step 4: Calculate the Value

Perform the division \(\frac{246}{1.03}\) to find \(x\): \[ x \approx 238.834 \] million.

Step 5: Round the Result

Round 238.834 to the nearest million to get the number of registered vehicles in 2002. The rounded value is 239 million.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are powerful tools for solving mathematical problems where you need to find unknown values. In this exercise, we looked into the world of percentage increase problems. We had an unknown number of vehicles in 2002, represented by \( x \).
To build an equation from the problem, we relied on the information provided that the number of vehicles increased by 3% from 2002 to 2007.
Thus, the equation \( 1.03x = 246 \) million was used. This equation encapsulates how the value in 2007 (246 million vehicles) is the result of a 3% increase applied to the value in 2002.
The equation allows you to solve for \( x \), revealing the number of vehicles initially on the road before the increase. Different problem types can demand setting up different equations, but understanding the relationship between your variables is key.

• Inside the equation, 1.03 stands for 100% (the initial amount) plus the 3% increase.
• Our task becomes solving for \( x \), the initial number before the increase, by isolating it in the equation.

Enhancing Problem-Solving Skills

Problem-solving is a crucial skill, especially when dealing with mathematical exercises such as percentage increase problems. Here are some strategies to enhance these skills:
First, thoroughly understand the problem details. What is being asked? What information is given? In our example, understanding the initial value and the percentage increase was critical.
Secondly, break the problem into manageable parts. This can often be seen as setting up an equation or using logical reasoning to work backward from the information provided. Our exercise required setting up an algebraic equation, which involves breaking down how the problem components relate to each other.
Finally, solving the equation and verifying your results is key. After determining the value of \( x \), confirming the calculation is done correctly ensures the solution is correct. Often, verifying involves checking if your result makes logical sense given the context.

• Detailed understanding prevents errors and improves accuracy.
• Breaking down problems helps in systematically approaching and solving complexities.
• Verification of results adds an extra layer of confidence in your answer.

Rounding Numbers with Precision

Rounding numbers to the right degree of precision is essential in real-world applications. In this exercise, we had to round the calculated value of vehicles to the nearest million. Rounding helps simplify numbers and can improve the clarity and usability of data.
Our computed value for \( x \) was approximately 238.834 million vehicles. By rounding this number, we aimed to give a value that is easier to communicate—239 million vehicles.
Here's how to round numbers accurately:

• Identify which digit is crucial for rounding. In our context, it was the number in the millions place.
• Look at the digit immediately to the right. If this digit is 5 or more, round up the target digit. If it's less than 5, keep the target digit the same.
• For our number, the tenths digit is 8, hence, we rounded 238.834 up to 239 million.

Rounding thus plays a role in reporting figures succinctly and correctly, enhancing the interpretation of results across different contexts.