Question

Use the following information. Jeff lives in a state in which speeders are fined \(\$ 20\) for each mile per hour (mi/h) over the speed limit. Jeff was given a ticket for \(\$ 260\) for speeding on a road where the speed limit is 45 miles per hour. Jeff wants to know how fast he was driving. Use mental math to solve the equation. What does the solution represent?

Short answer

Jeff was driving at 58 miles per hour.

Step-by-step solution

Understand the problem

Read the problem carefully. Each mile per hour over the speed limit is fined $20. Jeff was fined $260 for speeding. The speed limit where Jeff was driving was 45 mi/h. The task is to find out how fast Jeff was driving.

Set up the equation

Let's denote Jeff's speed as 's'. Each mile per hour over the speed limit is fine $20. So the amount of fine Jeff paid, $260, equals to $20 multiplied by how much his speed 's' exceeded the speed limit. The speed limit was 45 mi/h. So we have the following equation: \(20(s - 45) = 260\)

Solve the equation

To solve for 's', we first divide both sides of the equation by 20: \(s - 45 = 260 \, / \, 20\), which gives: \(s - 45 = 13\). Adding 45 to both sides of this equation gives: \(s = 13 + 45\), thus \(s = 58\)

Interpret the solution

The solution indicates the speed that Jeff was driving when he was given the ticket. In this case, Jeff was driving at 58 mi/h.

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They're fundamental in representing real-world situations in a form that we can manipulate to find solutions to various problems.

For instance, in the problem about Jeff's speeding fine, we convert the given situation into an algebraic expression. Here, Jeff's unknown speed is represented by 's', and the expression involves a simple linear equation, based on the relation between his speed, the speed limit, and the fine:
\(20(s - 45) = 260\).

Breaking down this expression, the '20' represents the cost per mile for speeding, \(s - 45\) symbolizes the unknown number of miles over the speed limit Jeff was driving, and '260' is the total fine he paid. Understanding how to build and interpret these expressions is crucial for converting real-life situations into algebraic problems that we can solve.

Mental Math

Mental math is the skill of performing calculations in your head without the use of external tools. This skill is particularly useful because it not only speeds up the calculation process but also enhances your numerical and problem-solving abilities.

When we approach Jeff's problem using mental math, we simplify the algebraic expression by dividing 260 by 20 to get the excess number of miles per hour he was driving over the limit. Mentally computing \(260 \/ 20\) gives us 13. Then, adding this to the speed limit of 45 mi/h, we quickly determine that Jeff was driving at 58 mi/h.

Practicing mental math can improve your algebraic intuition and make it easier to approach similar problems with confidence. Remember that some straightforward operations like the division and addition in Jeff's problem are ideal candidates for mental math computation.

Real-world Problem Solving

Real-world problem-solving involves using mathematics to tackle everyday challenges. This process requires a comprehensive understanding of how mathematical concepts apply to actual situations.

In the context of our example, Jeff's dilemma is a real-world problem that requires an understanding of algebra to resolve. By creating an equation that relates the fine he received to his speed over the speed limit, we are utilizing our mathematical knowledge to draw conclusions about his actual driving speed.

Moreover, real-world problems often require interpretation of the solution within the context of the situation. The solution \(s = 58\) mi/h not only tells us Jeff's speed but also has implications such as potential points on his driving record or increased insurance premiums, demonstrating how mathematical solutions translate into tangible outcomes.