Question

Match the problem with the formula needed to solve the problem. Then use the Guess, Check, and Revise strategy or another problem-solving strategy to solve the problem. Area of a rectangle \(\quad A=l w \quad\) Distance \(\quad d=r t\) Simple interest \(\quad I=P r t \quad\) Volume of a cube Temperature \(\quad C=\frac{5}{9}(F-32)\) Surface area of a cube \(S=6 s^{2}\) A car travels 60 miles per hour for a distance of 300 miles. How long did the trip take?

Short answer

The trip took 5 hours.

Step-by-step solution

Identify the formula

According to the problem, a car travels 60 miles per hour for a distance of 300 miles. The formula related to speed, distance, and time is \(d = rt\), where 'd' is distance, 'r' is rate (speed), and 't' is time.

Arrange the formula for desired variable

We have to calculate the time taken for the journey, so rearrange the formula to solve for 't' by dividing each side by 'r'. The rearranged formula will be \(t = \frac{d}{r}\).

Substitute values into formula

Substitute the given values into the rearranged formula. The rate 'r' is given as 60 miles per hour and the distance 'd' is given as 300 miles, so the equation becomes \(t = \frac{300}{60}\).

Solve the equation

Upon solving the equation \(t = \frac{300}{60}\), we get \(t = 5\). This means that the trip took 5 hours.

Key concepts

Algebra Problem Solving

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to the study of abstractions such as groups, rings, and fields.

When approaching algebra problem solving, it's crucial to first understand the question. In this case, we're dealing with a practical application of algebra: determining how long a car took to travel a specific distance. This requires identifying the correct formula that relates distance, rate, and time.

Before jumping into calculations, it's important to logically assess what is being asked and ensure that all variables required for computation are available. In our textbook example, the problem was simplified by matching it with the correct formula, which is fundamental in algebra. After identifying the relevant equation, the next step involves manipulating it, if necessary, to make the unknown the subject of the formula. In doing so, students practice key algebraic skills such as isolating variables and understanding the structural relationship between them.

Formula Manipulation

The process of formula manipulation involves rearranging and solving equations to find an unknown variable. It is a cornerstone of problem solving in various fields of science and mathematics. Mastery of this skill is essential for students, as it allows them to adapt formulas to solve for different variables depending on the problem at hand.

In the context of our example, the original distance formula used was \(d = rt\). This general formula relates distance, rate, and time. To find the time ('t'), the formula must be manipulated so that 't' is on one side of the equation by itself. The manipulation in this instance involves dividing both sides by the rate 'r', which yields \(t = \frac{d}{r}\).

It is imperative for students to get comfortable with such formula manipulations. They enable a fluid transition from a general formula to one that is specific to the given values. This is a fundamental skill in algebra that aids problem-solving efficiency and effectiveness.

Mathematical Equations

Mathematical equations are statements that assert the equality of two expressions. They are comprised of numbers, variables, and arithmetic operations. Equations are essential tools for modeling real-world situations, allowing us to render complex phenomena into manageable mathematical problems.

In the exercise involving the car's travel time, we applied the distance-rate-time equation. This is a linear equation – one of the simplest forms of mathematical equations. By substituting the known values into the equation \(t = \frac{300}{60}\), we transform it from an abstract representation into a specific, solvable algebraic statement. By engaging with equations in a variety of contexts, students develop the ability to translate real-world scenarios into mathematical language and solve them using algebraic techniques. Understanding the underlying principles of equations and their solutions is critical for students to progress in algebra and beyond.