Problem 3 - Algebra for College Students

Question

Tri traveled from Chicago to Des Moines, a distance of \(300 \mathrm{mi}\), in \(10 \mathrm{hr}\). What was his rate in miles per hour?

Step-by-Step Solution

Answer

Tri's rate was 30 miles per hour.

Step 1: Identify the given information

The problem provides the distance between Chicago and Des Moines as 300 miles and the travel time as 10 hours.

Step 2: Use the formula for rate

The formula to find the rate (speed) is: \[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} \]

Step 3: Substitute the given values

Substitute the distance (300 miles) and the time (10 hours) into the formula: \[ \text{Rate} = \frac{300 \text{ miles}}{10 \text{ hours}} \]

Step 4: Calculate the rate

Divide 300 miles by 10 hours: \[ \text{Rate} = 30 \text{ miles per hour} \]

Key Concepts

distance formula

In this section, we'll cover the 'distance formula,' which is fundamental in many kinematic problems. The distance formula relates three key variables: distance, speed (or rate), and time. They are connected through the equation:
\ \{ d = r \times t \ \}
where:

• d = distance
• r = rate or speed
• t = time

To understand this better, think of a car traveling on a highway. If you know how fast the car is going (rate) and how long it has been driving (time), you can determine the distance covered.
For instance, in the problem with Tri traveling from Chicago to Des Moines, the distance of 300 miles is given along with the time taken, 10 hours. To find out the speed, we rearrange the basic distance formula to solve for rate (r). This becomes: \[ r = \frac{d}{t} \]This rearranged formula is a critical tool in solving many problems related to travel and motion. Being familiar with the distance formula and knowing how to manipulate it for different variables will be very helpful.

speed calculation

Next, let's delve into 'speed calculation' using the information from our exercise. Speed, often called rate, is how fast an object is moving. The standard formula for speed is: \[ \text{Speed (rate)} = \frac{\text{Distance}}{\text{Time}} \] Speed is usually measured in units such as miles per hour (mph), kilometers per hour (kph), or meters per second (m/s), depending on the context.

Let's look at Tri's journey. You are given that the distance from Chicago to Des Moines is 300 miles and it took 10 hours to travel this distance. Using the formula provided:
\[ \text{rate} = \frac{300 \text{ miles}}{10 \text{ hours}} \]
When you divide 300 by 10, the result is 30 miles per hour. This means that, on average, Tri traveled at a speed of 30 miles per hour for the entire trip.

Understanding how to calculate speed can help in everyday situations, especially when estimating travel time and planning trips efficiently.

basic algebra

The third core concept is 'basic algebra,' which is essential for solving equations like the one in our exercise. Algebra allows you to manipulate mathematical expressions and solve for unknown variables.

In this example, solving for rate required basic algebraic steps:

• Identifying and substituting given values
• Rearranging formulas to solve for the desired variable
• Performing arithmetic operations like division

For the exercise, once we identified the distance (d = 300 miles) and time (t = 10 hours), we substituted these values into the rate formula: \[ r = \frac{d}{t} \]
Thus, we had: \[ r = \frac{300 \text{ miles}}{10 \text{ hours}} \]
Dividing 300 by 10, we find the rate (r) to be 30 miles per hour. This straightforward use of algebra lets us solve many similar real-world problems. Therefore, becoming proficient in basic algebra can make it easier to tackle various mathematical challenges, from everyday scenarios to more complex mathematical tasks.