Problem 75 - Elementary Algebra

Question

If the trip from Fresno to Sacramento can be made by car in \(21 / 2\) hours at an average speed of 67 miles per hour, then how far is Sacramento from Fresno?

Step-by-Step Solution

Answer

The distance from Fresno to Sacramento is 703.5 miles.

Step 1: Understand the Problem

We need to find the distance between Fresno and Sacramento. The problem gives us the time taken to travel and the average speed.

Step 2: Review the Formula for Distance

The formula to calculate distance is given by \( \text{Distance} = \text{Speed} \times \text{Time} \).

Step 3: Substitute the Given Values

We have the speed as 67 miles per hour and the time as \( \frac{21}{2} \) hours. Substitute these into the formula: \( \text{Distance} = 67 \times \frac{21}{2} \).

Step 4: Simplify the Expression

Calculate the product: \( \text{Distance} = 67 \times 10.5 = 703.5 \).

Step 5: Verify Units and Answer

The units of distance will be in miles as both speed and time were given in compatible units. Thus, the distance is 703.5 miles.

Key Concepts

Speed and Time Relationship

When we talk about speed, we're referring to how fast something is moving. Time, on the other hand, is how long it takes to get somewhere. Together, they help us figure out distances. Think of speed as a regular pace over time. For example, if you walk at the same pace, knowing your speed and the time helps you know how far you'll get.
Here's how they relate:

• Speed: The rate at which distance changes over time. It's usually in units like miles per hour (mph) or kilometers per hour (kph).
• Time: The duration for which the speed is maintained, measured in hours, minutes, etc.

By understanding both, you can determine distance using the formula: Distance = Speed x Time. This fundamental relationship guides many practical applications, from planning travel to understanding physics.

Distance Formula in Algebra

To solve problems like the one in our exercise, we use an algebraic formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). It's straightforward. You multiply speed by time, and you get distance. This is quite useful to know how far you have traveled or need to travel.
Here’s how it works:

• Identify Speed: Know the average speed during the journey (e.g., 67 mph).
• Determine Time: Find out how long you traveled (e.g., \( \frac{21}{2} \) hours).
• Multiplication: Using the formula, multiply them (67 × 10.5 = 703.5).

This method is highly reliable and accurate as long as the speed remains constant and the units are consistent.

Unit Conversion in Mathematics

When dealing with calculations related to distance, time, and speed, unit conversion becomes essential. In our example, we used speed in miles per hour and time in hours, which are compatible units for calculating distance in miles.
However, let's say you have:

• Speed in a different unit, like kilometers per hour (km/h).
• Time in minutes instead of hours.

In these cases, you need to convert them to compatible units:

• Converting Speed: If speed is in km/h but distance needs to be in miles, convert kilometers to miles (1 km = 0.6214 miles).
• Converting Time: To convert minutes to hours, divide by 60 since 1 hour = 60 minutes.

This keeps calculations correct and reduces chances of error. Always check your units before multiplying to ensure that your final answer makes sense.