Problem 76 - Elementary Algebra

Question

A high-speed train averages 170 miles per hour. How far can it travel in \(11 / 2\) hours?

Step-by-Step Solution

Answer

The train travels 935 miles in 5.5 hours.

Step 1: Understand the Problem

We are asked to find the distance a train can travel. We know its average speed is 170 miles per hour, and it travels for \( \frac{11}{2} \) hours. Use the formula for distance: \( \text{Distance} = \text{Speed} \times \text{Time} \).

Step 2: Set Up the Distance Formula

Using the formula from the previous step, set it up with the given values:\[\text{Distance} = 170 \text{ miles/hour} \times \frac{11}{2} \text{ hours}\]

Step 3: Simplify the Time

Convert \( \frac{11}{2} \) into a decimal to make the calculations easier:\[\frac{11}{2} = 5.5\]This means the train travels for 5.5 hours.

Step 4: Calculate the Distance

Now, we calculate the distance:\[\text{Distance} = 170 \times 5.5 = 935 \text{ miles}\]This means the train travels 935 miles in 5.5 hours.

Key Concepts

Average Speed

Understanding average speed is crucial when tackling problems involving travel over different time intervals. Average speed is a measure of the total distance covered divided by the total time taken. It helps us understand how fast an object is moving on average across a journey where the speed might vary.
To determine average speed, the basic formula we use is:

• \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)

In our exercise, we're given the average speed as 170 miles per hour (mph), which means that the train covers 170 miles in every hour on average. This consistent speed simplifies problems involving straightforward travel intervals, making it easier to predict distances over longer time periods.
Average speed is especially useful in calculating travel durations and distances when you don't have variations in speed to consider.

Time Calculation

Mastering time calculation is key in solving many algebraic problems, especially those involving movement. Time calculation allows us to find out how long it takes for an event to occur or an object to reach a particular destination when traveling at a constant speed. It involves converting time into a usable format when performing mathematical operations.
In our example, the train travels for \( \frac{11}{2} \) hours, which can be converted into a decimal, 5.5 hours, to simplify our calculations. This conversion is achieved by dividing the numerator by the denominator (11 divided by 2).
When calculating time, breaking fractions into decimals can make it easier to multiply by the given speed without dealing with fractions, thus speeding up the computation process. Correct time calculation ensures the distance is measured as accurately and straightforwardly as possible.

Algebraic Problem Solving

Algebraic problem solving involves using mathematical concepts and formulas to find unknown values or to make predictions based on known ones. In the context of our exercise, we need to find the distance the train travels in a specific amount of time at its average speed.
To solve this, we utilize the formula \( \text{Distance} = \text{Speed} \times \text{Time} \). By substituting the given average speed (170 miles per hour) and time (5.5 hours), the calculation becomes straightforward:

• \( \text{Distance} = 170 \times 5.5 = 935 \text{ miles} \)

This approach of breaking down the problem into known formulas and manageable steps is a key technique in algebraic problem solving. It helps in systematically reaching the solution by identifying the relationship between different elements of the problem and applying appropriate operations.