Problem 7 - Algebra 1: Concepts and Skills

Question

You are one mile from your home. You can walk at a speed of 4 miles per hour. The subway comes by every 15 minutes, and you heard one come by 3 minutes ago. The subway ride takes 8 minutes. Assign labels to your verbal. Use \(t\) to represent the unknown value.

Step-by-Step Solution

Answer

The solution involves calculating and comparing two different times: the time for walking home and the time for waiting and getting home by subway. The values are obtained by applying the relevant formulas and then converted to the same units before comparing.

Step 1: Calculate Walking Time

Firstly, calculate the time it would take to walk home. The formula to calculate time is distance divided by speed. In this case, the distance is one mile and the walking speed is 4 miles per hour. Thus, the walking time, denoted as \( t_{walk} \), can be calculated as follows: \( t_{walk} = \frac{1}{4} \) hours.

Step 2: Calculate Subway Time

Secondly, calculate the time it would take to wait for the subway and then ride it home. The next subway comes in every 15 minutes, but since one has already gone by 3 minutes ago, the waiting time for the next subway, denoted as \( t_{wait} \), is \( 15 - 3 \) minutes. After that, the subway ride home takes eight minutes, denoted as \( t_{ride} \). Therefore, the total time to get home by subway, denoted as \( t_{sub} \), can be calculated as follows: \( t_{sub} = t_{wait} + t_{ride} = (15 - 3 + 8) \) minutes.

Step 3: Compare Times

Lastly, compare the two times (walking and subway) to find out which is faster. To do this, the times must be in the same units. Convert the walking time from hours to minutes by multiplying by 60. Compare \( t_{walk} \times 60 \) and \( t_{sub} \).

Key Concepts

Calculating Time with Algebra

When faced with an everyday scenario, such as determining the fastest way to get home, algebra is a practical tool that helps solve such problems efficiently. In our example, using algebra to calculate time involves recognizing that time is typically the unknown variable one needs to find – often represented as \( t \).

Algebraic formulas, like the basic time formula \( t = \frac{d}{s} \) where \( d \) represents distance and \( s \) represents speed, come in handy. Calculating time using this formula requires having consistent units for distance and speed. Identifying the variables accurately, assigning the correct values to these variables, and manipulating the formula are crucial steps in problem-solving with algebra.

For example, when calculating the time it takes to walk a certain distance at a given speed, students must ensure that both the distance and speed are expressed in compatible units before performing the division. Understanding that time cannot be directly observed but rather deduced from known values is key in mastering the art of calculating time with algebra.

Distance-Speed-Time Relationship

The distance-speed-time relationship is a fundamental concept in physics, often expressed as \( d = s \cdot t \), which reveals the direct proportion between distance traveled and time, given a constant speed. This relationship is symmetric; knowing any two of the quantities allows you to solve for the third.

For instance, if you know the speed at which you can walk and the distance home, you can determine how long it will take you to get there. Conversely, if you know the distance and the time, you can calculate the speed necessary to arrive within that time frame.

• If the speed increases, the time to cover the same distance decreases.
• If the speed decreases, the time increases.
• If the distance to be covered becomes greater, the time will also increase assuming the speed stays consistent.

Understanding this intrinsic relationship allows students to solve problems involving travel and motion methodically, such as determining whether it is faster to walk or take a subway based on given speeds and distances.

Converting Units in Algebra

A key aspect of solving algebraic problems involving real-world scenarios—like travel time—is the ability to convert between different units. Conversions ensure that all terms in an equation are expressed in the same type of unit, which is essential for accurate calculation. The most common conversion in time-related problems is between hours and minutes.

To convert from hours to minutes in our example, we multiply the time in hours by 60 since there are 60 minutes in one hour. Simultaneously, if you have time in minutes that you wish to convert back into hours, you would divide by 60. This step is vital when comparing quantities like \( t_{walk} \) and \( t_{sub} \) from our exercise, which are initially in different units.

Being proficient in unit conversion allows students to handle diverse problems that not only include time but also distance, speed, and other dimensions, ensuring mathematical consistency and accuracy across their calculations.