Problem 56 - Elementary Algebra

Question

Mark and Janet carpooled home from college for the Thanksgiving holiday. They shared the driving, but Mark drove twice as far as Janet. If Janet drove 135 miles, then how many miles was the entire trip?

Step-by-Step Solution

Answer

The entire trip was 405 miles.

Step 1: Understand the problem

We need to find the total number of miles driven based on the fact that Mark drove twice the distance of Janet's and Janet drove 135 miles.

Step 2: Express Mark's distance

Since Mark drove twice as far as Janet, we represent the distance Mark drove as a function of Janet's distance, which is \(2 \times 135\).

Step 3: Calculate Mark's distance

Multiply 135 miles by 2 to find the distance Mark drove: \(2 \times 135 = 270\) miles.

Step 4: Find the total distance

Add the distance driven by Mark and Janet to find the total distance of the trip: \(135 + 270 = 405\) miles.

Key Concepts

Distance Problems

Distance problems are common in algebra and involve determining the distance traveled based on given conditions. Here, we have a scenario where two people, Mark and Janet, share the driving distance. Janet drives a known distance, while Mark drives twice that distance. To solve such problems:

• Identify the known quantities, such as Janet's distance being 135 miles.
• Determine relationships, like Mark driving twice as far as Janet.
• Use basic arithmetic to compute the unknowns, such as Mark's distance and then the total distance traveled.

These types of problems help improve analytical skills by requiring you to set up relationships and solve for unknowns. They also encourage logical thinking to apply given information effectively.

Step-by-Step Solutions

Providing a step-by-step solution is crucial for understanding and solving algebraic distance problems. Breaking down the problem into simpler steps helps prevent confusion and errors:

• Step 1: Understand the Problem
  Comprehend what the question is asking. Here, the focus is on finding the total distance based on Janet's drive and the relationship to Mark's drive.
• Step 2: Express Mark's Distance
  Translate the given fact that Mark drove twice as far into an equation or expression. This requires multiplying Janet's distance (135 miles) by 2.
• Step 3: Calculate Mark's Distance
  Do the arithmetic to find the exact distance Mark drove, which is 270 miles.
• Step 4: Find the Total Distance
  Add the distances driven by both to find the overall trip's mileage, totaling to 405 miles.

This methodical breakdown ensures each component of the problem is clear and simplifies complex calculations.

Math Problem Solving

Math problem solving, especially in distance-related problems, develops logical reasoning, comprehension, and practical application skills. To tackle these problems efficiently:

• Identify Variables and Equations
  Recognize what is known and unknown, then establish equations to express relationships. For example, knowing Janet's distance helps us find Mark's.
• Apply Arithmetic Operations
  Use addition, subtraction, multiplication, or division to manipulate numbers as needed by the situation, like calculating Mark's distance and total trip distance.
• Verify Your Solution
  Check your work by reviewing each step and ensuring every aspect of the problem is addressed. Confirmation, such as re-adding distances, ensures correctness.

These problems are not just about finding answers but understanding the process, improving critical thinking, and practicing real-world applications of math.