Problem 57 - Algebra for College Students

Question

Bret started on a 70 -mile bicycle ride at 20 miles per hour. After a time he became a little tired and slowed down to 12 miles per hour for the rest of the trip. The entire trip of 70 miles took \(4 \frac{1}{2}\) hours. How far had Bret ridden when he reduced his speed to 12 miles per hour?

Step-by-Step Solution

Answer

Bret rode 40 miles before reducing his speed.

Step 1: Understand the Total Journey

The total distance Bret has to cover is 70 miles, and the total time taken for this trip is given as \(4 \frac{1}{2}\) hours, which converts to \(4.5\) hours.

Step 2: Set up Equations for Each Segment

Let the distance traveled at 20 mph be \(d\) miles. Then, the time taken for this part of the journey is \(\frac{d}{20}\) hours. This means the remaining distance is \(70 - d\) miles, which he travels at 12 mph. The time for the second segment is \(\frac{70 - d}{12}\).

Step 3: Formulate the Equation for Time

Since the total time for the trip is 4.5 hours, we set up the equation: \[\frac{d}{20} + \frac{70 - d}{12} = 4.5\]

Step 4: Solve the Equation

To solve the equation, we first find a common denominator for the fractions, which is 60. Multiply the entire equation by 60 to eliminate the fractions:\[3d + 5(70 - d) = 270\] which simplifies tothe equation: \[3d + 350 - 5d = 270\]simplifying further to: \[-2d + 350 = 270\] Then solve for \(d\) to find: \[-2d = -80\] \[d = 40\]

Step 5: Interpret the Result

Bret had traveled 40 miles at the speed of 20 miles per hour before he reduced his speed to 12 miles per hour.

Key Concepts

Distance-Speed-Time Relation

The concept of distance-speed-time relation is fundamental in mathematics and particularly useful in solving word problems like the one involving Bret's bicycle trip. This relation is encapsulated in the simple formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). Understanding this equation helps in analyzing the components of any journey.
In Bret's case, the journey is divided into two parts: one at 20 miles per hour and the other at 12 miles per hour. Suppose Bret covers a distance \( d \) at 20 mph. The time taken for this section can be calculated as \( \frac{d}{20} \). For the remaining distance, \( 70 - d \), he traveled slower at 12 mph, taking \( \frac{70 - d}{12} \) hours.
By relating these sections with the total trip time of 4.5 hours, you get a well-structured equation. This equation represents the allocation of distances and speeds in parts of Bret's journey so that you can logically deduce his point of speed change.

Solving Equations

Solving equations is a critical skill in algebra, acting like a detective's tool to uncover unknowns. In Bret's problem, we set up an equation based on the information given:
\[ \frac{d}{20} + \frac{70 - d}{12} = 4.5 \]
This equation consists of two parts, each representing the time taken for parts of Bret's journey. The trick is to find a common denominator for simplifying fractions, which in this case is 60. By multiplying the whole equation by 60, it eliminates the fractions, converting it into:
\[ 3d + 5(70 - d) = 270 \]
Continuing with simplification and algebraic operations leads to unraveling the mystery. Each step taken from distributing multiplication to combining like terms brings us closer to discovering \(d\), the distance Bret rode before slowing down. Luckily, solving repetitive operations becomes second nature with practice.

Mathematical Problem Solving

Problem-solving in mathematics is about clever thinking and strategy. It's both an art and a science that lets you translate real-world situations into manageable mathematical expressions.
In this problem, assessing the different speeds and distances requires both careful reading and logical setup. Crafting a suitable equation is the starting point, but solving it demands persistence and methodical work.
Here are some critical steps in mathematical problem solving:

• Break the problem into parts, understanding each segment independently.
• Identify knowns and unknowns; establish relations between them.
• Translate word problems into equations using mathematical logic.

Each effort refines thought processes, trains the mind to articulate relations clearly, and improves confidence in tackling new challenges.