Question

Cyclists Two cyclists leave a city at the same time, one going east and the other going west. The westbound cyclist bikes 5 mph faster than the eastbound cyclist. After 6 hours they are 246 miles apart. How fast is each cyclist riding?

Short answer

The eastbound cyclist rides at 18 mph and the westbound cyclist rides at 23 mph.

Step-by-step solution

Define Variables

Let the speed of the eastbound cyclist be denoted as \( x \) mph. Then, the speed of the westbound cyclist is \( x + 5 \) mph.

Establish the Distance Formula

Distance traveled is equal to speed multiplied by time. Therefore, the distance traveled by the eastbound cyclist in 6 hours is \( 6x \) miles and the distance traveled by the westbound cyclist in 6 hours is \( 6(x + 5) \) miles.

Set Up the Equation

According to the problem, the total distance between the two cyclists after 6 hours is 246 miles. Hence, the equation will be \( 6x + 6(x + 5) = 246 \).

Simplify the Equation

Combine like terms in the equation: \( 6x + 6x + 30 = 246 \). This simplifies to \( 12x + 30 = 246 \).

Solve for \( x \)

Subtract 30 from both sides to get \( 12x = 216 \). Then, divide both sides by 12 to find \( x = 18 \).

Determine Each Cyclist's Speed

The speed of the eastbound cyclist is \( 18 \) mph, and the speed of the westbound cyclist is \( x + 5 = 18 + 5 = 23 \) mph.

Key concepts

Distance-Speed-Time Relationship

Understanding the relationship between distance, speed, and time is vital in solving relative motion problems like the one with the two cyclists. The basic formula to remember is:

• Distance = Speed × Time

In this problem, we have two cyclists and need to determine their speeds based on the information given about their travel times and distances. By knowing that the distance each cyclist travels can be determined by multiplying their speed by the time they traveled, we can set up equations to find unknown variables like speed.

Equation Solving

After setting up the relevant formulae, the next step involves solving equations to find the unknown variable. Here, we have the total distance between the two cyclists and their individual speeds as variables. We set up the equation:

distance traveled by eastbound cyclist + distance traveled by westbound cyclist = total distance

to solve for the speed of each cyclist. In the problem, this equation is represented as:

\(6x + 6(x + 5) = 246\).

By solving this equation step-by-step, we deduce each cyclist's speed accurately.

Algebraic Simplification

When solving for the variables, algebraic simplification plays a crucial role in making the equations more manageable. Here's a look at how we simplify:

Starting with the equation: \[ 6x + 6(x + 5) = 246 \]
We first distribute the 6: \[ 6x + 6x + 30 = 246 \]
Then, we combine like terms: \[ 12x + 30 = 246 \]
Next, we subtract 30 from both sides of the equation to isolate the term with the variable: \[ 12x = 216 \]
Finally, we divide both sides by 12 to solve for \( x \): \[ x = 18 \]

This process of simplification makes the complex equation easier to solve, revealing the speed of the eastbound cyclist.

Word Problems in Algebra

Word problems in algebra, like the two cyclists problem, require translating a real-life scenario into mathematical expressions. Begin by defining variables to represent unknown quantities. In this example, \( x \) is the speed of the eastbound cyclist.

Next, use known relationships to set up equations. Here, it's the distance, speed, and time relationship:

• Distance traveled by eastbound: \( 6x \)
• Distance traveled by westbound: \( 6(x + 5) \)

Add these distances to equal the given total distance (246 miles): \( 6x + 6(x + 5) = 246 \)
Lastly, proceed with algebraic simplifications and solving the equation. This structured approach helps tackle complex word problems step by step, aligning mathematical solutions with real-world contexts.