Problem 80 - Elementary Algebra

Question

Set up an algebraic equation and then solve. Two brothers leave from the same location, one in a car and the other on a bicycle, to meet up at their grandmother's house for dinner. If one brother averages 30 miles per hour in the car and the other averages 12 miles per hour on the bicycle, then it takes the brother on the bicycle 1 hour less than 3 times as long as the other in the car. How long does it take each of them to make the trip?

Step-by-Step Solution

Answer

The car takes 2 hours, and the bicycle takes 5 hours.

Step 1: Define Variables

Let \( t \) represent the time in hours it takes the brother driving the car to reach their grandmother's house.

Step 2: Express Bicycle Time

According to the problem, it takes the brother on the bicycle 1 hour less than 3 times as long as the brother in the car. Thus, the time for the bicyclist is represented as \( 3t - 1 \) hours.

Step 3: Setup Distance Equation

Both brothers travel the same distance. The distance for the car is expressed by \( 30t \) because the speed is 30 mph. Similarly, the distance for the bicycle is \( 12(3t - 1) \) because the speed is 12 mph. Thus, the equation is: \[ 30t = 12(3t - 1) \]

Step 4: Expand Bicycle Expression

Expand the expression for the bicyclist's distance: \[ 12(3t - 1) = 36t - 12 \].

Step 5: Solve the Equation

Substitute the expanded expression into the distance equation and solve for \( t \):\[ 30t = 36t - 12 \]Subtract \( 36t \) from both sides:\[ 30t - 36t = -12 \]\[ -6t = -12 \]Divide by \(-6\):\[ t = 2 \]

Step 6: Calculate Bicycle Time

Substitute \( t = 2 \) back into the expression for the bicyclist's time:\[ 3t - 1 = 3(2) - 1 = 6 - 1 = 5 \]

Step 7: Conclusion Step: Verify Solution

The brother in the car takes 2 hours, and the brother on the bicycle takes 5 hours. Calculating the distances: Car: \( 30 \times 2 = 60 \) miles, Bicycle: \( 12 \times 5 = 60 \) miles. Both are equal, confirming the solution is correct.

Key Concepts

Variable Definition

One of the first steps in solving algebraic equations is to define the variables involved. In this exercise, we began by letting \( t \) represent the time in hours that it takes the brother driving a car to reach their grandmother's house. By defining \( t \), it allows us to translate the given word problem into a mathematical form we can work with.
Another important variable in this exercise is the bicycle time, which is expressed as \( 3t - 1 \). This form arises from the information that the bicyclist's time is 1 hour less than three times the car's time. Defining clear variables at the start helps prevent confusion as you proceed to set up equations and solve the problem.

Distance Calculation

Calculating distance is a key part of solving this problem. We have two brothers, each traveling at a different uniform speed, yet covering the same distance.
For the brother in the car, the distance can be calculated using the formula: \( \text{distance} = \text{speed} \times \text{time} \). Since his speed is 30 mph and the time is \( t \), the distance he travels is \( 30t \).
Similarly, the brother on the bicycle travels at 12 mph. His time was defined as \( 3t - 1 \), leading to his distance being represented as \( 12(3t - 1) \). Both calculations are framed to illustrate that despite different speeds and times, the brothers travel the same distance to reach their destination.

Solving Equations

With our expressions for distance setup, we can now solve the equation. Since both brothers travel the same distance, we equate the two expressions:

• For the car: \( 30t \)
• For the bicycle: \( 12(3t - 1) \)

The equation is: \( 30t = 12(3t - 1) \).
Expanding the right-hand side gives us \( 36t - 12 \). Now, substitute this into the equation: \( 30t = 36t - 12 \).
By simplifying further, we move all terms to one side: \( 30t - 36t = -12 \) which simplifies to \( -6t = -12 \).
Solving for \( t \) by dividing both sides by \( -6 \) gives \( t = 2 \). Thus, the car journey takes 2 hours.

Speed and Time Relationship

The relationship between speed, time, and distance is fundamental to solving this problem. This relationship is encapsulated by the formula: \( \text{distance} = \text{speed} \times \text{time} \).
To better understand this, let's see how varying each element affects the others:

• If you increase speed while keeping distance constant, the time required decreases.
• If distance increases with speed constant, the required time increases.
• Keeping time constant but increasing speed leads to covering more distance.

In our problem, understanding this relationship helps clarify how the bicyclist's slower speed results in a longer travel time compared to his brother in the car. It also assists in verifying that both brothers covered the same distance even when their mode of transport and speeds were different.