Problem 56 - Algebra for College Students

Question

A car leaves a town at 60 kilometers per hour. How long will it take a second car, traveling at 75 kilometers per hour, to catch the first car if it leaves 1 hour later?

Step-by-Step Solution

Answer

The second car takes 4 hours to catch up with the first car.

Step 1: Determine Distance Covered by the First Car

First, calculate the distance that the first car travels in one hour, since it has an hour's head start. The speed of the first car is 60 km/h, so in one hour, the distance covered is calculated by the formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). Therefore, the first car travels \( 60 \times 1 = 60 \) kilometers in that hour.

Step 2: Set Up Equation for the Catch-Up Scenario

Let the time taken by the second car to catch up be \( t \) hours. In that time, the second car will travel \( 75t \) kilometers, while the first car will travel a total of \( 60 + 60t \) kilometers (the initial 60 kilometers plus 60 km for each hour).Since the second car catches up with the first car, the distance covered by both cars will be the same: \( 75t = 60 + 60t \).

Step 3: Solve the Equation for Time \( t \)

To find \( t \), solve the equation from Step 2:\[ 75t = 60 + 60t \]Subtract \( 60t \) from both sides to isolate \( t \):\[ 15t = 60 \]Divide both sides by 15:\[ t = \frac{60}{15} = 4 \]The second car takes 4 hours to catch up with the first car.

Step 4: Verify Solution

To verify, calculate the distance traveled by both cars in the catch-up time:- The second car travels \( 75 \times 4 = 300 \) kilometers.- The first car travels \( 60 + 60 \times 4 = 300 \) kilometers.Since both distances match, the calculated time is correct.

Key Concepts

Distance Rate Time Problems

Distance, rate, and time are three fundamental aspects in many real-world problems. Understanding how these three variables relate to each other can help solve various puzzles, like the one with the cars given. The basic equation connecting these quantities is:

• Distance = Rate × Time

This equation implies that if you know any two of the variables, you can find the third. In our problem, the first car has a head start, traveling at a constant speed of 60 km/h. This gives it a certain distance advantage. In contrast, the second car travels faster at a speed of 75 km/h, starting later. The challenge is to determine when the faster car will cover enough ground and time to catch the head start of the slower car. Understanding how to manipulate this important relationship between distance, rate, and time is key to solving such problems smoothly.

Linear Equations

Linear equations are vital tools in solving problems involving comparisons of different rates and distances, as showcased in the catching up scenario with the two cars. The equation in the example is set up to represent equal total distances traveled by each car at the time the faster car catches up with the slower car. We formulate the equation as follows:

• First car's total distance: 60 + 60t
• Second car's total distance: 75t

Since these distances must be equal at the catch-up point, the equation is:\(75t = 60 + 60t\)This is a linear equation because it results in a straight line when plotted on a graph. Solving involves simplifying it to find the value of \( t \) when both cars have traveled the same total distance.

Problem Solving Steps

Breaking down complex problems into smaller, manageable steps makes it easier to think clearly and arrive at the correct solutions. Let's revisit the steps used in our example:1. **Understand the problem**: Identify what is given, such as the speeds, and what we need to find, which is the time for the second car to catch up.2. **Set up the equations**: Use the known relationships (such as Distance = Rate × Time) to express what you're trying to find in terms you know.3. **Solve for the variable**: Rearrange the equation and solve for the unknown variable, like we did when finding the time \( t \).4. **Verify the solution**: Always check your solution by substituting your found value back into the scenario to ensure consistency and correctness.By systematically tackling problems using these steps, you can increase your chances of solving them correctly and efficiently.