Problem 42 - Elementary Algebra Within Reach

Question

Time A jogger leaves a point on a fitness trail running at a rate of 4 miles per hour. Ten minutes later, a second jogger leaves from the same location running at 5 miles per hour. How long will it take the second jogger to overtake the first? How far will each have run at that point?

Step-by-Step Solution

Answer

It will take \(\frac{2}{3}\) hours (or 40 minutes) for the second jogger to overtake the first jogger. Both will have run \(\frac{10}{3}\) miles (or approximately 3.33 miles) at that point.

Step 1: Determine the Time and Distance the First Jogger has run

The first jogger runs for 10 minutes (which is \(\frac{1}{6}\) hours) before the second jogger starts. So, the distance covered by first jogger before second jogger starts is Speed (S1) x Time (T1) = 4 x \(\frac{1}{6}\) = \(\frac{2}{3}\) miles.

Step 2: Formulate the Equal Distance Equation for when the Second Jogger Catches Up

When the second jogger catches up to the first, they would have covered the same distance. Hence, the distance covered by first jogger is S1 x (T1+ t) = 4 x (\(\frac{1}{6}\) + t) and distance covered by second jogger is S2 x t = 5t. The equal distance gives the equation 4(\(\frac{1}{6}\) + t) = 5t.

Step 3: Solve the Equation

Solve the equation for t. We get t = \(\frac{2}{3}\) hours. Hence it will take the second jogger \(\frac{2}{3}\) hours to overtake the first jogger.

Step 4: Find the Distance

The distance covered by each jogger when the second overtakes the first is S2 x t = 5 x \(\frac{2}{3}\) = \(\frac{10}{3}\) miles OR S1 x (T1 + t) = 4 x ( \(\frac{1}{6}\) + \(\frac{2}{3}\) ) = \(\frac{10}{3}\) miles.

Key Concepts

Rate Problems

Rate problems are a common type of question in algebra that deal with the relationship between distance, rate (or speed), and time. The key principle is that distance traveled is equal to the rate multiplied by time. This relationship is often expressed as the formula:

• Distance = Rate × Time

In our exercise, each jogger moves at a constant rate. Their speeds are given as 4 miles per hour for the first jogger and 5 miles per hour for the second jogger. Rate problems often involve comparing the rates of two or more moving objects and determining when one object will catch up to another or how far apart they will be after a certain time. Here, we need to find when the second jogger catches up to the first.
Using the given rates and the fact that the second jogger starts later, we create an equation based on the distance each jogger runs until they meet.

Algebraic Equations

An algebraic equation is a mathematical statement that uses variables, numbers, and operations to represent a relationship. In the context of our exercise, we use an equation to solve for the time at which the second jogger overtakes the first. We rely on the principle that when the second jogger catches up, both joggers will have traveled the same distance.
This leads us to the equation:

• 4\( \left( \frac{1}{6} + t \right) \) = 5t

Here, \( t \) represents the time the second jogger runs. We derived this equation by setting the expressions for the distances traveled by each jogger equal to each other. Rearranging and simplifying this equation allows us to solve for \( t \), which tells us how long it takes for the second jogger to catch up. Decomposition and simplification of such equations are key skills in solving algebraic problems.

Problem Solving Steps

When solving distance problems like this one, it is important to follow systematic steps that guide you to the solution. Let's break down the steps:
1. **Understand the Problem**: Identify what is given and what needs to be determined. Here, we're given the rates of the joggers and the delay in their starting times.
2. **Convert Units if Necessary**: Ensure all components are in consistent units. In our case, we convert 10 minutes to hours so that both time and speed relate correctly.
3. **Set up the Equation**: Formulate an equation based on the relationship (distance = rate x time). Given both joggers travel the same distance when the second overtakes the first, we derive our equation.
4. **Solve the Equation**: Use algebraic manipulation to find the unknown variable, \( t \).
5. **Verify the Solution**: Check your solution by substituting the value back into the problem to ensure it satisfies the condition of the joggers meeting.
Following these problem-solving steps helps ensure logic and consistency in reaching the correct solution.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve problems and justify solutions. It is essential when tackling algebraic equations and distance problems.
During problem-solving, it is crucial to:

• Identify relationships between given quantities and decide which information is relevant.
• Use prior knowledge to apply formulas and develop equations that represent these relationships.
• Deduce the unknown variable through logical steps that align with mathematical principles.

In our example, mathematical reasoning allowed us to identify that both joggers should have traveled the same distance when they meet, which led us to set up an equal distance equation. This logical application of mathematical concepts helps break down complex situations into solvable parts, making the problem manageable step by step. Sharpening these reasoning skills is fundamental for solving not just rate problems, but any mathematical challenge.