Calculate Marina's Running Speed: A Math Problem

by Alex Johnson 49 views

Let's dive into a classic problem involving speed, time, and distance! This scenario presents a situation where we need to figure out Marina's running speed, given information about her cycling speed and the distances she covers in equal amounts of time. This is a fun puzzle that combines basic math principles with a real-world scenario, making it both engaging and practical. So, let's put on our thinking caps and break down the problem step by step.

Understanding the Problem

At the heart of this question is understanding the relationship between speed, time, and distance. The fundamental formula we'll use is:

Distance = Speed × Time

Our goal is to find Marina's running speed. We know that Marina can bicycle 19.5 miles in the same time it takes her to run 6 miles. We also know that she bikes 9 miles per hour faster than she runs. This crucial piece of information is what allows us to set up an equation and solve for her running speed. It's like having a secret key that unlocks the puzzle! The challenge now is to translate these words into mathematical expressions that we can manipulate and solve. Think of it as translating a story into a language of numbers and symbols, which will then guide us to the final answer.

Setting up the Equations

Let's use variables to represent the unknowns. Let:

  • r = Marina's running speed (in miles per hour)
  • b = Marina's biking speed (in miles per hour)
  • t = The time it takes Marina to run 6 miles and bike 19.5 miles (in hours)

From the given information, we can establish two equations:

  1. b = r + 9 (Marina bikes 9 miles per hour faster than she runs)
  2. Time = Distance / Speed. Therefore:
    • Time running: t = 6 / r
    • Time biking: t = 19.5 / b

Because the time is the same for both activities, we can set the two time expressions equal to each other:

6 / r = 19.5 / b

Now we have a system of equations that we can solve! We've taken the initial word problem and transformed it into a set of mathematical statements. This is a significant step because it allows us to use the power of algebra to find the solution. The next step involves using these equations in tandem to eliminate one variable and solve for the other. It's like a detective piecing together clues to solve a mystery, each equation providing a new piece of the puzzle.

Solving the Equations

Now, we have two equations:

  1. b = r + 9
  2. 6 / r = 19.5 / b

We can substitute the first equation into the second equation to eliminate b:

6 / r = 19.5 / (r + 9)

Next, we cross-multiply to get rid of the fractions:

6(r + 9) = 19.5r

Expanding the left side gives us:

6r + 54 = 19.5r

Now, we need to isolate r. Subtract 6r from both sides:

54 = 13.5r

Finally, divide both sides by 13.5 to solve for r:

r = 54 / 13.5

r = 4

So, Marina's running speed is 4 miles per hour. We've successfully navigated through the algebraic steps to arrive at our answer! This process involved a bit of manipulation and careful attention to detail, but by breaking it down into smaller steps, we were able to isolate the variable we were looking for. This is the beauty of algebra – it provides us with the tools to solve complex problems in a systematic and logical way.

Verification

To be sure our answer is correct, let's verify it. If Marina runs at 4 miles per hour, then her biking speed is:

b = 4 + 9 = 13 miles per hour

Now, let's calculate the time it takes for each activity:

  • Time running: t = 6 miles / 4 mph = 1.5 hours
  • Time biking: t = 19.5 miles / 13 mph = 1.5 hours

The times are the same, so our answer is correct! It's always a good idea to check our work, especially in math problems. This step ensures that we haven't made any errors along the way and that our solution makes sense in the context of the original problem. In this case, the fact that the times are equal when we plug in our calculated running speed gives us confidence that we've found the right answer.

Conclusion

Marina runs at a speed of 4 miles per hour. This problem beautifully illustrates how we can use mathematical principles to solve real-world scenarios. By carefully setting up equations and using algebraic techniques, we were able to determine Marina's running speed. Remember, the key to solving such problems is to break them down into smaller, manageable steps and to always verify your solution. Math is like a puzzle, and each step is a piece that fits together to reveal the final picture.

For further exploration of speed, time, and distance problems, you might find valuable resources at websites like Khan Academy. They offer a wealth of information and practice exercises to help you master these concepts.