Calculating Travel Time: A Step-by-Step Guide For Mathematical Problem Solving

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Let's dive into the fascinating world of mathematical problem-solving, guys! Today, we're tackling a classic travel time question that requires us to choose the correct expression for calculating the number of hours it takes to travel a certain distance at a given speed. This is a super practical skill that you can use in everyday life, like planning road trips or estimating commute times. So, buckle up and let's get started!

Understanding the Problem: Speed, Distance, and Time

Before we jump into the specific question, let's refresh our understanding of the relationship between speed, distance, and time. These three concepts are interconnected, and the formula that binds them is quite simple: Distance = Speed × Time. This formula is the key to solving a variety of travel-related problems. To illustrate, imagine you are driving at a constant speed. The farther you drive, the longer it takes, directly proportional with each other. The faster you go, the less time it takes, inversly proportional with each other.

The Interplay of Distance, Speed, and Time

Think of it this way: if you know two of these variables, you can always find the third. For instance, if you know the distance you need to travel and the speed at which you're traveling, you can calculate the time it will take. Similarly, if you know the speed and the time, you can determine the distance covered.

To make this even clearer, let's break down the formula and explore how we can rearrange it to solve for different variables.

  • Finding Time: If we want to find the time it takes to travel a certain distance, we need to rearrange the formula. To do this, we divide both sides of the equation by speed: Time = Distance / Speed. This is the formula we'll be using to solve our problem today.
  • Finding Speed: If we want to find the average speed of a journey, we divide the total distance traveled by the total time taken: Speed = Distance / Time.
  • Finding Distance: As we mentioned earlier, to find the distance traveled, we simply multiply the speed by the time: Distance = Speed × Time.

Understanding these relationships is crucial for tackling various mathematical problems, not just those related to travel. It's like having a superpower for solving real-world scenarios!

Analyzing the Question: A Car's Journey

Now, let's get to the heart of the matter. Our question presents a scenario involving a car traveling a certain distance at a specific speed. The question asks: "If a car averages 50 rac{1}{3} miles per hour, how many hours will it take to go 47 rac{1}{2} miles?" To find the expression that equals the number of hours, we need to apply our understanding of the relationship between speed, distance, and time.

Identifying the Key Information

First, let's identify the key information provided in the question:

  • Speed: The car averages 50 rac{1}{3} miles per hour. This is the rate at which the car is traveling.
  • Distance: The car needs to travel 47 rac{1}{2} miles. This is the total length of the journey.
  • What We Need to Find: We need to find the time it will take for the car to complete the journey, measured in hours.

Applying the Correct Formula

As we discussed earlier, the formula for calculating time is Time = Distance / Speed. Now, we simply need to plug in the values from the question into this formula.

In our case:

  • Distance = 47 rac{1}{2} miles
  • Speed = 50 rac{1}{3} miles per hour

So, the expression for the time it will take is: Time = 47 rac{1}{2} / 50 rac{1}{3}.

This expression represents the number of hours it will take the car to travel the given distance at the specified speed. It's important to note that we haven't actually calculated the time yet; we've simply identified the correct expression to do so. The next step would be to perform the division to find the numerical answer. But for now, we've successfully chosen the expression that equals the number of hours!

Converting Mixed Numbers to Improper Fractions

Before we can perform the division, guys, we need to convert the mixed numbers in our expression into improper fractions. This will make the calculation much easier. Remember, a mixed number is a combination of a whole number and a fraction, like 47 rac{1}{2} or 50 rac{1}{3}. An improper fraction, on the other hand, has a numerator that is greater than or equal to its denominator, like rac{95}{2} or rac{151}{3}.

The Conversion Process

To convert a mixed number to an improper fraction, we follow these steps:

  1. Multiply the whole number by the denominator of the fraction.
  2. Add the result to the numerator of the fraction.
  3. Keep the same denominator.

Let's apply this to our mixed numbers:

  • Converting 47 rac{1}{2}:
    1. Multiply the whole number (47) by the denominator (2): 47 × 2 = 94
    2. Add the result to the numerator (1): 94 + 1 = 95
    3. Keep the same denominator (2): So, 47 rac{1}{2} = rac{95}{2}
  • Converting 50 rac{1}{3}:
    1. Multiply the whole number (50) by the denominator (3): 50 × 3 = 150
    2. Add the result to the numerator (1): 150 + 1 = 151
    3. Keep the same denominator (3): So, 50 rac{1}{3} = rac{151}{3}

Now that we've converted the mixed numbers to improper fractions, our expression looks like this: Time = rac{95}{2} / rac{151}{3}.

We're one step closer to finding the actual travel time! The next step is to divide these fractions, which is actually quite simple once you know the trick.

Dividing Fractions: Keep, Change, Flip

Alright, guys, let's conquer the division of fractions! Dividing fractions might seem a bit daunting at first, but there's a super handy trick that makes it a breeze: Keep, Change, Flip. This simple mnemonic will guide you through the process.

The Keep, Change, Flip Method

The Keep, Change, Flip method is a visual way to remember the steps involved in dividing fractions. Here's how it works:

  1. Keep: Keep the first fraction exactly as it is.
  2. Change: Change the division sign (÷) to a multiplication sign (×).
  3. Flip: Flip the second fraction (the divisor) by swapping its numerator and denominator. This is also known as finding the reciprocal of the fraction.

Let's apply this method to our expression: Time = rac{95}{2} / rac{151}{3}.

  1. Keep: Keep the first fraction: rac{95}{2}
  2. Change: Change the division to multiplication: ×
  3. Flip: Flip the second fraction: rac{3}{151}

Now our expression looks like this: Time = rac{95}{2} × rac{3}{151}.

See how much simpler that looks? We've transformed a division problem into a multiplication problem, which is much easier to handle.

Multiplying Fractions: Numerator Times Numerator, Denominator Times Denominator

Now that we've transformed our division problem into a multiplication problem, let's multiply the fractions. Multiplying fractions is straightforward: you simply multiply the numerators together and the denominators together.

The Multiplication Process

The rule for multiplying fractions is:

  • Multiply the numerators: Numerator1 × Numerator2
  • Multiply the denominators: Denominator1 × Denominator2

Applying this to our expression, Time = rac{95}{2} × rac{3}{151}, we get:

  • Numerator: 95 × 3 = 285
  • Denominator: 2 × 151 = 302

So, our result is: Time = rac{285}{302} hours.

This is the exact answer, expressed as a fraction. We could leave it like this, or we could simplify the fraction or convert it to a decimal to get a better sense of the actual travel time.

Simplifying the Fraction (Optional)

Simplifying a fraction means reducing it to its lowest terms. This involves finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by the GCF.

In our case, the fraction is rac{285}{302}. Finding the GCF of 285 and 302 can be a bit tricky, but it turns out that they don't share any common factors other than 1. This means that the fraction is already in its simplest form.

Converting to a Decimal (Optional)

If we want to get a decimal approximation of the travel time, we simply divide the numerator by the denominator: 285 ÷ 302.

Using a calculator, we find that rac{285}{302} is approximately equal to 0.9437 hours.

Interpreting the Result

So, the car will take approximately 0.9437 hours to travel 47 rac{1}{2} miles at an average speed of 50 rac{1}{3} miles per hour.

To get a better sense of this time, we can convert it to minutes. Since there are 60 minutes in an hour, we multiply 0.9437 by 60:

  1. 9437 hours × 60 minutes/hour ≈ 56.62 minutes

Therefore, it will take the car approximately 56.62 minutes to complete the journey.

Conclusion: Mastering the Art of Problem Solving

Guys, we've successfully navigated through this mathematical journey! We started with a word problem, identified the key information, applied the correct formula, and performed the necessary calculations. We even learned some handy tricks like converting mixed numbers to improper fractions and using the Keep, Change, Flip method for dividing fractions.

Key Takeaways

  • Understanding the relationship between speed, distance, and time is crucial for solving travel-related problems.
  • Converting mixed numbers to improper fractions makes calculations easier.
  • The Keep, Change, Flip method simplifies the division of fractions.
  • Multiplying fractions involves multiplying the numerators and the denominators.
  • Simplifying fractions and converting them to decimals can help us interpret the results more easily.

By mastering these skills, you'll be well-equipped to tackle a wide range of mathematical problems, both in the classroom and in the real world. So keep practicing, keep exploring, and keep challenging yourself! You've got this!