Calculating Victoria's Speed Step-by-Step Solution

In this article, we will delve into a fascinating mathematical problem involving distance, time, and speed. We aim to determine Victoria's speed, given that she drove $21^1 / 2$ miles in $2 / 3$ of an hour. This problem is a classic example of how we can apply fundamental concepts of mathematics to solve real-world scenarios. Understanding the relationship between distance, time, and speed is crucial in various fields, from everyday commuting to advanced physics. This article will not only provide a step-by-step solution but also explain the underlying principles, ensuring a comprehensive understanding of the topic. So, let's embark on this journey of mathematical exploration and uncover Victoria's speed!

Before we jump into calculations, let's break down the problem statement. Victoria drove a distance of $21^1 / 2$ miles, which is a mixed number representing 21 and a half miles. The time she took to cover this distance was $2 / 3$ of an hour. Our objective is to find her speed, which is the distance she covered per unit of time, typically measured in miles per hour (mph). The formula that connects these three quantities is:

Speed = Distance / Time

This formula is the cornerstone of our solution. It tells us that if we divide the total distance traveled by the time taken, we will obtain the average speed during that journey. However, before we can apply this formula, we need to ensure that our numbers are in the correct format. Specifically, we need to convert the mixed number ($21^1 / 2$) into an improper fraction and understand how to divide by a fraction. These are fundamental arithmetic skills that we will review in the following sections. By carefully understanding each step, we can confidently solve this problem and similar ones in the future. The problem seems straightforward, but it requires careful attention to detail and a solid understanding of fractions and division. Let's proceed step by step to ensure accuracy and clarity.

The first step in solving this problem is to convert the mixed number $21^1 / 2$ into an improper fraction. A mixed number is a combination of a whole number and a fraction, while an improper fraction has a numerator greater than or equal to its denominator. Converting to an improper fraction simplifies the division process later on. To convert $21^1 / 2$ to an improper fraction, we follow these steps:

1. Multiply the whole number (21) by the denominator of the fraction (2). This gives us $21 * 2 = 42$.
2. Add the result to the numerator of the fraction (1). This gives us $42 + 1 = 43$.
3. Place the result (43) over the original denominator (2). This gives us the improper fraction $43 / 2$.

So, $21^1 / 2$ miles is equivalent to $43 / 2$ miles. This conversion is crucial because it allows us to perform the division operation more easily. When dealing with fractions, it is often simpler to work with improper fractions rather than mixed numbers. This is because improper fractions represent a single numerical value, whereas mixed numbers combine a whole number and a fractional part. By converting to an improper fraction, we streamline the calculation process and minimize the risk of errors. This step sets the stage for the next phase of our solution, where we will apply the speed formula. Now that we have the distance in the form of an improper fraction, we can move on to the division operation.

Now that we have the distance as an improper fraction ($43 / 2$ miles) and the time as a fraction ($2 / 3$ of an hour), we can apply the speed formula:

Speed = Distance / Time

Substituting the values, we get:

Speed = ($43 / 2) / (2 / 3)

To divide by a fraction, we multiply by its reciprocal. The reciprocal of $2 / 3$ is $3 / 2$. Therefore, the equation becomes:

Speed = ($43 / 2) * (3 / 2)

Now, we multiply the numerators and the denominators:

Speed = (43 * 3) / (2 * 2)

Speed = 129 / 4

This gives us the speed as an improper fraction, $129 / 4$ miles per hour. However, to match the answer choices provided, we need to convert this improper fraction back into a mixed number. Converting improper fractions to mixed numbers is a necessary skill for presenting results in a more understandable format. The improper fraction $129 / 4$ represents the speed in a precise mathematical form, but a mixed number will give us a more intuitive sense of the value. This conversion will also help us to compare our result with the given options and select the correct answer. Let's proceed with the conversion to mixed number to complete our calculation.

To convert the improper fraction $129 / 4$ to a mixed number, we perform division. We divide 129 by 4:

1. Divide 129 by 4: $129 ÷ 4 = 32$ with a remainder of 1.
2. The quotient (32) becomes the whole number part of the mixed number.
3. The remainder (1) becomes the numerator of the fractional part.
4. The denominator (4) remains the same.

Therefore, $129 / 4$ is equal to $32^1 / 4$. So, Victoria's speed is $32^1 / 4$ miles per hour. This conversion is the final step in our calculation, and it provides the answer in a format that is easy to understand and compare with the given options. Converting back to a mixed number allows us to express the speed as a whole number and a fractional part, giving a clearer sense of Victoria's rate of travel. This completes the process of finding Victoria's speed, and we can now confidently select the correct answer from the multiple choices provided. The process of converting improper fractions to mixed numbers is a fundamental skill in arithmetic, and it is essential for expressing mathematical results in a practical and understandable manner.

Comparing our result, $32^1 / 4$ miles per hour, with the given options:

A) $32^1 / 3$ miles per hour B) $32^3 / 4$ miles per hour C) $32^1 / 4$ miles per hour D) $32^1 / 2$ miles per hour

We find that option C, $32^1 / 4$ miles per hour, matches our calculated speed. Therefore, Victoria's speed was $32^1 / 4$ miles per hour.

In conclusion, by carefully applying the formula Speed = Distance / Time and performing the necessary conversions between mixed numbers and improper fractions, we successfully determined Victoria's speed. This problem highlights the importance of understanding fundamental mathematical concepts and their application in solving real-world problems. The step-by-step approach we followed ensures clarity and accuracy in our calculations. From converting mixed numbers to improper fractions and then back again, each step played a crucial role in arriving at the correct solution. This exercise not only enhances our problem-solving skills but also reinforces our understanding of fractions, division, and the relationship between distance, time, and speed. Such skills are valuable not only in academic settings but also in everyday situations where we need to calculate rates, speeds, or distances. The ability to break down a problem into manageable steps and apply the correct formulas is a key attribute of a proficient problem solver. This is what we have demonstrated in this detailed solution.