Solving Proportional Relationships Nina's Dress Stitching Problem

#h1 Nina can stitch $\frac{2}{3}$ of a dress in 4 hours. If $d$ represents the number of dresses and $h$ represents the number of hours, which equation represents this proportional relationship?

This is a classic proportionality problem often encountered in mathematics, particularly in algebra and ratio and proportion scenarios. To solve it, we need to understand the relationship between the number of dresses Nina can stitch and the time she spends stitching. The core concept here is to find a constant rate of work, which will allow us to form an equation that links the variables $d$ (number of dresses) and $h$ (number of hours).

Understanding Proportional Relationships

In a proportional relationship, two quantities vary directly with each other. This means that as one quantity increases, the other increases at a constant rate. The relationship can be expressed in the form $y = kx$, where $y$ and $x$ are the two quantities, and $k$ is the constant of proportionality. This constant, $k$, represents the ratio between $y$ and $x$ and remains the same throughout the relationship. Identifying and calculating $k$ is crucial in solving proportionality problems.

In our scenario, the number of dresses stitched ($d$) is directly proportional to the number of hours spent stitching ($h$). Nina's stitching rate is constant, which means she stitches the same fraction of a dress every hour. Our goal is to find the constant rate and then express the relationship as an equation. Understanding this foundational concept of proportionality is the first step in tackling this problem. The ability to translate real-world scenarios into mathematical equations is a fundamental skill in algebra and is essential for problem-solving in various fields. Moreover, recognizing proportional relationships helps in making predictions and estimations in different contexts, from simple everyday calculations to more complex scientific applications.

Analyzing the Given Information

We are given that Nina can stitch $\frac{2}{3}$ of a dress in 4 hours. This is the crucial piece of information that allows us to determine Nina's stitching rate. To find the rate, we need to determine how much of a dress Nina can stitch in one hour. This involves dividing the fraction of the dress she completes (rac{2}{3}) by the number of hours it takes (4 hours). This calculation will give us the fraction of a dress stitched per hour, which is our constant rate of work. This rate will then become the constant of proportionality in our equation.

Calculating the stitching rate accurately is essential. An error in this step will lead to an incorrect equation and ultimately the wrong answer. Once we have the rate, we can set up the equation that represents the relationship between the number of dresses and the number of hours. This process of breaking down the given information into manageable steps is a key strategy in problem-solving. It allows us to focus on specific aspects of the problem before integrating them into a comprehensive solution. Furthermore, understanding how to extract relevant information from a problem statement and use it to find rates and constants is a skill that extends beyond mathematics and is valuable in various analytical tasks.

Calculating the Stitching Rate

To determine Nina's stitching rate, we need to divide the amount of work done (rac{2}{3} of a dress) by the time taken (4 hours). Mathematically, this is represented as:

Rate = $\frac{\frac{2}{3}}{4}$

To divide a fraction by a whole number, we can rewrite the whole number as a fraction with a denominator of 1. So, 4 becomes $\frac{4}{1}$. Now we have:

Rate = $\frac{\frac{2}{3}}{\frac{4}{1}}$

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{4}{1}$ is $\frac{1}{4}$. Therefore, the equation becomes:

Rate = $\frac{2}{3} \times \frac{1}{4}$

Multiplying the numerators and denominators, we get:

Rate = $\frac{2 \times 1}{3 \times 4} = \frac{2}{12}$

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

Rate = $\frac{2 \div 2}{12 \div 2} = \frac{1}{6}$

So, Nina stitches $\frac{1}{6}$ of a dress per hour. This rate is the constant of proportionality in our equation. The accurate calculation of this rate is a pivotal step in solving the problem, as it forms the basis for establishing the relationship between the number of dresses and the number of hours. This process also demonstrates the practical application of fraction operations in real-world scenarios. Mastering these calculations is essential for solving various mathematical problems involving rates and proportions.

Forming the Equation

Now that we know Nina stitches $\frac{1}{6}$ of a dress per hour, we can form the equation that represents the proportional relationship between the number of dresses ($d$) and the number of hours ($h$). In a proportional relationship, the equation is of the form $y = kx$, where $k$ is the constant of proportionality. In our case, $d$ represents the number of dresses (which is analogous to $y$), $h$ represents the number of hours (which is analogous to $x$), and $\frac{1}{6}$ is the constant of proportionality ($k$).

Therefore, the equation that represents the relationship is:

$d = \frac{1}{6}h$

This equation states that the number of dresses Nina can stitch is equal to $\frac{1}{6}$ times the number of hours she spends stitching. This equation accurately captures the proportional relationship between the two variables and allows us to predict how many dresses Nina can complete in a given number of hours, or conversely, how many hours she needs to stitch a certain number of dresses. The ability to translate a rate into a proportional equation is a fundamental skill in algebra and has wide-ranging applications in various fields, including physics, engineering, and economics. It allows us to model and analyze real-world situations where quantities vary in direct proportion to each other.

Evaluating the Answer Choices

We have derived the equation $d = \frac{1}{6}h$ that represents the proportional relationship between the number of dresses Nina stitches ($d$) and the number of hours she spends stitching ($h$). Now, we need to compare this equation with the given answer choices to identify the correct one.

The answer choices are:

A. $d = 6h$ B. $3d = 4h$ C. $d = \frac{1}{6}h$ D. $d = 4h$

By comparing our derived equation with the answer choices, we can see that option C, $d = \frac{1}{6}h$, exactly matches our equation. Therefore, option C is the correct answer. The other options represent different relationships between $d$ and $h$ and do not reflect the stitching rate we calculated based on the given information. This step of comparing the derived solution with the given options is crucial in problem-solving, as it ensures that the final answer aligns with the question's requirements. It also helps in verifying the correctness of the solution and identifying any potential errors in the process. Moreover, this comparison process enhances the understanding of the relationships between different mathematical expressions and equations.

Conclusion

In conclusion, by carefully analyzing the given information, calculating Nina's stitching rate, forming the proportional equation, and comparing it with the answer choices, we have determined that the equation that represents the relationship between the number of dresses ($d$) and the number of hours ($h$) is:

$d = \frac{1}{6}h$

This corresponds to answer choice C. This problem highlights the importance of understanding proportional relationships and how to translate real-world scenarios into mathematical equations. It also emphasizes the significance of accurate calculations and logical reasoning in problem-solving. The ability to solve such problems is not only essential in mathematics but also in various practical situations where understanding proportions and rates is crucial. This process demonstrates the interconnectedness of mathematical concepts and their application in real-world contexts. Furthermore, mastering these problem-solving techniques enhances critical thinking skills and the ability to approach complex situations with a structured and analytical mindset. By understanding the underlying principles of proportionality and practicing problem-solving strategies, one can confidently tackle similar challenges and apply these skills in various academic and professional pursuits.

#h2 Options Review

Let's analyze why the other options are incorrect:

• A. $d = 6h$: This equation suggests that Nina stitches 6 dresses per hour, which contradicts the given information that she stitches only $\frac{2}{3}$ of a dress in 4 hours. This equation represents a much faster stitching rate than what is provided in the problem statement.
• B. $3d = 4h$: To better understand this equation, we can rearrange it to solve for $d$: $d = \frac{4}{3}h$. This equation indicates that Nina stitches $\frac{4}{3}$ of a dress per hour. While this might seem plausible at first glance, it doesn't align with the given information. If we plug in 4 hours for $h$, we get $d = \frac{4}{3} \times 4 = \frac{16}{3}$, which means Nina would stitch 5rac{1}{3} dresses in 4 hours. This is inconsistent with the initial condition that she stitches $\frac{2}{3}$ of a dress in 4 hours. Therefore, this equation does not accurately represent the proportional relationship described in the problem.
• D. $d = 4h$: This equation implies that Nina stitches 4 dresses per hour, which is significantly faster than the actual rate. This option is far from the correct rate and can be easily ruled out by considering the information given in the problem.

#h2 Key Concepts Recap

• Proportional Relationships: Understanding that proportional relationships can be expressed in the form $y = kx$, where $k$ is the constant of proportionality.
• Calculating Rates: The ability to calculate rates by dividing the amount of work done by the time taken is crucial.
• Forming Equations: Translating a rate into a proportional equation that accurately represents the relationship between variables.
• Verifying Solutions: Checking the derived equation against the given information and answer choices to ensure accuracy.

#h2 Practical Applications

This type of problem has practical applications in various real-world scenarios, such as:

• Manufacturing: Calculating production rates and estimating the time required to complete a certain number of units.
• Construction: Estimating the amount of material needed for a project based on the rate of consumption.
• Service Industries: Calculating service times and scheduling appointments based on the rate of service provision.
• Everyday Life: Estimating travel times, calculating cooking times, and managing household tasks based on individual rates of work.

#h2 Further Practice

To further enhance your understanding of proportional relationships and problem-solving skills, consider practicing similar problems with varying scenarios and contexts. This will help you develop confidence in applying these concepts to different situations and improve your overall mathematical proficiency.

By mastering these fundamental concepts and practicing problem-solving strategies, you can confidently tackle various mathematical challenges and apply these skills in diverse real-world contexts. Remember, the key to success in mathematics lies in understanding the underlying principles and practicing consistently to build a strong foundation.