Janet's Fabric Project Calculating The Difference In Fabric Used

#h1 Janet's Fabric Project: Calculating the Difference in Blue and Yellow Fabric

In this article, we will delve into a practical problem involving fractions and subtraction. The problem revolves around Janet's fabric project, where she used different amounts of blue and yellow fabric to make a shirt. Our primary goal is to determine the difference in the amount of blue and yellow fabric used, expressing the answer as a fraction in its simplest form. This exercise provides a great opportunity to reinforce our understanding of fraction operations and problem-solving strategies in a real-world context.

Setting the Stage: Understanding the Problem

The problem states that Janet used $\frac{5}{8}$ yards of blue fabric and $\frac{1}{2}$ yards of yellow fabric to make a shirt. The core question we need to answer is: How many more yards of blue fabric did she use than yellow fabric? To solve this, we need to find the difference between the amount of blue fabric and the amount of yellow fabric. This involves subtracting the fraction representing the amount of yellow fabric from the fraction representing the amount of blue fabric.

The Key to Solving: Subtracting Fractions

To subtract fractions, a fundamental principle comes into play: we can only subtract fractions that have a common denominator. In our case, we have the fractions $\frac{5}{8}$ and $\frac{1}{2}$. The denominators are 8 and 2, respectively. These are different, so we need to find a common denominator before we can proceed with the subtraction.

Finding the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest multiple that the denominators of both fractions share. To find the LCD of 8 and 2, we can list the multiples of each number:

• Multiples of 2: 2, 4, 6, 8, 10, ...
• Multiples of 8: 8, 16, 24, ...

As we can see, the smallest multiple that both 2 and 8 share is 8. Therefore, the LCD is 8. This means we will rewrite both fractions so that they have a denominator of 8.

Converting Fractions to Equivalent Fractions with the LCD

The fraction $\frac{5}{8}$ already has a denominator of 8, so we don't need to change it. However, we need to convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8. To do this, we need to determine what number we can multiply the denominator (2) by to get 8. The answer is 4 (since 2 x 4 = 8). To maintain the value of the fraction, we must multiply both the numerator and the denominator by 4:

$$\frac{1}{2} \times \frac{4}{4} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$$

Now we have two fractions with the same denominator: $\frac{5}{8}$ and $\frac{4}{8}$.

Performing the Subtraction

Now that the fractions have a common denominator, we can subtract them. To subtract fractions with the same denominator, we subtract the numerators and keep the denominator the same:

$$\frac{5}{8} - \frac{4}{8} = \frac{5 - 4}{8} = \frac{1}{8}$$

So, Janet used $\frac{1}{8}$ yards more of blue fabric than yellow fabric.

Expressing the Answer in Simplest Form

The fraction $\frac{1}{8}$ is already in its simplest form. This is because the numerator (1) and the denominator (8) have no common factors other than 1. A fraction is in its simplest form when the greatest common factor (GCF) of the numerator and denominator is 1.

The Final Answer: Janet Used 1/8 Yards More Blue Fabric

Therefore, the final answer to our problem is that Janet used $\frac{1}{8}$ yards more of blue fabric than yellow fabric. This seemingly simple problem illustrates the importance of understanding how to subtract fractions, find common denominators, and simplify fractions. These are essential skills in mathematics and have practical applications in various real-life situations.

#h2 Real-World Applications of Fraction Subtraction

The application of subtracting fractions extends far beyond just solving textbook problems. It is a fundamental skill that we use in various everyday situations. Understanding how to work with fractions allows us to make informed decisions, solve practical problems, and gain a deeper understanding of the world around us. Let's explore some real-world scenarios where fraction subtraction comes into play.

Cooking and Baking: Precision in Measurements

In the culinary world, precision is key, and recipes often call for specific amounts of ingredients expressed as fractions. Imagine you're baking a cake, and the recipe requires $\frac{3}{4}$ cup of flour. However, you only have $\frac{1}{2}$ cup of flour available. To determine how much more flour you need, you would subtract the fraction representing the available amount from the fraction representing the required amount:

$$\frac{3}{4} - \frac{1}{2} = ?$$

To solve this, you would need to find a common denominator (which is 4 in this case) and convert the fractions:

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

This calculation tells you that you need an additional $\frac{1}{4}$ cup of flour. This simple example highlights the importance of fraction subtraction in ensuring accurate measurements in cooking and baking.

Home Improvement Projects: Measuring and Cutting Materials

Home improvement projects often involve measuring and cutting materials like wood, fabric, or tiles. These measurements are frequently expressed in fractions. For instance, you might need to cut a piece of wood that is $5\frac{1}{2}$ inches long from a board that is $8\frac{3}{4}$ inches long. To determine how much of the board will be left, you need to subtract the length of the piece you're cutting from the total length of the board:

$$8\frac{3}{4} - 5\frac{1}{2} = ?$$

This involves subtracting mixed numbers, which requires converting them to improper fractions or subtracting the whole numbers and fractions separately. The result will tell you the remaining length of the board, ensuring you have enough material for your project and minimize waste. This skill of subtracting fractions is invaluable in home improvement, ensuring accurate cuts and efficient use of materials.

Financial Planning: Budgeting and Tracking Expenses

Managing personal finances involves budgeting and tracking expenses, where fractions can help represent portions of income or spending. Let's say you allocate $\frac{1}{4}$ of your monthly income to rent and $\frac{1}{8}$ to utilities. To determine the total fraction of your income allocated to these two expenses, you would add the fractions. However, if you want to know how much more you spend on rent than utilities, you would subtract the fraction representing utilities from the fraction representing rent:

$$\frac{1}{4} - \frac{1}{8} = ?$$

Solving this subtraction helps you understand the relative proportions of your expenses and make informed financial decisions. Fraction subtraction can also be used to calculate savings goals, debt repayment plans, and investment strategies.

Time Management: Scheduling and Planning Activities

We often divide our time into fractions when scheduling and planning activities. For example, you might spend $\frac{1}{3}$ of your day working, $\frac{1}{4}$ sleeping, and $\frac{1}{6}$ on other activities. To calculate the fraction of the day you have left for leisure or personal time, you would need to subtract the fractions representing work, sleep, and other activities from the whole (which can be represented as 1 or $\frac{1}{1}$). This involves adding and subtracting fractions to determine the remaining portion of your day. Effective time management relies on the ability to calculate and compare fractions of time.

Measuring Progress: Tracking Goals and Achievements

Fractions are also useful for measuring progress towards goals and tracking achievements. Suppose you're training for a marathon and have run $\frac{2}{5}$ of the total distance. To determine how much further you need to run, you would subtract the fraction representing the distance you've run from the whole (1 or $\frac{5}{5}$):

$$\frac{5}{5} - \frac{2}{5} = \frac{3}{5}$$

This calculation tells you that you still have $\frac{3}{5}$ of the distance to cover. This ability to quantify progress using fractions is motivating and helps you stay on track towards your goals.

Conclusion: The Ubiquitous Nature of Fraction Subtraction

These examples illustrate that subtracting fractions is not just an abstract mathematical concept; it is a practical skill that we use in countless real-world situations. From cooking and home improvement to financial planning and time management, understanding fraction subtraction empowers us to make informed decisions, solve problems effectively, and navigate the world around us with greater confidence. By mastering this fundamental mathematical operation, we unlock a powerful tool for success in various aspects of life. The applications of fraction subtraction are truly ubiquitous, making it an essential skill for everyone to develop and refine.

#h2 Mastering Fraction Subtraction: A Step-by-Step Guide

Mastering fraction subtraction is a crucial skill in mathematics, providing a foundation for more advanced concepts and real-world applications. While the basic principle of subtracting fractions might seem straightforward, it involves a series of steps that need to be followed meticulously to arrive at the correct answer. This step-by-step guide will break down the process into manageable components, ensuring you have a clear understanding of how to subtract fractions effectively.

Step 1: Identifying the Fractions and the Operation

The first step in any fraction subtraction problem is to clearly identify the fractions involved and the operation you need to perform. This might seem obvious, but it's crucial to ensure you're subtracting the correct fractions in the right order. For instance, if the problem asks, "What is $\frac{3}{4}$ minus $\frac{1}{2}$?", you need to recognize that you're subtracting $\frac{1}{2}$ from $\frac{3}{4}$, which can be written as:

$$\frac{3}{4} - \frac{1}{2}$$

Clearly identifying the fractions and the operation sets the stage for the subsequent steps.

Step 2: Finding a Common Denominator

The most critical step in subtracting fractions is ensuring they have a common denominator. A common denominator is a shared multiple of the denominators of the fractions you're subtracting. You cannot directly subtract fractions with different denominators because the fractions represent parts of different wholes. To find a common denominator, you typically look for the least common multiple (LCM) of the denominators, also known as the least common denominator (LCD).

Finding the Least Common Denominator (LCD)

There are several methods to find the LCD, but one common approach is to list the multiples of each denominator until you find a common one. For example, if you're subtracting $\frac{1}{3}$ from $\frac{1}{2}$, the denominators are 3 and 2. Let's list their multiples:

• Multiples of 2: 2, 4, 6, 8, ...
• Multiples of 3: 3, 6, 9, 12, ...

The least common multiple (LCM) of 2 and 3 is 6, so the LCD is 6.

Step 3: Converting Fractions to Equivalent Fractions with the LCD

Once you've found the LCD, the next step is to convert each fraction into an equivalent fraction with the LCD as the new denominator. An equivalent fraction has the same value as the original fraction but a different numerator and denominator. To convert a fraction, you need to multiply both the numerator and the denominator by the same number. This number is determined by dividing the LCD by the original denominator.

For example, let's convert $\frac{1}{2}$ and $\frac{1}{3}$ to equivalent fractions with a denominator of 6:

• For $\frac{1}{2}$, divide the LCD (6) by the denominator (2): 6 ÷ 2 = 3. Multiply both the numerator and denominator of $\frac{1}{2}$ by 3:

  $$\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}$$

• For $\frac{1}{3}$, divide the LCD (6) by the denominator (3): 6 ÷ 3 = 2. Multiply both the numerator and denominator of $\frac{1}{3}$ by 2:

  $$\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}$$

Now you have the equivalent fractions $\frac{3}{6}$ and $\frac{2}{6}$, which have the same denominator.

Step 4: Subtracting the Numerators

With the fractions having a common denominator, you can now subtract the numerators. Subtract the numerator of the second fraction from the numerator of the first fraction, keeping the denominator the same. In our example, we have:

$$\frac{3}{6} - \frac{2}{6} = \frac{3 - 2}{6} = \frac{1}{6}$$

The result is a new fraction with the difference of the numerators as the numerator and the common denominator as the denominator.

Step 5: Simplifying the Resulting Fraction (If Necessary)

The final step is to simplify the resulting fraction, if possible. Simplifying a fraction means reducing it to its lowest terms. To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and the denominator and divide both by it. If the GCF is 1, the fraction is already in its simplest form.

In our example, the fraction is $\frac{1}{6}$. The numerator (1) and the denominator (6) have no common factors other than 1, so the fraction is already in its simplest form.

Example Requiring Simplification

Let's consider another example: $\frac{8}{12}$. The GCF of 8 and 12 is 4. Divide both the numerator and the denominator by 4:

$$\frac{8 \div 4}{12 \div 4} = \frac{2}{3}$$

So, the simplified fraction is $\frac{2}{3}$.

Step 6: Dealing with Mixed Numbers (If Applicable)

If the problem involves mixed numbers, you have two main approaches: convert the mixed numbers to improper fractions or subtract the whole numbers and fractions separately. Let's explore both methods:

Method 1: Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator. For example, let's convert $2\frac{1}{4}$ to an improper fraction:

• Multiply the whole number (2) by the denominator (4): 2 x 4 = 8
• Add the numerator (1): 8 + 1 = 9
• Keep the same denominator (4): The improper fraction is $\frac{9}{4}$

After converting the mixed numbers to improper fractions, follow the steps for subtracting fractions as outlined above. Once you get the result, you can convert the improper fraction back to a mixed number if needed.

Method 2: Subtracting Whole Numbers and Fractions Separately

This method involves subtracting the whole numbers first and then subtracting the fractions. If the fraction in the second mixed number is larger than the fraction in the first mixed number, you'll need to borrow 1 from the whole number in the first mixed number and convert it to a fraction with the common denominator before subtracting. This step can be more intuitive for some learners.

Conclusion: Practicing for Proficiency

Mastering fraction subtraction requires a thorough understanding of the steps involved and consistent practice. By following this step-by-step guide and working through various examples, you can build your confidence and proficiency in subtracting fractions. Remember to always identify the fractions, find a common denominator, convert the fractions, subtract the numerators, and simplify the result. With dedication and practice, you'll be able to tackle any fraction subtraction problem with ease. The key to success lies in consistent effort and a willingness to learn from mistakes. Practice consistently to make the learning process more engaging.

#h2 Common Mistakes to Avoid When Subtracting Fractions

Subtracting fractions, while a fundamental arithmetic operation, can often be a source of errors if certain common mistakes are not avoided. Understanding these pitfalls is crucial for achieving accuracy and building a solid foundation in mathematics. This section will explore some of the most frequent errors made when subtracting fractions, providing insights and strategies to prevent them. By recognizing these potential stumbling blocks and implementing the correct techniques, you can significantly improve your ability to subtract fractions accurately and efficiently. Avoiding common mistakes is vital for mastering any mathematical operation.

Mistake 1: Failing to Find a Common Denominator

Perhaps the most prevalent error in fraction subtraction is attempting to subtract fractions that do not have a common denominator. As emphasized earlier, fractions represent parts of a whole, and you cannot directly subtract parts of different wholes. You must first express the fractions with a common denominator, which serves as the common unit of measurement. For instance, trying to subtract $\frac{1}{3}$ from $\frac{1}{2}$ without finding a common denominator would lead to an incorrect answer.

Why This Mistake Occurs

This mistake often stems from a misunderstanding of the fundamental concept of fractions and their representation. Students may focus solely on the numerators without considering the denominators, leading them to perform the subtraction incorrectly.

How to Avoid This Mistake

Always ensure that the fractions have a common denominator before subtracting. Review the steps for finding the least common denominator (LCD) if needed. This is the cornerstone of accurate fraction subtraction. Finding a common denominator is a non-negotiable step.

Mistake 2: Subtracting Denominators as Well as Numerators

Another common error is incorrectly subtracting the denominators along with the numerators. When subtracting fractions with a common denominator, you subtract only the numerators; the denominator remains the same. Subtracting the denominators would change the size of the parts you are working with, leading to an incorrect result.

Why This Mistake Occurs

This mistake can arise from a misunderstanding of how fraction subtraction works. Students may mistakenly apply the subtraction operation to both the numerators and denominators, rather than recognizing that the denominator represents the size of the parts and should not be altered during subtraction.

How to Avoid This Mistake

Remember that when subtracting fractions with a common denominator, you only subtract the numerators. The denominator remains the same. This reinforces the idea that the denominator represents the common unit of measurement. Focus on subtracting numerators while keeping denominators constant.

Mistake 3: Not Simplifying the Resulting Fraction

While subtracting the fractions correctly is essential, the job isn't complete until the resulting fraction is simplified to its lowest terms. Failing to simplify the fraction means the answer is not in its most concise and understandable form. This can also lead to issues in subsequent calculations or when comparing fractions.

Why This Mistake Occurs

Sometimes, students may overlook the simplification step due to time constraints or a lack of understanding of the simplification process. They might be satisfied with obtaining a fraction as an answer without recognizing the need to reduce it further.

How to Avoid This Mistake

Always check if the resulting fraction can be simplified. Find the greatest common factor (GCF) of the numerator and denominator and divide both by it. This ensures the fraction is in its simplest form. Simplifying fractions is the final touch of accuracy.

Mistake 4: Incorrectly Handling Mixed Numbers

Subtracting mixed numbers introduces additional complexities, and errors can occur if they are not handled correctly. One common mistake is subtracting the whole numbers and fractions separately without proper borrowing or conversion to improper fractions.

Why This Mistake Occurs

This mistake can stem from a misunderstanding of how mixed numbers represent a combination of whole numbers and fractions. Students may not realize the need to borrow from the whole number if the fraction being subtracted is larger or may make errors in the borrowing process.

How to Avoid This Mistake

When subtracting mixed numbers, either convert them to improper fractions before subtracting or subtract the whole numbers and fractions separately, ensuring proper borrowing when necessary. Choose the method you're most comfortable with and practice it consistently. Mastering mixed number subtraction is a crucial skill.

Mistake 5: Making Arithmetic Errors in Basic Operations

Sometimes, errors in fraction subtraction are not due to a misunderstanding of fraction concepts but rather due to simple arithmetic mistakes in basic operations like multiplication, division, addition, or subtraction. These errors can occur when finding common denominators, converting fractions, or simplifying results.

Why This Mistake Occurs

Arithmetic errors can happen due to carelessness, time pressure, or a lack of fluency in basic arithmetic facts. They can derail the entire process of fraction subtraction, leading to an incorrect answer.

How to Avoid This Mistake

Double-check your calculations at each step of the process. Practice basic arithmetic facts to improve your speed and accuracy. Careful attention to detail is key. Double-check calculations to minimize errors.

Conclusion: Precision and Practice are Key

Avoiding these common mistakes is essential for mastering fraction subtraction. By understanding the potential pitfalls and implementing the strategies to prevent them, you can significantly enhance your accuracy and confidence in working with fractions. Remember that precision and practice are key. Take your time, double-check your work, and consistently practice fraction subtraction to solidify your understanding and skills. With diligence, you'll be able to overcome these challenges and excel in fraction arithmetic. The journey to mastery is paved with careful attention and persistent effort. Consistent practice leads to mastery.

#h2 Practice Problems: Sharpening Your Fraction Subtraction Skills

To truly sharpen your fraction subtraction skills, consistent practice is paramount. Working through a variety of problems helps you solidify your understanding of the concepts, identify areas where you might need further clarification, and build confidence in your abilities. This section provides a set of practice problems covering various scenarios, including fractions with common denominators, fractions with different denominators, and mixed numbers. By diligently working through these problems, you'll reinforce your knowledge and develop the fluency needed to tackle any fraction subtraction challenge.

Practice Problems

Here are some practice problems to test your understanding of subtracting fractions. Work through each problem carefully, showing your steps, and simplify your answers whenever possible.

1. $\frac{7}{8} - \frac{3}{8} = ?$
2. $\frac{5}{6} - \frac{1}{3} = ?$
3. $\frac{9}{10} - \frac{2}{5} = ?$
4. $\frac{1}{2} - \frac{3}{8} = ?$
5. $\frac{11}{12} - \frac{1}{4} = ?$
6. $3\frac{1}{2} - 1\frac{1}{4} = ?$
7. $5\frac{2}{3} - 2\frac{1}{3} = ?$
8. $4\frac{3}{4} - 1\frac{1}{2} = ?$
9. $6\frac{1}{5} - 2\frac{3}{10} = ?$
10. $8\frac{5}{8} - 3\frac{1}{4} = ?$

Detailed Solutions

Now, let's work through each problem step-by-step, providing detailed solutions and explanations.

Problem 1: $\frac{7}{8} - \frac{3}{8} = ?$

Solution:

Since the fractions have a common denominator of 8, we can directly subtract the numerators:

$$\frac{7}{8} - \frac{3}{8} = \frac{7 - 3}{8} = \frac{4}{8}$$

Now, simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 4:

$$\frac{4 \div 4}{8 \div 4} = \frac{1}{2}$$

Answer: $\frac{1}{2}$

Problem 2: $\frac{5}{6} - \frac{1}{3} = ?$

Solution:

The fractions have different denominators, so we need to find a common denominator. The least common multiple (LCM) of 6 and 3 is 6. Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of 6:

$$\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}$$

Now, subtract the fractions:

$$\frac{5}{6} - \frac{2}{6} = \frac{5 - 2}{6} = \frac{3}{6}$$

Simplify the fraction by dividing both the numerator and denominator by their GCF, which is 3:

$$\frac{3 \div 3}{6 \div 3} = \frac{1}{2}$$

Answer: $\frac{1}{2}$

Problem 3: $\frac{9}{10} - \frac{2}{5} = ?$

Solution:

The fractions have different denominators, so we need to find a common denominator. The LCM of 10 and 5 is 10. Convert $\frac{2}{5}$ to an equivalent fraction with a denominator of 10:

$$\frac{2}{5} \times \frac{2}{2} = \frac{4}{10}$$

Now, subtract the fractions:

$$\frac{9}{10} - \frac{4}{10} = \frac{9 - 4}{10} = \frac{5}{10}$$

Simplify the fraction by dividing both the numerator and denominator by their GCF, which is 5:

$$\frac{5 \div 5}{10 \div 5} = \frac{1}{2}$$

Answer: $\frac{1}{2}$

Problem 4: $\frac{1}{2} - \frac{3}{8} = ?$

Solution:

The fractions have different denominators, so we need to find a common denominator. The LCM of 2 and 8 is 8. Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8:

$$\frac{1}{2} \times \frac{4}{4} = \frac{4}{8}$$

Now, subtract the fractions:

$$\frac{4}{8} - \frac{3}{8} = \frac{4 - 3}{8} = \frac{1}{8}$$

Answer: $\frac{1}{8}$ (The fraction is already in its simplest form.)

Problem 5: $\frac{11}{12} - \frac{1}{4} = ?$

Solution:

The fractions have different denominators, so we need to find a common denominator. The LCM of 12 and 4 is 12. Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of 12:

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

Now, subtract the fractions:

$$\frac{11}{12} - \frac{3}{12} = \frac{11 - 3}{12} = \frac{8}{12}$$

Simplify the fraction by dividing both the numerator and denominator by their GCF, which is 4:

$$\frac{8 \div 4}{12 \div 4} = \frac{2}{3}$$

Answer: $\frac{2}{3}$

Problem 6: $3\frac{1}{2} - 1\frac{1}{4} = ?$

Solution:

Method 1: Convert mixed numbers to improper fractions:

• $3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$
• $1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{5}{4}$

Now, subtract the improper fractions. The LCM of 2 and 4 is 4. Convert $\frac{7}{2}$ to an equivalent fraction with a denominator of 4:

$$\frac{7}{2} \times \frac{2}{2} = \frac{14}{4}$$

Subtract the fractions:

$$\frac{14}{4} - \frac{5}{4} = \frac{14 - 5}{4} = \frac{9}{4}$$

Convert the improper fraction back to a mixed number:

$$\frac{9}{4} = 2\frac{1}{4}$$

Method 2: Subtract whole numbers and fractions separately:

$$3\frac{1}{2} - 1\frac{1}{4}$$

Subtract the whole numbers:

$$3 - 1 = 2$$

Subtract the fractions. Find a common denominator (LCM of 2 and 4 is 4):

$$\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}$$

Combine the results:

$$2 + \frac{1}{4} = 2\frac{1}{4}$$

Answer: $2\frac{1}{4}$

Problem 7: $5\frac{2}{3} - 2\frac{1}{3} = ?$

Solution:

Since the fractions have a common denominator, we can subtract the whole numbers and fractions separately:

Subtract the whole numbers:

$$5 - 2 = 3$$

Subtract the fractions:

$$\frac{2}{3} - \frac{1}{3} = \frac{2 - 1}{3} = \frac{1}{3}$$

Combine the results:

$$3 + \frac{1}{3} = 3\frac{1}{3}$$

Answer: $3\frac{1}{3}$

Problem 8: $4\frac{3}{4} - 1\frac{1}{2} = ?$

Solution:

Method 1: Convert mixed numbers to improper fractions:

• $4\frac{3}{4} = \frac{(4 \times 4) + 3}{4} = \frac{19}{4}$
• $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$

Now, subtract the improper fractions. The LCM of 4 and 2 is 4. Convert $\frac{3}{2}$ to an equivalent fraction with a denominator of 4:

$$\frac{3}{2} \times \frac{2}{2} = \frac{6}{4}$$

Subtract the fractions:

$$\frac{19}{4} - \frac{6}{4} = \frac{19 - 6}{4} = \frac{13}{4}$$

Convert the improper fraction back to a mixed number:

$$\frac{13}{4} = 3\frac{1}{4}$$

Method 2: Subtract whole numbers and fractions separately:

$$4\frac{3}{4} - 1\frac{1}{2}$$

Subtract the whole numbers:

$$4 - 1 = 3$$

Subtract the fractions. Find a common denominator (LCM of 4 and 2 is 4):

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

Combine the results:

$$3 + \frac{1}{4} = 3\frac{1}{4}$$

Answer: $3\frac{1}{4}$

Problem 9: $6\frac{1}{5} - 2\frac{3}{10} = ?$

Solution:

Method 1: Convert mixed numbers to improper fractions:

• $6\frac{1}{5} = \frac{(6 \times 5) + 1}{5} = \frac{31}{5}$
• $2\frac{3}{10} = \frac{(2 \times 10) + 3}{10} = \frac{23}{10}$

Now, subtract the improper fractions. The LCM of 5 and 10 is 10. Convert $\frac{31}{5}$ to an equivalent fraction with a denominator of 10:

$$\frac{31}{5} \times \frac{2}{2} = \frac{62}{10}$$

Subtract the fractions:

$$\frac{62}{10} - \frac{23}{10} = \frac{62 - 23}{10} = \frac{39}{10}$$

Convert the improper fraction back to a mixed number:

$$\frac{39}{10} = 3\frac{9}{10}$$

Method 2: Subtract whole numbers and fractions separately:

$$6\frac{1}{5} - 2\frac{3}{10}$$

Subtract the whole numbers:

$$6 - 2 = 4$$

Subtract the fractions. Find a common denominator (LCM of 5 and 10 is 10):

$$\frac{1}{5} - \frac{3}{10} = \frac{2}{10} - \frac{3}{10}$$

Since $\frac{2}{10}$ is smaller than $\frac{3}{10}$, we need to borrow 1 from the whole number (4), convert it to $\frac{10}{10}$, and add it to $\frac{2}{10}$:

$$4 \rightarrow 3 + 1 = 3 + \frac{10}{10}$$

So, we have:

$$3 + \frac{10}{10} + \frac{2}{10} - \frac{3}{10} = 3 + \frac{12}{10} - \frac{3}{10} = 3 + \frac{9}{10} = 3\frac{9}{10}$$

Answer: $3\frac{9}{10}$

Problem 10: $8\frac{5}{8} - 3\frac{1}{4} = ?$

Solution:

Method 1: Convert mixed numbers to improper fractions:

• $8\frac{5}{8} = \frac{(8 \times 8) + 5}{8} = \frac{69}{8}$
• $3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}$

Now, subtract the improper fractions. The LCM of 8 and 4 is 8. Convert $\frac{13}{4}$ to an equivalent fraction with a denominator of 8:

$$\frac{13}{4} \times \frac{2}{2} = \frac{26}{8}$$

Subtract the fractions:

$$\frac{69}{8} - \frac{26}{8} = \frac{69 - 26}{8} = \frac{43}{8}$$

Convert the improper fraction back to a mixed number:

$$\frac{43}{8} = 5\frac{3}{8}$$

Method 2: Subtract whole numbers and fractions separately:

$$8\frac{5}{8} - 3\frac{1}{4}$$

Subtract the whole numbers:

$$8 - 3 = 5$$

Subtract the fractions. Find a common denominator (LCM of 8 and 4 is 8):

$$\frac{5}{8} - \frac{1}{4} = \frac{5}{8} - \frac{2}{8} = \frac{3}{8}$$

Combine the results:

$$5 + \frac{3}{8} = 5\frac{3}{8}$$

Answer: $5\frac{3}{8}$

Conclusion: Consistent Practice for Mastery

These practice problems and their detailed solutions provide a comprehensive resource for honing your fraction subtraction skills. By working through these exercises diligently and reviewing the explanations, you can identify areas where you excel and areas that require further attention. Remember, the key to mastering fraction subtraction is consistent practice and a commitment to understanding the underlying concepts. The more you practice, the more confident and proficient you will become in handling any fraction subtraction challenge. Consistent effort transforms skill into mastery.

#repair-input-keyword How many more yards of blue fabric did Janet use than yellow fabric? Write your answer as a fraction in simplest form.

#title How Many More Yards of Blue Fabric Did Janet Use? A Fraction Problem