Simplify The Expression: ${ 13 \frac{7}{8} - 3 \frac{3}{8} }$

$$

Introduction to Mixed Numbers

When dealing with mixed numbers, it's essential to understand the concept of adding and subtracting fractions. A mixed number is a combination of a whole number and a fraction. In this case, we have two mixed numbers: $13 \frac{7}{8}$ and $3 \frac{3}{8}$. To simplify the expression, we need to find a common denominator for both fractions and then perform the subtraction.

Understanding the Concept of Common Denominator

A common denominator is the smallest multiple that both denominators can divide into evenly. In this case, the denominators are 8 and 8, which means we can use 8 as the common denominator. To do this, we need to convert both mixed numbers into improper fractions.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we need to multiply the whole number by the denominator and then add the numerator. For the first mixed number, $13 \frac{7}{8}$, we multiply 13 by 8 and add 7, which gives us 105 + 7 = 112. So, the improper fraction for $13 \frac{7}{8}$ is $\frac{112}{8}$. Similarly, for the second mixed number, $3 \frac{3}{8}$, we multiply 3 by 8 and add 3, which gives us 24 + 3 = 27. So, the improper fraction for $3 \frac{3}{8}$ is $\frac{27}{8}$.

Finding the Common Denominator

As mentioned earlier, the common denominator for both fractions is 8. Now that we have the improper fractions, we can rewrite the expression as $\frac{112}{8} - \frac{27}{8}$.

Subtracting Fractions

To subtract fractions, we need to have the same denominator. In this case, we already have the same denominator, which is 8. Now, we can subtract the numerators: 112 - 27 = 85. So, the result of the subtraction is $\frac{85}{8}$.

Converting the Result to a Mixed Number

To convert the improper fraction $\frac{85}{8}$ to a mixed number, we need to divide the numerator by the denominator. 85 ÷ 8 = 10 with a remainder of 5. So, the mixed number equivalent of $\frac{85}{8}$ is $10 \frac{5}{8}$.

Conclusion

In conclusion, to simplify the expression $13 \frac{7}{8} - 3 \frac{3}{8}$, we need to find a common denominator, convert the mixed numbers to improper fractions, subtract the fractions, and then convert the result back to a mixed number. The final result is $10 \frac{5}{8}$.

Tips and Tricks

• When dealing with mixed numbers, it's essential to find a common denominator before performing addition or subtraction.
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
• To subtract fractions, have the same denominator and subtract the numerators.
• To convert an improper fraction to a mixed number, divide the numerator by the denominator and write the remainder as the new numerator.

Real-World Applications

Mixed numbers are used in various real-world applications, such as:

• Measuring ingredients in cooking and baking
• Calculating distances and speeds in physics and engineering
• Determining areas and volumes in geometry and architecture
• Representing time and dates in scheduling and planning

Common Mistakes to Avoid

• Not finding a common denominator before performing addition or subtraction
• Not converting mixed numbers to improper fractions before performing operations
• Not subtracting fractions with the same denominator
• Not converting improper fractions to mixed numbers when necessary

Practice Problems

1. Simplify the expression: $15 \frac{2}{3} - 7 \frac{1}{3}$
2. Convert the mixed number $4 \frac{5}{6}$ to an improper fraction.
3. Subtract the fractions: $\frac{23}{6} - \frac{17}{6}$
4. Convert the improper fraction $\frac{37}{4}$ to a mixed number.

Solutions to Practice Problems

1. To simplify the expression, we need to find a common denominator, convert the mixed numbers to improper fractions, subtract the fractions, and then convert the result back to a mixed number. The final result is $8 \frac{1}{6}$.
2. To convert the mixed number $4 \frac{5}{6}$ to an improper fraction, we multiply 4 by 6 and add 5, which gives us 24 + 5 = 29. So, the improper fraction for $4 \frac{5}{6}$ is $\frac{29}{6}$.
3. To subtract the fractions, we need to have the same denominator. In this case, we already have the same denominator, which is 6. Now, we can subtract the numerators: 23 - 17 = 6. So, the result of the subtraction is $\frac{6}{6}$, which simplifies to 1.
4. To convert the improper fraction $\frac{37}{4}$ to a mixed number, we need to divide the numerator by the denominator. 37 ÷ 4 = 9 with a remainder of 1. So, the mixed number equivalent of $\frac{37}{4}$ is $9 \frac{1}{4}$.
   

$$

Q: What is the first step in simplifying the expression $13 \frac{7}{8} - 3 \frac{3}{8}$?

A: The first step is to find a common denominator for both fractions. In this case, the denominators are 8 and 8, which means we can use 8 as the common denominator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you need to multiply the whole number by the denominator and add the numerator. For example, to convert $13 \frac{7}{8}$ to an improper fraction, you would multiply 13 by 8 and add 7, which gives you 105 + 7 = 112. So, the improper fraction for $13 \frac{7}{8}$ is $\frac{112}{8}$.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: How do I subtract fractions with the same denominator?

A: To subtract fractions with the same denominator, you simply subtract the numerators. For example, to subtract $\frac{112}{8}$ and $\frac{27}{8}$, you would subtract 112 - 27 = 85. So, the result of the subtraction is $\frac{85}{8}$.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the remainder as the new numerator. For example, to convert $\frac{85}{8}$ to a mixed number, you would divide 85 by 8, which gives you 10 with a remainder of 5. So, the mixed number equivalent of $\frac{85}{8}$ is $10 \frac{5}{8}$.

Q: What are some real-world applications of mixed numbers?

A: Mixed numbers are used in various real-world applications, such as measuring ingredients in cooking and baking, calculating distances and speeds in physics and engineering, determining areas and volumes in geometry and architecture, and representing time and dates in scheduling and planning.

Q: What are some common mistakes to avoid when working with mixed numbers?

A: Some common mistakes to avoid when working with mixed numbers include not finding a common denominator before performing addition or subtraction, not converting mixed numbers to improper fractions before performing operations, not subtracting fractions with the same denominator, and not converting improper fractions to mixed numbers when necessary.

Q: How can I practice working with mixed numbers?

A: You can practice working with mixed numbers by trying the following problems:

• Simplify the expression: $15 \frac{2}{3} - 7 \frac{1}{3}$
• Convert the mixed number $4 \frac{5}{6}$ to an improper fraction.
• Subtract the fractions: $\frac{23}{6} - \frac{17}{6}$
• Convert the improper fraction $\frac{37}{4}$ to a mixed number.

Q: What are some tips for simplifying expressions with mixed numbers?

A: Some tips for simplifying expressions with mixed numbers include:

• Finding a common denominator before performing addition or subtraction
• Converting mixed numbers to improper fractions before performing operations
• Subtracting fractions with the same denominator
• Converting improper fractions to mixed numbers when necessary

Q: How can I use technology to help me work with mixed numbers?

A: You can use technology, such as calculators or computer software, to help you work with mixed numbers. For example, you can use a calculator to convert a mixed number to an improper fraction or to subtract fractions with the same denominator.

Q: What are some resources for learning more about mixed numbers?

A: Some resources for learning more about mixed numbers include:

• Online tutorials and videos
• Math textbooks and workbooks
• Online math communities and forums
• Math apps and software

Q: How can I apply what I've learned about mixed numbers to real-world problems?

A: You can apply what you've learned about mixed numbers to real-world problems by using mixed numbers to measure ingredients in cooking and baking, calculate distances and speeds in physics and engineering, determine areas and volumes in geometry and architecture, and represent time and dates in scheduling and planning.