Add The Following:$\[ 7 \frac{3}{5} + \frac{1}{4} \\]Answer As A Mixed Number In Simplest Form. Enter The Whole Number.

Understanding Mixed Numbers and Fractions

In mathematics, mixed numbers and fractions are two types of numbers that can be added together. A mixed number is a combination of a whole number and a fraction, while a fraction is a part of a whole. When adding mixed numbers and fractions, it's essential to follow a specific order of operations to ensure accurate results.

The Problem: Adding 7 3/5 and 1/4

The problem we're going to solve is:

${ 7 \frac{3}{5} + \frac{1}{4} }$

To add these two numbers, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step 1: Convert the Mixed Number to an Improper Fraction

To add the mixed number 7 3/5 to the fraction 1/4, we need to convert the mixed number to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

To convert 7 3/5 to an improper fraction, we multiply the whole number 7 by the denominator 5, and then add the numerator 3.

${ 7 \times 5 = 35 }$ ${ 35 + 3 = 38 }$

So, the improper fraction equivalent of 7 3/5 is 38/5.

Step 2: Find a Common Denominator

To add the fractions 38/5 and 1/4, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

The LCM of 5 and 4 is 20.

Step 3: Convert the Fractions to Have the Common Denominator

To convert the fractions to have the common denominator 20, we multiply the numerator and denominator of each fraction by the necessary factor.

For 38/5, we multiply the numerator and denominator by 4:

${ 38 \times 4 = 152 }$ ${ 5 \times 4 = 20 }$

So, the fraction 38/5 becomes 152/20.

For 1/4, we multiply the numerator and denominator by 5:

${ 1 \times 5 = 5 }$ ${ 4 \times 5 = 20 }$

So, the fraction 1/4 becomes 5/20.

Step 4: Add the Fractions

Now that both fractions have the common denominator 20, we can add them together.

${ 152/20 + 5/20 = 157/20 }$

Step 5: Convert the Improper Fraction to a Mixed Number

To convert the improper fraction 157/20 to a mixed number, we divide the numerator 157 by the denominator 20.

${ 157 ÷ 20 = 7 \text{ with a remainder of } 17 }$

So, the mixed number equivalent of 157/20 is 7 17/20.

The Final Answer

Therefore, the final answer to the problem is:

${ 7 \frac{3}{5} + \frac{1}{4} = 7 \frac{17}{20} }$

Conclusion

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

A: A mixed number is a combination of a whole number and a fraction, while a fraction is a part of a whole. For example, 3 1/2 is a mixed number, while 1/2 is a fraction.

Q: How do I add mixed numbers and fractions?

A: To add mixed numbers and fractions, you need to follow the order of operations, which is:

1. Convert the mixed number to an improper fraction.
2. Find a common denominator.
3. Convert the fractions to have the common denominator.
4. Add the fractions.
5. Convert the improper fraction to a mixed number.

Q: What is an improper fraction?

A: An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 3/2 is an improper fraction.

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

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. For example, to convert 3 1/2 to an improper fraction, you multiply 3 by 2 and add 1:

${ 3 \times 2 = 6 }$ ${ 6 + 1 = 7 }$

So, the improper fraction equivalent of 3 1/2 is 7/2.

Q: How do I find a common denominator?

A: To find a common denominator, you need to find the least common multiple (LCM) of the two denominators. For example, to find the common denominator of 2 and 3, you need to find the LCM of 2 and 3, which is 6.

Q: How do I convert fractions to have a common denominator?

A: To convert fractions to have a common denominator, you multiply the numerator and denominator of each fraction by the necessary factor. For example, to convert 1/2 and 1/3 to have a common denominator of 6, you multiply the numerator and denominator of each fraction by the necessary factor:

For 1/2, you multiply the numerator and denominator by 3:

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

So, the fraction 1/2 becomes 3/6.

For 1/3, you multiply the numerator and denominator by 2:

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

So, the fraction 1/3 becomes 2/6.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator and then add the fractions. For example, to add 1/2 and 1/3, you need to find a common denominator of 6 and then add the fractions:

${ 1/2 + 1/3 = 3/6 + 2/6 = 5/6 }$

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

A: To convert an improper fraction to a mixed number, you divide the numerator by the denominator and write the remainder as the new numerator. For example, to convert 5/2 to a mixed number, you divide 5 by 2 and write the remainder as the new numerator:

${ 5 ÷ 2 = 2 \text{ with a remainder of } 1 }$

So, the mixed number equivalent of 5/2 is 2 1/2.

Q: What are some common mistakes to avoid when adding mixed numbers and fractions?

A: Some common mistakes to avoid when adding mixed numbers and fractions include:

• Not following the order of operations
• Not converting mixed numbers to improper fractions
• Not finding a common denominator
• Not converting fractions to have a common denominator
• Not adding the fractions correctly

By avoiding these common mistakes, you can ensure accurate results when adding mixed numbers and fractions.