Problem 72 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ 7+\frac{1}{10} $$

Step-by-Step Solution

Answer

\(7 + \frac{1}{10} = \frac{71}{10}\)

Step 1: Identify the Operation

We need to perform the operation of adding a whole number \(7\) and a fraction \(\frac{1}{10}\). The result should be expressed as a fraction in its simplest form.

Step 2: Convert Whole Number to Fraction

Convert the whole number \(7\) into a fraction by writing it as \(\frac{7}{1}\). This allows us to work with both terms as fractions.

Step 3: Find a Common Denominator

Identify the least common denominator (LCD) for the fractions \(\frac{7}{1}\) and \(\frac{1}{10}\). The LCD is 10.

Step 4: Convert Fractions to Equivalent Fractions

Convert \(\frac{7}{1}\) to an equivalent fraction with a denominator of 10 by multiplying both the numerator and denominator by 10: \[\frac{7}{1} = \frac{70}{10}\].

Step 5: Add the Fractions

Now that both fractions have the same denominator, add them together: \[\frac{70}{10} + \frac{1}{10} = \frac{71}{10}\].

Step 6: Simplify the Fraction

Check if \(\frac{71}{10}\) can be reduced further. Since 71 is a prime number and does not divide evenly by 10, the fraction is already in its simplest form.

Key Concepts

Addition of Fractions

When adding fractions, it is crucial to work with fractions that share the same denominator. This commonality allows you to directly add or subtract the numerators while keeping the denominator unchanged. For example, given two fractions like \( \frac{3}{5} \) and \( \frac{4}{5} \), you can simply add the numerators:

• Add the numerators: 3 + 4 = 7
• Keep the denominator: 5
• The result is: \( \frac{7}{5} \)

Notice how the denominator stays the same. This happens because the fraction's division part does not change; only the counted units—the numerators—are combined. If the denominators are different, which is often the case, you must first convert them to have a common denominator before proceeding.

Least Common Denominator

The least common denominator (LCD) is the smallest multiple that is common to the denominators of two or more fractions. Identifying the LCD is crucial because it allows for the addition or subtraction of fractions with ease. To find the LCD:

• List the multiples of each denominator.
• Find the smallest multiple they share.

For example, to add \( \frac{1}{2} \) and \( \frac{1}{3} \):

• Multiples of 2: 2, 4, 6, 8, 10...
• Multiples of 3: 3, 6, 9, 12, 15...
• The smallest common multiple is 6.

Thus, 6 is the LCD. Using the LCD ensures that both fractions are expressed with a like denominator, simplifying the process of adding or subtracting them.

Converting Whole Numbers to Fractions

Converting whole numbers into fractions is an essential skill, especially when performing operations with fractions. A whole number can be written as a fraction by using the number itself as the numerator and 1 as the denominator. For example, the whole number 5 can be written as:

• \( \frac{5}{1} \)

This conversion does not change the value; it just represents the number in a form that can easily be used in addition, subtraction, multiplication, or division of fractions.
Consider adding a whole number and a fraction, such as 3 and \( \frac{1}{4} \). The whole number 3 becomes \( \frac{3}{1} \), making it easier to find a common denominator and perform the addition once the fractions have equivalent denominators. Converting whole numbers to fractions expands the flexibility and ease of performing arithmetic operations with mixed numbers and fractions.