Problem 81 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in lowest terms. (Skills Review p. 764) $$\frac{1}{8}+\frac{1}{5}$$

Step-by-Step Solution

Answer

The sum of the fractions is \(\frac{13}{40}\).

Step 1: Find the Least Common Denominator (LCD)

To solve this problem, first we need to find the least common denominator (LCD) for both fractions. The LCD of 8 and 5 is 40.

Step 2: Equivalent Fractions

Convert each fraction into an equivalent fraction with denominator 40. Multiply the numerator and the denominator of each fraction by the factor that makes the denominator equal to 40. The equivalent fraction for \(\frac{1}{8}\) is \(\frac{1 * 5}{8 * 5} = \frac{5}{40}\) and for \(\frac{1}{5}\) is \(\frac{1*8}{5*8} = \frac{8}{40}\).

Step 3: Add Fractions

Now, we add the two fractions \(\frac{5}{40} + \frac{8}{40}\). This gives us \(\frac{13}{40}\).

Step 4: Simplify the fraction

Next, we need to simplify the fraction if possible. However, 13 and 40 do not have any common factors (except 1), so the fraction \(\frac{13}{40}\) is already in its lowest terms.

Key Concepts

Least Common Denominator

When adding fractions, finding the Least Common Denominator (LCD) is crucial. The LCD allows us to convert all fractions involved into equivalent fractions with a common denominator, making it possible to add or subtract them easily. To find the LCD of two or more fractions:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in all lists.

In our example, the fractions are \( \frac{1}{8} \) and \( \frac{1}{5} \). The denominators are 8 and 5. We list the multiples:

• Multiples of 8: 8, 16, 24, 32, 40, ...
• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...

The smallest common multiple is 40, which is our LCD. This means both fractions can be expressed with a denominator of 40, facilitating their addition.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. To convert a fraction to an equivalent one, multiply the numerator and the denominator by the same non-zero number. This keeps the fraction's value unchanged.In our problem, we converted both fractions to have the LCD of 40:

• For \( \frac{1}{8} \): Multiply by \( \frac{5}{5} \) to get \( \frac{5}{40} \).
• For \( \frac{1}{5} \): Multiply by \( \frac{8}{8} \) to get \( \frac{8}{40} \).

By converting both fractions to \( \frac{5}{40} \) and \( \frac{8}{40} \), they become easy to add since they now have a common denominator. This common denominator is vital for fraction addition or subtraction to be performed properly.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form, where the numerator and denominator are as small as possible. This is done by dividing both the numerator and the denominator by their greatest common factor (GCF).After adding the fractions \( \frac{5}{40} + \frac{8}{40} \), we get \( \frac{13}{40} \). To simplify, we check if 13 and 40 have any common factors:

• The factors of 13 are 1 and 13 (since 13 is a prime number).
• The factors of 40 include 1, 2, 4, 5, 8, 10, 20, and 40.

The only common factor is 1, meaning \( \frac{13}{40} \) is already in its simplest form. Simplification is a good practice, especially to ensure the fraction is presented as neatly as possible without any reducible parts.