Problem 47 - Prealgebra

Question

Find the LCD of each pair of fractions. (lesson \(5-6\) ) $$\frac{5}{8}, \frac{3}{14}$$

Step-by-Step Solution

Answer

The LCD of \( \frac{5}{8} \) and \( \frac{3}{14} \) is 56.

Step 1: Identify the denominators

The given pair of fractions are \( \frac{5}{8} \) and \( \frac{3}{14} \). The denominators in these fractions are \( 8 \) and \( 14 \).

Step 2: Factor each denominator

Factor the numbers \( 8 \) and \( 14 \) into their prime factors.\(8 = 2^3 \) and \( 14 = 2 \times 7 \).

Step 3: Determine the highest powers of all prime factors

From the factorizations, the prime factors are \( 2 \) and \( 7 \). The highest power of \( 2 \) is \( 2^3 \) (from 8), and the highest power of \( 7 \) is \( 7^1 \) (from 14).

Step 4: Calculate the Least Common Denominator (LCD)

Multiply the highest powers of all prime factors to find the LCD: \[ 2^3 \times 7^1 = 8 \times 7 = 56 \] Thus, the LCD of \( \frac{5}{8} \) and \( \frac{3}{14} \) is 56.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest components, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To factor a number, we divide it by the smallest prime number and continue the process with the quotient until only prime numbers are left.
For example:

• The prime factorization of 8 is given by: \( 8 = 2^3 \). Here, the only prime number used is 2, and it is raised to the power of 3, because 2 is multiplied by itself three times.
• The prime factorization of 14 is \( 14 = 2 \times 7 \). It involves the prime numbers 2 and 7.

The prime factors are used to find commonalities between denominators in fractions, helping us determine the Least Common Denominator (LCD).
When you know how to break down numbers into their prime factors, it becomes easier to manage and compare multiple fractions by finding a common base, such as the LCD.

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator. The numerator indicates how many parts are being considered, while the denominator denotes the total number of equal parts.
For instance, in the fraction \( \frac{5}{8} \):

• The numerator is 5, meaning we are considering 5 parts.
• The denominator is 8, indicating there are a total of 8 equal parts.

Fractions are used to represent numbers that are not whole and are a critical component in math computation.
To work effectively with multiple fractions, especially when adding or comparing them, it is often necessary to convert them to have the same denominator. This is where finding the Least Common Denominator comes into play, allowing fractions to be precisely manipulated by aligning their base measurements, or denominators.

Denominators

Denominators are an essential component of fractions, as they specify the total number of equal parts that make up the whole. Understanding and manipulating denominators is vital, especially when dealing with operations involving multiple fractions.
Take, for example, the fractions \( \frac{5}{8} \) and \( \frac{3}{14} \):

• The denominator 8 shows that the whole is divided into 8 equal parts.
• Similarly, the denominator 14 shows the whole divided into 14 equal parts.

When performing operations like addition or comparison, these denominators need to be unified.
This unification is achieved through the calculation of the Least Common Denominator (LCD). By finding the smallest number that both denominators can evenly divide into, we simplify the process of fraction comparison or calculation, making complex arithmetic operations easier to manage.