Problem 48 - Prealgebra

Question

Find the LCD of each pair of fractions. (lesson \(5-6\) ) $$\frac{3}{15 t^{\prime}}, \frac{2}{5 t}$$

Step-by-Step Solution

Answer

The LCD of \(\frac{3}{15t'}\) and \(\frac{2}{5t}\) is \(15t't\).

Step 1: Identify Denominators

First, identify the denominators of the fractions given in the problem. The denominators are \(15t'\) for the first fraction and \(5t\) for the second fraction.

Step 2: Factor Each Denominator

Next, factor each denominator into its prime factors. For \(15t'\), the factors are \(3 \times 5 \times t'\). For \(5t\), the factors are \(5 \times t\).

Step 3: List All Unique Factors

List all unique prime factors from the denominators noted in the previous step. The unique factors are \(3\), \(5\), \(t'\), and \(t\).

Step 4: Form the LCD

Construct the Least Common Denominator (LCD) by taking the highest power of each unique factor identified. The LCD is \(3 \times 5 \times t' \times t\), which simplifies to \(15t't\).

Step 5: Verify the LCD

Ensure each denominator divides the LCD without leaving a remainder. Both \(15t'\) and \(5t\) divide \(15t't\) evenly.

Key Concepts

Prime Factorization

Prime factorization is the process of rewriting a number by breaking it down into its prime numbers. Prime numbers are numbers greater than 1 that can only be divided without a remainder by 1 and themselves, such as 2, 3, 5, 7, and so on. For example, the number 15 can be expressed as a product of prime numbers: \(15 = 3 \times 5\).
Similarly, when working with algebraic terms like \(t'\) and \(t\), we consider these variables part of the expression's factorization, even though they are not numerical primes. Breaking down a fraction's denominator into its prime factors is a crucial step in finding the least common denominator (LCD), because it allows us to clearly see what factors are involved. This makes it easier to combine and work with fractions that have different denominators.

Unique Factors

Unique factors are those prime factors that appear in the factorization of each denominator without repetition. When determining the least common denominator, it is important to consider all the unique prime factors present in any of the denominators.
For the original exercise, the denominators factored into \(3 \times 5 \times t'\) and \(5 \times t\). The unique factors, taking each distinct element from these, are \(3\), \(5\), \(t'\), and \(t\).
Considering each unique factor ensures that the LCD will be able to accommodate all individual parts of each denominator, which means the LCD will accurately allow the fractions to be added or subtracted.

Fractions

Fractions represent parts of a whole and consist of two pieces: the numerator (top number) and the denominator (bottom number). The numerator indicates how many parts we are considering, while the denominator indicates how many equal parts make up a whole.
In the context of the exercise, the fractions given are \(\frac{3}{15t'}\) and \(\frac{2}{5t}\).
To perform operations like addition and subtraction with fractions that have different denominators, it's crucial to find the least common denominator. This allows you to rewrite fractions in an equivalent form that facilitates direct operations, much like changing currency into the same denomination for easy comparison and calculation.

Denominators

Denominators are the component of a fraction found beneath the line, representing the total number of equal parts that make up a whole. In the exercise, the denominators identified are \(15t'\) and \(5t\).
Understanding denominators is essential for manipulating fractions. When fractions have different denominators, it is not possible to directly add or subtract them until they share a common denominator. This is why we compute the least common denominator.

• Common Denominator: A number that is a multiple of each original denominator.
• Least Common Denominator (LCD): The smallest possible common denominator using the highest power of each unique factor present in the denominators.

With the LCD determined to be \(15t't\), the fractions can be adjusted to join smoothly under the same denominator, allowing straightforward computation.