Problem 61 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of fractions. $$\frac{14}{54}, \frac{31}{81}$$

Step-by-Step Solution

Answer

The least common denominator of the given pair of fractions is 162.

Step 1: Identify the Fraction Denominators

Look at the fraction and identify the denominators, which are the bottom numbers in each fraction. In this case, the denominators are 54 and 81.

Step 2: Factor the Denominators

Now, factor each denominator into a product of prime numbers. The prime factorization of 54 is \(2 \times 3^3\) and 81 is \(3^4\).

Step 3: Identify the Common Factors

Identify the common factors in the factorization of both numbers. In this case, the common factorization is \(3^3\).

Step 4: Identify the Uncommon Factors

Identify the factors present in either denominator but not in both. In this case, the factors are 2 and \(3^1\).

Step 5: Calculate the LCD

The LCD is found by multiplying the common factors and any uncommon factors together. So the LCD in this case is \(2 \times 3^4 = 162\)

Key Concepts

Fraction Denominators

Understanding fraction denominators is fundamental when working with different fractions, especially when adding, subtracting, or comparing them. The denominator of a fraction tells us into how many parts the whole is divided. For example, in the fractions \(\frac{14}{54}\) and \(\frac{31}{81}\), 54 and 81 are the denominators.

In mathematics, when dealing with more than one fraction, having a common denominator simplifies the process, as it allows you to work with fractions directly without converting them. To combine fractions through addition or subtraction or to compare fractions, they must have the same denominator, which we call the least common denominator (LCD). The LCD is the smallest number that is a common multiple of both denominators. After finding the LCD, the fractions are converted so that they both have this common denominator, enabling calculations between them.

Prime Factorization

Prime factorization is a technique used to express a number as a product of its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. When we break down a composite number into a product of prime numbers, we have found its prime factorization.

For example, in the step-by-step solution provided, the denominators 54 and 81 can be broken down into their prime factors, \(2 \times 3^3\) and \(3^4\) respectively. This prime factorization is crucial when determining the least common denominator of two fractions, as it highlights both common and unique factors so that we can calculate the LCD efficiently. Remember that identifying prime factors needs to be done meticulously to ensure that we are working with the most basic building blocks of the numbers, facilitating the finding of their least common multiple.

Common Factors

Identifying common factors between two or more numbers is a key step in finding the least common denominator for fractions. A common factor is a factor that two or more numbers have in common. In prime factorization, these would be the prime numbers that appear in the factorization of each of the numbers under consideration.

In our case, the number \(3^3\) is a common factor of both 54 and 81. When finding the LCD, it's not only the shared factors that are important, but also the powers of those factors. The highest power of any common prime factors should be used in the calculation of the LCD. Of course, if there are factors in one number that are not in the factorization of the other, these too must be included in the LCD calculation to ensure that the number we derive is indeed a multiple of both original numbers.