Question

What is the least common denominator when adding the fractions \(\frac{a}{2}, \frac{b}{3}, \frac{c}{9},\) and \(\frac{d}{15} ?\) A. 45 B. 90 C. 135 D. 270 E. 810

Short answer

Answer: B. 90

Step-by-step solution

List the Prime Factors of Each Denominator

To find the LCM, we first need to list the prime factors of each denominator: 2: 2 3: 3 9: 3^2 15: 3 x 5

Identify the Unique Prime Factors and Their Highest Powers

Now that we have the prime factors, we must determine the unique prime factors and their highest powers in the four denominators. Those are: 2: 2^1 3: 3^2 5: 5^1

Calculate the Least Common Multiple (LCM)

To find the LCM, we multiply the unique prime factors and their highest powers together: LCM = 2^1 x 3^2 x 5^1 = 2 x 9 x 5 = 90

Identify the LCD from the Answer Choices

Since the least common multiple (LCM) is 90, the least common denominator (LCD) for the fractions is 90. Therefore, the correct answer is B. 90.

Key concepts

Least Common Multiple (LCM)

Understanding the Least Common Multiple (LCM) is crucial when working with fractions, particularly in the realm of fraction addition. The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. When we add, subtract, or compare fractions, LCM helps us find a common ground, which we call the least common denominator (LCD).

Finding the LCM can seem challenging, but it's like building a puzzle; we search for pieces that fit together just right. The key to this puzzle lies in breaking the numbers down into their most basic components - prime factors. Once we have the unique prime factors and their highest powers, we multiply them together. This product is our LCM, which, for the addition of fractions, becomes the LCD, ensuring that all fractions speak the same mathematical language.

Prime Factors

Diving deeper, we encounter prime factors, the building blocks of all numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. When we factor a number into primes, we're essentially uncovering the original pieces from which the number is made.

To find a number's prime factors, we can use a factor tree, which involves breaking down a number into two factors at a time until we can break it down no more - the leaves of the tree are the prime factors. For our exercise, the denominators are broken into their prime factors. Not only does this process reveal the DNA of a number, but these factors are also used to find the least common multiple, a critical player in the game of fraction addition.

Fraction Addition

When we add fractions, they must speak the same language, and this is where the concept of the LCD becomes a game-changer. Think of it as a diplomatic meeting where every fraction must use a common language to communicate. To add fractions effectively, they must have the same denominator.

To use the LCD for fraction addition, we first convert each fraction to an equivalent fraction with the LCD. We adjust the numerators accordingly, to ensure the value of the fractions remains unchanged. Once the fractions have the same denominator, we simply add the numerators while keeping the denominator the same. The LCM of the denominators, as discovered through prime factorization, allows us to find this common denominator and bring the fractions together harmoniously. With this approach, adding fractions becomes an intuitive and systematic process.