Problem 56 - Prealgebra

Question

Write two fractions whose least common denominator (LCD) is 35

Step-by-Step Solution

Answer

The fractions \( \frac{1}{5} \) and \( \frac{1}{7} \) have a least common denominator of 35.

Step 1: Understand the LCD Concept

The least common denominator (LCD) for fractions is the smallest number that is a multiple of the denominators of two or more fractions. For the exercise, we're asked to find two fractions whose LCD is 35.

Step 2: Identify the Factors of 35

35 can be broken down into its prime factors: 35 = 5 × 7. This means that any fraction with a denominator of either 5 or 7 (or a factor of 5 and 7) can potentially be used with another fraction to have an LCD of 35.

Step 3: Choose Denominators

For the two fractions, we can choose 5 and 7 as the denominators, since both numbers are factors of 35 and their least common multiple is 35.

Step 4: Write the Fractions

Select numerators for the fractions. To show the concept clearly, choose simple numerators like 1. This gives the fractions \( \frac{1}{5} \) and \( \frac{1}{7} \).

Step 5: Verify the LCD

Verify that the LCD of the chosen fractions \( \frac{1}{5} \) and \( \frac{1}{7} \) is 35. The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, ... and for 7 are 7, 14, 21, 28, 35, ... The smallest common multiple is 35.

Key Concepts

Fractions

Fractions are essential parts of everyday mathematics and represent a portion or part of a whole. A fraction consists of two numbers separated by a line: the numerator and the denominator.

• The numerator is the top number and indicates how many parts we have.
• The denominator is the bottom number that shows how many equal parts the whole is divided into.

For example, in the fraction \( \frac{1}{5} \), 1 is the numerator, and 5 is the denominator, indicating that the whole is divided into 5 equal parts, and we have 1 part of it. Whenever working with fractions, especially when comparing or adding them, it's important to understand their denominators. This is where the least common denominator comes into play. By finding the least common denominator, we can perform operations with fractions more smoothly.

Prime Factors

Prime factors are the building blocks of numbers. They are prime numbers that exactly divide a given number without leaving a remainder. To break down a number into its prime factors, you need to divide it by the smallest possible prime number and continue the process for the quotient. For example, the number 35 can be broken down as follows:

• Start with 2, which doesn't divide 35.
• Next try 3, still no luck.
• Then 5: well, 35 divided by 5 gives us 7. Now, 5 and 7 are both primes.

Thus, the prime factors of 35 are 5 and 7. Using prime factors, we can easily find the least common multiple and the least common denominator, which helps in solving fraction problems.

Least Common Multiple (LCM)

The least common multiple is a crucial concept when working with fractions, particularly when looking for a common denominator. The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. Finding the LCM involves using the prime factorization method:

• First, find the prime factors of each number involved.
• Then, select the highest power of each prime number present in the factorization.
• Multiply these together to get the LCM.

For example, the LCM of 5 and 7 is 35 because both are prime numbers, so they multiply directly to 35.

The idea of the LCM is vital because it ensures that when we add or subtract fractions, we are working with terms that have a common ground. By converting fractions to have the same denominator, arithmetic becomes straightforward, enhancing simplicity in calculations.