Problem 47 - Algebra 1: Concepts and Skills

Question

Find the least common denominator (LCD) of each pair of fractions. Then rewrite each pair with their LCD. (Skills Review p.762). $$ \frac{5}{7}, \frac{2}{3} $$

Step-by-Step Solution

Answer

The LCD for \( \frac{5}{7} \) and \( \frac{2}{3} \) is 21. \( \frac{5}{7} \) can be rewritten as \( \frac{15}{21} \) and \( \frac{2}{3} \) as \( \frac{14}{21} \).

Step 1: Determine LCD

First, find least common multiple (LCM) of the denominators (7 and 3). The easiest way to do that is to enumerate multiples of the larger number until a multiple of the smaller number is found. So, multiples of 7 are: 7, 14, 21. Among these, 21 is a multiple of 3 as well. Hence, the LCM of 7 and 3 is 21, so the least common denominator (LCD) is 21.

Step 2: Rewrite fractions with LCD

Next, rewrite the fractions to have the LCD. A fraction is equivalent to another if the numerator and denominator are multiplied by the same number. So multiply the numerator and denominator of \( \frac{5}{7} \) by 3 and of \( \frac{2}{3} \) by 7 to get them with LCD. Thus, \( \frac{5}{7} \) is equivalent to \( \frac{15}{21} \) and \( \frac{2}{3} \) is equivalent to \( \frac{14}{21} \).

Step 3: Final Answer

Both fractions \( \frac{5}{7} \) and \( \frac{2}{3} \) can be rewritten as \( \frac{15}{21} \) and \( \frac{14}{21} \) respectively, sharing the common denominator of 21.

Key Concepts

Least Common Multiple

The least common multiple, often abbreviated as LCM, is a crucial concept when working with fractions, especially when you need to find a common denominator. To find the LCM of two numbers, you look for the smallest number that is a multiple of both numbers. This method is particularly helpful in ensuring that fractions have the same denominator.

When finding the LCM, you can:

• List the multiples of each number.
• Identify the smallest number that appears in both lists.

For example, to find the LCM of 7 and 3, you can list their multiples. Multiples of 7 are 7, 14, 21, and so on, while multiples of 3 are 3, 6, 9, 12, 15, 18, 21, etc. The smallest common multiple here is 21, making it the LCM (and thus, the least common denominator when working with these particular fractions).

Fractions

Fractions are a way to express parts of a whole. They consist of a numerator, the top number, which shows how many parts we have, and a denominator, the bottom number, which indicates how many parts make up a whole.

Understanding fractions involves knowing:

• The numerator and denominator's roles.
• Operations that can be performed with fractions, like addition and subtraction, require a common denominator.

In our exercise, we dealt with the fractions \( \frac{5}{7} \) and \( \frac{2}{3} \). The primary challenge with these fractions is that they have different denominators. To add or subtract them efficiently, transforming them to have the same denominator is essential, which is where finding the least common denominator comes in. Each fraction shares a portion of the whole, yet it's described in different sized parts.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. They are an essential concept as they allow us to change fractions to have a common denominator while keeping their value the same.

To create equivalent fractions:

• Multiply both the numerator and denominator of a fraction by the same number.
• This operation ensures the fraction's value does not change.

For instance, in the exercise, we transformed \( \frac{5}{7} \) to \( \frac{15}{21} \) by multiplying both the numerator and the denominator by 3. Similarly, \( \frac{2}{3} \) became \( \frac{14}{21} \) by multiplying the numerator and denominator by 7. These transformations allowed both fractions to be rewritten with a least common denominator of 21, making them easy to add or compare.