Problem 48 - 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). $$ \frac{1}{2}, \frac{3}{7} $$

Step-by-Step Solution

Answer

The least common denominator (LCD) of the two fractions 1/2 and 3/7 is 14, so the fractions can be rewritten as 7/14 and 6/14 respectively.

Step 1 - Finding the least common denominator (LCD)

We can find the least common denominator of the two fractions by finding the least common multiple (LCM) of the denominators (2 and 7). The LCM of 2 and 7 is simply 2*7=14, because 2 and 7 have no common factors other than 1.

Step 2 - Rewriting the fractions

After finding the LCD, we rewrite each of the two fractions with the LCD 14. \[ \frac{1}{2} \] becomes: \( \frac{1*(14/2)}{2*(14/2)} = \frac{7}{14} \). And \[ \frac{3}{7} \] becomes: \( \frac{3*(14/7)}{7*(14/7)} = \frac{6}{14} \).

Key Concepts

Least Common Multiple

To understand how we find the Least Common Denominator (LCD), we first need to grasp the concept of the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. It is useful when adding, subtracting, or comparing fractions.

In our exercise, we are dealing with the denominators 2 and 7. Since 2 and 7 share no factors other than 1, their least common multiple is simply their product:

• The LCM of 2 and 7 is found by multiplying them: 2 × 7 = 14.
• This means 14 is the smallest number both 2 and 7 divide into without a remainder.

Understanding this will help us rewrite fractions with a common denominator.

Fractions

Fractions are a way to represent parts of a whole. A fraction consists of a numerator (top number) and a denominator (bottom number). The denominator tells us into how many equal parts the whole is divided, while the numerator indicates how many of these parts we have.

For example, in the fraction \(\frac{1}{2}\), the denominator is 2, meaning we have divided something into two parts, and the numerator is 1, indicating one part is considered.

In the context of rewriting fractions to have a least common denominator, understanding the role of numerators and denominators is crucial.

Denominators

Denominators are critical for understanding fractions, especially when performing operations like addition or subtraction. When denominators differ, we need a common denominator to carry out these operations. This is why finding the Least Common Denominator is essential.

The denominator dictates the number of equal parts the unit is divided into, and when comparing fractions, we align these parts.

• In our given fractions, the denominators are 2 and 7.
• We need these to be the same for any arithmetic operations, leading us to find their LCM, which is 14.

This alignment simplifies the process of mathematical operations involving fractions.

Rewriting Fractions

Rewriting fractions with a common denominator involves adjusting the numerators so the fractions remain equivalent.

In our example, once the least common denominator of 14 is found, we rewrite each fraction. Here's how:

• The fraction \(\frac{1}{2}\) becomes: \(\frac{1 \times 7}{2 \times 7} = \frac{7}{14}\).
• The fraction \(\frac{3}{7}\) becomes: \(\frac{3 \times 2}{7 \times 2} = \frac{6}{14}\).

The key is multiplying both the numerator and denominator by the same number to keep the value of the fractions unchanged. This ensures that they are rewritten with a common denominator, allowing easy comparison and calculation.