Problem 51 - 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). $$ \frac{12}{13}, \frac{5}{26} $$

Step-by-Step Solution

Answer

The least common denominator of the two fractions is 26. The fractions, when rewritten with the LCD, are \( \frac{24}{26} \) and \( \frac{5}{26} \).

Step 1: Find the LCD

First, note that the denominators are 13 and 26. Knowing that 26 is a multiple of 13, it can be understood that the least common denominator is 26.

Step 2: Rewrite the first fraction

Rewrite \( \frac{12}{13} \) with a denominator of 26. This is achieved by multiplying the numerator and denominator by 2, yielding \( \frac{24}{26} \).

Step 3: Rewrite the second fraction

The second fraction \( \frac{5}{26} \) is already expressed in terms of the LCD and thus requires no changes.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. Each fraction consists of two parts: the numerator and the denominator.
The numerator is the number on top and indicates the number of parts you have. The denominator is the number on the bottom and shows the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{12}{13} \), 12 is the numerator and 13 is the denominator.
Understanding fractions is crucial because they are used in everyday mathematics, particularly when dealing with parts of a whole or mixed numbers.

• The numerator tells you how many parts you have.
• The denominator tells you how many parts make up a whole.

Everyday examples of fractions include cutting a pizza into 8 slices where eating 3 slices would be \( \frac{3}{8} \) of the pizza. Similarly, fractions are used when calculating portions, measurements, and in many more real-world scenarios.

Finding Common Multiples

When dealing with fractions, particularly when you need to add, subtract, or compare them, finding common multiples is invaluable.
A common multiple is a number that is a multiple of two or more numbers. For the fractions \( \frac{12}{13} \) and \( \frac{5}{26} \), you need to find the least common multiple of 13 and 26 because these two numbers are the denominators in the fractions.

• 13 and 26 are denominators here.
• 26 is a multiple of 13, as \( 13 \times 2 = 26 \).

This makes 26 the least common denominator (LCD) for the fractions. Therefore, the least common denominator is essential to equate fractions to have a common base or denominator. This simplifies the process of comparing, adding, or subtracting fractions. Once fractions share the same denominator, arithmetic operations become much more straightforward.

Simplifying Fractions to Reduced Form

Simplifying fractions means expressing them in their simplest form where the numerator and the denominator have no other common divisor than 1.
This is called reducing a fraction. A reduced fraction is easier to understand and use in calculations.
For example, you can rewrite \( \frac{24}{26} \) as \( \frac{12}{13} \) because you can divide both the numerator and the denominator by 2. Reducing fractions to their simplest form:

• Divide the numerator and the denominator by their greatest common divisor (GCD).
• Keep reducing until no more common divisors exist between the numbers.

Finding the reduced form makes fractions easier to comprehend and apply in operations, ensuring the results are as straightforward as possible. Not only does this help in solving fraction problems correctly, but it also assists in maintaining clarity when interpreting mathematical expressions involving fractions.