Problem 78 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{57}{132}-\frac{13}{132} $$

Step-by-Step Solution

Answer

\( \frac{1}{3} \)

Step 1: Identify the Common Denominator

Both fractions in the expression \( \frac{57}{132} - \frac{13}{132} \) share a common denominator of 132. This allows us to combine the fractions directly by subtracting the numerators.

Step 2: Subtract the Numerators

Subtract the numerator of the second fraction from the numerator of the first fraction. This means calculating \( 57 - 13 = 44 \). Thus, the expression becomes \( \frac{44}{132} \).

Step 3: Simplify the Fraction

We need to simplify the fraction \( \frac{44}{132} \). Start by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 44 and 132 is 44. Divide both the numerator and the denominator by 44 to simplify the fraction. This gives us \( \frac{44 \div 44}{132 \div 44} = \frac{1}{3} \).

Step 4: Confirm the Final Answer is Simplified

Confirm that the fraction \( \frac{1}{3} \) is in its simplest form. Since the numerator is 1 and cannot be reduced further, \( \frac{1}{3} \) is indeed in its lowest terms.

Key Concepts

Simplifying Fractions

When you're working with fractions, you often encounter the task of simplifying them. Simplifying a fraction means rewriting it with the smallest possible numerator and denominator, while keeping the same value as the original fraction. It's about making it as simple as possible to understand and work with.

Here's how you can simplify fractions:

• Step 1: Identify the greatest common divisor (GCD) of the numerator and denominator.
• Step 2: Divide both the numerator and the denominator by their GCD.
• Step 3: Write the fraction with the new simplified numbers.

If you're simplifying \( \frac{44}{132} \), like in the original exercise, you find the GCD of 44 and 132, which is 44. Then you divide both the numerator and denominator by 44, resulting in the simplified fraction \( \frac{1}{3} \). This is because \( \frac{44}{132} \) and \( \frac{1}{3} \) are equivalent, but \( \frac{1}{3} \) is in the simplest form.

Common Denominator

A common denominator is essential when you're adding or subtracting fractions. It's the number that the denominators of two or more fractions share, allowing for the fractions to be easily combined.

When both fractions have the same denominator, like we had with \( \frac{57}{132} \) and \( \frac{13}{132} \), you can go ahead and perform the arithmetic on the numerators directly. This step simplifies the process:

• Add or subtract the numerators: Simply perform the operation on the numerators while keeping the common denominator unchanged.
• Keep the common denominator: Since your work with numerators doesn’t affect the denominator, it remains constant.

Using the same denominator minimizes the steps required in fraction operations and reduces confusion. Therefore, it's helpful to find a common denominator, especially when you're dealing with fractions that initially have different denominators.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is an important concept in math, especially when it comes to simplifying fractions. It is the largest number that divides two or more numbers without leaving a remainder.

To find the GCD of two numbers:

• List the factors: Write down all the factors of each number.
• Identify the common factors: Find what factors they share in common.
• Choose the largest factor: The biggest shared factor is the GCD.

For example, to simplify \( \frac{44}{132} \), first list out the factors of 44 and 132, then identify that 44 is the largest one they have in common. Therefore, dividing both by 44 will give the simplest form \( \frac{1}{3} \).

Using the GCD makes reducing fractions both reliable and efficient, ensuring you reach the simplest and most manageable form possible.