Problem 27 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{66}{48} $$

Step-by-Step Solution

Answer

The fraction \( \frac{66}{48} \) simplifies to \( \frac{11}{8} \).

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{66}{48} \), we need to find the greatest common divisor (GCD) of 66 and 48. We can find the GCD by listing the factors of both numbers. Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66.Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.The greatest common factor that appears in both lists is 6. Therefore, the GCD of 66 and 48 is 6.

Step 2: Divide Both Numerator and Denominator by GCD

Now that we have the GCD, we divide both the numerator and the denominator of the fraction \( \frac{66}{48} \) by 6.\[ \frac{66 \div 6}{48 \div 6} = \frac{11}{8} \]This gives us the simplified form of the fraction.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a concept that helps us simplify fractions by identifying the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD involves determining the greatest common factor that both numbers share.

• Example: For the numbers 66 and 48, their factors are:
• Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66
• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

In this case, the number 6 is the largest number that divides both 66 and 48, which makes it their greatest common divisor. The GCD is used to simplify fractions by ensuring that the fraction can be reduced to its simplest form, which makes calculations easier and results in the simplest proportional representation of a relationship.

Numerator

The numerator is the top part of a fraction and it represents the number of parts we have or are considering in a whole. In our example fraction \(\frac{66}{48}\), the number 66 is the numerator.

• It tells us how many parts of a whole are being discussed.
• The numerator influences the size of the fraction value: the larger the numerator, the larger the fraction's value, provided that the denominator remains constant.

When simplifying a fraction, the numerator may be reduced by dividing it by the GCD, which helps in expressing the same fraction in its simplest form. For example, by dividing the numerator 66 by the GCD 6, we simplify it to 11, providing us the simplified fraction of \(\frac{11}{8}\). Understanding the numerator is crucial for operations involving fractions like addition, subtraction, and simplification.

Denominator

The denominator is the bottom part of a fraction and it indicates the total number of equal parts that make up a whole. It is crucial for understanding the scale or the frame of reference of the fraction. In the fraction \(\frac{66}{48}\), the number 48 is the denominator.

• It tells us into how many parts the whole is divided.
• When the denominator increases, the fraction value decreases, assuming the numerator stays the same.

For simplification purposes, the denominator is divided by the GCD just like the numerator. In our example, dividing 48 by the GCD 6 gives us 8, resulting in the simplified fraction \(\frac{11}{8}\). Grasping the role of the denominator is essential, especially in performing fraction operations, making comparisons, or interpreting proportional relationships in mathematical expressions.