Problem 79 - Prealgebra

Question

Simplify. $$\frac{1+\frac{1}{2}}{1-\frac{1}{2}}$$

Step-by-Step Solution

Answer

The simplified form is 3.

Step 1: Simplify the Numerator

First, focus on simplifying the numerator: \(1 + \frac{1}{2}\). To combine these, the integer 1 is expressed as \(\frac{2}{2}\). Thus, the expression becomes \(\frac{2}{2} + \frac{1}{2} = \frac{3}{2}\).

Step 2: Simplify the Denominator

Next, simplify the denominator: \(1 - \frac{1}{2}\). Express the integer 1 as \(\frac{2}{2}\). Thus, it becomes \(\frac{2}{2} - \frac{1}{2} = \frac{1}{2}\).

Step 3: Division of Fractions

Now, divide the simplified numerator by the simplified denominator: \(\frac{3}{2} \div \frac{1}{2}\). Dividing by a fraction is the same as multiplying by its reciprocal, thus \(\frac{3}{2} \times \frac{2}{1}\).

Step 4: Multiply the Fractions

Multiply the fractions: \(\frac{3}{2} \times \frac{2}{1} = \frac{3 \times 2}{2 \times 1} = \frac{6}{2}\).

Step 5: Simplify the Result

Finally, simplify the fraction \(\frac{6}{2}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. Thus, \(\frac{6}{2} = 3\).

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. When looking at a fraction, it consists of two numbers separated by a line. This line indicates division. Imagine cutting a pizza into equal slices; each slice is a fraction of the whole pizza.
The top number in a fraction is known as the numerator. It tells you how many parts you have. The bottom number is the denominator, which says how many parts make up a whole.

• A larger denominator means smaller parts because the whole is divided into more pieces.
• A larger numerator means you have more parts of the whole pizza.

Understanding fractions is crucial as they are used in many areas of math to represent parts of a whole or to solve problems involving division.

Numerator and Denominator Explained

The terms 'numerator' and 'denominator' are central to understanding fractions. Let's break them down for clarity:

The numerator is the number above the fraction line. It indicates how many parts out of the whole are being considered or used. For example, in the fraction \(\frac{1}{4}\), the numerator is 1, indicating one part of a whole divided into four parts.
The denominator is the number below the line. It tells us into how many total parts the whole is divided. In \(\frac{3}{4}\), the denominator 4 indicates the whole is split into four equal parts.

In calculations, having a solid grasp of numerator and denominator helps simplify operations like addition, subtraction, and comparison of fractions.

• In an addition like \(\frac{1}{2} + \frac{1}{3}\), finding a common denominator is necessary.
• Understanding these roles can simplify solving problems involving rational numbers.

The Role of the Greatest Common Divisor

The greatest common divisor (GCD) is a key concept for simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Using the GCD makes it easier to reduce fractions to their simplest form; often called 'simplifying'.
When you simplify a fraction like \(\frac{6}{8}\), you divide both the top and bottom by their GCD. The GCD of 6 and 8 is 2, making the fraction \(\frac{6}{8}\) simplified to \(\frac{3}{4}\).

• When calculating, finding the GCD helps to reduce complexity.
• It ensures the fraction represents the same value but in a simplified form.

In problems across math, the GCD assists not just in fractions, but in solving equations and understanding ratios too.