Problem 1 - Prealgebra and Introductory Algebra

Question

Exercises 1 to 3 are the examples of complex fractions given at the beginning of Objective 11.4A. By what fraction would you multiply each complex fraction in order to simplify it? $$\frac{3}{2-\frac{1}{2}}$$

Step-by-Step Solution

Answer

The simplest form of the complex fraction \(\frac{3}{2-\frac{1}{2}}\) is 2. So, we would multiply the complex fraction by 2 to simplify it.

Step 1: Simplify the denominator

Begin by simplifying the denominator of the fraction, which is a simple fraction itself. Calculate \(2 - \frac{1}{2}\) which equals to \(1.5\). So, the whole complex fraction now becomes \(\frac{3}{1.5}\).

Step 2: Simplify the whole fraction

Now, we proceed to simplify the whole fraction by dividing the numerator by the denominator. So, \(\frac{3}{1.5} = 2\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This is helpful because it makes fractions much easier to understand and work with. To simplify a fraction, you need to divide the numerator and the denominator by their greatest common divisor (GCD). This process ensures that the fraction is as simple as possible.

For example, in simplifying the complex fraction \(\frac{3}{2-\frac{1}{2}}\), the denominator \(2-\frac{1}{2}\) becomes \(1.5\). The whole fraction becomes \(\frac{3}{1.5}\), which simplifies to 2, as both parts of the fraction are divisible evenly by 1.5. Simplifying makes calculation easier and helps in quickly understanding the relationship between numbers.

• Check if both numerator and denominator can be divided by the same number.
• Use the greatest common divisor (GCD) to reduce fractions.
• Always ensure final results are in the simplest form possible.

Fractions

Fractions represent parts of a whole and are expressed as a ratio between two numbers called the numerator and the denominator. Fractions can take many forms, including simple fractions and complex fractions.

Simple fractions are straightforward and consist of a single numerator over a single denominator, such as \(\frac{3}{4}\). Complex fractions, like \(\frac{3}{2-\frac{1}{2}}\), are more intricate because they have fractions in either their numerators, denominators, or both. In these cases, the objective often involves simplifying the fraction by dealing with the numerator and the denominator separately to form a simple fraction.

• Numerator: The top part of a fraction that signifies how many parts are considered.
• Denominator: The bottom part, indicating into how many parts the whole is divided.
• Complex fractions contain fractions within fractions and require additional steps for simplification.

Algebra

Algebra is particularly useful in simplifying fractions and solving equations involving unknowns. In algebra, you work with variables, expressions, and numbers to find unknown values or simplify expressions.

Simplifying complex fractions involves algebraic skills, like combining terms or finding common denominators. When handling expressions like \(\frac{3}{2-\frac{1}{2}}\), algebra helps guide us in reducing the fraction, especially when the expressions in the numerator and denominator become more complicated.

Algebraic techniques include:

• Solving linear equations to simplify expressions.
• Combining like terms to make expressions simpler.
• Using properties of operations like distributive, associative, and commutative properties to rearrange terms for simplification.

Understanding algebraic methods is crucial when managing complex fractions, leading to much easier and more accurate calculations.