Problem 23 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{2}{3}}{\frac{4}{3}}$$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{1}{2} \).

Step 1: Identify the Complex Fraction

The complex fraction given is \( \frac{\frac{2}{3}}{\frac{4}{3}} \). This means we have a fraction \( \frac{2}{3} \) divided by another fraction \( \frac{4}{3} \).

Step 2: Simplify by Multiplying by the Reciprocal

To simplify, multiply \( \frac{2}{3} \) by the reciprocal of \( \frac{4}{3} \). The reciprocal of \( \frac{4}{3} \) is \( \frac{3}{4} \). Thus, \[ \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{2}{3} \times \frac{3}{4} \].

Step 3: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \( 2 \times 3 = 6 \) and \( 3 \times 4 = 12 \). So, \[ \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} \].

Step 4: Reduce the Fraction to Lowest Terms

Divide the numerator and the denominator by their greatest common divisor, which is 6. Therefore, \( \frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \).

Key Concepts

Reciprocal

In the world of fractions, the term "reciprocal" is crucial when working with division problems. A reciprocal of a fraction simply swaps its numerator and denominator. For instance, the reciprocal of \( \frac{4}{3} \) is \( \frac{3}{4} \). This switch allows us to transform division into multiplication, which is often easier to manage.

• To find the reciprocal, flip the fraction upside down.
• This is particularly useful when simplifying complex fractions.
• When a fraction is multiplied by its reciprocal, the product is always 1.

For example, using the reciprocal helped in simplifying the fraction \( \frac{\frac{2}{3}}{\frac{4}{3}} \) by converting it to a multiplication problem \( \frac{2}{3} \times \frac{3}{4} \). Understanding reciprocals is therefore essential for manipulating and simplifying expressions involving fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest or lowest terms. This involves making sure the numerator and denominator have no common factors other than 1. Let's illustrate this using the calculations in the exercise.

• After multiplying \( \frac{2}{3} \times \frac{3}{4} \), we got the fraction \( \frac{6}{12} \).
• The simplification process requires us to divide both the numerator and denominator by their greatest common factor (GCF).
• The GCF of 6 and 12 is 6.

By dividing both 6 and 12 by 6, we reduce \( \frac{6}{12} \) to \( \frac{1}{2} \). Simplifying fractions aids not just in solving problems, but also in better understanding the relationships between different quantities.

Lowest Terms

A fraction is said to be in its lowest terms when the numerator and denominator have no common divisors other than 1. Achieving this form makes fractions easier to work with and understand.

• The process involves finding the greatest common divisor (GCD) of both numerator and denominator.
• For \( \frac{6}{12} \), the GCD is 6, making the fraction \( \frac{1}{2} \) in its simplest form.
• Working with fractions in their lowest terms facilitates easier addition, subtraction, and comparison.

Ensuring that fractions are in their lowest terms is a fundamental skill in mathematics that simplifies later operations and improves clarity.

Greatest Common Divisor

The greatest common divisor (GCD) is vital for simplifying fractions as it is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps in reducing fractions to their simplest form.

• To find the GCD of 6 and 12, list the divisors: 6 (1, 2, 3, 6) and 12 (1, 2, 3, 4, 6, 12).
• The highest number in both lists is 6, which is the GCD.
• We use the GCD to divide the numerator and denominator, reducing the fraction \( \frac{6}{12} \) to \( \frac{1}{2} \).

Understanding and finding the GCD is key to simplifying fractions and improving problem-solving efficiency. It is a foundational concept that supports numerous mathematical techniques and procedures.