Problem 49 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Objective 3. $$ \frac{12}{5} \div \frac{24}{45} $$

Step-by-Step Solution

Answer

\(\frac{12}{5} \div \frac{24}{45} = \frac{9}{2}\) after simplification.

Step 1: Understand the Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. This means that instead of dividing by \(\frac{24}{45}\), you can multiply \(\frac{12}{5}\) by the reciprocal of \(\frac{24}{45}\), which is \(\frac{45}{24}\).

Step 2: Set Up the Multiplication

Rewrite the division expression \(\frac{12}{5} \div \frac{24}{45}\) as a multiplication expression: \(\frac{12}{5} \times \frac{45}{24}\).

Step 3: Multiply the Numerators and Denominators

Using the equivalent multiplication expression, multiply the numerators together and the denominators together:\[\frac{12 \times 45}{5 \times 24}\]

Step 4: Simplify the Expression

First, calculate both the numerator and the denominator:- Numerator: \(12 \times 45 = 540\).- Denominator: \(5 \times 24 = 120\).Now, the expression is \(\frac{540}{120}\).

Step 5: Reduce the Fraction

To simplify \(\frac{540}{120}\), find the greatest common divisor (GCD) of 540 and 120, which is 60. Divide both numerator and denominator by the GCD:\[\frac{540}{60} = 9, \ \frac{120}{60} = 2\]This reduces the fraction to \(\frac{9}{2}\).

Key Concepts

Reciprocal of a Fraction

When we talk about the reciprocal of a fraction, we're referring to what you get when you "flip" the fraction over. This means swapping the numerator and the denominator. For example, the reciprocal of the fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).

This concept is particularly useful in division because dividing by a fraction is the same as multiplying by its reciprocal. Given our exercise, instead of dividing \( \frac{12}{5} \) by \( \frac{24}{45} \), we multiply \( \frac{12}{5} \) by \( \frac{45}{24} \). Using reciprocals helps simplify the complex task of division into basic multiplication, which is often easier to perform.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1.

To simplify a fraction like \( \frac{540}{120} \), you need to divide both the numerator and the denominator by their greatest common divisor (GCD).

This makes computations easier and fractions more understandable in real-world applications. For instance, by simplifying \( \frac{540}{120} \), we discover it equals \( \frac{9}{2} \), a much neater number.

• Always identify the common factors in the numerator and denominator.
• If possible, cancel out these common factors.
• Continue until you cannot find any more common factors besides 1.

This process can greatly assist students in visualizing and managing fractions correctly.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction. Understanding the GCD is crucial in mathematics, especially when you need to simplify complex fractions.

To find the GCD of two numbers, you can use several methods such as prime factorization or the Euclidean algorithm. In our example, the GCD of 540 and 120 is 60, as 60 is the largest number that divides both without leaving a remainder.

• List the factors of both numbers.
• Identify the largest factor that the numbers share.
• Use this factor to simplify the fraction further.

By using the GCD, we found \( \frac{540}{120} \) simplifies to \( \frac{9}{2} \). This makes handling and comparing large fractions far more convenient.