Problem 98 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{\frac{3}{4}}{\frac{1}{4}} $$

Step-by-Step Solution

Answer

3

Step 1: Understand the problem

The problem requires evaluating the expression \(\frac{\frac{3}{4}}{\frac{1}{4}}\). This is a complex fraction, and the first step is to rewrite it for simplification.

Step 2: Rewrite the complex fraction

Rewrite the complex fraction \(\frac{\frac{3}{4}}{\frac{1}{4}}\) as a division of two fractions: \(\frac{3}{4} \div \frac{1}{4}\).

Step 3: Apply the division rule for fractions

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. This means \(\frac{3}{4} \div \frac{1}{4} = \frac{3}{4} \times \frac{4}{1}\).

Step 4: Multiply the fractions

Multiply the numerators and the denominators: \(\frac{3 \times 4}{4 \times 1} = \frac{12}{4}\).

Step 5: Simplify the result

Simplify the fraction \(\frac{12}{4}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4: \(\frac{12}{4} = 3\).

Key Concepts

Division of Fractions

When you encounter a complex fraction, it's often best to break it down by using division of fractions. Division of fractions involves taking two fractions and finding their quotient. To do this, you convert the division into multiplication by flipping the second fraction, which is also known as taking its reciprocal. For example, with the fractions \(\frac{3}{4} \) and \(\frac{1}{4} \), you perform the operation \(\frac{3}{4} \div \frac{1}{4} \). This becomes \(\frac{3}{4} \times \frac{4}{1}\). Here's the process more clearly explained:

• Rewrite the division problem as a multiplication problem.
• Multiply the first fraction by the reciprocal of the second fraction.

By following these steps, you can easily handle complex fractions.

Reciprocal of a Fraction

The reciprocal of a fraction is a crucial concept in division of fractions. A reciprocal simply flips the numerator (top number) and the denominator (bottom number) of a fraction. For example, the reciprocal of \(\frac{1}{4}\) is \(\frac{4}{1}\). When taking the reciprocal, always remember:

• The product of a fraction and its reciprocal is always 1, such as \(\frac{1}{4} \times \frac{4}{1} = 1\).
• This property allows us to transform the division of fractions into a multiplication problem, which simplifies the calculation.

Using the previous example, \(\frac{3}{4} \div \frac{1}{4} \) turns into \(\frac{3}{4} \times \frac{4}{1}\).

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible. This means reducing the fraction so that the numerator and the denominator share no common factors other than 1. For the final step in the given problem, we get the fraction \(\frac{12}{4}\). To simplify \(\frac{12}{4}\), consider these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In this case, both 12 and 4 can be divided by their GCD, which is 4. So, \(\frac{12}{4} = 3\). Always ensure that your answer is in its simplest form.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. It is fundamental in simplifying fractions. Let's break down finding the GCD:

• List out all the factors of each number.
• Identify the largest common factor.

For example, for 12 and 4:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 4: 1, 2, 4
The common factors are 1, 2, 4 and the GCD is 4. By dividing both 12 and 4 by their GCD, we obtain the simplest form, \(\frac{12}{4} = 3\). Knowing how to find the GCD makes the process of simplifying fractions straightforward and easier to manage.