Problem 97 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{\frac{6}{1}}{\frac{1}{2}} $$

Step-by-Step Solution

Answer

12

Step 1 - Simplify the numerator

The numerator is already a simple fraction: \( \frac{6}{1} \). This simplifies to 6.

Step 2 - Identify the fraction division

The given expression is \( \frac{6}{\frac{1}{2}} \), which is essentially dividing 6 by \( \frac{1}{2} \).

Step 3 - Apply the division rule for fractions

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \( \frac{1}{2} \) is 2. So, \( \frac{6}{\frac{1}{2}} = 6 \times 2 \).

Step 4 - Calculate the result

Perform the multiplication: \( 6 \times 2 = 12 \).

Key Concepts

Simplifying Fractions

When dealing with fractions, simplifying them makes calculations easier. Simplifying means finding an equivalent fraction where the numerator and denominator are the smallest possible whole numbers. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For instance, if you have the fraction \(\frac{8}{12}\), the GCD of 8 and 12 is 4. Dividing both by 4, you get \(\frac{2}{3}\).
In the provided problem, the numerator \(\frac{6}{1}\) simplifies to 6 since 6 divided by 1 is 6.

Reciprocals

Understanding reciprocals is crucial in fraction operations, especially division. The reciprocal of a fraction is simply the fraction flipped. So, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This concept is handy because dividing by a fraction is the same as multiplying by its reciprocal.
In our exercise, the fraction to be divided by is \(\frac{1}{2}\). The reciprocal of \(\frac{1}{2}\) is \(2\), because we flip the fraction. This turns our division problem into a multiplication problem, making the math easier.

Multiplication of Fractions

Multiplying fractions is straightforward. Multiply the numerators together and the denominators together. For example, to multiply \(\frac{a}{b}\) by \(\frac{c}{d}\), you get \(\frac{a \times c}{b \times d}\). If you are multiplying a whole number by a fraction, like in our exercise, treat the whole number as a fraction with a denominator of 1.
In the exercise, after finding the reciprocal, we multiplied 6 by 2. Mathematically, this means: \[6 \times \frac{2}{1} = \frac{6 \times 2}{1} = 12\].
Hence, the result of \(\frac{6}{\frac{1}{2}}\) is 12.