Problem 2 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{1}{8}}{-\frac{5}{12}} $$

Step-by-Step Solution

Answer

The simplified fraction is \(-\frac{3}{10}\).

Step 1: Identify the Complex Fraction

The given complex fraction is \( \frac{\frac{1}{8}}{-\frac{5}{12}} \). Here, the numerator is \( \frac{1}{8} \) and the denominator is \( -\frac{5}{12} \).

Step 2: Invert the Denominator Fraction

To simplify the expression, we need to multiply the numerator by the reciprocal of the denominator. The reciprocal of \( -\frac{5}{12} \) is \( -\frac{12}{5} \).

Step 3: Multiply the Fractions

Now, multiply the numerator \( \frac{1}{8} \) by the reciprocal of the denominator \( -\frac{12}{5} \):\[ \frac{1}{8} \times -\frac{12}{5} = -\frac{1 \times 12}{8 \times 5} = -\frac{12}{40} \]

Step 4: Simplify the Fraction

To simplify \( -\frac{12}{40} \), find the greatest common divisor (GCD) of 12 and 40, which is 4. Divide both the numerator and the denominator by 4:\[ -\frac{12 \div 4}{40 \div 4} = -\frac{3}{10} \]

Key Concepts

Fraction Simplification

Simplifying fractions is crucial to making them easier to work with. When simplifying a fraction, you are essentially writing it in its reduced form, where both the numerator and the denominator are divided by their common factor.
This common factor that's shared between the numerator and the denominator helps to shrink the value of each, without changing the actual value of the fraction itself. By doing this, you are expressing the fraction in its simplest, smallest terms possible.
To simplify a fraction, you need to:

• Find a number that divides both the numerator and denominator.
• Divide both the top and bottom of the fraction by this number.
• Continue this process until no further division is possible except by 1.

Always remember, simplifying fractions is about reducing them to their simplest form without altering the actual value. It's about finding the most basic way to represent a given quantity.

Reciprocal of a Fraction

A reciprocal is a special kind of fraction. To get the reciprocal of a fraction, you simply flip the numerator and the denominator. This means the number at the bottom (the denominator) goes to the top, and the number at the top (the numerator) goes to the bottom.
A reciprocal is important because it turns division into multiplication. For example, dividing by a fraction is the same as multiplying by its reciprocal. This is a handy trick because multiplication is often simpler to execute and understand than division.
To find the reciprocal of a fraction:

• Take the value of the numerator and swap it with the denominator.
• Ensure the signs of the numbers are also taken into account correctly.

This concept aids significantly in simplifying complex fraction expressions, as multiplying fractions is generally more straightforward than dividing fractions.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes known as the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. It's a key concept in mathematics that helps us simplify fractions efficiently.
The GCD of the numerator and the denominator is a powerful tool because it allows us to break down fractions to their simplest form. This makes it easier to understand, compare, and compute with fractions.
To find the GCD:

• List all the factors of each number.
• Identify the common factors between the numbers.
• The largest number among these common factors is the greatest common divisor.

Finding the GCD is an essential step in the fraction simplification process. It allows for easier and quicker calculations by reducing numbers to their lowest terms.