Problem 8 - Algebra 1

Question

Find the reciprocal of the number. $$-\frac{3}{4}$$

Step-by-Step Solution

Answer

The reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\).

Step 1: Understanding the premise

The number provided here is \(-\frac{3}{4}\). The task requires finding the reciprocal of this number.

Step 2: Formulating the reciprocal

The reciprocal of a number is simply 1 divided by that number. Therefore, the reciprocal of \(-\frac{3}{4}\) would simply be \(\frac{1}{-\frac{3}{4}}\).

Step 3: Calculating the reciprocal

To resolve the equation, one would need to simplify \(\frac{1}{-\frac{3}{4}}\) to \(-\frac{4}{3}\). This is achieved by multiplying the top and bottom of the fraction by -4.

Key Concepts

Negative Fractions

When dealing with fractions, you might come across negative fractions like \(-\frac{3}{4}\). A negative fraction has either the numerator or the denominator as a negative number. In \(-\frac{3}{4}\), the negative sign is in front, indicating the entire fraction is negative.

Key points about negative fractions:

• The negative sign can be attached to the numerator, the denominator, or in front of the fraction itself.
• All three representations are equivalent, but typically, the negative sign is displayed at the front.
• Converting between them does not change the value of the fraction.

For example, \(-\frac{3}{4}\), \(\frac{-3}{4}\), and \(\frac{3}{-4}\) are the same.

This understanding is crucial when finding reciprocals of negative fractions, as the negative sign remains unaffected.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers.

The denominator must not be zero. These numbers include positive fractions, negative fractions, whole numbers, and zero itself.

• Whole numbers: These can be written as a fraction, e.g., \(5\) as \(\frac{5}{1}\).
• Negative fractions: These fall into the category of rational numbers. \(-\frac{3}{4}\) is an example of a rational number.
• Zero: Although it is a rational number, it cannot be in the denominator of a fraction.

Understanding rational numbers helps in identifying when a number is suitable for operations like finding reciprocals.

Always remember that any rational number, except zero, has a reciprocal.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, ensuring the numerator and the denominator share no common factors other than 1.

While finding the reciprocal of a fraction such as \(-\frac{3}{4}\), simplification can be vital, though \(-\frac{3}{4}\) is already in its simplest form.

Steps to simplify a fraction:

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

In our case, finding the reciprocal \(-\frac{4}{3}\) doesn’t require additional simplification as this result is already simplified. This rule holds true for any fraction derived from operations like reciprocals.