Problem 2 - Basic College Mathematics

Question

Find the reciprocal of each number. \(\frac{7}{8}\)

Step-by-Step Solution

Answer

The reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

Step 1: Understand the Reciprocal

The reciprocal of a number is defined as 1 divided by that number. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

Step 2: Identify the given fraction

The given number is \(\frac{7}{8}\).

Step 3: Apply the Reciprocal Formula

Switch the numerator and the denominator. The reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

Step 4: Confirm if the fraction is simplified

Check to see if \(\frac{8}{7}\) can be simplified. In this case, \(\frac{8}{7}\) is already in its simplest form.

Key Concepts

Finding the Reciprocal of a Fraction

To find the reciprocal of a number, divide 1 by that number. For fractions, it's even simpler. Just switch the numerator (top number) and denominator (bottom number). This process gives you the reciprocal.

Let’s consider the fraction \(\frac{7}{8}\). To find its reciprocal, switch the numerator and denominator. So, the reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

Remember, the reciprocal of a reciprocal returns to the original number. That means, the reciprocal of \(\frac{8}{7}\) is \(\frac{7}{8}\). This process is essential when working with division and complex equations.

Understanding Fractions

Fractions represent parts of a whole. A fraction consists of two numbers: a numerator and a denominator.

For example, in the fraction \(\frac{7}{8}\),
- The numerator is 7, representing the parts you have.
- The denominator is 8, representing the total parts available.

Fractions can be used to represent values less than one, equal to one, or greater than one. It’s useful to visualize fractions to understand these concepts. Think of a pizza cut into 8 slices; 7 out of 8 slices can be written as \(\frac{7}{8}\).

Simplification of Fractions

Simplification makes fractions easier to work with by reducing them to their simplest form. This process involves dividing the numerator and denominator by their greatest common divisor (GCD).

For instance, \(\frac{8}{7}\) is already in its simplest form because the numbers 8 and 7 have no common divisors other than 1. If a fraction’s numerator and denominator can’t be divided by the same number other than 1, it’s completely simplified.

Simplification can ensure calculations involving fractions are more manageable and accurate.