Problem 7 - Basic College Mathematics

Question

Find the reciprocal of each number. \(\frac{10}{3}\)

Step-by-Step Solution

Answer

\(\frac{3}{10}\)

Step 1: Understand the Reciprocal

The reciprocal of a number is found by exchanging the numerator and the denominator of a fraction.

Step 2: Reciprocal of a Fraction

Given the fraction \(\frac{10}{3}\), the reciprocal would be \(\frac{3}{10}\).

Step 3: Simplify if Necessary

Check if the reciprocal fraction \(\frac{3}{10}\) can be simplified. In this case, it is already in its simplest form.

Key Concepts

Fraction Basics

A fraction represents a part of a whole. It is written as one number over another, separated by a line. The top number is called the numerator, and the bottom number is called the denominator. For example, in the fraction \(\frac{10}{3}\), 10 is the numerator and 3 is the denominator.

Fractions have many applications in everyday life, from splitting a pizza to dividing up time. Understanding fractions is fundamental to grasping more complex mathematical concepts.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). For instance, if a fraction is \(\frac{4}{8}\), both 4 and 8 can be divided by 4. So, \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\).

Here are steps to simplify fractions:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the simplified fraction.

In our exercise, the reciprocal \(\frac{3}{10}\) cannot be simplified further as 3 and 10 do not have any common divisors other than 1.

Numerator and Denominator

The numerator and denominator are crucial parts of a fraction.

Numerator: This is the number above the line in a fraction. It represents how many parts of the whole are being considered. In \(\frac{10}{3}\), the numerator is 10.

Denominator: This is the number below the line in a fraction. It shows the total number of equal parts the whole is divided into. In \(\frac{10}{3}\), the denominator is 3.

When finding a reciprocal, the roles of the numerator and the denominator switch places. So the reciprocal of \(\frac{10}{3}\) becomes \(\frac{3}{10}\). It's important to understand these terms to avoid mistakes in mathematical operations.