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Chapter 1: Problem 81

Find the reciprocal of each number, if it exists. $$ \frac{51}{-10} $$

Short Answer

Expert verified
The reciprocal of \(\frac{51}{-10}\) is \(\frac{-10}{51}\).

Step by step solution

01

Understand the Reciprocal

The reciprocal of a number is obtained by swapping the numerator with the denominator. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
02

Identify the Numerator and Denominator

In the fraction \(\frac{51}{-10}\), the numerator is 51 and the denominator is -10.
03

Swap the Numerator and Denominator

To find the reciprocal, swap the numerator and denominator. This changes \(\frac{51}{-10}\) to \(\frac{-10}{51}\).
04

Simplify the Reciprocal

Check if the reciprocal can be simplified further. \(\frac{-10}{51}\) is already in its simplest form as 10 and 51 have no common factors other than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
A reciprocal is essentially the 'flipped' version of a number or a fraction. When you reciprocate a fraction, you swap its numerator and denominator. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). Reciprocals are key in algebra for tasks like division of fractions
because dividing by a fraction is the same as multiplying by its reciprocal.
To illustrate, consider the fraction \(\frac{51}{-10}\). To find its reciprocal, simply exchange the numerator (51) and the denominator (-10), giving you \(\frac{-10}{51}\). It's that simple!
Numerator and Denominator
A fraction consists of two parts: the numerator and the denominator. The numerator is the top part of the fraction, representing how many parts we have. The denominator is the bottom part, indicating the total number of equal parts.
In the fraction \(\frac{51}{-10}\), 51 is the numerator and -10 is the denominator.
These two components are essential for understanding and manipulating fractions.
Identifying them correctly is the first step towards finding reciprocals or simplifying fractions.
Fraction Simplification
Simplifying a fraction means reducing it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, if we have \(\frac{4}{8}\), both 4 and 8 can be divided by 4,
which simplifies the fraction to \(\frac{1}{2}\).
In the context of reciprocals, after swapping the numerator and the denominator, always check if the new fraction can be simplified further. However, the fraction \(\frac{-10}{51}\) does not simplify further
since 10 and 51 have no common factors other than 1.
This means \(\frac{-10}{51}\) is already in its simplest form.

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