Problem 97 - Algebra 1

Question

RECIPROCALS Find the reciprocal. $$ 1 $$

Step-by-Step Solution

Answer

The reciprocal of 1 is 1.

Step 1: Recognize the Structure

Recognize that any whole number can be written as that number over 1. Thus, 1 is expressed as 1/1.

Step 2: Find the Reciprocal

Find the reciprocal of a fraction by swapping the numerator and denominator. Therefore, the reciprocal of 1/1 is 1/1.

Step 3: Simplify If Needed

Simplify if possible. In this case, the reciprocal of 1 is 1.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole number. Imagine a pizza: if you cut it into 4 equal pieces, each piece is a fraction of the pizza, or \( \frac{1}{4} \). Fractions consist of two numbers:

• Numerator: This is the top number and it tells us how many parts of the whole we have.
• Denominator: This is the bottom number and indicates how many parts the whole is divided into.

Let's assume we have a fraction like \( \frac{3}{5} \). Here, 3 is the numerator, indicating three parts, and 5 is the denominator, showing that the whole is divided into five parts. One way to work with fractions is by using reciprocals, which involves switching the numerator and the denominator. This concept is helpful in many math operations like division.

Numerator and Denominator

Understanding the numerator and denominator is crucial in working with fractions. Think of the fraction \( \frac{a}{b} \), where:

• Numerator (\( a \)): Tells us how many parts we have.
• Denominator (\( b \)): Tells us the total number of equal parts in a whole.

In any fraction, the numerator and denominator play essential roles. They give fractions their value and meaning. If you want to find the reciprocal of a fraction, you simply switch these numbers. For example, the reciprocal of \( \frac{7}{3} \) is \( \frac{3}{7} \). This swap is particularly useful when dividing fractions, as multiplying by the reciprocal is a standard method in these calculations. It's good to remember that the denominator can never be zero, as division by zero is undefined.

Simplification in Mathematics

Simplification is a technique that makes calculations easier. By simplifying, you express numbers in their most basic form. When it comes to fractions, simplification often means reducing a fraction to its lowest terms, where the numerator and denominator are as small as possible and still retain the same value. For example, to simplify \( \frac{8}{12} \), you find the greatest common divisor (GCD) of 8 and 12, which is 4, and divide both the numerator and denominator by it, resulting in \( \frac{2}{3} \).But in some cases, like the number '1',

• '1' is already a whole number, often represented as \( \frac{1}{1} \) in fraction form.
• The reciprocal of \( \frac{1}{1} \) is still \( \frac{1}{1} \).

Thus, '1' is already simple and doesn’t require further simplification. Mastering this concept aids in solving more complex mathematical problems effortlessly.