Problem 78 - Algebra 1: Concepts and Skills

Question

Find the reciprocal. \(\frac{5}{6}\)

Step-by-Step Solution

Answer

\(\frac{6}{5}\) is the reciprocal of \(\frac{5}{6}\).

Step 1: Identify the current fraction

The fraction given is \(\frac{5}{6}\).

Step 2: Find the reciprocal

To find the reciprocal of a fraction, we simply flip the numerator and denominator. This gives us \(\frac{6}{5}\) as the reciprocal of the given fraction.

Key Concepts

Fraction Basics

Fractions are a way of representing parts of a whole. They are a vital part of mathematics and appear everywhere, from cooking recipes to scientific measurements. A fraction consists of two numbers separated by a line. This line is known as the fraction bar. The number on top of the line is the numerator and the number below the line is the denominator.
Understanding fractions helps in performing various mathematical operations like addition, subtraction, multiplication, and division. Fractions also play a key role when finding their reciprocals. A reciprocal of a fraction is simply the fraction turned upside down, meaning the numerator and denominator swap positions. For example, the reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\). This concept applies to all fractions, except those with a zero because division by zero is undefined.

• Basic format: \(\frac{a}{b}\)
• Related operations: addition, subtraction, multiplication, division
• Reciprocal: swap the numerator and denominator

Understanding Numerator

The numerator is the top part of a fraction. It indicates how many parts of the whole we are considering. In the fraction \(\frac{5}{6}\), the number 5 is the numerator. This tells us that we have 5 parts out of the total, which is indicated by the denominator.
In general, if the numerator is equal to the denominator, the fraction represents one whole. For example, \(\frac{6}{6} = 1\). If it's less, such as \(\frac{3}{6}\), it means we have a portion of the whole. Numerators are essential for determining the value of a fraction as it directly impacts the size of the fraction.

• Numerator represents parts of the whole
• In \(\frac{5}{6}\), numerator is 5
• A larger numerator means a larger portion of the whole

Role of Denominator

The denominator is the bottom part of a fraction and is crucial in determining what the fraction stands for. In \(\frac{5}{6}\), the number 6 is the denominator. This means that the whole is divided into 6 equal parts.
The denominator tells us how many parts the whole is split into, and thus affects the size or value of those parts. A smaller denominator means larger pieces of the whole, such as \(\frac{1}{2}\), while a larger denominator means smaller pieces, like \(\frac{1}{8}\).When working with reciprocals, understanding denominators is key because when you flip the fraction, the denominator becomes the numerator. In our example, the reciprocal of \(\frac{5}{6}\) becomes \(\frac{6}{5}\). This changes its role in how we perceive the size of the new fraction.

• Denominator represents the total parts
• In \(\frac{5}{6}\), denominator is 6
• Larger denominators mean smaller parts