Problem 54 - Fundamentals of Mathematics

Question

Find the reciprocal of \(8 \frac{5}{6}\).

Step-by-Step Solution

Answer

The reciprocal of \(8 \frac{5}{6}\) is \(\frac{6}{53}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number into an improper fraction. A mixed number consists of a whole number and a fraction. The mixed number here is \(8 \frac{5}{6}\). To convert it, multiply the whole number by the fraction's denominator and add the numerator: \(8 \times 6 + 5 = 48 + 5 = 53\). So, the improper fraction is \(\frac{53}{6}\).

Step 2: Find the Reciprocal

To find the reciprocal of a fraction, invert the numerator and the denominator. The reciprocal of \(\frac{53}{6}\) is \(\frac{6}{53}\).

Key Concepts

Mixed Number to Improper Fraction Conversion

A mixed number is a combination of a whole number and a fraction. When you need to work with these numerals in calculations, you often convert them to improper fractions. This is because it's easier to apply arithmetic operations on improper fractions.
To convert a mixed number like \(8 \frac{5}{6}\) into an improper fraction, follow these steps:

• Multiply the whole number by the denominator. Here, 8 multiplied by 6 equals 48.
• Add this result to the numerator of the fraction. That's 48 plus 5, which equals 53.
• The improper fraction thus becomes \(\frac{53}{6}\).

This transformation swaps the mixed number for a fraction, making it suitable for reciprocal finding and other calculations.

Finding Reciprocals

Finding the reciprocal of a fraction is a common requirement in mathematics, especially when dealing with division and fraction inversion.
The reciprocal of a fraction is simply found by swapping its numerator with its denominator. The idea is straightforward:

• Take the fraction \(\frac{a}{b}\) and invert it to become \(\frac{b}{a}\).

Applying this to our improper fraction \(\frac{53}{6}\), by inverting we get \(\frac{6}{53}\).
This new fraction is the reciprocal. When you multiply a fraction by its reciprocal, they cancel out to give 1, which is why reciprocals are useful.

Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division frequently involve fractions and whole numbers.
For instance, when converting mixed numbers for calculation, you are essentially using multiplication and addition:

• The multiplication of the whole number by the denominator transforms part of the expression into fraction form.
• Adding the numerator includes the extra fractional part of the mixed number.

These operations help streamline various mathematical processes, ensuring accuracy and simplicity in problem-solving.