Problem 62 - Fundamentals of Mathematics

Question

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction. $$72 \frac{605}{666}$$

Step-by-Step Solution

Answer

\(\frac{48557}{666}\)

Step 1: Understand the Problem

We need to convert the mixed number \(72 \frac{605}{666}\) into an improper fraction.

Step 2: Identify the Components

The mixed number \(72 \frac{605}{666}\) has three parts: the whole number part, which is 72, and the fractional part, which is \(\frac{605}{666}\).

Step 3: Convert the Whole Number to a Fraction

Convert the whole number 72 into a fraction with the denominator 666. This is calculated as \(72 = \frac{72 \times 666}{666}\).

Step 4: Calculate the Equivalent Fraction

Calculate \(72 \times 666\) to get the equivalent fraction of the whole number: \(72 \times 666 = 47952\). Thus, the fraction is \(\frac{47952}{666}\).

Step 5: Add the Fractions

Add the fractions \(\frac{47952}{666}\) and \(\frac{605}{666}\) to get the improper fraction: \(\frac{47952 + 605}{666}\).

Step 6: Solve the Addition

Add the numerators: \(47952 + 605 = 48557\). The improper fraction is \(\frac{48557}{666}\).

Step 7: Final Check

We verified each step by ensuring the calculations are correct and ensuring the final result maintains the same denominator, confirming that the improper fraction \(\frac{48557}{666}\) represents the mixed number.

Key Concepts

Improper Fraction

An improper fraction is a type of fraction where the numerator is larger than or equal to the denominator. This means the fraction represents a value greater than or equal to one whole unit.
In the context of our exercise, we started with the mixed number \(72 \frac{605}{666}\). A mixed number is a whole number paired with a proper fraction. By converting this into an improper fraction, we simplify calculations such as multiplication or division with other fractions.
This conversion helps in providing a unified format for conducting operations with fractions, especially when comparing or adding fractions.

Fraction Addition

Fraction addition involves combining the numerators while keeping a common denominator. This operation is crucial when changing mixed numbers into improper fractions. Once the whole number is converted to a fraction with the same denominator as the fractional part, you can easily sum the two fractions.
For example, in our problem, adding the fractions \(\frac{47952}{666}\) and \(\frac{605}{666}\) is straightforward because both fractions have the same denominator.

• Add the numerators: \(47952 + 605 = 48557\).
• The common denominator remains \(666\).

The result is the improper fraction \(\frac{48557}{666}\).

Whole Number to Fraction Conversion

To work efficiently with mixed numbers, converting the whole number part to an equivalent fraction is essential.
Consider the whole number \(72\) from \(72 \frac{605}{666}\). This whole number needs conversion to a fraction with the same denominator as the fractional part of the mixed number.
You achieve this by multiplying the whole number with the denominator and placing it over the same denominator:

• Calculate \(72 \times 666\) to get \(47952\).
• Express as a fraction: \(\frac{47952}{666}\).

This method ensures consistency in combining the fractions and converting the mixed number precisely into an improper fraction.

Denominator

The denominator is the lower part of a fraction, signifying how many equal parts a whole is divided into. It is crucial as it determines the size of each part. When working with fractions, maintaining a common denominator is key for addition, subtraction, and conversion to improper fractions.
In our example of the mixed number \(72 \frac{605}{666}\), the denominator is \(666\), which remains unchanged during the conversion process.
By ensuring that the whole number is converted using the same denominator, we preserve the fraction's value and facilitate arithmetic operations easily.