Problem 41 - Prealgebra

Question

Change to improper fractions. $$2 \frac{4}{5}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{14}{5}\).

Step 1: Understand the Concept of Improper Fractions

An improper fraction has a numerator that is greater than or equal to its denominator. Our task is to convert the given mixed number, which is a combination of a whole number and a fraction, into an improper fraction.

Step 2: Define the Mixed Number

The mixed number given is \(2 \frac{4}{5}\). Here, 2 is the whole number, and \(\frac{4}{5}\) is the fractional part.

Step 3: Multiply the Whole Number by the Denominator

To convert a mixed number to an improper fraction, first multiply the whole number by the denominator of the fractional part. This means calculating \(2 \times 5 = 10\).

Step 4: Add the Result to the Numerator of the Fractional Part

Add the product from step 3 to the numerator of the fraction. Thus, \(10 + 4 = 14\).

Step 5: Write the Expression as an Improper Fraction

The result from step 4 becomes the numerator of the improper fraction, while the denominator remains the same as that of the fractional part. Therefore, the improper fraction is \(\frac{14}{5}\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction. It is used to represent numbers that are more than a whole number but less than two whole numbers. For example, the mixed number \(2 \frac{4}{5}\) consists of the whole number 2 and the fraction \(\frac{4}{5}\). Mixed numbers are practical in everyday life because they allow us to express quantities more naturally, such as saying "two and four-fifths of a pie," rather than using only fractions.

Numerator

The numerator is the top part of a fraction and it represents how many parts of a whole are being considered. In the fraction \(\frac{4}{5}\), 4 is the numerator. It tells us that, out of a whole divided into 5 parts, 4 parts are being taken into account or used. When converting mixed numbers to improper fractions, the numerator changes based on calculations involving both the whole number and the fractional part.

Denominator

The denominator is the bottom part of a fraction and it indicates into how many equal parts the whole is divided. For \(\frac{4}{5}\), the denominator is 5, meaning the whole is divided into 5 equal parts. In any fraction, the denominator remains constant when converting between mixed numbers and improper fractions, because the division of the whole does not change. It's crucial to understand that the denominator determines the size of each part in fractions.

Converting Fractions

Converting a mixed number to an improper fraction is a simple yet systematic process.To convert, follow these steps:

• Multiply the whole number by the denominator of the fractional part. This gives you the number of parts in whole units: \(2 \times 5 = 10\).
• Add this result to the numerator of the fractional part to find the new numerator: \(10 + 4 = 14\).
• The denominator remains the same, so you write the improper fraction as \(\frac{14}{5}\).

This process helps in mathematical operations such as addition, subtraction, and multiplication involving mixed numbers and improper fractions.