Problem 38 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{19}{2}$$

Step-by-Step Solution

Answer

\( \frac{19}{2} \) as a mixed number is 9 \( \frac{1}{2} \).

Step 1: Understanding Improper Fractions

An improper fraction has a numerator that is greater than or equal to its denominator. In this case, \( \frac{19}{2} \) is an improper fraction because 19 (numerator) is greater than 2 (denominator).

Step 2: Dividing the Numerator by the Denominator

Divide 19 (numerator) by 2 (denominator) to find the whole number part of the mixed number. Performing the division, we get 19 ÷ 2 = 9 with a remainder of 1.

Step 3: Expressing the Mixed Number

A mixed number is written with the quotient from the division as the whole number and the remainder as the new numerator over the original denominator. Therefore, the mixed number equivalent of \( \frac{19}{2} \) is 9 \( \frac{1}{2} \).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are another way to represent improper fractions in a simpler form. For example, if we have an improper fraction like \( \frac{19}{2} \), it can be expressed as a mixed number.

• The whole number part comes from dividing the numerator by the denominator.
• The fraction part is the remainder over the original denominator.

In our example, \( \frac{19}{2} \) becomes 9 \( \frac{1}{2} \), where 9 is the quotient (whole number), and \( \frac{1}{2} \) is the leftover part expressed as a fraction. Using mixed numbers can make it easier to understand and visualize how much of a whole you have beyond just looking at improper fractions.

Numerator and Denominator

In fractions, the numerator and the denominator play distinct roles:

• The numerator is the number above the fraction bar. It shows how many parts we have.
• The denominator is the number below the fraction bar. It indicates the total number of equal parts the whole is divided into.

For instance, in our improper fraction \( \frac{19}{2} \): - "19" is the numerator, indicating 19 parts.- "2" is the denominator, implying that each unit is divided into 2 equal parts.Understanding how the numerator and denominator work together is crucial for grasping the concept of fractions, whether they are proper, improper, or mixed numbers. It also helps in calculations such as converting improper fractions to mixed numbers.

Division in Fractions

Dividing fractions involves a few straightforward steps and is essential for converting improper fractions to mixed numbers.In the case of \( \frac{19}{2} \), our aim is to divide the numerator (19) by the denominator (2).

• Perform the division: 19 divided by 2 equals 9 with a remainder of 1.
• The result of division gives you two components:
  • The quotient (9) becomes the whole number part of the mixed number.
  • The remainder (1) is part of the fractional component \( \frac{1}{2} \).

This division process helps to easily convert any improper fraction to its corresponding mixed number form. It is the basic arithmetic operation that bridges both formats.