Problem 15 - Prealgebra

Question

Change each improper fraction to a mixed number. $$\frac{19}{4}$$

Step-by-Step Solution

Answer

\( \frac{19}{4} = 4\frac{3}{4} \).

Step 1: Understand the Problem

We need to convert the improper fraction \( \frac{19}{4} \) into a mixed number. An improper fraction has a numerator larger than the denominator, whereas a mixed number consists of a whole number and a proper fraction.

Step 2: Divide the Numerator by the Denominator

Divide 19 by 4. The division yields a quotient and a remainder. Starting the division, 4 goes into 19 a total of 4 times (since \(4 \times 4 = 16\)), leaving a remainder.

Step 3: Calculate the Remainder

The remainder is what is left after dividing 19 by 4. In this case, \(19 - 16 = 3\). Therefore, the remainder is 3.

Step 4: Write as a Mixed Number

The quotient from Step 2 becomes the whole number part of the mixed number, and the remainder from Step 3 is the numerator of the fraction part with the original denominator. Thus, \( \frac{19}{4} = 4\frac{3}{4} \).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are used to express values that are more than one whole unit but not quite reaching the next whole number. A mixed number displays both the complete parts of a value and the remaining portion that doesn't form a complete whole.
For example, consider the improper fraction \( \frac{19}{4} \). When changed into a mixed number, it reads as \( 4\frac{3}{4} \). Here, "4" represents the entire or whole units, and \( \frac{3}{4} \) is the remaining part, showing that we have 3 parts out of a possible 4 to complete the next whole number.
This format provides a more intuitive understanding for many situations, especially in everyday life, like understanding measurements or quantities.

Numerator and Denominator Explained

The terms numerator and denominator are fundamental in understanding fractions. The numerator is the top number in a fraction and represents how many parts we have, while the denominator is the bottom number and indicates how many equal parts make up a whole.
In the improper fraction \( \frac{19}{4} \):

• The numerator "19" shows the total number of parts being considered.
• The denominator "4" tells us that these parts are divided into groups or sections, each equivalent to one quarter of a whole.

Analyzing the fraction with these two components is essential when converting improper fractions into mixed numbers. Dividing the numerator by the denominator gives you the whole number, while the remainder stays as part of the fraction.

The Importance of Remainder

Understanding the remainder is critical when converting improper fractions into mixed numbers. The remainder is what is left after you divide the numerator by the denominator.
When we divide 19 by 4:

• The quotient is 4, which tells us that 4 whole parts can be taken from the fraction.
• The remainder is 3, which is the part of the fraction that doesn't fit into whole numbers.

The remainder directly becomes the numerator of the fractional part in mixed numbers. Just like in the example, after dividing 19 by 4, you have a remainder of 3. This remainder is placed over the original denominator to form the fraction in the mixed number representation. Hence, the improper fraction \( \frac{19}{4} \) can be rewritten as \( 4\frac{3}{4} \), with 3 being the crucial remainder part.