Problem 19 - Prealgebra

Question

Change each improper fraction to a mixed number. $$\frac{13}{4}$$

Step-by-Step Solution

Answer

The improper fraction \( \frac{13}{4} \) converts to the mixed number \( 3 \frac{1}{4} \).

Step 1: Understand the Problem

We have the improper fraction \( \frac{13}{4} \). An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Our goal is to convert this improper fraction into a mixed number.

Step 2: Divide the Numerator by the Denominator

To convert \( \frac{13}{4} \) into a mixed number, start by dividing the numerator (13) by the denominator (4). Perform the division: 13 divided by 4 equals 3 with a remainder of 1. This division tells us that 13 is equal to 4 multiplied by 3 plus 1, or in equation form: \( 13 = 4 \times 3 + 1 \).

Step 3: Write the Quotient as the Whole Number

In the division performed, we got a quotient of 3. This will be the whole number part of the mixed number.

Step 4: Write the Remainder as the Fraction

The remainder from the division was 1. This becomes the numerator of the fraction part of the mixed number. The original denominator (4) stays the same. Thus, the fraction part is \( \frac{1}{4} \).

Step 5: Combine Whole Number and Fraction

Combine the whole number and the fraction to write the mixed number. Therefore, \( \frac{13}{4} \) as a mixed number is \( 3 \frac{1}{4} \).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fractional part. They are used to express values greater than or equal to 1 in a way that shows both the whole and the remaining parts separately. Imagine you have 3 whole pies and a quarter pie left; you could describe this as 3 mixed with a \(\frac{1}{4}\).
A mixed number consists of:

• A whole number (from the quotient of the division)
• A proper fraction (comprised of the remainder and the original denominator)

Mixed numbers make it easier to understand quantities in everyday contexts, such as in cooking or when dealing with measurements like feet and inches.

Division

Division is a vital mathematical operation used to break down numbers into equal parts. When you divide, you are essentially finding how many times one number (the divisor) fits into another (the dividend).
In the context of converting improper fractions to mixed numbers, division helps determine:

• The quotient, which becomes the whole number part of the mixed number
• The remainder, which forms the fractional part of the mixed number

For example, when dividing 13 by 4, you see that 4 fits into 13 a total of 3 times with a remainder of 1. Thus, this operation is critical in transforming improper fractions into more easily understandable mixed numbers.

Fractions

Fractions represent parts of a whole and are written as two numbers separated by a line. They consist of a numerator and a denominator.
An improper fraction, like \(\frac{13}{4}\), has a numerator larger than the denominator, indicating that the value is greater than one whole.
Once an improper fraction is converted into a mix of a whole number and a proper fraction, understanding the concept of fractions helps picture how much of the pie (or the whole) is represented. Proper fractions, which have numerators smaller than denominators, show a value less than one.
Recognizing different types of fractions and their components is essential for performing arithmetic operations and solving real-world problems.

Numerator and Denominator

The numerator and denominator are the two crucial parts of a fraction that define its value. The numerator is the top number, representing how many parts of the whole are considered. The denominator, on the bottom, shows how many total parts the whole is divided into.
In \(\frac{13}{4}\), 13 is the numerator, indicating that we have 13 parts, while 4 is the denominator, meaning each whole is divided into 4 parts.
When converting improper fractions like \(\frac{13}{4}\) into mixed numbers, one performs division where:

• The remainder becomes the new numerator
• The original denominator stays the same

This understanding helps in simplifying improper fractions and facilitates smoother calculations and comparisons across computations.