Problem 11 - Prealgebra

Question

Change each mixed number to an improper fraction. $$12 \frac{31}{33}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{427}{33}\).

Step 1: Understand Mixed Numbers

A mixed number is a combination of a whole number and a fraction. Here, in the number \(12 \frac{31}{33}\), \(12\) is the whole number, and \(\frac{31}{33}\) is the fractional part.

Step 2: Calculate the Denominator

The denominator of the improper fraction will remain the same as the denominator of the fraction part of the mixed number. Therefore, for \(12 \frac{31}{33}\), the denominator of the improper fraction is \(33\).

Step 3: Multiply the Whole Number by the Denominator

Multiply the whole number (\(12\)) by the denominator (\(33\)): \(12 \times 33 = 396\). This operation converts the whole part of the mixed number into an equivalent fraction with the same denominator.

Step 4: Add the Numerator of the Fractional Part

To get the new numerator for the improper fraction, add the result from Step 3 to the numerator of the fraction part. So, \(396 + 31 = 427\).

Step 5: Formulate the Improper Fraction

Combine the new numerator from Step 4 with the denominator from Step 2 to form the improper fraction. Hence, the improper fraction is \(\frac{427}{33}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine a whole number with a fraction. They provide an easy way to represent values greater than one in a more intuitive and comprehensive view. For example, in the mixed number \(12 \frac{31}{33}\), the whole number part is \(12\), and the fractional part is \(\frac{31}{33}\).
These two components together help to illustrate parts of a whole beyond just integers, making them practical in many real-world applications where exact amounts are essential. Here, it means 12 whole units and 31 parts of another 33.
Recognizing a mixed number is important in various mathematical operations, especially when needing to perform conversions or simplify calculations.

Improper Fractions

Improper fractions are fractions in which the numerator is larger than or equal to the denominator. This concept may seem unusual at first, but it is incredibly useful.
For instance, in the improper fraction \(\frac{427}{33}\), the numerator (427) is larger than the denominator (33).
Improper fractions can be thought of as a way to describe the same quantity as a mixed number but without the explicit whole number part. They're quite handy when performing arithmetic operations, like addition or subtraction, because they simplify the processes by eliminating the complexity of handling whole numbers separately.

Numerator and Denominator

Every fraction is composed of two main components: the numerator and the denominator. Understanding these terms is key to grasping fractional concepts, whether improper or mixed.

• The numerator is the top number of the fraction. It represents how many parts out of the whole you have.
• The denominator, on the other hand, is the bottom number. It shows how many equal parts the whole is divided into.

In the process of converting a mixed number to an improper fraction, both parts play crucial roles. For example, in the fraction part \(\frac{31}{33}\) of the mixed number \(12 \frac{31}{33}\), 31 is the numerator and 33 is the denominator. These numbers maintain their roles when performing conversions.

Fraction Conversion

Converting mixed numbers into improper fractions involves a specific sequence of steps that ensure the quantity remains the same, only the form changes.
Here's a concise version of the process:

• Keep the denominator of the fraction part unchanged. This ensures the whole fraction represents the parts of the same whole.
• Convert the whole number into fraction form by multiplying it by the denominator.
• Add the original numerator to this product to form the new numerator.

So, the conversion of \(12 \frac{31}{33}\) follows this: keep 33 as the denominator, convert 12 to \(396\) by multiplying with \(33\), and add 31 resulting in a numerator of \(427\). This manipulation changes the representation from a mixed number to an improper fraction, \(\frac{427}{33}\), without altering the quantity represented.