Problem 32 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{121}{11}$$

Step-by-Step Solution

Answer

The mixed number is 11.

Step 1: Understand the Problem

We have an improper fraction \(\frac{121}{11}\), which means the numerator (121) is larger than the denominator (11). The goal is to convert it into a mixed number, which consists of a whole number and a proper fraction.

Step 2: Divide the Numerator by the Denominator

To convert the improper fraction to a mixed number, divide the numerator by the denominator: \(121 \div 11\).

Step 3: Perform the Division

Divide 121 by 11. After doing the division, you will find that 11 goes into 121 exactly 11 times, with no remainder.

Step 4: Write the Mixed Number

Since there is no remainder, the improper fraction \(\frac{121}{11}\) is exactly equal to the whole number 11. Thus, the mixed number is 11.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. Proper fractions have numerators smaller than their denominators, such as \( \frac{1}{4} \) or \( \frac{2}{3} \).
Mixed numbers arise when you need to express improper fractions in a more intuitive way. For example, \( \frac{121}{11} \) as a mixed number is 11, without needing a fractional part because the division is exact.
Why Use Mixed Numbers?
- They simplify the representation of whole and part quantities.
- Easier to interpret in real-world contexts, like recipes or measurements.
Mixed numbers provide a clear picture when visualizing amounts that are not entirely whole, making them a valuable tool in mathematical expressions and real-life applications.

Division Process

The division process is crucial in converting improper fractions to mixed numbers. Here's how it works:
To convert \( \frac{121}{11} \) into a mixed number, begin by dividing the numerator (121) by the denominator (11).
Steps in the Division Process:

• Determine how many times the denominator fits into the numerator.
• The quotient is the whole number part of the mixed number.
• If there is a remainder, it becomes the numerator of the proper fraction part.
• The denominator remains unchanged for the fraction part.

Performing \( 121 \div 11 \) yields 11 with no remainder, meaning \( \frac{121}{11} \) is exactly 11. This means the division was straightforward and resulted in a whole number without additional fractional parts.

Converting Fractions to Mixed Numbers

Converting fractions to mixed numbers enhances their understandability. The process is simple and involves a few steps:
Conversion Steps:

• Divide the numerator by the denominator to get the whole number part.
• Calculate the remainder from the division, which helps form the fractional part.
• Construct the mixed number with the quotient as the whole number and the remainder as the numerator.
• The denominator remains the same.

In the exercise \( \frac{121}{11} \), dividing yields 11 with zero remainder, so the mixed number is simply 11.
When the Remainder Appears:
For fractions like \( \frac{14}{5} \), dividing results in 2 with a remainder of 4, creating the mixed number 2\( \frac{4}{5} \). Converting helps in simplifying calculations and achieving clearer, more manageable forms of expressions.