Problem 33 - Prealgebra

Question

Write each decimal as a mixed number. $$1.22$$

Step-by-Step Solution

Answer

\(1.22\) as a mixed number is \(1\frac{11}{50}\).

Step 1: Recognize Decimal Components

Identify the whole number and the decimal part of the number. For the decimal \(1.22\), the whole number is \(1\), and the decimal part is \(.22\).

Step 2: Convert Decimal to Fraction

Convert the decimal \(.22\) to a fraction. Recognize that \(.22\) has two decimal places, so it is equivalent to \(\frac{22}{100}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{22}{100}\). The greatest common divisor of 22 and 100 is 2, so divide both the numerator and the denominator by 2, resulting in \(\frac{11}{50}\).

Step 4: Compose the Mixed Number

Combine the whole number with the simplified fraction. The decimal \(1.22\) as a mixed number is \(1\frac{11}{50}\).

Key Concepts

Understanding Decimal Components

When converting decimals to mixed numbers, it's crucial to understand what a decimal consists of. Decimals have two main parts: the whole number and the fractional part. For instance, in the decimal 1.22, the whole number is 1. This represents the complete units. The fractional part, in this case, is .22, which is less than one full unit.
Understanding these components not only helps in visualization but also aids in the conversion process.

• Whole Number: The part of the number before the decimal point.
• Decimal Part: The part after the decimal point, often expressed as a fraction.

Grasping these two components is foundational when you begin the process of transforming decimals into mixed numbers.

Simplifying Fractions in Conversions

A vital step in converting decimals to mixed numbers is simplifying fractions. When you transform the decimal part, like .22, into a fraction, it initially becomes \(\frac{22}{100}\). However, to make the mixed number simpler and more elegant, this fraction needs to be reduced.
Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator. For \(\frac{22}{100}\), the GCD is 2. By dividing both the numerator (22) and the denominator (100) by 2, we get \(\frac{11}{50}\).

• Find GCD: Determine the largest number that evenly divides both the numerator and the denominator.
• Divide Both: Simplify by dividing the numerator and denominator by this common factor.

This reduction makes the fraction easier to understand and work with when combined back into a mixed number.

Forming Mixed Numbers from Decimals

Once the decimal components are broken down and simplified, we merge them to form a mixed number. A mixed number combines the straightforward completeness of a whole number with the exactitude of a fraction. For example, turning 1.22 into a mixed number involves a few simple steps.
First, take the whole number part, which is 1. Then, add the simplified fraction \(\frac{11}{50}\) that represents the decimal part.

• Combine Whole and Fraction: 1 + \(\frac{11}{50}\) forms the mixed number.
• Express it Simply: Write it as \(1 \frac{11}{50}\).

This result, \(1 \frac{11}{50}\), captures the value of 1.22 fully in an easily understandable form. Mixed numbers like these are useful both in academic studies and everyday problem solving.