Problem 30 - Elementary Algebra

Question

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places. \(\frac{2}{9}\)

Step-by-Step Solution

Answer

Question: Convert the fraction 2/9 to a decimal rounded to 3 decimal places. Answer: The decimal representation of 2/9 is approximately 0.222.

Step 1: Perform the division

Divide 2 by 9 to determine the decimal form. \(2 \div 9 = 0.22222...\) The division result in a nonterminating decimal.

Step 2: Round to 3 decimal places

Round the nonterminating decimal to 3 decimal places. \(0.22222... \approx 0.222\) So, the decimal fraction of \(\frac{2}{9}\) is approximately \(0.222\).

Key Concepts

Understanding Nonterminating Decimals

When you convert a fraction into a decimal by dividing the numerator by the denominator, you sometimes encounter a decimal that never seems to end. These are called nonterminating decimals. In our given example, when we divided 2 by 9, we arrived at 0.22222..., which didn't end or repeat because there are no exact whole numbers that give a complete result.
\( \frac{2}{9} \) is a fascinating example because the decimal goes on indefinitely as 0.22222... but never quite concludes. Such infinite decimals can be challenging because it's impossible to write them entirely. Instead, we approximate them or use special notation to represent them succinctly. In mathematics, nonterminating decimals can be either repeating, like 0.333..., or non-repeating, but in our example, it repeats the digit 2.

Importance of Rounding Decimals

Rounding decimals is an essential skill in mathematics to make long numbers simpler and more manageable. When you encounter a nonterminating decimal, approximation becomes necessary. For example, after dividing 2 by 9, you get a decimal that goes on continuously as 0.22222... But for practical purposes, you can round it.
To round a decimal to a specific number of places, look at the digit immediately following your last desired place. If that digit is 5 or more, you round up the last desired place by one. Otherwise, you keep it the same. In our case, rounding 0.22222... to three decimal places gives us 0.222. This method of approximation allows you to represent these values with less precision while staying close to the actual number. Rounding is particularly useful for maintaining clarity when working with irrational or lengthy decimal numbers.

Understanding Decimal Fractions

Decimal fractions are a way of expressing numbers that lie between whole numbers. They use digits to the right of the decimal point to express values less than one. Decimal fractions are a convenient way to write fractions, particularly those that stem from common divisors like 10, 100, or 1000.
Converting a simple fraction, like \( \frac{2}{9} \), into a decimal fraction involves simple division. Although this fraction results in a nonterminating decimal, for purposes like analysis and calculations, it's often represented as 0.222 when rounded. Decimal fractions help in presenting numbers in an easy-to-read manner, which is especially helpful in real-world applications like currency, measurement, and statistical data where precision might be required.