Problem 85 - Prealgebra

Question

Write as a percent. Write the remainder in fractional form. $$\frac{2}{9}$$

Step-by-Step Solution

Answer

\( \frac{2}{9} \) is 22.22...%

Step 1: Divide the Fraction

To convert the fraction \( \frac{2}{9} \) into a percent, we first need to perform the division \( 2 \div 9 \). This will give us a decimal result. When you divide 2 by 9, the result is approximately 0.222... (repeating).

Step 2: Convert Decimal to Percent

To convert a decimal into a percent, you multiply it by 100. Thus, \( 0.222... \times 100 = 22.22...\% \). This repeating decimal means the percent is a repeating percentage.

Step 3: Identify the Percent and the Remainder

The repeating decimal suggests the percent is \( 22.22\% \) with the decimal repeating. To express the remainder from dividing 2 by 9 in fractional form, we recognize that the decimal 0.222... represents \( \frac{2}{9} \), so there is no remainder when expressing it in fraction with respect to 9.

Key Concepts

Fractions to Decimals

Converting fractions to decimals is a fundamental mathematical skill. A fraction is a part of a whole, represented as two numbers divided by a line: the numerator (top number) and the denominator (bottom number). To convert a fraction like \( \frac{2}{9} \) to a decimal, you simply divide the numerator by the denominator.

• Numerator: The number above the line in a fraction (in this case, 2).
• Denominator: The number below the line in a fraction (in this case, 9).
• Operation: Division of 2 by 9.

When you divide 2 by 9, the result is approximately 0.222...; this means the number 2 repeats indefinitely. Dividing fractions gives you a way to express them in a different form, which can be very helpful. For instance, when performing calculations in financial or scientific contexts, decimals are often preferred. Understanding how to switch between these forms makes it easier to perform various mathematical operations.

Repeating Decimals

Decimals can either terminate or repeat. A terminating decimal has a finite number of digits after the decimal point. On the other hand, a repeating decimal has one or more digits that repeat infinitely. In the case of \( \frac{2}{9} \), when you do the division, you end up with the repeating decimal 0.222.... It's important to know how to recognize and write repeating decimals. Typically, repeating decimals are written with a line over the repeating digit(s), known as a vinculum or bar notation. For example, 0.222... can be represented as \(0.\overline{2}\).Repeating decimals can be converted back to fractions using algebraic methods, but in the case of simple fractions like \( \frac{2}{9} \), it's easy to see the relation between the fraction and its decimal form. This conversion shows that fractions can lead to repeating decimals and how both forms represent the same value.

Division of Fractions

When dealing with fractions, understanding how to perform division is crucial. Division of fractions might seem complicated at first, but it's straightforward with practice. To divide fractions, you multiply by the reciprocal. However, when converting a single fraction like \( \frac{2}{9} \) to a decimal or percentage, you're focusing on division similar to long division.

• Long Division: Using long division, you divide the numerator by the denominator to convert a fraction to a decimal.
• Reciprocal Method: When dividing fractions by another fraction, turn the divisor upside down and then multiply.

In practical terms, such as converting to a percentage, this division helps illustrate how fractions interact with whole numbers. Once \(0.222...\) is attained from \( \frac{2}{9} \), understanding it as a repeating decimal ensures the fraction has been broken down accurately into its decimal form before turning it into a percentage. The knowledge of fraction division is important in many real-world applications including financial calculations and data analysis.