Fractional Form Of Repeating Decimal 0.02: Step-by-Step Solution
In the realm of mathematics, converting repeating decimals into fractions is a fundamental skill. This article delves into the process of transforming the repeating decimal 0.02 into its fractional form, providing a step-by-step guide and illuminating the underlying mathematical principles. We will explore the concept of repeating decimals, understand the significance of place value, and apply algebraic techniques to arrive at the accurate fractional representation. Furthermore, we will analyze the given options, eliminating incorrect answers and justifying the selection of the correct one. By the end of this exploration, you will have a solid grasp of how to convert repeating decimals to fractions, empowering you to tackle similar problems with confidence.
Understanding Repeating Decimals
Repeating decimals, also known as recurring decimals, are decimal numbers in which one or more digits repeat infinitely. These decimals arise when a fraction's denominator has prime factors other than 2 and 5. The repeating pattern is denoted by a bar over the repeating digits or by writing the digits followed by an ellipsis (...). Understanding repeating decimals is crucial in various mathematical contexts, including algebra, calculus, and number theory. These decimals represent rational numbers, which can be expressed as a fraction of two integers. The conversion process involves setting up an equation, manipulating it algebraically, and isolating the repeating decimal part to eliminate it. This process allows us to express the repeating decimal as a fraction, providing a precise representation of its value.
The Significance of Place Value
Place value is a fundamental concept in mathematics that dictates the value of a digit based on its position in a number. In the decimal system, each position to the right of the decimal point represents a negative power of 10. For instance, the first position is the tenths place (10⁻¹), the second is the hundredths place (10⁻²), and so on. Understanding place value is crucial for comprehending the magnitude of digits in a decimal number and is particularly important when dealing with repeating decimals. In the repeating decimal 0.02, the digits 02 repeat infinitely. The first 0 is in the tenths place, the 2 is in the hundredths place, and the repetition continues in the thousandths, ten-thousandths, and subsequent places. Accurately identifying the place values allows us to set up the correct algebraic equation for conversion.
Algebraic Techniques for Conversion
To convert a repeating decimal to a fraction, we employ algebraic techniques to eliminate the repeating part. Let's denote the repeating decimal as x. In this case, x = 0.02. The repeating part consists of two digits, 02. To eliminate the repeating part, we multiply x by 100, which shifts the decimal point two places to the right, resulting in 100x = 2.02. Now, we subtract the original equation (x = 0.02) from the multiplied equation (100x = 2.02). This subtraction aligns the repeating parts, allowing them to cancel each other out: 100x - x = 2.02 - 0.02. Simplifying the equation, we get 99x = 2. Finally, we solve for x by dividing both sides by 99: x = 2/99. This algebraic approach effectively converts the repeating decimal 0.02 into its fractional form, 2/99. Understanding these algebraic manipulations is essential for converting any repeating decimal into a fraction.
Step-by-Step Conversion of 0.02
Let's break down the conversion of the repeating decimal 0.02 into a fraction step by step:
- Assign a variable: Let x = 0.02.
- Multiply to shift the decimal: Since two digits repeat, multiply both sides of the equation by 100: 100x = 2.02.
- Subtract the original equation: Subtract the original equation (x = 0.02) from the multiplied equation (100x = 2.02): 100x - x = 2.02 - 0.02.
- Simplify: This simplifies to 99x = 2.
- Solve for x: Divide both sides by 99: x = 2/99.
This step-by-step process demonstrates the methodical approach to converting repeating decimals into fractions. Each step is crucial in isolating the repeating part and expressing the decimal as a ratio of two integers.
Analyzing the Given Options
Now, let's analyze the given options to identify the correct fractional form of 0.02:
- A. 2/99: This matches the result we obtained through the algebraic conversion.
- B. 1/11: This fraction is equivalent to 0.09, which is not the same as 0.02.
- C. 2/100: This fraction is equivalent to 0.02, but it is not the fractional form of the repeating decimal 0.02.
- D. 1/12: This fraction is approximately equal to 0.083, which is not the same as 0.02.
By comparing each option with the converted fraction, we can confidently eliminate the incorrect answers and pinpoint the correct one.
Selecting the Correct Answer
Based on our analysis and the step-by-step conversion, the correct answer is A. 2/99. This fraction accurately represents the repeating decimal 0.02. The other options do not match the value of the repeating decimal and can be ruled out. Understanding the conversion process and the significance of repeating decimals allows us to confidently select the correct fractional representation.
Conclusion
Converting repeating decimals to fractions is a fundamental mathematical skill with practical applications in various fields. In this article, we explored the process of converting the repeating decimal 0.02 into its fractional form. We discussed the concept of repeating decimals, the significance of place value, and the application of algebraic techniques to eliminate the repeating part. Through a step-by-step conversion, we arrived at the fractional representation of 2/99. By analyzing the given options, we confirmed that option A, 2/99, is the correct answer. This comprehensive guide empowers you to confidently convert repeating decimals into fractions and apply this knowledge to solve mathematical problems.
Which of the following is the fractional form of the repeating decimal $0 . \overline{02}$? A. $ rac{2}{99}$ B. $ rac{1}{11}$ C. $ rac{2}{100}$ D. $ rac{1}{12}$
