Convert Recurring Decimals to Fractions: The p/q Form Explained
In mathematics, recurring decimals can be converted to fractions in the form of p/q. This process is useful for various applications in mathematics, including simplifying calculations and improving precision. In this article, we will explore how to convert the recurring decimal 0.3u0305 (where the digit 3 repeats indefinitely) to its p/q form. We will also provide a detailed explanation for a few related examples, ensuring a comprehensive understanding of the concept.
Step-by-Step Guide to Converting 0.3u0305 to p/q
Let's denote x 0.3u0305. Next, multiply both sides by 10, which gives us 1 3.3u0305. Subtract the original equation (1) from the new equation (2): 1 - x 3.3u0305 - 0.3u0305 9x 3 Solving for x, we get: x 3/9 x 1/3Thus, the p/q form of 0.3u0305 is 1/3.
Additional Examples
Let's explore a few more examples to further solidify the concept:
Example: Convert 0.324u0305 to p/q Form
Let y 0.324u0305 (where 24 repeats indefinitely). Multiply both sides of the equation by 1, 10, and 1000 to get: 1000y 324.324u0305 y 0.324u0305 10y 3.24u0305 Subtract the second equation from the first: 1000y - 10y 324.324u0305 - 3.24u0305 990y 321 y 321/990 Simplify the fraction: y 107/330Example: Convert 0.3u0305 to p/q Form
Let z 0.3u0305. Multiply both sides by 10 to get: 10z 3.3u0305 Subtract the original equation from the new equation: 10z - z 3.3u0305 - 0.3u0305 9z 3 Solve for z: z 3/9 z 1/3Thus, the p/q form of 0.3u0305 is 1/3 again.
Conclusion
Converting recurring decimals to fractions is a valuable skill in mathematics. By following a systematic approach, we can easily convert any recurring decimal to its equivalent p/q form. This method is not only useful for academic purposes but also in practical applications where precision is crucial.