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Understanding how to transform fractions into percentages is essential for interpreting data, comparing values, and solving real-world problems. The term 'percent' translates to 'per one hundred,' which implies that it is a fraction where the denominator is 100. The conversion process involves two primary steps: turning the fraction into a decimal and then converting that decimal into a percent.

To do this, as seen in the provided example, you first divide the numerator of the fraction by the denominator. For the fraction \(\frac{1}{3}\), when we divide 1 by 3, we get the decimal 0.3333, after rounding to four places. The next step is to multiply the decimal by 100 to find the percentage equivalent. So, 0.3333 times 100 gives us 33.33. Lastly, we add the percent symbol, %, to denote that the number is a percentage, not a decimal or any other form of number.

When converting fractions to decimals, we employ division. The top number (numerator) is divided by the bottom number (denominator) of the fraction. This process may result in a terminating decimal, where the numbers after the decimal point come to a natural end, or a repeating decimal, where one or more digits repeat infinitely.

In the case of \(\frac{1}{3}\), we achieve a repeating decimal of 0.3333... where the threes continue indefinitely. It is crucial to remember that while working with these repeating decimals, especially in calculations and conversions, rounding is often used to make the numbers more manageable. However, precise calculations may require keeping several decimal places or using the repeating decimal notation, like 0.\bar{3}, to indicate the infinite repetition of the digit 3.

Mathematical notation is the system of symbols and signs used to express mathematical ideas succinctly and precisely. Conventions in notation help ensure that mathematical language is universally understood among mathematicians and students alike. For instance, in our example, the fraction \(\frac{1}{3}\) expresses the division of 1 by 3. The percentage symbol (%) immediately lets us know that the number is out of 100 and the decimal point indicates a value less than one.

To write fractions, decimals, and percentages clearly, we use the symbols '/', '.', and '%', respectively. Each has a specific role in helping us understand and convert between different numerical representations. So, when you see 33.33%, it tells you that for every 100 parts, 33.33 parts are being considered. Similarly, the notation 0.\bar{3} conveys that the number 3 repeats indefinitely after the decimal point, serving as a clear and efficient way to express a repeating decimal.