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When it comes to understanding ratios and their relation to fractions, the first step is converting a given ratio into fraction form. How is this done? Quite simply, actually. A ratio, such as the one presented in the exercise, is given as '12 to 30' or in ratio notation as 12:30. To convert this into a fraction, you would place the first number, known as the numerator, above the second number, the denominator, with a division sign between them—much like you would write out a division problem. This gives us \(\frac{12}{30}\).This fraction represents the same proportional relationship as the original ratio but is now in a format that can be easily simplified and used in further mathematical calculations, such as finding a part of a whole or comparing two quantities.

To simplify a fraction, you first need to find the greatest common factor (GCF) of the numerator and the denominator. But what is the GCF? It is the largest number that evenly divides both the numerator and the denominator without leaving a remainder.For the given example, where we have the fraction \(\frac{12}{30}\), we look for the highest number that can divide both 12 and 30 completely. Through either listing the factors of each or using a method like prime factorization, we find that the GCF is 6. Knowing the GCF is crucial because it's the key to reducing the fraction to its simplest form; it tells you by how much you can 'scale down' both the top and bottom of the fraction to make it cleaner and easier to work with.

The final step, after converting the ratio to a fraction and finding the GCF, is to simplify the fraction. Simplifying a fraction means reducing it to its lowest terms, making it easier to understand and use. To simplify the fraction \(\frac{12}{30}\), we divide both the numerator and the denominator by the GCF, which is 6, as identified in the previous step.By dividing 12 by 6, we arrive at 2, and by dividing 30 by 6, we get 5. Hence, the fraction simplifies to \(\frac{2}{5}\).It cannot be simplified any further because 2 and 5 share no common factors other than 1. Simplifying fractions helps in presenting data or results in a more understandable way and is often required in mathematical solutions to ensure they are in the simplest and most standardized form.