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The prime factors of a number are the basic building blocks that combine to form that number, using only multiplication. A prime number is a number greater than 1, which cannot be divided by any other numbers except for 1 and itself. When it comes to determining if a fraction converts into a terminating decimal, the prime factors of the denominator are crucial.
To have a terminating decimal, the denominator must be simplified to the product of powers of only 2 and/or 5. This means that after complete factorization, no other prime factors should be present besides 2 and 5.
For example, if you were working with a denominator such as 10, its prime factors would be 2 and 5 (since 10 = 2 × 5). Similarly, for the fraction \(\frac{5}{2}\), the denominator is already a prime factor, as 2 is a prime number. This is an essential step because it lets you predict whether a fraction will result in a terminating decimal.

Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1. This step might be necessary when working to determine the presence of only 2s and 5s as prime factors in the denominator.
To simplify, divide both the numerator and the denominator by their greatest common divisor. Once simplified, you should review the denominator to ensure it can only be composed of 2s or 5s.
In the case of \(\frac{5}{2}\), the fraction is already in its simplest form. The denominator is 2, which is one of our desired prime factors. When exploring a more complex fraction like \(\frac{20}{50}\), you simplify it to \(\frac{2}{5}\) by dividing both by 10, your greatest common divisor. This simplification process is essential to clearly see if the fraction meets the criteria for a terminating decimal.

Division transforms a fraction into its decimal form, providing you a clear and precise value. This is the final step after confirming the fraction has a denominator composed only of 2s and/or 5s.
For example, when dividing the fraction \(\frac{5}{2}\), you divide 5 by 2. Here’s how it works out:

• 5 divided by 2 equals 2 with a remainder of 1.
• Bring down a 0 to make it 10, and then divide by 2 again, which equals 5.

Thus, completing this division gives us the decimal 2.5 which is a terminating decimal. This process confirms that such a fraction, with the proper prime factors in the denominator, yields a neat, finite decimal.