As an expert in the field of mathematics and numerical representation, I can provide a comprehensive explanation of the decimal 0.25 and its various equivalent forms.

The decimal system is a base-10 system, which means that each place value is a power of 10. When we look at the decimal 0.25, we can break it down into its constituent parts. The '2' is in the tenths place, which means it represents 2 times 0.1, and the '5' is in the hundredths place, which means it represents 5 times 0.01. Mathematically, this can be expressed as:

\[ 0.25 = 2 \times 0.1 + 5 \times 0.01 = \frac{2}{10} + \frac{5}{100} \]

Now, let's explore the concept of equivalent fractions. A fraction is a way to represent a part of a whole, and it consists of a numerator (the top number) and a denominator (the bottom number). The decimal 0.25 can be expressed as a fraction by considering the place values as the denominator:

\[ 0.25 = \frac{25}{100} \]

This fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that number. The GCD of 25 and 100 is 25, so we divide both by 25:

\[ \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \]

So, the decimal 0.25 is equivalent to the fraction \( \frac{1}{4} \). This is a simplified form of the fraction, and it is one of the most common ways to represent the decimal 0.25.

Now, let's delve into the relationship between decimals and fractions. Decimals are a convenient way to represent fractions with denominators that are powers of 10. As mentioned earlier, the decimal 0.25 can be expressed as \( \frac{25}{100} \), but it can also be represented with different denominators that are powers of 10. For example:

- \( \frac{1}{4} \) is equivalent to \( \frac{25}{100} \) because \( 25 \times 4 = 100 \) and \( 1 \times 25 = 25 \).
- \( \frac{2}{8} \) is another equivalent fraction because \( 2 \times 4 = 8 \) and \( 8 \times \frac{1}{4} = 2 \).
- \( \frac{4}{16} \) is also equivalent because \( 4 \times 4 = 16 \) and \( 16 \times \frac{1}{4} = 4 \).

These are just a few examples of how the decimal 0.25 can be represented as fractions with different denominators. The key is that the value remains the same, but the representation can change.

It's also worth noting that decimals can be converted to percentages by multiplying by 100. In the case of 0.25, this would be:

\[ 0.25 \times 100 = 25\% \]

This percentage can be used in various contexts, such as representing proportions, discounts, or any other situation where a part of a whole is expressed as a percentage.

In conclusion, the decimal 0.25 is a versatile number that can be represented in multiple ways, including as a fraction, a percentage, or even in terms of other mathematical concepts like ratios and proportions. Understanding these different representations is crucial for anyone studying mathematics or working in fields that require numerical literacy.

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The decimal 0.25 represents the fraction 25/100. Decimal fractions always have a denominator based on a power of 10. We know that 5/10 is equivalent to 1/2 since 1/2 times 5/5 is 5/10. Therefore, the decimal 0.5 is equivalent to 1/2 or 2/4, etc.read more >>