Problem 95 - Algebra 1: Concepts and Skills

Question

Write the decimal as a fraction in simplest form. $$ 0.25 $$

Step-by-Step Solution

Answer

The fraction in simplest form is \(\frac{1}{4}\).

Step 1: Identify the Place Value

The given decimal is 0.25. The last digit, 5, is in the hundredths place, so we can write the decimal as \(\frac{25}{100}\). Remember, the first digit after a decimal point represents tenths and the second digit represents hundredths.

Step 2: Reduce the Fraction to Simplest Form

To reduce \(\frac{25}{100}\) to simplest form, find the greatest common factor (GCF) of 25 and 100. The GCF is 25. Divide both the numerator and the denominator by the GCF to give \(\frac{1}{4}\).

Key Concepts

Place Value

Place value is crucial when converting decimals into fractions. It helps determine the fraction's denominator. Consider decimal numbers as representations of parts of entire wholes. Their position, or place value, tells us the size of those parts.
For example, in the decimal 0.25, each digit has its own place value.

• The digit immediately after the decimal point, 2, represents tenths (i.e., 2/10).
• The second digit, 5, represents hundredths (i.e., 5/100).

These place values help in constructing the fraction. Here, 0.25 becomes \( \frac{25}{100} \). When you combine the values, each decimal place represents a power of ten. Knowing this, decimals can be easily rewritten as fractions with base 10 denominators like 10, 100, or 1000, depending on the number's complexity.

Simplifying Fractions

Simplifying fractions means reducing them to their most basic form. This involves dividing the numerator and the denominator by a common factor until you cannot go any further while still keeping whole numbers.
When you have a fraction like \( \frac{25}{100} \), you should break it down to see if both parts share any common divisors.
Start by dividing by smaller common factors and increasingly higher until you reach the greatest one.

• If the numerator and denominator have common factors, you can simplify them by removing these factors to get the simplest form, \( \frac{1}{4} \) in this case.

Simplifying not only makes fractions easier to work with but also helps in understanding their exact values relative to others.

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCF is an essential step in reducing fractions.
Let's understand how this works with \( \frac{25}{100} \). Both 25 and 100 can be divided by 25:

• 25 divided by 25 is 1
• 100 divided by 25 is 4

So, \( \frac{25}{100} \) can be simplified to \( \frac{1}{4} \) using this GCF.
This technique helps in transforming complex-looking fractions into neat and easily understandable numbers, which simplifies calculations and comparisons in mathematical problems.