Problem 43 - Prealgebra

Question

Change each decimal to a fraction, and then reduce to lowest terms. $$0.25$$

Step-by-Step Solution

Answer

The decimal 0.25 as a fraction in lowest terms is \( \frac{1}{4} \).

Step 1: Understanding the Decimal

0.25 is a decimal number with 2 digits after the decimal point, which indicates it is in the hundredths place.

Step 2: Converting Decimal to Fraction

To convert 0.25 to a fraction, recognize that it is equivalent to 25 hundredths, which can be written as \( \frac{25}{100} \).

Step 3: Finding the Greatest Common Divisor (GCD)

To reduce the fraction to its lowest terms, we need to find the greatest common divisor (GCD) of 25 and 100. The GCD of 25 and 100 is 25.

Step 4: Reducing the Fraction

Divide both the numerator and the denominator by the GCD (25). This gives us \( \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \).

Key Concepts

Reducing Fractions

Reducing fractions is a vital step in simplifying them to their simplest form. This makes calculations easier and fractions easier to read and understand. When a fraction is expressed in its lowest terms, the numerator and denominator have no common divisors other than 1. To reduce a fraction, follow these simple steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

Let's take an example of reducing the fraction \( \frac{25}{100} \). Start by finding the GCD of 25 and 100, which is 25. Then, divide both 25 and 100 by their GCD:\[\frac{25 \div 25}{100 \div 25} = \frac{1}{4}\]This process makes understanding and comparing fractions much easier, helping you work with these numbers more comfortably in math problems.

Greatest Common Divisor (GCD)

To effectively reduce fractions, you first need to find the Greatest Common Divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator of the fraction without leaving a remainder. Identifying the GCD helps simplify fractions by reducing them to their simplest form.Aquiress GCD using simple steps:

• List out the factors of both the numerator and the denominator.
• Identify the largest factor that both numbers have in common.

In our example of the fraction \( \frac{25}{100} \), the factors of 25 are 1, 5, and 25, while the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The greatest factor they share is 25, making it the GCD. Once you have the GCD, you can reduce the fraction to its simplest form, making computation much easier.

Decimal Place Value

Understanding decimal place value is a key concept in converting decimals to fractions. Decimal numbers are written using a base 10 system, which means each position in the number represents a power of 10. As you move to the right of the decimal point, the place values get smaller:

• First place right of the decimal is tenths \((10^{-1})\).
• Second place is hundredths \((10^{-2})\).
• Third place is thousandths \((10^{-3})\), and so on.

For example, in the decimal number 0.25, the 2 is in the tenths place and the 5 is in the hundredths place. This means 0.25 can be thought of as 25 hundredths, which converts directly to the fraction \( \frac{25}{100} \). Understanding this concept helps in accurately transforming decimals into fractions by recognizing their place values. Getting familiar with place values allows for more effortless conversion and simplification of numbers in fraction form.