Problem 45 - Prealgebra

Question

Change each decimal to a fraction, and then reduce to lowest terms. $$0.125$$

Step-by-Step Solution

Answer

0.125 as a fraction in simplest form is \(\frac{1}{8}\).

Step 1: Understand the Decimal

The given decimal is 0.125, which means it is read as 125 thousandths or \(\frac{125}{1000}\). The idea here is to express the decimal as a fraction where its denominator is a power of 10.

Step 2: Express as a Fraction

Convert the decimal to a fraction: 0.125 becomes \(\frac{125}{1000}\). This is because 125 is in the thousandths place.

Step 3: Reduce the Fraction to Lowest Terms

To reduce \(\frac{125}{1000}\), find the greatest common divisor (GCD) of 125 and 1000. The GCD of 125 and 1000 is 125. Divide both the numerator and denominator by 125: \(\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}\).

Step 4: Verify Your Result

Check if the fraction is in its simplest form. Since 1 and 8 have no common factors other than 1, \(\frac{1}{8}\) is the simplest form of \(\frac{125}{1000}\).

Key Concepts

Greatest Common Divisor

The concept of the Greatest Common Divisor (GCD) is essential in understanding how to simplify fractions. The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder. To find the GCD, you can use a simple algorithm: the Euclidean Algorithm. This involves repeated division of the given two numbers until the remainder is zero. The last non-zero remainder is the GCD.

However, for smaller numbers, like in the case of converting decimals to fractions, listing out the factors of both numbers can be helpful and quick:

• List the factors of the numerator.
• List the factors of the denominator.
• Identify the largest factor common to both lists.

For example, in our exercise with \(rac{125}{1000}\), the GCD is 125.
This means both the numerator and denominator can be evenly divided by 125, allowing us to simplify the fraction directly.

Reducing Fractions

Reducing fractions, or simplifying them, is all about making them simpler while keeping the same value. Once you know the GCD, reducing a fraction is straightforward: divide both the numerator and denominator by their GCD.

• This step is crucial to ensure that the fraction is as simple as possible for calculations or comparisons.
• It makes the fraction easier to understand and work with.
• Fractions that aren't reduced can be confusing, especially when dealing with large numbers.

In the example of \(rac{125}{1000}\), dividing both by 125 gives \(rac{1}{8}\).This process doesn't change the actual value of the fraction, it simply presents it in a simpler form. This reduced form is easier to interpret and use in further mathematical operations.

Simplest Form of Fractions

The simplest form of a fraction is when the fraction is as reduced as possible, meaning there are no common factors other than 1 between its numerator and denominator. A fraction in its simplest form is the most refined way to present it while maintaining its original value.

A crucial advantage of expressing fractions in their simplest form is that it makes them easier to compare with other fractions because you work with the smallest terms. This can be particularly useful in both arithmetic and algebra, helping to avoid unnecessary complexity.

After reducing \(rac{125}{1000}\) to \(rac{1}{8}\), you'll notice that \(1\) and \(8\) share no common divisor other than \(1\). Therefore, \(rac{1}{8}\) is already in its simplest form. This reduced form guarantees that any mathematical operation using this fraction will be as straightforward as possible.