Problem 23 - Prealgebra

Question

Write each decimal as a fraction in lowest terms. $$0.15$$

Step-by-Step Solution

Answer

The decimal 0.15 is equal to the fraction \( \frac{3}{20} \) in lowest terms.

Step 1: Express the Decimal as a Fraction

Convert the decimal number 0.15 into a fraction. 0.15 can be written as \( \frac{15}{100} \) because there are two digits after the decimal point, indicating the hundredths place.

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{15}{100} \) to its lowest terms. First, identify the greatest common divisor (GCD) of 15 and 100. The GCD of 15 and 100 is 5.

Step 3: Divide the Numerator and Denominator by their GCD

Divide both the numerator and denominator of \( \frac{15}{100} \) by the GCD, which is 5.\[ \frac{15 \, \div \, 5}{100 \, \div \, 5} = \frac{3}{20} \]

Step 4: Conclusion: Verify the Simplified Fraction

Verify that \( \frac{3}{20} \) is in its simplest form, meaning no integer greater than 1 divides both 3 and 20.Since 3 and 20 have no common divisors other than 1, \( \frac{3}{20} \) is indeed in its lowest terms.

Key Concepts

Understanding Fractions

Fractions are numerical expressions used to represent parts of a whole or a relationship between a part and a whole. A fraction consists of two parts:

• The numerator, which is the top number, indicates how many parts of the whole are being considered.
• The denominator, which is the bottom number, signifies the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator, meaning that we are considering 3 parts out of a total of 4. When you convert a decimal to a fraction, you're essentially identifying how many portions make up a fraction of 10, 100, or another power of ten. This allows us to express decimals as fractions.

Simplifying Fractions

Simplifying a fraction means making it into an equivalent fraction where the numerator and the denominator are as small as possible. This process involves reducing a fraction to its lowest terms. Here's how to simplify:

• Identify: Look for a common factor that both numerator and denominator share.
• Divide: Divide both numbers by this common factor until no further simplification is possible without affecting the value of the fraction.

For instance, if we have the fraction \( \frac{15}{100} \), we simplify by finding a number that evenly divides both 15 and 100, which would allow us to express the fraction in a simpler form. This simplification is crucial in making computations and comparisons easier.

The Role of the Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides two numbers without leaving a remainder. It is a fundamental concept in simplifying fractions. To find the GCD, you can use a couple of methods:

• Prime Factorization: Break down both numbers into their prime factors and multiply the common factors.
• Euclidean Algorithm: Use subtraction or the division of numbers until reaching the GCD.

In practice, finding the GCD gives us the number to divide both the numerator and denominator by, thereby simplifying the fraction as much as possible. For \( \frac{15}{100} \), the GCD is 5, which is used to reduce it to \( \frac{3}{20} \).

Expressing Fractions in Lowest Terms

A fraction is in its lowest terms when the numerator and the denominator have no common factor other than 1. Achieving lowest terms is a sign that the fraction is fully simplified. To ensure a fraction is in its lowest terms, check for divisors:

• If after dividing by the GCD, no other number except 1 divides both parts, then the fraction is in its lowest terms.
• Repeating the process, as needed, can confirm that no further reduction is possible.

For example, once \( \frac{15}{100} \) is divided by 5, we arrive at \( \frac{3}{20} \), which is verified to be in lowest terms as 3 and 20 share no common divisors other than 1. This ensures the fraction reflects the simplest form of parts out of a whole.