Problem 22 - Elementary Algebra

Question

For the following problems, convert each decimal fraction to a fraction. 0.115

Step-by-Step Solution

Answer

Question: Convert the decimal fraction 0.115 to a common fraction form. Answer: \(\frac{23}{200}\)

Step 1: Rewrite the decimal as a fraction over 1

We need to express the given decimal value as a fraction. We can do this by rewriting 0.115 as a fraction with a numerator equal to the given decimal value, and the denominator as 1. So, we have: $$ \frac{0.115}{1} $$

Step 2: Multiply the numerator and denominator by a power of 10

Now, we need to eliminate the decimals in the fraction. The given decimal has three decimal places, so we'll multiply the numerator and the denominator by a power of 10 equal to the number of decimal places, which, in this case, is \(10^3\) or 1000. $$ \frac{0.115 \times 1000}{1 \times 1000} = \frac{115}{1000} $$

Step 3: Simplify the fraction

The next step is to simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both the numerator and the denominator by the GCD. The GCD of 115 and 1000 is 5, so simplifying the fraction is as follows: $$ \frac{115}{1000} = \frac{115 \div 5}{1000 \div 5} = \frac{23}{200} $$ The decimal fraction 0.115 has been successfully converted to the common fraction \(\frac{23}{200}\).

Key Concepts

Understanding Numerator and Denominator

When we talk about fractions, we often come across the terms **numerator** and **denominator**. But what do they really mean?
The numerator is the top part of a fraction. It represents the number of equal parts we are considering. For example, in the fraction \( \frac{3}{4} \), the numerator is 3, meaning we have 3 parts out of a whole.
The denominator, on the other hand, is the bottom part of a fraction. It tells us the total number of equal parts the whole is divided into. In \( \frac{3}{4} \), the denominator is 4, indicating that the whole is divided into 4 equal parts.
Remember:

• Numerator = number of parts considered
• Denominator = total number of equal parts in a whole

Knowing how to read and write numerators and denominators is essential for understanding and working with fractions effectively.

The Process of Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.
To simplify a fraction, follow these steps:

• Identify the common factors of the numerator and denominator.
• Divide both the numerator and the denominator by their greatest common factor (GCF).

For example, let's simplify \( \frac{24}{36} \):
- Find the common factors: 2, 3, 4, 6, 12- The GCF is 12
Now, divide both by 12:\[ \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \]
Our fraction is now simplified. Simplifying fractions helps in making arithmetic operations like addition, subtraction, multiplication, and division easier.

Finding the Greatest Common Divisor

The **greatest common divisor (GCD)** is an essential concept in simplifying fractions. It's the largest number that can divide both the numerator and the denominator without leaving a remainder.
Here's how to find the GCD:

• List all the divisors of the numerator.
• List all the divisors of the denominator.
• Identify the largest number that appears in both lists.

Consider finding the GCD of 48 and 180:
Divisors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Divisors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
The largest common divisor is 12, so the GCD is 12.
Using the GCD, we can simplify fractions efficiently, ensuring that they are in their simplest form possible.