Problem 7 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.14 $$

Step-by-Step Solution

Answer

The fraction is \( \frac{7}{50} \).

Step 1: Write the Decimal as a Fraction

Convert the decimal number 0.14 to a fraction by writing it as \( \frac{14}{100} \). This is because 0.14 means 14 over 100.

Step 2: Simplify the Fraction

Simplify \( \frac{14}{100} \) by finding the greatest common divisor (GCD) of 14 and 100. The GCD of 14 and 100 is 2.

Step 3: Divide Numerator and Denominator by GCD

Divide both the numerator and the denominator of \( \frac{14}{100} \) by their GCD, which is 2: \( \frac{14 \div 2}{100 \div 2} \). This simplifies to \( \frac{7}{50} \).

Key Concepts

Simplifying Fractions

To simplify a fraction, our main goal is to make it as simple as possible. This means reducing the numerator (the top number) and the denominator (the bottom number) to their smallest possible values.
You accomplish this by dividing both numbers by their greatest common divisor (GCD).
This process ensures the fraction is in its simplest form. For instance, in our example, the fraction was simplified from \(\frac{14}{100}\) to \(\frac{7}{50}\).
Breaking it down further, you divide both the numerator and the denominator by their GCD, which is 2 in this case. Therefore:

• \(14 \div 2 = 7\)
• \(100 \div 2 = 50\)

This results in the simplified fraction \(\frac{7}{50}\).
Understanding this concept is crucial when dealing with fractions because it helps in comparing, adding, subtracting, and multiplying fractions more easily.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder.
In simpler terms, it's the biggest number that fits into both numbers evenly. Finding the GCD is essential for simplifying fractions.
To find the GCD of two numbers like 14 and 100, you can use different methods:

• Prime Factorization: Break down both numbers into their prime factors. The common prime factors are then used to find the GCD.
• Euclidean Algorithm: A faster method where you repeatedly subtract the smaller number from the larger one until you get the GCD.
• Listing Factors: Write down all factors of both numbers and identify the greatest one they have in common. This method is more manual but can be very effective for smaller numbers.

In our example, the GCD of 14 and 100 is 2. Once we find this divisor, we can use it to simplify the fraction.

Numerator and Denominator

Understanding the terms numerator and denominator is fundamental when working with fractions.
The numerator is the top number of a fraction and represents how many parts we have.
The denominator is the bottom number and represents how many parts make up a whole.
For example, in the fraction \(\frac{14}{100}\), 14 is the numerator and 100 is the denominator.
Knowing this helps us both create and simplify fractions.
To convert a decimal to a fraction, you write the decimal number as the numerator and the denominator based on the place value of the last digit.
Take 0.14; since the 4 is in the hundredths place, our fraction is \(\frac{14}{100}\).
From there, we simplify by finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by this number.
In our example, we divided both by 2, resulting in the simplified fraction \(\frac{7}{50}\).
Keeping these concepts clear in your mind makes dealing with any fraction problem much more manageable.