Problem 94 - Algebra 1: Concepts and Skills

Question

Write the decimal as a fraction in simplest form. $$ 0.14 $$

Step-by-Step Solution

Answer

The decimal 0.14 written as a fraction in its simplest form is \(\frac{7}{50}\).

Step 1: Write the decimal as a fraction

Write the decimal 0.14 as a fraction. Since there are 2 digits after the decimal, the fraction would be \(\frac{14}{100}\).

Step 2: Simplify the fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 14 and 100. The GCD is 2. Now divide the numerator and the denominator by the GCD to simplify the fraction. This yields a simplified fraction of \(\frac{7}{50}\).

Key Concepts

Simplifying Fractions

When you begin working with fractions, you often encounter complex forms. Simplifying fractions means transforming them into a more understandable form without changing their value. The process involves dividing both the numerator and the denominator by a common factor.
A fraction is in its simplest form if the only common factor between the numerator and the denominator is 1. For example, the fraction \(\frac{14}{100}\) can be simplified because both numbers are divisible by a greater common factor. Simplifying involves these steps:

• Identify any common factors of the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common divisor or GCD.
• Continue the process until no further simplification is possible.

This makes comparisons and arithmetic operations easier and clearer.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a vital concept in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder.
To find the GCD, follow these steps:

• List all factors of the numerator.
• List all factors of the denominator.
• Identify the largest factor that appears in both lists.

For example, the GCD of 14 and 100 is 2. By using the GCD, you ensure the fraction is reduced to its simplest form.
Calculating the GCD efficiently handles larger numbers, making simplification less time-consuming.

Decimals to Fractions

Converting decimals to fractions is a fundamental exercise in mathematics. It allows you to express parts of a whole in two different numeric forms. Here’s how you can go about this conversion:
Begin with the decimal, for example, 0.14. The steps to convert it into a fraction involve:

• Count the number of digits after the decimal point. There are two in 0.14.
• Place the decimal digits over "1" followed by as many zeros as there are digits: \(\frac{14}{100}\).
• Simplify the fraction by finding the GCD of the numerator and the denominator and divide both by this number.

The final result is the fraction that represents the decimal in simplest form.
Converting between these forms helps in understanding the relationship between parts and wholes in various contexts.