Problem 68 - Prealgebra

Question

Convert the given decimal to a fraction. Reduce your answer to lowest terms. 0.78

Step-by-Step Solution

Answer

0.78 as a fraction is \( \frac{39}{50} \).

Step 1: Express the Decimal as a Fraction

To convert 0.78 to a fraction, we acknowledge that 0.78 means 78 hundredths. Therefore, we can initially express it as the fraction \( \frac{78}{100} \).

Step 2: Simplify the Fraction

Next, we simplify the fraction \( \frac{78}{100} \) by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 78 and 100 is 2. We divide both by their GCD to simplify. \( \frac{78 \div 2}{100 \div 2} = \frac{39}{50} \).

Step 3: Verification of Simplified Form

Verify that \( \frac{39}{50} \) is in its simplest form. Check if there is any common factor other than 1 between 39 and 50. Since both are co-prime (no common divisor other than 1), the fraction is indeed in its simplest form.

Key Concepts

Simplifying Fractions

When you have a fraction, the goal is often to make it as simple as possible: this is called simplifying the fraction. This means getting the numerator (the top number) and the denominator (the bottom number) to the smallest numbers that still represent the same value.
For instance, given our fraction \(\frac{78}{100}\), we see both numbers can be divided by a common value, which makes it simpler to write. By simplifying, the fraction becomes easier to work with, particularly for adding, subtracting, or comparing with other fractions.
The first step in simplification is finding if there's a number that can divide both the top and the bottom of the fraction with no remainder. By dividing both by this number, known as the greatest common divisor (GCD), you'll have the fraction in its simplest form. In our example, dividing 78 and 100 by 2 simplifies it to \(\frac{39}{50}\). This is much neater and straightforward.

Greatest Common Divisor

The greatest common divisor (GCD) is a very useful tool when simplifying fractions. It is the largest number that can divide two different numbers without leaving a remainder.
To simplify the fraction \(\frac{78}{100}\), you begin by finding the GCD of 78 and 100. This involves listing out the factors of each number.

• The factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78.
• The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100.

By comparing these lists, the largest common factor is 2. That means 2 is the biggest number that can evenly divide both 78 and 100.
Once you find this GCD, you can divide the numerator and denominator by it, giving you a simplified version of the original fraction, which is \(\frac{39}{50}\) here. Using the GCD ensures you get the fraction in its simplest form with minimal effort.

Co-prime Numbers

Co-prime numbers, sometimes called relatively prime, are numbers that do not have any common factors other than 1. When simplifying fractions, it's important to spot when two numbers are co-prime because this means the fraction can't be simplified any further.
After simplifying \(\frac{78}{100}\) to \(\frac{39}{50}\), we check if 39 and 50 share any common factors. A quick look at their divisors shows:

• 39 has the factors 1, 3, 13, 39.
• 50 has the factors 1, 2, 5, 10, 25, 50.

The only common factor they have is 1, confirming they are co-prime.
This tells us \(\frac{39}{50}\) is fully reduced; \(\frac{39}{50}\) is as simple as this fraction can get. Recognizing co-prime numbers helps ensure your fraction is in its lowest possible terms, saving time and reducing errors in math problems.