Problem 46 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$63.6 \%$$

Step-by-Step Solution

Answer

The fraction in lowest terms is \( \frac{159}{250} \).

Step 1: Convert Percent to Fraction

Start by expressing the percent as a fraction by writing it over 100. Therefore, \( 63.6\% \) becomes \( \frac{63.6}{100} \).

Step 2: Remove the Decimal

To remove the decimal from \( \frac{63.6}{100} \), multiply the numerator and the denominator by 10 to make both whole numbers. This gives \( \frac{636}{1000} \).

Step 3: Simplify the Fraction

We need to simplify \( \frac{636}{1000} \). Find the greatest common divisor (GCD) of 636 and 1000, which is 4. Divide both the numerator and denominator by 4 to simplify the fraction. This results in \( \frac{159}{250} \).

Step 4: Verify Simplification

Lastly, ensure \( \frac{159}{250} \) is in its simplest form. Check that 159 and 250 have no common factors aside from 1. Thus, \( \frac{159}{250} \) is indeed in lowest terms.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we're trying to make a fraction as simple as possible. This means making sure the numerator (the top number) and the denominator (the bottom number) do not have any common factors other than 1. A simplified fraction is often easier to understand and work with.

To simplify any fraction:

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

The GCF is the largest number that can divide both numbers without a remainder. For example, in the exercise solution, 636 and 1000 share the GCF of 4. Hence, dividing them by 4 results in a simplified fraction, making the problem-solving process much easier.

Greatest Common Divisor

The greatest common divisor, usually known as the GCD, is a handy tool when working with fractions. It helps us find the largest number that can evenly divide both the numerator and the denominator of a fraction. This concept ensures fractions are in their simplest form.

Here's how to find the GCD:

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

The GCD is the backbone of fraction simplification. In our example, the numbers 636 and 1000 were analyzed, and their GCD was determined to be 4. Dividing both numbers by this GCD resulted in the fraction being simplified to its lowest terms. Calculating the GCD correctly is crucial for efficient fraction simplification.

Removing Decimals in Fractions

Removing decimals from fractions is an important step when converting percentages to fractions because this ensures that calculations are dealing with whole numbers, which are easier to work with.

Here are the steps to get rid of decimals:

• Identify where the decimal is located in the fraction.
• Multiply the numerator and the denominator by a power of 10 that matches the decimal position. For a single decimal place, you multiply by 10; for two decimal places, multiply by 100, and so forth.

In the initial conversion from a percent to a fraction, you might end up with decimals, like with the fraction \( \frac{63.6}{100} \). To remove the decimal, multiplying both by 10 turns the fraction to \( \frac{636}{1000} \), making both numbers whole. This strategy makes it much easier to then simplify the fraction.