Problem 43 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$26.5 \%$$

Step-by-Step Solution

Answer

26.5% as a fraction in lowest terms is \( \frac{53}{200} \).

Step 1: Understand the Percent

The given value is a percentage: 26.5%. We can interpret this as 26.5 per 100, meaning 26.5 out of every 100 parts.

Step 2: Convert Percent to Fraction

To convert a percent to a fraction, we divide the percent by 100. Therefore, 26.5% becomes \( \frac{26.5}{100} \).

Step 3: Eliminate the Decimal

Since 26.5 has a decimal, we can multiply both the numerator and the denominator by 10 to eliminate it, giving \( \frac{265}{1000} \).

Step 4: Simplify the Fraction

Now, we simplify \( \frac{265}{1000} \). First, find the greatest common divisor (GCD) of 265 and 1000, which is 5. Divide both the numerator and the denominator by 5: \( \frac{265 \div 5}{1000 \div 5} = \frac{53}{200} \).

Step 5: Verify Simplest Form

Check if \( \frac{53}{200} \) is in its simplest form. Since 53 is a prime number and does not divide 200, the fraction is already simplified to its lowest terms.

Key Concepts

Simplifying Fractions

Simplifying fractions is a key mathematical skill that involves reducing fractions to their lowest terms. A fraction is simplified when the numerator (top number) and the denominator (bottom number) cannot be divided by the same number anymore, except for 1. By simplifying a fraction, we make it easier to work with in calculations.
To simplify a fraction:

• Find the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and the denominator by this number.

For example, if we have the fraction \( \frac{265}{1000} \), we check for common factors. Since their greatest common divisor is 5, dividing them gives us the simplified fraction \( \frac{53}{200} \). This means 53 and 200 have no common factors other than 1.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can divide two numbers without leaving a remainder. Finding the GCD is essential for simplifying fractions because it ensures the fraction is reduced to its simplest form.
To find the GCD:

• List the prime factors of each number.
• Identify the common factors both numbers share.
• Select the largest of these common factors.

For example, to find the GCD of 265 and 1000: - 265 can be factored as 5 and 53 (since 53 is a prime number). - 1000 includes factors like 2, 5, and more. The GCD is 5, as it's the largest factor that divides both 265 and 1000.

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. This means it cannot be formed by multiplying two smaller natural numbers. Understanding prime numbers is crucial for factorization tasks like finding the greatest common divisor or simplifying fractions.
Some basic prime numbers include 2, 3, 5, 7, 11, and so forth.
Knowing prime numbers is helpful because:

• They only have two distinct positive divisors.
• They are used to determine the prime factorization of numbers.
• They help in identifying the simplest form of fractions effectively.

In scenarios like our exercise, recognizing that 53 is a prime number reassures us that \( \frac{53}{200} \) is indeed simplified, as 53 has no other divisors than 1 and itself, and it does not divide 200.