Problem 44 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$34.2 \%$$

Step-by-Step Solution

Answer

The fraction is \( \frac{171}{500} \).

Step 1: Convert Percent to a Fraction

To convert 34.2% to a fraction, start by writing it as a fraction with a denominator of 100. This gives us: \( \frac{34.2}{100} \). Since fractions should typically not have decimals in the numerator, we multiply both the numerator and the denominator by 10 to remove the decimal, resulting in \( \frac{342}{1000} \).

Step 2: Simplify the Fraction by Finding GCD

Next, simplify the fraction \( \frac{342}{1000} \). To do this, find the greatest common divisor (GCD) of 342 and 1000. The GCD of 342 and 1000 is 2.

Step 3: Divide Numerator and Denominator by GCD

Using the GCD found in the previous step, divide both the numerator and the denominator by 2. \( \frac{342 \div 2}{1000 \div 2} = \frac{171}{500} \). Thus, the fraction is simplified to \( \frac{171}{500} \).

Step 4: Verify the Simplified Fraction

Check that \( \frac{171}{500} \) is in its simplest form by ensuring there are no common factors between 171 and 500 other than 1. Both numbers are not divisible by any common numbers except 1, confirming the fraction is in lowest terms.

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental concept in mathematics, where we reduce a fraction to its simplest form. A fraction is simplified when the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. To simplify a fraction:

• Identify a common factor of both numbers.
• Divide the numerator and the denominator by the common factor repeatedly.
• Continue this process until no further common factors exist, except 1.

Simplification helps in comparing fractions, performing arithmetic operations, and interpreting mathematical problems efficiently. When dealing with decimals in a fraction, like in our exercise with 34.2%, start by eliminating the decimal to convert the fraction to whole numbers. This often involves scaling up by multiplying the numerator and denominator by powers of 10, paving the way for simplification.

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. It plays an essential role in simplifying fractions by identifying the largest factor that both numbers share. To find the GCD:

• List the factors of each number.
• Identify the largest common factor in both lists.
• This common factor is your GCD.

Another efficient method is using the Euclidean algorithm, which involves a series of divisions until a remainder of 0 is reached. The last non-zero remainder is the GCD. In our example of reducing 34.2% to a fraction, the GCD of 342 and 1000 is 2. By dividing both numbers by this GCD, we achieve the fraction in its simplest form.

Fractions in Simplest Form

A fraction is said to be in simplest form when both the numerator and the denominator have no common factors other than 1. This form is the most basic version of the fraction, making it easier to work with in calculations and comparisons. To determine if a fraction is in its simplest form:

• Check for any common divisors of the numerator and denominator.
• If the only common divisor is 1, then the fraction is in simplest form.
• Use simplification or prime factorization to verify.

In our exercise, after simplifying \( \frac{342}{1000} \) using the GCD, we found the fraction \( \frac{171}{500} \). By ensuring no further common factors exist, we confirm it is indeed in simplest form. Fractions in simplest form help provide clarity in mathematical expressions and ensure results are straightforward and easy to understand.