Problem 41 - Elementary Algebra

Question

For exercises 39-46, rewrite the percent as a fraction. Simplify the fraction into lowest terms. $$ 12 \% $$

Step-by-Step Solution

Answer

12% as a fraction is \( \frac{3}{25} \).

Step 1: Convert Percent to Fraction

Start by writing the percentage as a fraction. For 12%, write it as \( \frac{12}{100} \).

Step 2: Simplify the Fraction

Next, simplify the fraction \( \frac{12}{100} \). Both the numerator and the denominator can be divided by their greatest common divisor, which is 4. Thus, \( \frac{12 \, \div \, 4}{100 \, \div \, 4} = \frac{3}{25} \).

Step 3: Final Fraction in Lowest Terms

The fraction \( \frac{3}{25} \) is in its simplest form because 3 and 25 do not have any common divisors other than 1.

Key Concepts

percent to fraction conversion

To convert a percentage to a fraction, begin by understanding what a percent represents. The word 'percent' simply means 'per hundred.' So, when you have a number followed by the percent symbol (like 12%), it's equivalent to that number over 100. For example, 12% is written as the fraction \( \frac{12}{100} \). This means 12 parts out of 100 parts in total. This step is crucial because it lays the foundation for further simplification. Remember, converting percents to fractions always uses 100 as the denominator initially. So, if you encounter 45%, it would convert to \( \frac{45}{100} \).

simplifying fractions

Once you have your fraction from a percent, the next step is to simplify it. Simplifying involves reducing the fraction to its smallest equivalent form. Let's revisit our example of \( \frac{12}{100} \). To simplify this fraction, look for the greatest common divisor (GCD) of the numerator (12) and the denominator (100). The GCD is the largest number that can divide both 12 and 100 without leaving a remainder. For 12 and 100, the GCD is 4. Divide both the numerator and the denominator by 4:
\( \frac{12 \div 4}{100 \div 4} = \frac{3}{25} \).
Now you have a simplified fraction. It captures the same value but in a more compact form.

lowest terms in fractions

A fraction is in its lowest terms when the numerator and the denominator have no common divisors other than 1. Simplifying fractions involves breaking them down until they can't be reduced further. For instance, in the fraction \( \frac{3}{25} \), the numbers 3 and 25 don't share any common factors other than 1. Therefore, \( \frac{3}{25} \) is already in its simplest form, or lowest terms. Ensuring your fraction is in its lowest terms makes it easier to understand and work with. Always check your simplified fraction to make sure no further reduction is possible. If no further simplification is needed, your fraction is in the lowest terms.