Problem 46 - Elementary Algebra

Question

For exercises 39-46, rewrite the percent as a fraction. Simplify the fraction into lowest terms. $$ 117 \% $$

Step-by-Step Solution

Answer

\( \frac{117}{100} \)

Step 1: Convert the percent to a fraction

First, write the given percent as a fraction by placing it over 100. So, \(117\%\) becomes \( \frac{117}{100} \).

Step 2: Simplify the fraction

Next, simplify the fraction \( \frac{117}{100} \). In this case, 117 and 100 have no common factors other than 1, so the fraction is already in its simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in mathematics. It involves reducing a fraction to its simplest form where the numerator (top number) and the denominator (bottom number) are as small as possible while maintaining the same value. To simplify a fraction, we look for the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For instance, with the fraction \(\frac{117}{100}\), since 117 and 100 share no common factors besides 1, it is already in its simplest form. However, if you had a fraction like \(\frac{50}{100}\), you could simplify it by dividing both the numerator and the denominator by their GCD, which is 50, resulting in \(\frac{50}{50} = 1\).

Percent to Fraction

Converting a percent to a fraction is straightforward. Percent literally means 'per hundred,' so we can always write the percentage as a fraction with a denominator of 100. For example, to convert 117% to a fraction, we write it as \(\frac{117}{100}\). This step translates the percent into a form that can be further manipulated mathematically. If needed, we can simplify the resulting fraction in the next step. Getting comfortable with this conversion helps in understanding proportions, ratios, and even certain algebraic concepts more deeply.

Lowest Terms

Expressing a fraction in its lowest terms means making the numerator and the denominator as small as possible while keeping the fraction equivalent in value. After converting a percent to a fraction, we should always check if it can be simplified. To simplify a fraction, find the GCD of the numerator and the denominator and divide both by this number. For instance, with 117%, we converted it to \(\frac{117}{100}\). Since the GCD of 117 and 100 is 1, the fraction is already in its lowest terms. If there were a common factor, like in \(\frac{50}{100}\), we'd divide both by the GCD (50) to get \(\frac{1}{2}\). This process ensures our answer is as precise and simple as possible.