Problem 65 - Algebra 1: Concepts and Skills

Question

Write the percent as a fraction or as a mixed number in simplest form. (Skills Review p. 768 ) $$ 91 \% $$

Step-by-Step Solution

Answer

The percent 91% as a fraction in its simplest form is 91/100.

Step 1: Convert Percentage to Fraction

Express 91% as the fraction 91/100. Every percentage can be represented as a fraction by placing the percentage number over 100.

Step 2: Simplify fraction

In this case, the fraction 91/100 cannot be simplified furher because 91 and 100 do not have any common divisors other than 1.

Step 3: Write the final answer

The final answer is the fraction we got in the previous step, which is 91/100. It is already in its simplest form, so no further steps are needed.

Key Concepts

Percentage to Fraction

Converting a percentage to a fraction is a fundamental math skill necessary for many real-life and academic applications. When we talk about percentages, we're referring to a value out of 100. To convert a percentage to a fraction, simply use the percentage value as the numerator and 100 as the denominator. For example, take the problem of converting 91% into a fraction.

Here's how you do it:

• First, write down the percentage value without the percent sign as the numerator of the fraction. In our case, 91 becomes the numerator.
• Next, use 100 as the denominator. This is because 'percent' literally means 'per hundred'.
• Thus, 91% can be written as the fraction \( \frac{91}{100} \) .

This conversion is the initial step to understand percentages in terms of fractions, which allows for further mathematical manipulation and simplification when possible.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of finding an equivalent fraction that has the smallest possible numerator and denominator. To simplify a fraction, you need to find the greatest common divisor (GCD) of both the numerator and the denominator and then divide both of them by this number.

Here are the steps for simplifying fractions:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD to get the simplified fraction.

For instance, if you have the fraction \( \frac{50}{100} \) , the GCD of 50 and 100 is 50, so when you divide both numbers by 50, you get \( \frac{1}{2} \) which is the simplified form. However, with our original problem \( \frac{91}{100} \) , since 91 and 100 do not share any common divisors other than 1, the fraction is already in its simplest form.

Fractions in Simplest Form

A fraction is in its simplest form when no number other than one can divide both the numerator and the denominator without leaving a remainder. This means that the numerator and the denominator are relatively prime – they do not have any common factors other than 1.

To check if a fraction is in its simplest form, follow these steps:

• Look for any common factors between the numerator and the denominator.
• If there are common factors, the fraction can be simplified further by dividing both the numerator and the denominator by the common factor.
• If there are no common factors, other than 1, the fraction is already in its simplest form.

Using our example \( \frac{91}{100} \) , we observe that 91 is a prime number, meaning it can only be divided by itself and 1. Since 100 is not divisible by 91, there are no common factors that would allow us to simplify the fraction further. Hence, \( \frac{91}{100} \) is already in the simplest form.