Problem 18 - Elementary Algebra Within Reach

Question

Convert the percent to a fraction. $$85 \%$$

Step-by-Step Solution

Answer

The fraction form of 85% is \(\frac{17}{20}\).

Step 1: Express the Percentage as a Fraction Over 100

Since any percentage is a fraction of 100, we can initially present 85% as the fraction \(\frac{85}{100}\). This is done by placing the percentage number over 100.

Step 2: Simplify the Fraction

The fraction \(\frac{85}{100}\) must be reduced to the simplest form, meaning the numerator and denominator should be divided by their greatest common divisor (GCD). In this case, the GCD of 85 and 100 is 5. When you divide both numbers by 5, it becomes \(\frac{17}{20}\)

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we're referring to the process of reducing a fraction to its simplest form. This means rewriting it so that the numerator (the top number) and the denominator (the bottom number) are as small as possible. To do this, you find the greatest common divisor (GCD) of both numbers. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

• First, identify the numerator and denominator.
• Find the GCD for these two numbers.
• Divide both the numerator and the denominator by the GCD.
• The fraction you now have is fully simplified.

Simplifying fractions makes them easier to understand and work with, especially in mathematical problems. For example, in the original exercise, the fraction \(\frac{85}{100}\) can be simplified by dividing both 85 and 100 by their GCD, which is 5, resulting in the simplified fraction \(\frac{17}{20}\). This process is crucial for making mathematical expressions more manageable.

Percentages

Percentages are a way to express numbers as a fraction of 100. By converting numbers to percentages, you can easily compare different quantities or determine proportions. In everyday contexts, percentages are often used to convey discounts, interest rates, and statistics.
To express a percentage as a fraction, you place the percentage number over 100. In this way, a percentage can be directly converted into a fraction. For example, 85 percent becomes \(\frac{85}{100}\). This fraction represents the part of a whole signified by 85 out of 100 parts.

• Write the percentage number over 100 to create a fraction.
• If needed, simplify the fraction.
• Use the resulting fraction in further mathematical calculations or expressions.

This conversion is incredibly useful in mathematics for solving problems where dealing with fractions is more practical or when performing operations that need exact values as opposed to rounded percentages.

Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a key concept in mathematics. It refers to the largest whole number that can divide two or more integers without leaving a remainder. Finding the GCD is essential for simplifying fractions, as it allows you to reduce fractions to their simplest form by eliminating any common factors.
To find the GCD of two numbers like in \(\frac{85}{100}\), follow these steps:

• List the factors of each number.
• Identify the largest factor that both numbers share.
• This shared factor is the GCD.

For example, when simplifying \(\frac{85}{100}\), the factors of 85 are 1, 5, 17, 85, and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. The greatest common divisor they share is 5.
Using the GCD to simplify fractions not only helps in making calculations easier but also provides a clearer understanding of the mathematical concepts involved. It allows you to work with smaller numbers, which are often easier to handle and more intuitive.