Problem 110 - Fundamentals of Mathematics

Question

A ream of paper contains 500 sheets. What fraction of a ream of paper is 200 sheets? Be sure to reduce.

Step-by-Step Solution

Answer

200 sheets is \( \frac{2}{5} \) of a ream of paper.

Step 1: Understand the Problem

The problem asks us to find out what fraction 200 sheets is of a 500-sheet ream of paper. To solve it, we need to set up the fraction using the given values.

Step 2: Set Up the Fraction

Start by writing the fraction as \( \frac{200}{500} \), where 200 represents the number of sheets, and 500 represents the total number of sheets in the ream.

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{200}{500} \) by finding the greatest common divisor (GCD) of 200 and 500. The GCD of 200 and 500 is 100.

Step 4: Divide by the GCD

Divide both the numerator and the denominator by their GCD. \( \frac{200}{500} = \frac{200 \div 100}{500 \div 100} = \frac{2}{5} \). The simplified fraction is \( \frac{2}{5} \).

Key Concepts

Understanding Simplification

Simplification of fractions is a crucial process in mathematics. It helps make fractions easier to read and work with. When we simplify a fraction, we're reducing it to its simplest form without changing its value. This means:

• Finding the largest number that divides both the numerator (the top number) and the denominator (the bottom number) without leaving a remainder.
• Dividing both the numerator and the denominator by this number.

For example, starting with the fraction \( \frac{200}{500} \), we can find that both 200 and 500 are divisible by 100. This step allows us to transform \( \frac{200}{500} \) into \( \frac{2}{5} \), which is far simpler and easier to understand. Through simplification, calculations become more manageable, helping us see the true proportion more clearly.

Exploring Fraction Conversion

Fraction conversion helps in transforming fractions into different forms. Often, we convert fractions to mixed numbers, decimals, or percents depending on the need. However, the type of conversion discussed here focuses on reducing or translating fractions into simpler terms.

• Converting fractions to a simpler form through common divisors involves identifying numbers that both numerator and denominator share as factors.
• This conversion is not about changing the essence of the fraction, but simplifying the expression without altering its value.

For the given example of \( \frac{200}{500} \), conversion through simplification helps us represent the same value as \( \frac{2}{5} \). This process helps maintain clarity while dealing with fractions in various mathematical operations.

The Role of the Greatest Common Divisor

The greatest common divisor (GCD) plays a pivotal role in the simplification of fractions. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder.Finding the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest factor they have in common.

For instance, the GCD of 200 and 500 is 100, because 100 is the largest number that can divide both 200 and 500 evenly.
Using the GCD is crucial because it ensures that you're simplifying the fraction as much as possible, to its most basic form. Without using the GCD, fractions might appear more complicated or larger than they actually are. In our example, by dividing both 200 and 500 by their GCD, we condensed \( \frac{200}{500} \) into \( \frac{2}{5} \). This step underlines the fraction’s true proportion, simplifying the arithmetic processes in which this fraction could later be involved.