Problem 65 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. Explain. \(\frac{250}{350}=\frac{2}{3}\)

Step-by-Step Solution

Answer

The statement is false.

Step 1: Simplify the first fraction

The first fraction has 250 as the numerator and 350 as the denominator. On simplifying this fraction using their Greatest Common Divisor (GCD), we get \(\frac{5}{7}\). The GCD of 250 and 350 is 50. So we divide both the numerator and denominator by 50 to get \(\frac{5}{7}\).

Step 2: Check the second fraction

The second fraction \(\frac{2}{3}\) is already in its simplest form. We do not need to simplify it further.

Step 3: Compare the simplified fractions

In this step we will compare the simplest form of the first fraction with the second fraction. Since the simplest form of the first fraction \(\frac{5}{7}\) is not equal to the second fraction \(\frac{2}{3}\), the original statement is false.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its smallest form, where the numerator (top number) and denominator (bottom number) cannot be divided further by the same number except 1. This process helps make fractions easier to understand and compare.
To simplify a fraction, follow these steps:

• Identify the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

This will give you the simplest form of the fraction. For example, in the fraction \(\frac{250}{350}\), the GCD is 50. By dividing both 250 and 350 by 50, the simplified fraction becomes \(\frac{5}{7}\). Simplifying fractions helps in comparing them more easily and understanding the underlying relationship between the numbers.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest positive integer that divides two numbers without leaving a remainder. Finding the GCD is an essential step when simplifying fractions.
To find the GCD of two numbers:

• List all the factors of each number.
• Identify the largest factor that appears in both lists.

In the example with 250 and 350, the GCD is 50 because 50 is the largest number that can divide both 250 and 350 evenly.
Using the GCD ensures that you are reducing the fraction by the largest possible amount, which gives you the simplest form. Understanding how to find the GCD is crucial for working with fractions efficiently.

Comparing Fractions

Comparing fractions involves determining which fraction is larger, smaller, or if they are equivalent. When fractions are already in their simplest form, comparing them becomes straightforward. There are a few methods to compare fractions:

• If the fractions have the same denominator, simply compare the numerators.
• If the denominators are different, convert the fractions to equivalent fractions with a common denominator or compare them in their simplest forms.

In the exercise given, \(\frac{5}{7}\) and \(\frac{2}{3}\) are compared by recognizing that they do not equal each other when simplified. Thus, the statement \(\frac{250}{350} = \frac{2}{3}\) is false. Comparing fractions helps in making decisions where relative size matters, such as in estimating or solving real-world problems related to proportions.