Problem 63 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. Explain. \(\frac{2}{11}=\frac{10}{55}\)

Step-by-Step Solution

Answer

The statement \(\frac{2}{11} = \(\frac{10}{55}\) is true.

Step 1: Understand the problem

The task is to determine if \(\frac{2}{11}\) is equal to \(\frac{10}{55}\). This involves understanding how fractions work and how the numerators and denominators relate to each other.

Step 2: Simplify the fraction on the right

To compare the two fractions, we can simplify \(\frac{10}{55}\) to its lowest terms. This is done by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of 10 and 55 is 5. Dividing both 10 and 55 by 5 we get \(\frac{2}{11}\).

Step 3: Compare the two fractions

Now that both fractions are simplified, we can see that \(\frac{2}{11} = \frac{2}{11}\). This means that the initial statement is true.

Key Concepts

Equivalent Fractions

Equivalent fractions are fractions that may look different but represent the same value. Although their numerators (top numbers) and denominators (bottom numbers) are different, they simplify to the same simplest form.
When working with equivalent fractions, you can multiply or divide both the numerator and the denominator by the same non-zero number without changing the value of the fraction.
For example, the fractions \(\frac{2}{11}\) and \(\frac{10}{55}\) are equivalent because they both simplify to the same simplest form, \(\frac{2}{11}\). This is why understanding equivalent fractions is crucial when comparing or simplifying fractions.

• They are particularly useful in fraction addition, subtraction, and comparison.
• Recognizing equivalent fractions can help simplify calculations significantly.

Knowing how to identify and work with equivalent fractions will make you adept at handling various math problems involving fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. This is achieved by dividing both parts of the fraction by the greatest common divisor (GCD).
Simplifying helps make fractions easier to understand and work with. In our example, \(\frac{10}{55}\) is simplified to \(\frac{2}{11}\) by dividing both the numerator and denominator by 5.
Here are a few reasons why simplifying fractions is important:

• Makes fractions easier to compare and perform operations on.
• Reveals the true value of the fraction in its simplest form.
• Essential for solving equations and real-world problems involving fractions.

Understanding how to simplify fractions ensures you deal with the simplest and most manageable numbers in your calculations.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides exactly into two or more numbers. When simplifying fractions, finding the GCD helps in reducing the fraction to its simplest terms.
For instance, to simplify the fraction \(\frac{10}{55}\), we first find the GCD of 10 and 55. The largest number that divides both 10 and 55 without leaving a remainder is 5.
With the GCD found, we can perform the division:

• Divide both the numerator and the denominator by the GCD.
• This results in the simplest form of the fraction.

The GCD is an essential tool not only for simplifying fractions but also for many mathematical calculations and problem-solving tasks. Understanding and correctly identifying the GCD fosters your ability to simplify fractions quickly and accurately.