Problem 3 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{5}{11}$$

Step-by-Step Solution

Answer

The fraction \( \frac{5}{11} \) is already in simplest form.

Step 1: Determine Common Factors

Identify if the numerator and denominator share any common factors other than 1. The numerator is 5 and the denominator is 11. Since 5 and 11 are both prime numbers, the only common factor they share is 1.

Step 2: Simplify the Fraction

Since 5 and 11 do not share any factors other than 1, \( \frac{5}{11} \) is already in its simplest form.

Key Concepts

Prime Numbers

Prime numbers are fundamental in the world of mathematics. They are the building blocks for all other numbers because of their unique properties. A prime number is a number greater than 1 that has no divisors other than 1 and itself. This means that a prime number can't be divided by any other number without leaving a remainder. For instance, the numbers 2, 3, 5, 7, and 11 are all prime numbers.

Understanding prime numbers is crucial when simplifying fractions. In the given exercise, both the numerator (5) and denominator (11) are prime numbers. Since they share no factors other than 1, the fraction is already in its simplest form. Recognizing whether numbers are prime helps determine if a fraction can be simplified further.

• Prime numbers are only divisible by 1 and themselves.
• They help in identifying the simplest form of a fraction.

Being able to quickly identify prime numbers makes the task of simplifying fractions much easier.

Common Factors

Common factors of two numbers are the numbers that can evenly divide both without leaving a remainder. Finding common factors is a critical step when simplifying fractions. It’s all about finding the greatest common factor (GCF) that can simplify the fraction.

For the fraction \( \frac{5}{11} \), we determine that both 5 and 11 are prime, meaning they have no common factors other than 1. This means you can’t simplify the fraction any further. If the numbers were not prime, you would try to find the largest number that divides into both the numerator and the denominator that isn't 1.

• Common factors are used to simplify fractions.
• The greatest common factor is the largest number that divides two numbers.

Knowing how to identify common factors helps solidify the understanding of when a fraction is already at its simplest form.

Numerator and Denominator

Understanding what a numerator and a denominator are is essential in the realm of fractions. The numerator is the top number of a fraction and represents how many parts of a whole we have. The denominator is the bottom number in a fraction and indicates into how many parts the whole is divided.

In our example \( \frac{5}{11} \), 5 is the numerator and 11 is the denominator. The goal of simplifying fractions is to make the numbers smaller while keeping the same ratio. This is done by dividing both the numerator and the denominator by their greatest common factor.

• The numerator shows the number of parts considered.
• The denominator shows the total number of equal parts.

Comprehending the roles of the numerator and denominator is key to mastering fraction simplification efficiently.