Problem 9 - Elementary Algebra

Question

For exercises 1-12, simplify. $$ \frac{5}{365} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{73}\).

Step 1: Identify the Problem

The task is to simplify the fraction \(\frac{5}{365}\). Simplifying means reducing the fraction to its lowest terms.

Step 2: Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of 5 and 365. Since 5 is a prime number, check if 5 is a factor of 365.

Step 3: Divide Both Numerator and Denominator by the GCD

The GCD of 5 and 365 is 5. Divide both the numerator and the denominator by 5: \(\frac{5 \div 5}{365 \div 5} = \frac{1}{73}\).

Step 4: Verify the Simplified Fraction

Check if \(\frac{1}{73}\) can be simplified further. Since 1 is relatively prime to any number, \(\frac{1}{73}\) is already in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

Understanding the greatest common divisor (GCD) is essential for simplifying fractions. The GCD of two numbers is the largest number that can exactly divide both of them without leaving a remainder. To find the GCD, you can use several methods, but one simple way is to list the factors of each number and look for the greatest one they share.
For example, for the numbers 12 and 18, the factors are:

• 12: 1, 2, 3, 4, 6, 12
• 18: 1, 2, 3, 6, 9, 18

The common factors are 1, 2, 3, and 6. The greatest of these is 6, so the GCD of 12 and 18 is 6.
In our fraction example, we were simplfiyng \(\frac{5}{365}\). Since 5 is a prime number, it can divide 365 without a remainder (giving us the GCD as 5).

Reducing Fractions

Reducing fractions is the process of simplifying them to their smallest possible form. This means altering the fraction so that the numerator and denominator are reduced but still represent the same value. To reduce a fraction:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

In our example, we started with \(\frac{5}{365}\). First, we determined the GCD of 5 and 365, which is 5. Then, we divided both the numerator and the denominator by this GCD:

• Numerator: 5 divided by 5 equals 1.
• Denominator: 365 divided by 5 equals 73.

This leaves us with the simplified fraction \(\frac{1}{73}\), which is already in its lowest terms.

Prime Numbers

Prime numbers are fundamental in simplifying fractions. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it can’t be divided evenly by any other numbers.
For instance, the number 5 is prime because its only divisors are 1 and 5. In our example, we used the fact that 5 is a prime number to simplify \(\frac{5}{365}\). We checked if 5 is a factor of 365. Since it is, we knew that the GCD of 5 and 365 is 5.
Recognizing prime numbers can significantly ease the process of simplifying fractions. Some common prime numbers to remember are 2, 3, 5, 7, 11, 13, 17, and so on.
Remember, if either the numerator or the denominator is a prime number, it can help to quickly identify the common factors and simplify the fraction more easily.