Problem 14 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{15}{21}$$

Step-by-Step Solution

Answer

The fraction \( \frac{15}{21} \) simplifies to \( \frac{5}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{15}{21} \), first identify the greatest common divisor (GCD) of the numerator 15 and the denominator 21. List the factors of 15: 1, 3, 5, 15 and the factors of 21: 1, 3, 7, 21. The greatest common factor they share is 3.

Step 2: Divide Both the Numerator and Denominator by the GCD

Divide the numerator and the denominator by their GCD. So, \( \frac{15}{21} \) becomes \( \frac{15 \div 3}{21 \div 3} \).

Step 3: Perform the Division

Calculate the result of the division. For the numerator, 15 divided by 3 is 5. For the denominator, 21 divided by 3 is 7. So, \( \frac{15}{21} \) simplifies to \( \frac{5}{7} \).

Step 4: Verify if the Fraction is in Simplest Form

Check if the simplified fraction \( \frac{5}{7} \) can be further reduced. Since 5 and 7 share no common factors other than 1, \( \frac{5}{7} \) is in its simplest form.

Key Concepts

Greatest Common Divisor

To simplify fractions effectively, understanding the concept of the Greatest Common Divisor (GCD) is essential. The GCD is the largest number that can exactly divide both the numerator and the denominator of a fraction. It acts like a magical key that unlocks the simplest form of any fraction.

Here's how you can find the GCD:

• List all factors of the numerator and the denominator. For example, with 15 and 21, the factors are:
  • Factors of 15: 1, 3, 5, 15
  • Factors of 21: 1, 3, 7, 21
• Identify the common factors. Both 15 and 21 share the numbers 1 and 3.
• The greatest number in the common set of factors is the GCD. Here it's 3.

Using the GCD simplifies the fraction by dividing both the numerator and denominator by the GCD. This initial step forms the basis for reducing fractions effectively.

Numerator and Denominator

A fraction is built from two main parts: the numerator and the denominator. These terms define the essence and value of any fraction, making comprehension of them crucial. Think of a fraction, such as \( \frac{15}{21} \):

- **Numerator**: This is the number on the top. It indicates how many parts of a whole we are considering, in this case, 15 parts.- **Denominator**: This is the number below the line. It represents the total number of equal parts that make up the whole, here totaling 21.The numerator is like the counting piece, while the denominator tells us how big each piece really is in relation to the whole. Understanding these roles helps when you simplify fractions as it involves ensuring both numbers are expressed as the smallest possible whole number ratio, clarified through dividing each by the GCD.

Simplest Form

Achieving the simplest form of a fraction means expressing it with the smallest possible whole numbers while maintaining the same value or proportion. Simplifying \( \frac{15}{21} \) involves a step-by-step process:

• First, we found the GCD, which was 3.
• Next, both 15 (numerator) and 21 (denominator) were divided by 3, the GCD.
• This division resulted in \( \frac{5}{7} \), a simpler fraction.

Now let's verify if \( \frac{5}{7} \) is really in its simplest form. We check if there are any common factors, but 5 and 7 have no factors in common except 1. This confirms \( \frac{5}{7} \) is indeed the simplest form.

By always simplifying to the simplest form, fractions become easier to interpret, calculate, and compare. It's the neatest way of presenting fractions in mathematics!