Problem 47 - Basic College Mathematics

Question

Simplify. $$ \frac{20}{20} $$

Step-by-Step Solution

Answer

1

Step 1: Identify the Fraction.

The given fraction is \( \frac{20}{20} \).

Step 2: Understand the Concept of Division.

Any number divided by itself is 1. This is based on the principle that the numerator and the denominator are equal, so their ratio is 1.

Step 3: Simplify the Fraction.

Since 20 divided by 20 equals 1, \( \frac{20}{20} = 1 \).

Key Concepts

Basic Division

When it comes to simplifying fractions, understanding basic division is crucial. Division is the process of determining how many times one number is contained within another. In our example, we have the fraction \( \frac{20}{20} \). Here, 20 is being divided by 20. According to the rules of basic division, any number divided by itself equals 1. This rule applies universally, so the fraction \( \frac{20}{20} \) simplifies to 1.

To further grasp this concept:

• Imagine 20 apples divided equally into 20 groups. Each group will have exactly one apple.
• This demonstrates that the quotient of a number divided by itself is always 1.

Mastering basic division makes fraction simplification straightforward.

Numerator and Denominator

In fractions, understanding the numerator and denominator is key. The fraction we are simplifying is \( \frac{20}{20} \).

Both the numerator and denominator here are 20.

• The numerator is the top number, representing how many parts we have.
• The denominator is the bottom number, indicating how many equal parts the whole is divided into.

When these two numbers are the same, effectively dividing them shows how many times one fits into the other. Here, 20 fits into 20 exactly once. This simplifies the fraction to 1.

So, remembering these definitions:

• Numerator: top number
• Denominator: bottom number

can help you easily simplify similar fractions.

Principle of Equality

The principle of equality is fundamental in fraction simplification. It states that if the numerator and denominator are equal, their ratio is 1. In our example, \( \frac{20}{20} = 1 \). This is because both the numerator and denominator are the same number, making their division result 1.

Here are a few more important points to consider:

• This principle holds for any numbers. For example, \( \frac{15}{15} \) and \( \frac{32}{32} \) both equal 1.
• This concept is often used to simplify calculations and solve equations.

By relying on the principle of equality, understanding fraction simplification becomes a lot easier.

Remember, equal numerators and denominators always simplify to 1. That makes it simpler to deal with complex fractions and equations.