Problem 18 - Elementary Algebra

Question

Simplify the following problems. $$ \frac{-25}{5} $$

Step-by-Step Solution

Answer

Question: Simplify the given fraction: \(\frac{-25}{5}\) Answer: -5

Step 1: Identifying the terms in the fraction

In the given problem, we have the fraction \(\frac{-25}{5}\). The numerator is -25, and the denominator is 5.

Step 2: Identify common factors

Now, we must look for common factors both in the numerator and denominator. The number 5 divides both the numerator and the denominator.

Step 3: Simplify the fraction

Now we can divide both the numerator and the denominator by 5 to cancel out this common factor. When we do this, we get: $$ \frac{-25}{5} = \frac{-25 \div 5}{5 \div 5} = \frac{-5}{1} $$

Step 4: Final result

As the denominator is now equal to 1, the simplified fraction is equal to the numerator alone. Therefore, our final simplified form is: $$ -5 $$

Key Concepts

Numerator and Denominator

In any fraction, the numerator and denominator play significant roles. The numerator is the number on top of the fraction. It represents how many parts we have. In the exercise, the numerator is -25.
The denominator is the number on the bottom. It shows into how many parts the whole is divided. For our problem, the denominator is 5. Understanding the roles of these components is crucial when simplifying a fraction.
Fractions can represent many types of quantities, and often we simplify them by dividing both these top and bottom numbers by a common value, called a common factor.

Common Factors

Finding common factors is a key part of simplifying fractions. A common factor is a number that can divide two numbers without leaving a remainder.
In our exercise, we looked at the numbers -25 and 5. The number 5 is a common factor because both -25 and 5 can be divided by 5 exactly.

• -25 divided by 5 is equal to -5
• 5 divided by 5 is equal to 1.

This step is important because dividing by the common factor "simplifies" the fraction, making it easier to work with or understand.
Always start by identifying the greatest common factor for the most efficient simplification.

Division in Fractions

Division is the main operation used in simplifying fractions. When we divide both the numerator and the denominator by their greatest common factor, we simplify the fraction.
In the given exercise, this process turned \(-\frac{25}{5}\) into \(-\frac{5}{1}\) by dividing both terms by the number 5.
It is crucial to note that any number divided by itself equals 1, which is why our denominator ends up being 1 in this instance. This leaves us with just the numerator as the simplified answer.

• Always check your division carefully to ensure accuracy.
• This process will make the fraction represent the same quantity in a simpler form.
• Seeing the fraction as a division problem helps understand that fractions themselves are a type of division.

Remember, a simplified fraction is often easier to interpret and work with in further mathematical operations or real-life applications.