Problem 25 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{16}{20}+\frac{1}{20}+\frac{2}{20}\)

Step-by-Step Solution

Answer

Answer: The sum of the given fractions is \(\frac{19}{20}\).

Step 1: Identifying the problem

We are given three fractions to add: \(\frac{16}{20}+\frac{1}{20}+\frac{2}{20}\). All the fractions have the same denominator, which is 20. Since they already have the same denominator, we can proceed to add the numerators.

Step 2: Adding the numerators

Add the numerators of the three fractions while keeping the denominator the same. So, we have: \((\frac{16}{20}+\frac{1}{20}+\frac{2}{20}) = \frac{16+1+2}{20}\)

Step 3: Simplifying the result

Now we need to simplify the result by adding the numerators and reducing the fraction to its simplest form: \(\frac{16+1+2}{20} = \frac{19}{20}\) The fraction is already in its simplest form, since 19 is a prime number and cannot be divided by any other number except 1 and itself.

Step 4: Final answer

Thus, the sum of the given fractions is: \(\frac{16}{20}+\frac{1}{20}+\frac{2}{20} = \frac{19}{20}\)

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when adding or subtracting fractions. Let's imagine each fraction is like a slice of pizza. For easy addition, the slices need to be the same size, meaning the fractions need a common denominator.
In the example given, the fractions \( \frac{16}{20}, \frac{1}{20}, \frac{2}{20} \) already share a common denominator, which is 20. This makes the addition straightforward:

• All fractions have the same denominator: 20.
• Add the numerators directly: \( 16 + 1 + 2 \).
• Keep the common denominator: 20.

This results in \(\frac{19}{20}\). When fractions don't have a common denominator, you have to find one, usually the least common multiple of the denominators. This ensures all fractions describe equal parts of a whole, allowing easy addition or subtraction.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. This makes fractions easier to understand and compare.
For simplification, we divide the numerator and denominator by their greatest common divisor (GCD). However, if the numerator is a prime number, as in our example \(\frac{19}{20}\), it simplifies the process because prime numbers only divide evenly by 1 and themselves.

• Find the GCD of the numerator and the denominator.
• Divide both by the GCD.
• If the numerator is a prime number and it doesn’t share factors with the denominator, the fraction is already simplified.

This step is vital in ensuring your final answer is presented in the cleanest form possible, making it more concise and readable.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of the number system, much like atoms in chemistry.
In fractions, a prime numerator simplifies the process of reducing a fraction. Why? Because if a number is prime, it seemingly cannot be broken down any further through division, unless paired with the number 1.
In our example with \(\frac{19}{20}\), the number 19 is a prime number thus ensuring that our fraction cannot be simplified further. This helps immediately prove that the fraction \(\frac{19}{20}\) is in its simplest form:

• No common factors with the denominator.
• No need for further division steps.

Understanding prime numbers help you recognize these scenarios where simplification is straightforward.