Problem 21 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{5}{12}+\frac{7}{12}\)

Step-by-Step Solution

Answer

Answer: The sum of the fractions \(\frac{5}{12}\) and \(\frac{7}{12}\) is \(1\).

Step 1: Identify the common denominator

Both fractions have a common denominator: \(12\). This means we can directly add the numerators.

Step 2: Add the numerators

Add the numerators of the fractions: \(5+7=12\).

Step 3: Write the sum as a fraction

Write the sum of the numerators over the common denominator: \(\frac{12}{12}\).

Step 4: Simplify the fraction, if possible

In this case, the fraction \(\frac{12}{12}\) can be simplified to \(1\), because both the numerator and the denominator have the same value. So, the final answer is \(1\).

Key Concepts

Common Denominator

When adding fractions, having a common denominator is crucial. It simplifies the process by aligning both fractions to have the same base. This base, or denominator, is essentially the number at the bottom of the fraction.
In the exercise, both fractions \(\frac{5}{12}\) and \(\frac{7}{12}\) already share a common denominator of 12. This makes it easy to add their numerators directly.

• Having a common denominator means aligning the different parts to a standard value.
• This is a vital step because it ensures you're combining equivalent parts of the whole.

There are cases when fractions don't start with the same denominator. In such situations, you need to find the Least Common Denominator (LCD) by determining the smallest multiple that is shared. Once the denominators match, you can proceed with other operations like addition or subtraction.

Fraction Simplification

Simplifying fractions is about reducing them to their smallest form without changing their value. This involves finding a number that can evenly divide both the numerator (the top part) and the denominator (the bottom part).
In the provided problem, the resulting fraction after adding the numerators was \(\frac{12}{12}\).
This fraction simplifies to 1, because both top and bottom are the same, meaning you have a whole. Here are some quick steps to simplify any fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this number.
• If they divide fully without leaving a remainder, you have their simplest form.

This is a very useful process, especially in complex problems, ensuring your answers are as neat and clean as possible.

Elementary Algebra

Elementary algebra deals with solving basic mathematical operations, and understanding how to work with unknowns or variables. In this context, we're focusing on the arithmetic operation of addition but through the lens of fractions, as seen in the exercise.
The main operation involves adding like terms, which in this case are the numerators, since the denominators are already the same.

• First, inspect the denominators to ensure they're aligned.
• If they are, like here, add the numerators directly, like adding regular numbers.

Elementary algebra aids in building a strong foundation for more complex mathematical tasks. Proper understanding of operations like these is not just essential for obtaining the correct answer, but also for grasping more advanced algebraic concepts later on. This ensures students can transition smoothly from foundational arithmetic to handling expressions that involve variables, like \(x\) or \(y\). Understanding these basics is crucial for future success in mathematics.