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Chapter 4: Problem 263

In the following exercises, find each sum. $$\frac{y}{3}+\frac{2}{3}$$

Short Answer

Expert verified
\(\frac{y + 2}{3}\)

Step by step solution

01

Identify the common denominator

The common denominator for the terms \(\frac{y}{3}\) and \(\frac{2}{3}\) is already 3. This simplifies our task since both fractions are over the same denominator.
02

Combine the numerators

Add the numerators together. The expression \(\frac{y}{3}\) plus \(\frac{2}{3}\) can be written as: \[\frac{y + 2}{3}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When adding fractions, one of the first steps is to find the common denominator. The denominator is the bottom number in a fraction; it tells us how many equal parts the whole is divided into.

If the denominators of the fractions are the same, they are called common denominators. In the given problem, both fractions already have the same denominator (3). This makes our work easier because we don't need to adjust the fractions before adding them.

By ensuring the denominators are the same, combining fractions becomes a straightforward process. Remember, changing denominators is only necessary when they are different.
Combining Numerators
Once the denominators are the same, the next step is to combine the numerators. The numerator is the top number in a fraction and shows how many parts you are considering.

In our exercise, the fractions are: \(\frac{y}{3}\) and \(\frac{2}{3}\). Note that their denominators are already 3, so we directly add the numerators. Combine the numerators as follows:
  • Take the first numerator, which is y.
  • Take the second numerator, which is 2.
  • Add them together: y + 2.

The combined result is \(\frac{y + 2}{3}\). Therefore, the numerators y and 2 form a single numerator in the new fraction.
Simplifying Fractions
Simplifying a fraction means reducing it to its simplest form where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with.

However, in our example, \(\frac{y + 2}{3}\) is already simplified because:
  • The numerator (y + 2) cannot be simplified further unless specific values for y make it possible.
  • The denominator (3) is a prime number and cannot be factored further.

Thus, \(\frac{y + 2}{3}\) remains as it is, representing the simplest form of the given fractions' sum. Always check if you can further reduce a fraction for clarity.
Basic Algebra
Basic algebra involves using symbols and letters to represent numbers and quantities in equations and expressions. It is essential in simplifying and solving mathematical problems. In our fraction addition problem, we use basic algebraic ideas.

Here鈥檚 how:
  • Identify variables: y is a variable that represents an unknown number.
  • Use addition to combine like terms: We added y and 2, the numerators, using basic addition.
  • Keep the structure of fractions intact: We maintained the denominator while combining the numerators.

Algebra helps in structuring and solving problems systematically. Understanding these foundational concepts will aid in tackling more complex equations and expressions in future math tasks.

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