Problem 84 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{2}{3}+\frac{1}{6}-\frac{1}{3}\)

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{2}{3}+\frac{1}{6}-\frac{1}{3} \) is \( \frac{1}{2} \).

Step 1: Identify the common denominator

In this case, 2/3, 1/6 and 1/3 are fractions to be simplified. The common denominator for these fractions is 6 as 6 is divisible by both 2 and 3.

Step 2: Convert fractions to common denominator

Change the fractions to have the common denominator. The fractions become \( \frac{4}{6}+\frac{1}{6}-\frac{2}{6} \). This is done by dividing 6 by the original denominator and multiplying the numerator by the resultant factor.

Step 3: Perform the operations

Now add and subtract the fractions part by part by only looking at the numerators because the denominators are the same. This results into \( \frac{4+1-2}{6}=\frac{3}{6} \).

Step 4: Simplify the result

You simplify fractions to their simplest form by dividing both the numerator and the denominator by their greatest common divisor. In this case, 3 can divide both 3 and 6, so the simplest form of the fraction is \( \frac{1}{2} \).

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when working with fractions. It's the backbone of adding, subtracting, or comparing fractions. A common denominator refers to a shared multiple of the denominators of two or more fractions. In other words, it's a common ground upon which different fractions can easily be added, subtracted, or compared.

For instance, the exercise \(\frac{2}{3}+\frac{1}{6}-\frac{1}{3}\) required finding a number that is divisible by both 3 and 6, which is 6. To work with fractions effectively, always identify a common denominator, as it will enable you to manipulate the fractions to perform various arithmetic operations.

Mixed Numbers

Mixed numbers combine a whole number with a fraction, exemplifying numbers that stand between two whole numbers. For example, 1 and a half is written as 1 1/2 in mixed number form.

Though mixed numbers weren't directly involved in the given exercise, it's important to know how to handle them when they appear in problems. If, after performing operations, the fractional part is improper (where the numerator is larger than the denominator), you convert it to a mixed number. This makes the result more understandable and relatable to real-life quantities.

Simplest Form

The simplest form, also known as reduced form, of a fraction occurs when the numerator and denominator have no common factor other than 1. Simplifying makes the fraction as simple as it can be. To get a fraction in its simplest form, divide both the numerator and the denominator by their greatest common divisor (GCD).

The exercise demonstrates this principle. After adding and subtracting the fractions, we obtain \(\frac{3}{6}\). To simplify it, we note that 3 is the GCD of both the numerator and the denominator, so dividing both by 3 gives us \(\frac{1}{2}\), which is the simplest form of the fraction.

Numerators and Denominators

Fractions consist of two parts: the numerator and the denominator. The numerator, located above the fraction bar, indicates how many parts are considered, while the denominator, beneath the fraction bar, shows the total number of equal parts in a whole.

In the context of the given exercise, after converting to a common denominator, we observe the operations on the numerators: \(\frac{4}{6}+\frac{1}{6}-\frac{2}{6}\). Since the denominators are equal, we can focus on adding and subtracting the numerators to find the solution. This process illuminates the roles numerators and denominators play in arithmetic operations involving fractions.