Question

In each case, simplify the given expression, if possible. $\frac{1}{3}x+\frac{1}{2}x$

Short answer

Question: Simplify the mathematical expression $\frac{1}{3}x+\frac{1}{2}x$. Answer: $\frac{5}{6}x$

Step-by-step solution

Identify the denominators

In the given expression, $\frac{1}{3}x+\frac{1}{2}x$, the denominators are 3 and 2.

Find the least common denominator (LCD)

Here, we have to find the least common multiple (LCM) of the two denominators (3 and 2). In this case, the LCM of 3 and 2 is 6, which makes it the least common denominator (LCD) for the fractions.

Convert the fractions into equivalent fractions using the LCD

Now, we need to write the fractions with the LCD of 6: $$\frac{1}{3}x=\frac{1\cdot 2}{3\cdot 2}x=\frac{2}{6}x$$ and $$\frac{1}{2}x=\frac{1\cdot 3}{2\cdot 3}x=\frac{3}{6}x$$

Combine and simplify the fractions

Now that both fractions have the same denominator, we can combine the numerators and simplify: $$\frac{2}{6}x+\frac{3}{6}x=\frac{2+3}{6}x=\frac{5}{6}x$$ The simplified expression is $\frac{5}{6}x$.

Key concepts

Least Common Denominator

In algebra, dealing with fractions often requires the identification of the least common denominator (LCD) when adding, subtracting, or comparing fractions. The LCD is the smallest number that each of the denominators of multiple fractions can divide into without leaving a remainder. It's essentially the least common multiple (LCM) of the denominators.

Finding the LCD helps in transforming fractions with different denominators into equivalent fractions with the same denominator, which is a crucial step before any addition or subtraction of the fractions can be done. As seen in the solved exercise, with denominators of 3 and 2, the LCD is determined to be 6. This step ensures that the fractions are compatible for summation or subtraction by creating a common ground.

Equivalent Fractions

Finding Equivalent Fractions

To add fractions with different denominators, we must first convert them into equivalent fractions with a common denominator. Equivalent fractions are fractions that represent the same portion of a whole, even though they may appear different at a glance due to having different numerators and denominators. The procedure involves multiplying both the numerator and the denominator by the same number.

For example, if the LCD is 6 and the original denominator is 3, we multiply both the numerator and the denominator by 2 to achieve the equivalent fraction with a denominator of 6. The process is similarly followed for other fractions until all have the common denominator, allowing for straightforward arithmetic operations between them.

Algebraic Expressions

Algebraic expressions are made up of variables, numbers, and arithmetic operations. In expressions like the one given in the exercise, variables are denoted by letters such as 'x'. In the context of simplifying algebraic fractions, the variables will follow the typical arithmetic rules, just like numerical fractions.

Hence, when we have $\frac{1}{3}x$ and $\frac{1}{2}x$, we understand that 'x' is present in both terms; therefore, we only need to focus on the coefficients (numerical factors) and the denominators to perform arithmetic operations.

Simplification of Expressions

Combining Like Terms

Simplification is about combining like terms and reducing expressions to their simplest form. In the context of algebraic fractions, once we have equivalent fractions with a common denominator, we combine the numerators while keeping the denominator constant. This is possible because the fractions represent parts of the same whole.

The final step is to simplify the fraction if possible. For the exercise given, $\frac{5}{6}x$ cannot be simplified further as 5 and 6 are relatively prime (they have no common factors other than 1). Therefore, this is the simplest form of the expression. Always check for a common factor that you could divide both the numerator and denominator by to reduce the fraction to its simplest terms.