Question

Combine and simplify. Try some by calculator. $$\frac{1}{a}+\frac{5}{a}$$

Short answer

\(\frac{6}{a}\)

Step-by-step solution

Identify Common Denominator

Since both fractions have the same denominator, we can directly combine them by adding their numerators.

Combine Numerators

Add the numerators of the two fractions while keeping the common denominator. So, we get \(\frac{1}{a} + \frac{5}{a} = \frac{1+5}{a}\).

Simplify the Fraction

The sum of the numerators is 6. Thus, the simplified result is \(\frac{6}{a}\).

Key concepts

Common Denominator

When working with fractions, it's critical to understand the concept of a common denominator.
A common denominator refers to a shared multiple of two or more fractions' denominators, which is essential for performing operations such as addition and subtraction. The lowest common denominator is the smallest number that each denominator can divide into without leaving a remainder.

In our exercise, simplifying the expression \(\frac{1}{a} + \frac{5}{a}\) starts with the recognition that both fractions already share a common denominator, which is \(a\). This commonality allows us to easily combine the fractions.
The presence of a common denominator simplifies the process and makes it straightforward: since the denominators are the same, the focus shifts to directly combining the numerators.

Combining Numerators

Once a common denominator has been identified, the next step is combining numerators.
The numerator is the top number of a fraction and represents the number of parts being considered. When fractions share a common denominator, their numerators can be added or subtracted directly. This is a basic rule of the arithmetic of fractions.

In our sample problem, \(\frac{1}{a} + \frac{5}{a}\), we simply add together the numerators 1 and 5, giving us the combined numerator of 6.
Thus, the combined fractions yield a new fraction: \(\frac{1+5}{a}\), which is \(\frac{6}{a}\). This is the essence of combining numerators; it's a simple but powerful tool in fraction arithmetic.

Arithmetic of Fractions

Arithmetic of fractions involves addition, subtraction, multiplication, and division of numbers expressed as fractions.
Each operation has specific rules, but a common thread in dealing with fractions is managing numerators and denominators appropriately. For addition and subtraction, a common denominator must be found. For multiplication, multiply numerators together and denominators together. For division, invert the second fraction and multiply.

In the example \(\frac{1}{a} + \frac{5}{a}\), we are focusing on addition, which requires a common denominator to combine fractions. As aforementioned, since \(a\) serves as that common denominator, we are simply left with the task of adding the numerators.
The beauty of fraction arithmetic lies in the simplicity of such rules; once the principle of operation is understood, executing it—even without a calculator—becomes far less intimidating.