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Adding fractions involves finding a common denominator, which is crucial for ensuring that both parts of the fraction have a shared base. This commonality allows us to directly sum the numerators. For example, when adding \( \frac{2}{3} \) and \( \frac{3}{5} \), we first need to find the least common denominator (LCD) for 3 and 5. The LCD here is 15.

Next, we convert each fraction so they both have the denominator of 15:

• \( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)
• \( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)

Now, the fractions can be easily added together by summing their numerators while keeping the denominator the same:
\( \frac{10}{15} + \frac{9}{15} = \frac{19}{15} \).
It’s important to simplify the fractions when necessary, though in this case, \( \frac{19}{15} \) is already in its simplest form.

Subtraction of fractions follows a very similar process to addition, but instead of adding, we subtract the numerators once a common denominator is established. Let’s look at subtracting the imaginary parts of the given complex numbers: \( -\frac{1}{5} i \) and \( -\frac{3}{4} i \).

The first step here is to find the common denominator for the fractions \( \frac{1}{5} \) and \( \frac{3}{4} \). The least common denominator is 20.

We then rewrite each fraction so the denominators match:

• \( -\frac{1}{5} i = -\frac{4}{20} i \)
• \( -\frac{3}{4} i = -\frac{15}{20} i \)

With a shared denominator, subtraction becomes straightforward:
\( -\frac{4}{20} i - \frac{15}{20} i = -\frac{19}{20} i \).
The process requires careful attention to the signs of the fractions being subtracted, especially since they are part of a complex number, ensuring calculations remain accurate.

Imaginary numbers arise when we find the square root of a negative number. Instead of a real number, the square root of -1 is represented as \( i \). This concept is especially useful in complex numbers and various engineering fields. In our case, the given complex expression involves imaginary components: \( -\frac{1}{5} i \) and \( -\frac{3}{4} i \).

Complex numbers combine real and imaginary numbers in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. When working with any imaginary number, it behaves just like a normal algebraic term, except that \( i^2 = -1 \).
Complex arithmetic, such as addition and subtraction of these components, follows standard operations:

• Add/subtract the real parts)
• Add/subtract the imaginary parts)

Each part is addressed separately within computations, and the final result combines them into a single, unified solution as shown in the exercise: \( \frac{19}{15} - \frac{19}{20} i \).