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The first step in many fraction problems is simplification. Simplifying a fraction makes it much easier to work with. It involves breaking the fraction down to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the given exercise, we start with \(\frac{135}{510}\). To simplify this, we need to find the GCD of the numerator (135) and the denominator (510).
Once we have the GCD, we divide both the numerator and the denominator by this number. This reduces the fraction to its simplest form, making it much easier to handle in subsequent steps.

The greatest common divisor (GCD) is the largest positive integer that divides both numbers without leaving a remainder. To find the GCD of 135 and 510, you can use the Euclidean algorithm or prime factorization methods, but here we'll use the Euclidean algorithm, which is efficient.
1. Divide 510 by 135 to get a quotient and remainder.
2. Take the divisor (135) and divide it by the remainder from the first division.
3. Repeat the process until the remainder is 0. The divisor at this step will be the GCD.
In this exercise, the GCD of 135 and 510 is found to be 15. We then divide both the numerator and the denominator by 15 to simplify the fraction to \(\frac{9}{34}\).

Long division is a method to divide two numbers, which can provide a decimal result when the division is not exact. In the given problem, after simplifying the fraction to \(\frac{9}{34}\), we perform long division to convert the fraction into decimal form.
Here is how long division works with \(\frac{9}{34}\):
1. Set up 9 (the numerator) as the dividend inside the division bracket and 34 (the denominator) as the divisor outside the bracket.
2. Determine how many times 34 can fit into 9. Since 34 doesn't fit into 9, place a zero and a decimal point.
3. Add a zero to the right of 9, making it 90. Determine how many times 34 fits into 90, which is 2. Write 2 above the line after the decimal point.
4. Multiply 34 by 2 to get 68, then subtract 68 from 90 to get a remainder of 22.
5. Bring down another 0 to make it 220 and repeat the division process. This continues until you reach the desired level of accuracy (in this case, approximately 0.2647).