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When subtracting fractions, the first step is ensuring both fractions have a common denominator. This is essential because fractions can only be directly added or subtracted when they refer to the same whole, which is what the denominator represents. The common denominator is typically the least common multiple of the two denominators, allowing for the smallest possible equivalent fraction. This prevents the answer from being unnecessarily large and makes it easier to simplify later on.
For instance, if we have fractions with denominators of 4 and 6, the common denominator would be based on their least common multiple, which is 12. Thus, any fractions we deal with would need to have 12 as the denominator before we proceed to subtraction or addition.

The Least Common Multiple (LCM) plays a vital role in finding a common denominator for fraction subtraction and addition. It is the smallest number that is a multiple of both denominators. To calculate the LCM, list out the multiples of each denominator and then identify the smallest number that appears in both lists.
However, it's more efficient to use prime factorization or the ladder method when dealing with larger numbers. Once the LCM is identified, you can convert the fractions into equivalent fractions with the LCM as their new denominator. This step ensures that the fractions now talk about 'parts' of the same sized 'whole', enabling hassle-free arithmetic operations on them.

To subtract fractions with different denominators, we need to express them as equivalent fractions with a common denominator. Equivalent fractions are different fractions that represent the same part of a whole. You can find an equivalent fraction by multiplying or dividing the numerator and denominator by the same number.
For example, in the given exercise, the fraction \(\frac{9}{20}\) was converted to \(\frac{27}{60}\) by multiplying both the numerator and the denominator by 3. This didn't change the value of the fraction but made it comparable with \(\frac{1}{30}\), which in turn was converted to \(\frac{2}{60}\). Thus, equivalent fractions enable you to adjust fractions in a way that facilitates the subtraction process, leading eventually to the simplified difference between them.