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When we talk about quotients, we are referring to the result obtained when one number or variable is divided by another. In essence, division allows us to determine how many times one number can 'fit' into another or, in a more abstract sense, how a certain quantity relates to another in terms of scale or proportion.

For example, if you have 10 pieces of fruit and you want to share them equally among 2 friends, you would perform the division 10 divided by 2. The quotient here would be 5, indicating that each friend gets 5 pieces of fruit. The same principle applies when working with variables such as in our exercise wherein the quotient of (-a) and (b) is expressed as the fraction (-a/b).

The concept of negative variables can be initially confusing but plays a crucial role in mathematics, particularly in equations and inequalities. A negative variable simply means that the variable represents a value that is less than zero. This is denoted by a minus sign '-' preceding the variable.

When dividing by a negative variable or number, it's important to remember the rules about the signs: A negative divided by a positive yields a negative result, a positive divided by a negative also gives a negative result, and of course, a negative divided by a negative gives a positive result. Thus, in our exercise, the quotient of the negative variable (-a) and the positive variable (b) is a negative fraction, (-a/b).

Fractions are an essential component in mathematics, representing a piece of a whole. They consist of two parts: the numerator, which is the number on top, and the denominator, which is the number at the bottom. The numerator signifies how many parts are being considered, while the denominator indicates into how many total parts that whole was divided.

In the context of our exercise, the fraction (-a/b) shows how we divide something represented by (-a) into (b) parts. In the world of algebra, the variables within a fraction behave just like numbers do. If either the numerator or the denominator is negative, the entire fraction is negative. It's also worth noting that fractions can sometimes be simplified or converted into decimal numbers to make them easier to work with or to understand the size of the quantity they represent.