The equivalent of "abd plus one fifth a" can be expressed as "a + (1/5)a" or "6/5a". This is derived by combining the term "abd" with the fraction "1/5a". The term "abd" is likely a placeholder for a variable or a specific value, and when combined with "1/5a", it results in a new expression that represents the sum of these two components.

To understand why this is the case, let's break down the components. The term "abd" can be thought of as "a" multiplied by some coefficient, which we'll call "b". Therefore, "abd" can be rewritten as "ba". When we add "one fifth a" to "abd", we are essentially adding "1/5a" to "ba". This can be represented as:

ba + 1/5a

To simplify this expression, we can find a common denominator, which in this case is 5. Multiplying "ba" by 5/5 gives us "5ba/5". Now we can add the fractions:

5ba/5 + 1/5a = (5ba + a) / 5

Since "5ba" is equivalent to "5a" (because "b" is a coefficient), we can further simplify the expression:

(5a + a) / 5 = 6a / 5

This is the equivalent of "abd plus one fifth a" expressed as a single fraction. The reason for this simplification is based on the principles of algebra, specifically the distributive property and the ability to combine like terms. According to the distributive property, a(b + c) = ab + ac. In this case, "a" is distributed across "b" and "1/5a".

Data supporting this concept can be found in various algebra textbooks and educational resources. For instance, the distributive property is a fundamental concept in algebra that is typically introduced in middle school mathematics. Resources such as the National Council of Teachers of Mathematics (NCTM) provide guidelines and examples that illustrate the use of the distributive property in solving algebraic expressions.

In conclusion, the equivalent of "abd plus one fifth a" is "6/5a", which is derived from the principles of algebra and the distributive property. This simplification is a standard practice in algebraic manipulation and is supported by educational resources and textbooks.