What is the product?
$\frac{a-3}{15 a} \cdot \frac{5}{a-3}$
We are presented with the product of two rational expressions, \(\frac{a-3}{15 a} \cdot \frac{5}{a-3}\). To simplify this, we must first note that the expression is defined only when the denominators are non-zero. Thus, we have the conditions \(a eq 0\) and \(a eq 3\). The task is to multiply these two fractions and reduce the resulting expression to its simplest form.
To multiply the two fractions, we multiply the numerators together and the denominators together:
\[ \frac{a-3}{15 a} \cdot \frac{5}{a-3} = \frac{(a-3) \cdot 5}{15a \cdot (a-3)} \]
Since \(a
eq 3\), the term \((a-3)\) is not equal to zero. Therefore, we can safely divide both the numerator and the denominator by the common factor \((a-3)\), which effectively cancels them out:
\[ \frac{5}{15a} \]
Now, we simplify the fraction \(\frac{5}{15a}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
\[ \frac{5 \div 5}{15a \div 5} = \frac{1}{3a} \]
This is the simplest form of the expression. Looking at the given options, the result corresponds to the second choice.
Imagine you are analyzing the efficiency of a manufacturing process. Suppose the total cost of producing a certain batch of items is represented by the fraction \(\frac{a-3}{15a}\) dollars per item, and you are buying a quantity of items represented by \(\frac{5}{a-3}\). To find the total cost of your purchase, you multiply these two expressions. Simplifying algebraic expressions like this allows engineers and economists to quickly determine the overall cost or efficiency without dealing with unnecessary complexity, making it easier to optimize production plans.