Divide.
$\frac{2 b}{5 c} \div \frac{6 b c}{10 c}$
Simplify your answer as much as possible.
The problem asks to simplify the expression involving the division of two rational algebraic fractions: \(\frac{2b}{5c} \div \frac{6bc}{10c}\).
To perform this division, we use the property that dividing by a fraction is equivalent to multiplying by its reciprocal:
\[\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}\]
where the dividend is \(\frac{2b}{5c}\) and the divisor is \(\frac{6bc}{10c}\). Assuming \(b
eq 0\) and \(c
eq 0\), we can proceed with the simplification.
First, rewrite the division as multiplication by the reciprocal:
\[\frac{2b}{5c} \div \frac{6bc}{10c} = \frac{2b}{5c} \times \frac{10c}{6bc}\]
Now, combine these two fractions into a single product by multiplying their numerators and denominators:
\[\frac{2b \times 10c}{5c \times 6bc}\]
Multiply the terms in the numerator and denominator:
\[\frac{20bc}{30bc^2}\]
Next, we simplify this resulting fraction by canceling common factors. The term \(10bc\) is common to both the numerator and the denominator:
\[\frac{20bc}{30bc^2} = \frac{2 \times (10bc)}{3c \times (10bc)} = \frac{2}{3c}\]
Thus, the expression simplifies to \(\frac{2}{3c}\).
This algebraic simplification is essential in various fields, such as engineering or physics, where you often deal with complex formulas containing multiple variables. For instance, if you are calculating the total resistance in a circuit or the velocity of an object given specific mass-to-volume ratios, you might encounter similar rational expressions. Simplifying these expressions allows you to obtain a clear, concise result, making it easier to analyze the relationship between the variables and make informed decisions or predictions based on your model.