Answered

Which expression is equivalent to $\frac{2 a+1}{10 a-5} \div \frac{10 a}{4 a^2-1}$ ?


A$\frac{2 a}{(2 a-1)^2}$
B$\frac{50 a}{(2 a+1)^2}$
C$\frac{(2 a-1)^2}{2 a}$
D$\frac{(2 a+1)^2}{50 a}$

Verified Answer :

  • Rewrite the expression as multiplication by the reciprocal: \(\frac{2a+1}{10a-5} \cdot \frac{4a^2-1}{10a}\).
  • Factor terms: \(10a-5 = 5(2a-1)\) and \(4a^2-1 = (2a-1)(2a+1)\).
  • Cancel common factors like \((2a-1)\) to simplify: \(\frac{2a+1}{5(2a-1)} \cdot \frac{(2a-1)(2a+1)}{10a} = \frac{(2a+1)^2}{50a}\).
  • The equivalent expression is \(\boxed{\frac{(2a+1)^2}{50a}}\).

Explanation

  1. Converting Division to Multiplication

To simplify the given rational expression \(\frac{2a+1}{10a-5} \div \frac{10a}{4a^2-1}\), we begin by converting the division of fractions into the multiplication of the first fraction by the reciprocal of the second. This gives us:
\[\frac{2a+1}{10a-5} \cdot \frac{4a^2-1}{10a}\]

  1. Factoring the Expressions

Next, we factor the components of the rational expressions.

  • In the denominator \(10a - 5\), we can factor out a 5 to get \(5(2a - 1)\).
  • The numerator \(4a^2 - 1\) is a difference of squares, which factors as \((2a - 1)(2a + 1)\).

Substituting these back into the expression, we have:
\[\frac{2a+1}{5(2a-1)} \cdot \frac{(2a-1)(2a+1)}{10a}\]

  1. Simplifying the Expression

Now, we combine the fractions and cancel out common factors. Combining them into one fraction, we get:
\[\frac{(2a+1)(2a-1)(2a+1)}{5(2a-1)(10a)}\]
We observe that \((2a - 1)\) appears in both the numerator and the denominator, so we can cancel it out (provided \(a eq 1/2\)):
\[\frac{(2a+1)(2a+1)}{5(10a)} = \frac{(2a+1)^2}{50a}\]

  1. Conclusion and Final Answer

After simplifying, we compare our result, \(\frac{(2a+1)^2}{50a}\), to the given options. The expression matches the fourth option. Therefore, the equivalent expression is \(\frac{(2a+1)^2}{50a}\).

Examples

Rational expressions are frequently used in engineering and physics to model rates and ratios. For example, if you are calculating the efficiency of a system where multiple gears are linked, the ratio of input speed to output speed can often be represented as a product of rational expressions. Simplifying these expressions allows engineers to quickly determine how changes in one variable, such as the number of teeth on a gear, affect the overall performance of the machine.

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