Simplify the complex fraction. [tex]$\frac{\frac{4}{p+4}-1}{\frac{4}{p+4}+1}$[/tex]
Answer (1)
Multiply both the numerator and the denominator by p + 4 .
Distribute and simplify the numerator: 4 − ( p + 4 ) = − p .
Distribute and simplify the denominator: 4 + ( p + 4 ) = p + 8 .
The simplified complex fraction is p + 8 − p .
Explanation
Problem Analysis We are asked to simplify the complex fraction:
Complex Fraction p + 4 4 + 1 p + 4 4 − 1
Multiply by LCD To simplify this complex fraction, we can multiply both the numerator and the denominator by the least common denominator (LCD) of the inner fractions, which is p + 4 . This will eliminate the inner fractions.
Multiplication Multiplying the numerator and denominator by p + 4 , we get:
Expression ( p + 4 4 + 1 ) ( p + 4 ) ( p + 4 4 − 1 ) ( p + 4 )
Distribution Distribute ( p + 4 ) in both the numerator and the denominator:
Distributing (p+4) p + 4 4 ( p + 4 ) + 1 ( p + 4 ) p + 4 4 ( p + 4 ) − 1 ( p + 4 )
Simplification Simplify the expression:
Simplified Expression 4 + ( p + 4 ) 4 − ( p + 4 )
Combining Like Terms Now, simplify the numerator and the denominator by combining like terms:
Simplifying Numerator and Denominator 4 + p + 4 4 − p − 4
Final Simplification p + 8 − p
Final Answer So, the simplified complex fraction is p + 8 − p .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios, such as calculating electrical circuit resistance or analyzing rates of change in physics. For instance, when dealing with parallel resistors, the total resistance is often expressed as a complex fraction. Simplifying these fractions allows engineers to quickly determine the overall resistance and optimize circuit performance. Similarly, in physics, complex fractions can arise when modeling the motion of objects under varying forces, where simplifying the fraction helps in predicting the object's trajectory or velocity.
