Simplify the complex fraction. Use either method.

$\frac{\frac{t}{t+4}}{\frac{5}{t^2-16}}$

Rewrite the complex fraction as a division problem.
Rewrite the division as multiplication by the reciprocal.
Factor the quadratic term.
Cancel the common factor and simplify the expression: 5 t 2 − 4 t ​ ​

Explanation

Understanding the Problem We are asked to simplify the complex fraction t 2 − 16 5 ​ t + 4 t ​ ​ . This involves dividing one fraction by another, which we can accomplish by multiplying by the reciprocal of the denominator.

Rewriting as Division First, we rewrite the complex fraction as a division problem: t 2 − 16 5 ​ t + 4 t ​ ​ = t + 4 t ​ ÷ t 2 − 16 5 ​

Multiplying by the Reciprocal Next, we rewrite the division as multiplication by the reciprocal: t + 4 t ​ ÷ t 2 − 16 5 ​ = t + 4 t ​ ⋅ 5 t 2 − 16 ​

Factoring the Quadratic Now, we factor the quadratic term in the numerator: t 2 − 16 = ( t + 4 ) ( t − 4 )

Substituting the Factored Form Substitute the factored form into the expression: t + 4 t ​ ⋅ 5 t 2 − 16 ​ = t + 4 t ​ ⋅ 5 ( t + 4 ) ( t − 4 ) ​

Canceling Common Factors Cancel the common factor ( t + 4 ) from the numerator and denominator: t + 4 t ​ ⋅ 5 ( t + 4 ) ( t − 4 ) ​ = 5 t ( t − 4 ) ​

Simplifying the Expression Finally, we simplify the expression by distributing t in the numerator: 5 t ( t − 4 ) ​ = 5 t 2 − 4 t ​


Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For instance, when calculating the average speed of a journey with varying speeds over different segments, or when dealing with ratios of ratios in mixture problems, simplifying complex fractions becomes essential. Imagine you're a chef adjusting a recipe that involves ratios of ingredients; simplifying complex fractions helps you maintain the correct proportions and achieve the desired taste.

Answered by GinnyAnswer | 2025-07-05

To simplify the complex fraction t 2 − 16 5 ​ t + 4 t ​ ​ , we rewrite it as a division, multiply by the reciprocal, and factor where necessary. After canceling common factors, we arrive at the result 5 t 2 − 4 t ​ . Thus, the simplified form is ( \boxed{\frac{t^2 - 4t}{5}} .
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Answered by Anonymous | 2025-07-14