Solve the following equation for $ x $:$$\frac{3x}{1 + \frac{1}{1 + \frac{x}{1-x}}} = 12$$

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Examoogle · Accepted Answer

The correct answer is Option A: Not provided in the source image. Simplify the nested fraction starting from the bottom: $1 + \frac{x}{1-x} = \frac{1-x+x}{1-x} = \frac{1}{1-x}$. Then $1 + \frac{1}{1/(1-x)} = 1 + (1-x) = 2-x$. Finally solve $\frac{3x}{2-x} = 12$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Simplify the nested fraction starting from the bottom: $1 + \frac{x}{1-x} = \frac{1-x+x}{1-x} = \frac{1}{1-x}$. Then $1 + \frac{1}{1/(1-x)} = 1 + (1-x) = 2-x$. Finally solve $\frac{3x}{2-x} = 12$.

Formula Used: $$\frac{3x}{1 + \frac{1}{1 + \frac{x}{1-x}}} = 12$$

Given: An equation with a nested fraction equal to 12. Equation: 3x / (1 + 1 / (1 + x / (1 - x))) = 12

Steps: Simplify the innermost denominator: First, evaluate $1 + \frac{x}{1-x}$. Finding a common denominator, we get $\frac{1-x+x}{1-x}$, which simplifies to $\frac{1}{1-x}$.. Substitute back into the fraction: Now substitute this result into the original expression: $1 + \frac{1}{1/(1-x)} = 1 + (1-x) = 2-x$.. Solve for x: The original equation becomes $\frac{3x}{2-x} = 12$. Multiply both sides by $(2-x)$: $3x = 12(2-x) \Rightarrow 3x = 24 - 12x$. Adding $12x$ to both sides gives $15x = 24$.

Answer: A is correct because the step-by-step simplification of the continued fraction results in 1.6.

Exam Tip: This concept of nested fractions frequently appears in simplification-based arithmetic questions where solving from inside out prevents calculation errors.

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Option B: Not provided in the source image

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Option C: Not provided in the source image

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Option D: Not provided in the source image

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