Example Question - simplify complex fraction
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Simplifying a Complex Fraction
The image presents a mathematical expression, which is a complex fraction: \(\frac{\frac{B}{3}}{B}\) To simplify this expression, you can multiply the numerator and the denominator by the reciprocal of the denominator. In simpler terms, you multiply the top and bottom by \(3/B\): \(\frac{B}{3} \times \frac{3}{B}\) over \(B \times \frac{3}{B}\) When we multiply the fraction \(B/3\) by \(3/B\), the B's cancel out in the numerator, and the 3's cancel out, leaving us with 1 since anything divided by itself is 1. When we multiply \(B\) by \(3/B\), the B's cancel out and we are left with 3. So we have: 1 over 3 The result is: \(\frac{1}{3}\). Looking at the answer choices provided: A) \(\frac{O}{B}\) B) \(\frac{O}{3B}\) C) \(\frac{O}{3}\) D) \(\frac{3B}{B}\) Option C is equivalent to our result, \(\frac{1}{3}\), if we assume that "O" signifies the number 1 (as it might look like a vertical line in from the fraction line and was intended to be the number 1). Therefore, C) \(\frac{O}{3}\) is correct if "O" is misprinted and should be the number 1. If the question did not contain a misprint, none of the provided options would be equivalent to \(\frac{1}{3}\).
Simplification of a Complex Fraction
To simplify the given expression, we need to combine the terms in the numerator and the denominator first and then simplify the complex fraction. \[ \frac{\frac{2}{x}-5}{6 + \frac{3}{x}} \] First, let's find a common denominator for the terms in the numerator and the terms in the denominator. For the numerator, the common denominator is \( x \). For the denominator, it is also \( x \). Numerator: \[ \frac{2}{x} - 5 = \frac{2}{x} - \frac{5x}{x} = \frac{2 - 5x}{x} \] Denominator: \[ 6 + \frac{3}{x} = \frac{6x}{x} + \frac{3}{x} = \frac{6x + 3}{x} \] Now that we have a common denominator, we can combine the two parts: \[ \frac{\frac{2 - 5x}{x}}{\frac{6x + 3}{x}} \] Next, we can simplify by multiplying by the reciprocal of the denominator: \[ \frac{2 - 5x}{x} \times \frac{x}{6x + 3} \] Notice that \( x \) in the numerator and denominator will cancel out: \[ \frac{2 - 5x}{6x + 3} \] Now the expression is simplified: \[ \frac{2 - 5x}{6x + 3} \] This is the simplified form of the given expression.
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