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Write the following as a single rational expression. \( \frac{170+50 x}{x} \div \frac{120+30 x}{x} \) a) \( \frac{120+30 x}{170+50 x} \) b) \( \frac{2(25+10 x)}{x^{2}} \) c) \( \frac{50+20 x}{x^{2}} \) d) \( \frac{17+5 x}{12+3 x} \) (

Ask by Warren Burns. in the United States
Jan 26,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The simplified single rational expression is \( \frac{17+5x}{12+3x} \), which corresponds to option d).

Solution

To simplify the expression \(\frac{170 + 50x}{x} \div \frac{120 + 30x}{x}\) into a single rational expression, follow these steps: 1. **Convert Division to Multiplication:** \[ \frac{170 + 50x}{x} \div \frac{120 + 30x}{x} = \frac{170 + 50x}{x} \times \frac{x}{120 + 30x} \] 2. **Simplify the Expression:** \[ \frac{170 + 50x}{x} \times \frac{x}{120 + 30x} = \frac{170 + 50x}{120 + 30x} \] 3. **Factor Out Common Terms:** - In the numerator, factor out 10: \[ 170 + 50x = 10(17 + 5x) \] - In the denominator, factor out 30: \[ 120 + 30x = 30(4 + x) = 30(x + 4) \] 4. **Reduce the Fraction:** \[ \frac{10(17 + 5x)}{30(x + 4)} = \frac{17 + 5x}{3(x + 4)} = \frac{17 + 5x}{3x + 12} \] So, the simplified single rational expression is: **Answer:** d) \( \frac{17+5 x}{12+3 x} \)

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The Deep Dive

First, let's rewrite the division of fractions as multiplication by the reciprocal. This means we can rewrite your expression as follows: \[ \frac{170 + 50x}{x} \div \frac{120 + 30x}{x} = \frac{170 + 50x}{x} \cdot \frac{x}{120 + 30x} \] Notice the \(x\) in the denominator and numerator cancels out, leading us to: \[ \frac{170 + 50x}{120 + 30x} \] Now, we can simplify this expression. Divide both the numerator and denominator by 10 for clarity: \[ \frac{17 + 5x}{12 + 3x} \] So the answer is: d) \( \frac{17+5 x}{12+3 x} \)

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