To estimate the sum of the fractions \( \frac{31}{4} + \frac{17}{3} + \frac{8}{18} \), we can first convert these fractions to decimal approximations to make the addition simpler.
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Calculate \( \frac{31}{4} \): \[ \frac{31}{4} = 7.75 \]
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Calculate \( \frac{17}{3} \): \[ \frac{17}{3} \approx 5.67 \quad (\text{since } 3 \text{ goes into } 17, 5 \text{ times with a remainder of } 2) \]
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Calculate \( \frac{8}{18} \): \[ \frac{8}{18} = \frac{4}{9} \approx 0.44 \]
Now, let's sum these approximate values: \[ 7.75 + 5.67 + 0.44 \approx 7.75 + 5.67 = 13.42 \] Then adding \( 0.44 \): \[ 13.42 + 0.44 \approx 13.86 \]
This sum approaches about 14 when rounded to the nearest whole number.
Thus, the best estimate of the sum of the fractions is: \[ \boxed{14\text{ and } \frac{1}{2}} \]
Considering the options available (12 and one-half, 13, 14 and one-half, 15), the closest answer is \( 14 \text{ and one-half} \).
The correct answer is \( 14 \text{ and one-half} \).