To determine how many pounds of turnips Marco bought, we first need to calculate the total weight of the turnips based on the weight of the parsnips.
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Weight of the Parsnips: The weight of the parsnips is given as \(3 \frac{5}{6}\) pounds.
To convert this mixed number into an improper fraction, we start by converting \(3\) to sixths: \[ 3 = \frac{3 \times 6}{6} = \frac{18}{6}. \] Now, add \(\frac{5}{6}\): \[ 3 \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6}. \]
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Weight of the Turnips: It is stated that the turnips weigh \(4 \frac{1}{2}\) times as much as the parsnips.
First, we convert \(4 \frac{1}{2}\) into an improper fraction: \[ 4 = \frac{4 \times 2}{2} = \frac{8}{2}. \] Adding \(\frac{1}{2}\): \[ 4 \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2}. \]
The weight of the turnips can now be calculated as: \[ \text{Weight of turnips} = \frac{9}{2} \times \frac{23}{6}. \]
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Multiplying the Fractions: We multiply the numerators and the denominators: \[ = \frac{9 \times 23}{2 \times 6} = \frac{207}{12}. \]
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Simplifying the Fraction: Now we can simplify \(\frac{207}{12}\). First, we calculate the greatest common divisor (GCD) of 207 and 12. The factors of 12 are 1, 2, 3, 4, 6, 12.
Checking divisibility:
- \(207 \div 3 = 69\) and \(12 \div 3 = 4\).
We can simplify the fraction by dividing both the numerator and denominator by 3: \[ \frac{207 \div 3}{12 \div 3} = \frac{69}{4}. \]
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Converting to Mixed Number: We can convert \(\frac{69}{4}\) into a mixed number. Dividing \(69\) by \(4\):
- \(69 \div 4 = 17\) remainder \(1\).
Thus, \[ \frac{69}{4} = 17 \frac{1}{4}. \]
So, the final weight of the turnips that Marco bought is \[ \boxed{17 \frac{1}{4}} \text{ pounds.} \]