To find out how many jars the raspberry jam will fill, we first need to calculate the amount of raspberry jam Mariana made in relation to the strawberry jam she prepared.
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We know that Mariana made enough strawberry jam to fill \(2 \frac{1}{2}\) jars. We can convert this mixed number to an improper fraction: \[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \] So, the amount of strawberry jam is \(\frac{5}{2}\) jars.
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Mariana made \(3 \frac{1}{3}\) times as much raspberry jam as strawberry jam. First, we convert \(3 \frac{1}{3}\) to an improper fraction: \[ 3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]
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Now, we calculate the amount of raspberry jam: \[ \text{Amount of raspberry jam} = \frac{10}{3} \times \text{Amount of strawberry jam} \] Substituting the amount of strawberry jam: \[ \text{Amount of raspberry jam} = \frac{10}{3} \times \frac{5}{2} \]
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We multiply the fractions: \[ \frac{10 \times 5}{3 \times 2} = \frac{50}{6} \]
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We simplify \(\frac{50}{6}\): \[ \frac{50 \div 2}{6 \div 2} = \frac{25}{3} \]
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Finally, we convert \(\frac{25}{3}\) to a mixed number: \[ \frac{25}{3} = 8 \frac{1}{3} \]
Thus, the raspberry jam will fill \(8 \frac{1}{3}\) jars.
The final answer is: \[ \boxed{8 \frac{1}{3}} \text{ jars} \]
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