To find the length of Brianna's picture, we start by determining how long it is based on its height and the ratio given.
The height of the picture is \( 7 \frac{1}{3} \) inches. We can convert this mixed number into an improper fraction for easier calculations.
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Convert \( 7 \frac{1}{3} \) to an improper fraction: \[ 7 \frac{1}{3} = 7 + \frac{1}{3} = \frac{21}{3} + \frac{1}{3} = \frac{22}{3} \]
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Now we know that the length of the picture is \( 8 \frac{3}{4} \) times its height. First, we also need to convert \( 8 \frac{3}{4} \) into an improper fraction: \[ 8 \frac{3}{4} = 8 + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{35}{4} \]
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The length \( L \) of the picture can be calculated by multiplying the height by this ratio: \[ L = \text{height} \times \text{ratio} = \frac{22}{3} \times \frac{35}{4} \]
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Now, we perform the multiplication of the fractions: \[ L = \frac{22 \times 35}{3 \times 4} = \frac{770}{12} \]
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Next, we simplify this fraction:
- The greatest common divisor of \( 770 \) and \( 12 \) is \( 2 \).
- Thus, we divide the numerator and the denominator by \( 2 \): \[ L = \frac{770 \div 2}{12 \div 2} = \frac{385}{6} \]
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Now we convert \( \frac{385}{6} \) into a mixed number:
- Dividing \( 385 \) by \( 6 \):
- \( 6 \) goes into \( 385 \) a total of \( 64 \) times because \( 6 \times 64 = 384 \).
- The remainder is \( 1 \), so we can write: \[ \frac{385}{6} = 64 \frac{1}{6} \]
- Dividing \( 385 \) by \( 6 \):
Therefore, the length of Brianna's picture is \( \boxed{64 \frac{1}{6}} \) inches.
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