To find Gary's brother's age, we need to subtract the age difference from Gary's age.
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Convert Gary's age to a fraction: Gary is 8 and 3/4 years old. \[ 8 \frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \]
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Convert the age difference to a fraction: Gary's brother is 3 and 7/12 years younger than Gary. \[ 3 \frac{7}{12} = \frac{3 \times 12 + 7}{12} = \frac{36 + 7}{12} = \frac{43}{12} \]
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Subtract the age difference from Gary's age: \[ \text{Age of Gary's brother} = \text{Gary's age} - \text{Age difference} \] \[ = \frac{35}{4} - \frac{43}{12} \]
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Find a common denominator to perform the subtraction: The least common multiple of 4 and 12 is 12.
Convert \(\frac{35}{4}\) to have a denominator of 12: \[ \frac{35}{4} = \frac{35 \times 3}{4 \times 3} = \frac{105}{12} \]
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Now perform the subtraction: \[ \frac{105}{12} - \frac{43}{12} = \frac{105 - 43}{12} = \frac{62}{12} \]
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Simplify the fraction: \[ \frac{62}{12} = \frac{31}{6} \]
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Convert \(\frac{31}{6}\) to a mixed number: \[ \frac{31}{6} = 5 \frac{1}{6} \]
Therefore, Gary’s brother is 5 and 1/6 years old.