To find out how old Gary's brother is, we need to subtract the age difference from Gary's age.
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Convert Gary's age into an improper fraction:
- \( 8 \frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \) years old.
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Convert his brother's age difference into an improper fraction:
- \( 3 \frac{7}{12} = \frac{3 \times 12 + 7}{12} = \frac{36 + 7}{12} = \frac{43}{12} \) years.
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Now we need to find Gary's brother's age by subtracting the age difference from Gary's age:
- \( \text{Gary's brother's age} = \frac{35}{4} - \frac{43}{12} \).
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To perform the subtraction, we need a common denominator. The least common multiple of 4 and 12 is 12.
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Convert \( \frac{35}{4} \) to a fraction with a denominator of 12:
- \( \frac{35}{4} = \frac{35 \times 3}{4 \times 3} = \frac{105}{12} \).
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Now we can subtract:
- \( \frac{105}{12} - \frac{43}{12} = \frac{105 - 43}{12} = \frac{62}{12} \).
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Simplifying \( \frac{62}{12} \):
- Divide both the numerator and the denominator by 2:
- \( \frac{62 \div 2}{12 \div 2} = \frac{31}{6} \).
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Convert \( \frac{31}{6} \) back into a mixed number:
- \( 31 \div 6 = 5 \) with a remainder of \( 1 \).
- Thus, \( \frac{31}{6} = 5 \frac{1}{6} \).
Therefore, Gary's brother is \( 5 \frac{1}{6} \) years old.