Evaluate \( 5 \frac{7}{15}-\frac{2}{3}+1 \frac{5}{12} \)

To solve the expression \( 5 \frac{7}{15} - \frac{2}{3} + 1 \frac{5}{12} \), we first convert the mixed numbers into improper fractions. 1. Convert \( 5 \frac{7}{15} \): \[ 5 \frac{7}{15} = \frac{5 \times 15 + 7}{15} = \frac{75 + 7}{15} = \frac{82}{15} \] 2. Convert \( 1 \frac{5}{12} \): \[ 1 \frac{5}{12} = \frac{1 \times 12 + 5}{12} = \frac{12 + 5}{12} = \frac{17}{12} \] Now, the expression becomes: \[ \frac{82}{15} - \frac{2}{3} + \frac{17}{12} \] Next, we need a common denominator for the fractions. The denominators are 15, 3, and 12. The least common multiple of these numbers is 60. Now, we convert each fraction: - For \( \frac{82}{15} \): \[ \frac{82}{15} = \frac{82 \times 4}{15 \times 4} = \frac{328}{60} \] - For \( \frac{2}{3} \): \[ \frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60} \] - For \( \frac{17}{12} \): \[ \frac{17}{12} = \frac{17 \times 5}{12 \times 5} = \frac{85}{60} \] Now we can rewrite the original expression: \[ \frac{328}{60} - \frac{40}{60} + \frac{85}{60} \] Combine the fractions: \[ \frac{328 - 40 + 85}{60} = \frac{328 - 40 = 288}{60} + \frac{85}{60} = \frac{288 + 85}{60} = \frac{373}{60} \] Now, we can simplify \( \frac{373}{60} \): Since 373 is already a prime number, we can't simplify further. Thus, we can write it as a mixed number: - \( 373 \div 60 = 6 \) remainder \( 13 \). Therefore, we can express \( \frac{373}{60} \) as: \[ 6 \frac{13}{60} \] Hence, the final result is: \[ \boxed{6 \frac{13}{60}} \]