To evaluate the expression \[ \frac{\frac{1}{10}\times\frac{2}{3}+\frac{1}{4}}{\frac{\frac{1}{2}}{\frac{3}{5}}-\frac{1}{4}}, \] we will break it down step-by-step. ### Step 1: Evaluate the Numerator The numerator is \[ \frac{1}{10} \times \frac{2}{3} + \frac{1}{4}. \] **Calculating \(\frac{1}{10} \times \frac{2}{3}\):** \[ \frac{1}{10} \times \frac{2}{3} = \frac{1 \times 2}{10 \times 3} = \frac{2}{30} = \frac{1}{15}. \] **Now, add \(\frac{1}{15}\) and \(\frac{1}{4}\):** To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 4 is 60. **Convert \(\frac{1}{15}\) to a fraction with a denominator of 60:** \[ \frac{1}{15} = \frac{1 \times 4}{15 \times 4} = \frac{4}{60}. \] **Convert \(\frac{1}{4}\) to a fraction with a denominator of 60:** \[ \frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}. \] **Now, add the two fractions:** \[ \frac{4}{60} + \frac{15}{60} = \frac{4 + 15}{60} = \frac{19}{60}. \] ### Step 2: Evaluate the Denominator The denominator is \[ \frac{\frac{1}{2}}{\frac{3}{5}} - \frac{1}{4}. \] **Calculating \(\frac{\frac{1}{2}}{\frac{3}{5}}\):** Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{1 \times 5}{2 \times 3} = \frac{5}{6}. \] **Now, subtract \(\frac{1}{4}\) from \(\frac{5}{6}\):** Again, we need a common denominator. The LCM of 6 and 4 is 12. **Convert \(\frac{5}{6}\) to a fraction with a denominator of 12:** \[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}. \] **Convert \(\frac{1}{4}\) to a fraction with a denominator of 12:** \[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}. \] **Now, subtract the two fractions:** \[ \frac{10}{12} - \frac{3}{12} = \frac{10 - 3}{12} = \frac{7}{12}. \] ### Step 3: Combine the Results Now we have: - Numerator: \(\frac{19}{60}\) - Denominator: \(\frac{7}{12}\) We need to evaluate: \[ \frac{\frac{19}{60}}{\frac{7}{12}}. \] **Dividing by a fraction:** This is the same as multiplying by the reciprocal: \[ \frac{19}{60} \div \frac{7}{12} = \frac{19}{60} \times \frac{12}{7} = \frac{19 \times 12}{60 \times 7}. \] **Calculating the numerator and denominator:** \[ 19 \times 12 = 228, \] \[ 60 \times 7 = 420. \] So we have: \[ \frac{228}{420}. \] ### Step 4: Simplify the Fraction To simplify \(\frac{228}{420}\), we need to find the greatest common divisor (GCD) of 228 and 420. **Finding the GCD:** - The prime factorization of 228 is \(2^2 \times 3 \times 19\). - The prime factorization of 420 is \(2^2 \times 3 \times 5 \times 7\). The GCD is \(2^2 \times 3 = 12\). **Now divide both the numerator and denominator by 12:** \[ \frac{228 \div 12}{420 \div 12} = \frac{19}{35}. \] ### Final Answer Thus, the final answer is \[ \frac{19}{35}. \] ### Explanation of Options - **Option A: \(\frac{7}{12}\)** - This is incorrect because it does not represent the final value of the expression. - **Option B: \(\frac{19}{35}\)** - This is the correct answer as shown in our calculations. - **Option C: \(\frac{2}{25}\)** - This is incorrect as it does not match any part of our calculations. - **Option D: \(\frac