1. **State the problem:** Calculate the value of the expression $$\frac{1}{\frac{0.3}{0.33} + \frac{0.5}{0.55} + \frac{0.7}{0.77}}$$ and select the correct answer from the options A) 1, B) 1.1, C) 11, D) 22, E) 44. 2. **Recall the formula and rules:** When dividing decimals, convert the division into fraction form and simplify each fraction. 3. **Calculate each fraction in the denominator:** $$\frac{0.3}{0.33} = \frac{30/100}{33/100} = \frac{30}{33} = \frac{10}{11} \approx 0.9091$$ $$\frac{0.5}{0.55} = \frac{50/100}{55/100} = \frac{50}{55} = \frac{10}{11} \approx 0.9091$$ $$\frac{0.7}{0.77} = \frac{70/100}{77/100} = \frac{70}{77} = \frac{10}{11} \approx 0.9091$$ 4. **Sum the fractions in the denominator:** $$0.9091 + 0.9091 + 0.9091 = 3 \times 0.9091 = 2.7273$$ 5. **Calculate the entire expression:** $$\frac{1}{2.7273} \approx 0.3667$$ 6. **Compare with the options:** None of the options exactly match 0.3667, but since the problem likely expects a simplified fraction, let's check the exact fraction form: Each fraction is exactly $\frac{10}{11}$, so sum is: $$3 \times \frac{10}{11} = \frac{30}{11}$$ Therefore: $$\frac{1}{\frac{30}{11}} = \frac{11}{30} \approx 0.3667$$ 7. **Conclusion:** The value is approximately 0.3667, which is not listed among the options. Possibly the problem expects the reciprocal of the sum of fractions, which is $\frac{11}{30}$. **Final answer:** None of the given options A) 1, B) 1.1, C) 11, D) 22, E) 44 matches the exact value $\frac{11}{30}$. The closest interpretation is that the answer is $\frac{11}{30}$ or approximately 0.367. If the problem expects the sum of the fractions instead, that is $\frac{30}{11} \approx 2.7273$, which also does not match options. Hence, the correct evaluation is $\boxed{\frac{11}{30}}$ or approximately 0.367.