1. **Problem Statement:** We have a list $T$ of 30 positive decimals, none of which is an integer. The sum of these decimals is $S$. We define an estimated sum $E$ by rounding each decimal in $T$ as follows: - If the tenths digit of the decimal is even, round up to the nearest integer. - If the tenths digit is odd, round down to the nearest integer. Given that exactly one-third of the decimals have an even tenths digit, we want to find which of the values $E - S$ can take from the options: $-16$, $6$, and $10$. 2. **Understanding the rounding rule:** - For decimals with even tenths digit, rounding up means $E_i = \lceil x_i \rceil$. - For decimals with odd tenths digit, rounding down means $E_i = \lfloor x_i \rfloor$. 3. **Notation and variables:** - Total decimals: $30$ - Number with even tenths digit: $n_e = \frac{1}{3} \times 30 = 10$ - Number with odd tenths digit: $n_o = 30 - 10 = 20$ 4. **Expressing $E - S$:** Let the decimals be $x_1, x_2, ..., x_{30}$. For even tenths digit decimals (10 numbers): $$E_i - x_i = \lceil x_i \rceil - x_i = 1 - f_i$$ where $f_i$ is the fractional part of $x_i$ (since $x_i$ is not an integer, $0 < f_i < 1$). For odd tenths digit decimals (20 numbers): $$E_i - x_i = \lfloor x_i \rfloor - x_i = -f_i$$ 5. **Sum over all decimals:** $$E - S = \sum_{i=1}^{10} (1 - f_i) + \sum_{j=1}^{20} (-f_j) = 10 - \sum_{i=1}^{10} f_i - \sum_{j=1}^{20} f_j = 10 - \sum_{k=1}^{30} f_k$$ 6. **Range of fractional parts:** Each fractional part $f_k$ satisfies $0 < f_k < 1$. Therefore, $$0 < \sum_{k=1}^{30} f_k < 30$$ 7. **Range of $E - S$:** $$E - S = 10 - \sum_{k=1}^{30} f_k$$ Since $\sum f_k$ is between 0 and 30, $$-20 < E - S < 10$$ 8. **Checking the options:** - $-16$ is within $(-20, 10)$, so possible. - $6$ is within $(-20, 10)$, so possible. - $10$ is not possible because $E - S$ must be strictly less than 10. 9. **Answer choice:** Only options I ($-16$) and II ($6$) are possible. **Final answer:** B. I and II only