Convert the following numbers between bases: i) 1101001.10112 to Base 10. ii) 236.72510 to Base 7.

$$
1101001_2 = (1 \times 2^6) + (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)
$$
$$
= (1 \times 64) + (1 \times 32) + (0 \times 16) + (1 \times 8) + (0 \times 4) + (0 \times 2) + (1 \times 1)
$$
$$
= 64 + 32 + 0 + 8 + 0 + 0 + 1 = 105_{10}
$$

$$
0.1011_2 = (1 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3}) + (1 \times 2^{-4})
$$
$$
= (1 \times 0.5) + (0 \times 0.25) + (1 \times 0.125) + (1 \times 0.0625)
$$
$$
= 0.5 + 0 + 0.125 + 0.0625 = 0.6875_{10}
$$

$$
105_{10} + 0.6875_{10} = 105.6875_{10}
$$
The converted number is $\boxed{105.6875_{10}}$.

$$
236 \div 7 = 33 \text{ remainder } 5
$$
$$
33 \div 7 = 4 \text{ remainder } 5
$$
$$
4 \div 7 = 0 \text{ remainder } 4
$$
Reading the remainders from bottom up gives $455_7$.

$$
0.725 \times 7 = 5.075 \implies \text{integer part is } 5
$$
$$
0.075 \times 7 = 0.525 \implies \text{integer part is } 0
$$
$$
0.525 \times 7 = 3.675 \implies \text{integer part is } 3
$$
$$
0.675 \times 7 = 4.725 \implies \text{integer part is } 4
$$
(We can stop here or continue, as the pattern 0.725 repeats, leading to 5034...)  
Reading the integer parts from top down gives $0.5034..._7$.

$$
236.725_{10} \approx 455.5034_7
$$
The converted number is approximately $\boxed{455.5034_7}$.

$$
1_{16} + F_{16} = 1 + 15 = 16_{10}
$$
$$
16_{10} = 1 \times 16 + 0 \implies \text{write down } 0, \text{ carry over } 1
$$

$$
C_{16} + B_{16} + 1 (\text{carry}) = 12 + 11 + 1 = 24_{10}
$$
$$
24_{10} = 1 \times 16 + 8 \implies \text{write down } 8, \text{ carry over } 1
$$

$$
6_{16} + 7_{16} + 1 (\text{carry}) = 6 + 7 + 1 = 14_{10}
$$
$$
14_{10} = E_{16} \implies \text{write down } E, \text{ carry over } 0
$$

$$
4_{16} + 7_{16} + 0 (\text{carry}) = 4 + 7 + 0 = 11_{10}
$$
$$
11_{10} = B_{16} \implies \text{write down } B
$$

Combining the results:
$$
46C1_{16} + 77BF_{16} = BE80_{16}
$$
The sum is $\boxed{BE80_{16}}$.