Question: What is the measure of angle <AOD?
Statement 1:
$$\angle EOB=\frac{4}{5}\cdot\angle AOD$$
$$\angle AOD=\frac{\angle EOB}{\frac{4}{5}}$$
$$\angle AOD=\angle EOB\cdot\frac{5}{4}$$
$$But\ \angle EOB\ is\ unknown,\ hence,\ statement\ 1\ is\ NOT\ SUFFICIENT$$

Statement 2:
$$\angle DOE=\frac{\angle AOB}{2}$$
$$\angle AOB=angle\ on\ a\ straight\ line=180^0$$
$$\angle DOE=\frac{180^0}{2}=90^0$$
$$\angle AOB=\angle BOA=180^0$$
$$\angle AOC=\angle BOC=90^0$$
$$\angle DOE=90^0\ then\ \angle DOC=45^0$$
$$Since\ \angle AOC=90^0,\ therefore,\ \angle AOD=\frac{90^0}{2}=45^0$$
Hence, statement 2 alone is SUFFICIENT.

Answer = option B

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Assume < AOD = x, < DOE = y and < EOB = z.
Then we have x + y + z = 180.

Since we have 3 variables (x, y and z) and 1 equation (x + y + z = 180), C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have z = (4/5)x from condition 1) and y = 90 from condition 2).
Then we have x + z = 90 and x + (4/5)x = 90, or (9/5)x = 90.
Then, x = 50.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.