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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Assume x, y and z are the number of watermelons, pears, and apples, respectively.
Then we have 5x + y + 0.5z = 25 or 10x + 2y + z = 50.

Since we have 3 variables (x, y, and z) and 1 equation, 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 x + y + z = 10 and 5x + y + 0.5z = 25 or 10x + 2y + z = 50. Subtracting the first equation from the second equation gives us (10x + 2y + z) – (x + y + z) = 50 - 10, or 9x + y = 40.
Then y = 40 – 9x.
Then the possible values of (x, y) are (1, 31), (2, 22), (3, 13), and (4, 4).
If x = 1, and y = 31, then z = 10 – x – y = -22 doesn’t make sense, since z must be positive.
If x = 2, and y = 22, then z = 10 – x – y = -14 doesn’t make sense, since z must be positive.
If x = 3, and y = 13, then z = 10 – x – y = -6 doesn’t make sense, since z must be positive.
If x = 4, and y = 4, then we have z = 10 – x – y = 2.

Since both conditions together yield a unique solution, they are sufficient.

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.

The total money spent on fruit = $25
Price of Watermelon = $5
Price of Pear = $5
Price of Apple = $5
How many Apples did he buy?
Statement 1: He bought 10 pieces of fruit, and he didn't get any change back.
If the quantity of fruit is 4 watermelons, 4 pears, and 2 apples
Total quantity = 10 pieces, and
The total money spent = (5*4) + (1*4) + (2*0.5) = $25, and apple = 2
There are many variations of numbers that add up to be = 10 but the only one that meets the requirements for the total amount spent on 10 fruits without collecting any change is for 4 watermelons, 4 pears, and 2 apples.
So, Thomas bought 2 apples from the grocery store. Hence, statement 1 is SUFFICIENT.

Statement 2: He had atleast one fruit each.
The quantity of fruit could be = 4 watermelons, 4 pears, and 2 apples, or 3 watermelons, 8 pears, and 4 apples, or any variation that will make the total amount spent on fruit = $25.
With this information, we cannot get the exact number of apple as it changes with tested values. Since, we cannot get the definite number of apple that Thomas bought, then statement 2 is NOT SUFFICIENT.

Therefore, only statement 1 is SUFFICIENT, thus, option A is the correct answer.

Thanks