A team played more than 50 and less than 60 matches in an entire year.

Solution:

Number of matches played \(50<x<60\)

We are asked if \(x>55\) or not

Statement 1: The team won \(\frac{2}{3}\) of the first \(\frac{1}{3}\) of its matches played during the year

According to this statement out of \(\frac{50}{3}<\frac{x}{3}<\frac{60}{3}\) i.e., out of either 17, 18 or 19 mathes, the team won \(\frac{1}{3}\times 17\), \(\frac{1}{3}\times 18\) or \(\frac{1}{3}\times 20\) matches

We see that only \(\frac{1}{3}\times 18=6\) is an integer, which means \(\frac{x}{3}=18\) or \(x=54 \)

Thus we can say that x is not greater than 55 and statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: Out of its latter half matches played during the year, the team lost exactly \(\frac{1}{3}\) matches

We can use a similar approach here

According to this statement out of \(\frac{50}{2}<\frac{x}{2}<\frac{60}{2}\) i.e., out of either 26, 27, 28 or 29 mathes, the team lost \(\frac{1}{3}\times 26\), \(\frac{1}{3}\times 27\), \(\frac{1}{3}\times 28\) or \(\frac{1}{3}\times 29\) matches

We see that only \(\frac{1}{3}\times 27=9\) is an integer, which means \(\frac{x}{2}=27\) or \(x=54 \)

Thus we can say that x is not greater than 55 and statement 2 alone is also sufficient

Hence the right answer is Option D