To determine the percentage of people who play at least two sports in the academy, we need to carefully break down the given information using the principles of set theory or by constructing a mental Venn diagram.

Sports Academy Player Analysis

The problem provides detailed information about the number of people playing various sports in an academy with a total of 300 individuals. We are given the following data:

• Total people in the sports academy = 300
• Number of people who play only cricket = 105
• Number of people who play only hockey = 70
• Number of people who play only football = 50
• Number of people who play both cricket and hockey = 25
• Number of people who play both hockey and football = 15
• Number of people who play both cricket and football = 30

The problem states that the "rest of them play all three sports." This implies that the figures for "both cricket and hockey," "both hockey and football," and "both cricket and football" refer to those who play *only* those two specific sports (and not the third sport). This is a common interpretation in such problems when an "all three" category is separately mentioned as the remainder.

Cricket, Hockey, and Football Player Data Breakdown

Let's categorize the players based on the number of sports they play:

• People playing exactly one sport:
  • Only cricket = 105
  • Only hockey = 70
  • Only football = 50
  • Total playing exactly one sport = $105 + 70 + 50 = 225$ people
• People playing exactly two sports:
  • Only cricket and hockey = 25
  • Only hockey and football = 15
  • Only cricket and football = 30
  • Total playing exactly two sports = $25 + 15 + 30 = 70$ people

Determining All Three Sports Players

We know the total number of people in the academy is 300. The people who play all three sports are the remaining individuals after accounting for those who play exactly one or exactly two sports.

Let 'X' be the number of people who play all three sports.

The total number of people in the academy can be expressed as the sum of those playing exactly one sport, exactly two sports, and all three sports:

$\text{Total people} = (\text{People playing exactly one sport}) + (\text{People playing exactly two sports}) + (\text{People playing all three sports})$

$300 = 225 + 70 + X$

First, sum the known categories:

$225 + 70 = 295$

Now, substitute this back into the equation:

$300 = 295 + X$

To find X, subtract 295 from 300:

$X = 300 - 295$

$X = 5$

So, there are 5 people who play all three sports.

Calculating At Least Two Sports Players

We need to find the number of people who play "at least two sports". This category includes individuals who play exactly two sports AND individuals who play exactly three sports.

• People playing exactly two sports = 70 (calculated previously from $25 + 15 + 30$)
• People playing all three sports = 5 (calculated above)

Number of people who play at least two sports = (People playing exactly two sports) + (People playing all three sports)

Number of people who play at least two sports = $70 + 5$

Number of people who play at least two sports = $75$

Percentage of Sports Players

Finally, to find the percentage of people who play at least two sports, we use the formula:

$\text{Percentage} = \left( \frac{\text{Number of people playing at least two sports}}{\text{Total people in the academy}} \right) \times 100\%$

Substitute the calculated values:

$\text{Percentage} = \left( \frac{75}{300} \right) \times 100\%$

Simplify the fraction:

$\frac{75}{300} = \frac{1}{4}$

Now, calculate the percentage:

$\text{Percentage} = \frac{1}{4} \times 100\%$

$\text{Percentage} = 25\%$

Thus, 25.00% of the people in the sports academy play at least two sports.