Understanding the Sports Participation Data

This problem involves calculating ratios based on student participation in sports, specifically Hockey and Football. We are given percentages and relationships between different groups of students. Our goal is to find the ratio of students who participated in Hockey to those who participated in only Football.

Defining Variables for Calculation

Let's define the key variables to help us solve this problem:

• Let $N$ represent the total number of students in college B.
• Let $H_{only}$ represent the number of students who participated in only Hockey.
• Let $F_{only}$ represent the number of students who participated in only Football.
• Let $Both$ represent the number of students who participated in both Hockey and Football.

Step-by-Step Calculation of Student Numbers

We are provided with the following key pieces of information:

• The percentage of students participating in both sports is 36%. This can be expressed mathematically as: $Both = 0.36 \times N$.
• The number of students who participated in only Football ($F_{only}$) is 120% more than those who participated in only Hockey ($H_{only}$). This relationship can be written as: $F_{only} = H_{only} + 1.20 \times H_{only}$, which simplifies to $F_{only} = 2.20 \times H_{only}$.
• The total number of students participating in only one sport is 960. This means the sum of students in only Hockey and only Football is 960: $H_{only} + F_{only} = 960$.

Calculating Students in Only Hockey and Only Football

We can use the equations derived from the given information to find the exact numbers of students in each category. Substitute the relationship $F_{only} = 2.20 \times H_{only}$ into the equation for students participating in only one sport:

$H_{only} + (2.20 \times H_{only}) = 960$

Combine the terms involving $H_{only}$:

$3.20 \times H_{only} = 960$

Now, we solve for $H_{only}$ by dividing 960 by 3.20:

$H_{only} = \frac{960}{3.20}$

$H_{only} = \frac{9600}{32}$

$H_{only} = 300$

With the value of $H_{only}$ calculated as 300, we can now find the number of students who participated in only Football ($F_{only}$):

$F_{only} = 2.20 \times H_{only} = 2.20 \times 300 = 660$

As a check, the sum of students in only one sport is $300 + 660 = 960$, which correctly matches the given information.

Calculating the Total Number of Students

The total number of students ($N$) in college B is the sum of those who participated in only Hockey, only Football, and both sports.

$N = H_{only} + F_{only} + Both$

We know that $H_{only} + F_{only} = 960$ and $Both = 0.36 \times N$. Substituting these into the equation for $N$ gives:

$N = 960 + 0.36 \times N$

To find the total number of students $N$, we rearrange the equation:

$N - 0.36 \times N = 960$

$0.64 \times N = 960$

Now, we solve for $N$:

$N = \frac{960}{0.64}$

$N = \frac{96000}{64}$

$N = 1500$

Therefore, the total number of students in college B is 1500.

Calculating Students in Both Sports

Now that we know the total number of students, we can calculate the number of students who participated in both Hockey and Football:

$Both = 0.36 \times N = 0.36 \times 1500 = 540$

Calculating Students Participating in Hockey

The question asks for the ratio of students who participated in Hockey to those in only Football. First, we need to find the total number of students who participated in Hockey. This includes students who played only Hockey and those who played both sports.

Number of students in Hockey $= H_{only} + Both$

Number of students in Hockey $= 300 + 540 = 840$

Determining the Ratio of Hockey Participants to Only Football Participants

The final step is to find the ratio of the number of students who participated in Hockey to the number of students who participated in only Football ($F_{only}$).

Ratio = (Number of students in Hockey) : ($F_{only}$)

Using the values we calculated:

Ratio = $840 : 660$

Simplifying the Ratio

To simplify the ratio $840:660$, we find the greatest common divisor (GCD) or simplify step-by-step:

First, divide both numbers by 10:

$840 \div 10 = 84$

$660 \div 10 = 66$

The ratio becomes $84:66$.

Now, we divide both numbers by 6, which is their GCD:

$84 \div 6 = 14$

$66 \div 6 = 11$

The simplified ratio is 14:11.

Comparing this result with the given options:

• 22:21
• 14:11

• 29:49

• 41:31

Our calculated ratio of 14:11 matches the second option.