MATHEMATICS In a Secondary school. 80 Students play Hockey or Football. The number that play Football is five more than twice the number that play Hockey. If 15 students play both games and every student in the school plays at least one game. Find (i) the number of student that play football (ii). The number of students that play Hockey

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Let's break down the problem step by step.

### **Given:**
- Total number of students playing at least one game (Hockey or Football): **80**
- Number of students playing both games: **15**
- The number of students playing Football is five more than twice the number of students playing Hockey.

### **Let:**
- $$ H $$ = Number of students playing **Hockey**
- $$ F $$ = Number of students playing **Football**

### **Equations:**
1. **Total students playing at least one game:**
   $$
   H + F - \text{Both} = 80
   $$
   Substituting the number who play both:
   $$
   H + F - 15 = 80 \quad \Rightarrow \quad H + F = 95 \quad \text{(Equation 1)}
   $$

2. **Relationship between Football and Hockey players:**
   $$
   F = 2H + 5 \quad \text{(Equation 2)}
   $$

### **Solving the Equations:**

Substitute Equation 2 into Equation 1:
$$
H + (2H + 5) = 95
$$
$$
3H + 5 = 95
$$
$$
3H = 90
$$
$$
H = 30
$$

Now, substitute $$ H = 30 $$ back into Equation 2:
$$
F = 2(30) + 5 = 60 + 5 = 65
$$

### **Answers:**
1. **Number of students that play Football:** $$ \boxed{65} $$
2. **Number of students that play Hockey:** $$ \boxed{30} $$
There are 30 students who play Hockey and 65 students who play Football.

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