in a senior secondary school,80 students play hockey or football. The number that play football is 5 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 with the aid of using venn diagram :the number of students that play football
Let's assign variables to the different quantities mentioned in the problem:
Let h = number of students that play hockey
Let f = number of students that play football
We are given that 80 students play hockey or football, so we can write the equation:
h + f = 80 (Equation 1)
We are also given that the number of students that play football is 5 more than twice the number that play hockey, so we can write the equation:
f = 2h + 5 (Equation 2)
We are also given that 15 students play both games, so we can write the equation:
f + h - 15 = 80 (Equation 3) (This accounts for the overlap between football and hockey players)
Now, we can use these equations to solve for the number of students that play football.
From Equation 1, we can express h in terms of f:
h = 80 - f
Substituting this into Equations 2 and 3, we have:
f = 2h + 5
f = 2(80 - f) + 5
f = 160 - 2f + 5
3f = 165
f = 55
Therefore, the number of students that play football is 55.
To solve this problem using a Venn diagram, we first need to understand the given information. Let's break it down.
We are given that in a senior secondary school, there are a total of 80 students who play hockey or football. Let's represent this with a circle labeled "Total Students."
We are also given that the number of students who play football is 5 more than twice the number that play hockey. Let's represent this with a circle labeled "Football Players." Since the number of students who play football is dependent on the number of students who play hockey, we will use a variable to represent the number of hockey players, let's call it 'h.' Therefore, the number of football players can be represented as (2h + 5).
Lastly, we are given that 15 students play both hockey and football. This means there is an overlap between the "Total Students" circle and the "Football Players" circle. Let's represent this overlap with a common region labeled "Both Hockey and Football."
Now, to find the number of students that play football, we need to add up the number of students in the "Football Players" circle and the "Both Hockey and Football" region.
Since the total number of students who play hockey or football is 80, we can write the equation:
Total Students = Football Players + Hockey Players - Both Hockey and Football
Writing this equation using our representations:
80 = (2h + 5) + h - 15
Simplifying the equation:
80 = 3h - 10
Adding 10 to both sides:
90 = 3h
Dividing both sides by 3:
h = 30
Now that we have found the value of 'h,' which represents the number of students who play hockey, we can substitute it back into our equation to find the number of students who play football:
Football Players = 2h + 5
Football Players = 2(30) + 5
Football Players = 60 + 5
Football Players = 65
Therefore, the number of students that play football is 65.
To solve this problem using a Venn diagram, you can follow these steps:
Step 1: Draw two overlapping circles to represent hockey and football.
Step 2: Label one circle as "Hockey" and the other as "Football."
Step 3: Write the given information in the appropriate sections of the Venn diagram.
Let's denote the number of students playing hockey as 'h' and the number of students playing football as 'f.'
According to the given information:
- 80 students play hockey or football, so the total number of students in both sections of the Venn diagram is 80.
- The number that plays football is five more than twice the number that plays hockey, which can be written as: f = 2h + 5.
- 15 students play both games, so this value should be placed in the overlapping region of the Venn diagram.
Step 4: Substitute the value of f in terms of h, which is f = 2h + 5, into the equation that represents the total number of students: h + f - 15 = 80.
- Replace f with 2h + 5: h + (2h + 5) - 15 = 80.
- Combine like terms: 3h - 10 = 80.
- Simplify the equation: 3h = 90.
- Divide both sides by 3: h = 30.
Now, we know that there are 30 students playing hockey.
Step 5: Determine the value of f by substituting the value of h into the equation f = 2h + 5:
- f = 2(30) + 5.
- Simplify: f = 60 + 5.
- f = 65.
Therefore, there are 65 students playing football.
So, the number of students that play football is 65.
Show it with Venn diagram
Apologies for the confusion. Here is how the Venn diagram should look like based on the given information:
The left circle represents hockey players, the right circle represents football players, and the overlapping region represents students who play both hockey and football.
Label the left circle as "Hockey" and write "h" inside it, to represent the number of students playing hockey.
Label the right circle as "Football" and write "f" inside it, to represent the number of students playing football.
In the overlapping region, write "15" to represent the number of students playing both hockey and football.
Now, we know that the total number of students who play hockey or football is 80. So, write "80" outside the circles to represent the total number of students.
To find the number of students playing football, we need to determine the value of "f". To do this, we will use the given information that the number of students playing football is 5 more than twice the number playing hockey.
Let's say there are 30 students playing hockey (h=30).
Using the given information, we can calculate the number of students playing football:
f = 2h + 5 = 2(30) + 5 = 60 + 5 = 65.
So, we now know that there are 65 students playing football.
In the Venn diagram, you can write "65" inside the Football circle to indicate the number of students playing football.
This Venn diagram represents the given information and visually represents the number of students playing hockey and football.