In a class of 50 students.the number of students who offer accounting is twice as the number who offer economics.10 students offer neither of the two subjects and 5 students offer both subjects.Illustrate the information on a Venn diagram.How many students offer accounting?How many students offer economics?How many students offer both subjects?
Within a rectangle, which represents the universal set of 50 students,
draw two intersecting circles, label them A and E
- place 5 in the intersection of the two circles and 10 inside the rectangle, but outside the circles.
- label the A only part as x and the E only part as y
We are told that x + 5 = 2(y + 5)
x = 2y + 5
So change the label of x to 2y+5
From the diagram:
2y+5 + 5 + y + 10 = 50
3y = 30
y = 10
and
x = 20+5 = 25
Now you can re-number all the regions, and therefore all the
questions can be answered
To illustrate the information on a Venn diagram, we can create two overlapping circles. Let's label one circle "Accounting" and the other circle "Economics."
According to the given information:
- The number of students who offer accounting is twice as the number who offer economics.
- 10 students offer neither of the two subjects.
- 5 students offer both subjects.
Let's represent these values on the Venn diagram:
```
---------------------
| |
| Accounting |
| |
---------------------
/ \
/ \
------------------------------------------------------
| |
| |
| |
| Both Subjects |
| |
| |
| |
-------------------------------------------------------
\ /
\ /
---------------------
| |
| Economics |
| |
---------------------
```
Now, we can answer the questions.
1. How many students offer accounting?
Looking at the Venn diagram, the number of students offering accounting is represented by the number in the "Accounting" circle. Let's call this number "A."
Since the number of students who offer accounting is twice as the number who offer economics, we can represent this as: A = 2E.
2. How many students offer economics?
Similarly, the number of students offering economics is represented by the number in the "Economics" circle. Let's call this number "E."
3. How many students offer both subjects?
The number of students offering both subjects is represented by the number in the overlapping region of the circles. Let's call this number "B."
Additional information:
- The total number of students in the class is 50.
- 10 students offer neither of the two subjects.
Using this information, we can create equations to solve for A, E, and B.
A + E + B + 10 = 50 (Total number of students)
A = 2E (Number of accounting students is twice the number of economics students)
Now, let's solve these equations to find the values.
Substituting A = 2E in the first equation:
2E + E + B + 10 = 50
3E + B + 10 = 50
3E + B = 40 ---------- Equation 1
Now, we have two equations:
A = 2E
3E + B = 40
To solve these equations, we need one more equation. We can use the fact that 5 students offer both subjects:
B = 5 ---------- Equation 2
Substituting Equation 2 into Equation 1:
3E + 5 = 40
3E = 35
E = 35/3
Since the number of students cannot be fractional, we round this value to the nearest whole number:
E ≈ 11
Using this value, we can find the number of students who offer accounting:
A = 2E
A = 2(11)
A = 22
Therefore, there are 22 students who offer accounting, 11 students who offer economics, and 5 students who offer both subjects.
To answer these questions, let's first start by creating a Venn diagram.
A Venn diagram is a visual representation of a set of overlapping circles, where each circle represents a set of items or elements. In this case, we have two sets: "accounting" and "economics."
Now, let's design the Venn diagram based on the given information:
1. We are told that the number of students who offer accounting is twice as many as the number who offer economics. Let's represent the number of students who offer accounting as "A" and the number of students who offer economics as "E."
2. We are also given that 10 students offer neither accounting nor economics. We can represent this as a region outside both sets, which we'll call "Neither" or "N."
3. Finally, we know that 5 students offer both accounting and economics. We'll represent this as the overlapping region between the "A" and "E" circles.
Based on these details, our Venn diagram should look as follows:
```
A E
--------------- ---------------
| | | |
| | | |
| Both | | |
| (5) | | |
| | | |
| | | |
--------------- ---------------
N N
```
Now, let's answer the questions based on the Venn diagram:
1. How many students offer accounting?
We can see that the total number of students who offer accounting is the sum of the "A" circle and the overlapping "Both" region. So, the number of students who offer accounting is 5 (from the "Both" region) + A (still unknown).
2. How many students offer economics?
Similarly, the total number of students who offer economics is the sum of the "E" circle and the overlapping "Both" region. So, the number of students who offer economics is 5 (from the "Both" region) + E (still unknown).
3. How many students offer both subjects?
We already know that 5 students offer both accounting and economics. This is represented by the overlapping "Both" region.
To determine the values of A and E, we need an additional piece of information.