Question

In a class of 50 students, the number of students who offer accounting is twice as the number who offer economics . 10 student offer neither of the two subjects and 5 students offer both subjects. 1. Illustrate the information on a venn diagram. 2. How many students offer accounting. 3. How many students offer economics. 4. How many students offer only one subject

Answer 1

Let's break down the problem and solve it step by step.

### Step 1: Define Variables
- Let \( E \) be the number of students who offer Economics.
- Let \( A \) be the number of students who offer Accounting.

According to the problem:
- \( A = 2E \) (The number of students who offer Accounting is twice those who offer Economics)
- 10 students offer neither subject.
- 5 students offer both Accounting and Economics.

### Step 2: Set Up the Equation
In a class of 50 students, those offering at least one of the two subjects can be expressed as:
\[
50 - 10 = 40 \quad \text{(students offering at least one subject)}
\]

### Step 3: Use the Venn Diagram Variables
From the Venn diagram notation, we have:
- Students offering only Accounting: \( A - 5 \)
- Students offering only Economics: \( E - 5 \)
- Students offering both: 5
- Total students offering at least one subject:
\[
(A - 5) + (E - 5) + 5 = 40
\]
This simplifies to:
\[
A + E - 5 = 40
\]
Therefore:
\[
A + E = 45 \quad \quad \text{(1)}
\]

### Step 4: Substitute for A
Using \( A = 2E \) in equation (1):
\[
2E + E = 45
\]
This gives:
\[
3E = 45
\]
Thus:
\[
E = 15
\]

### Step 5: Find Number of Students in Accounting
Now, substituting back to find \( A \):
\[
A = 2E = 2 \times 15 = 30
\]

### Step 6: Summary of Results
Now, we have:
- Students who offer Accounting (\( A \)): **30**
- Students who offer Economics (\( E \)): **15**

### Step 7: Finding Students Offering Only One Subject
- Students who offer only Accounting:
\[
A - 5 = 30 - 5 = 25
\]
- Students who offer only Economics:
\[
E - 5 = 15 - 5 = 10
\]

### Step 8: Total Students Offering Only One Subject
Total students offering only one subject:
\[
(30 - 5) + (15 - 5) = 25 + 10 = 35
\]

Thus, the answers to all parts of the question are as follows:

1. **Venn Diagram Illustration**: Place in a Venn diagram with circles for Accounting and Economics, marking:
- "5" in the intersection (students offering both),
- "25" in the Accounting circle (only Accounting),
- "10" in the Economics circle (only Economics).

2. **Number of Students Who Offer Accounting**: **30**

3. **Number of Students Who Offer Economics**: **15**

4. **Number of Students Who Offer Only One Subject**: **35**