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: $$50-10=40\text{(students offering at least one subject)}$$

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: $$A+E=45\text{(1)}$$

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