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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

1 answer

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