Question

In a class of 40 students who study mathematics, physics and chemistry, the number of students studying mathematics is 2 more than 40% of those studying chemistry, while 20 of the students study physics. Three less than one fifth of the total students in the class study all three subjects. The number of students studying only physics is 2 less than the number of students studying mathematics. The number of students studying only mathematics and only chemistry is 3 and 15 respectively. The number of students studying mathematics as well as physics is the same as the number of students studying mathematics as well as chemistry. How many students are studying both mathematics and chemistry but not all the three subjects?

• A. 25
• B. 15
• C. 10
• D. 2

Hint:

Set up variables for the pairwise-only overlaps and use the fact that the two given intersections are equal.

Answer 1

The Correct Option is D

Step 1: List the given information. Total students = 40. Only Mathematics = 3, Only Chemistry = 15. One fifth of 40 is 8, and three less than that is 5, so 5 students study all three subjects. Physics total = 20.
Step 2: Assign variables to the unknown overlap regions. Let x = students studying Mathematics and Physics only (not Chemistry), y = students studying Mathematics and Chemistry only (not Physics), z = students studying Physics and Chemistry only (not Mathematics). Since the Mathematics-Physics total equals the Mathematics-Chemistry total, and both totals include the same group of 5 triple-overlap students, the only-two regions must also be equal, so x = y.
Step 3: Write the Mathematics and Chemistry totals. Mathematics total = Only Mathematics + x + y + All three = 3 + x + y + 5 = 8 + 2x, using y = x. Chemistry total = Only Chemistry + y + z + All three = 15 + x + z + 5 = 20 + x + z.
Step 4: Use the percentage condition. Mathematics = 2 + 40% of Chemistry, so \(8 + 2x = 2 + 0.4(20 + x + z)\). Expanding gives \(8 + 2x = 10 + 0.4x + 0.4z\), so \(1.6x - 0.4z = 2\), which simplifies to \(4x - z = 5\).
Step 5: Use the Only Physics condition. Only Physics = Mathematics total minus 2 = \(6 + 2x\). Physics total = Only Physics + x + z + All three = 20, so \((6 + 2x) + x + z + 5 = 20\), which simplifies to \(3x + z = 9\).
Step 6: Solve the two equations. Adding \(4x - z = 5\) and \(3x + z = 9\) gives \(7x = 14\), so \(x = 2\). Then \(z = 9 - 6 = 3\) and \(y = x = 2\).
Step 7: Verify against the total of 40. Only Mathematics(3) + Only Physics(10) + Only Chemistry(15) + Mathematics-Physics only(2) + Mathematics-Chemistry only(2) + Physics-Chemistry only(3) + All three(5) = 40, which checks out.
Step 8: State the answer. Students studying both Mathematics and Chemistry but not all three subjects is exactly y.\[\boxed{2}\]