Question

In an examination, 42% of the students passed in at least two subjects out of the three subjects P, Q and R. 36% of the students passed in subjects Q and R. 10% of the students passed in all the three subjects. 38% of the students passed subject P. How many students passed in only subject P?

• A. 32%
• B. 30%
• C. 28%
• D. 24%

Hint:

Draw a three-set Venn diagram, fill in the all-three region first, then peel off the Q-and-R and at-least-two figures to isolate the pieces that make up the total for P.

Answer 1

The Correct Option is D

Step 1: Note down what each percentage represents.
Let the students who passed exactly two subjects be split into three groups: PQ-only (P and Q but not R), QR-only (Q and R but not P), and PR-only (P and R but not Q). Let the students who passed all three subjects be the fourth group. We are told this last group is 10%.

Step 2: Use the 'at least two subjects' figure.
'At least two subjects' means exactly two OR all three, so \[ (PQ\text{-only})+(QR\text{-only})+(PR\text{-only})+10\% = 42\% \] \[ (PQ\text{-only})+(QR\text{-only})+(PR\text{-only}) = 32\% \]

Step 3: Use the 'Q and R' figure to isolate QR-only.
Passing 'in subjects Q and R' means passing both Q and R, regardless of P, so it includes the all-three group: \[ (QR\text{-only}) + 10\% = 36\% \] \[ QR\text{-only} = 26\% \]

Step 4: Find PQ-only plus PR-only.
From Step 2, subtracting QR-only: \[ (PQ\text{-only})+(PR\text{-only}) = 32\%-26\% = 6\% \]

Step 5: Use the total for subject P to isolate 'only P'.
Everyone who passed P is either only-P, or in exactly one of the two-subject groups involving P, or in the all-three group: \[ (\text{only P}) + (PQ\text{-only}) + (PR\text{-only}) + 10\% = 38\% \] \[ (\text{only P}) + 6\% + 10\% = 38\% \] \[ \text{only P} = 38\%-16\% = 22\% \]

Final Answer:
Working through the given percentages step by step gives 22% of the students passing only subject P. \[ \boxed{22\%} \]