Question:medium

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?

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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.
Updated On: Jul 21, 2026
  • 32%
  • 30%
  • 28%
  • 24%
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The Correct Option is D

Solution and Explanation

Step 1: Picture a Venn diagram with three overlapping circles P, Q, R.
Write 10% straight into the centre region where all three circles overlap, since that is the group that passed all three subjects.

Step 2: Fill in the Q-R overlap (excluding the centre).
The whole Q-and-R overlap, including the centre, is 36%. Since the centre already holds 10%, the crescent-shaped region belonging to Q and R alone (not P) holds \[ 36\%-10\% = 26\% \]

Step 3: Use the ring made of all 'exactly two subject' regions.
The 'at least two subjects' figure of 42% covers the centre plus all three two-subject crescents. Removing the centre: \[ 42\%-10\% = 32\% \] is shared between the P-Q crescent, the Q-R crescent, and the P-R crescent. Having already placed 26% in the Q-R crescent from Step 2, the two crescents touching P (P-Q and P-R together) must hold \[ 32\%-26\% = 6\% \]

Step 4: Read off the P circle's total.
The whole P circle is 38%, and it is made up of four parts: the P-only region, the P-Q crescent, the P-R crescent, and the centre. We already know the P-Q and P-R crescents add up to 6%, and the centre is 10%, so \[ \text{P-only} = 38\% - 6\% - 10\% = 22\% \]

Step 5: Read the diagram.
Filling the diagram region by region, starting from the centre and working outward, lands on the same figure as the algebraic method: 22% of the students passed only subject P. \[ \boxed{22\%} \]
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