Chapter 5 · Topic 5.3 · Prelims + Mains3 of 3
General Mental Ability: Venn diagrams: both, neither and exactly one
Key points
- Two groups: at least one = first + second − both; neither = total − at least one.
- Exactly one = at least one − both.
- Three groups: add singles, subtract pairs, add back all three.
3 PYQs 2016–2025
Prelims box: facts and traps
- Answers: 2019 Q52 = 9 (b); 2018 Q42 = 6 (c); 2015 Q36 = 40 (a); 2015 Q31 = 20% (c); 2021 Q35 = 30 or more (d).
Venn diagrams: both, neither and exactly one
Like counting guests at two birthday parties on the same day: anyone who went to both must be counted only once.
The rules.
- Draw two (or three) overlapping circles inside a box (the box is the whole group).
- At least one = first + second − both.
- Neither = total − at least one (the space outside the circles).
- Exactly one = at least one − both.
CSAT 2019, Q52. 15 people; 7 read French, 8 read English, 3 read neither. How many read exactly one?
- At least one = 15 − 3 = 12.
- Both = 7 + 8 − 12 = 3.
- Exactly one = 12 − 3 = 9. Answer (b).
In a group of 15 people, 7 can read French, 8 can read English while 3 of them can read neither of these two languages. The number of people who can read exactly one language is
- 10
- 9
- 5
- 4
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Answer: (b) · UPSC official key
CSAT 2018, Q42. 19 boys; 11 wear hockey shirts, 14 wear hockey pants; nobody wears neither. Full uniform means both.
- Everyone wears at least one, so both = 11 + 14 − 19 = 6. Answer (c).
19 boys turn out for playing hockey. Of these, 11 are wearing hockey shirts and 14 are wearing hockey pants. There are no boys without shirts and/or pants. What is the number of boys wearing full uniform?
- 3
- 5
- 6
- 8
Show answer
Answer: (c) · UPSC official key
CSAT 2015, Q36. 130 students; 62 failed English, 52 failed Mathematics, 24 failed both.
- Failed in at least one = 62 + 52 − 24 = 90.
- Passed = 130 − 90 = 40. Answer (a).
Out of 130 students appearing in an examination, 62 failed in English, 52 failed in Mathematics, whereas 24 failed in both English and Mathematics. The number of students who passed finally is
- 40
- 50
- 55
- 60
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Answer: (a) · UPSC official key
CSAT 2015, Q31 (three circles). Add the singles, subtract the pairs, add back all three.
- 45 + 55 + 40 − 30 − 15 − 25 + 10 = 80% read at least one.
- None = 100 − 80 = 20%. Answer (c).
In a town, 45% population read magazine A, 55% read magazine B, 40% read magazine C, 30% read magazines A and B, 15% read magazines B and C, 25% read magazines A and C; and 10% read all the three magazines. What percentage do not read any magazine ?
- 10%
- 15%
- 20%
- 25%
Show answer
Answer: (c) · UPSC official key
CSAT 2021, Q35 (a range, not one number). 120 persons; 80 Indians, so 40 foreigners; 70 speak English.
- Even if all 40 foreigners speak English, at least 70 − 40 = 30 English speakers are Indians.
- There could be more. Answer (d), 30 or more.
In a group of 120 persons, 80 are Indians and rest are foreigners. Further, 70 persons in the group can speak English. The number of Indians who can speak English is
- 20
- 30
- 30 or less
- 30 or more
Show answer
Answer: (d) · UPSC official key
Common trap: forgetting the “neither” group, or adding the overlap twice.
What UPSC asks: Prelims: draw circles; first + second − both. Next likely angle: “neither” sits outside.
Practice
Practice MCQs (8)
Which one of the following best describes the relationship among Mammals, Dogs and Fish?
Answer (b) & explanation
Step 1: Every dog is a mammal, so the Dogs circle lies completely inside the Mammals circle. Step 2: No fish is a mammal and no fish is a dog, so the Fish circle is separate from both. Step 3: That is one circle inside another with the third apart, which is (b).
What is the angle between the hour hand and the minute hand of a clock at 7:20?
Answer (a) & explanation
Step 1: Use the formula angle = |30H − 5.5M|, where H is the hour and M the minutes. Step 2: 30 × 7 = 210 and 5.5 × 20 = 110. Step 3: |210 − 110| = 100°. Step 4: It is less than 180°, so it is already the smaller angle. Check: the minute hand is at 4 (120°); the hour hand has moved a third of the way from 7 to 8, i.e. 210° + 10° = 220°; 220 − 120 = 100°.
The six faces of a cube are numbered 1 to 6 and laid out flat as shown below. In the layout, '-' marks an empty cell, and squares in the same column of adjacent rows share an edge. Row 1: - 1 - - Row 2: 2 3 4 5 Row 3: - 6 - - When the layout is folded into a cube, which number will be opposite the face numbered 2?
Answer (b) & explanation
Step 1: In a straight row of squares in a net, faces with exactly one square between them end up opposite each other. Step 2: Row 2 reads 2, 3, 4, 5; so 2 is opposite 4, and 3 is opposite 5. Step 3: The two remaining faces, 1 and 6, are opposite each other. Step 4: Hence the face opposite 2 is 4.
Which one of the following best describes the relationship among Nurses, Women and Mothers?
Answer (b) & explanation
Step 1: Every mother is a woman, so Mothers lies completely inside Women. Step 2: Some nurses are women and some are men, so Nurses overlaps Women without lying inside it. Step 3: Some nurses are mothers and some mothers are not nurses, so Nurses also overlaps Mothers. Step 4: That is a small circle inside a larger one with a third overlapping both: (b).
In a survey of 100 students, 50 study Mathematics, 40 study Physics and 30 study Chemistry. 15 study both Mathematics and Physics, 10 study both Physics and Chemistry, 12 study both Mathematics and Chemistry, and 5 study all three. How many students study none of the three subjects?
Answer (d) & explanation
Step 1: Use inclusion-exclusion: union = 50 + 40 + 30 − 15 − 10 − 12 + 5. Step 2: 120 − 37 + 5 = 88 students study at least one subject. Step 3: None = 100 − 88 = 12.
Using the same survey (100 students; Mathematics 50, Physics 40, Chemistry 30; Mathematics and Physics 15, Physics and Chemistry 10, Mathematics and Chemistry 12; all three 5), how many students study exactly two of the three subjects?
Answer (a) & explanation
Step 1: Each pair figure (15, 10, 12) includes the 5 students who study all three. Step 2: Exactly two for each pair: 15 − 5 = 10, 10 − 5 = 5, 12 − 5 = 7. Step 3: Total exactly two = 10 + 5 + 7 = 22.
A clock gains 5 minutes every hour. It is set to the correct time at 8:00 a.m. What is the correct time when this clock shows 2:00 p.m. on the same day?
Answer (d) & explanation
Step 1: The faulty clock shows 65 minutes for every 60 true minutes. Step 2: From 8:00 a.m. to 2:00 p.m. it shows 6 hours = 360 minutes. Step 3: True time passed = 360 × 60/65 = 4320/13 = 332 4/13 minutes = 5 hours 32 4/13 minutes. Step 4: 8:00 a.m. + 5 h 32 4/13 min = 1:32 4/13 p.m.
Three different positions of the same cube are described below. Its six faces carry the letters P, Q, R, S, T and U. Position 1: P is on top, R faces the viewer and U is on the right. Position 2: Q is on top, R faces the viewer and T is on the right. Position 3: S is on top, T faces the viewer and P is on the right.
Which letter is on the face opposite P?
Answer (d) & explanation
Step 1: Faces seen together in one position share edges, so they are adjacent; a face is never adjacent to its opposite. Step 2: Position 1 shows P next to R and U. Position 3 shows P next to S and T. Step 3: So P is adjacent to R, U, S and T, four different faces. Step 4: The only face never seen with P is Q, so Q is opposite P.
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Prelims PYQs on this topic (3)
Three teams P, Q, R participated in a tournament in which the teams play with one another exactly once. A win fetches a team 2 points and a draw 1 point. A team gets no point for a loss. Each team scored exactly one goal in the tournament. The team P got 3 points, Q got 2 points and R got 1 point. Which of the following statements is/are correct?
- I. The result of the match between P and Q is a draw with the score 0-0.
- II. The number of goals scored by R against Q is 1.
Which of the statements given above is/are correct?
Answer (c) & explanation
Both statements are correct (c). Step 1: points: P = 3 (a win and a draw), Q = 2, R = 1 (a draw and a loss). There are three matches: P–Q, P–R and Q–R. Step 2: suppose P beat Q and drew R. Then Q must beat R for 2 points. P has only one goal, so P–Q was 1–0 and P–R 0–0; R's one goal comes against Q, so Q needs 2 goals to win, but Q has only one. Impossible. So P drew Q, P beat R, Q drew R. Step 3: each team scored exactly one goal. P's single goal beat R 1–0, so P–Q was 0–0 (I true). Q's goal came against R in a 1–1 draw, so R scored 1 against Q (II true). Options (a), (b) and (d) fail.
Tip: in points tables, list each team's possible results first, then match goals.
Answer as per the official UPSC answer key.
There are four letters and four envelopes and exactly one letter is to be put in exactly one envelope with the correct address. If the letters are randomly inserted into the envelopes, then consider the following statements :
- 1. It is possible that exactly one letter goes into an incorrect envelope.
- 2. There are only six ways in which only two letters can go into the correct envelopes.
Which of the statements given above is/are correct?
Answer (b) & explanation
Only statement 2 is correct (b). Step 1: if three letters are in the right envelopes, only one envelope is left, and it is the right one for the fourth letter. So exactly one wrong letter is impossible. Statement 1 is false. Step 2: for exactly two right letters, choose which two are right: AB, AC, AD, BC, BD, CD = 6 ways. Step 3: the other two must both be wrong, so they swap, in only 1 way. Total = 6 × 1 = 6. Statement 2 is true. Options (a), (c) and (d) fail.
Tip: in matching problems, "exactly one wrong" can never happen. A wrong letter always pushes at least one other letter out of place.
Answer as per the official UPSC answer key.
In a group of 15 people, 7 can read French, 8 can read English while 3 of them can read neither of these two languages. The number of people who can read exactly one language is
Answer (b) & explanation
9 people can read exactly one language (b). Step 1: 3 read neither, so 15 − 3 = 12 read at least one. Step 2: both = 7 + 8 − 12 = 3. Step 3: exactly one = those who read at least one − those who read both = 12 − 3 = 9. Check: French only = 7 − 3 = 4; English only = 8 − 3 = 5; 4 + 5 = 9. Option (a), 10, forgets to subtract correctly. Option (c), 5, and (d), 4, give only one language's "only" group.
Tip: "exactly one" = (A only) + (B only). Always remove the "neither" group before using the overlap formula.
Answer as per the official UPSC answer key.
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In 150 words, explain the significance of General Mental Ability with one recent example.
Flashcards (4)
Two-group rule?
At least one = first + second − both.
Three-group rule?
Add singles, subtract pairs, add all three.
15 people, 7 French, 8 English, 3 neither:
exactly one? 9.
120 people, 40 foreigners, 70 speak
English: Indians who speak English? 30 or more.
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