Logical Reasoning — Notes

1. Number & Letter Series

Key formulas & rules

Letter positions (learn this table)

Shortcuts

Solved examples

Example 1 (Easy). Find the next term: 3, 7, 13, 21, 31, ?

1. Differences: 7 − 3 = 4, 13 − 7 = 6, 21 − 13 = 8, 31 − 21 = 10.
2. The differences go up by 2 each time: 4, 6, 8, 10, so the next difference is 12.
3. Next term = 31 + 12 = 43.

Answer: 43.

Example 2 (Easy). Find the next term: 2, 5, 10, 17, 26, ?

1. Each term is one more than a perfect square: 1 + 1, 4 + 1, 9 + 1, 16 + 1, 25 + 1.
2. So the rule is n² + 1. The 6th term is 36 + 1.

Answer: 37. Faster way: differences are 3, 5, 7, 9 (odd numbers), so add 11: 26 + 11 = 37. Both methods agree, which is a good check.

Example 3 (Medium). Find the next term: 5, 6, 10, 19, 35, ?

1. Differences: 1, 4, 9, 16. These are 1², 2², 3², 4².
2. Next difference = 5² = 25.
3. Next term = 35 + 25 = 60.

Answer: 60.

Example 4 (Medium). Find the next term: 4, 10, 8, 14, 12, 18, ?

1. The series goes up and down, so suspect an alternating rule.
2. Steps: +6, −2, +6, −2, +6. The next step is −2.
3. Next term = 18 − 2 = 16.

Answer: 16. Another view: odd places 4, 8, 12, 16 and even places 10, 14, 18 are two separate +4 series.

Example 5 (Medium). Find the next term: A, C, F, J, O, ? Also: Z, W, S, N, ?

1. Convert: A = 1, C = 3, F = 6, J = 10, O = 15. Differences: +2, +3, +4, +5. Next is +6: 15 + 6 = 21 = U.
2. Second series: Z = 26, W = 23, S = 19, N = 14. Differences: −3, −4, −5. Next is −6: 14 − 6 = 8 = H.

Answer: U and H. EJOTY helps here: O = 15 is an anchor, and U is one after T (20).

Example 6 (Hard). Find the wrong term: 2, 5, 11, 23, 47, 96

1. Terms roughly double, so test "×2 + something".
2. 2 × 2 + 1 = 5, 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47, 47 × 2 + 1 = 95.
3. The last term should be 95, not 96.

Answer: 96 is wrong (should be 95). Shortcut: in a ×2 + 1 series every term after the first is odd. 96 is the only even term, so it stands out at once.

Practice set

2. Analogy & Odd One Out

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). Doctor : Hospital :: Teacher : ? (a) Book (b) Student (c) School (d) Class

1. Make the sentence: "A doctor works in a hospital."
2. Apply it: "A teacher works in a ___." The workplace is a school.
3. "Student" and "Book" are linked to teachers, but not by "works in".

Answer: (c) School.

Example 2 (Easy). 3 : 27 :: 5 : ?

1. 27 = 3 × 3 × 3 = 3³.
2. Same rule for 5: 5³ = 125.

Answer: 125.

Example 3 (Medium). ACE : BDF :: GIK : ?

1. A → B, C → D, E → F: each letter moves +1.
2. G → H, I → J, K → L.

Answer: HJL.

Example 4 (Medium). 7 : 50 :: 9 : ?

1. 50 is close to 49 = 7². So the rule is n² + 1.
2. 9² + 1 = 81 + 1 = 82.

Answer: 82.

Example 5 (Medium). Find the odd one out: 3, 5, 7, 9, 11, 13 — and — 121, 144, 169, 196, 210

1. First set: all are primes except 9 (= 3 × 3). Odd one: 9.
2. Second set: 121 = 11², 144 = 12², 169 = 13², 196 = 14². 210 is not a perfect square. Odd one: 210.

Answer: 9 and 210. Notice all six numbers in the first set are odd, so "odd/even" does not separate them. The property that splits exactly one item away is "prime".

Example 6 (Hard). Find the odd one out: BDF, HJL, MOQ, RTW

1. Convert gaps: B→D→F is +2, +2. H→J→L is +2, +2. M→O→Q is +2, +2.
2. R (18) → T (20) → W (23) is +2, +3.

Answer: RTW.

Practice set

3. Coding–Decoding

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). In a code, MANGO is written NBOHP. How is APPLE written?

1. M→N, A→B, N→O, G→H, O→P. Every letter moves +1.
2. APPLE: A→B, P→Q, P→Q, L→M, E→F.

Answer: BQQMF.

Example 2 (Easy). If TABLE is coded ELBAT, how is CHAIR coded?

1. ELBAT is TABLE written backwards.
2. CHAIR backwards is RIAHC.

Answer: RIAHC.

Example 3 (Medium). If GOLD is written TLOW, how is SILVER written?

1. G (7) → T (20): 7 + 20 = 27. O (15) → L (12): 27. So each letter becomes its opposite.
2. S (19) → 8 = H, I (9) → 18 = R, L (12) → 15 = O, V (22) → 5 = E, E (5) → 22 = V, R (18) → 9 = I.

Answer: HROEVI. Shortcut: use the opposite-pair list (HS, IR, LO, EV) instead of computing.

Example 4 (Medium). If BAD is written 214, how is FACE written?

1. B = 2, A = 1, D = 4, written side by side.
2. F = 6, A = 1, C = 3, E = 5.

Answer: 6135.

Example 5 (Medium). If sky is called sea, sea is called water, water is called air and air is called cloud, what do we drink?

1. In real life we drink water.
2. In this code, water is called "air".

Answer: air.

Example 6 (Hard). In a code language, "pit na sa" means "you are good", "na ho pit" means "they are good", and "ka sa" means "you go". What is the code for "they"?

1. Sentences 1 and 2 share "are good" and the codes "pit na". So {are, good} = {pit, na}.
2. Sentence 2 is "they are good" = "na ho pit". Remove pit and na: they = ho.
3. (Also: sentences 1 and 3 share "you" and "sa", so you = sa, and go = ka.)

Answer: ho. Note that we cannot tell which of pit / na is "are" and which is "good" — exam options avoid asking that.

Example 7 (Hard). If ABC is written BDF, how is CODE written?

1. A→B is +1, B→D is +2, C→F is +3. The shift equals the letter's place in the word.
2. C +1 = D, O +2 = Q, D +3 = G, E +4 = I.

Answer: DQGI.

Practice set

4. Blood Relations

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). A is B's brother. C is A's mother. D is C's father. How is D related to B?

1. A and B are siblings, so C is the mother of both A and B.
2. D is C's father, so D is the father of B's mother.
3. Mother's father = grandfather.

Answer: D is B's grandfather (maternal).

Example 2 (Medium). Pointing to a photograph, a man said, "I have no brother or sister, but that man's father is my father's son." Who is in the photo?

1. "My father's son" — the speaker has no brother, so his father's only son is the speaker himself.
2. So "that man's father is me".
3. The man in the photo is the speaker's son.

Answer: his son. Many students answer "himself" — but the speaker is the man's father, not the man.

Example 3 (Medium). Pointing to a woman, Meena said, "She is the daughter of my mother's only son." How is the woman related to Meena?

1. Meena is female, so "my mother's only son" is Meena's brother.
2. The woman is the daughter of Meena's brother.
3. Brother's daughter = niece.

Answer: niece.

Example 4 (Medium). If A + B means A is the brother of B, A − B means A is the sister of B, A × B means A is the father of B, and A ÷ B means A is the mother of B, what does P × Q + R − S tell us about P and S?

1. P × Q: P is the father of Q.
2. Q + R: Q is the brother of R. So P is also R's father.
3. R − S: R is the sister of S. So S is another child of P.

Answer: P is the father of S. We do not know whether S is male or female, and we do not need to.

Example 5 (Medium). X is the husband of Y. Y is the sister of Z. Z is the father of W. How is X related to W?

1. Draw X (male) married to Y (female).
2. Y and Z are siblings: same row, joined by a bar.
3. W is one row below Z. Y is W's father's sister, that is W's aunt.
4. The husband of an aunt is an uncle.

Answer: X is W's uncle.

Example 6 (Hard). Introducing a man, a woman said, "His wife is the only daughter of my father." How is the woman related to the man?

1. "The only daughter of my father", said by a woman, is the woman herself.
2. So "his wife is me".

Answer: she is his wife.

Practice set

5. Direction Sense

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). Ram walks 3 km North, turns right and walks 4 km. How far is he from the start, and in which direction?

1. Facing North, a right turn means East. So he is 3 km North and 4 km East of the start.
2. Distance = √(3² + 4²) = √25 = 5 km.
3. He is North and East of the start: North-East.

Answer: 5 km, North-East.

Example 2 (Easy). A girl walks 10 m East, turns left and walks 5 m, then turns left again and walks 10 m. Where is she now?

1. East 10 m. Left from East = North, 5 m. Left from North = West, 10 m.
2. East–West: +10 − 10 = 0. North–South: +5.

Answer: 5 m North of the start.

Example 3 (Medium). One morning after sunrise, Rahul and Sita stand facing each other. Rahul's shadow falls exactly to his right. Which direction is Rahul facing?

1. Morning: the sun is in the East, so shadows fall to the West.
2. The shadow is on Rahul's right, so his right hand points West.
3. If your right hand points West, you are facing South.

Answer: South (and Sita, facing him, faces North).

Example 4 (Medium). A man walks 6 km South, turns left and walks 8 km, then turns left and walks 12 km. How far and in which direction is he from the start?

1. South 6. Left from South = East, 8 km. Left from East = North, 12 km.
2. North–South: −6 + 12 = 6 km North. East–West: 8 km East.
3. Distance = √(6² + 8²) = √100 = 10 km (triple 6-8-10).

Answer: 10 km, North-East.

Example 5 (Medium). A person faces North. He turns 90° clockwise, then 135° anticlockwise. Which direction does he face now?

1. Net turn = +90 − 135 = −45°, that is 45° anticlockwise.
2. 45° anticlockwise from North is North-West.

Answer: North-West.

Example 6 (Hard). Priya walks 10 m North, turns right and walks 8 m, turns right again and walks 25 m. How far is she from the start, and in which direction?

1. North 10. Right from North = East, 8 m. Right from East = South, 25 m.
2. North–South: +10 − 25 = −15, so 15 m South. East–West: 8 m East.
3. Distance = √(15² + 8²) = √(225 + 64) = √289 = 17 m (triple 8-15-17).

Answer: 17 m, South-East.

Practice set

6. Ranking & Order

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). Ravi is 7th from the left and 12th from the right in a row. How many people are in the row?

1. Left + right counts Ravi twice.
2. Total = 7 + 12 − 1 = 18.

Answer: 18.

Example 2 (Easy). In a row of 40 students, Mona is 15th from the left. What is her position from the right?

1. Right = Total − Left + 1 = 40 − 15 + 1.

Answer: 26th. Check: 15 + 26 − 1 = 40.

Example 3 (Medium). In a class of 35, A is 10th from the top and B is 20th from the bottom. How many students are between them?

1. Convert B to "from the top": 35 − 20 + 1 = 16th.
2. A is 10th, B is 16th. Between = 16 − 10 − 1 = 5.

Answer: 5.

Example 4 (Medium). In a row of 25 children, A is 15th from the left and B is 18th from the right. How many children are between A and B?

1. 15 + 18 = 33 is more than 25, so the counts overlap: B is to the left of A.
2. B from the left = 25 − 18 + 1 = 8.
3. Between = 15 − 8 − 1 = 6.

Answer: 6.

Example 5 (Hard). In a row, A is 12th from the left and B is 18th from the right. They swap places, and now A is 20th from the left. How many people are in the row?

1. After the swap, A stands where B stood.
2. That spot is 20th from the left (A's new position) and 18th from the right (B's old position).
3. Total = 20 + 18 − 1 = 37.

Answer: 37. Note that A's old position (12th) is not needed. It is there to confuse you, or for a follow-up question (B is now 12th from the left).

Example 6 (Hard). Among five friends, A is taller than B but shorter than C. D is taller than C. E is shorter than B. Who is in the middle by height?

1. A > B and C > A gives C > A > B.
2. D > C gives D > C > A > B.
3. E < B gives D > C > A > B > E.
4. The 3rd of five is A.

Answer: A.

Practice set

7. Alphabet & Word Tests

Key formulas & rules

Shortcuts

Solved examples

Example 1 (Easy). Arrange in dictionary order: Plan, Planet, Plane, Plain, Place. Which word comes third?

1. All start "Pla". Next letters: n, n, n, i, c. So Place (c) comes first, then Plain (i).
2. Plan, Plane, Planet: Plan is the start of the other two, so it comes first; Plane is the start of Planet.
3. Order: Place, Plain, Plan, Plane, Planet.

Answer: Plan.

Example 2 (Easy). Which letter is 7th to the right of the 14th letter from the left?

1. 14 + 7 = 21. The 21st letter is U (T = 20, then U).

Answer: U.

Example 3 (Easy). If the alphabet is written in reverse order, which letter is 8th from the left?

1. 8th from the left in reverse = 8th from the right in normal order = 27 − 8 = 19th.
2. The 19th letter is S.

Answer: S.

Example 4 (Medium). How many pairs of letters in WONDER have as many letters between them in the word as in the alphabet?

1. Positions: W 23, O 15, N 14, D 4, E 5, R 18.
2. Neighbours (gap 1): O–N differ by 1, yes. D–E differ by 1, yes. W–O (8), N–D (10), E–R (13), no.
3. Gap 2 to 4: no pair has a matching difference (for example O–D is 2 apart but differs by 11).
4. Gap 5: W–R, 23 − 18 = 5, yes.

Answer: 3 pairs (ON, DE, WR).

Example 5 (Medium). Which word cannot be formed from the letters of EXAMINATION? (a) NATION (b) ANIMATE (c) MOTION (d) TAXI

1. Letter counts: A2, I2, N2, E1, X1, M1, T1, O1.
2. MOTION needs two O's, but EXAMINATION has only one O.
3. NATION (N2, A, T, I, O), ANIMATE (A2, N, I, M, T, E) and TAXI all fit.

Answer: (c) MOTION.

Example 6 (Hard). If the letters of PRODUCT are arranged in alphabetical order, how many letters stay in the same position?

1. Original: P R O D U C T.
2. Sorted: C D O P R T U.
3. Compare place by place: only the 3rd letter (O) matches.

Answer: 1.

Practice set

8. Linear Seating Arrangement

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). Five friends P, Q, R, S and T sit in a row facing north. R sits in the middle. P sits to the immediate left of R. T sits at the right end. Q is not a neighbour of T. Who sits at the left end?

1. Middle of 5 seats = seat 3, so R = 3.
2. Immediate left of R (facing north) = seat 2, so P = 2.
3. Right end = seat 5, so T = 5. Seats 1 and 4 are left for Q and S.
4. Seat 4 is next to T. Q is not next to T, so Q = 1 and S = 4.

Row: Q P R S T. Answer: Q.

Example 2 (full set). Seven people A, B, C, D, E, F and G sit in a row facing north.

• D sits exactly in the middle of the row.
• Only two people sit between D and E.
• C sits to the immediate left of E.
• A sits third to the left of C.
• B does not sit next to A.
• F sits to the immediate right of B.

Questions: (i) Who sits exactly between A and G? (ii) How many people sit between B and C? (iii) Who sits second to the right of F? (iv) Who sits at the right end?

1. Definite clue first. Middle of 7 = seat 4, so D = 4.
2. Gap clue. Two people between D and E means the seats differ by 3. E = 4 − 3 = 1 or E = 4 + 3 = 7. Two cases.
3. Kill a case. C must be on E's immediate left (seat E − 1). If E = 1, there is no seat 0, so Case 1 dies. E = 7 and C = 6.
4. Relative clue. A is third to the left of C: 6 − 3 = 3, so A = 3.
5. Negative clue. Free seats are 1, 2 and 5. B is not next to A (seats 2 and 4), so B = 1 or 5. F must be on B's immediate right. If B = 5, F = 6, but C is there. So B = 1, F = 2, and G takes the last seat, 5.

Final row: B F A D G C E. Answers: (i) seats 3 and 5 have seat 4 between them: D. (ii) B = 1, C = 6, so 6 − 1 − 1 = 4 people. (iii) F = 2, two seats to the right = seat 4: D. (iv) E.

Faster way: the moment you read "C sits to the immediate left of E" together with "E is 3 seats from the middle", you know E cannot be at the left end. Look for such "no room" eliminations first.

Example 3 (facing south). Five people J, K, L, M and N sit in a row facing south. M sits in the middle. K sits second to the left of M. N sits to the immediate right of K. J sits at the extreme right end of the row. Who sits to the immediate left of L?

1. Facing south: left = +1 on paper, right = −1.
2. M = 3. K is second to the left of M: 3 + 2 = 5, so K = 5.
3. N is immediately right of K: 5 − 1 = 4, so N = 4.
4. The extreme right end of south-facing people is seat 1 (the page left). So J = 1, and L = 2.
5. Immediate left of L = seat 2 + 1 = 3 = M.

Row on paper: J L M N K. Answer: M.

Example 4 (some face north, some face south). Six people U, V, W, X, Y and Z sit in a row. Some face north and some face south.

• X sits at an extreme end and faces south.
• W sits second to the right of X, and W faces the same direction as X.
• Z sits to the immediate right of X.
• U sits at an extreme end.
• Y sits to the immediate right of W. W sits to the immediate left of Y.
• The immediate neighbours of Y face opposite directions.
• U faces the same direction as V. Z faces the direction opposite to X.

Question: Who sits second to the left of Z, and how many people face north?

1. X faces south, so X's right is −1. "W is second to the right of X" needs two seats on the −1 side, so X cannot be seat 1. X = 6, W = 4, Z = 5.
2. W faces south (same as X). W's right is −1, so Y = 3.
3. "W is to the immediate left of Y": W is at 4 = Y + 1, so Y's left is +1. That means Y faces south.
4. U is at an end; seat 6 is taken, so U = 1 and V = 2.
5. Y's neighbours are V (seat 2) and W (south). They face opposite ways, so V faces north. U faces like V: north. Z faces opposite to X: north.

Z faces north and sits in seat 5, so its left is −1: second to the left = seat 3 = Y. North-facing people: U, V, Z = 3.

Example 5 (two parallel rows). P, Q, R sit in Row 1 facing south. A, B, C sit in Row 2 facing north, so each person faces someone in the other row. Q sits in the middle of Row 1. A faces the person to the immediate right of Q. R sits to the immediate left of Q. C does not face R. Who faces P, and who sits to the immediate right of C?

1. Row 1 faces south, so right = −1 on paper. Q = 2, immediate right of Q = seat 1. Immediate left of Q = seat 3 = R, so P = 1.
2. A faces seat 1, so A = 1 in Row 2.
3. C does not face R (seat 3), so C = 2 and B = 3.
4. A faces P. C faces north, so C's right is +1: seat 3 = B.

Answer: A faces P; B sits to the immediate right of C.

Practice set

9. Circular Seating Arrangement

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). Six friends A, B, C, D, E and F sit around a round table facing the centre. A sits opposite D. B sits to the immediate left of A. C sits second to the right of B. E is not an immediate neighbour of C. Who sits opposite E?

1. Put A at seat 0. Opposite = 0 + 3 = seat 3, so D = 3.
2. Facing the centre, left = clockwise (+1). B = seat 1.
3. Right = anticlockwise (−1). C = 1 − 2 = −1, which is seat 5.
4. Seats 2 and 4 are left. Seat 4 is next to C (seat 5), so E = 2 and F = 4.
5. Opposite E (seat 2) = seat 5 = C.

Clockwise from A: A B E D F C. Answer: C.

Example 2 (full set). Eight people P, Q, R, S, T, U, V and W sit around a circular table facing the centre.

• P and T sit opposite each other.
• Q sits second to the left of P.
• R is an immediate neighbour of T, but R is not an immediate neighbour of Q.
• V sits to the immediate right of P.
• U sits between Q and T.
• S is not an immediate neighbour of P.

Questions: (i) Who sits opposite Q? (ii) Who sits third to the left of V? (iii) Counting clockwise from W, how many people sit between W and R? (iv) Who sits between S and P?

1. Anchor. P = seat 0, T = seat 4 (opposite).
2. Left = clockwise. Q = 0 + 2 = seat 2.
3. Right = anticlockwise. V = 0 − 1 = seat 7.
4. Two cases for R. Neighbours of T are seats 3 and 5. Seat 3 is next to Q (seat 2), so R = 5.
5. U sits between Q (2) and T (4): U = 3.
6. Seats 1 and 6 are left for S and W. Seat 1 is next to P, so S = 6 and W = 1.

Answers: (i) Q = 2, opposite = 6: S. (ii) V = 7, left = clockwise, 7 + 3 = 10 → seat 2: Q. (iii) W = 1, R = 5, clockwise seats 2, 3, 4 lie between: 3 people. (iv) S = 6 and P = 0 (8): the seat between is 7: V.

Example 3 (facing outside). G, H, I, J, K and L sit around a circle facing outside. H sits second to the left of G. J sits opposite H. K sits to the immediate right of J. L is not an immediate neighbour of G. Who sits to the immediate left of I?

1. Facing outside: left = anticlockwise (−1), right = clockwise (+1).
2. G = 0. H = 0 − 2 = seat 4. J opposite H = seat 1.
3. K = J + 1 = seat 2.
4. Seats 3 and 5 are left. Seat 5 is next to G, so L = 3 and I = 5.
5. Immediate left of I = 5 − 1 = seat 4 = H.

Clockwise from G: G J K L H I. Answer: H.

Example 4 (some face in, some face out). U, V, W, X, Y and Z sit around a circle. Three face the centre and three face outside. U faces the centre, and X sits opposite U. V sits to the immediate left of U. W sits second to the left of V and faces the same direction as V. X faces outside, and Y sits to the immediate left of X. Z faces the centre and V faces outside. Who sits to the immediate right of Z, and which way does Y face?

1. U = 0, faces in. X = 3 (opposite), faces out.
2. V is on U's left; U faces in, so left = clockwise: V = 1. V faces out.
3. W is second to V's left; V faces out, so left = anticlockwise: W = 1 − 2 = seat 5. W faces out (same as V).
4. Y is on X's left; X faces out: Y = 3 − 1 = seat 2. Z takes seat 4.
5. Facing the centre so far: U, Z. Facing out: V, W, X. Three must face in, so Y faces the centre.
6. Z faces in, so right = anticlockwise: 4 − 1 = seat 3 = X.

Answer: X sits to the immediate right of Z; Y faces the centre.

Example 5 (how many people?). In a circle of people facing the centre, M is 4th to the left of N and also 7th to the right of N. How many people sit in the circle?

Going left 4 seats and going right 7 seats reach the same person, so together they cover the full circle once: n = 4 + 7 = 11.

Practice set

10. Puzzles: Floors, Boxes & Scheduling

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy floor). Five people A, B, C, D and E live on floors 1 to 5 of a building (1 = bottom). B lives on the lowest floor. Only one person lives between B and C. D lives immediately above A. Who lives on the floor just below C?

1. B = 1. One person between B and C means C = 3.
2. D = A + 1. Free floors are 2, 4, 5. The only pair of consecutive free floors is 4 and 5, so A = 4, D = 5.
3. E takes floor 2, which is just below C.

Answer: E.

Example 2 (floors + a second detail, with cases). Six people A–F live on floors 1 to 6, one on each floor. Each likes a different fruit: Apple, Banana, Cherry, Grape, Mango, Orange.

• Only two people live between the Apple-lover and B.
• C lives immediately above E. F lives immediately above C.
• E likes Orange. C likes Banana.
• D lives on the top floor and does not like Grape.
• The Grape-lover lives immediately below D.
• B does not like Mango.

Questions: (i) Who likes Grape? (ii) Who likes Apple? (iii) What does B like?

1. Definite: D = 6. Grape = floor 5.
2. Glue block: F-C-E sit on three consecutive floors (E lowest) among floors 1–5. Cases: E-C-F on 1-2-3, 2-3-4, or 3-4-5.
3. Case 1 (E=1, C=2, F=3): A and B are on 4 and 5. Apple must be at floor 1 (E, Orange) or 2 (C, Banana) or beyond the building. Fails.
4. Case 3 (E=3, C=4, F=5): A and B are on 1 and 2. The Apple-lover is 3 floors from B, so at floor 4 or 5. Floor 4 is C (Banana), floor 5 is Grape. Fails.
5. Case 2 (E=2, C=3, F=4): A and B are on 1 and 5. If B = 5, Apple would be at floor 2 (E, Orange): fails. So B = 1, A = 5, and Apple is at floor 4 = F.
6. A is on floor 5, so A likes Grape. Cherry and Mango remain for B and D; B does not like Mango, so B = Cherry, D = Mango.

Answers: (i) A (ii) F (iii) Cherry.

Example 3 (box stacking). Six boxes K, L, M, N, P and Q are kept one above the other (position 1 = bottom). Two boxes are kept between K and P. L is kept immediately above P. M is kept at the bottom. N is kept somewhere above K. Q is kept immediately below K. Which box is on top?

1. M = 1. Blocks: P-L (L just above P) and Q-K (Q just below K).
2. K and P differ by 3. Try P = 2: L = 3, K = 5, Q = 4, and N must be above K: N = 6. Works.
3. Try P = 3: L = 4, K = 6, Q = 5, and N has no place above K. Fails.
4. Try P above K instead: K = 2 needs Q = 1 (M is there). K = 3 gives P = 6, L = 7: no room. So the first case is the only one.

Stack from bottom: M, P, L, Q, K, N. Answer: N is on top.

Example 4 (days). Six people A–F each have an appointment on a different day from Monday to Saturday. C's appointment is on Monday. F's is on the day immediately after C's. Only two people have appointments between A and D. B's appointment is on the day immediately before D's. On which day is E's appointment?

1. Mon = C, Tue = F.
2. A and D are 3 days apart and B is the day before D. Try A before D: A = Wed, D = Sat, B = Fri. Works.
3. Try D before A: D = Wed would need B = Tue (F's day); D = Thu would need A = Sun (not allowed). No other case.
4. Thursday is the only free day, so E = Thursday.

Order: C, F, A, E, B, D. Answer: Thursday.

Example 5 (months). Six people P, Q, R, S, T and U have exams in six different months from January to June. Q's exam is in the month with the fewest days. P's exam is in the month immediately after Q's. R and T both have exams in 30-day months. U's exam is in the month immediately before T's. In which month is S's exam?

1. Fewest days = February, so Q = Feb, P = Mar.
2. 30-day months between Jan and Jun: April and June. R and T share them.
3. If T = Apr, U = Mar, but P is in March. So T = Jun, U = May, R = Apr.
4. The only month left is January.

Answer: January.

Example 6 (table-grid). Arun, Bina, Chetan and Divya live in different cities (Delhi, Pune, Goa, Jaipur) and play different sports (Cricket, Chess, Tennis, Hockey). Arun lives in Pune but does not play Chess. The Tennis player lives in Goa. Bina plays Hockey. Divya lives in neither Goa nor Jaipur. Who plays Chess, and where does that person live?

1. Divya: not Goa, not Jaipur, not Pune (Arun) → Delhi.
2. Goa and Jaipur remain for Bina and Chetan. Goa's person plays Tennis, but Bina plays Hockey, so Bina = Jaipur and Chetan = Goa (Tennis).
3. Arun is not Chess, not Hockey, not Tennis → Cricket. Divya gets Chess.

Answer: Divya plays Chess and lives in Delhi.

Practice set

11. Syllogisms

Key formulas & rules

Shortcuts

Solved examples

Example 1. Statements: All dogs are animals. All animals are living things. Conclusions: I. All dogs are living things. II. Some living things are dogs.

1. Dogs sit inside animals, which sit inside living things.
2. I: every dog is inside the big circle. True in every picture.
3. II: dogs exist and are inside living things, so some living things are dogs.

Answer: Both I and II follow.

Example 2. Statements: Some pens are books. All books are bags. Conclusions: I. Some pens are bags. II. All bags are pens.

1. Some + All = Some: the pens that are books are also bags. I follows.
2. II: draw bags much bigger than pens. Nothing forbids it. II fails.

Answer: Only I follows.

Example 3. Statements: No cat is a dog. All dogs are pets. Conclusions: I. No pet is a cat. II. Some pets are not cats.

1. Dogs are inside pets. Cats are away from dogs.
2. I: the cats circle can still overlap pets outside the dogs region. Then some pets are cats, so I fails.
3. II: the dogs region of pets never contains a cat, so some pets (the dogs) are not cats. II follows.

Answer: Only II follows. Shortcut: No + All = "Some C are not A" (Some pets are not cats).

Example 4 (either–or). Statements: Some A are B. Some B are C. Conclusions: I. Some A are C. II. No A is C.

1. Some + Some gives no definite link between A and C, so I alone fails and II alone fails.
2. Same two terms (A, C), and the pair is Some / No: one of them must always be true.

Answer: Either I or II follows.

Example 5 (possibility). Statements: All rings are gold. No gold is silver. Conclusions: I. No ring is silver. II. All gold being rings is a possibility.

1. All + No = No: rings are inside gold, gold is away from silver, so no ring is silver. I follows.
2. II: can the gold circle be exactly the same as the rings circle? Yes, nothing forbids it. II follows as a possibility.

Answer: Both I and II follow.

Example 6 (three statements). Statements: Some A are B. All B are C. No C is D. Conclusions: I. Some A are not D. II. Some C are A.

1. Some A are B + All B are C → Some A are C (Some + All = Some).
2. Those A that are C cannot be D (No C is D). So Some A are not D. I follows.
3. Some A are C reversed = Some C are A. II follows.

Answer: Both I and II follow.

Practice set

For each question choose: (a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither I nor II follows (e) Both I and II follow.

12. Statements: Conclusions & Assumptions

Key formulas & rules

Shortcuts

Solved examples

Example 1 (conclusion). Statement: Of all the students who took the test, only 40% passed. The rest must repeat the course. Conclusions: I. 60% of the students will repeat the course. II. The test was very hard.

1. I: 100% − 40% = 60% failed, and "the rest must repeat". Directly follows.
2. II: a low pass rate could be due to poor preparation, strict marking, anything. The statement does not say why. Does not follow.

Answer: Only I follows.

Example 2 (conclusion). Statement: Morning walks are good for health. Conclusions: I. All healthy people take morning walks. II. Evening walks are harmful.

1. I: "good for health" does not mean it is the only way to be healthy. Extreme word "all". Fails.
2. II: nothing is said about evening walks. Comparison trap. Fails.

Answer: Neither follows.

Example 3 (assumption). Statement: "Use Brand X toothpaste for whiter teeth" — an advertisement. Assumptions: I. People want whiter teeth. II. Other toothpastes do not whiten teeth.

1. I: negate it: "people do not care about white teeth". Then the advert is pointless. So I is assumed.
2. II: the advert praises X; it does not need other brands to fail. Comparison trap.

Answer: Only I is implicit.

Example 4 (assumption). Statement: The school has asked parents to send their children in warm clothes from next week. Assumptions: I. The weather is expected to turn cold next week. II. Most parents will follow the school's request.

1. I: why ask for warm clothes? Because cold is expected. Negate it and the request makes no sense. Implicit.
2. II: any request assumes it will be followed; otherwise the school would not bother. Implicit.

Answer: Both are implicit.

Example 5 (either–or conclusion). Statement: The committee will either approve the new bridge this month or drop the project completely. Conclusions: I. The committee will approve the new bridge this month. II. The committee will drop the bridge project.

1. Neither is certain alone.
2. The statement gives exactly these two outcomes, and one of them must happen.

Answer: Either I or II follows.

Practice set

Options for each: (a) Only I (b) Only II (c) Either I or II (d) Neither I nor II (e) Both I and II.

13. Cause & Effect, Course of Action

Key formulas & rules

Shortcuts

Solved examples

Example 1 (cause & effect). I. The prices of vegetables in the city have risen sharply this month. II. Heavy rain has destroyed crops in the nearby farming region.

1. Crops destroyed comes first (it reduces supply).
2. Less supply leads straight to higher prices.

Answer: (b) II is the cause and I is the effect.

Example 2 (cause & effect). I. Schools in the district were closed for the day. II. Several trains to the district were cancelled.

1. Closing schools does not cancel trains, and cancelled trains do not close schools.
2. Both fit a single event such as a flood or a heavy snowfall in the district.

Answer: (e) Both are effects of some common cause.

Example 3 (cause & effect). I. The price of petrol has gone up this week. II. The sale of umbrellas has gone up this week.

Two results with no believable shared trigger (fuel prices and rain are unrelated). Answer: (d) Both are effects of independent causes.

Example 4 (course of action). Statement: Many students of a college failed their final exams, and most of them had very poor attendance. Courses of action: I. The college should set a minimum attendance rule for sitting the exam. II. The college should be shut down.

1. I is practical, within the college's power, and addresses the link between attendance and failure. Follows.
2. II is extreme and punishes everyone, including students who passed. Does not follow.

Answer: Only I follows.

Example 5 (strong / weak argument). Should mobile phones be banned inside classrooms? I. Yes, phones distract students during lessons and lower attention. II. No, students in many rich countries carry phones to class.

1. I gives a real, relevant reason tied to learning. Strong.
2. II copies another country's practice without showing it helps learning. Weak.

Answer: Only argument I is strong.

Practice set

Cause & effect options: (a) I is cause, II is effect (b) II is cause, I is effect (c) both independent causes (d) both effects of independent causes (e) both effects of a common cause.

14. Data Sufficiency

Key formulas & rules

Shortcuts

Solved examples

Example 1. What is Ravi's age? I. Ravi is 4 years older than Sita. II. Sita is 18 years old.

1. I alone: Ravi = Sita + 4, but Sita's age is unknown. Not enough.
2. II alone: says nothing about Ravi. Not enough.
3. Together: Ravi = 18 + 4 = 22.

Answer: (d) both together.

Example 2. Is the integer n odd? I. 5n is odd. II. n + 3 is even.

1. I: 5n is odd only when n is odd. Definite yes.
2. II: n + 3 even means n is odd. Definite yes.

Answer: (c) either alone.

Example 3. What is the value of the integer x? I. 2x + 3 = 11. II. x² = 16.

1. I: 2x = 8, x = 4. One value. Sufficient.
2. II: x = 4 or x = −4. Two values. Not sufficient.

Answer: (a) I alone.

Example 4. How many students stand in the row? I. Amit is 10th from the left end. II. Amit is 15th from the right end.

Each alone gives only one side. Together: total = 10 + 15 − 1 = 24. Answer: (d) both together.

Example 5. In which direction is P from Q? I. P is to the north of R. II. R is to the east of Q.

1. I alone: Q is not mentioned. Not enough.
2. II alone: P is not mentioned. Not enough.
3. Together: go east from Q to R, then north to P. Whatever the distances, P is north-east of Q.

Answer: (d) both together.

Example 6. Among A, B, C and D, who is the tallest? I. A is taller than B and C. II. D is shorter than C.

1. I alone: A beats B and C, but D is unknown. Not enough.
2. II alone: only C and D are compared. Not enough.
3. Together: A > C > D and A > B, so A is tallest.

Answer: (d) both together.

Practice set

Options for each: (a) I alone (b) II alone (c) Either alone (d) Both together (e) Not sufficient even together.

15. Inequalities

Key formulas & rules

Shortcuts

Solved examples

Example 1. Statement: A > B ≥ C = D < E. Conclusions: I. A > D. II. E > B.

1. I: A > B ≥ C = D, all signs point one way, one is strict. A > D is true.
2. II: from B to E the path is B ≥ C = D < E. The "<" points the other way. No relation between E and B.

Answer: Only I follows.

Example 2 (either–or). Statement: P ≥ Q = R > S ≤ T. Conclusions: I. P > R. II. P = R.

1. P ≥ Q = R gives P ≥ R. So P > R is not certain, and P = R is not certain.
2. Same letters, and ">" plus "=" together make exactly "≥", which is true.

Answer: Either I or II follows.

Example 3 (combining statements). Statements: J ≥ K, K > L, L = M, M ≤ N. Conclusions: I. J > M. II. N ≥ L.

1. Join: J ≥ K > L = M ≤ N.
2. I: J ≥ K > L = M, same direction, one strict: J > M. True.
3. II: L = M ≤ N, read backwards: N ≥ M = L, so N ≥ L. True.

Answer: Both follow.

Example 4 (coded). "A @ B" means A ≥ B, "A # B" means A > B, "A $ B" means A = B, "A % B" means A < B, "A & B" means A ≤ B. Statements: F # G, G $ H, H & K. Conclusions: I. F # H. II. K % G.

1. Decode: F > G, G = H, H ≤ K. Join: F > G = H ≤ K.
2. I decodes to F > H: F > G = H. True.
3. II decodes to K < G: but K ≥ H = G, so K < G is false.

Answer: Only I follows.

Example 5 (no relation). Statement: A < B ≤ C > D. Conclusions: I. A < D. II. A > D.

1. A < B ≤ C and then C > D: the path from A to D changes direction at C. No relation.
2. Same letters, but I and II leave out "A = D", so they do not cover every case. Not either–or.

Answer: Neither follows. If II had been "A ≥ D", the pair would cover all cases and the answer would be "Either I or II".

Practice set

Options: (a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither follows (e) Both follow.

16. Clocks

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). Find the angle between the hands at 3:40 and at 7:20.

1. 3:40 → θ = |30 × 3 − 5.5 × 40| = |90 − 220| = 130°. It is under 180°, so that is the answer.
2. 7:20 → θ = |30 × 7 − 5.5 × 20| = |210 − 110| = 100°.

Answer: 130° and 100°. Why it works: from 12, the hour hand at 3:40 has moved 3 × 30 + 40 × 0.5 = 110°, and the minute hand 40 × 6 = 240°. The gap is 130°, the same as the formula.

Example 2 (easy). At what time between 4 and 5 o'clock are the hands together?

1. At 4:00 the hour hand is 120° ahead of the minute hand (4 × 30).
2. The minute hand gains 5.5° every minute, so it needs 120 ÷ 5.5 = 240/11 minutes to catch up.
3. 240/11 = 21 9/11 minutes.

Answer: 21 9/11 minutes past 4. Shortcut: 60H/11 = 60 × 4 / 11 = 21 9/11.

Example 3 (medium). At what time between 2 and 3 are the hands at right angles?

1. Use M = (2/11)(30H ± 90) with H = 2, so 30H = 60.
2. With the plus sign: (2/11)(150) = 300/11 = 27 3/11 minutes. This lies between 0 and 60, so it is valid.
3. With the minus sign: (2/11)(−30) is negative, so it is not valid. The next right angle is exactly at 3:00, which belongs to the next hour.

Answer: 27 3/11 minutes past 2 (only once in that hour). This is exactly why there are 44 right angles a day and not 48.

Example 4 (medium). At what time between 7 and 8 are the hands in opposite directions?

1. At 7:00 the hour hand is 210° ahead. For them to be opposite, the minute hand must be 180° behind, so it has to close a gap of 210 − 180 = 30°.
2. Time = 30 ÷ 5.5 = 60/11 = 5 5/11 minutes.

Answer: 5 5/11 minutes past 7. Formula check: (2/11)(210 − 180) = 60/11.

Example 5 (medium). The mirror image of a clock shows 3:40. What is the actual time?

1. Mirror and real time always add up to 11:60 (that is, 12:00).
2. Actual = 11:60 − 3:40 = 8:20.

Answer: 8:20.

Example 6 (hard). A clock is set right at 8 a.m. It gains 10 minutes in 24 hours. What is the true time when it shows 1 p.m. the next day?

1. From 8 a.m. to 1 p.m. the next day the clock shows 29 hours have passed.
2. When 24 true hours pass, the clock shows 24 h 10 min = 24 1/6 h = 145/6 h.
3. So true time = 29 × 24 ÷ (145/6) = 29 × 144 / 145 = 28.8 hours = 28 h 48 min.
4. 8 a.m. + 28 h 48 min = 12:48 p.m. the next day.

Answer: 12:48 p.m. A gaining clock is ahead, so the true time must be earlier than what it shows: a quick sense check.

Example 7 (hard). The hands of a faulty clock coincide every 64 minutes of true time. How much does it gain or lose per day?

1. In a correct clock the hands meet every 720/11 = 65 5/11 minutes of clock time.
2. This clock completes that same clock-time in only 64 true minutes, so it runs fast: it gains 65 5/11 − 64 = 16/11 min every 64 min.
3. In a day (1440 min): 16/11 × 1440 / 64 = 360/11 = 32 8/11 minutes.

Answer: it gains 32 8/11 minutes per day.

Practice set

17. Calendars

Key formulas & rules

These gaps hold inside one century; if a non-leap century year (like 2100) falls in between, count odd days directly.

The odd-day method, step by step

Shortcuts

Solved examples

Example 1 (easy). 1 January 2023 was a Sunday. What day was 1 January 2024?

1. 2023 is an ordinary year, so it has 1 odd day.
2. Sunday + 1 = Monday.

Answer: Monday.

Example 2 (easy). 10 March 2020 was a Tuesday. What day was 10 March 2021?

1. Between 10 March 2020 and 10 March 2021 the February is February 2021, which has 28 days. The leap day, 29 February 2020, is before the start date.
2. So the gap is 365 days = 1 odd day. Tuesday + 1 = Wednesday.

Answer: Wednesday. Most students add 2 because "2020 is a leap year". That leap day was already behind us.

Example 3 (medium). What day of the week was 15 August 1947?

1. Complete years = 1946 = 1600 + 300 + 46.
2. 1600 years → 0. 300 years → 1.
3. 46 years: 11 leap years (1904 … 1944) and 35 ordinary. Odd days = 22 + 35 = 57 → 57 mod 7 = 1.
4. Jan–Jul 1947 (not leap): 3 + 0 + 3 + 2 + 3 + 2 + 3 = 16 → 2.
5. Date: 15 → 1.
6. Total = 0 + 1 + 1 + 2 + 1 = 5 → Friday.

Answer: Friday.

Example 4 (medium). What day was 26 January 1950?

1. Complete years = 1949 = 1600 + 300 + 49.
2. 0 + 1 so far. 49 years: 12 leap (1904 … 1948) and 37 ordinary → 24 + 37 = 61 → 5.
3. No complete month. Date 26 → 26 mod 7 = 5.
4. Total = 0 + 1 + 5 + 5 = 11 → 4 → Thursday.

Answer: Thursday.

Example 5 (medium). The calendar of 2019 will be repeated in which year?

1. We need the running total of odd days to hit a multiple of 7, and the new year must be ordinary like 2019.
2. Odd days: 2019 (1), 2020 (2), 2021 (1), 2022 (1), 2023 (1), 2024 (2), 2025 (1), 2026 (1), 2027 (1), 2028 (2), 2029 (1).
3. Running total: 1, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14. It reaches 14 at the end of 2029.
4. So 1 January 2030 falls on the same weekday as 1 January 2019, and 2030 is ordinary.

Answer: 2030. Shortcut: 2019 is leap + 3, so add 11.

Example 6 (hard). What day will 1 January 2100 be?

1. Complete years = 2099 = 2000 + 99.
2. 2000 years → 0 (five 400s).
3. 99 years: 24 leap years (2004 … 2096) and 75 ordinary → 48 + 75 = 123 → 123 mod 7 = 4.
4. Date 1 → 1. Total = 5 → Friday.

Answer: Friday. Note 2100 itself is not a leap year, so its calendar is not the same as 2072's even though that is 28 years earlier.

Practice set

18. Cubes & Dice

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). A cube painted on all faces is cut into 27 equal cubes. How many small cubes have 0, 1, 2 and 3 faces painted?

1. 27 = 3³, so n = 3 and n − 2 = 1.
2. 3 faces: 8. 2 faces: 12 × 1 = 12. 1 face: 6 × 1² = 6. 0 faces: 1³ = 1.
3. Check: 8 + 12 + 6 + 1 = 27.

Answer: 1, 6, 12, 8.

Example 2 (easy). A 4 cm cube is painted and cut into 1 cm cubes. How many cubes have at least one face painted?

1. n = 4, total 64. Unpainted core = (4 − 2)³ = 8.
2. At least one face = 64 − 8 = 56.

Answer: 56.

Example 3 (medium). A painted cube is cut into 125 equal cubes. How many have exactly two faces painted, and how many exactly one?

1. n = 5, n − 2 = 3.
2. Two faces: 12 × 3 = 36. One face: 6 × 9 = 54.

Answer: 36 and 54.

Example 4 (medium). Two views of a die are shown. Which number is opposite 3?

1. 6 is on top in both views, so only the four side faces moved.
2. In View 1, 2 is at front-left. In View 2, 2 has moved to front-right. So the die turned one step, and the new front-left face (4) came from the hidden back-left position of View 1.
3. In View 1, the back-left face is exactly opposite the front-right face, which shows 3.

Answer: 4. What we cannot say: whether 2 is opposite 1 or 5 (both fit). A good exam question only asks what the views decide.

Example 5 (hard). Three views of one die are shown. Find all three opposite pairs.

1. Views 1 and 2 share two numbers (1 and 2). By the two-common-numbers rule, the third numbers, 3 and 6, are opposite.
2. Neighbours of 2 seen so far: 1, 3 (View 1), 6 (View 2), 4 (View 3). That is 4 neighbours, so the leftover number, 5, is opposite 2.
3. The last two numbers, 1 and 4, must be opposite each other.

Answer: 1–4, 2–5, 3–6. (A brute-force check over all 720 ways to number a cube confirms only this pairing fits all three views.)

Example 6 (medium). Which faces are opposite in these two nets?

1. Cross net: in the row B C D E, B and D are alternate, so opposite; C and E are alternate, so opposite. In the column A C F, A and F are the ends, so opposite.
2. Staircase net: fold C as the floor. B folds up as the back wall, D as the right wall, A hangs off B to become the left wall, E hangs off D to become the front wall, and F then becomes the lid.
3. So A (left) faces D (right), B (back) faces E (front), C (floor) faces F (lid).

Answer: cross A–F, B–D, C–E; staircase A–D, B–E, C–F.

Practice set

19. Mirror, Water Images & Patterns

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). How does the word TOMATO look in a vertical mirror?

1. A mirror reverses the order: O T A M O T.
2. Now flip each letter left–right. T, O, M and A all have a vertical line of symmetry, so they look the same after flipping.

Answer: OTAMOT, with every letter looking normal.

Example 2 (easy). Which word looks exactly the same in a vertical mirror? (a) MOM (b) DAD (c) NOON (d) BOB

1. It must read the same backwards (a palindrome): all four are.
2. Every letter must also be vertically symmetric. D, N and B are not (their mirror images face the other way).
3. Only M and O pass.

Answer: (a) MOM.

Example 3 (easy). How does the word BOX look in water?

1. A water image keeps the order: B O X.
2. B, O and X each have a horizontal line of symmetry, so each looks the same upside down.

Answer: BOX, unchanged.

Example 4 (medium). Find the next figure in the series.

1. Compare figure 1 with figure 2: the arrow turned 45° clockwise.
2. Check figures 2 → 3, 3 → 4, 4 → 5: 45° clockwise each time. The rule holds.
3. Figure 5 points down (180° from up). Figure 6 = 180° + 45° = 225°, which points down-left.

Answer: an arrow pointing down-left.

Example 5 (medium). The figures in a series have 1, 3, 6 and 10 dots arranged as triangles. How many dots are in the next figure?

1. The differences are +2, +3, +4, so the next difference is +5.
2. 10 + 5 = 15.

Answer: 15 (the triangular numbers n(n + 1)/2).

Example 6 (hard). What is the mirror image of the water image of the letter b?

1. Water image of b flips it top-to-bottom: the loop stays on the right but the stem now points down, giving p.
2. Mirror image of p flips it left-to-right: the loop moves to the left, giving q.
3. Check with the 180° rule: turning b upside down (half a turn) also gives q.

Answer: q.

Practice set

20. Venn Diagrams

Key formulas & rules

Shortcuts

Solved examples

Example 1 (easy). Which diagram shows Doctors, Women and Mothers? And India, Asia, Earth?

1. All mothers are women, so the Mothers circle goes inside the Women circle.
2. Some doctors are women and some are not (male doctors), so the Doctors circle overlaps Women and also sticks out.
3. Some mothers are doctors and some are not, so Doctors also overlaps Mothers.
4. India lies inside Asia, which lies inside Earth: three nested circles.

Example 2 (easy). In a class of 60, 35 like tea, 30 like coffee and 10 like both. How many like neither?

1. Tea or coffee = 35 + 30 − 10 = 55.
2. Neither = 60 − 55 = 5.

Answer: 5.

Example 3 (medium). In a survey, 70% like Maths and 80% like Physics. At least what percentage like both?

1. If the overlap were smaller, the two groups together would need more than 100% of people.
2. Minimum overlap = 70 + 80 − 100 = 50%.

Answer: 50%. (Maximum overlap would be 70%, when all Maths lovers also like Physics.)

Example 4 (hard). Out of 100 students, 50 take Maths (M), 40 Physics (P), 45 Chemistry (C). 15 take M and P, 12 take P and C, 18 take M and C, and 7 take all three. Find (i) how many take none, (ii) exactly two, (iii) exactly one, (iv) only Maths.

1. Centre: all three = 7.
2. Only-two regions: M and P only = 15 − 7 = 8; P and C only = 12 − 7 = 5; M and C only = 18 − 7 = 11. Exactly two = 8 + 5 + 11 = 24.
3. Only regions: only M = 50 − 8 − 11 − 7 = 24; only P = 40 − 8 − 5 − 7 = 20; only C = 45 − 11 − 5 − 7 = 22. Exactly one = 66.
4. Union = 66 + 24 + 7 = 97 (formula check: 135 − 45 + 7 = 97). None = 100 − 97 = 3.

Answer: none 3, exactly two 24, exactly one 66, only Maths 24.

Example 5 (hard). Of 50 people, 30 like apples and 25 like bananas. What are the maximum and minimum numbers who like neither?

1. Neither is largest when the union is smallest, which happens when the overlap is as big as possible: all 25 banana lovers also like apples. Union = 30, neither = 20.
2. Neither is smallest when the union is largest: overlap as small as possible = 30 + 25 − 50 = 5. Union = 50, neither = 0.

Answer: maximum 20, minimum 0.

Practice set

Quick Rules Cheat Sheet

One screen per topic group. Read this the night before the test and again 10 minutes before it starts. Anything highlighted is a must-know.

Letters and numbers you must know by heart

Part 1: Verbal reasoning (topics 1–7)

Part 2: Analytical reasoning (topics 8–15)

Part 3: Non-verbal & logical (topics 16–20)

Time-saving tips

Test Strategy & Learning Path

How to approach the section

1. First 60 seconds: scan. Flip through the section and note which question types appear. Mark any puzzle or seating set (4–5 questions on one arrangement).
2. Round 1: quick singles. Series, analogy, odd one out, coding, alphabet, direction, ranking, clocks, calendars, painted cubes, mirror images, numeric Venn. Target 30–45 seconds each.
3. Round 2: logic singles. Syllogisms, inequalities, blood relations, data sufficiency, statements. Target 45–60 seconds each.
4. Round 3: sets. Seating and puzzles. Choose the set with the most definite clues first. Budget 5–7 minutes for a 4–5 question set; if the arrangement is not fixed after 3 minutes, move on.
5. Last 2 minutes: return to marked questions. Guess only if there is no negative marking, or if you have eliminated at least two options.

6-week plan

About 75–90 minutes a day, 6 days a week, with a timed test on the 7th day. Each topic day follows the same routine: read the notes section (15 min), redo its solved examples without looking (15 min), do its practice set timed (20 min), then 15–20 extra questions from any question bank, then update your error log.

A sample day

The error log

Keep one page per topic. For every wrong or slow question, write a single line: what type it was, why you missed it (rule forgotten, misread, calculation slip, too slow) and the rule that fixes it. Read the whole log every Sunday. By week 6 most mistakes repeat; the log shows you exactly which three or four rules you keep forgetting.