A cooperative society with 600 members elected its chairperson in two stages. Every member voted in both stages, and every vote was valid.
Stage 1: Five candidates — P, Q, R, S and T — contested, and each member voted for exactly one of them. The two candidates with the most votes advanced to Stage 2; the other three were eliminated.
Stage 2: Each member voted for one of the two finalists.
The following information is available:
- In Stage 1, the numbers of votes received by the five candidates, arranged in decreasing order, formed an arithmetic progression, and the candidate with the fewest votes received exactly one-third as many votes as the candidate with the most votes.
- P received the most votes in Stage 1 and Q advanced to Stage 2. S received fewer votes than R but more than T.
- In Stage 2, every member who had voted for P or Q in Stage 1 voted for the same candidate again.
- T's Stage-1 voters split equally between the finalists in Stage 2, while S's Stage-1 voters split in the ratio 2 : 1 in favour of P.
- P won Stage 2 by a margin of 20 votes.
Stage 1: Five candidates — P, Q, R, S and T — contested, and each member voted for exactly one of them. The two candidates with the most votes advanced to Stage 2; the other three were eliminated.
Stage 2: Each member voted for one of the two finalists.
The following information is available:
- In Stage 1, the numbers of votes received by the five candidates, arranged in decreasing order, formed an arithmetic progression, and the candidate with the fewest votes received exactly one-third as many votes as the candidate with the most votes.
- P received the most votes in Stage 1 and Q advanced to Stage 2. S received fewer votes than R but more than T.
- In Stage 2, every member who had voted for P or Q in Stage 1 voted for the same candidate again.
- T's Stage-1 voters split equally between the finalists in Stage 2, while S's Stage-1 voters split in the ratio 2 : 1 in favour of P.
- P won Stage 2 by a margin of 20 votes.
Question 1
Numerical answer
How many votes did Q receive in Stage 2?
Show answer and solution
Answer: 290
Deduction. Let the Stage-1 votes be a + 4d, a + 3d, a + 2d, a + d, a. Sum = 5a + 10d = 600 → a + 2d = 120. Also a = (a + 4d)/3 → 2a = 4d → a = 2d. So 4d = 120, d = 30, a = 60: votes are 180, 150, 120, 90, 60.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
Q's Stage-2 tally = 150 (own) + 80 (from R) + 30 (from S) + 30 (from T) = 290. Check: 310 + 290 = 600 ✓.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
Q's Stage-2 tally = 150 (own) + 80 (from R) + 30 (from S) + 30 (from T) = 290. Check: 310 + 290 = 600 ✓.
Question 2
Single correct option
What fraction of R's Stage-1 voters voted for Q in Stage 2?
- A. 1/3
- B. 1/2
- C. 2/3
- D. 3/4
Show answer and solution
Answer: (C) 2/3
Deduction. Let the Stage-1 votes be a + 4d, a + 3d, a + 2d, a + d, a. Sum = 5a + 10d = 600 → a + 2d = 120. Also a = (a + 4d)/3 → 2a = 4d → a = 2d. So 4d = 120, d = 30, a = 60: votes are 180, 150, 120, 90, 60.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
Of R's 120 voters, 40 chose P and 80 chose Q. Fraction for Q = 80/120 = 2/3.
A common slip is to report 1/3 — that is the fraction that went to P.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
Of R's 120 voters, 40 chose P and 80 chose Q. Fraction for Q = 80/120 = 2/3.
A common slip is to report 1/3 — that is the fraction that went to P.
Question 3
Numerical answer
How many votes did T receive in Stage 1?
Show answer and solution
Answer: 60
Let the votes in decreasing order be a + 4d, a + 3d, a + 2d, a + d, a.
Sum: 5a + 10d = 600 → a + 2d = 120 (the middle term is the average, 600/5 = 120).
Lowest = one-third of highest: 3a = a + 4d → a = 2d. Then 2d + 2d = 120 → d = 30, a = 60.
Votes: 180, 150, 120, 90, 60. T has the fewest (T < S < R, and P, Q are the top two), so T received 60 votes.
Sum: 5a + 10d = 600 → a + 2d = 120 (the middle term is the average, 600/5 = 120).
Lowest = one-third of highest: 3a = a + 4d → a = 2d. Then 2d + 2d = 120 → d = 30, a = 60.
Votes: 180, 150, 120, 90, 60. T has the fewest (T < S < R, and P, Q are the top two), so T received 60 votes.
Question 4
Single correct option
Suppose instead that all of S's Stage-1 voters had voted for Q in Stage 2, with everything else unchanged. What would the Stage-2 result have been?
- A. P wins by 20 votes
- B. Q wins by 40 votes
- C. Q wins by 100 votes
- D. The result is a tie
Show answer and solution
Answer: (C) Q wins by 100 votes
Deduction. Let the Stage-1 votes be a + 4d, a + 3d, a + 2d, a + d, a. Sum = 5a + 10d = 600 → a + 2d = 120. Also a = (a + 4d)/3 → 2a = 4d → a = 2d. So 4d = 120, d = 30, a = 60: votes are 180, 150, 120, 90, 60.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
In the new scenario, the 60 S-voters who went to P move to Q.
P = 310 − 60 = 250; Q = 290 + 60 = 350.
Q would win by 100 votes. Shortcut: shifting 60 votes from one side to the other changes the margin by 2 × 60 = 120, turning P's +20 into Q's +100.
P is highest (180) and Q advanced, so Q = 150; R > S > T gives R = 120, S = 90, T = 60.
Stage 2: T's 60 split 30–30; S's 90 split 60 (P) – 30 (Q). Let x of R's 120 voters choose P.
P = 180 + x + 60 + 30 = 270 + x; Q = 150 + (120 − x) + 30 + 30 = 330 − x.
P − Q = 2x − 60 = 20 → x = 40. So P = 310, Q = 290.
In the new scenario, the 60 S-voters who went to P move to Q.
P = 310 − 60 = 250; Q = 290 + 60 = 350.
Q would win by 100 votes. Shortcut: shifting 60 votes from one side to the other changes the margin by 2 × 60 = 120, turning P's +20 into Q's +100.
Saved on this device. Sign in or create a free account to keep it on every device.