In an election of 3 candidates A, B and C, A gets 50% more votes than B. A also beats C by 18,000 votes. If it is known that B gets 5% more votes than C, then find the number of voters on the voting list, given that 90% of the voters on the voting list voted and no votes were illegal.

RPSC deleted this question in its final answer key, so no option is marked correct.

The stem links three candidates' votes by percentages and a margin, then asks for the number of voters on the voting list. The method: write every count in terms of one unknown, use the margin to find it, add the votes, and scale up for turnout.

Step 1 — write the votes in terms of C's votes, c.
"B gets 5% more votes than C": B = 1.05c
"A gets 50% more votes than B": A = 1.5 × 1.05c = 1.575c

Step 2 — use the margin.
"A also beats C by 18,000 votes": A − C = 0.575c = 18,000
c = 18,000 ÷ 0.575 ≈ 31,304.3

Step 3 — total votes polled.
No votes were illegal, so votes polled = A + B + C
= 1.575c + 1.05c + c = 3.625c ≈ 1,13,478

Step 4 — scale up for turnout.
90% of the voters on the list voted, so list = votes polled ÷ 0.9.

The idea to remember: percentage increases in a chain multiply (1.5 × 1.05 = 1.575), and a list is found from votes polled by dividing by the turnout.

Why the others are wrong

• (1)
  1,00,000 — Option (1) offers 1,00,000 voters on the list. RPSC's final key marks no option.

  At 90% turnout a list of 1,00,000 means 1,00,000 × 0.9 = 90,000 votes polled, which is the figure printed in option (3).

• (2)
  81,000 — Option (2) offers 81,000 voters on the list. RPSC's final key marks no option.

  81,000 is 90% of 90,000, the figure in option (3). As a list, it would mean 81,000 × 0.9 = 72,900 votes polled.

• (3)
  90,000 — Option (3) offers 90,000 voters on the list. RPSC's final key marks no option.

  90,000 is 90% of 1,00,000, the figure in option (1). As a list, it would mean 90,000 × 0.9 = 81,000 votes polled, the figure in option (2).

• (4)
  1,10,000 — Option (4) offers 1,10,000 voters on the list. RPSC's final key marks no option.

  At 90% turnout a list of 1,10,000 means 1,10,000 × 0.9 = 99,000 votes polled, with all of them valid under the stem's condition.

"X is p% more than Y" means X = Y × (1 + p/100). "X is p% less than Y" means X = Y × (1 − p/100).

Increases in a chain multiply rather than add. 50% more than a value that is itself 5% more than C gives 1.5 × 1.05 = 1.575 times C, a 57.5% increase.

A margin between two candidates is the difference of their multiples of the common unknown, so it fixes that unknown.

Turnout links votes and voters: votes polled = turnout × voters on the list, so voters on the list = votes polled ÷ turnout.

Key facts

• X is p% more than Y means X = Y × (1 + p/100).
• Chained increases multiply: 50% more than a value that is 5% more than C is 1.5 × 1.05 = 1.575 times C.
• As multiples of C's votes c: A = 1.575c, B = 1.05c, C = c; together 3.625c.
• A's margin over C is 1.575c − c = 0.575c.
• With a 90% turnout and no invalid votes, voters on the list = total votes ÷ 0.9.

Common traps

• Adding chained percentages. 50% more than B, where B is 5% more than C, is 57.5% more than C, not 55%.
• Scaling turnout the wrong way. The list is larger than the votes polled: list = votes ÷ 0.9, not votes × 0.9.
• Taking the margin between the wrong pair. The 18,000 is A's lead over C, not over B.

• practice — not a real PYQ

  In an election between two candidates, 80% of the voters on the list voted and all votes were valid. The winner got 60% of the votes polled and won by 8,000 votes. How many voters were on the list?

  • (a)40,000
  • (b)50,000
  • (c)32,000
  • (d)48,000
  Answer(2) — The winner's lead is 60% − 40% = 20% of the votes polled, so votes polled = 8,000 ÷ 0.2 = 40,000, and the list = 40,000 ÷ 0.8 = 50,000.

  Option (1) is the votes polled; option (3) multiplies by 0.8 instead of dividing; option (4) adds 20% to the votes polled.

• practice — not a real PYQ

  P's salary is 20% more than Q's, and Q's salary is 10% more than R's. By what percentage is P's salary more than R's?

  • (a)30%
  • (b)32%
  • (c)10%
  • (d)20%
  Answer(2) — P = 1.2 × Q = 1.2 × 1.1 × R = 1.32R, so P is 32% more than R. Option (1) adds the two percentages; options (3) and (4) take only one of the two increases.