In an election between Sunita and Mamata, 85% of voters cast their votes but 2% of votes were invalid. If Sunita got 64,974 votes which is 78% of valid votes, then the total number of voters enrolled in the election was:
- (a)1,65,000
- (b)1,00,000
- (c)1,25,000
- (d)1,80,000
Answer
Why
Correct — B. Work backwards from Sunita's 64,974 votes, undoing one percentage at a time.
Sunita's votes are 78% of the valid votes:
valid = 64,974 ÷ 0.78 = 83,300
Valid votes are 98% of the votes cast, because 2% were invalid:
cast = 83,300 ÷ 0.98 = 85,000
Votes cast are 85% of the enrolled voters:
enrolled = 85,000 ÷ 0.85 = 1,00,000 → option (b)
Why the others are wrong
- (a)1,65,000 — 1,65,000 fails the forward check: 1,65,000 × 0.85 × 0.98 × 0.78 ≈ 1,07,207 votes for Sunita, far above the 64,974 the question states.
- (c)1,25,000 — 1,25,000 yields 1,25,000 × 0.85 × 0.98 × 0.78 ≈ 81,218 votes — still well over 64,974, so the enrolment must be smaller than this.
- (d)1,80,000 — 1,80,000 moves the wrong way entirely, giving about 1,16,953 votes for Sunita — nearly double the figure the question fixes.
Concept
Election-percentage questions stack percentages on different bases, and the base changes at every step.
The 85% is a share of the enrolled voters. The 2% invalid is a share of the votes actually cast, not of the enrolment. The 78% is a share of the valid votes alone.
So the chain runs enrolled → cast → valid → candidate. Going forward you multiply by each fraction; going backwards from a known count you divide by the same fractions in reverse order.
The question hands you the last number in the chain, so division is the natural direction. The clean intermediate values (78% of 83,300 is exactly 64,974) are the sign that the item was built forward from a round enrolment figure.
Key facts
- The 2% invalid rate applies to the votes cast, so valid votes are 98% of votes cast and not 98% of the enrolment.
- Sunita's 64,974 votes are 78% of the 83,300 valid votes.
- The whole chain checks out as 1,00,000 × 0.85 × 0.98 × 0.78 = 64,974.
Study next
Common traps
- Applying the 2% invalid rate to the enrolled voters instead of to the votes actually cast
- Stopping at 85,000, the votes cast, which is a real link in the chain but not the enrolment
- Multiplying by 0.78 instead of dividing when working backwards
An item of this shape can hand you any one link in the enrolled → cast → valid → candidate chain and ask for another, so name the base of every percentage before you compute.
The same shift runs a chain of successive percentages on price at Quant Q.3, where three discounts of 20%, 30% and 10% are applied one after another.
Related PYQs
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