Karthikeyan arranged a party to celebrate his birthday. 20 of his close…
2024
Karthikeyan arranged a party to celebrate his birthday. 20 of his close friends were invited. One friend left before the cake was cut due to an urgent phone call. The big cake was drawing the attention of the party visitors. This cake is to be divided among the friends, and of course Karthikeyan will get his share. A man eats 3 pieces, a woman eats two pieces, and a child eats half a piece of cake. Including Karthikeyan, count the number of men, women, and children at the party. There are 20 pieces of cake in all.
- A.
6 women, 2 men, 12 children
- B.
7 women, 1 man, 12 children
- C.
5 women, 1 man, 14 children
- D.
4 women, 2 men, 14 children
Show answer & explanation
Correct answer: C
Concept: when a word problem fixes both a total headcount and a total consumption for an unknown mix of groups, the mix must satisfy two simultaneous linear equations at once — the headcount equation and the consumption equation — not just one of them. The quickest way to identify the valid mix among a set of candidate splits is to substitute each candidate into both equations.
Application: including Karthikeyan, the party has 20 people (20 friends invited, one left, plus Karthikeyan), so headcount = men + women + children = 20. Cake consumption gives 3×(men) + 2×(women) + 0.5×(children) = 20, since a man eats 3 pieces, a woman eats 2 pieces, and a child eats half a piece. Checking each candidate split against both equations:
6 women, 2 men, 12 children → headcount 6+2+12 = 20 (holds); pieces 3×2 + 2×6 + 0.5×12 = 6+12+6 = 24 (fails the 20-piece equation).
7 women, 1 man, 12 children → headcount 7+1+12 = 20 (holds); pieces 3×1 + 2×7 + 0.5×12 = 3+14+6 = 23 (fails the 20-piece equation).
5 women, 1 man, 14 children → headcount 5+1+14 = 20 (holds); pieces 3×1 + 2×5 + 0.5×14 = 3+10+7 = 20 (satisfies both equations).
4 women, 2 men, 14 children → headcount 4+2+14 = 20 (holds); pieces 3×2 + 2×4 + 0.5×14 = 6+8+7 = 21 (fails the 20-piece equation).
Cross-check: only the split of 5 women, 1 man, and 14 children satisfies both constraints simultaneously — re-adding the heads (5+1+14 = 20) and the pieces (3+10+7 = 20) independently confirms it, and every other candidate breaks the piece-consumption total while still matching the headcount, which is exactly why checking both equations together is necessary.
