In a room, there are some chairs and some people. If on each chair, only one person is seated, then there is no chair for exactly one person. If on each chair, two persons sit, then there is one vacant chair. What is the number of chairs in the room?

Correct — D. Let there be c chairs and p people.

One person per chair leaves exactly one person standing: p = c + 1
Two people per chair leaves one chair empty, so the p people fill c − 1 chairs: p = 2(c − 1)

Equate the two: c + 1 = 2c − 2
c = 3

Check with 3 chairs and 4 people. One each seats 3 and leaves 1 standing, two each fills 2 chairs and leaves 1 chair vacant → option (d).

Why the others are wrong

• (a)
  6 — 6 chairs would mean 7 people. Seated two to a chair they cannot fill 5 chairs exactly, and 2(6 − 1) = 10 people would be needed to leave just one vacant.
• (b)
  7 — 7 chairs would mean 8 people, and 8 people two to a chair occupy 4 chairs, leaving 3 vacant rather than one.
• (c)
  4 — 4 chairs is the near miss: 5 people cannot sit two to a chair, since 5 is odd. The condition needs p = 2(4 − 1) = 6 people, but p = c + 1 gives 5.

Two seating conditions give two equations in the same pair of unknowns, and the whole difficulty is in the translation.

No chair for exactly one person means the people outnumber the chairs by one, p = c + 1.

One vacant chair with two to a chair means only c − 1 chairs are in use and each of them holds two, so p = 2(c − 1). Solving the pair gives 3 chairs and 4 people.

Key facts

• No chair for exactly one person translates to p = c + 1.
• One vacant chair when two sit per chair translates to p = 2(c − 1), because only c − 1 chairs are used.
• The solution is 3 chairs and 4 people, and both conditions have to be checked back against those numbers.

Common traps

• Writing the second condition as p = 2c − 1 instead of p = 2(c − 1).
• Answering 4, the number of people, when the question asks for chairs.
• Reading one vacant chair as one empty seat on an otherwise occupied chair.

No directly related past PYQ was found.