There are some nectar-filled flowers on a tree and some bees are hovering on it. If one bee lands on each flower, one bee will be left out. If two bees land on each flower, one flower will be left out. The number of flowers and bees respectively are :

(a) 2 and 4
(b) 3 and 2
(c) 3 and 4 ✓ UPSC's answer
(d) 4 and 3

Why the answer is (c)

• Let the number of flowers be $f$ and the number of bees be $b$.

• The first condition states that if one bee lands on each flower, one bee is left out, which gives the equation $b = f + 1$.

• The second condition states that if two bees land on each flower, one flower is left out, meaning the bees are sufficient to cover $f - 1$ flowers with 2 bees each, so $b = 2(f - 1)$.

• Substituting the first equation into the second gives $f + 1 = 2f - 2$.

• Solving for $f$ yields $f = 3$.

• Substituting $f = 3$ back into $b = f + 1$ gives $b = 4$, so the number of flowers and bees are 3 and 4 respectively.

Why the other options are wrong

(a) 2 and 4: Option (a) implies 2 flowers and 4 bees, which violates the second condition as 4 bees would exactly fill 2 flowers with 2 bees each, leaving no flower out.
(b) 3 and 2: Option (b) implies 3 flowers and 2 bees, which violates the first condition as 2 bees cannot cover 3 flowers with one bee each, leaving a flower empty rather than a bee out.
(d) 4 and 3: Option (d) implies 4 flowers and 3 bees, which violates the first condition as 3 bees cannot cover 4 flowers with one bee each, leaving a flower empty rather than a bee out.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.