Initial stock of toffees

CAT 2020 Slot 1 · QA · Hard · Linear Equations

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A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

(TITA)

Answer

62

Explanation

Work backwards from the 5th child:

Let SkS_k be the stock before treating the kk-th child. When stock SS is given, remaining stock is S2−1\frac{S}{2} - 1. Hence, Sprev=2(Scurr+1)S_{\text{prev}} = 2(S_{\text{curr}} + 1).

  • After 5th child, stock =0  ⟹  S5=2(0+1)=2= 0 \implies S_5 = 2(0 + 1) = 2.
  • Before 4th child: S4=2(2+1)=6S_4 = 2(2 + 1) = 6.
  • Before 3rd child: S3=2(6+1)=14S_3 = 2(6 + 1) = 14.
  • Before 2nd child: S2=2(14+1)=30S_2 = 2(14 + 1) = 30.
  • Before 1st child: S1=2(30+1)=62S_1 = 2(30 + 1) = 62.

Initial stock of toffees was 62.

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