Question

A woman sells to the first customer half her stock of apples and half an apple, to the second customer she sells half her remaining stock and half an apple, and so on to the third and to the fourth customer. She finds that she has now $15$ apples left. How many apples did she have before she started selling?

• A. $300$
• B. $285$
• C. $255$
• D. $225$

Answer 1

Answer: (C)

$255$

Suppose she had $x$ apples in the beginning
Sold to the first customer $=$ $\frac{x}{2}+\frac{1}{2}=\frac{x+1}{2}$
Remaining stock $=$ $x-\frac{x+1}{2}=\frac{2x-x-1}{2}=\frac{x-1}{2}$
Sold to the second customer $=$ $\frac{1}{2}\times \frac{x-1}{2}+\frac{1}{2}=\frac{x-1}{4}+\frac{1}{2}=\frac{x-1+2}{4}=\frac{x+1}{4}$
Remaining stock $=$ $\left(\frac{x-1}{2}\right)-\left(\frac{x-1}{4}\right)=\frac{2x-2-x-1}{4}=\frac{x-3}{4}$
Sold to the third customer $=$ $\frac{1}{2}\times \frac{x-3}{4}+\frac{1}{2}=\frac{x-3}{8}+\frac{1}{2}=\frac{x-3+4}{8}=\frac{x+1}{8}$
Remaining stock $=$ $\left(\frac{x-3}{4}\right)-\left(\frac{x+1}{8}\right)=\frac{2x-6-x-1}{8}=\frac{x-7}{8}$
Sold to the fourth cutomer $=$ $\frac{1}{2}\times \frac{x-7}{8}+\frac{1}{2}=\frac{x-7+8}{16}=\frac{x+1}{16}$
Therefore, $x-$ $[\frac{x+1}{2}+\frac{x+1}{4}+\frac{x+1}{8}+\frac{x+1}{16}]=15$

$\Rightarrow$ $x-$ $\left[\frac{8x+8+4x+4+2x+2+x+1}{16}\right]=15$

$\Rightarrow$ $x-$ $\left[\frac{15x+15}{16}\right]=15$ $\Rightarrow$ $x=240+15=255$
Therefore, she had $255$ apples before she started selling.