A women 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?
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Answer:
[tex]\huge\underline\bold {Answer:} [/tex]
Suppose she had x apples in the beginning.
Sold to the first customer
[tex] \frac{x}{2} + \frac{1}{2} = \frac{x + 1}{2} \\ [/tex]
Remaining stock
[tex] = x - \frac{x + 1}{2} = \frac{2x - x - 1}{2} \\ = \frac{x - 1}{2} [/tex]
Sold to the second customer
[tex] = \frac{1}{2} \times \frac{x - 1}{2} + \frac{1}{2} \\ = \frac{x - 1}{4} + \frac{1}{2} [/tex]
[tex] = \frac{x - 1 + 2}{4} = \frac{x + 1}{4} [/tex]
Remaining stock
[tex] = ( \frac{x - 1}{2}) - ( \frac{x + 1}{4} ) \\ = \frac{2x - 2 - x - 1}{4} \\ = \frac{x - 3}{4} [/tex]
Sold to the third customer
[tex] = \frac{1}{2} \times \frac{x - 3}{4} + \frac{1}{2} \\ = \frac{x - 3 + 4}{8} \\ = \frac{x + 1}{8} [/tex]
Remaining stock
[tex] = ( \frac{x - 3}{4} ) - ( \frac{x + 1}{8} ) \\ = \frac{2x - 6 - x - 1}{8} = \frac{x - 7}{8} [/tex]
Sold to the fourth customer
[tex] = \frac{1}{2} \times \frac{x - 7}{16} + \frac{1}{2} \\ = \frac{x - 7}{16} + \frac{1}{2} \\ = \frac{x - 7 + 8}{16} \\ = \frac{x + 1}{16} [/tex]
Therefore,
[tex]x - ( \frac{x + 1}{2} + \frac{x + 1}{4} + \frac{x + 1}{8} + \frac{x + 1}{16} ) \\ = 15[/tex]
[tex] = > x - ( \frac{8x + 8 + 4x + 4 + 2x + 2 + x + 1}{16} ) \\ = 15[/tex]
[tex] = > x - ( \frac{15x + 15}{16} ) = 15 \\ = > \frac{16 x - 15x - 15}{16} \\ = 15 \\ = >x - 15 = 16 \times 15 = 240 \\ = > x = 240 + 15 = 255.[/tex]
Therefore, she had 225 apples before she started selling.
Answer:
Therefore, she had 225 apples before she started selling.