Question

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?​

Answer 1

Answer:

$\huge\underline\bold {Answer:}$

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

$= ( \frac{x - 1}{2}) - ( \frac{x + 1}{4} ) \\ = \frac{2x - 2 - x - 1}{4} \\ = \frac{x - 3}{4}$

Sold to the third customer

$= \frac{1}{2} \times \frac{x - 3}{4} + \frac{1}{2} \\ = \frac{x - 3 + 4}{8} \\ = \frac{x + 1}{8}$

Remaining stock

$= ( \frac{x - 3}{4} ) - ( \frac{x + 1}{8} ) \\ = \frac{2x - 6 - x - 1}{8} = \frac{x - 7}{8}$

Sold to the fourth customer

$= \frac{1}{2} \times \frac{x - 7}{16} + \frac{1}{2} \\ = \frac{x - 7}{16} + \frac{1}{2} \\ = \frac{x - 7 + 8}{16} \\ = \frac{x + 1}{16}$

Therefore,

$x - ( \frac{x + 1}{2} + \frac{x + 1}{4} + \frac{x + 1}{8} + \frac{x + 1}{16} ) \\ = 15$

$= > x - ( \frac{8x + 8 + 4x + 4 + 2x + 2 + x + 1}{16} ) \\ = 15$

$= > x - ( \frac{15x + 15}{16} ) = 15 \\ = > \frac{16 x - 15x - 15}{16} \\ = 15 \\ = >x - 15 = 16 \times 15 = 240 \\ = > x = 240 + 15 = 255.$

Therefore, she had 225 apples before she started selling.

Answer 2

Answer:

Sold to the first customer

\begin{gathered}\frac{x}{2} + \frac{1}{2} = \frac{x + 1}{2} \\\end{gathered}

2

x

+

2

1

=

2

x+1

Remaining stock

\begin{gathered}= x - \frac{x + 1}{2} = \frac{2x - x - 1}{2} \\ = \frac{x - 1}{2}\end{gathered}

=x−

2

x+1

=

2

2x−x−1

=

2

x−1

Sold to the second customer

\begin{gathered}= \frac{1}{2} \times \frac{x - 1}{2} + \frac{1}{2} \\ = \frac{x - 1}{4} + \frac{1}{2}\end{gathered}

=

2

1

×

2

x−1

+

2

1

=

4

x−1

+

2

1

= \frac{x - 1 + 2}{4} = \frac{x + 1}{4}=

4

x−1+2

=

4

x+1

Remaining stock

\begin{gathered}= ( \frac{x - 1}{2}) - ( \frac{x + 1}{4} ) \\ = \frac{2x - 2 - x - 1}{4} \\ = \frac{x - 3}{4}\end{gathered}

=(

2

x−1

)−(

4

x+1

)

=

4

2x−2−x−1

=

4

x−3

Sold to the third customer

\begin{gathered}= \frac{1}{2} \times \frac{x - 3}{4} + \frac{1}{2} \\ = \frac{x - 3 + 4}{8} \\ = \frac{x + 1}{8}\end{gathered}

=

2

1

×

4

x−3

+

2

1

=

8

x−3+4

=

8

x+1

Remaining stock

\begin{gathered}= ( \frac{x - 3}{4} ) - ( \frac{x + 1}{8} ) \\ = \frac{2x - 6 - x - 1}{8} = \frac{x - 7}{8}\end{gathered}

=(

4

x−3

)−(

8

x+1

)

=

8

2x−6−x−1

=

8

x−7

Sold to the fourth customer

\begin{gathered}= \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}\end{gathered}

=

2

1

×

16

x−7

+

2

1

=

16

x−7

+

2

1

=

16

x−7+8

=

16

x+1

Therefore,

\begin{gathered}x - ( \frac{x + 1}{2} + \frac{x + 1}{4} + \frac{x + 1}{8} + \frac{x + 1}{16} ) \\ = 15\end{gathered}

x−(

2

x+1

+

4

x+1

+

8

x+1

+

16

x+1

)

=15

\begin{gathered}= > x - ( \frac{8x + 8 + 4x + 4 + 2x + 2 + x + 1}{16} ) \\ = 15\end{gathered}

=>x−(

16

8x+8+4x+4+2x+2+x+1

)

=15

\begin{gathered}= > x - ( \frac{15x + 15}{16} ) = 15 \\ = > \frac{16 x - 15x - 15}{16} \\ = 15 \\ = > x - 15 = 16 \times 15 = 240 \\ = > x = 240 + 15 = 255.\end{gathered}

=>x−(

16

15x+15

)=15

=>

16

16x−15x−15

=15

=>x−15=16×15=240

=>x=240+15=255.