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

Step-by-step explanation:

In the given question the woman is selling the apples through 2 different processes to different customers.

Do the reverse of what woman is doing & start from 15 apples.

Step 1: Reverse of “Giving away(selling) half an apple and half of her remaining stock” : Double the no. of apples & add half apple.

15 x 2 + .5 = 30.5

30.5 is the no. of apples she had before selling to the 4th customer.

Step 2: Reverse of “Giving away half her stock of apples and half an apple” : Add half apple & double the stock.

(30.5 + .5) x 2 = 62

62 is the no. of apples she had before selling to the 3rd customer.

Step 3: Repeat step 1

62 x 2 + .5 = 124.5

124.5 is the no. of apples she had before selling to the 2nd customer.

Step 4: Repeat step 2

(124.5+.5) x 2 = 250

250 is the no. of apples she had before selling to the 1st customer.

So the answer is 250

Or through options

C: 250

If she begins with 250 apples

Apples Left

250 ------> 124.5 : After giving 125 and then .5 apple

124.5------> 62 : After giving .5 and then 124/2 =62 apples

62 ------> 30.5 : After giving 125 and then .5 apple

30.5--------> 15 : After giving .5 and then 15 apples