Two friends went out for pizza and soda after work one day. They each ordered an

Question

Two friends went out for pizza and soda after work one day. They each ordered and drank one large soda and ate part of their pizza. They then decided they were still thirsty, so Molly went up to get another soda. Nancy didn't have any money, so she asked if she could share some of Molly's soda. Molly then poured half of her soda into Nancy's glass. Nancy protested, saying it was too much, and she poured one-third of the soda in her glass back into Molly's glass. Molly insisted that Nancy have some more, so she poured one-fourth of the soda in her glass back into Nancy's glass. Of course, Nancy would not take so much soda, so she poured one-fifth of the soda in her glass back into Molly's glass. This went on for quite a while, with each pouring increasing the denominator of the fraction by 1. After 99 pourings, they decided to stop. At this point, how full was each glass of soda?

Explanation / Answer

Solution:

Let us assume that

M = Molly's glass amount of water and N = Nancy's glass amount of water

Molly begins with 1 full glass, and Nancy with 0.

therefore

M = 1, N = 0

after first pouring

M = 1/2 , N = 1/2

Second pouring:

One third of Nancy's (1/2 * 1/3 = 1/6) is poured back into Molly's glass.
Subtract 1/6 from Nancy, add 1/6 to Molly.

N = 1/2 -1/6 = 1/3, and M = 1/2 + (1/6) = 2/3

third pouring

One quarter of Molly's (2/3 * 1/4 = 1/6) is poured into Nancy's glass.
Subtract 1/6 from Molly, add 1/6 to Nancy.
M = 2/3 - 1/6 = 1/2, and N = 1/3 + (1/6) = 1/2

after forth pouring

One fifth of Nancy's (1/2 * 1/5 = 1/10) is poured into Molly's glass.
Subtract 1/10 from Nancy, add 1/10 to Molly.

M = 1/2 -1/10 = 3/5, and N = 1/2-1/10 = 2/5

after 5th pouring

One sixth of Molly's (3/5 * 1/6 = 1/10) is poured into Nancy's glass.

M = 3/5 - 1/10 = 1/2 and N = 2/5 +1/10 = 1/2

Now from above we can see that

after every odd pouring, they are each at 1/2 again therefore after 99 pourings each glass are half filled ie 1/2 soad in each glass

Answer : 1/2 in each

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