Question-58663

Question Number 58663 by mr W last updated on 27/Apr/19

Answered by ajfour last updated on 27/Apr/19

{100 T}_{n + 1} = (n + 1) (T_{n} + T_{n - 1} + \dots + T_{1})

and T_{1} + T_{2} + \dots . + T_{n} + \dots + T_{N} = 1

\dots .

Answered by MJS last updated on 27/Apr/19

guest 10 gets \frac{245 373 636 545 037}{39 062 500 000 000} \approx 6 . 28157 %

interestingly this is 1.99948 π

guest 25 is the last one to get at least 1 %

if the pie is a big one for 100 persons, say

it weighs 25 kg then guest 40 is the last one

to get at least 1 g

the first 11 guests get almost 50 %

the guests 29 - 100 have to share \approx 1 %

Commented by peter frank last updated on 30/Apr/19

h o w d i d y o u g e t t h o s e

l a r g e n u m b e r ? ? ?

Commented by MJS last updated on 01/May/19

by using + - \times /

Commented by peter frank last updated on 01/May/19

s o r r y s i r e x p l a i n f i r s t

l i n e

Commented by MJS last updated on 01/May/19

pie 100 %; {guest}_{1} 1 %

pie = 100 % - 1 % = 99 %; {guest}_{2} 2 % of 99 % = \frac{99}{50} % = 1 . 98 %

pie = 99 % - \frac{99}{50} % = \frac{4 851}{50} %; {guest}_{3} 3 % of \frac{4 851}{50} % = \frac{14 553}{5 000} % \approx 2 . 91 %

p = \frac{470 547}{5 000} %; g_{4} = \frac{470 547}{125 000} % \approx 3 . 76 %

p = \frac{1 411 641}{15 625} %; g_{5} = \frac{1 411 641}{312 500} % \approx 4 . 52 %

p = \frac{26 821 179}{312 500} %; g_{6} = \frac{80 463 537}{15 625 000} % \approx 5 . 15 %

p = \frac{1 260 595 413}{15 625 000} %; g_{7} = \frac{8 824 167 891}{1 562 500 000} % \approx 5 . 65 %

p = \frac{117 235 373 409}{1 562 500 000} %; g_{8} = \frac{117 235 373 409}{19 531 250 000} % \approx 6 . 00 %

p = \frac{2 696 413 588 407}{39 062 500 000} %; g_{9} = \frac{24 267 722 295 663}{3 906 250 000 000} % \approx 6 . 21 %

p = \frac{245 373 636 545 037}{3 906 250 000 000} %; g_{10} = \frac{245 373 636 545 037}{39 062 500 000 000} % \approx 6 . 28 %

p = \frac{2 208 362 728 905 333}{39 062 500 000 000} %; g_{11} = \frac{24 291 990 017 958 663}{3 906 250 000 000 000} % \approx 6 . 22 %

Commented by mr W last updated on 06/May/19

t h a n k s f o r t h e s o l u t i o n s i r! i h a v e t r i e d

t o u s e t h e s a m e m e t h o d t o g e t a

g e n e r a l f o r m .

Answered by mr W last updated on 07/May/19

g u e s t 1 g e t s \frac{1}{100},

r e m a i n i n g p i e : (1 - \frac{1}{100}) \times 1 = \frac{99}{100}

g u e s t 2 g e t s \frac{2}{100} \times \frac{99}{100},

r e m a i n i n g p i e : (1 - \frac{2}{100}) \times \frac{99}{100} = \frac{98}{100} \times \frac{99}{100}

g u e s t 3 g e t s \frac{3}{100} \times \frac{98}{100} \times \frac{99}{100},

r e m a i n i n g p i e : (1 - \frac{3}{100}) \times \frac{98}{100} \times \frac{99}{100} = \frac{97}{100} \times \frac{98}{100} \times \frac{99}{100}

\dots

g u e s t n g e t s \frac{n}{100} \times \frac{99 - n + 2}{100} \times \dots \times \frac{97}{100} \times \frac{98}{100} \times \frac{99}{100}

r e m a i n i n g p i e : \frac{100 - n}{100} \times \frac{99 - n + 2}{100} \times \dots \times \frac{97}{100} \times \frac{98}{100} \times \frac{99}{100}

\dots

T_{n} = \frac{n}{100} \times \frac{99 - n + 2}{100} \times \dots \times \frac{97}{100} \times \frac{98}{100} \times \frac{99}{100}

T_{n} = \frac{n (99 - n + 2) \dots 97 \times 98 \times 99}{100^{n}}

T_{n} = \frac{n (99 - n + 1)! (99 - n + 2) \dots 97 \times 98 \times 99 \times 100}{100^{n + 1} (99 - n + 1)!}

\Rightarrow T_{n} = \frac{100! n}{100^{n + 1} (100 - n)!}

T_{n} i s m a x i m u m w h e n T_{n - 1} < T_{n} > T_{n + 1}, i . e .

\frac{100! n}{100^{n + 1} (100 - n)!} > \frac{100! (n + 1)}{100^{n + 2} (100 - n - 1)!}

\frac{n}{100 - n} > \frac{n + 1}{100}

100 n > (n + 1) (100 - n)

100 n > 100 n + 100 - n^{2} - n

0 > 100 - n^{2} - n

n^{2} + n - 100 > 0

\Rightarrow n > \frac{- 1 + \sqrt{1 + 400}}{2} \approx 9.5

i . e . t h e 10 t h g u e s t g e t s t h e l a r g e s t p i e c e .

T_{10} = \frac{100! 10}{100^{11} 90!} = \frac{99 \times 97 \times \dots \times 92 \times 91}{10^{19}}

= \frac{62815650955529472}{10000000000000000000}

\approx 6 . 28 %

Commented by MJS last updated on 06/May/19

great!

Commented by peter frank last updated on 19/May/19

t h a n k y o u