Megapizza

Megapizza will be divided among 100 people. First gets 1%, 2nd 2% of the remainder, 3rd 3% of the remainder, etc. Last 100th 100% of the remainder. Which person got the biggest portion?

Correct answer:

n =  10

Step-by-step explanation:

$$\begin{gathered} r_1={\displaystyle \frac{1}{100}}\cdot 1=0.01 \\ {r}_2={\displaystyle \frac{2}{100}}\cdot 0.99={\displaystyle \frac{99}{5000}}=0.0198 \\ {r}_3={\displaystyle \frac{3}{100}}\cdot 0.9702\doteq 0.0291 \\ {r}_4={\displaystyle \frac{4}{100}}\cdot 0.941094\doteq 0.0376 \\ {r}_5={\displaystyle \frac{5}{100}}\cdot 0.90345024\doteq 0.0452 \\ {r}_6={\displaystyle \frac{6}{100}}\cdot 0.858277728\doteq 0.0515 \\ {r}_7={\displaystyle \frac{7}{100}}\cdot 0.80678106432\doteq 0.0565 \\ {r}_8={\displaystyle \frac{8}{100}}\cdot 0.7503063898176\doteq 0.06 \\ {r}_9={\displaystyle \frac{9}{100}}\cdot 0.69028187863219\doteq 0.0621 \\ {r}_{10}={\displaystyle \frac{10}{100}}\cdot 0.62815650955529\doteq 0.0628 \\ {r}_{11}={\displaystyle \frac{11}{100}}\cdot 0.56534085859977\doteq 0.0622 \\ {r}_{12}={\displaystyle \frac{12}{100}}\cdot 0.50315336415379\doteq 0.0604 \\ {r}_{13}={\displaystyle \frac{13}{100}}\cdot 0.44277496045534\doteq 0.0576 \\ {r}_{14}={\displaystyle \frac{14}{100}}\cdot 0.38521421559614\doteq 0.0539 \\ {r}_{15}={\displaystyle \frac{15}{100}}\cdot 0.33128422541268\doteq 0.0497 \\ {r}_{16}={\displaystyle \frac{16}{100}}\cdot 0.28159159160078\doteq 0.0451 \\ {r}_{17}={\displaystyle \frac{17}{100}}\cdot 0.23653693694466\doteq 0.0402 \\ {r}_{18}={\displaystyle \frac{18}{100}}\cdot 0.19632565766406\doteq 0.0353 \\ {r}_{19}={\displaystyle \frac{19}{100}}\cdot 0.16098703928453\doteq 0.0306 \\ {r}_{20}={\displaystyle \frac{20}{100}}\cdot 0.13039950182047\doteq 0.0261 \\ {r}_{21}={\displaystyle \frac{21}{100}}\cdot 0.10431960145638\doteq 0.0219 \\ {r}_{22}={\displaystyle \frac{22}{100}}\cdot 0.082412485150538\doteq 0.0181 \\ {r}_{23}={\displaystyle \frac{23}{100}}\cdot 0.06428173841742\doteq 0.0148 \\ {r}_{24}={\displaystyle \frac{24}{100}}\cdot 0.049496938581413\doteq 0.0119 \\ {r}_{25}={\displaystyle \frac{25}{100}}\cdot 0.037617673321874\doteq 0.0094 \\ {r}_{26}={\displaystyle \frac{26}{100}}\cdot 0.028213254991405\doteq 0.0073 \\ {r}_{27}={\displaystyle \frac{27}{100}}\cdot 0.02087780869364\doteq 0.0056 \\ {r}_{28}={\displaystyle \frac{28}{100}}\cdot 0.015240800346357\doteq 0.0043 \\ {r}_{29}={\displaystyle \frac{29}{100}}\cdot 0.010973376249377\doteq 0.0032 \\ {r}_{30}={\displaystyle \frac{30}{100}}\cdot 0.0077910971370578\doteq 0.0023 \\ {r}_{31}={\displaystyle \frac{31}{100}}\cdot 0.0054537679959405\doteq 0.0017 \\ {r}_{32}={\displaystyle \frac{32}{100}}\cdot 0.0037630999171989\doteq 0.0012 \\ {r}_{33}={\displaystyle \frac{33}{100}}\cdot 0.0025589079436953\doteq 0.0008 \\ {r}_{34}={\displaystyle \frac{34}{100}}\cdot 0.0017144683222758\doteq 0.0006 \\ {r}_{35}={\displaystyle \frac{35}{100}}\cdot 0.001131549092702\doteq 0.0004 \\ {r}_{36}={\displaystyle \frac{36}{100}}\cdot 0.00073550691025633\doteq 0.0003 \\ {r}_{37}={\displaystyle \frac{37}{100}}\cdot 0.00047072442256405\doteq 0.0002 \\ {r}_{38}={\displaystyle \frac{38}{100}}\cdot 0.00029655638621535\doteq 0.0001 \\ {r}_{39}={\displaystyle \frac{39}{100}}\cdot 0.00018386495945352\doteq 0.0001 \\ {r}_{40}={\displaystyle \frac{40}{100}}\cdot 0.00011215762526665\doteq 4.4863\cdot 10^{-5} \\ {r}_{41}={\displaystyle \frac{41}{100}}\cdot 6.7294575159988\cdot 10^{-5}\doteq 2.7591\cdot 10^{-5} \\ {r}_{42}={\displaystyle \frac{42}{100}}\cdot 3.9703799344393\cdot 10^{-5}\doteq 1.6676\cdot 10^{-5} \\ {r}_{43}={\displaystyle \frac{43}{100}}\cdot 2.3028203619748\cdot 10^{-5}\doteq 9.9021\cdot 10^{-6} \\ {r}_{44}={\displaystyle \frac{44}{100}}\cdot 1.3126076063256\cdot 10^{-5}\doteq 5.7755\cdot 10^{-6} \\ {r}_{45}={\displaystyle \frac{45}{100}}\cdot 7.3506025954235\cdot 10^{-6}\doteq 3.3078\cdot 10^{-6} \\ {r}_{46}={\displaystyle \frac{46}{100}}\cdot 4.0428314274829\cdot 10^{-6}\doteq 1.8597\cdot 10^{-6} \\ {r}_{47}={\displaystyle \frac{47}{100}}\cdot 2.1831289708408\cdot 10^{-6}\doteq 1.0261\cdot 10^{-6} \\ {r}_{48}={\displaystyle \frac{48}{100}}\cdot 1.1570583545456\cdot 10^{-6}\doteq 5.5539\cdot 10^{-7} \\ {r}_{49}={\displaystyle \frac{49}{100}}\cdot 6.0167034436372\cdot 10^{-7}\doteq 2.9482\cdot 10^{-7} \\ {r}_{50}={\displaystyle \frac{50}{100}}\cdot 3.068518756255\cdot 10^{-7}\doteq 1.5343\cdot 10^{-7} \\ {r}_{51}={\displaystyle \frac{51}{100}}\cdot 1.5342593781275\cdot 10^{-7}\doteq 7.8247\cdot 10^{-8} \\ {r}_{52}={\displaystyle \frac{52}{100}}\cdot 7.5178709528247\cdot 10^{-8}\doteq 3.9093\cdot 10^{-8} \\ {r}_{53}={\displaystyle \frac{53}{100}}\cdot 3.6085780573558\cdot 10^{-8}\doteq 1.9125\cdot 10^{-8} \\ {r}_{54}={\displaystyle \frac{54}{100}}\cdot 1.6960316869572\cdot 10^{-8}\doteq 9.1586\cdot 10^{-9} \\ {r}_{55}={\displaystyle \frac{55}{100}}\cdot 7.8017457600033\cdot 10^{-9}\doteq 4.291\cdot 10^{-9} \\ {r}_{56}={\displaystyle \frac{56}{100}}\cdot 3.5107855920015\cdot 10^{-9}\doteq 1.966\cdot 10^{-9} \\ {r}_{57}={\displaystyle \frac{57}{100}}\cdot 1.5447456604807\cdot 10^{-9}\doteq 8.8051\cdot 10^{-10} \\ {r}_{58}={\displaystyle \frac{58}{100}}\cdot 6.6424063400668\cdot 10^{-10}\doteq 3.8526\cdot 10^{-10} \\ {r}_{59}={\displaystyle \frac{59}{100}}\cdot 2.7898106628281\cdot 10^{-10}\doteq 1.646\cdot 10^{-10} \\ {r}_{60}={\displaystyle \frac{60}{100}}\cdot 1.1438223717595\cdot 10^{-10}\doteq 6.8629\cdot 10^{-11} \\ {r}_{61}={\displaystyle \frac{61}{100}}\cdot 4.575289487038\cdot 10^{-11}\doteq 2.7909\cdot 10^{-11} \\ {r}_{62}={\displaystyle \frac{62}{100}}\cdot 1.7843628999448\cdot 10^{-11}\doteq 1.1063\cdot 10^{-11} \\ {r}_{63}={\displaystyle \frac{63}{100}}\cdot 6.7805790197904\cdot 10^{-12}\doteq 4.2718\cdot 10^{-12} \\ {r}_{64}={\displaystyle \frac{64}{100}}\cdot 2.5088142373224\cdot 10^{-12}\doteq 1.6056\cdot 10^{-12} \\ {r}_{65}={\displaystyle \frac{65}{100}}\cdot 9.0317312543608\cdot 10^{-13}\doteq 5.8706\cdot 10^{-13} \\ {r}_{66}={\displaystyle \frac{66}{100}}\cdot 3.1611059390263\cdot 10^{-13}\doteq 2.0863\cdot 10^{-13} \\ {r}_{67}={\displaystyle \frac{67}{100}}\cdot 1.0747760192689\cdot 10^{-13}\doteq 7.201\cdot 10^{-14} \\ {r}_{68}={\displaystyle \frac{68}{100}}\cdot 3.5467608635875\cdot 10^{-14}\doteq 2.4118\cdot 10^{-14} \\ {r}_{69}={\displaystyle \frac{69}{100}}\cdot 1.134963476348\cdot 10^{-14}\doteq 7.8312\cdot 10^{-15} \\ {r}_{70}={\displaystyle \frac{70}{100}}\cdot 3.5183867766788\cdot 10^{-15}\doteq 2.4629\cdot 10^{-15} \\ {r}_{71}={\displaystyle \frac{71}{100}}\cdot 1.0555160330036\cdot 10^{-15}\doteq 7.4942\cdot 10^{-16} \\ {r}_{72}={\displaystyle \frac{72}{100}}\cdot 3.0609964957105\cdot 10^{-16}\doteq 2.2039\cdot 10^{-16} \\ {r}_{73}={\displaystyle \frac{73}{100}}\cdot 8.5707901879895\cdot 10^{-17}\doteq 6.2567\cdot 10^{-17} \\ {r}_{74}={\displaystyle \frac{74}{100}}\cdot 2.3141133507572\cdot 10^{-17}\doteq 1.7124\cdot 10^{-17} \\ {r}_{75}={\displaystyle \frac{75}{100}}\cdot 6.0166947119686\cdot 10^{-18}\doteq 4.5125\cdot 10^{-18} \\ {r}_{76}={\displaystyle \frac{76}{100}}\cdot 1.5041736779922\cdot 10^{-18}\doteq 1.1432\cdot 10^{-18} \\ {r}_{77}={\displaystyle \frac{77}{100}}\cdot 3.6100168271812\cdot 10^{-19}\doteq 2.7797\cdot 10^{-19} \\ {r}_{78}={\displaystyle \frac{78}{100}}\cdot 8.3030387025167\cdot 10^{-20}\doteq 6.4764\cdot 10^{-20} \\ {r}_{79}={\displaystyle \frac{79}{100}}\cdot 1.8266685145537\cdot 10^{-20}\doteq 1.4431\cdot 10^{-20} \\ {r}_{80}={\displaystyle \frac{80}{100}}\cdot 3.8360038805627\cdot 10^{-21}\doteq 3.0688\cdot 10^{-21} \\ {r}_{81}={\displaystyle \frac{81}{100}}\cdot 7.6720077611254\cdot 10^{-22}\doteq 6.2143\cdot 10^{-22} \\ {r}_{82}={\displaystyle \frac{82}{100}}\cdot 1.4576814746138\cdot 10^{-22}\doteq 1.1953\cdot 10^{-22} \\ {r}_{83}={\displaystyle \frac{83}{100}}\cdot 2.6238266543049\cdot 10^{-23}\doteq 2.1778\cdot 10^{-23} \\ {r}_{84}={\displaystyle \frac{84}{100}}\cdot 4.4605053123183\cdot 10^{-24}\doteq 3.7468\cdot 10^{-24} \\ {r}_{85}={\displaystyle \frac{85}{100}}\cdot 7.1368084997093\cdot 10^{-25}\doteq 6.0663\cdot 10^{-25} \\ {r}_{86}={\displaystyle \frac{86}{100}}\cdot 1.0705212749564\cdot 10^{-25}\doteq 9.2065\cdot 10^{-26} \\ {r}_{87}={\displaystyle \frac{87}{100}}\cdot 1.498729784939\cdot 10^{-26}\doteq 1.3039\cdot 10^{-26} \\ {r}_{88}={\displaystyle \frac{88}{100}}\cdot 1.9483487204206\cdot 10^{-27}\doteq 1.7145\cdot 10^{-27} \\ {r}_{89}={\displaystyle \frac{89}{100}}\cdot 2.3380184645048\cdot 10^{-28}\doteq 2.0808\cdot 10^{-28} \\ {r}_{90}={\displaystyle \frac{90}{100}}\cdot 2.5718203109553\cdot 10^{-29}\doteq 2.3146\cdot 10^{-29} \\ {r}_{91}={\displaystyle \frac{91}{100}}\cdot 2.5718203109553\cdot 10^{-30}\doteq 2.3404\cdot 10^{-30} \\ {r}_{92}={\displaystyle \frac{92}{100}}\cdot 2.3146382798597\cdot 10^{-31}\doteq 2.1295\cdot 10^{-31} \\ {r}_{93}={\displaystyle \frac{93}{100}}\cdot 1.8517106238878\cdot 10^{-32}\doteq 1.7221\cdot 10^{-32} \\ {r}_{94}={\displaystyle \frac{94}{100}}\cdot 1.2961974367214\cdot 10^{-33}\doteq 1.2184\cdot 10^{-33} \\ {r}_{95}={\displaystyle \frac{95}{100}}\cdot 7.7771846203287\cdot 10^{-35}\doteq 7.3883\cdot 10^{-35} \\ {r}_{96}={\displaystyle \frac{96}{100}}\cdot 3.8885923101643\cdot 10^{-36}\doteq 3.733\cdot 10^{-36} \\ {r}_{97}={\displaystyle \frac{97}{100}}\cdot 1.5554369240657\cdot 10^{-37}\doteq 1.5088\cdot 10^{-37} \\ {r}_{98}={\displaystyle \frac{98}{100}}\cdot 4.6663107721972\cdot 10^{-39}\doteq 4.573\cdot 10^{-39} \\ {r}_{99}={\displaystyle \frac{99}{100}}\cdot 9.3326215443944\cdot 10^{-41}\doteq 9.2393\cdot 10^{-41} \\ {r}_{100}={\displaystyle \frac{100}{100}}\cdot 9.3326215443945\cdot 10^{-43}\doteq 9.3326\cdot 10^{-43} \\ {r}_{10}>\mathrm{\forall }r \\ {r}_{10}\dots max \\ n=10=10 \end{gathered}$$

Try another example

Showing 1 comment:

Dr Math

Perhaps a more elegant solution: The Kth person gets 99! / (99 - (k-1)! * K / 100exp (k). K + 1. The person gets 99! / (99 - k)! * (K + 1) / 100exp (k + 1) The ratio m (k + 1) / m (k) must be greater than one, ie: the next person in the order will receive a larger portion than the previous one.Applies to k = 1, 2, … 9. The 10th person gets the largest portion.

6 months ago  Like

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