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 result:

n =  10

Solution:

$$\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 0.750306389818\doteq 0.06 \\ {r}_9={\displaystyle \frac{9}{100}}\cdot 0.690281878632\doteq 0.0621 \\ {r}_{10}={\displaystyle \frac{10}{100}}\cdot 0.628156509555\doteq 0.0628 \\ {r}_{11}={\displaystyle \frac{11}{100}}\cdot 0.5653408586\doteq 0.0622 \\ {r}_{12}={\displaystyle \frac{12}{100}}\cdot 0.503153364154\doteq 0.0604 \\ {r}_{13}={\displaystyle \frac{13}{100}}\cdot 0.442774960455\doteq 0.0576 \\ {r}_{14}={\displaystyle \frac{14}{100}}\cdot 0.385214215596\doteq 0.0539 \\ {r}_{15}={\displaystyle \frac{15}{100}}\cdot 0.331284225413\doteq 0.0497 \\ {r}_{16}={\displaystyle \frac{16}{100}}\cdot 0.281591591601\doteq 0.0451 \\ {r}_{17}={\displaystyle \frac{17}{100}}\cdot 0.236536936945\doteq 0.0402 \\ {r}_{18}={\displaystyle \frac{18}{100}}\cdot 0.196325657664\doteq 0.0353 \\ {r}_{19}={\displaystyle \frac{19}{100}}\cdot 0.160987039285\doteq 0.0306 \\ {r}_{20}={\displaystyle \frac{20}{100}}\cdot 0.13039950182\doteq 0.0261 \\ {r}_{21}={\displaystyle \frac{21}{100}}\cdot 0.104319601456\doteq 0.0219 \\ {r}_{22}={\displaystyle \frac{22}{100}}\cdot 0.0824124851505\doteq 0.0181 \\ {r}_{23}={\displaystyle \frac{23}{100}}\cdot 0.0642817384174\doteq 0.0148 \\ {r}_{24}={\displaystyle \frac{24}{100}}\cdot 0.0494969385814\doteq 0.0119 \\ {r}_{25}={\displaystyle \frac{25}{100}}\cdot 0.0376176733219\doteq 0.0094 \\ {r}_{26}={\displaystyle \frac{26}{100}}\cdot 0.0282132549914\doteq 0.0073 \\ {r}_{27}={\displaystyle \frac{27}{100}}\cdot 0.0208778086936\doteq 0.0056 \\ {r}_{28}={\displaystyle \frac{28}{100}}\cdot 0.0152408003464\doteq 0.0043 \\ {r}_{29}={\displaystyle \frac{29}{100}}\cdot 0.0109733762494\doteq 0.0032 \\ {r}_{30}={\displaystyle \frac{30}{100}}\cdot 0.00779109713706\doteq 0.0023 \\ {r}_{31}={\displaystyle \frac{31}{100}}\cdot 0.00545376799594\doteq 0.0017 \\ {r}_{32}={\displaystyle \frac{32}{100}}\cdot 0.0037630999172\doteq 0.0012 \\ {r}_{33}={\displaystyle \frac{33}{100}}\cdot 0.0025589079437\doteq 0.0008 \\ {r}_{34}={\displaystyle \frac{34}{100}}\cdot 0.00171446832228\doteq 0.0006 \\ {r}_{35}={\displaystyle \frac{35}{100}}\cdot 0.0011315490927\doteq 0.0004 \\ {r}_{36}={\displaystyle \frac{36}{100}}\cdot 0.000735506910256\doteq 0.0003 \\ {r}_{37}={\displaystyle \frac{37}{100}}\cdot 0.000470724422564\doteq 0.0002 \\ {r}_{38}={\displaystyle \frac{38}{100}}\cdot 0.000296556386215\doteq 0.0001 \\ {r}_{39}={\displaystyle \frac{39}{100}}\cdot 0.000183864959454\doteq 0.0001 \\ {r}_{40}={\displaystyle \frac{40}{100}}\cdot 0.000112157625267\doteq 4.4863\cdot 10^{-5} \\ {r}_{41}={\displaystyle \frac{41}{100}}\cdot 6.729457516\cdot 10^{-5}\doteq 2.7591\cdot 10^{-5} \\ {r}_{42}={\displaystyle \frac{42}{100}}\cdot 3.97037993444\cdot 10^{-5}\doteq 1.6676\cdot 10^{-5} \\ {r}_{43}={\displaystyle \frac{43}{100}}\cdot 2.30282036197\cdot 10^{-5}\doteq 9.9021\cdot 10^{-6} \\ {r}_{44}={\displaystyle \frac{44}{100}}\cdot 1.31260760633\cdot 10^{-5}\doteq 5.7755\cdot 10^{-6} \\ {r}_{45}={\displaystyle \frac{45}{100}}\cdot 7.35060259542\cdot 10^{-6}\doteq 3.3078\cdot 10^{-6} \\ {r}_{46}={\displaystyle \frac{46}{100}}\cdot 4.04283142748\cdot 10^{-6}\doteq 1.8597\cdot 10^{-6} \\ {r}_{47}={\displaystyle \frac{47}{100}}\cdot 2.18312897084\cdot 10^{-6}\doteq 1.0261\cdot 10^{-6} \\ {r}_{48}={\displaystyle \frac{48}{100}}\cdot 1.15705835455\cdot 10^{-6}\doteq 5.5539\cdot 10^{-7} \\ {r}_{49}={\displaystyle \frac{49}{100}}\cdot 6.01670344364\cdot 10^{-7}\doteq 2.9482\cdot 10^{-7} \\ {r}_{50}={\displaystyle \frac{50}{100}}\cdot 3.06851875625\cdot 10^{-7}\doteq 1.5343\cdot 10^{-7} \\ {r}_{51}={\displaystyle \frac{51}{100}}\cdot 1.53425937813\cdot 10^{-7}\doteq 7.8247\cdot 10^{-8} \\ {r}_{52}={\displaystyle \frac{52}{100}}\cdot 7.51787095282\cdot 10^{-8}\doteq 3.9093\cdot 10^{-8} \\ {r}_{53}={\displaystyle \frac{53}{100}}\cdot 3.60857805736\cdot 10^{-8}\doteq 1.9125\cdot 10^{-8} \\ {r}_{54}={\displaystyle \frac{54}{100}}\cdot 1.69603168696\cdot 10^{-8}\doteq 9.1586\cdot 10^{-9} \\ {r}_{55}={\displaystyle \frac{55}{100}}\cdot 7.80174576\cdot 10^{-9}\doteq 4.291\cdot 10^{-9} \\ {r}_{56}={\displaystyle \frac{56}{100}}\cdot 3.510785592\cdot 10^{-9}\doteq 1.966\cdot 10^{-9} \\ {r}_{57}={\displaystyle \frac{57}{100}}\cdot 1.54474566048\cdot 10^{-9}\doteq 8.8051\cdot 10^{-10} \\ {r}_{58}={\displaystyle \frac{58}{100}}\cdot 6.64240634007\cdot 10^{-10}\doteq 3.8526\cdot 10^{-10} \\ {r}_{59}={\displaystyle \frac{59}{100}}\cdot 2.78981066283\cdot 10^{-10}\doteq 1.646\cdot 10^{-10} \\ {r}_{60}={\displaystyle \frac{60}{100}}\cdot 1.14382237176\cdot 10^{-10}\doteq 6.8629\cdot 10^{-11} \\ {r}_{61}={\displaystyle \frac{61}{100}}\cdot 4.57528948704\cdot 10^{-11}\doteq 2.7909\cdot 10^{-11} \\ {r}_{62}={\displaystyle \frac{62}{100}}\cdot 1.78436289994\cdot 10^{-11}\doteq 1.1063\cdot 10^{-11} \\ {r}_{63}={\displaystyle \frac{63}{100}}\cdot 6.78057901979\cdot 10^{-12}\doteq 4.2718\cdot 10^{-12} \\ {r}_{64}={\displaystyle \frac{64}{100}}\cdot 2.50881423732\cdot 10^{-12}\doteq 1.6056\cdot 10^{-12} \\ {r}_{65}={\displaystyle \frac{65}{100}}\cdot 9.03173125436\cdot 10^{-13}\doteq 5.8706\cdot 10^{-13} \\ {r}_{66}={\displaystyle \frac{66}{100}}\cdot 3.16110593903\cdot 10^{-13}\doteq 2.0863\cdot 10^{-13} \\ {r}_{67}={\displaystyle \frac{67}{100}}\cdot 1.07477601927\cdot 10^{-13}\doteq 7.201\cdot 10^{-14} \\ {r}_{68}={\displaystyle \frac{68}{100}}\cdot 3.54676086359\cdot 10^{-14}\doteq 2.4118\cdot 10^{-14} \\ {r}_{69}={\displaystyle \frac{69}{100}}\cdot 1.13496347635\cdot 10^{-14}\doteq 7.8312\cdot 10^{-15} \\ {r}_{70}={\displaystyle \frac{70}{100}}\cdot 3.51838677668\cdot 10^{-15}\doteq 2.4629\cdot 10^{-15} \\ {r}_{71}={\displaystyle \frac{71}{100}}\cdot 1.055516033\cdot 10^{-15}\doteq 7.4942\cdot 10^{-16} \\ {r}_{72}={\displaystyle \frac{72}{100}}\cdot 3.06099649571\cdot 10^{-16}\doteq 2.2039\cdot 10^{-16} \\ {r}_{73}={\displaystyle \frac{73}{100}}\cdot 8.57079018799\cdot 10^{-17}\doteq 6.2567\cdot 10^{-17} \\ {r}_{74}={\displaystyle \frac{74}{100}}\cdot 2.31411335076\cdot 10^{-17}\doteq 1.7124\cdot 10^{-17} \\ {r}_{75}={\displaystyle \frac{75}{100}}\cdot 6.01669471197\cdot 10^{-18}\doteq 4.5125\cdot 10^{-18} \\ {r}_{76}={\displaystyle \frac{76}{100}}\cdot 1.50417367799\cdot 10^{-18}\doteq 1.1432\cdot 10^{-18} \\ {r}_{77}={\displaystyle \frac{77}{100}}\cdot 3.61001682718\cdot 10^{-19}\doteq 2.7797\cdot 10^{-19} \\ {r}_{78}={\displaystyle \frac{78}{100}}\cdot 8.30303870252\cdot 10^{-20}\doteq 6.4764\cdot 10^{-20} \\ {r}_{79}={\displaystyle \frac{79}{100}}\cdot 1.82666851455\cdot 10^{-20}\doteq 1.4431\cdot 10^{-20} \\ {r}_{80}={\displaystyle \frac{80}{100}}\cdot 3.83600388056\cdot 10^{-21}\doteq 3.0688\cdot 10^{-21} \\ {r}_{81}={\displaystyle \frac{81}{100}}\cdot 7.67200776113\cdot 10^{-22}\doteq 6.2143\cdot 10^{-22} \\ {r}_{82}={\displaystyle \frac{82}{100}}\cdot 1.45768147461\cdot 10^{-22}\doteq 1.1953\cdot 10^{-22} \\ {r}_{83}={\displaystyle \frac{83}{100}}\cdot 2.6238266543\cdot 10^{-23}\doteq 2.1778\cdot 10^{-23} \\ {r}_{84}={\displaystyle \frac{84}{100}}\cdot 4.46050531232\cdot 10^{-24}\doteq 3.7468\cdot 10^{-24} \\ {r}_{85}={\displaystyle \frac{85}{100}}\cdot 7.13680849971\cdot 10^{-25}\doteq 6.0663\cdot 10^{-25} \\ {r}_{86}={\displaystyle \frac{86}{100}}\cdot 1.07052127496\cdot 10^{-25}\doteq 9.2065\cdot 10^{-26} \\ {r}_{87}={\displaystyle \frac{87}{100}}\cdot 1.49872978494\cdot 10^{-26}\doteq 1.3039\cdot 10^{-26} \\ {r}_{88}={\displaystyle \frac{88}{100}}\cdot 1.94834872042\cdot 10^{-27}\doteq 1.7145\cdot 10^{-27} \\ {r}_{89}={\displaystyle \frac{89}{100}}\cdot 2.3380184645\cdot 10^{-28}\doteq 2.0808\cdot 10^{-28} \\ {r}_{90}={\displaystyle \frac{90}{100}}\cdot 2.57182031096\cdot 10^{-29}\doteq 2.3146\cdot 10^{-29} \\ {r}_{91}={\displaystyle \frac{91}{100}}\cdot 2.57182031096\cdot 10^{-30}\doteq 2.3404\cdot 10^{-30} \\ {r}_{92}={\displaystyle \frac{92}{100}}\cdot 2.31463827986\cdot 10^{-31}\doteq 2.1295\cdot 10^{-31} \\ {r}_{93}={\displaystyle \frac{93}{100}}\cdot 1.85171062389\cdot 10^{-32}\doteq 1.7221\cdot 10^{-32} \\ {r}_{94}={\displaystyle \frac{94}{100}}\cdot 1.29619743672\cdot 10^{-33}\doteq 1.2184\cdot 10^{-33} \\ {r}_{95}={\displaystyle \frac{95}{100}}\cdot 7.77718462033\cdot 10^{-35}\doteq 7.3883\cdot 10^{-35} \\ {r}_{96}={\displaystyle \frac{96}{100}}\cdot 3.88859231016\cdot 10^{-36}\doteq 3.733\cdot 10^{-36} \\ {r}_{97}={\displaystyle \frac{97}{100}}\cdot 1.55543692407\cdot 10^{-37}\doteq 1.5088\cdot 10^{-37} \\ {r}_{98}={\displaystyle \frac{98}{100}}\cdot 4.6663107722\cdot 10^{-39}\doteq 4.573\cdot 10^{-39} \\ {r}_{99}={\displaystyle \frac{99}{100}}\cdot 9.33262154439\cdot 10^{-41}\doteq 9.2393\cdot 10^{-41} \\ {r}_{100}={\displaystyle \frac{100}{100}}\cdot 9.33262154439\cdot 10^{-43}\doteq 9.3326\cdot 10^{-43} \\ {r}_{10}>\mathrm{\forall }r \\ {r}_{10}\dots max \\ n=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.

today  Like

Related math problems and questions:

• Three shapes
  1/5 of a circle is shaded. The ratio of area if square to the sum of area of rectangle and that of the circle is 1:2. 60% of the square is shaded and 1/3 of the rectangle is shaded. What is the ratio of the area of circle to that of the rectangle?
• Average mark
  Calculate average mark in English, if a student got a couple of 3's, 20% less 2 than 3, and 50% more 1's than the 2's.
• Three figures - numbers
  The sum of three numbers, if each is 10% larger than the previous one, is 662. Determine the figures.
• Percent calculation
  Calculate 51% if 99% is 79.
• Crowns
  1180 crowns is divided into three people that A got 20% less than B and C by 15% more than B. How many got A,B and C?
• Final exam
  At the final exam, the student answers from three areas, which are evaluated in a ratio of 1: 2: 2. What grade will John receive if he answered as follows: 3,1,2.
• Acid solution
  By adding 250 grams of a 96% solution of sulfuric acid to its 3% solution its initial concentration was changed to 25%. How many grams of 3% of the acid were used for dilution?
• Two doctors
  Doctor A will determine the correct diagnosis with a probability of 93% and doctor B with a probability of 79%. Calculate the probability of proper diagnosis if both doctors diagnose the patient.
• Geography tests
  On three 150-point geography tests, you earned grades of 88%, 94%, and 90%. The final test is worth 250 points. What percent do you need on the final to earn 93% of the total points on all tests?
• Percentages above 100%
  What is 122% of 185? What is the meaning of percentages above 100%?
• Water in vegetables
  Tomatoes in the store contain 99% water. After being transported to the shop, they were slightly dried and contained only 98% of water. How many kgs of tomatoes are in the store if there were 300 kg in stock?`
• Disinfecting solution
  How much distilled water is necessary to pour into 500 ml of 33% hydrogen peroxide solution to obtain 3% disinfecting solution?
• 1200 screws
  1200 screws are to be divided into 3 groups so that in the 1st group there are 300 screws more than in the 2nd group and in the 2nd group 150 screws less than in the 3rd group. How many screws will be in each group?
• Mixture 2
  How many liters of water must be added to 7 liters of a 20% solution to obtain a 10% solution?
• Unknown number 23
  Find 2/3 of unknown number, which is two thirds of the 99.
• Associative law multiplication
  In a warehouse, you obtain a 20% discount but you must pay a 15% sales tax. Which would you prefer to have calculated first: discount or tax? Explain. (result write as: 1 = first discount, 2 = first tax, 3 = no matter what first)
• Grams
  How many grams of 5% solution do we need to add to 100g of 50% solution to get a 20% solution?