[{"data":1,"prerenderedAt":2090},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fgrundwert":546,"formula-related-de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fgrundwert":2082},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":65,"checks":2027,"conditions":2047,"conditionsHtml":2048,"deprecated":2049,"description":2021,"extension":2050,"figure":2049,"figureCaption":2049,"figureHtml":2049,"formulaId":2051,"latex":2052,"latexHtml":2053,"level":2054,"meta":2055,"navigation":2056,"path":2057,"redirectFrom":2058,"related":2059,"seo":2063,"stem":2064,"subject":581,"summary":2065,"tags":2066,"variables":2071,"__hash__":2081},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Farithmetic\u002Fgrundwert.md","Grundwert",[550,551,552,553,554],"Grundwert berechnen","100 Prozent berechnen","das Ganze berechnen","G = W durch p mal 100","Ausgangswert berechnen",{"type":556,"value":557,"toc":2020},"minimark",[558,563,643,994,997,1001,1074,1239,1328,1496,1634,1638],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,570,571,634,635,638,639,642],"p",{},"Der ",[568,569,548],"strong",{}," ist das Ganze, also ",[572,573,576,608],"span",{"className":574},[575],"katex",[572,577,580],{"className":578},[579],"katex-mathml",[581,582,584],"math",{"xmlns":583},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[585,586,587,603],"semantics",{},[588,589,590,594,598],"mrow",{},[591,592,593],"mn",{},"100",[595,596,597],"mtext",{}," ",[599,600,602],"mi",{"mathvariant":601},"normal","%",[604,605,607],"annotation",{"encoding":606},"application\u002Fx-tex","100\\,\\%",[572,609,613],{"className":610,"ariaHidden":612},[611],"katex-html","true",[572,614,617,622,626,631],{"className":615},[616],"base",[572,618],{"className":619,"style":621},[620],"strut","height:0.8056em;vertical-align:-0.0556em;",[572,623,593],{"className":624},[625],"mord",[572,627],{"className":628,"style":630},[629],"mspace","margin-right:0.1667em;",[572,632,602],{"className":633},[625],". Man sucht ihn, wenn ein Anteil (der ",[568,636,637],{},"Prozentwert",") und sein ",[568,640,641],{},"Prozentsatz"," bekannt sind.",[564,644,645,646,824,825,853,854,894,895,924,925,963,964,993],{},"Die Formel entsteht aus ",[572,647,649,681],{"className":648},[575],[572,650,652],{"className":651},[579],[581,653,654],{"xmlns":583},[585,655,656,678],{},[588,657,658,661,665,668,671],{},[599,659,660],{},"W",[662,663,664],"mo",{},"=",[599,666,667],{},"G",[662,669,670],{},"⋅",[672,673,674,676],"mfrac",{},[599,675,564],{},[591,677,593],{},[604,679,680],{"encoding":606},"W = G \\cdot \\frac{p}{100}",[572,682,684,707,727],{"className":683,"ariaHidden":612},[611],[572,685,687,691,696,700,704],{"className":686},[616],[572,688],{"className":689,"style":690},[620],"height:0.6833em;",[572,692,660],{"className":693,"style":695},[625,694],"mathnormal","margin-right:0.1389em;",[572,697],{"className":698,"style":699},[629],"margin-right:0.2778em;",[572,701,664],{"className":702},[703],"mrel",[572,705],{"className":706,"style":699},[629],[572,708,710,713,716,720,724],{"className":709},[616],[572,711],{"className":712,"style":690},[620],[572,714,667],{"className":715},[625,694],[572,717],{"className":718,"style":719},[629],"margin-right:0.2222em;",[572,721,670],{"className":722},[723],"mbin",[572,725],{"className":726,"style":719},[629],[572,728,730,734],{"className":729},[616],[572,731],{"className":732,"style":733},[620],"height:1.0925em;vertical-align:-0.345em;",[572,735,737,742,820],{"className":736},[625],[572,738],{"className":739},[740,741],"mopen","nulldelimiter",[572,743,745],{"className":744},[672],[572,746,750,811],{"className":747},[748,749],"vlist-t","vlist-t2",[572,751,754,806],{"className":752},[753],"vlist-r",[572,755,759,780,791],{"className":756,"style":758},[757],"vlist","height:0.7475em;",[572,760,762,767],{"style":761},"top:-2.655em;",[572,763],{"className":764,"style":766},[765],"pstrut","height:3em;",[572,768,774],{"className":769},[770,771,772,773],"sizing","reset-size6","size3","mtight",[572,775,777],{"className":776},[625,773],[572,778,593],{"className":779},[625,773],[572,781,783,786],{"style":782},"top:-3.23em;",[572,784],{"className":785,"style":766},[765],[572,787],{"className":788,"style":790},[789],"frac-line","border-bottom-width:0.04em;",[572,792,794,797],{"style":793},"top:-3.4461em;",[572,795],{"className":796,"style":766},[765],[572,798,800],{"className":799},[770,771,772,773],[572,801,803],{"className":802},[625,773],[572,804,564],{"className":805},[625,694,773],[572,807,810],{"className":808},[809],"vlist-s","​",[572,812,814],{"className":813},[753],[572,815,818],{"className":816,"style":817},[757],"height:0.345em;",[572,819],{},[572,821],{"className":822},[823,741],"mclose",", wenn man nach ",[572,826,828,841],{"className":827},[575],[572,829,831],{"className":830},[579],[581,832,833],{"xmlns":583},[585,834,835,839],{},[588,836,837],{},[599,838,667],{},[604,840,667],{"encoding":606},[572,842,844],{"className":843,"ariaHidden":612},[611],[572,845,847,850],{"className":846},[616],[572,848],{"className":849,"style":690},[620],[572,851,667],{"className":852},[625,694]," umstellt. Man kann auch im Kopf rechnen: Erst auf ",[572,855,857,876],{"className":856},[575],[572,858,860],{"className":859},[579],[581,861,862],{"xmlns":583},[585,863,864,873],{},[588,865,866,869,871],{},[591,867,868],{},"1",[595,870,597],{},[599,872,602],{"mathvariant":601},[604,874,875],{"encoding":606},"1\\,\\%",[572,877,879],{"className":878,"ariaHidden":612},[611],[572,880,882,885,888,891],{"className":881},[616],[572,883],{"className":884,"style":621},[620],[572,886,868],{"className":887},[625],[572,889],{"className":890,"style":630},[629],[572,892,602],{"className":893},[625]," schließen (durch ",[572,896,898,911],{"className":897},[575],[572,899,901],{"className":900},[579],[581,902,903],{"xmlns":583},[585,904,905,909],{},[588,906,907],{},[599,908,564],{},[604,910,564],{"encoding":606},[572,912,914],{"className":913,"ariaHidden":612},[611],[572,915,917,921],{"className":916},[616],[572,918],{"className":919,"style":920},[620],"height:0.625em;vertical-align:-0.1944em;",[572,922,564],{"className":923},[625,694]," teilen), dann auf ",[572,926,928,945],{"className":927},[575],[572,929,931],{"className":930},[579],[581,932,933],{"xmlns":583},[585,934,935,943],{},[588,936,937,939,941],{},[591,938,593],{},[595,940,597],{},[599,942,602],{"mathvariant":601},[604,944,607],{"encoding":606},[572,946,948],{"className":947,"ariaHidden":612},[611],[572,949,951,954,957,960],{"className":950},[616],[572,952],{"className":953,"style":621},[620],[572,955,593],{"className":956},[625],[572,958],{"className":959,"style":630},[629],[572,961,602],{"className":962},[625]," (mal ",[572,965,967,980],{"className":966},[575],[572,968,970],{"className":969},[579],[581,971,972],{"xmlns":583},[585,973,974,978],{},[588,975,976],{},[591,977,593],{},[604,979,593],{"encoding":606},[572,981,983],{"className":982,"ariaHidden":612},[611],[572,984,986,990],{"className":985},[616],[572,987],{"className":988,"style":989},[620],"height:0.6444em;",[572,991,593],{"className":992},[625],").",[564,995,996],{},"Der Prozentsatz darf nicht null sein, weil man sonst durch null teilen würde.",[559,998,1000],{"id":999},"beispiele","Beispiele",[564,1002,1003,1032,1033,1073],{},[572,1004,1006,1020],{"className":1005},[575],[572,1007,1009],{"className":1008},[579],[581,1010,1011],{"xmlns":583},[585,1012,1013,1018],{},[588,1014,1015],{},[591,1016,1017],{},"18",[604,1019,1017],{"encoding":606},[572,1021,1023],{"className":1022,"ariaHidden":612},[611],[572,1024,1026,1029],{"className":1025},[616],[572,1027],{"className":1028,"style":989},[620],[572,1030,1017],{"className":1031},[625]," Kinder einer Klasse fahren mit dem Rad zur Schule. Das sind ",[572,1034,1036,1055],{"className":1035},[575],[572,1037,1039],{"className":1038},[579],[581,1040,1041],{"xmlns":583},[585,1042,1043,1052],{},[588,1044,1045,1048,1050],{},[591,1046,1047],{},"60",[595,1049,597],{},[599,1051,602],{"mathvariant":601},[604,1053,1054],{"encoding":606},"60\\,\\%",[572,1056,1058],{"className":1057,"ariaHidden":612},[611],[572,1059,1061,1064,1067,1070],{"className":1060},[616],[572,1062],{"className":1063,"style":621},[620],[572,1065,1047],{"className":1066},[625],[572,1068],{"className":1069,"style":630},[629],[572,1071,602],{"className":1072},[625]," der Klasse. Wie viele Kinder hat die Klasse?",[572,1075,1078],{"className":1076},[1077],"katex-display",[572,1079,1081,1115],{"className":1080},[575],[572,1082,1084],{"className":1083},[579],[581,1085,1087],{"xmlns":583,"display":1086},"block",[585,1088,1089,1112],{},[588,1090,1091,1093,1095,1107,1109],{},[599,1092,667],{},[662,1094,664],{},[672,1096,1097,1105],{},[588,1098,1099,1101,1103],{},[591,1100,1017],{},[662,1102,670],{},[591,1104,593],{},[591,1106,1047],{},[662,1108,664],{},[591,1110,1111],{},"30",[604,1113,1114],{"encoding":606},"G = \\frac{18 \\cdot 100}{60} = 30",[572,1116,1118,1136,1230],{"className":1117,"ariaHidden":612},[611],[572,1119,1121,1124,1127,1130,1133],{"className":1120},[616],[572,1122],{"className":1123,"style":690},[620],[572,1125,667],{"className":1126},[625,694],[572,1128],{"className":1129,"style":699},[629],[572,1131,664],{"className":1132},[703],[572,1134],{"className":1135,"style":699},[629],[572,1137,1139,1143,1221,1224,1227],{"className":1138},[616],[572,1140],{"className":1141,"style":1142},[620],"height:2.0074em;vertical-align:-0.686em;",[572,1144,1146,1149,1218],{"className":1145},[625],[572,1147],{"className":1148},[740,741],[572,1150,1152],{"className":1151},[672],[572,1153,1155,1209],{"className":1154},[748,749],[572,1156,1158,1206],{"className":1157},[753],[572,1159,1162,1174,1182],{"className":1160,"style":1161},[757],"height:1.3214em;",[572,1163,1165,1168],{"style":1164},"top:-2.314em;",[572,1166],{"className":1167,"style":766},[765],[572,1169,1171],{"className":1170},[625],[572,1172,1047],{"className":1173},[625],[572,1175,1176,1179],{"style":782},[572,1177],{"className":1178,"style":766},[765],[572,1180],{"className":1181,"style":790},[789],[572,1183,1185,1188],{"style":1184},"top:-3.677em;",[572,1186],{"className":1187,"style":766},[765],[572,1189,1191,1194,1197,1200,1203],{"className":1190},[625],[572,1192,1017],{"className":1193},[625],[572,1195],{"className":1196,"style":719},[629],[572,1198,670],{"className":1199},[723],[572,1201],{"className":1202,"style":719},[629],[572,1204,593],{"className":1205},[625],[572,1207,810],{"className":1208},[809],[572,1210,1212],{"className":1211},[753],[572,1213,1216],{"className":1214,"style":1215},[757],"height:0.686em;",[572,1217],{},[572,1219],{"className":1220},[823,741],[572,1222],{"className":1223,"style":699},[629],[572,1225,664],{"className":1226},[703],[572,1228],{"className":1229,"style":699},[629],[572,1231,1233,1236],{"className":1232},[616],[572,1234],{"className":1235,"style":989},[620],[572,1237,1111],{"className":1238},[625],[564,1240,1241,1242,1282,1283,1327],{},"Ein Rabatt von ",[572,1243,1245,1264],{"className":1244},[575],[572,1246,1248],{"className":1247},[579],[581,1249,1250],{"xmlns":583},[585,1251,1252,1261],{},[588,1253,1254,1257,1259],{},[591,1255,1256],{},"15",[595,1258,597],{},[599,1260,602],{"mathvariant":601},[604,1262,1263],{"encoding":606},"15\\,\\%",[572,1265,1267],{"className":1266,"ariaHidden":612},[611],[572,1268,1270,1273,1276,1279],{"className":1269},[616],[572,1271],{"className":1272,"style":621},[620],[572,1274,1256],{"className":1275},[625],[572,1277],{"className":1278,"style":630},[629],[572,1280,602],{"className":1281},[625]," macht ",[572,1284,1286,1301],{"className":1285},[575],[572,1287,1289],{"className":1288},[579],[581,1290,1291],{"xmlns":583},[585,1292,1293,1298],{},[588,1294,1295],{},[591,1296,1297],{},"4,50",[604,1299,1300],{"encoding":606},"4{,}50",[572,1302,1304],{"className":1303,"ariaHidden":612},[611],[572,1305,1307,1311,1315,1323],{"className":1306},[616],[572,1308],{"className":1309,"style":1310},[620],"height:0.8389em;vertical-align:-0.1944em;",[572,1312,1314],{"className":1313},[625],"4",[572,1316,1318],{"className":1317},[625],[572,1319,1322],{"className":1320},[1321],"mpunct",",",[572,1324,1326],{"className":1325},[625],"50"," € aus. Wie teuer war die Ware vorher?",[572,1329,1331],{"className":1330},[1077],[572,1332,1334,1367],{"className":1333},[575],[572,1335,1337],{"className":1336},[579],[581,1338,1339],{"xmlns":583,"display":1086},[585,1340,1341,1364],{},[588,1342,1343,1345,1347,1360,1362],{},[599,1344,667],{},[662,1346,664],{},[672,1348,1349,1358],{},[588,1350,1351,1354,1356],{},[591,1352,1353],{},"4,5",[662,1355,670],{},[591,1357,593],{},[591,1359,1256],{},[662,1361,664],{},[591,1363,1111],{},[604,1365,1366],{"encoding":606},"G = \\frac{4{,}5 \\cdot 100}{15} = 30",[572,1368,1370,1388,1487],{"className":1369,"ariaHidden":612},[611],[572,1371,1373,1376,1379,1382,1385],{"className":1372},[616],[572,1374],{"className":1375,"style":690},[620],[572,1377,667],{"className":1378},[625,694],[572,1380],{"className":1381,"style":699},[629],[572,1383,664],{"className":1384},[703],[572,1386],{"className":1387,"style":699},[629],[572,1389,1391,1394,1478,1481,1484],{"className":1390},[616],[572,1392],{"className":1393,"style":1142},[620],[572,1395,1397,1400,1475],{"className":1396},[625],[572,1398],{"className":1399},[740,741],[572,1401,1403],{"className":1402},[672],[572,1404,1406,1467],{"className":1405},[748,749],[572,1407,1409,1464],{"className":1408},[753],[572,1410,1412,1423,1431],{"className":1411,"style":1161},[757],[572,1413,1414,1417],{"style":1164},[572,1415],{"className":1416,"style":766},[765],[572,1418,1420],{"className":1419},[625],[572,1421,1256],{"className":1422},[625],[572,1424,1425,1428],{"style":782},[572,1426],{"className":1427,"style":766},[765],[572,1429],{"className":1430,"style":790},[789],[572,1432,1433,1436],{"style":1184},[572,1434],{"className":1435,"style":766},[765],[572,1437,1439,1442,1448,1452,1455,1458,1461],{"className":1438},[625],[572,1440,1314],{"className":1441},[625],[572,1443,1445],{"className":1444},[625],[572,1446,1322],{"className":1447},[1321],[572,1449,1451],{"className":1450},[625],"5",[572,1453],{"className":1454,"style":719},[629],[572,1456,670],{"className":1457},[723],[572,1459],{"className":1460,"style":719},[629],[572,1462,593],{"className":1463},[625],[572,1465,810],{"className":1466},[809],[572,1468,1470],{"className":1469},[753],[572,1471,1473],{"className":1472,"style":1215},[757],[572,1474],{},[572,1476],{"className":1477},[823,741],[572,1479],{"className":1480,"style":699},[629],[572,1482,664],{"className":1483},[703],[572,1485],{"className":1486,"style":699},[629],[572,1488,1490,1493],{"className":1489},[616],[572,1491],{"className":1492,"style":989},[620],[572,1494,1111],{"className":1495},[625],[564,1497,1498,1499,1527,1528,1566,1567,1595,1596,1633],{},"Die Ware kostete ",[572,1500,1502,1515],{"className":1501},[575],[572,1503,1505],{"className":1504},[579],[581,1506,1507],{"xmlns":583},[585,1508,1509,1513],{},[588,1510,1511],{},[591,1512,1111],{},[604,1514,1111],{"encoding":606},[572,1516,1518],{"className":1517,"ariaHidden":612},[611],[572,1519,1521,1524],{"className":1520},[616],[572,1522],{"className":1523,"style":989},[620],[572,1525,1111],{"className":1526},[625]," €. Probe: ",[572,1529,1531,1548],{"className":1530},[575],[572,1532,1534],{"className":1533},[579],[581,1535,1536],{"xmlns":583},[585,1537,1538,1546],{},[588,1539,1540,1542,1544],{},[591,1541,1256],{},[595,1543,597],{},[599,1545,602],{"mathvariant":601},[604,1547,1263],{"encoding":606},[572,1549,1551],{"className":1550,"ariaHidden":612},[611],[572,1552,1554,1557,1560,1563],{"className":1553},[616],[572,1555],{"className":1556,"style":621},[620],[572,1558,1256],{"className":1559},[625],[572,1561],{"className":1562,"style":630},[629],[572,1564,602],{"className":1565},[625]," von ",[572,1568,1570,1583],{"className":1569},[575],[572,1571,1573],{"className":1572},[579],[581,1574,1575],{"xmlns":583},[585,1576,1577,1581],{},[588,1578,1579],{},[591,1580,1111],{},[604,1582,1111],{"encoding":606},[572,1584,1586],{"className":1585,"ariaHidden":612},[611],[572,1587,1589,1592],{"className":1588},[616],[572,1590],{"className":1591,"style":989},[620],[572,1593,1111],{"className":1594},[625]," € sind ",[572,1597,1599,1612],{"className":1598},[575],[572,1600,1602],{"className":1601},[579],[581,1603,1604],{"xmlns":583},[585,1605,1606,1610],{},[588,1607,1608],{},[591,1609,1297],{},[604,1611,1300],{"encoding":606},[572,1613,1615],{"className":1614,"ariaHidden":612},[611],[572,1616,1618,1621,1624,1630],{"className":1617},[616],[572,1619],{"className":1620,"style":1310},[620],[572,1622,1314],{"className":1623},[625],[572,1625,1627],{"className":1626},[625],[572,1628,1322],{"className":1629},[1321],[572,1631,1326],{"className":1632},[625]," €.",[559,1635,1637],{"id":1636},"typische-fehler","Typische Fehler",[1639,1640,1641,1709,1875],"ul",{},[1642,1643,1644,1645,1708],"li",{},"Den Prozentwert mit dem Prozentsatz multiplizieren statt teilen: ",[572,1646,1648,1667],{"className":1647},[575],[572,1649,1651],{"className":1650},[579],[581,1652,1653],{"xmlns":583},[585,1654,1655,1664],{},[588,1656,1657,1659,1661],{},[591,1658,1017],{},[662,1660,670],{},[591,1662,1663],{},"0,6",[604,1665,1666],{"encoding":606},"18 \\cdot 0{,}6",[572,1668,1670,1688],{"className":1669,"ariaHidden":612},[611],[572,1671,1673,1676,1679,1682,1685],{"className":1672},[616],[572,1674],{"className":1675,"style":989},[620],[572,1677,1017],{"className":1678},[625],[572,1680],{"className":1681,"style":719},[629],[572,1683,670],{"className":1684},[723],[572,1686],{"className":1687,"style":719},[629],[572,1689,1691,1694,1698,1704],{"className":1690},[616],[572,1692],{"className":1693,"style":1310},[620],[572,1695,1697],{"className":1696},[625],"0",[572,1699,1701],{"className":1700},[625],[572,1702,1322],{"className":1703},[1321],[572,1705,1707],{"className":1706},[625],"6"," ergibt nicht die Klassengröße.",[1642,1710,1711,1712,1715,1716,1754,1755,1795,1796,1833,1834,1874],{},"Den Grundwert mit einem veränderten Wert verwechseln: Kostet eine Ware ",[568,1713,1714],{},"nach"," ",[572,1717,1719,1736],{"className":1718},[575],[572,1720,1722],{"className":1721},[579],[581,1723,1724],{"xmlns":583},[585,1725,1726,1734],{},[588,1727,1728,1730,1732],{},[591,1729,1256],{},[595,1731,597],{},[599,1733,602],{"mathvariant":601},[604,1735,1263],{"encoding":606},[572,1737,1739],{"className":1738,"ariaHidden":612},[611],[572,1740,1742,1745,1748,1751],{"className":1741},[616],[572,1743],{"className":1744,"style":621},[620],[572,1746,1256],{"className":1747},[625],[572,1749],{"className":1750,"style":630},[629],[572,1752,602],{"className":1753},[625]," Rabatt ",[572,1756,1758,1773],{"className":1757},[575],[572,1759,1761],{"className":1760},[579],[581,1762,1763],{"xmlns":583},[585,1764,1765,1770],{},[588,1766,1767],{},[591,1768,1769],{},"25,50",[604,1771,1772],{"encoding":606},"25{,}50",[572,1774,1776],{"className":1775,"ariaHidden":612},[611],[572,1777,1779,1782,1786,1792],{"className":1778},[616],[572,1780],{"className":1781,"style":1310},[620],[572,1783,1785],{"className":1784},[625],"25",[572,1787,1789],{"className":1788},[625],[572,1790,1322],{"className":1791},[1321],[572,1793,1326],{"className":1794},[625]," €, dann sind ",[572,1797,1799,1812],{"className":1798},[575],[572,1800,1802],{"className":1801},[579],[581,1803,1804],{"xmlns":583},[585,1805,1806,1810],{},[588,1807,1808],{},[591,1809,1769],{},[604,1811,1772],{"encoding":606},[572,1813,1815],{"className":1814,"ariaHidden":612},[611],[572,1816,1818,1821,1824,1830],{"className":1817},[616],[572,1819],{"className":1820,"style":1310},[620],[572,1822,1785],{"className":1823},[625],[572,1825,1827],{"className":1826},[625],[572,1828,1322],{"className":1829},[1321],[572,1831,1326],{"className":1832},[625]," € nur ",[572,1835,1837,1856],{"className":1836},[575],[572,1838,1840],{"className":1839},[579],[581,1841,1842],{"xmlns":583},[585,1843,1844,1853],{},[588,1845,1846,1849,1851],{},[591,1847,1848],{},"85",[595,1850,597],{},[599,1852,602],{"mathvariant":601},[604,1854,1855],{"encoding":606},"85\\,\\%",[572,1857,1859],{"className":1858,"ariaHidden":612},[611],[572,1860,1862,1865,1868,1871],{"className":1861},[616],[572,1863],{"className":1864,"style":621},[620],[572,1866,1848],{"className":1867},[625],[572,1869],{"className":1870,"style":630},[629],[572,1872,602],{"className":1873},[625]," des Grundwerts (siehe Änderungsfaktor).",[1642,1876,1877,1878,2019],{},"Die Probe vergessen: Mit ",[572,1879,1881,1906],{"className":1880},[575],[572,1882,1884],{"className":1883},[579],[581,1885,1886],{"xmlns":583},[585,1887,1888,1904],{},[588,1889,1890,1892,1894,1896,1898],{},[599,1891,660],{},[662,1893,664],{},[599,1895,667],{},[662,1897,670],{},[672,1899,1900,1902],{},[599,1901,564],{},[591,1903,593],{},[604,1905,680],{"encoding":606},[572,1907,1909,1927,1945],{"className":1908,"ariaHidden":612},[611],[572,1910,1912,1915,1918,1921,1924],{"className":1911},[616],[572,1913],{"className":1914,"style":690},[620],[572,1916,660],{"className":1917,"style":695},[625,694],[572,1919],{"className":1920,"style":699},[629],[572,1922,664],{"className":1923},[703],[572,1925],{"className":1926,"style":699},[629],[572,1928,1930,1933,1936,1939,1942],{"className":1929},[616],[572,1931],{"className":1932,"style":690},[620],[572,1934,667],{"className":1935},[625,694],[572,1937],{"className":1938,"style":719},[629],[572,1940,670],{"className":1941},[723],[572,1943],{"className":1944,"style":719},[629],[572,1946,1948,1951],{"className":1947},[616],[572,1949],{"className":1950,"style":733},[620],[572,1952,1954,1957,2016],{"className":1953},[625],[572,1955],{"className":1956},[740,741],[572,1958,1960],{"className":1959},[672],[572,1961,1963,2008],{"className":1962},[748,749],[572,1964,1966,2005],{"className":1965},[753],[572,1967,1969,1983,1991],{"className":1968,"style":758},[757],[572,1970,1971,1974],{"style":761},[572,1972],{"className":1973,"style":766},[765],[572,1975,1977],{"className":1976},[770,771,772,773],[572,1978,1980],{"className":1979},[625,773],[572,1981,593],{"className":1982},[625,773],[572,1984,1985,1988],{"style":782},[572,1986],{"className":1987,"style":766},[765],[572,1989],{"className":1990,"style":790},[789],[572,1992,1993,1996],{"style":793},[572,1994],{"className":1995,"style":766},[765],[572,1997,1999],{"className":1998},[770,771,772,773],[572,2000,2002],{"className":2001},[625,773],[572,2003,564],{"className":2004},[625,694,773],[572,2006,810],{"className":2007},[809],[572,2009,2011],{"className":2010},[753],[572,2012,2014],{"className":2013,"style":817},[757],[572,2015],{},[572,2017],{"className":2018},[823,741]," lässt sich das Ergebnis schnell kontrollieren.",{"title":2021,"searchDepth":2022,"depth":2022,"links":2023},"",2,[2024,2025,2026],{"id":561,"depth":2022,"text":562},{"id":999,"depth":2022,"text":1000},{"id":1636,"depth":2022,"text":1637},[2028,2030,2033,2037,2039,2041,2044],{"lhs":2029,"rhs":667},"(G*p\u002F100)*100\u002Fp",{"lhs":2031,"rhs":2032},"W*100\u002Fp","W\u002F(p\u002F100)",{"lhs":2031,"rhs":1111,"where":2034},{"W":2035,"p":2036},18,60,{"lhs":2038,"rhs":1111},"18\u002F0.6",{"lhs":2040,"rhs":1111},"4.5*100\u002F15",{"lhs":2042,"rhs":2043},"30*15\u002F100","4.5",{"lhs":2045,"rhs":2046},"30*85\u002F100","25.5","p \\neq 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p \\neq 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","base-value","G = \\frac{W \\cdot 100}{p}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>G\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>W\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmn>100\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">G = \\frac{W \\cdot 100}{p}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">G\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3603em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">100\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8804em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","lower",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Farithmetic\u002Fgrundwert",[],[2060,2061,2062],"percentage-value","percentage-rate","change-factor",{"title":548,"description":2021},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fgrundwert","Kennt man einen Anteil und seinen Prozentsatz, berechnet man das Ganze mit G = W · 100 \u002F p.",[2067,2068,2069,2070],"prozent","prozentrechnung","grundwert","prozentwert",[2072,2075,2078],{"symbol":667,"meaning":2073,"symbolHtml":2074},"Grundwert, das Ganze (entspricht 100 %)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>G\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">G\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">G\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":660,"meaning":2076,"symbolHtml":2077},"Prozentwert, der bekannte Anteil","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>W\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">W\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":564,"meaning":2079,"symbolHtml":2080},"Prozentsatz als Zahl, z.B. p = 60 bei 60 %","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","-FpeyMm_XQPVTL76pjbB9tM8yeTrPtal9_e3VLDeu2Y",[2083,2086,2088],{"stem":2084,"title":2085,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Faenderungsfaktor","Vermehrter und verminderter Grundwert",{"stem":2087,"title":641,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fprozentsatz",{"stem":2089,"title":637,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fprozentwert",1793824259610]