{"id":1696,"date":"2020-09-07T21:07:59","date_gmt":"2020-09-07T19:07:59","guid":{"rendered":"http:\/\/matma.com.pl\/?p=1696"},"modified":"2020-09-08T22:13:40","modified_gmt":"2020-09-08T20:13:40","slug":"iii-ulamki","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=1696","title":{"rendered":"III. U\u0142amki"},"content":{"rendered":"<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Zacznijmy od tego jak wygl\u0105da <span style=\"color: #339966;\"><strong>u\u0142amek zwyk\u0142y<\/strong><\/span><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\" wp-image-687 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u%C5%82amek-300x130.png\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u\u0142amek-300x130.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u\u0142amek.png 454w\" alt=\"\" width=\"286\" height=\"124\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #339966;\"><strong>U\u0142amek w\u0142a\u015bciwy<\/strong><\/span>\u00a0\u2013 licznik jest mniejszy od mianownika<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{15}\" alt=\"\\frac{2}{15}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #339966;\"><strong>U\u0142amek\u00a0niew\u0142a\u015bciwy<\/strong><\/span>\u00a0\u2013 licznik jest wi\u0119kszy od mianownika<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{15}{2}\" alt=\"\\frac{15}{2}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Liczba mieszana<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?3\\frac{1}{7}\" alt=\"3\\frac{1}{7}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Por\u00f3wnywanie u\u0142amk\u00f3w<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, u\u0142amki maj\u0105 takie same liczniki, to\u00a0wi\u0119kszy jest\u00a0ten, kt\u00f3rego mianownik jest mniejszy.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1091 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u%C5%82amki-licznik.png\" alt=\"\" width=\"71\" height=\"66\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, u\u0142amki maj\u0105 takie same mianowniki to wi\u0119kszy jest\u00a0ten,\u00a0kt\u00f3rego licznik jest wi\u0119kszy.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1092 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u%C5%82amki-mianownik.png\" alt=\"\" width=\"68\" height=\"60\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, u\u0142amki maj\u0105 r\u00f3\u017cne liczniki i mianowniki to sprowadzamy je do wsp\u00f3lnego licznika lub mianownika i dopiero wtedy por\u00f3wnujemy.<\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Dzia\u0142ania na u\u0142amkach zwyk\u0142ych<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Dodawanie i odejmowanie<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, u\u0142amki w\u0142a\u015bciwe maj\u0105 takie same mianowniki to dodajemy lub odejmujemy liczniki, a mianownik pozostaje bez zmian.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{11}{25}+\\frac{6}{25}=\\frac{17}{25}\" alt=\"\\frac{11}{25}+\\frac{6}{25}=\\frac{17}{25}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{11}{25}-\\frac{7}{25}=\\frac{4}{25}\" alt=\"\\frac{11}{25}-\\frac{7}{25}=\\frac{4}{25}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Przed dodawaniem i odejmowaniem u\u0142amk\u00f3w o r\u00f3\u017cnych mianownikach sprowadzamy u\u0142amki do wsp\u00f3lnego mianownika. Nast\u0119pnie dodajemy u\u0142amki.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}+\\frac{2}{3}=\\frac{3\\cdot&amp;space;3}{5\\cdot&amp;space;3}+\\frac{2\\cdot&amp;space;5}{3\\cdot&amp;space;5}=\\frac{9}{15}+\\frac{10}{15}=\\frac{19}{15}=1\\frac{4}{15}\" alt=\"\\frac{3}{5}+\\frac{2}{3}=\\frac{3\\cdot 3}{5\\cdot 3}+\\frac{2\\cdot 5}{3\\cdot 5}=\\frac{9}{15}+\\frac{10}{15}=\\frac{19}{15}=1\\frac{4}{15}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{3}-\\frac{3}{5}=\\frac{2\\cdot&amp;space;5}{3\\cdot&amp;space;5}-\\frac{3\\cdot&amp;space;3}{5\\cdot&amp;space;3}=\\frac{10}{15}-\\frac{9}{15}=\\frac{1}{15}\" alt=\"\\frac{2}{3}-\\frac{3}{5}=\\frac{2\\cdot 5}{3\\cdot 5}-\\frac{3\\cdot 3}{5\\cdot 3}=\\frac{10}{15}-\\frac{9}{15}=\\frac{1}{15}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Skracanie u\u0142amk\u00f3w<\/span><\/strong>\u00a0\u2013 dzielimy licznik i mianownik przez taka sama liczb\u0119, r\u00f3\u017cna od zera.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{20}{25}=\\frac{4\\cdot&amp;space;\\cancel5}{5\\cdot&amp;space;\\cancel5}=\\frac{4}{5}\" alt=\"\\frac{20}{25}=\\frac{4\\cdot \\cancel5}{5\\cdot \\cancel5}=\\frac{4}{5}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Rozszerzanie u\u0142amk\u00f3w<\/span><\/strong>\u00a0\u2013 mno\u017cymy licznik i mianownik przez taka sam\u0105 liczb\u0119 r\u00f3\u017cn\u0105 od zera.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}=\\frac{3\\cdot&amp;space;5}{7\\cdot&amp;space;5}=\\frac{15}{35}\" alt=\"\\frac{3}{7}=\\frac{3\\cdot 5}{7\\cdot 5}=\\frac{15}{35}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Mno\u017cenie<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Aby pomno\u017cy\u0107 dwa u\u0142amki mno\u017cymy liczniki u\u0142amk\u00f3w i mianowniki u\u0142amk\u00f3w. Przy mno\u017ceniu dw\u00f3ch u\u0142amk\u00f3w mo\u017cemy skraca\u0107 licznik jednego z mianownikiem drugiego u\u0142amka.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\\cdot&amp;space;\\frac{8}{9}=\\frac{\\cancel3}{\\cancel4}\\cdot&amp;space;\\frac{2\\cdot&amp;space;\\cancel4}{\\cancel3\\cdot&amp;space;3}=\\frac{2}{3}\" alt=\"\\frac{3}{4}\\cdot \\frac{8}{9}=\\frac{\\cancel3}{\\cancel4}\\cdot \\frac{2\\cdot \\cancel4}{\\cancel3\\cdot 3}=\\frac{2}{3}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Dzielenie<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Aby podzieli\u0107 dwa u\u0142amki nale\u017cy pierwszy u\u0142amek pomno\u017cy\u0107 przez odwrotno\u015b\u0107 drugiego u\u0142amka.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{6}{7}:\\frac{6}{35}=\\frac{\\cancel6}{7}\\cdot&amp;space;\\frac{35}{\\cancel6}=\\frac{5\\cdot&amp;space;\\cancel7}{\\cancel7}=\\frac{5}{1}=5\" alt=\"\\frac{6}{7}:\\frac{6}{35}=\\frac{\\cancel6}{7}\\cdot \\frac{35}{\\cancel6}=\\frac{5\\cdot \\cancel7}{\\cancel7}=\\frac{5}{1}=5\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, mamy wykona\u0107 mno\u017cenie lub dzielenie liczb mieszanych musimy je najpierw zamieni\u0107 na u\u0142amki niew\u0142a\u015bciwe.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #339966;\"><strong>Liczba odwrotna<\/strong><\/span> powstaje przez zamian\u0119 miejscami licznika z mianownikiem<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Liczb\u0105 odwrotn\u0105 do <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\" alt=\"\\frac{3}{4}\" align=\"absmiddle\" \/>\u00a0 jest <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{3}\" alt=\"\\frac{4}{3}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Iloczyn liczb odwrotnych jest r\u00f3wny 1<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #339966;\"><strong>U\u0142amki dziesi\u0119tne<\/strong><\/span>, to liczby naturalne zapisane z u\u017cyciem przecinka. Cz\u0119\u015b\u0107 po lewej stronie przecinka to\u00a0cz\u0119\u015b\u0107 ca\u0142kowita,\u00a0a po prawej stronie to\u00a0cz\u0119\u015b\u0107 u\u0142amkowa.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Jak poprawnie przeczyta\u0107 taki u\u0142amek?<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1701 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/cz\u0119sci-u\u0142amka-dziesi\u0119tnego-300x142.png\" alt=\"\" width=\"330\" height=\"156\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/cz\u0119sci-u\u0142amka-dziesi\u0119tnego-300x142.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/cz\u0119sci-u\u0142amka-dziesi\u0119tnego.png 593w\" sizes=\"(max-width: 330px) 100vw, 330px\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">czytamy: trzyna\u015bcie ca\u0142ych pi\u0119\u0107set pi\u0119\u0107dziesi\u0105t cztery tysi\u0119czne.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli dopiszemy zero (jedno lub kilka) na ko\u0144cu do naszego u\u0142amka na przyk\u0142ad\u00a0<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?13,5540\" alt=\"13,5540\" align=\"absmiddle\" \/>\u00a0 nie zmienia to warto\u015bci tej liczby.<\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Por\u00f3wnywanie u\u0142amk\u00f3w dziesi\u0119tnych<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Por\u00f3wnujemy w pierwszej kolejno\u015bci ca\u0142o\u015bci, je\u015bli s\u0105 takie same, to przechodzimy kolejno do cz\u0119\u015bci dziesi\u0119tnych, setnych i tak dalej.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1096 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/por%C3%B3w-u%C5%82-dzies.png\" alt=\"\" width=\"156\" height=\"49\" \/><\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Dzia\u0142ania na u\u0142amkach dziesi\u0119tnych<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Dodawanie i odejmowanie<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Przy dodawaniu i odejmowaniu u\u0142amk\u00f3w dziesi\u0119tnych dodajemy cz\u0119\u015bci ca\u0142kowite do ca\u0142kowitych, cz\u0119\u015bci u\u0142amkowe do u\u0142amkowych. Musimy pami\u0119ta\u0107, \u017ce 10 cz\u0119\u015bci dziesi\u0105tych to jedna ca\u0142o\u015b\u0107. Dziesi\u0119\u0107 cz\u0119\u015bci setnych to jedna dziesi\u0105ta. Dziesi\u0119\u0107 cz\u0119\u015bci tysi\u0105cznych to jedna setna i tak dalej.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1097 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/dod-u%C5%82amk-dzies.png\" alt=\"\" width=\"171\" height=\"48\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Mo\u017cemy te\u017c dodawa\u0107 i odejmowa\u0107 u\u0142amki dziesi\u0119tne pod kresk\u0119. Nale\u017cy jednak pami\u0119ta\u0107, aby przecinek jednego u\u0142amka by\u0142a pod przecinkiem drugiego u\u0142amka. Je\u015bli jeden z u\u0142amk\u00f3w jest kr\u00f3tszy to musimy rozszerzy\u0107 taki u\u0142amek dopisuj\u0105c zero na ko\u0144cu.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1098 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/dodaw-u%C5%82amk-dzies.png\" alt=\"\" width=\"79\" height=\"114\" \/><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1099 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odejmow-u%C5%82-dzies.png\" alt=\"\" width=\"92\" height=\"104\" \/><\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Mno\u017cenie<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">U\u0142amki dziesi\u0119tne mno\u017cymy tak jak liczby naturalne. Nale\u017cy jednak pami\u0119ta\u0107, aby w wyniku oddzieli\u0107 przecinkiem tyle cyfr na ko\u0144cu ile by\u0142o \u0142\u0105cznie w obu u\u0142amkach.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,5\\cdot&amp;space;0,3=0,15\" alt=\"0,5\\cdot 0,3=0,15\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Dzielenie<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Najpierw rozszerzamy u\u0142amki, tak aby dzielnik by\u0142 liczb\u0105 ca\u0142kowit\u0105. Nast\u0119pnie wykonujemy dzielenie, tak jak dzielenie liczb naturalnych.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,12:0,02=12:2=6\" alt=\"0,12:0,02=12:2=6\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,52:0,2=5,2:2=2,6\" alt=\"0,52:0,2=5,2:2=2,6\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, chcemy pomno\u017cy\u0107 lub podzieli\u0107 u\u0142amek dziesi\u0119tny przez liczb\u0119 10 nale\u017cy odpowiednio przesun\u0105\u0107 przecinek. Przy mno\u017ceniu o jedno miejsce w praw\u0105 stron\u0119, przy dzieleniu o jedno miejsce w lew\u0105 stron\u0119. Przy mno\u017ceniu przez 100 post\u0119pujemy analogicznie tylko przesuwamy przecinek o dwa miejsca. Przy mno\u017ceniu przez 1000 o trzy miejsca. Im wi\u0119cej zer, tym o wi\u0119cej miejsc przesuwamy przecinek.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,152\\cdot&amp;space;10=11,52\" alt=\"1,152\\cdot 10=11,52\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,152\\cdot&amp;space;100=115,2\" alt=\"1,152\\cdot 100=115,2\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,152:10=0,1152\" alt=\"1,152:10=0,1152\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,152:100=0,01152\" alt=\"1,152:100=0,01152\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>U\u0142amki zwyk\u0142e i dziesi\u0119tne<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Je\u015bli, mamy wykona\u0107 dzia\u0142anie, w kt\u00f3rym jest u\u0142amek zwyk\u0142y i dziesi\u0119tny to musimy oba zapisa\u0107 w takiej samej postaci (zwyk\u0142ej lub dziesi\u0119tnej). Ka\u017cdy u\u0142amek dziesi\u0119tny mo\u017cemy zamieni\u0107 na u\u0142amek zwyk\u0142y. Jednak, nie zawsze mo\u017cemy zamieni\u0107 u\u0142amek zwyk\u0142y na dziesi\u0119tny. Musimy pami\u0119ta\u0107, \u017ce aby to zrobi\u0107 nale\u017cy u\u0142amek rozszerzy\u0107 tak, aby w mianowniku by\u0142o 10, 100 lub 1000 i tak dalej.<\/span><\/p>\n<p><span style=\"color: #339966; font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong>Rozwini\u0119cia dziesi\u0119tne u\u0142amk\u00f3w zwyk\u0142ych<\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Aby u\u0142amek zwyk\u0142y zapisa\u0107 w postaci dziesi\u0119tnej mo\u017cemy poda\u0107 jego rozwini\u0119cie dziesi\u0119tne. Jak to zrobi\u0107? Wystarczy podzieli\u0107 licznik przez mianownik. Jedne u\u0142amki b\u0119d\u0105 mia\u0142y rozwini\u0119cie dziesi\u0119tne sko\u0144czone inne niesko\u0144czone (dzielenie nie ma ko\u0144ca).\u00a0Zajmiemy si\u0119 tymi z rozwini\u0119ciem niesko\u0144czonym. Poniewa\u017c, te z rozwini\u0119ciem dziesi\u0119tnym sko\u0144czonym mo\u017cna zamieni\u0107 na u\u0142amki zwyk\u0142e rozszerzaj\u0105c mianownik do 10, 100 lub 1000 i tak dalej.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Spr\u00f3bujmy podzieli\u0107 1 przez 3:<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1100 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u%C5%82-niesko%C5%84czony-121x300.png\" sizes=\"(max-width: 121px) 100vw, 121px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u\u0142-niesko\u0144czony-121x300.png 121w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/u\u0142-niesko\u0144czony.png 203w\" alt=\"\" width=\"113\" height=\"278\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Czyli mo\u017cemy zapisa\u0107:<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{3}=0,3333...\" alt=\"\\frac{1}{3}=0,3333...\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">W przypadku, kiedy u\u0142amek ma rozwini\u0119cie dziesi\u0119tne niesko\u0144czone sprawdzamy czy jaka\u015b liczba lub kilka liczb si\u0119 nie powtarza w niesko\u0144czono\u015b\u0107. Je\u015bli tak b\u0119dzie, to u\u0142amek b\u0119dzie mia\u0142\u00a0rozwini\u0119cie dziesi\u0119tne niesko\u0144czone okresowe.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{3}=0,3333...=0,\\left&amp;space;(&amp;space;3&amp;space;\\right&amp;space;)\" alt=\"\\frac{1}{3}=0,3333...=0,\\left ( 3 \\right )\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #339966;\">Zaokr\u0105glanie rozwini\u0119cia dziesi\u0119tnego<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Aby zaokr\u0105gli\u0107 liczb\u0119 pos\u0142ugujemy si\u0119 symbolem\u201d\u00a0<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\approx\" alt=\"\\approx\" align=\"absmiddle\" \/>\u201c, kt\u00f3ry czytamy: r\u00f3wny w przybli\u017ceniu.<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Najpierw musimy wybra\u0107 do, kt\u00f3rego miejsca zaokr\u0105glamy nasz\u0105 liczb\u0119 (pozycja cyfry w liczbie).<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">Nast\u0119pnie zaokr\u0105glamy liczb\u0119 do danego miejsca, odrzucaj\u0105c wszystkie cyfry na prawo od naszej cyfry. W miejsca odrzuconych cyfr wstawiamy zera. Je\u015bli pierwsza z odrzuconych cyfr b\u0119dzie r\u00f3wna:<\/span><\/p>\n<ul>\n<li><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">0,1,2,3 lub 4 to nasz\u0105 cyfr\u0119 zostawiamy bez zmian (<span style=\"color: #339966;\">przybli\u017cenie z niedomiarem<\/span>)<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}=0,\\left&amp;space;(&amp;space;428571&amp;space;\\right&amp;space;)\\approx&amp;space;0,4\" alt=\"\\frac{3}{7}=0,\\left ( 428571 \\right )\\approx 0,4\" align=\"absmiddle\" \/><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{9}=0,\\left&amp;space;(&amp;space;2&amp;space;\\right&amp;space;)\\approx&amp;space;0,22\\approx&amp;space;0,2\" alt=\"\\frac{2}{9}=0,\\left ( 2 \\right )\\approx 0,22\\approx 0,2\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\">5,6,7,8 lub 9 to nasz\u0105 cyfr\u0119 zwi\u0119kszamy o jeden (<span style=\"color: #339966;\">przybli\u017cenie z nadmiarem<\/span>)<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{7}=0,\\left&amp;space;(&amp;space;571428&amp;space;\\right&amp;space;)\\approx&amp;space;0,6\" alt=\"\\frac{4}{7}=0,\\left ( 571428 \\right )\\approx 0,6\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{7}{9}=0,\\left&amp;space;(&amp;space;7&amp;space;\\right&amp;space;)\\approx&amp;space;0,78\\approx&amp;space;0,8\" alt=\"\\frac{7}{9}=0,\\left ( 7 \\right )\\approx 0,78\\approx 0,8\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Przyk\u0142ad 1<\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Zamie\u0144 liczb\u0119 mieszana na u\u0142amek niew\u0142a\u015bciwy<\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?3\\frac{4}{5}=\\frac{3\\cdot&amp;space;5+4}{5}=\\frac{19}{4}\" alt=\"3\\frac{4}{5}=\\frac{3\\cdot 5+4}{5}=\\frac{19}{4}\" align=\"absmiddle\" \/>\u00a0<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zadanie 1<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zamie\u0144 liczb\u0119 mieszana na u\u0142amek niew\u0142a\u015bciwy:<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?4\\frac{1}{5}\" alt=\"4\\frac{1}{5}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?11\\frac{1}{11}\" alt=\"11\\frac{1}{11}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?5\\frac{1}{8}\" alt=\"5\\frac{1}{8}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ol>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Przyk\u0142ad 2<\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Por\u00f3wnaj u\u0142amki u\u017cywaj\u0105c znak\u00f3w &gt;, &lt;, =<\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{5}\" alt=\"\\frac{1}{5}\" align=\"absmiddle\" \/>\u00a0 i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{6}\" alt=\"\\frac{1}{6}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{5}&gt;&amp;space;\\frac{1}{6}\" alt=\"\\frac{1}{5}&gt; \\frac{1}{6}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?13,44\" alt=\"13,44\" align=\"absmiddle\" \/>\u00a0 i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?13,54\" alt=\"13,54\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?13,44&lt;&amp;space;13,54\" alt=\"13,44&lt; 13,54\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\" alt=\"\\frac{3}{4}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}\" alt=\"\\frac{3}{7}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}&gt;&amp;space;\\frac{3}{7}\" alt=\"\\frac{3}{4}&gt; \\frac{3}{7}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}\" alt=\"\\frac{3}{5}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{5}\" alt=\"\\frac{4}{5}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}&lt;&amp;space;\\frac{4}{5}\" alt=\"\\frac{3}{5}&lt; \\frac{4}{5}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{3}\" alt=\"\\frac{3}{3}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{7}{7}\" alt=\"\\frac{7}{7}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{3}=\\frac{7}{7}\" alt=\"\\frac{3}{3}=\\frac{7}{7}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}\" alt=\"\\frac{3}{5}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{7}{8}\" alt=\"\\frac{7}{8}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}=\\frac{24}{40}\" alt=\"\\frac{3}{5}=\\frac{24}{40}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{7}{8}=\\frac{35}{40}\" alt=\"\\frac{7}{8}=\\frac{35}{40}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}&lt;&amp;space;\\frac{7}{8}\" alt=\"\\frac{3}{5}&lt; \\frac{7}{8}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,25\" alt=\"0,25\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,2\" alt=\"0,2\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,2=0,20\" alt=\"0,2=0,20\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,25&gt;&amp;space;0,2\" alt=\"0,25&gt; 0,2\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zadanie 2<\/span><\/p>\n<p><span style=\"color: #000000; font-family: 'book antiqua', palatino, serif;\">Por\u00f3wnaj u\u0142amki u\u017cywaj\u0105c znak\u00f3w &gt;, &lt;, =<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{5}\" alt=\"\\frac{2}{5}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{5}\" alt=\"\\frac{3}{5}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,99\" alt=\"1,99\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1,999\" alt=\"1,999\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{7}\" alt=\"\\frac{2}{7}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{9}\" alt=\"\\frac{2}{9}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{11}{11}\" alt=\"\\frac{11}{11}\" align=\"absmiddle\" \/>\u00a0 i\u00a0\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{3}\" alt=\"\\frac{3}{3}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?12,234\" alt=\"12,234\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?12,432\" alt=\"12,432\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{5}\" alt=\"\\frac{1}{5}\" align=\"absmiddle\" \/>\u00a0 i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}\" alt=\"\\frac{3}{7}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ol>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Przyk\u0142ad 3<\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Por\u00f3wnaj u\u0142amki u\u017cywaj\u0105c znak\u00f3w &gt;, &lt;<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,11\" alt=\"-0,11\" align=\"absmiddle\" \/> i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,01\" alt=\"0,01\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,11&lt;&amp;space;0,1\" alt=\"-0,11&lt; 0,1\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{1}{4}\" alt=\"-\\frac{1}{4}\" align=\"absmiddle\" \/>\u00a0 \u00a0i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{1}{3}\" alt=\"-\\frac{1}{3}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{1}{4}&gt;&amp;space;-\\frac{1}{3}\" alt=\"-\\frac{1}{4}&gt; -\\frac{1}{3}\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,51\" alt=\"-0,51\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,5\" alt=\"-0,5\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,51&lt;&amp;space;-0,5\" alt=\"-0,51&lt; -0,5\" align=\"absmiddle\" \/><\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{2}{3}\" alt=\"-\\frac{2}{3}\" align=\"absmiddle\" \/>\u00a0 i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{4}{5}\" alt=\"-\\frac{4}{5}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{2}{3}=-\\frac{10}{15}\" alt=\"-\\frac{2}{3}=-\\frac{10}{15}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{4}{5}=-\\frac{12}{15}\" alt=\"-\\frac{4}{5}=-\\frac{12}{15}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{2}{3}&gt;&amp;space;-\\frac{4}{5}\" alt=\"-\\frac{2}{3}&gt; -\\frac{4}{5}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zadanie 3<\/span><\/p>\n<p><span style=\"color: #808080; font-family: 'book antiqua', palatino, serif;\">Por\u00f3wnaj u\u0142amki u\u017cywaj\u0105c znak\u00f3w &gt;, &lt;<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,99\" alt=\"-0,99\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,01\" alt=\"0,01\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{3}{4}\" alt=\"-\\frac{3}{4}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{3}{8}\" alt=\"-\\frac{3}{8}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,33\" alt=\"-0,33\" align=\"absmiddle\" \/>\u00a0 i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,3\" alt=\"-0,3\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{11}{12}\" alt=\"-\\frac{11}{12}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{7}{12}\" alt=\"-\\frac{7}{12}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{4}{9}\" alt=\"-\\frac{4}{9}\" align=\"absmiddle\" \/>\u00a0 i\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-\\frac{1}{7}\" alt=\"-\\frac{1}{7}\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ol>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Przyk\u0142ad 4<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Wykonaj dzia\u0142anie <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\\cdot&amp;space;4+0,5:\\frac{1}{2}-0,75\" alt=\"\\frac{3}{4}\\cdot 4+0,5:\\frac{1}{2}-0,75\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\\cdot&amp;space;4+0,5:\\frac{1}{2}-0,75=\\frac{3}{{\\cancel{4}}}\\cdot&amp;space;\\cancel{4}+\\frac{1}{\\cancel{2}}\\cdot&amp;space;\\cancel{2}-0,75=3+1-0,75=3,25\" alt=\"\\frac{3}{4}\\cdot 4+0,5:\\frac{1}{2}-0,75=\\frac{3}{{\\cancel{4}}}\\cdot \\cancel{4}+\\frac{1}{\\cancel{2}}\\cdot \\cancel{2}-0,75=3+1-0,75=3,25\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zadanie 4<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Wykonaj dzia\u0142ania:<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{8}:0,125+3-(-3)\\cdot&amp;space;\\frac{1}{3}\" alt=\"\\frac{1}{8}:0,125+3-(-3)\\cdot \\frac{1}{3}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,26:2-1\\cdot&amp;space;(-10)+\\frac{3}{4}:0,75-0,75\" alt=\"0,26:2-1\\cdot (-10)+\\frac{3}{4}:0,75-0,75\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-0,05+5\\cdot&amp;space;0,01-(-5)\\cdot&amp;space;0,2\" alt=\"-0,05+5\\cdot 0,01-(-5)\\cdot 0,2\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ol>\n<p><span style=\"font-family: 'book antiqua', palatino, serif; color: #808080;\">Przyk\u0142ad 5<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif; color: #808080;\">Zaokr\u0105glij u\u0142amek do cz\u0119\u015bci setnych i okre\u015bl czy jest to przybli\u017cenie z nadmiarem czy niedomiarem<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif; color: #808080;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{7}=0,\\left&amp;space;(&amp;space;571428&amp;space;\\right&amp;space;)\\approx&amp;space;0,57\" alt=\"\\frac{4}{7}=0,\\left ( 571428 \\right )\\approx 0,57\" align=\"absmiddle\" \/>\u00a0 przybli\u017cenie z niedomiarem<\/span><\/p>\n<p><span style=\"font-size: 12pt; font-family: 'book antiqua', palatino, serif; color: #808080;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}=0,\\left&amp;space;(&amp;space;428571&amp;space;\\right&amp;space;)\\approx&amp;space;0,43\" alt=\"\\frac{3}{7}=0,\\left ( 428571 \\right )\\approx 0,43\" align=\"absmiddle\" \/> przybli\u017cenie z nadmiarem<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zadanie 5<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zaokr\u0105glij u\u0142amek do cz\u0119\u015bci setnych i okre\u015bl czy jest to przybli\u017cenie z nadmiarem czy niedomiarem<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,12345\" alt=\"0,12345\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{8}\" alt=\"\\frac{1}{8}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{5}{7}\" alt=\"\\frac{5}{7}\" align=\"absmiddle\" \/><\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,(51)\" alt=\"0,(51)\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Zacznijmy od tego jak wygl\u0105da u\u0142amek zwyk\u0142y U\u0142amek w\u0142a\u015bciwy\u00a0\u2013 licznik jest mniejszy od mianownika U\u0142amek\u00a0niew\u0142a\u015bciwy\u00a0\u2013 licznik jest wi\u0119kszy od mianownika Liczba mieszana Por\u00f3wnywanie u\u0142amk\u00f3w Je\u015bli, u\u0142amki maj\u0105 takie same liczniki, to\u00a0wi\u0119kszy jest\u00a0ten, kt\u00f3rego mianownik jest mniejszy. Je\u015bli, u\u0142amki maj\u0105 takie same mianowniki to wi\u0119kszy jest\u00a0ten,\u00a0kt\u00f3rego licznik jest wi\u0119kszy. Je\u015bli, u\u0142amki maj\u0105 r\u00f3\u017cne liczniki i mianowniki [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[45],"tags":[],"class_list":["post-1696","post","type-post","status-publish","format-standard","hentry","category-egzamin-osmoklasisty"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>III. U\u0142amki - Matma.com.pl<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matma.com.pl\/?p=1696\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"III. U\u0142amki - Matma.com.pl\" \/>\n<meta property=\"og:description\" content=\"Zacznijmy od tego jak wygl\u0105da u\u0142amek zwyk\u0142y U\u0142amek w\u0142a\u015bciwy\u00a0\u2013 licznik jest mniejszy od mianownika U\u0142amek\u00a0niew\u0142a\u015bciwy\u00a0\u2013 licznik jest wi\u0119kszy od mianownika Liczba mieszana Por\u00f3wnywanie u\u0142amk\u00f3w Je\u015bli, u\u0142amki maj\u0105 takie same liczniki, to\u00a0wi\u0119kszy jest\u00a0ten, kt\u00f3rego mianownik jest mniejszy. Je\u015bli, u\u0142amki maj\u0105 takie same mianowniki to wi\u0119kszy jest\u00a0ten,\u00a0kt\u00f3rego licznik jest wi\u0119kszy. 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