127 frações e seus decimais
Todas as frações irredutíveis com denominador até 20. Os decimais vêm da divisão longa, então o período é exato: 0,1(6), não 0,1666…
Por denominador
Um denominador feito só de 2 e 5 dá decimal exato; se entrar um 3 ou um 7, ele passa a repetir. É porque 10 = 2 × 5.
/2Exato
1/20,550%
/3Periódico
1/30,(3)33,(3)%2/30,(6)66,(6)%
/4Exato
1/40,2525%3/40,7575%
/5Exato
1/50,220%2/50,440%3/50,660%4/50,880%
/6Periódico
1/60,1(6)16,(6)%5/60,8(3)83,(3)%
/7Periódico
1/70,(142857)14,(285714)%2/70,(285714)28,(571428)%3/70,(428571)42,(857142)%4/70,(571428)57,(142857)%5/70,(714285)71,(428571)%6/70,(857142)85,(714285)%
/8Exato
1/80,12512,5%3/80,37537,5%5/80,62562,5%7/80,87587,5%
/9Periódico
1/90,(1)11,(1)%2/90,(2)22,(2)%4/90,(4)44,(4)%5/90,(5)55,(5)%7/90,(7)77,(7)%8/90,(8)88,(8)%
/10Exato
1/100,110%3/100,330%7/100,770%9/100,990%
/11Periódico
1/110,(09)9,(09)%2/110,(18)18,(18)%3/110,(27)27,(27)%4/110,(36)36,(36)%5/110,(45)45,(45)%6/110,(54)54,(54)%7/110,(63)63,(63)%8/110,(72)72,(72)%9/110,(81)81,(81)%10/110,(90)90,(90)%
/12Periódico
1/120,08(3)8,(3)%5/120,41(6)41,(6)%7/120,58(3)58,(3)%11/120,91(6)91,(6)%
/13Periódico
1/130,(076923)7,(692307)%2/130,(153846)15,(384615)%3/130,(230769)23,(076923)%4/130,(307692)30,(769230)%5/130,(384615)38,(461538)%6/130,(461538)46,(153846)%7/130,(538461)53,(846153)%8/130,(615384)61,(538461)%9/130,(692307)69,(230769)%10/130,(769230)76,(923076)%11/130,(846153)84,(615384)%12/130,(923076)92,(307692)%
/14Periódico
1/140,0(714285)7,(142857)%3/140,2(142857)21,(428571)%5/140,3(571428)35,(714285)%9/140,6(428571)64,(285714)%11/140,7(857142)78,(571428)%13/140,9(285714)92,(857142)%
/15Periódico
1/150,0(6)6,(6)%2/150,1(3)13,(3)%4/150,2(6)26,(6)%7/150,4(6)46,(6)%8/150,5(3)53,(3)%11/150,7(3)73,(3)%13/150,8(6)86,(6)%14/150,9(3)93,(3)%
/16Exato
1/160,06256,25%3/160,187518,75%5/160,312531,25%7/160,437543,75%9/160,562556,25%11/160,687568,75%13/160,812581,25%15/160,937593,75%
/17Periódico
1/170,(0588235294117647)5,(8823529411764705)%2/170,(1176470588235294)11,(7647058823529411)%3/170,(1764705882352941)17,(6470588235294117)%4/170,(2352941176470588)23,(5294117647058823)%5/170,(2941176470588235)29,(4117647058823529)%6/170,(3529411764705882)35,(2941176470588235)%7/170,(4117647058823529)41,(1764705882352941)%8/170,(4705882352941176)47,(0588235294117647)%9/170,(5294117647058823)52,(9411764705882352)%10/170,(5882352941176470)58,(8235294117647058)%11/170,(6470588235294117)64,(7058823529411764)%12/170,(7058823529411764)70,(5882352941176470)%13/170,(7647058823529411)76,(4705882352941176)%14/170,(8235294117647058)82,(3529411764705882)%15/170,(8823529411764705)88,(2352941176470588)%16/170,(9411764705882352)94,(1176470588235294)%
/18Periódico
1/180,0(5)5,(5)%5/180,2(7)27,(7)%7/180,3(8)38,(8)%11/180,6(1)61,(1)%13/180,7(2)72,(2)%17/180,9(4)94,(4)%
/19Periódico
1/190,(052631578947368421)5,(263157894736842105)%2/190,(105263157894736842)10,(526315789473684210)%3/190,(157894736842105263)15,(789473684210526315)%4/190,(210526315789473684)21,(052631578947368421)%5/190,(263157894736842105)26,(315789473684210526)%6/190,(315789473684210526)31,(578947368421052631)%7/190,(368421052631578947)36,(842105263157894736)%8/190,(421052631578947368)42,(105263157894736842)%9/190,(473684210526315789)47,(368421052631578947)%10/190,(526315789473684210)52,(631578947368421052)%11/190,(578947368421052631)57,(894736842105263157)%12/190,(631578947368421052)63,(157894736842105263)%13/190,(684210526315789473)68,(421052631578947368)%14/190,(736842105263157894)73,(684210526315789473)%15/190,(789473684210526315)78,(947368421052631578)%16/190,(842105263157894736)84,(210526315789473684)%17/190,(894736842105263157)89,(473684210526315789)%18/190,(947368421052631578)94,(736842105263157894)%
/20Exato
1/200,055%3/200,1515%7/200,3535%9/200,4545%11/200,5555%13/200,6565%17/200,8585%19/200,9595%
/21Periódico
1/210,(047619)4,(761904)%2/210,(095238)9,(523809)%4/210,(190476)19,(047619)%5/210,(238095)23,(809523)%8/210,(380952)38,(095238)%10/210,(476190)47,(619047)%11/210,(523809)52,(380952)%13/210,(619047)61,(904761)%16/210,(761904)76,(190476)%17/210,(809523)80,(952380)%19/210,(904761)90,(476190)%20/210,(952380)95,(238095)%
/22Periódico
1/220,0(45)4,(54)%3/220,1(36)13,(63)%5/220,2(27)22,(72)%7/220,3(18)31,(81)%9/220,4(09)40,(90)%13/220,5(90)59,(09)%15/220,6(81)68,(18)%17/220,7(72)77,(27)%19/220,8(63)86,(36)%21/220,9(54)95,(45)%
/23Periódico
1/230,(0434782608695652173913)4,(3478260869565217391304)%2/230,(0869565217391304347826)8,(6956521739130434782608)%3/230,(1304347826086956521739)13,(0434782608695652173913)%4/230,(1739130434782608695652)17,(3913043478260869565217)%5/230,(2173913043478260869565)21,(7391304347826086956521)%6/230,(2608695652173913043478)26,(0869565217391304347826)%7/230,(3043478260869565217391)30,(4347826086956521739130)%8/230,(3478260869565217391304)34,(7826086956521739130434)%9/230,(3913043478260869565217)39,(1304347826086956521739)%10/230,(4347826086956521739130)43,(4782608695652173913043)%11/230,(4782608695652173913043)47,(8260869565217391304347)%12/230,(5217391304347826086956)52,(1739130434782608695652)%13/230,(5652173913043478260869)56,(5217391304347826086956)%14/230,(6086956521739130434782)60,(8695652173913043478260)%15/230,(6521739130434782608695)65,(2173913043478260869565)%16/230,(6956521739130434782608)69,(5652173913043478260869)%17/230,(7391304347826086956521)73,(9130434782608695652173)%18/230,(7826086956521739130434)78,(2608695652173913043478)%19/230,(8260869565217391304347)82,(6086956521739130434782)%20/230,(8695652173913043478260)86,(9565217391304347826086)%21/230,(9130434782608695652173)91,(3043478260869565217391)%22/230,(9565217391304347826086)95,(6521739130434782608695)%
/24Periódico
1/240,041(6)4,1(6)%5/240,208(3)20,8(3)%7/240,291(6)29,1(6)%11/240,458(3)45,8(3)%13/240,541(6)54,1(6)%17/240,708(3)70,8(3)%19/240,791(6)79,1(6)%23/240,958(3)95,8(3)%
Como ler
- Os algarismos entre parênteses se repetem sem fim: 0,1(6) é 0,16666…
- Se o denominador não tem outros fatores primos além de 2 e 5, o decimal termina.
- Frações iguais não ganham página própria: procurar 2/4 leva a 1/2.
- A porcentagem é o mesmo número vezes 100; os graus, o mesmo número vezes 360.
Perguntas frequentes
QQuanto é 0,375 em fração?
Três oitavos. Decimais exatos vêm de denominadores feitos só de 2 e 5, então um decimal de três casas como 0,375 fica sobre 8.
QPor que não existe 2/4?
Porque é o mesmo número que 1/2. Dois endereços para um valor dividiriam uma única resposta em duas páginas.
QO que significa a notação 0,(3)?
Que o que está entre parênteses se repete sem fim. 0,(3) é 0,3333… e 0,1(6) é 0,16666…
QQuais frações dão decimais periódicos?
Aquelas cujo denominador tem algum fator primo além de 2 e 5, porque 10 = 2 × 5.
QPor que parar no denominador 20?
Porque as frações com que as pessoas chegam cabem aí: as potências de dois (2, 4, 8, 16), o 5, 10 e 20, e os corriqueiros 3, 6, 7, 9 e 12.