127 fractions and their decimals
Every reduced fraction with a denominator up to 20. The decimals come from long division, so repeating parts are exact — 0.1(6), not 0.1666…
By denominator
A denominator made only of 2s and 5s gives a terminating decimal; bring in a 3 or a 7 and it repeats. That is because 10 = 2 × 5.
/2Terminating
1/20.550%
/3Repeating
1/30.(3)33.(3)%2/30.(6)66.(6)%
/4Terminating
1/40.2525%3/40.7575%
/5Terminating
1/50.220%2/50.440%3/50.660%4/50.880%
/6Repeating
1/60.1(6)16.(6)%5/60.8(3)83.(3)%
/7Repeating
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)%
/8Terminating
1/80.12512.5%3/80.37537.5%5/80.62562.5%7/80.87587.5%
/9Repeating
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)%
/10Terminating
1/100.110%3/100.330%7/100.770%9/100.990%
/11Repeating
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)%
/12Repeating
1/120.08(3)8.(3)%5/120.41(6)41.(6)%7/120.58(3)58.(3)%11/120.91(6)91.(6)%
/13Repeating
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)%
/14Repeating
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)%
/15Repeating
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)%
/16Terminating
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%
/17Repeating
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)%
/18Repeating
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)%
/19Repeating
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)%
/20Terminating
1/200.055%3/200.1515%7/200.3535%9/200.4545%11/200.5555%13/200.6565%17/200.8585%19/200.9595%
/21Repeating
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)%
/22Repeating
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)%
/23Repeating
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)%
/24Repeating
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)%
How to read this
- The digits in brackets repeat for ever — 0.1(6) means 0.16666…
- If the denominator has no prime factors besides 2 and 5, the decimal stops.
- Equal fractions do not get separate pages: looking up 2/4 brings you to 1/2.
- A percentage is the same number times 100; degrees are the same number times 360.
Frequently asked questions
QWhat is 0.375 as a fraction?
Three eighths. Decimals that stop come from denominators built only of 2s and 5s, so a three-place decimal like 0.375 sits over 8.
QWhy is there no 2/4?
Because it is the same number as 1/2. Two addresses for one value would split a single answer across two pages.
QWhat does the 0.(3) notation mean?
Whatever is inside the brackets repeats for ever. 0.(3) is 0.3333… and 0.1(6) is 0.16666…
QWhich fractions give repeating decimals?
Any whose denominator has a prime factor other than 2 or 5 — because 10 = 2 × 5.
QWhy stop at a denominator of 20?
Because the fractions people arrive with fall inside it: the powers of two (2, 4, 8, 16), 5, 10 and 20, and the everyday 3, 6, 7, 9 and 12.