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Exact drive ratios of bicycle hub gears, with improper fractions
Brand Speeds Model[1] Shifter position
-4 -3 -2 -1 (near) unity +1[2] +2[3] +3[4] +4[5]
Ratio
Bendix 2-speeds Red Band, Yellow Band         15/22 1        
Blue Band         1 22/15      
3-speeds (See Sachs, SRAM)         47/64 1 64/47      
Rohloff Speedhub, 14-speed Each value in this row is the one directly below it, times the third-stage ratio of 221/541. 3315[6] 5304[7] 34255[8]  221/541 38896[9] 6851[10] 4862[11]  
11902[12] 16771[13] 95216[14] 83855[15] 12984[16] 8115[17]
For these ratios, the third stage is locked at unity.   15/22   24/31  155/176 1 176/155 31/24 22/15  
Sachs, SRAM 2-speeds Doppel-torpedo         16/21 1        
Automatic, Duomatic         1 64/47      
Orbit         48/65 1        
3-speeds 25,29        3/5     3/4  1        
Universal Dreigang          3/4  1 4/3      
53          3/4  1 4/3      
55          3/4  1 4/3      
All 3-speeds 1970-, DualDrive         47/64 1 64/47      
4-speed Universal Torpedo        3/5     3/4  1 4/3      
5-speeds Sachs H 5100, 5113, 5120, 5213        2/3     7/9  1 9/7 3/2    
SRAM P5 5-speeds       19/30   57/73 1 73/57 30/19    
7-speeds Sachs Torpedo H 7203, -13, -20    273/460   21/31   75/92 1 92/75 31/21 460/273  
Sachs Super 7, SRAM S7, MH7205, -15, -25  252/439   21/31   72/89 1 89/72 31/21 439/252  
9-speed SRAM I-Motion 9[18]   13/24   18/29    8/11   64/75 1 75/64 11/8 29/18 24/13[19]
12-speed Sachs Elan 12-speed Speeds 1 through 6: ? ? 1[20] ? ? ?  
Speeds 7 through 12: ? ? ? ? ? ?  
Shimano 3-speeds 333, SG-3S21, G-3S23, F, G, 3SC, 3CC          3/4  1 4/3      
Nexus Inter 3         11/15 1 15/11      
4-speed Nexus Inter 4         1 97/78 3/2 623/338  
7-speeds Nexus 7-speed     12/19   20/27 3348[21]  620/627[22] 340/297 279/209 17/11  
3971[23]
8-speeds Nexus, Alfine 8-speed   49/93 [24] 22589[25] 8194[26]  343/403[27] 1 461/377 166/117 21/13  
35061[28] 10881[29]
Sturmey-Archer (Also licensees and copies: Brampton, Hercules, NK, Schwinn-Approved, Sears, Styre, SunTour for the AW, BSA for the X.) 2-speeds S2          5/7  1        
T,TB,TBC,TBF,TBFC,TF          3/4  1        
TC close-ratio fixed         45/52 1        
3-speeds[30] Original 1902          4/5  1 5/4      
A, C, F, FN, FX, N, S, V, X, BSA         16/21 1 21/16      
AB, AB3, AG, AT, AT3, AW, AW3, AWC, C30, K, KB, KC, KT, KTC, QS-RC3, RD-RB3, RS-RF3, RS-RR3, RX-RD3, S30, S3B, S3C, SAB3, SBR, S-RB3, S-RC3, S-RF3, SX-RB3, SX-RK3, TCW, TS-RC3, TS-RF3, XL-RD3, X-RD3          3/4  1 4/3      
AC         12/13 1 16/15      
AF, ASC fixed        3/4     9/10 1        
AM, AM Tandem         45/52 1 52/45      
AR         69/74 1 74/69      
KS          8/9  1 9/8      
KSW          6/7  1 7/6      
S3X     5/8 3/4 1        
SW         13/18 1 18/13      
4-speeds FB, FG, FW        2/3    15/19 1 19/15      
FC        3/4     9/10 1 12/11      
FM        2/3     6/7  1 9/8      
5-speeds AB5, AT5, S5, S5/1, S5/2, S5C, 5-Speed Alloy, 5 Star, Sprinter 5, XRD5, XRF5         2/3    15/19 1 19/15 3/2    
S50(W), S5C(W), S5S(W), S-RC5(W), S-RF5(W), XL-RD5(W), X-RC5(W), X-RD5(W), X-RF5(W)        5/8     3/4  1 4/3 8/5    
7-speeds AT7, S7, S7C, X-RD7, Sprinter 7[31]      3/5    20/29  115/143 1 143/115 29/20 5/3  
8-speeds X-RD8, X-RF8, X-RK8, X-RR8[32] This hub only gears up.  Gears 1 through 4: 1 41/32 29/20 120/73  
Gears 5 through 8: 1189/640 615/292 174/73 3567/1168[33]  
S80(W), X-RD8(W), X-RF8(W), X-RK8(W) This hub only gears up.  Gears 1 through 4:   1[34] ? ? ?  
Gears 5 through 8:   ? ? ? ?  
[1]
John Allen:
Lines in pink: need verification. Lines in blue: insufficient information or no information.
[2]
John Allen:
Unless otherwise indicated, fractions > 1 are formatted as text; Excel does not offer improper fractions as a formatting option.
[3]
John Allen:
Unless otherwise indicated, fractions > 1 are formatted as text; Excel does not offer improper fractions as a formatting option.
[4]
John Allen:
Unless otherwise indicated, fractions > 1 are formatted as text; Excel does not offer improper fractions as a formatting option.
[5]
John Allen:
Unless otherwise indicated, fractions > 1 are formatted as text; Excel does not offer improper fractions as a formatting option.
[6]
John Allen:
Numerator entered as an integer
[7]
John Allen:
Numerator entered as an integer
[8]
John Allen:
Numerator entered as an integer
[9]
John Allen:
Numerator entered as an integer
[10]
John Allen:
Numerator entered as an integer
[11]
John Allen:
Numerator entered as an integer
[12]
John Allen:
Denominator entered as an integer.
[13]
John Allen:
Denominator entered as an integer.
[14]
John Allen:
Denominator entered as an integer.
[15]
John Allen:
Denominator entered as an integer.
[16]
John Allen:
Denominator entered as an integer.
[17]
John Allen:
Denominator entered as an integer.
[18]
John Allen:
Reverse-engineered.
[19]
John Allen:
SRAM literature says 1.844 but calculation says 1.846.
[20]
John Allen:
This is unity ratio. Manufacturer only give other ratios to 2 decmial places.
[21]
John Allen:
Numerator entered as an integer
[22]
John Allen:
Ratio is achieved by gearing down, then up.
[23]
John Allen:
Denominator entered as an integer
[24]
John Allen:
This ratio is that of the fifth speed, times the first-stage ratio of 49/93.
[25]
John Allen:
Numerator entered as an integer. Ratio is that of the 6th speed times the first-stage ratio of 49/93.
[26]
John Allen:
Numerator entered as an integer. Ratio is that of the 7th speed times the first-stage ratios of 49/93
[27]
John Allen:
Ratio is that of the 8th speed, times the first-stage ratio of 0.527
[28]
John Allen:
Denominator entered as an integer. Ratio is that of the 6th speed times the first-stage ratio of 49/93.
[29]
John Allen:
Denominator entered as an integer. Ratio is that of the 7th speed times the first-stage ratio of 49/93.
[30]
John Allen:
[31]
John Allen:
Ratios are almost certainly correct, reverse engineered.
[32]
John Allen:
Ratios are derived from tooth counts in Sturmey-Archer technical information.
[33]
John Allen:
Number inserted as text; Excel will not generate fraction with 4-digit denominator.
[34]
John Allen:
Fractional ratios are unavailable. This is the only one which can be determined exactly.