2195:
1374:, the premier textbook in the 17th century, introduces its discussion of the rule of three with the problem "If 4 yards of cloth cost 12 shillings, what will 6 yards cost at that rate?" The rule of three gives the answer to this problem directly; whereas in modern arithmetic, we would solve it by introducing a variable
1807:
1689:
2163:
Pike's System of arithmetick abridged: designed to facilitate the study of the science of numbers, comprehending the most perspicuous and accurate rules, illustrated by useful examples: to which are added appropriate questions, for the examination of scholars, and a short system of
1543:
1449:
273:
971:
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For example, suppose we want to know how far a car will travel in 7 hours, if we know that its speed is constant and that it already travelled 90 miles in the last 3 hours. Converting the word problem into ratios, we get
1047:
661:
1591:
393:
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we can multiply the terms on each side by the same number, and the terms will remain equal. Therefore, if we multiply the fraction on each side by the product of the denominators of both sides—
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840:
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100:
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In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over:
1340:
891:
493:
1159:
53:
The method is also occasionally known as the "cross your heart" method because lines resembling a heart outline can be drawn to remember which things to multiply together.
1899:
Ubiratan D'Ambrósio; Joseph W. Dauben; Karen Hunger
Parshall (2014). "Mathematics Education in America in the Premodern Period". In Alexander Karp; Gert Schubring (eds.).
1384:
208:
1802:{\displaystyle x={\frac {20\ {\text{houses}}\times 100\ {\text{days}}\times 6\ {\text{builders}}}{8\ {\text{houses}}\times 10\ {\text{builders}}}}=150\ {\text{days}}.}
903:
779:
442:
982:
1684:{\displaystyle {\frac {\frac {8\ {\text{houses}}}{100\ {\text{days}}}}{6\ {\text{builders}}}}={\frac {\frac {20\ {\text{houses}}}{x}}{10\ {\text{builders}}}},}
1548:
An anonymous manuscript dated 1570 said: "Multiplication is vexation, / Division is as bad; / The Rule of three doth puzzle me, / And
Practice drives me mad."
597:
1242:
was a historical shorthand version for a particular form of cross-multiplication that could be taught to students by rote. It was considered the height of
796:
This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction. If we have an equation
1583:
If 6 builders can build 8 houses in 100 days, how many days would it take 10 builders to build 20 houses at the same rate?
1819:" includes the lines "He thought he saw a Garden-Door / That opened with a key: / He looked again, and found it was / A double Rule of Three".
1246:
maths education and still figures in the French national curriculum for secondary education, and in the primary education curriculum of Spain.
1908:
2182:
Rudyard
Kipling: You can work it out by Fractions or by simple Rule of Three, But the way of Tweedle-dum is not the way of Tweedle-dee.
278:
The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation
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This rule was already known to
Chinese mathematicians prior to the 2nd century CE, though it was not used in Europe until much later.
337:
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119:
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2199:
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1554:
refers to his use of the rule of three in estimating the number of species in a newly discerned genus. In a letter to
1538:{\displaystyle x={\frac {12\ {\text{shillings}}\times 6\ {\text{yards}}}{4\ {\text{yards}}}}=18\ {\text{shillings}}.}
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in 1855, Charles Darwin declared “I have no faith in anything short of actual measurement and the Rule of Three.”
1947:"Real Decreto 126/2014, de 28 de febrero, por el que se establece el currículo básico de la Educación Primaria"
1218:
1217:
Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the
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39:
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788:. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students.
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457:
31:
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1298:
where the variable to be evaluated is in the right-hand denominator, the rule of three states that
1990:
1444:{\displaystyle {\frac {4\ {\text{yards}}}{12\ {\text{shillings}}}}={\frac {6\ {\text{yards}}}{x}}}
849:
is a variable we are interested in solving for, we can use cross-multiplication to determine that
268:{\displaystyle {\frac {a}{b}}\nwarrow {\frac {c}{d}},\quad {\frac {a}{b}}\nearrow {\frac {c}{d}}.}
2042:
1555:
1223:
183:
35:
1578:, which involved finding an unknown value where five rather than three other values are known.
966:{\displaystyle {\frac {x}{7\ {\text{hours}}}}={\frac {90\ {\text{miles}}}{3\ {\text{hours}}}}.}
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Another justification of cross-multiplication is as follows. Starting with the given equation
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and we can divide both sides of the equation by any of the elements—in this case we will use
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1042:{\displaystyle x={\frac {7\ {\text{hours}}\times 90\ {\text{miles}}}{3\ {\text{hours}}}},}
1984:
656:{\displaystyle {\frac {a}{b}}\times {\frac {d}{d}}={\frac {c}{d}}\times {\frac {b}{b}},}
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1949:. Ministerio de Educación, Cultura y Deporte. 28 February 2014. pp. 19349–19420
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We can reduce the fractions to lowest terms by noting that the two occurrences of
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27:
2014:
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Each step in these procedures is based on a single, fundamental property of
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The Rule of Three as applied by
Michael of Rhodes in the fifteenth century
1378:
to stand for the cost of 6 yards of cloth, writing down the equation
785:
2129:
Merriam-Webster's Guide to
Everyday Math: A Home and Business Reference
1881:
1970:
The Nine
Chapters on the Mathematical Art: Companion and Commentary
1562:
adopted this declaration as the motto of his newly founded journal
187:
388:{\displaystyle {\frac {a}{b}}\times bd={\frac {c}{d}}\times bd.}
50:
to simplify the equation or determine the value of a variable.
172:{\displaystyle ad=bc\quad {\text{or}}\quad a={\frac {bc}{d}}.}
1968:
Shen
Kangshen; John N. Crossley; Anthony W.-C. Lun (1999).
402:
on the left-hand side cancel, as do the two occurrences of
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are solved using cross-multiplication, since the missing
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the same calculation can be achieved by considering the
1871:, though that usage is rare compared to other uses of
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2089:(Illustrated ed.). Harvard University Press.
2064:(Illustrated ed.). Harvard University Press.
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1454:and then using cross-multiplication to calculate
1927:. French ministry of education. 30 December 2012
1901:Handbook on the History of Mathematics Education
2003:Concise Oxford Dictionary of Quotations, 1964.
1694:which, with cross-multiplication twice, gives
1288:{\displaystyle {\frac {a}{b}}={\frac {c}{x}},}
1207:{\displaystyle {\frac {a}{1}}={\frac {x}{d}}.}
835:{\displaystyle {\frac {x}{b}}={\frac {c}{d}},}
537:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
317:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
95:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
8:
2019:Studies in History and Philosophy of Science
113:are not zero, one can cross-multiply to get
1867:This was sometimes also referred to as the
1574:An extension to the rule of three was the
2015:"Charles Darwin as a statistical thinker"
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2087:The Seven Pillars of Statistical Wisdom
2062:The Seven Pillars of Statistical Wisdom
1886:Brewer's Dictionary of Phrase and Fable
1860:
1581:An example of such a problem might be
1085:{\displaystyle x=210\ {\text{miles}}.}
2156:Abraham Lincoln and the Rule of Three
1125:Note that even simple equations like
7:
2085:Stigler, Stephen M (March 7, 2016).
2060:Stigler, Stephen M (March 7, 2016).
2168:- facsimile of the relevant section
1972:. Oxford: Oxford University Press.
1925:"Socle de connaissances, pilier 3"
1335:{\displaystyle x={\frac {bc}{a}}.}
886:{\displaystyle x={\frac {bc}{d}}.}
488:{\displaystyle a={\frac {bc}{d}}.}
14:
2177:The Rule Of Three in Mother Goose
1903:. Springer Science. p. 177.
2193:
1989:. London: John Hawkins. p.
1154:{\displaystyle a={\frac {x}{d}}}
406:on the right-hand side, leaving
38:, given an equation between two
1168:term is implicitly equal to 1:
238:
144:
138:
1888:. Philadelphia: Henry Altemus.
735:Cancel the common denominator
249:
222:
1:
1585:, and this can be set up as
1249:For an equation of the form
2031:10.1016/j.shpsa.2022.08.005
2237:
1121:So, 30mph×7hours=210miles.
591:= 1 on the right, getting
18:
2131:. Merriam-Webster, 1998,
976:Cross-multiplying yields
1219:least common denominator
19:Not to be confused with
2216:Fractions (mathematics)
1817:The Mad Gardener's Song
1353:of the proportion, and
569:= 1 on the left and by
56:Given an equation like
1983:Edward Cocker (1702).
1803:
1685:
1539:
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1349:is referred to as the
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1221:. This step is called
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774:{\displaystyle ad=cb.}
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437:{\displaystyle ad=bc,}
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16:Mathematical technique
2013:Ariew, André (2022).
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32:elementary arithmetic
2202:at Wikimedia Commons
2200:Cross-multiplication
1986:Cocker's Arithmetick
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1570:Double rule of three
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30:, specifically in
2198:Media related to
1910:978-1-4614-9155-2
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427:
424:
421:
418:
396:
395:
384:
381:
378:
375:
370:
367:
362:
359:
356:
353:
348:
345:
325:
324:
313:
308:
305:
300:
295:
292:
276:
275:
264:
259:
256:
251:
246:
243:
237:
232:
229:
224:
219:
216:
199:
196:
180:
179:
168:
163:
159:
156:
150:
147:
137:
134:
131:
128:
125:
103:
102:
91:
86:
83:
78:
73:
70:
48:cross-multiply
15:
13:
10:
9:
6:
4:
3:
2:
2233:
2222:
2219:
2217:
2214:
2213:
2211:
2201:
2196:
2192:
2191:
2187:
2183:
2180:
2178:
2175:
2173:
2170:
2167:
2165:
2164:book-keeping.
2160:
2158:
2157:
2152:
2150:
2149:
2148:Rule of Three
2144:
2142:
2138:
2137:9780877796213
2134:
2130:
2126:
2125:
2121:
2114:, Chapter 12.
2113:
2112:
2106:
2103:
2098:
2092:
2088:
2081:
2078:
2073:
2067:
2063:
2056:
2053:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2009:
2006:
2000:
1997:
1992:
1988:
1987:
1979:
1976:
1971:
1964:
1961:
1948:
1942:
1939:
1926:
1920:
1917:
1912:
1906:
1902:
1895:
1892:
1887:
1883:
1882:"Golden Rule"
1879:
1874:
1870:
1864:
1861:
1854:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1826:
1822:
1820:
1818:
1814:
1813:Lewis Carroll
1796:
1785:
1782:
1768:
1765:
1754:
1741:
1738:
1727:
1724:
1713:
1707:
1704:
1697:
1696:
1695:
1678:
1664:
1659:
1647:
1640:
1626:
1612:
1599:
1588:
1587:
1586:
1584:
1579:
1577:
1569:
1567:
1565:
1561:
1557:
1553:
1549:
1532:
1521:
1518:
1504:
1491:
1488:
1477:
1471:
1468:
1461:
1460:
1459:
1436:
1424:
1418:
1404:
1391:
1381:
1380:
1379:
1373:
1369:
1366:
1364:
1352:
1329:
1324:
1320:
1317:
1311:
1308:
1301:
1300:
1299:
1282:
1277:
1274:
1269:
1264:
1261:
1252:
1251:
1250:
1247:
1245:
1241:
1240:rule of three
1234:Rule of three
1233:
1231:
1228:
1226:
1225:
1220:
1201:
1196:
1193:
1188:
1183:
1180:
1171:
1170:
1169:
1146:
1143:
1138:
1135:
1128:
1127:
1126:
1123:
1122:
1118:
1117:
1098:
1097:
1079:
1068:
1065:
1062:
1055:
1054:
1053:
1036:
1022:
1009:
1006:
995:
989:
986:
979:
978:
977:
960:
946:
933:
927:
913:
909:
900:
899:
898:
880:
875:
871:
868:
862:
859:
852:
851:
850:
829:
824:
821:
816:
811:
808:
799:
798:
797:
791:
789:
787:
768:
765:
762:
759:
756:
753:
746:
745:
744:
719:
713:
710:
705:
702:
696:
690:
687:
682:
679:
669:
668:
667:
650:
645:
642:
637:
632:
629:
624:
619:
616:
611:
606:
603:
594:
593:
592:
585:
577:
563:
555:
531:
526:
523:
518:
513:
510:
501:
500:
499:
482:
477:
473:
470:
464:
461:
454:
453:
452:
431:
428:
425:
422:
419:
416:
409:
408:
407:
382:
379:
376:
373:
368:
365:
360:
357:
354:
351:
346:
343:
334:
333:
332:
311:
306:
303:
298:
293:
290:
281:
280:
279:
262:
257:
254:
244:
241:
235:
230:
227:
217:
214:
205:
204:
203:
197:
195:
193:
189:
185:
166:
161:
157:
154:
148:
145:
135:
132:
129:
126:
123:
116:
115:
114:
89:
84:
81:
76:
71:
68:
59:
58:
57:
54:
51:
49:
45:
41:
37:
33:
29:
22:
21:Cross product
2162:
2155:
2147:
2128:
2109:
2105:
2086:
2080:
2061:
2055:
2022:
2018:
2008:
1999:
1985:
1978:
1969:
1963:
1951:. Retrieved
1941:
1931:24 September
1929:. Retrieved
1919:
1900:
1894:
1885:
1868:
1863:
1844:Turn (angle)
1811:
1693:
1582:
1580:
1575:
1573:
1560:Karl Pearson
1550:
1547:
1453:
1370:
1367:
1362:
1350:
1344:
1297:
1248:
1239:
1237:
1229:
1222:
1216:
1163:
1124:
1120:
1119:
1100:
1099:
1095:
1094:
1051:
975:
895:
844:
795:
783:
734:
665:
583:
575:
561:
553:
547:multiply by
546:
497:
446:
397:
326:
277:
201:
190:as those of
181:
104:
55:
52:
47:
25:
2154:'Dr Math',
2146:'Dr Math',
2025:: 215–223.
1873:Golden Rule
1869:golden rule
1829:Cross-ratio
28:mathematics
2221:Arithmetic
2210:Categories
1855:References
1839:Trairāśika
1834:Odds ratio
1564:Biometrika
743:, leaving
46:, one can
2047:252246047
1766:×
1739:×
1725:×
1529:shillings
1489:×
1485:shillings
1412:shillings
1007:×
786:equations
638:×
612:×
451:—getting
374:×
352:×
250:↗
223:↖
198:Procedure
40:fractions
2039:36113233
1880:(1898).
1823:See also
1776:builders
1749:builders
1672:builders
1634:builders
1244:Colonial
331:—we get
2139:, pp.
1351:extreme
1114:
1105:90miles
1102:
1052:and so
666:and so
588:
572:
566:
550:
2166:, 1827
2141:85-101
2135:
2093:
2068:
2045:
2037:
1953:10 May
1907:
1875:. See
1789:
1772:
1762:houses
1758:
1745:
1731:
1721:houses
1717:
1668:
1655:houses
1651:
1630:
1616:
1607:houses
1603:
1525:
1508:
1495:
1481:
1428:
1408:
1395:
1116:=30mph
1111:3hours
1072:
1026:
1013:
999:
950:
937:
917:
845:where
188:ratios
105:where
2043:S2CID
1512:yards
1499:yards
1432:yards
1399:yards
1363:means
1076:miles
1030:hours
1017:miles
1003:hours
954:hours
941:miles
921:hours
2133:ISBN
2091:ISBN
2066:ISBN
2035:PMID
1955:2022
1933:2015
1905:ISBN
1815:'s "
1793:days
1735:days
1620:days
1357:and
1238:The
109:and
34:and
2027:doi
1991:103
1786:150
1728:100
1613:100
1069:210
792:Use
182:In
42:or
26:In
2212::
2041:.
2033:.
2023:95
2021:.
2017:.
1884:.
1769:10
1714:20
1665:10
1648:20
1566:.
1522:18
1478:12
1458::
1405:12
1365:.
1227:.
1010:90
934:90
741:db
739:=
737:bd
329:bd
194:.
141:or
2099:.
2074:.
2049:.
2029::
1993:.
1957:.
1935:.
1913:.
1797:.
1783:=
1755:8
1742:6
1708:=
1705:x
1679:,
1660:x
1641:=
1627:6
1600:8
1533:.
1519:=
1505:4
1492:6
1472:=
1469:x
1456:x
1437:x
1425:6
1419:=
1392:4
1376:x
1359:c
1355:b
1347:a
1330:.
1325:a
1321:c
1318:b
1312:=
1309:x
1283:,
1278:x
1275:c
1270:=
1265:b
1262:a
1202:.
1197:d
1194:x
1189:=
1184:1
1181:a
1166:b
1147:d
1144:x
1139:=
1136:a
1108:/
1080:.
1066:=
1063:x
1037:,
1023:3
996:7
990:=
987:x
961:.
947:3
928:=
914:7
910:x
881:.
876:d
872:c
869:b
863:=
860:x
847:x
830:,
825:d
822:c
817:=
812:b
809:x
769:.
766:b
763:c
760:=
757:d
754:a
720:.
714:b
711:d
706:b
703:c
697:=
691:d
688:b
683:d
680:a
651:,
646:b
643:b
633:d
630:c
625:=
620:d
617:d
607:b
604:a
584:b
580:/
576:b
562:d
558:/
554:d
532:,
527:d
524:c
519:=
514:b
511:a
483:.
478:d
474:c
471:b
465:=
462:a
449:d
432:,
429:c
426:b
423:=
420:d
417:a
404:d
400:b
383:.
380:d
377:b
369:d
366:c
361:=
358:d
355:b
347:b
344:a
312:,
307:d
304:c
299:=
294:b
291:a
263:.
258:d
255:c
245:b
242:a
236:,
231:d
228:c
218:b
215:a
167:.
162:d
158:c
155:b
149:=
146:a
136:c
133:b
130:=
127:d
124:a
111:d
107:b
90:,
85:d
82:c
77:=
72:b
69:a
23:.
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