2206:
1385:, 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
1818:
1700:
2174:
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
1554:
1460:
284:
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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
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672:
1602:
404:
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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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553:
333:
111:
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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:
1351:
902:
504:
1170:
64:
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.
1910:
Ubiratan D'Ambrósio; Joseph W. Dauben; Karen Hunger
Parshall (2014). "Mathematics Education in America in the Premodern Period". In Alexander Karp; Gert Schubring (eds.).
1395:
219:
1813:{\displaystyle x={\frac {20\ {\text{houses}}\times 100\ {\text{days}}\times 6\ {\text{builders}}}{8\ {\text{houses}}\times 10\ {\text{builders}}}}=150\ {\text{days}}.}
914:
790:
453:
993:
1695:{\displaystyle {\frac {\frac {8\ {\text{houses}}}{100\ {\text{days}}}}{6\ {\text{builders}}}}={\frac {\frac {20\ {\text{houses}}}{x}}{10\ {\text{builders}}}},}
1559:
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."
608:
1253:
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
807:
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
1594:
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?
1830:" 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".
1257:
maths education and still figures in the French national curriculum for secondary education, and in the primary education curriculum of Spain.
1919:
2193:
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.
289:
The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation
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2080:
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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.
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130:
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1565:
refers to his use of the rule of three in estimating the number of species in a newly discerned genus. In a letter to
1549:{\displaystyle x={\frac {12\ {\text{shillings}}\times 6\ {\text{yards}}}{4\ {\text{yards}}}}=18\ {\text{shillings}}.}
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1185:
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in 1855, Charles Darwin declared “I have no faith in anything short of actual measurement and the Rule of Three.”
1958:"Real Decreto 126/2014, de 28 de febrero, por el que se establece el currículo básico de la Educación Primaria"
1229:
1228:
Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the
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50:
2182:
799:. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students.
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where the variable to be evaluated is in the right-hand denominator, the rule of three states that
2001:
1455:{\displaystyle {\frac {4\ {\text{yards}}}{12\ {\text{shillings}}}}={\frac {6\ {\text{yards}}}{x}}}
860:
is a variable we are interested in solving for, we can use cross-multiplication to determine that
279:{\displaystyle {\frac {a}{b}}\nwarrow {\frac {c}{d}},\quad {\frac {a}{b}}\nearrow {\frac {c}{d}}.}
2053:
1566:
1234:
194:
46:
1589:, which involved finding an unknown value where five rather than three other values are known.
977:{\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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1053:{\displaystyle x={\frac {7\ {\text{hours}}\times 90\ {\text{miles}}}{3\ {\text{hours}}}},}
1995:
667:{\displaystyle {\frac {a}{b}}\times {\frac {d}{d}}={\frac {c}{d}}\times {\frac {b}{b}},}
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1960:. Ministerio de Educación, Cultura y Deporte. 28 February 2014. pp. 19349–19420
17:
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We can reduce the fractions to lowest terms by noting that the two occurrences of
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38:
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Each step in these procedures is based on a single, fundamental property of
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2183:
The Rule of Three as applied by
Michael of Rhodes in the fifteenth century
1389:
to stand for the cost of 6 yards of cloth, writing down the equation
796:
2140:
Merriam-Webster's Guide to
Everyday Math: A Home and Business Reference
1892:
1981:
The Nine
Chapters on the Mathematical Art: Companion and Commentary
1573:
adopted this declaration as the motto of his newly founded journal
198:
399:{\displaystyle {\frac {a}{b}}\times bd={\frac {c}{d}}\times bd.}
61:
to simplify the equation or determine the value of a variable.
183:{\displaystyle ad=bc\quad {\text{or}}\quad a={\frac {bc}{d}}.}
1979:
Shen
Kangshen; John N. Crossley; Anthony W.-C. Lun (1999).
413:
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
1882:, though that usage is rare compared to other uses of
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2100:(Illustrated ed.). Harvard University Press.
2075:(Illustrated ed.). Harvard University Press.
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1465:and then using cross-multiplication to calculate
1938:. French ministry of education. 30 December 2012
1912:Handbook on the History of Mathematics Education
2014:Concise Oxford Dictionary of Quotations, 1964.
1705:which, with cross-multiplication twice, gives
1299:{\displaystyle {\frac {a}{b}}={\frac {c}{x}},}
1218:{\displaystyle {\frac {a}{1}}={\frac {x}{d}}.}
846:{\displaystyle {\frac {x}{b}}={\frac {c}{d}},}
548:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
328:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
106:{\displaystyle {\frac {a}{b}}={\frac {c}{d}},}
8:
2030:Studies in History and Philosophy of Science
124:are not zero, one can cross-multiply to get
1878:This was sometimes also referred to as the
1585:An extension to the rule of three was the
2026:"Charles Darwin as a statistical thinker"
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2098:The Seven Pillars of Statistical Wisdom
2073:The Seven Pillars of Statistical Wisdom
1897:Brewer's Dictionary of Phrase and Fable
1871:
1592:An example of such a problem might be
1096:{\displaystyle x=210\ {\text{miles}}.}
2167:Abraham Lincoln and the Rule of Three
1136:Note that even simple equations like
7:
2096:Stigler, Stephen M (March 7, 2016).
2071:Stigler, Stephen M (March 7, 2016).
2179:- facsimile of the relevant section
1983:. Oxford: Oxford University Press.
1936:"Socle de connaissances, pilier 3"
1346:{\displaystyle x={\frac {bc}{a}}.}
897:{\displaystyle x={\frac {bc}{d}}.}
499:{\displaystyle a={\frac {bc}{d}}.}
25:
2188:The Rule Of Three in Mother Goose
1914:. Springer Science. p. 177.
2204:
2000:. London: John Hawkins. p.
1165:{\displaystyle a={\frac {x}{d}}}
417:on the right-hand side, leaving
49:, given an equation between two
1179:term is implicitly equal to 1:
249:
155:
149:
1899:. Philadelphia: Henry Altemus.
746:Cancel the common denominator
260:
233:
1:
1596:, and this can be set up as
1260:For an equation of the form
2042:10.1016/j.shpsa.2022.08.005
2248:
1132:So, 30mph×7hours=210miles.
602:= 1 on the right, getting
29:
2142:. Merriam-Webster, 1998,
987:Cross-multiplying yields
1230:least common denominator
30:Not to be confused with
2227:Fractions (mathematics)
1828:The Mad Gardener's Song
1364:of the proportion, and
580:= 1 on the left and by
67:Given an equation like
1994:Edward Cocker (1702).
1814:
1696:
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1360:is referred to as the
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1232:. This step is called
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785:{\displaystyle ad=cb.}
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448:{\displaystyle ad=bc,}
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27:Mathematical technique
2024:Ariew, André (2022).
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43:elementary arithmetic
2213:at Wikimedia Commons
2211:Cross-multiplication
1997:Cocker's Arithmetick
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1587:double rule of three
1581:Double rule of three
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18:Cross multiplication
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47:elementary algebra
41:, specifically in
2209:Media related to
1921:978-1-4614-9155-2
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2199:External links
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2138:Brian Burell:
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531:
526:
523:
507:
506:
495:
490:
486:
483:
477:
474:
456:
455:
444:
441:
438:
435:
432:
429:
407:
406:
395:
392:
389:
386:
381:
378:
373:
370:
367:
364:
359:
356:
336:
335:
324:
319:
316:
311:
306:
303:
287:
286:
275:
270:
267:
262:
257:
254:
248:
243:
240:
235:
230:
227:
210:
207:
191:
190:
179:
174:
170:
167:
161:
158:
148:
145:
142:
139:
136:
114:
113:
102:
97:
94:
89:
84:
81:
59:cross-multiply
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2244:
2233:
2230:
2228:
2225:
2224:
2222:
2212:
2207:
2203:
2202:
2198:
2194:
2191:
2189:
2186:
2184:
2181:
2178:
2176:
2175:book-keeping.
2171:
2169:
2168:
2163:
2161:
2160:
2159:Rule of Three
2155:
2153:
2149:
2148:9780877796213
2145:
2141:
2137:
2136:
2132:
2125:, Chapter 12.
2124:
2123:
2117:
2114:
2109:
2103:
2099:
2092:
2089:
2084:
2078:
2074:
2067:
2064:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2020:
2017:
2011:
2008:
2003:
1999:
1998:
1990:
1987:
1982:
1975:
1972:
1959:
1953:
1950:
1937:
1931:
1928:
1923:
1917:
1913:
1906:
1903:
1898:
1894:
1893:"Golden Rule"
1890:
1885:
1881:
1875:
1872:
1865:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1837:
1833:
1831:
1829:
1825:
1824:Lewis Carroll
1807:
1796:
1793:
1779:
1776:
1765:
1752:
1749:
1738:
1735:
1724:
1718:
1715:
1708:
1707:
1706:
1689:
1675:
1670:
1658:
1651:
1637:
1623:
1610:
1599:
1598:
1597:
1595:
1590:
1588:
1580:
1578:
1576:
1572:
1568:
1564:
1560:
1543:
1532:
1529:
1515:
1502:
1499:
1488:
1482:
1479:
1472:
1471:
1470:
1447:
1435:
1429:
1415:
1402:
1392:
1391:
1390:
1384:
1380:
1377:
1375:
1363:
1340:
1335:
1331:
1328:
1322:
1319:
1312:
1311:
1310:
1293:
1288:
1285:
1280:
1275:
1272:
1263:
1262:
1261:
1258:
1256:
1252:
1251:rule of three
1245:Rule of three
1244:
1242:
1239:
1237:
1236:
1231:
1212:
1207:
1204:
1199:
1194:
1191:
1182:
1181:
1180:
1157:
1154:
1149:
1146:
1139:
1138:
1137:
1134:
1133:
1129:
1128:
1109:
1108:
1090:
1079:
1076:
1073:
1066:
1065:
1064:
1047:
1033:
1020:
1017:
1006:
1000:
997:
990:
989:
988:
971:
957:
944:
938:
924:
920:
911:
910:
909:
891:
886:
882:
879:
873:
870:
863:
862:
861:
840:
835:
832:
827:
822:
819:
810:
809:
808:
802:
800:
798:
779:
776:
773:
770:
767:
764:
757:
756:
755:
730:
724:
721:
716:
713:
707:
701:
698:
693:
690:
680:
679:
678:
661:
656:
653:
648:
643:
640:
635:
630:
627:
622:
617:
614:
605:
604:
603:
596:
588:
574:
566:
542:
537:
534:
529:
524:
521:
512:
511:
510:
493:
488:
484:
481:
475:
472:
465:
464:
463:
442:
439:
436:
433:
430:
427:
420:
419:
418:
393:
390:
387:
384:
379:
376:
371:
368:
365:
362:
357:
354:
345:
344:
343:
322:
317:
314:
309:
304:
301:
292:
291:
290:
273:
268:
265:
255:
252:
246:
241:
238:
228:
225:
216:
215:
214:
208:
206:
204:
200:
196:
177:
172:
168:
165:
159:
156:
146:
143:
140:
137:
134:
127:
126:
125:
100:
95:
92:
87:
82:
79:
70:
69:
68:
65:
62:
60:
56:
52:
48:
44:
40:
33:
32:Cross product
19:
2173:
2166:
2158:
2139:
2120:
2116:
2097:
2091:
2072:
2066:
2033:
2029:
2019:
2010:
1996:
1989:
1980:
1974:
1962:. Retrieved
1952:
1942:24 September
1940:. Retrieved
1930:
1911:
1905:
1896:
1879:
1874:
1855:Turn (angle)
1822:
1704:
1593:
1591:
1586:
1584:
1571:Karl Pearson
1561:
1558:
1464:
1381:
1378:
1373:
1361:
1355:
1308:
1259:
1250:
1248:
1240:
1233:
1227:
1174:
1135:
1131:
1130:
1111:
1110:
1106:
1105:
1062:
986:
906:
855:
806:
794:
745:
676:
594:
586:
572:
564:
558:multiply by
557:
508:
457:
408:
337:
288:
212:
201:as those of
192:
115:
66:
63:
58:
36:
2165:'Dr Math',
2157:'Dr Math',
2036:: 215–223.
1884:Golden Rule
1880:golden rule
1840:Cross-ratio
39:mathematics
2232:Arithmetic
2221:Categories
1866:References
1850:Trairāśika
1845:Odds ratio
1575:Biometrika
754:, leaving
57:, one can
2058:252246047
1777:×
1750:×
1736:×
1540:shillings
1500:×
1496:shillings
1423:shillings
1018:×
797:equations
649:×
623:×
462:—getting
385:×
363:×
261:↗
234:↖
209:Procedure
51:fractions
2050:36113233
1891:(1898).
1834:See also
1787:builders
1760:builders
1683:builders
1645:builders
1255:Colonial
342:—we get
2150:, pp.
1362:extreme
1125:
1116:90miles
1113:
1063:and so
677:and so
599:
583:
577:
561:
2177:, 1827
2152:85-101
2146:
2104:
2079:
2056:
2048:
1964:10 May
1918:
1886:. See
1800:
1783:
1773:houses
1769:
1756:
1742:
1732:houses
1728:
1679:
1666:houses
1662:
1641:
1627:
1618:houses
1614:
1536:
1519:
1506:
1492:
1439:
1419:
1406:
1127:=30mph
1122:3hours
1083:
1037:
1024:
1010:
961:
948:
928:
856:where
199:ratios
116:where
2054:S2CID
1523:yards
1510:yards
1443:yards
1410:yards
1374:means
1087:miles
1041:hours
1028:miles
1014:hours
965:hours
952:miles
932:hours
2144:ISBN
2102:ISBN
2077:ISBN
2046:PMID
1966:2022
1944:2015
1916:ISBN
1826:'s "
1804:days
1746:days
1631:days
1368:and
1249:The
120:and
45:and
2038:doi
2002:103
1797:150
1739:100
1624:100
1080:210
803:Use
193:In
53:or
37:In
2223::
2052:.
2044:.
2034:95
2032:.
2028:.
1895:.
1780:10
1725:20
1676:10
1659:20
1577:.
1533:18
1489:12
1469::
1416:12
1376:.
1238:.
1021:90
945:90
752:db
750:=
748:bd
340:bd
205:.
152:or
2110:.
2085:.
2060:.
2040::
2004:.
1968:.
1946:.
1924:.
1808:.
1794:=
1766:8
1753:6
1719:=
1716:x
1690:,
1671:x
1652:=
1638:6
1611:8
1544:.
1530:=
1516:4
1503:6
1483:=
1480:x
1467:x
1448:x
1436:6
1430:=
1403:4
1387:x
1370:c
1366:b
1358:a
1341:.
1336:a
1332:c
1329:b
1323:=
1320:x
1294:,
1289:x
1286:c
1281:=
1276:b
1273:a
1213:.
1208:d
1205:x
1200:=
1195:1
1192:a
1177:b
1158:d
1155:x
1150:=
1147:a
1119:/
1091:.
1077:=
1074:x
1048:,
1034:3
1007:7
1001:=
998:x
972:.
958:3
939:=
925:7
921:x
892:.
887:d
883:c
880:b
874:=
871:x
858:x
841:,
836:d
833:c
828:=
823:b
820:x
780:.
777:b
774:c
771:=
768:d
765:a
731:.
725:b
722:d
717:b
714:c
708:=
702:d
699:b
694:d
691:a
662:,
657:b
654:b
644:d
641:c
636:=
631:d
628:d
618:b
615:a
595:b
591:/
587:b
573:d
569:/
565:d
543:,
538:d
535:c
530:=
525:b
522:a
494:.
489:d
485:c
482:b
476:=
473:a
460:d
443:,
440:c
437:b
434:=
431:d
428:a
415:d
411:b
394:.
391:d
388:b
380:d
377:c
372:=
369:d
366:b
358:b
355:a
323:,
318:d
315:c
310:=
305:b
302:a
274:.
269:d
266:c
256:b
253:a
247:,
242:d
239:c
229:b
226:a
178:.
173:d
169:c
166:b
160:=
157:a
147:c
144:b
141:=
138:d
135:a
122:d
118:b
101:,
96:d
93:c
88:=
83:b
80:a
34:.
20:)
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