1343:, which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparison works only when values being compared are consistent, like always expressing width in relation to height.
2939:
45:
1211:
concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason.
2908:
720:
and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
761:, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities
2070:(as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. The probability of success is 30%. In every ten trials, there are expected to be three wins and seven losses.
1210:
If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice
707:
developed a theory of ratio and proportion as applied to numbers. The
Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding
102:
contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount
1271:
If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In
1214:
Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare the quantities of any two of the
719:
The existence of multiple theories seems unnecessarily complex since ratios are, to a large extent, identified with quotients and their prospective values. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology
1263:
If a mixture contains substances A, B, C and D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total
800:
Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's
2782:, above, is known as rate, and illustrates a comparison between two variables with difference units. (...) A ratio of this sort produces a unique, new concept with its own entity, and this new concept is usually not considered a ratio, per se, but a rate or density."
2111:
ratios are usually expressed as weight/volume fractions. For example, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions.
1358:
Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3. The verbal equivalent is "40 is to 60 as 2 is to 3."
642:
For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement.
748:
Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity
1381:
is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4) Alternatively, it can be written as 0.8:1 (dividing both sides by 5).
756:
Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. Euclid defines a ratio as between two quantities
1355:(as fractions are) by dividing each quantity by the common factors of all the quantities. As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers.
740:
of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning
1813:
588:
646:
The meaning of such a proportion of ratios with more than two terms is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side.
106:
The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be
637:
732:
has 18 definitions, all of which relate to ratios. In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a
2043:
1938:
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Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities
1252:
If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the
1696:
2546:
Decimal fractions are frequently used in technological areas where ratio comparisons are important, such as aspect ratios (imaging), compression ratios (engines or data storage), etc.
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is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at
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has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive
1827:: even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio.
1385:
Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it a
1647:
684:("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used the word
164:
with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero)
716:. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.
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have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle.
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If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older
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550:
2762:"Velocity" can be defined as the ratio... "Population density" is the ratio... "Gasoline consumption" is measure as the ratio...
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As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms
2155:
602:
964:. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".
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entities covered by the ratio: for example, from a ratio of 2:3:7 we can infer that the quantity of the second entity is
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A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in
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derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is
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1330:, or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.
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Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a
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Ratios are sometimes used with three or even more terms, e.g., the proportion for the edge lengths of a "
2805:, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
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1701:
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the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.
712:) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to
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592:(unplaned measurements; the first two numbers are reduced slightly when the wood is planed smooth)
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446:. This latter form, when spoken or written in the English language, is often expressed as
8:
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Since all information is expressed in terms of ratios (the individual numbers denoted by
2098:. Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2.
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In general, a comparison of the quantities of a two-entity ratio can be expressed as a
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745:) is a part that, when multiplied by an integer greater than one, gives the quantity.
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2832:. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. 1908. pp. 112ff.
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also used to denote division or scale; for that mathematical use 2236 ∶ is preferred
310:). This can be expressed as a simple or a decimal fraction, or as a percentage, etc.
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1268:: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10).
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Euclid collected the results appearing in the
Elements from earlier sources. The
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A ratio may be specified either by giving both constituting numbers, written as "
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can be reduced by changing the first value to 60 seconds, so the ratio becomes
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Ginn and
Company (1925) pp. 477ff. Reprinted 1958 by Dover Publications.
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quantities (quantities whose ratio, as value of a fraction, amounts to an
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Consequently, a ratio may be considered as an ordered pair of numbers, a
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2784:, "Ratio and Proportion: Research and Teaching in Mathematics Teachers"
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On the other hand, there are non-dimensionless quotients, also known as
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of a quantity is another quantity that "measures" it and conversely, a
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Ratio and
Proportion: Research and Teaching in Mathematics Teachers
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Definition 6 says that quantities that have the same ratio are
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1808:{\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.}
655:
It is possible to trace the origin of the word "ratio" to the
343:, although Unicode also provides a dedicated ratio character,
37:"is to" redirects here. For the grammatical construction, see
2501:"ISO 80000-1:2022(en) Quantities and units — Part 1: General"
2423:
2082:, as in the case they relate quantities in units of the same
671:
666:
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a good concrete mix (in volume units) is sometimes quoted as
38:
583:{\displaystyle {\text{thickness : width : length }}=2:4:10;}
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90:
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632:{\displaystyle {\text{cement : sand : gravel }}=1:2:4.}
2744:
David Ben-Chaim; Yaffa Keret; Bat-Sheva Ilany (2012).
2005:
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Sometimes it is useful to write a ratio in the form 1:
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199:. A quotient of two quantities that are measured with
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The locations of points relative to a triangle with
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This equation has the positive, irrational solution
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274:as denominator that represents the quotient (i.e.,
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1478:'s circumference to its diameter, which is called
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1098:. Definitions 9 and 10 apply this, saying that if
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394:A statement expressing the equality of two ratios
302:
1409:). The earliest discovered example, found by the
27:Relationship between two numbers of the same kind
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2090:are initially different. For example, the ratio
2038:{\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},}
801:definition of equality is that given quantities
700:("proportionality") for the equality of ratios.
1933:{\displaystyle a:b=(2a+b):a\quad (=(2+b/a):1),}
2829:The thirteen books of Euclid's Elements, vol 2
2549:
2147:are often expressed in extended ratio form as
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1413:, is the ratio of the length of the diagonal
509:. The equality of three or more ratios, like
2813:2nd ed. (1916) Dodd Mead & Co. pp270-271
2045:so again at least one of the two quantities
1750:which has the positive, irrational solution
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2107:(sometimes also as ratios). In chemistry,
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833:if and only if, for any positive integers
2750:. Springer Science & Business Media.
2555:
2115:
1691:{\displaystyle x=1+{\tfrac {1}{x}}\quad }
322:, the two-dot character is sometimes the
172:, and may sometimes be natural numbers.
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2713:Encyclopædia Britannica Eleventh Edition
2194:, and therefore the ratio of weights at
2053:in the silver ratio must be irrational.
670:). Early translators rendered this into
650:
485:are called the terms of the proportion.
125:", or by giving just the value of their
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2811:New International Encyclopedia, Vol. 19
1401:Ratios may also be established between
723:
544:" that is ten inches long is therefore
14:
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2579:
175:A more specific definition adopted in
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2820:Fundamentals of practical mathematics
1507:, which is defined by the proportion
692:("proportion") to indicate ratio and
1613:{\displaystyle \quad a:b=(1+b/a):1.}
1396:
1301:, or 40% of the whole is apples and
1170:Number of terms and use of fractions
314:When a ratio is written in the form
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2277:, and therefore distances to sides
1623:Taking the ratios as fractions and
24:
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2594:The Unicode Standard, Version 15.0
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1740:{\displaystyle \quad x^{2}-x-1=0,}
1474:Another example is the ratio of a
1467:{\displaystyle a:d=1:{\sqrt {2}}.}
25:
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2646:Belle Group concrete mixing hints
1554:{\displaystyle a:b=(a+b):a\quad }
1248:Proportions and percentage ratios
1054:. This is extended to four terms
793:. This condition is known as the
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2617:from the Encyclopædia Britannica
556:thickness : width : length
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48:The ratio of width to height of
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1323:{\displaystyle {\tfrac {3}{5}}}
1294:{\displaystyle {\tfrac {2}{5}}}
1237:{\displaystyle {\tfrac {3}{7}}}
1200:{\displaystyle {\tfrac {2}{3}}}
999:if there are positive integers
303:{\displaystyle {\tfrac {A}{B}}}
2636:New International Encyclopedia
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2852:History of Mathematics, vol 2
2778:. The first type defined by
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2364:Proportionality (mathematics)
1981:{\displaystyle x^{2}-2x-1=0.}
1842:is defined by the proportion
2701:Heath, reference for section
2096:60 seconds : 40 seconds
2092:one minute : 40 seconds
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608:cement : sand : gravel
7:
2399:Rule of three (mathematics)
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2217:, a point with coordinates
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1207:that of the second entity.
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2324:Displacement–length ratio
1254:lowest common denominator
1038:to be in proportion when
2656:Penny Cyclopædia, p. 307
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1499:of two (mostly) lengths
1417:to the length of a side
949:(dividing both terms by
213:Notation and terminology
2818:"Ratio and Proportion"
2359:Price–performance ratio
2156:barycentric coordinates
2056:
1495:Also well known is the
1146:are in proportion then
1110:are in proportion then
195:measured with the same
3092:Elementary mathematics
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2329:Dimensionless quantity
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2801:The Penny Cyclopædia
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1642:{\displaystyle a:b}
795:Archimedes property
193:physical quantities
2914:Division and ratio
2557:Weisstein, Eric W.
2456:www.mathsisfun.com
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1196:
1187:
1156:triplicate ratio
948:
946:
945:
940:
937:
928:
926:
925:
920:
917:
759:of the same type
698:
696:proportionalitas
690:
682:
663:
662:
638:
636:
635:
630:
610:
607:
589:
587:
586:
581:
558:
555:
355:
352:
349:
347:
342:
339:
336:
334:
309:
307:
306:
301:
299:
290:
170:rational numbers
150:
148:
146:
145:
140:
137:
97:
96:
93:
92:
87:
86:
81:
80:
77:
74:
71:
21:
3122:
3121:
3117:
3116:
3115:
3113:
3112:
3111:
3082:
3081:
3080:
3075:
3046:Just intonation
2973:
2963:
2960:
2957:
2956:
2954:
2953:
2942:
2938:
2933:
2911:
2900:
2891:
2861:
2835:
2834:
2826:
2795:
2793:Further reading
2790:
2776:Ratio as a Rate
2772:
2768:
2758:
2742:
2738:
2733:
2729:
2724:
2720:
2709:
2705:
2700:
2696:
2691:
2687:
2682:
2678:
2673:
2669:
2664:
2660:
2655:
2651:
2644:
2640:
2635:
2631:
2626:
2622:
2615:
2611:
2599:
2597:
2589:
2585:
2584:
2580:
2571:
2569:
2554:
2550:
2545:
2541:
2536:
2532:
2522:
2518:
2509:
2507:
2498:
2494:
2485:
2483:
2473:
2469:
2460:
2458:
2450:
2449:
2445:
2440:
2436:
2432:
2384:Ratio (Twitter)
2374:Ratio estimator
2334:Financial ratio
2310:
2269:) in the ratio
2249:) in the ratio
2118:
2095:
2091:
2076:
2065:
2059:
2025:
2004:
1996:
1993:
1992:
1951:
1947:
1945:
1942:
1941:
1907:
1850:
1847:
1846:
1839:
1835:
1830:Similarly, the
1788:
1781:
1778:
1763:
1755:
1752:
1751:
1710:
1706:
1703:
1700:
1699:
1675:
1661:
1658:
1657:
1650:
1628:
1625:
1624:
1593:
1566:
1563:
1562:
1515:
1512:
1511:
1504:
1500:
1480:
1454:
1434:
1431:
1430:
1425:, which is the
1418:
1414:
1403:incommensurable
1399:
1349:
1308:
1306:
1303:
1302:
1279:
1277:
1274:
1273:
1250:
1222:
1220:
1217:
1216:
1185:
1183:
1180:
1179:
1172:
1120:duplicate ratio
941:
938:
933:
932:
930:
921:
918:
913:
912:
910:
901:both positive,
726:
653:
606:
604:
601:
600:
554:
552:
549:
548:
505:are called its
493:are called its
353:
350:
345:
344:
340:
337:
332:
331:
288:
286:
283:
282:
215:
179:(especially in
166:natural numbers
141:
138:
133:
132:
130:
129:
89:
83:
68:
64:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3120:
3110:
3109:
3104:
3099:
3094:
3077:
3076:
3074:
3073:
3068:
3063:
3058:
3053:
3048:
3043:
3042:
3041:
3031:
3026:
3025:
3024:
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2978:
2975:
2974:
2972:
2971:
2950:
2948:
2944:
2943:
2936:
2934:
2932:
2931:
2917:
2915:
2912:
2905:
2902:
2901:
2890:
2889:
2882:
2875:
2867:
2860:
2859:External links
2857:
2856:
2855:
2848:
2824:
2815:
2806:
2794:
2791:
2789:
2788:
2766:
2756:
2736:
2727:
2718:
2703:
2694:
2685:
2676:
2667:
2658:
2649:
2638:
2629:
2620:
2609:
2578:
2559:(2022-11-04).
2548:
2539:
2530:
2516:
2492:
2467:
2443:
2433:
2431:
2428:
2427:
2426:
2421:
2416:
2411:
2406:
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2331:
2326:
2321:
2319:Dilution ratio
2316:
2309:
2306:
2298:α, β, γ, x, y,
2117:
2114:
2078:Ratios may be
2075:
2072:
2061:Main article:
2058:
2055:
2034:
2029:
2024:
2021:
2018:
2012:
2009:
2003:
2000:
1989:
1988:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1954:
1950:
1929:
1926:
1923:
1920:
1917:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1891:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1804:
1798:
1792:
1787:
1784:
1777:
1771:
1768:
1762:
1759:
1748:
1747:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1713:
1709:
1683:
1680:
1674:
1671:
1668:
1665:
1638:
1635:
1632:
1621:
1620:
1609:
1606:
1603:
1600:
1596:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1463:
1458:
1453:
1450:
1447:
1444:
1441:
1438:
1398:
1395:
1351:Ratios can be
1348:
1345:
1316:
1313:
1287:
1284:
1249:
1246:
1230:
1227:
1193:
1190:
1171:
1168:
725:
722:
652:
649:
640:
639:
628:
625:
622:
619:
616:
613:
594:
593:
590:
579:
576:
573:
570:
567:
564:
561:
533:, is called a
468:
467:
312:
311:
296:
293:
260:
242:
237:
214:
211:
202:
26:
18:Ratio analysis
9:
6:
4:
3:
2:
3119:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3089:
3087:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3040:
3037:
3036:
3035:
3032:
3030:
3027:
3023:
3020:
3019:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2979:
2976:
2952:
2951:
2949:
2945:
2930:
2926:
2922:
2919:
2918:
2916:
2909:
2903:
2899:
2895:
2888:
2883:
2881:
2876:
2874:
2869:
2868:
2865:
2853:
2849:
2845:
2839:
2831:
2830:
2825:
2823:
2821:
2816:
2814:
2812:
2809:"Proportion"
2807:
2804:
2802:
2797:
2796:
2786:
2783:
2781:
2775:
2770:
2763:
2759:
2757:9789460917844
2753:
2749:
2748:
2740:
2731:
2722:
2715:
2714:
2707:
2698:
2692:Smith, p. 480
2689:
2683:Heath, p. 113
2680:
2674:Heath, p. 112
2671:
2665:Smith, p. 478
2662:
2653:
2647:
2642:
2633:
2627:Heath, p. 126
2624:
2618:
2613:
2606:
2595:
2588:
2582:
2568:
2567:
2562:
2558:
2552:
2543:
2534:
2528:
2525:
2520:
2506:
2502:
2496:
2482:
2478:
2471:
2457:
2453:
2447:
2438:
2434:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2409:Scale (ratio)
2407:
2405:
2402:
2400:
2397:
2395:
2394:Relative risk
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2311:
2305:
2303:
2299:
2294:
2292:
2288:
2285:in the ratio
2284:
2280:
2276:
2272:
2268:
2265:(across from
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2231:perpendicular
2228:
2224:
2220:
2216:
2211:
2209:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2152:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2123:
2113:
2110:
2106:
2105:
2099:
2089:
2085:
2081:
2071:
2069:
2064:
2054:
2052:
2048:
2032:
2027:
2022:
2019:
2016:
2010:
2007:
2001:
1998:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1952:
1948:
1927:
1921:
1918:
1912:
1908:
1904:
1901:
1898:
1892:
1885:
1882:
1876:
1873:
1870:
1867:
1861:
1858:
1855:
1852:
1845:
1844:
1843:
1833:
1828:
1826:
1822:
1818:
1802:
1796:
1790:
1785:
1782:
1775:
1769:
1766:
1760:
1757:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1711:
1707:
1681:
1678:
1672:
1669:
1666:
1663:
1656:
1655:
1654:
1636:
1633:
1630:
1607:
1604:
1598:
1594:
1590:
1587:
1584:
1578:
1575:
1572:
1569:
1547:
1544:
1538:
1535:
1532:
1526:
1523:
1520:
1517:
1510:
1509:
1508:
1498:
1493:
1491:
1487:
1483:
1477:
1461:
1456:
1451:
1448:
1445:
1442:
1439:
1436:
1428:
1424:
1412:
1408:
1404:
1394:
1392:
1388:
1383:
1380:
1376:
1372:
1367:
1365:
1364:simplest form
1360:
1356:
1354:
1344:
1342:
1341:
1336:
1331:
1314:
1311:
1285:
1282:
1269:
1267:
1261:
1259:
1255:
1245:
1228:
1225:
1212:
1208:
1191:
1188:
1177:
1167:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1024:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
965:
963:
962:in proportion
959:
954:
952:
944:
936:
924:
916:
908:
904:
900:
896:
892:
891:Dedekind cuts
888:
884:
880:
876:
872:
868:
865:according as
864:
860:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
798:
796:
792:
788:
784:
780:
776:
772:
768:
764:
760:
754:
752:
746:
744:
739:
735:
731:
721:
717:
715:
711:
706:
701:
699:
697:
691:
689:
683:
681:
680:
673:
669:
668:
658:
657:Ancient Greek
648:
644:
626:
623:
620:
617:
614:
611:
599:
598:
597:
591:
577:
574:
571:
568:
565:
562:
559:
547:
546:
545:
543:
538:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
476:
472:
465:
461:
457:
453:
449:
448:
447:
445:
441:
437:
433:
429:
425:
421:
417:
414:, written as
413:
409:
405:
401:
397:
392:
390:
389:
384:
380:
379:
374:
370:
366:
362:
357:
329:
325:
321:
317:
294:
291:
281:
277:
273:
269:
265:
261:
258:
254:
250:
246:
243:
241:
238:
236:
232:
229:the ratio of
228:
227:
226:
224:
220:
210:
208:
207:
200:
198:
194:
190:
189:dimensionless
186:
182:
178:
173:
171:
167:
163:
158:
156:
155:
144:
136:
128:
124:
120:
116:
111:
109:
104:
101:
95:
62:
58:
51:
46:
40:
33:
19:
2897:
2851:
2850:D.E. Smith,
2828:
2819:
2810:
2800:
2777:
2773:
2769:
2761:
2746:
2739:
2734:Heath p. 125
2730:
2721:
2711:
2706:
2697:
2688:
2679:
2670:
2661:
2652:
2641:
2632:
2623:
2612:
2604:
2598:. Retrieved
2593:
2581:
2570:. Retrieved
2564:
2551:
2542:
2533:
2523:
2519:
2508:. Retrieved
2504:
2495:
2484:. Retrieved
2480:
2470:
2459:. Retrieved
2455:
2446:
2437:
2301:
2297:
2295:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2226:
2222:
2218:
2212:
2207:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2153:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2119:
2103:
2100:
2077:
2067:
2066:
2050:
2046:
1990:
1832:silver ratio
1829:
1820:
1816:
1749:
1622:
1497:golden ratio
1494:
1411:Pythagoreans
1400:
1384:
1378:
1374:
1370:
1368:
1361:
1357:
1350:
1340:aspect ratio
1338:
1332:
1270:
1262:
1251:
1213:
1209:
1173:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1027:
1025:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
966:
961:
958:proportional
957:
955:
950:
942:
934:
922:
914:
906:
902:
898:
894:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
846:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
799:
790:
786:
782:
778:
774:
770:
766:
762:
758:
755:
750:
747:
743:aliquot part
737:
733:
727:
718:
705:Pythagoreans
702:
693:
685:
675:
665:
654:
645:
641:
595:
539:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
470:
469:
463:
459:
455:
451:
443:
439:
435:
431:
427:
423:
419:
415:
411:
410:is called a
407:
403:
399:
395:
393:
386:
382:
376:
372:
368:
364:
360:
359:The numbers
358:
319:
315:
313:
279:
275:
271:
267:
256:
252:
248:
244:
239:
234:
230:
222:
218:
216:
205:
184:
174:
159:
153:
152:
142:
134:
122:
118:
114:
112:
105:
60:
54:
3034:Irreducible
2964:Denominator
2780:Freudenthal
2725:Heath p.114
2404:Scale (map)
2339:Fold change
2314:Cross ratio
2241:) and side
1429:, formally
1337:have a 4:3
1335:televisions
1272:this case,
1266:percentages
542:two by four
278:divided by
57:mathematics
3086:Categories
3066:Percentage
3061:Paper size
2970:= Quotient
2600:2022-11-26
2572:2022-11-26
2510:2023-07-23
2486:2020-08-22
2481:Purplemath
2461:2020-08-22
2430:References
2389:Rate ratio
2349:Odds ratio
2135:and sides
1391:multiplier
1377::1, where
905:stands to
777:such that
728:Book V of
412:proportion
388:consequent
385:being the
378:antecedent
375:being the
330:, this is
154:proportion
3107:Quotients
3039:Reduction
2997:Continued
2982:Algebraic
2958:Numerator
2894:Fractions
2838:cite book
2566:MathWorld
2414:Sex ratio
2261:and side
2084:dimension
1967:−
1958:−
1723:−
1717:−
1347:Reduction
893:as, with
688:proportio
201:different
181:metrology
3012:Egyptian
2947:Fraction
2929:Quotient
2921:Dividend
2799:"Ratio"
2477:"Ratios"
2452:"Ratios"
2308:See also
2289: :
2273: :
2253: :
2225: :
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2190: :
2174: :
2122:vertices
2080:unitless
1488:, but a
1176:fraction
1007:so that
829: :
751:measures
738:multiple
495:extremes
351:∶
264:fraction
162:fraction
127:quotient
108:positive
3097:Algebra
3029:Integer
3002:Decimal
2967:
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2925:Divisor
2803:vol. 19
2561:"Colon"
2505:iso.org
2160:α, β, γ
1353:reduced
1258:percent
1154:is the
1130:and if
1118:is the
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371:, with
328:Unicode
187:is the
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3102:Ratios
3022:Silver
3017:Golden
3007:Dyadic
2992:Binary
2987:Aspect
2898:ratios
2754:
2202:being
2186:being
2170:being
2143:, and
2131:, and
1476:circle
1423:square
1387:factor
497:, and
462:is to
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2424:Slope
2104:rates
2074:Units
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881:, or
857:, or
679:ratio
672:Latin
667:logos
661:λόγος
507:means
354:RATIO
341:COLON
324:colon
280:B, or
266:with
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61:ratio
39:am to
3071:Unit
2896:and
2844:link
2752:ISBN
2300:and
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2166:and
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2063:Odds
2057:Odds
2049:and
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