1332:, 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.
2928:
34:
1200:
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.
2897:
709:
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.
750:, 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
2059:(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.
1199:
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
696:
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
91:
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
1260:
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
1203:
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
708:
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
1252:
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
789:
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
2771:, 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."
2100:
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.
1347:
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."
631:
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.
737:
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
1370:
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).
745:
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
1344:(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.
729:
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
1802:
577:
635:
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.
95:
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
626:
721:
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
2032:
1927:
956:
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
1241:
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
1685:
2535:
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
1816:: 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.
1374:
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
1636:
673:("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
153:
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)
705:. 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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1322:
If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older
1742:
539:
2751:"Velocity" can be defined as the ratio... "Population density" is the ratio... "Gasoline consumption" is measure as the ratio...
1015:
As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms
2144:
591:
953:. 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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1351:
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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1983:
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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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1319:, 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 "
2794:, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
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the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.
701:) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to
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581:(unplaned measurements; the first two numbers are reduced slightly when the wood is planed smooth)
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2513:"The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)"
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435:. 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
2087:. 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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2821:. 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
299:). This can be expressed as a simple or a decimal fraction, or as a percentage, etc.
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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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2773:, "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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1797:{\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.}
644:
It is possible to trace the origin of the word "ratio" to the
332:, although Unicode also provides a dedicated ratio character,
26:"is to" redirects here. For the grammatical construction, see
2490:"ISO 80000-1:2022(en) Quantities and units — Part 1: General"
2412:
2071:, as in the case they relate quantities in units of the same
660:
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585:
a good concrete mix (in volume units) is sometimes quoted as
27:
572:{\displaystyle {\text{thickness : width : length }}=2:4:10;}
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67:
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621:{\displaystyle {\text{cement : sand : gravel }}=1:2:4.}
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David Ben-Chaim; Yaffa Keret; Bat-Sheva Ilany (2012).
1994:
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Sometimes it is useful to write a ratio in the form 1:
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188:. 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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1087:. Definitions 9 and 10 apply this, saying that if
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383:A statement expressing the equality of two ratios
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1398:). The earliest discovered example, found by the
16:Relationship between two numbers of the same kind
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2079:are initially different. For example, the ratio
2027:{\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},}
790:definition of equality is that given quantities
689:("proportionality") for the equality of ratios.
1922:{\displaystyle a:b=(2a+b):a\quad (=(2+b/a):1),}
2818:The thirteen books of Euclid's Elements, vol 2
2538:
2136:are often expressed in extended ratio form as
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1402:, is the ratio of the length of the diagonal
498:. The equality of three or more ratios, like
2802:2nd ed. (1916) Dodd Mead & Co. pp270-271
2034:so again at least one of the two quantities
1739:which has the positive, irrational solution
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2096:(sometimes also as ratios). In chemistry,
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822:if and only if, for any positive integers
2739:. Springer Science & Business Media.
2544:
2104:
1680:{\displaystyle x=1+{\tfrac {1}{x}}\quad }
311:, the two-dot character is sometimes the
161:, and may sometimes be natural numbers.
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2702:Encyclopædia Britannica Eleventh Edition
2183:, and therefore the ratio of weights at
2042:in the silver ratio must be irrational.
659:). Early translators rendered this into
639:
474:are called the terms of the proportion.
114:", or by giving just the value of their
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2800:New International Encyclopedia, Vol. 19
1390:Ratios may also be established between
712:
533:" that is ten inches long is therefore
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164:A more specific definition adopted in
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2809:Fundamentals of practical mathematics
1496:, which is defined by the proportion
681:("proportion") to indicate ratio and
1602:{\displaystyle \quad a:b=(1+b/a):1.}
1385:
1290:, or 40% of the whole is apples and
1159:Number of terms and use of fractions
303:When a ratio is written in the form
2488:
2266:, and therefore distances to sides
1612:Taking the ratios as fractions and
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2583:The Unicode Standard, Version 15.0
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1729:{\displaystyle \quad x^{2}-x-1=0,}
1463:Another example is the ratio of a
1456:{\displaystyle a:d=1:{\sqrt {2}}.}
14:
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2635:Belle Group concrete mixing hints
1543:{\displaystyle a:b=(a+b):a\quad }
1237:Proportions and percentage ratios
1043:. This is extended to four terms
782:. This condition is known as the
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1312:{\displaystyle {\tfrac {3}{5}}}
1283:{\displaystyle {\tfrac {2}{5}}}
1226:{\displaystyle {\tfrac {3}{7}}}
1189:{\displaystyle {\tfrac {2}{3}}}
988:if there are positive integers
292:{\displaystyle {\tfrac {A}{B}}}
2625:New International Encyclopedia
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2841:History of Mathematics, vol 2
2767:. The first type defined by
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2353:Proportionality (mathematics)
1970:{\displaystyle x^{2}-2x-1=0.}
1831:is defined by the proportion
2690:Heath, reference for section
2085:60 seconds : 40 seconds
2081:one minute : 40 seconds
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597:cement : sand : gravel
7:
2388:Rule of three (mathematics)
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2313:Displacement–length ratio
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1488:of two (mostly) lengths
1406:to the length of a side
938:(dividing both terms by
202:Notation and terminology
2807:"Ratio and Proportion"
2348:Price–performance ratio
2145:barycentric coordinates
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1484:Also well known is the
1135:are in proportion then
1099:are in proportion then
184:measured with the same
3081:Elementary mathematics
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1631:{\displaystyle a:b}
784:Archimedes property
182:physical quantities
2903:Division and ratio
2546:Weisstein, Eric W.
2445:www.mathsisfun.com
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937:
935:
934:
929:
926:
917:
915:
914:
909:
906:
748:of the same type
687:
685:proportionalitas
679:
671:
652:
651:
627:
625:
624:
619:
599:
596:
578:
576:
575:
570:
547:
544:
344:
341:
338:
336:
331:
328:
325:
323:
298:
296:
295:
290:
288:
279:
159:rational numbers
139:
137:
135:
134:
129:
126:
86:
85:
82:
81:
76:
75:
70:
69:
66:
63:
60:
3111:
3110:
3106:
3105:
3104:
3102:
3101:
3100:
3071:
3070:
3069:
3064:
3035:Just intonation
2962:
2952:
2949:
2946:
2945:
2943:
2942:
2931:
2927:
2922:
2900:
2889:
2880:
2850:
2824:
2823:
2815:
2784:
2782:Further reading
2779:
2765:Ratio as a Rate
2761:
2757:
2747:
2731:
2727:
2722:
2718:
2713:
2709:
2698:
2694:
2689:
2685:
2680:
2676:
2671:
2667:
2662:
2658:
2653:
2649:
2644:
2640:
2633:
2629:
2624:
2620:
2615:
2611:
2604:
2600:
2588:
2586:
2578:
2574:
2573:
2569:
2560:
2558:
2543:
2539:
2534:
2530:
2525:
2521:
2511:
2507:
2498:
2496:
2487:
2483:
2474:
2472:
2462:
2458:
2449:
2447:
2439:
2438:
2434:
2429:
2425:
2421:
2373:Ratio (Twitter)
2363:Ratio estimator
2323:Financial ratio
2299:
2258:) in the ratio
2238:) in the ratio
2107:
2084:
2080:
2065:
2054:
2048:
2014:
1993:
1985:
1982:
1981:
1940:
1936:
1934:
1931:
1930:
1896:
1839:
1836:
1835:
1828:
1824:
1819:Similarly, the
1777:
1770:
1767:
1752:
1744:
1741:
1740:
1699:
1695:
1692:
1689:
1688:
1664:
1650:
1647:
1646:
1639:
1617:
1614:
1613:
1582:
1555:
1552:
1551:
1504:
1501:
1500:
1493:
1489:
1469:
1443:
1423:
1420:
1419:
1414:, which is the
1407:
1403:
1392:incommensurable
1388:
1338:
1297:
1295:
1292:
1291:
1268:
1266:
1263:
1262:
1239:
1211:
1209:
1206:
1205:
1174:
1172:
1169:
1168:
1161:
1109:duplicate ratio
930:
927:
922:
921:
919:
910:
907:
902:
901:
899:
890:both positive,
715:
642:
595:
593:
590:
589:
543:
541:
538:
537:
494:are called its
482:are called its
342:
339:
334:
333:
329:
326:
321:
320:
277:
275:
272:
271:
204:
168:(especially in
155:natural numbers
130:
127:
122:
121:
119:
118:
78:
72:
57:
53:
31:
24:
17:
12:
11:
5:
3109:
3099:
3098:
3093:
3088:
3083:
3066:
3065:
3063:
3062:
3057:
3052:
3047:
3042:
3037:
3032:
3031:
3030:
3020:
3015:
3014:
3013:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2967:
2964:
2963:
2961:
2960:
2939:
2937:
2933:
2932:
2925:
2923:
2921:
2920:
2906:
2904:
2901:
2894:
2891:
2890:
2879:
2878:
2871:
2864:
2856:
2849:
2848:External links
2846:
2845:
2844:
2837:
2813:
2804:
2795:
2783:
2780:
2778:
2777:
2755:
2745:
2725:
2716:
2707:
2692:
2683:
2674:
2665:
2656:
2647:
2638:
2627:
2618:
2609:
2598:
2567:
2548:(2022-11-04).
2537:
2528:
2519:
2505:
2481:
2456:
2432:
2422:
2420:
2417:
2416:
2415:
2410:
2405:
2400:
2395:
2390:
2385:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2308:Dilution ratio
2305:
2298:
2295:
2287:α, β, γ, x, y,
2106:
2103:
2067:Ratios may be
2064:
2061:
2050:Main article:
2047:
2044:
2023:
2018:
2013:
2010:
2007:
2001:
1998:
1992:
1989:
1978:
1977:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1943:
1939:
1918:
1915:
1912:
1909:
1906:
1903:
1899:
1895:
1892:
1889:
1886:
1883:
1880:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1793:
1787:
1781:
1776:
1773:
1766:
1760:
1757:
1751:
1748:
1737:
1736:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1702:
1698:
1672:
1669:
1663:
1660:
1657:
1654:
1627:
1624:
1621:
1610:
1609:
1598:
1595:
1592:
1589:
1585:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1452:
1447:
1442:
1439:
1436:
1433:
1430:
1427:
1387:
1384:
1340:Ratios can be
1337:
1334:
1305:
1302:
1276:
1273:
1238:
1235:
1219:
1216:
1182:
1179:
1160:
1157:
714:
711:
641:
638:
629:
628:
617:
614:
611:
608:
605:
602:
583:
582:
579:
568:
565:
562:
559:
556:
553:
550:
522:, is called a
457:
456:
301:
300:
285:
282:
249:
231:
226:
203:
200:
191:
15:
9:
6:
4:
3:
2:
3108:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3078:
3076:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3029:
3026:
3025:
3024:
3021:
3019:
3016:
3012:
3009:
3008:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2968:
2965:
2941:
2940:
2938:
2934:
2919:
2915:
2911:
2908:
2907:
2905:
2898:
2892:
2888:
2884:
2877:
2872:
2870:
2865:
2863:
2858:
2857:
2854:
2842:
2838:
2834:
2828:
2820:
2819:
2814:
2812:
2810:
2805:
2803:
2801:
2798:"Proportion"
2796:
2793:
2791:
2786:
2785:
2775:
2772:
2770:
2764:
2759:
2752:
2748:
2746:9789460917844
2742:
2738:
2737:
2729:
2720:
2711:
2704:
2703:
2696:
2687:
2681:Smith, p. 480
2678:
2672:Heath, p. 113
2669:
2663:Heath, p. 112
2660:
2654:Smith, p. 478
2651:
2642:
2636:
2631:
2622:
2616:Heath, p. 126
2613:
2607:
2602:
2595:
2584:
2577:
2571:
2557:
2556:
2551:
2547:
2541:
2532:
2523:
2517:
2514:
2509:
2495:
2491:
2485:
2471:
2467:
2460:
2446:
2442:
2436:
2427:
2423:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2398:Scale (ratio)
2396:
2394:
2391:
2389:
2386:
2384:
2383:Relative risk
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2300:
2294:
2292:
2288:
2283:
2281:
2277:
2274:in the ratio
2273:
2269:
2265:
2261:
2257:
2254:(across from
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2221:
2220:perpendicular
2217:
2213:
2209:
2205:
2200:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2141:
2139:
2135:
2131:
2127:
2123:
2119:
2115:
2112:
2102:
2099:
2095:
2094:
2088:
2078:
2074:
2070:
2060:
2058:
2053:
2043:
2041:
2037:
2021:
2016:
2011:
2008:
2005:
1999:
1996:
1990:
1987:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1941:
1937:
1916:
1910:
1907:
1901:
1897:
1893:
1890:
1887:
1881:
1874:
1871:
1865:
1862:
1859:
1856:
1850:
1847:
1844:
1841:
1834:
1833:
1832:
1822:
1817:
1815:
1811:
1807:
1791:
1785:
1779:
1774:
1771:
1764:
1758:
1755:
1749:
1746:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1700:
1696:
1670:
1667:
1661:
1658:
1655:
1652:
1645:
1644:
1643:
1625:
1622:
1619:
1596:
1593:
1587:
1583:
1579:
1576:
1573:
1567:
1564:
1561:
1558:
1536:
1533:
1527:
1524:
1521:
1515:
1512:
1509:
1506:
1499:
1498:
1497:
1487:
1482:
1480:
1476:
1472:
1466:
1450:
1445:
1440:
1437:
1434:
1431:
1428:
1425:
1417:
1413:
1401:
1397:
1393:
1383:
1381:
1377:
1372:
1369:
1365:
1361:
1356:
1354:
1353:simplest form
1349:
1345:
1343:
1333:
1331:
1330:
1325:
1320:
1303:
1300:
1274:
1271:
1258:
1256:
1250:
1248:
1244:
1234:
1217:
1214:
1201:
1197:
1180:
1177:
1166:
1156:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1013:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
959:
954:
952:
951:in proportion
948:
943:
941:
933:
925:
913:
905:
897:
893:
889:
885:
881:
880:Dedekind cuts
877:
873:
869:
865:
861:
857:
854:according as
853:
849:
845:
841:
837:
833:
829:
825:
821:
817:
813:
809:
805:
801:
797:
793:
787:
785:
781:
777:
773:
769:
765:
761:
757:
753:
749:
743:
741:
735:
733:
728:
724:
720:
710:
706:
704:
700:
695:
690:
688:
686:
680:
678:
672:
670:
669:
662:
658:
657:
647:
646:Ancient Greek
637:
633:
615:
612:
609:
606:
603:
600:
588:
587:
586:
580:
566:
563:
560:
557:
554:
551:
548:
536:
535:
534:
532:
527:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
454:
450:
446:
442:
438:
437:
436:
434:
430:
426:
422:
418:
414:
410:
406:
403:, written as
402:
398:
394:
390:
386:
381:
379:
378:
373:
369:
368:
363:
359:
355:
351:
346:
318:
314:
310:
306:
283:
280:
270:
266:
262:
258:
254:
250:
247:
243:
239:
235:
232:
230:
227:
225:
221:
218:the ratio of
217:
216:
215:
213:
209:
199:
197:
196:
189:
187:
183:
179:
178:dimensionless
175:
171:
167:
162:
160:
156:
152:
147:
145:
144:
133:
125:
117:
113:
109:
105:
100:
98:
93:
90:
84:
51:
47:
40:
35:
29:
22:
2886:
2840:
2839:D.E. Smith,
2817:
2808:
2799:
2789:
2766:
2762:
2758:
2750:
2735:
2728:
2723:Heath p. 125
2719:
2710:
2700:
2695:
2686:
2677:
2668:
2659:
2650:
2641:
2630:
2621:
2612:
2601:
2593:
2587:. Retrieved
2582:
2570:
2559:. Retrieved
2553:
2540:
2531:
2522:
2512:
2508:
2497:. Retrieved
2493:
2484:
2473:. Retrieved
2469:
2459:
2448:. Retrieved
2444:
2435:
2426:
2290:
2286:
2284:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2215:
2211:
2207:
2201:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2142:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2108:
2092:
2089:
2066:
2056:
2055:
2039:
2035:
1979:
1821:silver ratio
1818:
1809:
1805:
1738:
1611:
1486:golden ratio
1483:
1400:Pythagoreans
1389:
1373:
1367:
1363:
1359:
1357:
1350:
1346:
1339:
1329:aspect ratio
1327:
1321:
1259:
1251:
1240:
1202:
1198:
1162:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1014:
1009:
1005:
1001:
997:
993:
989:
985:
981:
977:
973:
969:
965:
961:
957:
955:
950:
947:proportional
946:
944:
939:
931:
923:
911:
903:
895:
891:
887:
883:
875:
871:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
823:
819:
815:
811:
807:
803:
799:
795:
791:
788:
779:
775:
771:
767:
763:
759:
755:
751:
747:
744:
739:
736:
732:aliquot part
726:
722:
716:
707:
694:Pythagoreans
691:
682:
674:
664:
654:
643:
634:
630:
584:
528:
523:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
458:
452:
448:
444:
440:
432:
428:
424:
420:
416:
412:
408:
404:
400:
399:is called a
396:
392:
388:
384:
382:
375:
371:
365:
361:
357:
353:
349:
348:The numbers
347:
308:
304:
302:
268:
264:
260:
256:
245:
241:
237:
233:
228:
223:
219:
211:
207:
205:
194:
173:
163:
148:
142:
141:
131:
123:
111:
107:
103:
101:
94:
49:
43:
3023:Irreducible
2953:Denominator
2769:Freudenthal
2714:Heath p.114
2393:Scale (map)
2328:Fold change
2303:Cross ratio
2230:) and side
1418:, formally
1326:have a 4:3
1324:televisions
1261:this case,
1255:percentages
531:two by four
267:divided by
46:mathematics
3075:Categories
3055:Percentage
3050:Paper size
2959:= Quotient
2589:2022-11-26
2561:2022-11-26
2499:2023-07-23
2475:2020-08-22
2470:Purplemath
2450:2020-08-22
2419:References
2378:Rate ratio
2338:Odds ratio
2124:and sides
1380:multiplier
1366::1, where
894:stands to
766:such that
717:Book V of
401:proportion
377:consequent
374:being the
367:antecedent
364:being the
319:, this is
143:proportion
3096:Quotients
3028:Reduction
2986:Continued
2971:Algebraic
2947:Numerator
2883:Fractions
2827:cite book
2555:MathWorld
2403:Sex ratio
2250:and side
2073:dimension
1956:−
1947:−
1712:−
1706:−
1336:Reduction
882:as, with
677:proportio
190:different
170:metrology
3001:Egyptian
2936:Fraction
2918:Quotient
2910:Dividend
2788:"Ratio"
2466:"Ratios"
2441:"Ratios"
2297:See also
2278: :
2262: :
2242: :
2214: :
2210: :
2195: :
2179: :
2163: :
2111:vertices
2069:unitless
1477:, but a
1165:fraction
996:so that
818: :
740:measures
727:multiple
484:extremes
340:∶
253:fraction
151:fraction
116:quotient
97:positive
3086:Algebra
3018:Integer
2991:Decimal
2956:
2944:
2914:Divisor
2792:vol. 19
2550:"Colon"
2494:iso.org
2149:α, β, γ
1342:reduced
1247:percent
1143:is the
1119:and if
1107:is the
936:
920:
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900:
360:, with
317:Unicode
176:is the
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3091:Ratios
3011:Silver
3006:Golden
2996:Dyadic
2981:Binary
2976:Aspect
2887:ratios
2743:
2191:being
2175:being
2159:being
2132:, and
2120:, and
1465:circle
1412:square
1376:factor
486:, and
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327::
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2705:p682.
2579:(PDF)
2413:Slope
2093:rates
2063:Units
1410:of a
870:, or
846:, or
668:ratio
661:Latin
656:logos
650:λόγος
496:means
343:RATIO
330:COLON
313:colon
269:B, or
255:with
174:ratio
50:ratio
28:am to
3060:Unit
2885:and
2833:link
2741:ISBN
2289:and
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2171:and
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2052:Odds
2046:Odds
2038:and
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1492:and
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