Can You Explain Why the Answers to Parts B and C Are Not the Same

Relationship betwixt two numbers of the aforementioned kind

For non-dimensionless ratios, run across Rates.

"is to" redirects here. For the grammatical construction, see am to.

In mathematics, a ratio indicates how many times one number contains another. For instance, if there are viii oranges and half dozen lemons in a bowl of fruit, and then the ratio of oranges to lemons is eight to half dozen (that is, 8:6, which is equivalent to the ratio 4:iii). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:fourteen (or iv:7).

The numbers in a ratio may be quantities of any kind, such equally counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.

A ratio may exist specified either past giving both constituting numbers, written equally "a to b" or "a:b", or by giving just the value of their caliber a / b .^{[1] [2] [iii]} Equal quotients stand for to equal ratios.

Consequently, a ratio may be considered as an ordered pair of numbers, a fraction 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-nothing) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit of measurement, as is often the instance, their ratio is a dimensionless number. A caliber of two quantities that are measured with different units is called a rate.^[4]

Notation and terminology [edit]

The ratio of numbers A and B can be expressed every bit:^[five]

• the ratio of A to B
• A:B
• A is to B (when followed by "equally C is to D "; see below)
• a fraction with A as numerator and B as denominator that represents the quotient (i.e., A divided by B, or ${\displaystyle {\tfrac {A}{B}}}$ ). This can exist expressed as a simple or a decimal fraction, or as a pct, etc.^[6]

A colon (:) is often used in place of the ratio symbol, Unicode U+2236 (:).

The numbers A and B are sometimes called terms of the ratio, with A being the ancestor and B beingness the consequent.^[7]

A statement expressing the equality of two ratios A:B and C:D is called a proportion,^[eight] written equally A:B = C:D or A:B∷C:D. This latter form, when spoken or written in the English linguistic communication, is oftentimes expressed every bit

(A is to B) every bit (C is to D).

A, B, C and D are called the terms of the proportion. A and D are called its extremes, and B and C are called its means. The equality of three or more ratios, similar A:B = C:D = E:F, is called a continued proportion.^[ix]

Ratios are sometimes used with 3 or fifty-fifty more terms, eastward.yard., the proportion for the border lengths of a "two by four" that is ten inches long is therefore

${\displaystyle {\text{thickness : width : length }}=2:iv:10;}$

(unplaned measurements; the commencement ii numbers are reduced slightly when the woods is planed shine)

a adept concrete mix (in volume units) is sometimes quoted as

${\displaystyle {\text{cement : sand : gravel }}=one:2:four.}$ ^[10]

For a (rather dry out) mixture of four/1 parts in volume of cement to water, information technology could exist said that the ratio of cement to water is iv:1, that at that place is 4 times as much cement as water, or that there is a quarter (1/4) every bit much water as cement.

The meaning of such a proportion of ratios with more than two terms is that the ratio of whatsoever two terms on the left-hand side is equal to the ratio of the respective two terms on the right-paw side.

History and etymology [edit]

Information technology is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the give-and-take "rational"). A more modern interpretation^{[ compared to? ]} of Euclid's significant is more akin to computation or reckoning.^[11] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.^[12]

Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as practical to numbers.^[13] The Pythagoreans' conception of number included but what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, equally the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book Seven of The Elements reflects the earlier theory of ratios of commensurables.^[14]

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 insufficiently recent development, as can be seen from the fact that modern geometry textbooks notwithstanding apply distinct terminology and note for ratios and quotients. The reasons for this are twofold: showtime, there was the previously mentioned reluctance to accept irrational numbers as truthful numbers, and 2d, the lack of a widely used symbolism to supervene upon the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.^[15]

Euclid'southward definitions [edit]

Volume V of Euclid's Elements has 18 definitions, all of which chronicle to ratios.^[xvi] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The beginning two definitions say that a role of a quantity is some other quantity that "measures" it and conversely, a multiple of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied past an integer greater than one—and a part of a quantity (meaning aliquot part) is a function that, when multiplied by an integer greater than 1, gives the quantity.

Euclid does not define the term "measure" as used here, Still, one may infer that if a quantity is taken as a unit of measurement, and a 2d quantity is given as an integral number of these units, then the first quantity measures the second. These definitions are repeated, nearly word for discussion, as definitions three and 5 in book Vii.

Definition three describes what a ratio is in a full general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.^[17] Euclid defines a ratio every bit betwixt 2 quantities of the aforementioned type, so past this definition the ratios of two lengths or of two areas are divers, but not the ratio of a length and an expanse. 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 p and q, if there exist integers m and n such that mp>q and nq>p. This status is known as the Archimedes property.

Definition five is the virtually complex and difficult. It defines what information technology ways for 2 ratios to be equal. Today, this can be washed past merely stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modernistic note, Euclid's definition of equality is that given quantities p, q, r and s, p:q∷r :s if and but if, for any positive integers thousand and northward, np<mq, np=mq, or np>mq according as nr<ms, nr=ms, or nr>ms, respectively.^[18] This definition has affinities with Dedekind cuts as, with north and q both positive, np stands to mq as p / q stands to the rational number m / due north (dividing both terms by nq).^[19]

Definition 6 says that quantities that have the same ratio are proportional or in proportion. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".

Definition 7 defines what information technology means for 1 ratio to be less than or greater than another and is based on the ideas present in definition 5. In modernistic note it says that given quantities p, q, r and s, p:q>r:s if in that location are positive integers m and due north so that np>mq and nr≤ms.

Equally with definition iii, definition 8 is regarded by some as being a later insertion by Euclid'south editors. It defines three terms p, q and r to be in proportion when p:q∷q:r. This is extended to four terms p, q, r and s as p:q∷q:r∷r:south, and so on. Sequences that take the property that the ratios of consecutive terms are equal are chosen geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion and so p:r is the duplicate ratio of p:q and if p, q, r and s are in proportion and then p:due south is the triplicate ratio of p:q.

Number of terms and use of fractions [edit]

In general, a comparison of the quantities of a ii-entity ratio can exist expressed equally a fraction derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is ${\displaystyle {\tfrac {two}{three}}}$ that of the second entity.

If in that location are 2 oranges and iii apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is two:5. These ratios can too exist expressed in fraction form: at that place are 2/3 as many oranges every bit apples, and two/5 of the pieces of fruit are oranges. If orangish juice concentrate is to be diluted with water in the ratio one:4, then i function of concentrate is mixed with four parts of water, giving five parts full; the corporeality of orange juice concentrate is 1/4 the corporeality of water, while the amount of orange juice concentrate is ane/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.

Fractions can also exist inferred from ratios with more two entities; however, a ratio with more than 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 entities covered by the ratio: for example, from a ratio of ii:iii:7 we can infer that the quantity of the 2nd entity is ${\displaystyle {\tfrac {3}{7}}}$ that of the 3rd entity.

Proportions and per centum ratios [edit]

If nosotros multiply all quantities involved in a ratio by the aforementioned number, the ratio remains valid. For example, a ratio of 3:ii is the same every bit 12:viii. Information technology is usual either to reduce terms to the everyman common denominator, or to express them in parts per hundred (percent).

If a mixture contains substances A, B, C and D in the ratio 5:nine:iv:ii then in that location are five parts of A for every ix parts of B, four parts of C and 2 parts of D. Every bit 5+9+4+two=20, the full mixture contains 5/twenty of A (5 parts out of 20), 9/20 of B, iv/20 of C, and 2/twenty of D. If we divide all numbers past the total and multiply by 100, we have converted to percentages: 25% A, 45% B, 20% C, and x% D (equivalent to writing the ratio as 25:45:twenty:x).

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 3 oranges and no other fruit is fabricated upwards of 2 parts apples and three parts oranges. In this case, ${\displaystyle {\tfrac {2}{v}}}$ , or 40% of the whole is apples and ${\displaystyle {\tfrac {3}{five}}}$ , or threescore% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.

If the ratio consists of only ii values, it can exist represented as a fraction, in particular as a decimal fraction. For example, older televisions have a iv:three aspect ratio, which means that the width is 4/iii of the height (this can as well be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More than recent widescreen TVs take a xvi:nine aspect ratio, or one.78 rounded to ii decimal places. 1 of the pop widescreen movie formats is 2.35:one or only 2.35. Representing ratios as decimal fractions simplifies their comparing. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider paradigm. Such a comparison works just when values being compared are consistent, like always expressing width in relation to top.

Reduction [edit]

Ratios can be reduced (as fractions are) by dividing each quantity past the mutual 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.

Thus, the ratio forty:threescore is equivalent in meaning to the ratio ii:iii, the latter existence obtained from the former past dividing both quantities by twenty. Mathematically, we write 40:60 = 2:3, or equivalently forty:sixty∷2:iii. The verbal equivalent is "forty is to sixty equally 2 is to 3."

A ratio that has integers for both quantities and that cannot be reduced any farther (using integers) is said to be in simplest form or lowest terms.

Sometimes it is useful to write a ratio in the form 1:x or x:1, where x is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:five can be written every bit i:ane.25 (dividing both sides by 4) Alternatively, it can be written as 0.viii:1 (dividing both sides by five).

Where the context makes the pregnant clear, a ratio in this class is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it a cistron or multiplier.

Irrational ratios [edit]

Ratios may also be established betwixt incommensurable quantities (quantities whose ratio, equally value of a fraction, amounts to an irrational number). The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal d to the length of a side s of a square, which is the square root of 2, formally ${\displaystyle a:d=ane:{\sqrt {two}}.}$ Another example is the ratio of a circle'southward circumference to its diameter, which is chosen π, and is not only an algebraically irrational number, but a transcendental irrational.

Also well known is the golden ratio of 2 (mostly) lengths a and b, which is divers by the proportion

${\displaystyle a:b=(a+b):a\quad }$ or, equivalently ${\displaystyle \quad a:b=(1+b/a):1.}$

Taking the ratios equally fractions and ${\displaystyle a:b}$ equally having the value ten, yields the equation

${\displaystyle x=one+{\tfrac {1}{10}}\quad }$ or ${\displaystyle \quad x^{2}-x-i=0,}$

which has the positive, irrational solution ${\displaystyle x={\tfrac {a}{b}}={\tfrac {i+{\sqrt {5}}}{2}}.}$ Thus at least one of a and b has to be irrational for them to be in the golden ratio. An example of an occurrence of the gold ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers: even though all these ratios are ratios of ii integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio.

Similarly, the silver ratio of a and b is defined by the proportion

${\displaystyle a:b=(2a+b):a\quad (=(ii+b/a):one),}$ respective to ${\displaystyle x^{2}-2x-1=0.}$

This equation has the positive, irrational solution ${\displaystyle 10={\tfrac {a}{b}}=1+{\sqrt {two}},}$ so again at least one of the 2 quantities a and b in the silver ratio must be irrational.

Odds [edit]

Primary article: Odds

Odds (every bit in gambling) are expressed equally a ratio. For example, odds of "7 to 3 against" (7:three) 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, in that location are expected to exist three wins and seven losses.

Units [edit]

Ratios may be unitless, every bit in the instance they relate quantities in units of the same dimension, even if their units of measurement are initially different. For case, the ratio i minute : forty seconds tin can be reduced past changing the showtime value to lx seconds, so the ratio becomes threescore seconds : 40 seconds. Once the units are the same, they can exist omitted, and the ratio can be reduced to 3:2.

On the other hand, there are non-dimensionless ratios, also known as rates.^{[20] [21]} In chemistry, mass concentration ratios are usually expressed equally weight/volume fractions. For example, a concentration of 3% w/five usually ways 3 thou of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, equally in weight/weight or volume/book fractions.

Triangular coordinates [edit]

The locations of points relative to a triangle with vertices A, B, and C and sides AB, BC, and CA are ofttimes expressed in extended ratio form as triangular coordinates.

In barycentric coordinates, a point with coordinates α, β, γ is the point upon which a weightless sheet of metallic 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 A and B being α : β, the ratio of the weights at B and C existence β : γ, and therefore the ratio of weights at A and C being α : γ.

In trilinear coordinates, a point with coordinates x :y :z has perpendicular distances to side BC (beyond from vertex A) and side CA (beyond from vertex B) in the ratio x :y, distances to side CA and side AB (across from C) in the ratio y :z, and therefore distances to sides BC and AB in the ratio x :z.

Since all information is expressed in terms of ratios (the private numbers denoted by α, β, γ, x, y, and z have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle.

Meet too [edit]

• Dilution ratio
• Deportation–length ratio
• Dimensionless quantity
• Financial ratio
• Fold modify
• Interval (music)
• Odds ratio
• Parts-per notation
• Price–performance ratio
• Proportionality (mathematics)
• Ratio distribution
• Ratio reckoner
• Rate (mathematics)
• Charge per unit ratio
• Relative adventure
• Dominion of three (mathematics)
• Calibration (map)
• Scale (ratio)
• Sex ratio
• Superparticular ratio
• Slope

References [edit]

1. ^ New International Encyclopedia
2. ^ "Ratios". www.mathsisfun.com . Retrieved 2020-08-22 .
3. ^ Stapel, Elizabeth. "Ratios". Purplemath . Retrieved 2020-08-22 .
4. ^ "The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)", "The Mathematics Dictionary" [1]
5. ^ New International Encyclopedia
6. ^ Decimal fractions are often used in technological areas where ratio comparisons are of import, such as aspect ratios (imaging), compression ratios (engines or data storage), etc.
7. ^ from the Encyclopædia Britannica
8. ^ Heath, p. 126
9. ^ New International Encyclopedia
10. ^ Belle Group physical mixing hints
11. ^ Penny Cyclopædia, p. 307
12. ^ Smith, p. 478
13. ^ Heath, p. 112
14. ^ Heath, p. 113
15. ^ Smith, p. 480
16. ^ Heath, reference for section
17. ^ "Geometry, Euclidean" Encyclopædia Britannica Eleventh Edition p682.
18. ^ Heath p.114
19. ^ Heath p. 125
20. ^ "'Velocity' tin can be defined as the ratio... 'Population density' is the ratio... 'Gasoline consumption' is measure as the ratio...", "Ratio and Proportion: Research and Education in Mathematics Teachers" [2]
21. ^ "Ratio equally a Rate. The first type [of ratio] defined by Freudenthal, above, is known equally rate, and illustrates a comparing between two variables with deviation units. (...) A ratio of this sort produces a unique, new concept with its own entity, and this new concept is unremarkably non considered a ratio, per se, only a rate or density.", "Ratio and Proportion: Research and Teaching in Mathematics Teachers" [3]

Further reading [edit]

• "Ratio" The Penny Cyclopædia vol. 19, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
• "Proportion" New International Encyclopedia, Vol. 19 2nd ed. (1916) Dodd Mead & Co. pp270-271
• "Ratio and Proportion" Fundamentals of applied mathematics, George Wentworth, David Eugene Smith, Herbert Druery Harper (1922) Ginn and Co. pp. 55ff
• The xiii books of Euclid's Elements, vol 2. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Printing. 1908. pp. 112ff. {{cite volume}}: CS1 maint: others (link)
• D.E. Smith, History of Mathematics, vol 2 Ginn and Visitor (1925) pp. 477ff. Reprinted 1958 by Dover Publications.

External links [edit]

Look upwards ratio in Wiktionary, the free lexicon.

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Source: https://en.wikipedia.org/wiki/Ratio

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