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9 and 8 being the given differences, we have only to multiply 9 by 8, and 8 by 9 to obtain equal products.
This process constitutes the basis of the rule in what is termed Alligation Alternate, which rule, where the proportions of two ingredients only are concerned, may be stated briefly as follows :--Take the difference between the rate of each ingredient and the mean rate, and place it opposite to the other rate. The operation is usually conducted as follows:Mean Rates of
1 7d. .................... 8
9 at 24d. = 216d. 8 at 7d. = 56d. 17
32722.(16d. mean rate. Having obtained in this manner, one set of proportional quantities, which are shown to fulfil the conditions of the question, we may deduce from them an infinite variety of other proportions, either whole or fractional, by simply multiplying or dividing both quantities by any number whatever : e. g.,
9 multiplied by 2, 3, 4, 5, 1, }, o, &c. = 18, 27, 36, 45, 41, 14, L. &c.
8 , by 2, 3, 4, 5, 1, 1, I, &c. = 16, 24, 32, 40, 4, 11, 4. &c. It will be seen on trial that any two of these proportions will produce a compound of the required value. The rule of Alligation, therefore, furnishes one very ready solution of the question, from which other solutions may be had at pleasure.
When more than two ingredients are proposed, an extension of the same method is still applicable. For if we select any two of the rates, one greater and one less than the mean, and proceed as above indicated with these alone, we shall obtain proportions giving the mean value required, apart from the influence of the other ingredients. And if we repeat the operation on two other rates respectively greater and less than the mean, until all are exhausted, we shall thus form, as it were, so many independent additional compounds each of the necessary value, the sum or admixture of which together cannot affect the average of any single pair of ingredients. Should an odd number of rates be given, or should those which are greater than the mean exceed in number those which are less, or the reverse, we may unite the game rate several times with others of the group, using the sum of the alternate differences thus set out opposite to each rate. The reason of this will best appear from an example.
Example (2.) Form a spirituous mixture of the strength of 24 per cent., U. P., from ingredients respectively 18, 22, 28, 33, and 54 per cent., U. P.
Diffs. , 18. . 30 +4. . 34
4 + 9 Mean
22 . . . . . 9 30 + 4 = 34 4 + 30 = 34 Strength. 28
| 33 . . . . . 2
* When the given strengths are all below or all above Proof, it is unnecessary to take the differences between the rates per cent, and 100, as in other cases.
The three sets of proportional quantities here presented are the results of different modes of coupling the given rates, but all will produce a mixture of the desired strength, as may readily be verified on taking the average value in each case. It is evident that several other sets of differences can be obtained by vary. ing the manner of pairing the rates. An account of the process of deriving the first set of differences will sufficiently explain this part of the subject.
The five given rates are formed into the following pairs:-18 and 54; 22 and 33; 18 (again) and 28; where each pair consists of a rate greater and a rate less than the mean. The difference between 18 and the mean is placed opposite to 54, and that between 54 and the mean opposite to 18. A similar operation is gone through with the remaining pairs, and the two differences standing opposite to 18 are made into a total, as 34 at 18 combined with 6 at 54, and 6 at 28, must have the same collective value as 30 at 18, and 4 at 18, combined separately with 6 at 54, and 6 at 28.
In working questions under the rule of Alligation, it has, until recently, been the practice to connect the several rates together in pairs by means of curves or rectangular figures, for the purpose, no doubt, of indicating more clearly where the alternate differences should be placed. Hence the name, Alligation, as has already been remarked. But the use of such devices is now almost discarded, as having an awkward appearance, and serving rather to confuse than assist the eye.
When the compound is limited to a certain quantity, that is, when the question requires that the ingredients shall amount to a stated sum, it is necessary after finding the alternate differences in the manner just exemplified, to apply the rule of division into proportional parts. (See page 118.)
Example (3.) It is required to compound 865 gallons at 7.5 per cent. U. P., from spirits at 34.6 per cent. O. P.; Proof ; 10.5 per cent., U. P., and water. What should be the quantity of each ingredient ?
( 134.6 . . . . . . 92-5 99.5 100. . . . . . . . . 3.0 89-5 . . . . . .
When the quantity of one of the ingredients is given, we have, as before, to take the difference between each rate and the mean, and then say by the Rule of Three,-As the difference standing against the rate of the ingredient, whose quantity is specified, is to each of the other differences, so is the given quantity to each of the required quantities.
The reason of the latter part of this rule needs no explanation.
Example (4.) How much spirits at 13s. 6d., 128. 9d., and 10s. 6d. per gallon, respectively, must be added to 38 gallons at 9s. 5d. per gallon, so that the mixture shall be worth 11s. per gallon ?
* For an easier method of performing the work of division into proportional parts, sec page 119, example 3.
( 162d. . . . 19 132d. / 153d.... 6
(126d. . . . 21
113d. . .. 30 30 : 19 :: 38 : 24.07
24.07 galls. at 138. 6d. 30 : 6 :: 38 : 760 Answer. 7.60 „ at 12s. 9d. 30 : 21 :: 38 : 26.60
26 60 „ at 10s. 6d. As the first and the third terms are the same in the three statements, the shorter way would be to form at once the quotient of 38 • 30, and employ it as a common multiplier. Thus, 88 = 1.267. And 1.267 x 19 = 24.07 )
1.267 x 6 = 7.60
1.267 x 21 = 26.60 In most questions of this class, as only one of the component quantities is limited, various answers may be obtained, according to the number of the ingredients, and the mode in which the rates are coupled together. A different treatment of the last example gives, for instance, as a second solution,
10 86 galls. at 13s. Od.)
54:28 galls. at 10s. 6d. It is left to the officer to verify the forgoing results by compounding the several rates and quantities, and finding the average value of the mixture.
Example (4.) In a distillery vessel there are 5650 gallons of worts at 24° gravity. How many gallons of worts at 70° must be added to raise the former gravity to 500 ?
mean 24o..... 20
50, 70° ..... 26 20 : 26 :: 5650 7345. Answer 7345 galls. If by the terms of the question, more than one of the ingredients should be limited as to quantity, the obvious course will be, first to compute the average value of all the ingredients so limited, and then substitute the total quantity at this value for its several component parts.
Example (5.) Required the quantities of worts at 18°, 26°, and 32°, gravity respectively, which will form with 856 gallons at 44°, and 972 gallons at 55°, a mixture having the gravity of 40°.
856 x 44 = 37664
limited ingredients. 18 ................ 9.85
9.85 32 ................ 9.85
( 49.85. . 22 + 14 + 8 = 44:
44 : 9.85 :: 1828 : 409.22 Answer, 409.22 galls. at each of the gravities 18°, 26°, and 32°, respectively.
EXERCISES IN AVERAGES. (1.) The dry malt obtained from 1200 quarters of barley, exceeded that quantity by 5.1 per cent. In another instance, the increase on 1800 quarters amounted to 81 per cent. What was the total or average increase per cent. ?
Answer, 7.15. (2.) Compute the average gravity of 262,580 gallons of worts at 55° ; 359,845 gallons at 51°; 76,283 gallons at 47°, and 56,781 gallons at 35° gravity.*
Answer, 50.7o. (3.) What is the strength of a mixture of 322 gallons of spirits at 22 per cent. 0.P. ; 150 gallons at 80 per cent. U.P., and 14 gallons of water ?
Answer, 13 per cent. U.P. (4.) In what proportions should spirits at the strengths respectively of 50 per cent. O.P., and 11 per cent. O.P. be mixed, so as to produce a compound at the strength of 25 per cent. O.P. ? Answer. In the proportion of 14 measures at 50 O.P. to 25 equal measures at 11 O.P.
(5.) From spirits worth respectively 11s. 9d., 10s. 3d., 8s. 6d., and 6s. 4d. a gallon, prepare a compound worth 9s. a gallon.
Answer. Take 32 galls. at 11s. 9d.
6 galls. at 10s, 3d. 15 galls. at 8s. 6d.
33 galls. at 6s. 4d. (6.) A vessel is found to contain 16,800 gallons of wash at the gravity of 45°. A subsequent account shows an increase of 5 per cent. in the quantity and 2 degrees in the gravity. What was the gravity of the wort that must have been added ?
Answer. 87° (7.) What quantity of spirits worth 14s. per gallon should be blended with 41 gallons of other spirits at 9s. 6d., and 59 gallons at 10s. 8d., so that the mixture shall be worth 11s. 6d. per gallon ?
Answer. 52 15 galls. (8.) A cask of the content of 120 gallons is filled with spirits at 22 per cent. O.P. But this being too weak, what quantity must be taken out and replaced with spirits at 60 per cent. O.P., so that the strength of the whole shall be 25 per cent. O.P.
Answer. 9.47 gallons or 9 gallons 3 pints.
(10.) EVOLUTION.-In treating of this subject it is necessary first to explain the meaning and notation of the powers of numbers. Evolution or the extracting of roots, is the reverse of Involution or the raising of powers. By a power of a number is signified the product formed when that number is multiplied once or several times successively into itself. Thus, 5 x 5 = 25. Here two equal factors are multiplied together, and their product, 25, is termed accordingly the second power of 5. Again, 125 is the third power of 5, since 5 x 5 x 5 = 125, whero three equal factors are concerned ; 625 is the fourth power of 5, for 5 x 5 x 5 x 5 (four factors) = 625, and so on of all other numbers, whole and fractional, the different powers in each case taking their names from the number of equal factors required to produce them respectively.
* A device by wbich the labour of finding the average of large quantities at numerous rates may be greatly abridged, is described in the Inland Revenue Almanack for 1859, page 15. According to the inexact method now deemed sufficient for the purpose of the aunual distillery returns, the mean gravity in Example (2) above, would be shown 47°, instead of 50.7.
It should be observed that any power of a fraction proper, * is necessarily less than the fraction itself. For instance, .4 x 4= .16, and .16 is not so great as •4, since 4-10ths, of 4-10ths can be only a fraction of .4. Similarly, XX X gives a product, which is only ths of . (See page 72.)
Powers are briefly denoted by placing an appropriate small figure called the index or exponent, close to the right hand upper corner of the number to be raised to the given power. Thus, 52 stands for the second power of 5; 59 for the third power of 5, &c. The index shows how often the number to which it is annexed, is to be repeated as factor.
It is customary to employ the word square instead of second power, and cube instead of third power, from the connection which exists between the process of finding the area of a square figure and the second power of a number, and from the similar relation between the content or volume of a solid body and the third power of a number. The fourth power is sometimes called also the biquadratic power, but there is no short term in general use for powers higher than the square and cube.
To complete the scale of powers, a number is said to be its own first power. 5° = 5, that is, 5 raised to the 1st power is 5.
There is nothing in the method of Involution that differs from ordinary multiplication. The object is always to find the actual product of a number multiplied continually by itself, until the number of factors equals the number of units in the index of the given power. If it were required to involve 7 to the fifth power, as indicated by the expression 79 the process would be to multiply 7 four times into itself, that is, to multiply five sevens together. In computing powers beyond the cube, certain abbreviations are possible, which will readily occur to any person from the following considerations. 5 x 5 x 5 x 5 = (5 x 5) x 5 x 5) = 5°x 5 = 5, that is, the 4th
power is the 2nd power squared, or multiplied by itself. 5 x 5 x 5 x 5 x 5 = (5 x 5 x 5) x (5 x 5) = 5°x 5° = 5, or the
5th power is the cube x the square. 5 x 5 x 5 x 5 x 5 = (5 x 5) x (5 x 5) x 5= 5 x 5 x 5 = 5,
or the 5th power is also the square x the square x the given number. It is evident that the same principle may be extended to any number of factors.
By a root of a number or power, is meant such a number as when multiplied a stated number of times into itself will produce the given number or power. Thus, as 25 is the square of 5, so 5 is the square root of 25, As .000512 is the cube of .08, 80 .08 is the cube root of .000512, and similarly with regard to all other powers and their corresponding roots.
Roots are usually denoted in arithmetic by the sign / called the radical sign, prefixed to the number of which a certain root is to be extracted. The square
* That is a fraction, the numerator of which is less than the denominator, and not a whole number or a whole number and fraction merely written in the form of a fraction. See page 66.