100

(7.) If the prime cost of leaf tobacco, inclusive of the duty, amount to £21 16s. 9 60 d. per cwt. what weight of water must be added to each cwt., so that the manufactured tobacco may be sold, without any advance on the cost-price, at a profit of 2d. per lb. ?

Answer 5 lbs.

Compound Proportion.-A ratio is said to be compounded of two or more ratios, when it is formed by multiplying together the antecedents and consequents, that is, the corresponding terms, respectively, of these ratios. Thus, the ratio 60; 72 is compounded of the ratios 6: 8 and 10: 9, for 6 x 10 gives the first term, and 8 x 9 the second term of the compound ratio. The ratio of 60 to 72 may also be decompounded into the ratios of 5: 3, 4: 6, and 3: 4, since 60-5 x 4 x 3, and 72-3x6 × 4. In the same way a compound fraction expresses the ratio formed of the products of its several component numerators and denominators respectively.

For instance,

where the numerators represent the antecedents, and the denominators the consequents of the ratios, and

If in two or more sets of proportionals, the corresponding terms be multiplied together, the products will be proportionals.

For, by the given proportions, = §, and § = . Hence, on the principle that, if equals be multiplied by equals the results must be equal, x = × 10 that is, 1998, or 10: 12: 60: 72. For the same reason, any number of sets of proportionals, if multiplied together, or compounded, in this manner will remain proportionals.

The subject of compound proportion finds its most important application in what is called the Double Rule of Three, a name which implies that every question coming under this rule consists of at least two related questions in the Single Rule of Three, or simple proportion. These questions, or rather the data contained in them, can always be combined so as to form a case of compound pro>portion, which may then be solved by the general methods laid down in the Double Rule of Three; but as such problems are of little practical value, and seldom occur in actual business, except in the adjustment of certain mercantile accounts-exchanges of money, merchandise, &c.-it is best to regard them merely as useful arithmetical exercises, and to reason out the result in each instance by the aid of simple proportion, dispensing with any special rule or formula for the solution.

Example (1.) 90 gallons of spirits, 11 per cent. O. P. are bought, duty paid, at 12s. 6d. per proof gallon. 5 bulk gallons are lost by waste in store, and the strength decreases to 8 O. P.; at what price must the remainder be sold to realise a profit of 2s. 6d. per proof gallon on the original quantity?

Here, we first consider that if no deficiency took place either in bulk or strength, each proof gallon of the spirits should sell for 15s. to give a profit of 2s. 6d. on

the cost price. But as the bulk becomes reduced to 85 gallons, a proportionally greater price must be charged to afford the required profit, and we say accordingly,

We next take into account the falling off in strength which, of course, makes it necessary to increase still further the selling price, and we have evidently the proportion,

The answer is, therefore, 16s. 315d. per proof gallon.

To prove the correctness of this result, we may treat the question in a different way;-90 gallons at 11 per cent. O. P. 99 9 gallons at proof, and the amount of 99.9 gallons, at 15s. the gallon, is £74 18s. 6d. But, 85 gallons at 8 per cent. O. P. = 91.8 gallons at proof; and to find the price at which this quantity should be sold so as to yield the sum of £74 18s. 6d. and thus secure the profit of 2s. 6d. a gallon, we have only to divide £74 18s. 6d. by 91·8. The quotient, 16s. 315d., corroborates the answer previously obtained.

In working by two statements in the Rule of Three, it will save much labour if we merely indicate the component parts of the fourth term of the first proportion, and employ the expression of that result, instead of the developed result itself, as the third term of the second statement. We thus avoid the introduction

of fractions until the end of the process, e. g.

From the expression,

15 x 90 x 111
85 × 108

:

= = 16 315

34

85 × 108.

which furnishes the answer in this case, it may be seen how the methcd usually prescribed in the Double Rule of Three is to be applied to the solution of questions involving a compound ratio. We have first the ratio of 90 gallons to 85 gallons, and secondly the ratio of 111 degrees of strength to 108 degrees of strength. Now, the cost price added to the required profit, (= 15s.) must be increased in the ratio of 90: 85, and again in the ratio of 111 108, that is, in the compound ratio of 90 x 111: The operation is accordingly to multiply 15 by the fraction or ratio, Whatever number of different ratios may be given, the quantity which is placed as the third term of the proportion, must be multiplied by each of these ratios successively, or by the single ratio compounded of their products. A little consideration will point out the proper manner of forming each ratio from the data of the question, that is, which of the given numbers or quantities should be used as multipliers, and which as divisors of the third term.

90 × 111
85 × 108'

Example (2.) Suppose it asked,-If 3 officiators writing 8 hours a day can together fill up 1250 forms in 5 days, in how many days will 4 officiators, writing with equal despatch for 6 hours a day, fill up 1475 such forms?

Here, according to the Rule of Three, we set 5 days, which is the term of the same kind as the answer, in the third place; and then excluding for the time all the other conditions of the question, we reason that if 3 persons can do a certain amount of work in 5 days, 4 such persons can do it in ths of that time. We have, therefore, for the first multiplier or ratio, 2. Secondly, we consider, that a person working only 6 hours a day will require ths of the time that he would need to execute the same task if he worked 8 hours a day; whence we have for the second ratio, g. Thirdly, as 1475 forms have to be filled up instead of 1250, an equal number of persons will take a longer time than 5 days to do it in, and we have for the third ratio, 1473. Accordingly, the compound ratio by which 5 days should be multiplied, consists of the product of the ratios, xx 1178, and the answer is, 5×3×8×1475 4x6x1250

= = 58

1250

5 days of 6 hours each, or 5 days, 5 hours, 24 minutes.

10

A great variety of well-contrived questions in the Double Rule of Three, and in Compound Proportion generally, may be obtained from most of the modern treatises on arithmetic. These questions, if properly analysed and solved by a true perception of the manner in which the several data are related to each other, afford very useful exercises in numerical reasoning, but if treated as so many applications of an arbitrary rule, the principle of which is not inquired into or understood, they are valueless as a means of instruction or improvement.

Proportional Parts.—To divide a given number or quantity into parts which shall have to one another any assigned proportion, we proceed as follows;-RULE. Add together the numbers which express the required proportion, and say by the Rule of Three-As this sum is to each of its parts successively, so is the given number or quantity to each of the parts into which it is to be divided.

Example (1.) Divide 350 into two parts which shall be to each other in the ratio of 16: 9.

The correctness of the answer is proved by the proportion, 224; 126: 16: 9, for

224 × 9126 × 16.

With respect to the principle of the rule in this case, it will be evident that the given quantity is itself the sum of certain parts which are to bear to one another a specified ratio, and as the sum of the numbers composing that ratio is to each of its parts, so must the given quantity or sum be to each of its parts. Whatever fraction each term of the given ratio is of the sum of its terms, the same fraction must be each of the required proportional parts of the sum of those parts,-that is, of the quantity to be divided: 18 = 128; 25 224

350

=126

350

Example (2.) Divide £78 in the proportion of 2, 5, 7, and 10.

And generally, any two of the given parts of 24 are to each other in the same ratio as are any two corresponding parts of £78.

When the proportional parts are numerous, or the required division involves the use of fractions, it will be found the most convenient plan to divide the third term by the first-the given quantity by the sum of the numbers which express the ratios-and then to multiply each of these numbers in succession by the quotient thus obtained, as in the following example.

Example (3.) A distiller mashes from a grist made up of the following materials :— malt, 188 bush.; barley, 376 bush.; oats, 240 bush.; rye, 324 bush.; and locust beans, 232 bush. What proportion of each of these ingredients is there in 100 parts of the mixture?

The total number of bushels is 1360, and five distinct statements are necessary, in each of which 1360 is the first term or divisor, and 100 the third term, or constant factor of the dividend. It will save much labour, therefore, to form at once the quotient of 100 1360, and to multiply each of the given quantities successively by this number.

Thus, 100136007353 common multiplier.

13.82 malt, in 100 parts.

07353 x 188

The trouble of performing the last multiplication may be avoided in these cases by deducting the sum of all the preceding results from the given number, but then the adding of the several parts together will not prove the accuracy of the work, and it is always desirable to obtain a test of correctness.

EXERCISES IN PROPORTIONAL PARTS.

Exercises.-(1.) Proof spirit is composed of 57.06 measures of alcohol, and 42.94 equal measures of water. What proportion of each ingredient is contained in 95 gallons of proof spirit?

Answer. 54 21 alcohol. 40.79 water.

(2.) A distiller uses 3ths malt, 18ths barley, 1th oats, ths rye, andths 옳 maize. Required the quantity of each in 2000 bushels of grist ? Answer. Malt, 240 bush.; barley, 760 bush.; oats, 200 bush.; rye, 320 bush ; maize, 480 bush.

(3.) Divide 5ths in the proportion of 7, 8, and 9.

Answer. 35

40 45 384 384' 384

6. PRACTICE.-Practice is the name given in mercantile arithmetic to a short process of compound multiplication, which is effected chiefly by means of what are called Aliquot Parts. One number or fraction is said to be an aliquot part of another number or fraction, when the first is contained an exact number of times in the second. Thus, 3, 2, 14, &c., are aliquots parts of 6, being contained in it respectively, 2, 3, 4, &c. times. 10s., 6s. 8d., 5s., 4s., 3s. 4d., &c., are aliquot parts of a pound, since 10s.£1, 6s. 8d.=£}, 5s.=£1, 4s.=£1, 3s. 4d. =£1, &c. Accordingly, every fraction of a unit, which has 1 for its numerator, may be called an aliquot part of that unit. Agreeably to this definition, 4d. is an aliquot part (rd) of a shilling, but 8d. is not, since 8d. is not contained in a shilling a whole number of times.

It is plain, therefore, that the terms measure and aliquot part are identical in meaning. The latter term, however, is specially applied to certain fractions of compound quantities, the use of which facilitates the performance of the calculations commonly required in business. Only the more convenient and easily managed fractions are employed as aliquot parts, those with large denominators not being suited to the powers of ordinary computers. The general practice is to avoid any aliquot part, which involves division by a number greater than 12;-10d., for instance, when referred to a pound as the unit, can hardly be regarded as an available aliquot part, since few persons are able at one operation to divide a sum of money by a number so high as 24, without having recourse to the process of long division, and this would be inconsistent with the rapidity of working which is mainly the object sought to be attained by the use of aliquot parts. The expedient adopted in most cases, is either to select such fractions of the integer as fall within the limits of ordinary short division, or else to decompose, if possible, the higher aliquot parts into two or more factors each of which furnishes a ready divisor, and to apply these successively, striking out all but the final result before the several quotients are added together.

Questions in simple proportion or the rule of three, where the first term of the statement is unity, may be expeditiously solved by Practice. The computation of prices and values generally, at given rates, may be performed with advantage in the same manner. Nearly all mental reckonings relating to compound quantities are best carried on by the help of aliquot parts, as will presently be exemplified.

It is the distinctive feature of this mode of computing that it preserves all such values as £ s. d., cwt., qrs., lbs., &c., in their highest given denomination throughout