and the number of days, resulting from the concurrence of all these circumstances, is 24 by 9 by 8 by 327 by 18 by 7 71 by 11 by 65 by 13 by 5 Performing the multiplications and divisions, we get the answer required, 21 days. 123. This number is equal to 24 multiplied by the fractional quantity 9 by 8 by 327 by 18 by 7 71 by 11 by 65 by 13 by 5 but this last quantity, which expresses the relation of the number of days given, to the number of days required, is itself the product of the following fractions; Now going back to the denominations given to these numbers in the statement of the question, we see that these fractions are the ratios of the men and the hours, of the lengths, the breadths, and the depths of the two ditches; hence it follows, that the ratio of the number of days given, to the number of days sought, is equal to the product of all the ratios, which result from a comparison of the terms relating to each circumstance of the question.. This may be resolved in a simple manner by first finding the value of each of these ratios; for, by multiplying together the fractions that express them, we form a fraction which represents the ratio of the quantity required to the given quantity of the same kind. This fraction, which will be the product of all the ratios in the question, will have for its numerator the product of all the antecedents, and for its denominator, that of all the consequents. A ratio resulting, in this manner, from the multiplication of several others, is called a compound ratio. By means of the fractional expression Arith. 9 by 8 by 327 by 18 by 7 15 and the given number of days, 24, we make the following proportion; 71 by 11 by 65 by 13 by 5: 9 by 8 by 327 by 18 by 7 : : 24 : x, which may also be represented in this manner, as was the preceding example; 71: 9 65: 327 :: 24: x. 5: 7 Our remarks may be summed up in this rule; Make the number which is of the same kind with the required answer, the third term ; and of the remaining numbers, take any two that are of the same kind, and place one for a first term and the other for a second term according to the directions in simple proportion; then any other two of the same kind, and so on, till all are used; lastly, multiply the third term by the product of the second terms, and divide the result by the product of the first terms, and the quotient will be the fourth term, or answer required. Examples for practice. If 100. in one year gain 5l. interest, what will be the interest of 7501. for 7 years? Ans. 2621. 10s. What principal will gain 2621. 10s. in 7 years, at 51. per cent. per annum? Ans. 7501. If a footman travel 130 miles in 3 days, when the days are 12 hours long; in how many days, of 10 hours each, may he travel 360 miles? Ans. 9 how many Ans. 1021 days. If 120 bushels of corn can serve 14 horses 56 days; days will 94 bushels serve 6 horses? If 7oz. 5dwt. of bread be bought at 43d. when corn is at 4s. 2d. per bushel, what weight of it may be bought for 1s. 2d. when the price per bushel is 5s. 6d. ? Ans. 1lb. 4oz. 341dwt. If the transportation of 13cwt. 1qr. 72 miles be 2l. 10s. 6d. what will be the transportation of 7cwt. 3qrs. 112 miles? Ans. 21. 5s. 11d. 177q. 159 A wall, to be built to the height of 27 feet, was raised to the height of 9 feet by 12 men in 6 days; how many men must be employed to finish the wall in 4 days at the same rate of working? Ans. 36 men. If a regiment of soldiers, consisting of 939 men, consume 351 quarters of wheat in 7 months; how many soldiers will consume 1464 quarters in 5 months, at that rate? Ans. 5483. If 248 men, in 5 days of 11 hours each, dig a trench 230 yards long, 3 wide, and 2 deep; in how many days of 9 hours in length, will 24 men dig a trench of 420 yards long, 5 wide, and 3 deep? Ans. 2887 Fellowship. 124. The object of this rule is to divide a number into parts, which shall have a given relation to each other; we shall see in the following example its origin, and whence it has its name. Three merchants formed a company for the purpose of trade; the first advanced 25000 dollars, the second 18000, and the third 42000; after some time they separated, and wished to divide the joint profit, which amounted to 57225 dollars; how much ought each one to have? To resolve this question we must consider, that each man's gain ought to have the same relation to the whole gain, as the money he advanced has to the whole sum advanced; for he who furnishes a half or third of this sum, ought plainly to have a half or third of the profit. In the present example, the whole sum being 85000 dollars, the particular sums will be respectively 8, 1988, 43888 of it; and if we multiply these fractions by the whole gain, 57225, we shall obtain the gain belonging to each man. It is moreover evident, that the sum of the parts will be equal to the whole gain, because the sum of the above fractions, having its numerator equal to its denominator, is necessarily an unit. 25000 18000 42000 85000 25000 :: 57225 to the first man's gain, which may be enunciated thus ; The whole sum advanced to each man's particular sum :: the whole gain to each man's particular gain. By simplifying the first ratio of each of these proportions we have $ 85: 25:: 57225: to the gain of the 1st. or $1683015, 85: 18 :: 57225 85:42 : 57225 859 If all the sums advanced had been equal, the operation would have been reduced to dividing the whole gain by the number of sums advanced; we may reduce the question to this in the present case by supposing the whole sum, $85000, divided into 85 partial sums, or stocks of $1000 each; the gain of each of these sums will evidently be the 85th part of the whole gain; and nothing remains to be done, except to multiply this part severally by 25, 18, and 42, considering the sums 25000, 18000, and 42000 as the amounts of 25 shares, 18 shares, and 42 shares. In commercial language the money advanced is called the capital or stock, and the gain to be divided, the dividend. The following question is very similar to that just resolved. 125. It is required to divide an estate of 67250 dollars among 3 heirs, in such a manner, that the share of the second shall be of that of the first, and the share of the third of that of the second. 2 30 201 It is plain that the share of the third, compared with that of the first, will be of of it, or; then the three parts required will be to each other in the proportion of the numbers 1, and Reducing these to a common denominator, we find them ,, and, and have the three numbers 20, 8, and 7, which are proportional to the first; but as their sum is 35, it is plain, that if we take three parts expressed by the fractions 3, 3, and, they will be in the required proportion. The question will then be resolved by taking, then, and then 7 of 67250 dollars, which will give the sums due to the heirs, according to the manner prescribed, namely; $38428, $153711, and $13450. 126. Again, here are two fountains, the first of which will fill a certain reservoir in 2 hours, and the second will fill the same reservoir in 33 hours; how much time will be required to fill the reservoir, by means of both fountains running at the same time? We must first ascertain what part of the reservoir will be filled by the first fountain in any given time, an hour for instance. It is evident that, if we take the content of the reservoir for unity, we have only to divide 1 by 24, or, which gives us for the part filled in one hour by the first fountain. In the same manner, dividing 1 by 33, or 5, we obtain for the part of the reservoir filled in an hour by the second fountain; consequently, the two fountains, flowing together for an hour, will fill to, or of the reservoir. If we now divide 1, or the content of the reservoir, by 1, we shall obtain the number of hours necessary for filling it at this rate; and shall find it to be 15 or an hour and an half. added Authors who have written upon arithmetic, have multiplied and varied these questions in many ways, and have reduced to rules the processes which serve to resolve them; but this multiplication of precepts is, at least, useless, because a question of the kind we have been considering may always be solved with facility by one who perceives what follows from the enunciation, especially when he can avail himself of the aid of algebra; we shall therefore proceed to another subject. 127. Besides the proportions composed of four numbers, one of the two first of which contains the other as many times as the corresponding one of the two last contains the other; it has been usual to consider as such the assemblage of four numbers, such as 2, 7, 9, 14, of which one of the two first exceeds the other as much as the corresponding one of the two last exceeds the other. These numbers, which may be called equidifferent, possess a remarkable property, analogous to that of proportion; for the sum of the extreme terms, 2 and 14, is equal to the sum of the means 7 and 9.* * The ancients kept the theory of proportions very distinct from the operations of arithmetic. Euclid gives this theory in the fifth book of his elements, and as he applies the proportions to lines, it is apparent, that we thence derive the name of geometrical proportion; |