$500 to the latter for his management of the business. How much did each receive?

Ans. The one $76663, the other $43334. 216. Divide $1925 into three portions, so that the second may be $150 greater than the first, and the third $125 greater than the second. Ans. $500, $650, $775.

217. A man died leaving an estate of $7500 to his widow, two sons, and three daughters, in such a manner that each son was to receive twice as much as each daughter, and the widow $500 more than all the children together. How much is the share of each person?

Ans. The widow $4000, each son $1000, each daughter $500. 218. On New Year's day a father presented $100 to his five children, dividing it so that each received $2 more than the next younger child. What was the share of the youngest? Ans. $16.

219. A man left his property by will in such a manner that his widow was to receive one half of it, less $3000, his son one third, less $1000, and his daughter one fourth, more $800. What was the amount of the estate, and what the share of each legatee? Ans. $38,400, $16,200, $11,800, $10,400.

Suggestive Question.-How much is one twelfth of the estate? 220. A certain property valued at $5600 is to be divided among 5 persons as follows:-B is to receive twice as much as A, and $200 more; C three times as much as A, less $400; D half of what B and C receive together, and $150 more; and E one fourth of what all the others receive, and $475 more. What is the share of each ?

Ans. A $500, B $1200, C $1100, D $1300, and E $1500. 221. If I have a certain number of dollars in my purse, and by adding 24 to it the sum becomes $80, how many dollars were in it at first? Prove. By what process was it found?

222. I have a certain number of dollars in my purse, and, after having subtracted $24 from it for the payment of expenses, there remain $56. How many dollars were in at first? Prove. What was the process?

223. A certain number, multiplied by 24, gives 1800 for product. What was the original number? Prove. What was the process?

224. A certain number divided by 24, gives 75 for quotient. What was the original number? Prove. What was the process?

225. How many dollars did you pay for that? says William to John. If you multiply the number by 7, replied John, add 3 to the product, divide this by 2, subtract 4 from the quotient, the remainder will be 15. How much was paid? Prove. See the last 4 examples.

226. A man whose age is 30 years has a son aged 10. In how many years will this man, who is now three times, be only twice as old? Prove.

227. The boy mentioned in the preceding exercise has a brother aged 6. In how many years will the ages of both the boys together equal that of their father? Prove.

228. A cistern has three pipes. By the first it can be filled in 2 hours, by the second in 3 hours, and by the third in 4 hours. In what time can it bé filled when all three run together? Ans. in 55,5 minutes.

229. A cistern of 365 cubic feet has three pipes. The first discharges 52 cubic feet in 3 minutes; the second 51 cubic feet in 32 minutes; and the third 27 cubic feet in 21 minutes. In what time can the cistern be filled if they all run at once? Ans. In 8,85 minutes.

172

230. A contribution being necessary for a purpose in which a number of persons were equally interested, 80 cents each was first proposed, which was found to produce $10 too much, and then 60 cents each was tried, which was found to produce $10 too little. What was the amount wanted, and how many were the contributors? Prove by trial.

231. A merchant, having a piece of unsalable silk, disposed of it to a lady at prime cost. Having sent it home to her without measurement, and being unable to find the original bill, he could only ascertain the length and wholesale price by recollecting that if he had sold it for $1.25 per yard his profit would have been $12, whereas, at $1 per yard, it would have netted him only $6. What was the number of yards, and their prime cost? Prove by trial.

232. The income of two brothers taken together is $1000 per annum. If the income of the elder was increased sixfold

and that of the younger fourfold, their joint income would be $4800. What was the income of each ?

Ans. Elder's income $400, younger's $600.

233. Find two numbers such that if the first be taken 4 times and the second 3 times, their sum is 2760; and if the first be taken 5 times and the second 7 times, their sum is 4490. Prove by trial.

234. A person has a certain number of pieces of money which he wishes to arrange in the form of a square. At the first trial there were 190 over; but, when the side of the square was enlarged by 4 more pieces there only remained 14. How many pieces of money had he? Draw a square on the slate, and enlarge it by 4 additions. Prove by trial.

235. A person has 330 coins, consisting of eagles and dollars. Their value amounts to $1500. How many are there of each? Prove by trial.

Suggestive Questions.-How many coins would there be if the money was all in eagles? How many more should there be? Every eagle changed into dollars gives how many additional coins? Then how many eagles must be changed into dollars to make 330 in all?

236. What number is that, which, if you multiply it by 4g, take 19 from the product, multiply it again by 31, and take away 13, nothing will remain? Prove by trial.

237. A boy having a basket of apples, sold half of them and 3 more to one of his neighbors, and half the remainder and 5 more at the next house. Finally, he sold half of what still remained and three more, and then found he had only three left. How many had he at first? Prove by trial.

238. A man bequeathed, by will, to his widow $250 and half the remainder of his property; to his son a fifth of the residue and $525 more; and the remainder, amounting to $1375, to his daughter. What was the whole amount of the property? Prove by trial.

239. A person being asked how many dollars he had, replied, if you add to them their third part and 176 more, and then multiply by 21, the sum will as much exceed $1000 as it now falls short of it. Prove by trial.

CHAPTER V.

PRACTICAL APPLICATION

OF THE ELEMENTS OF ARITHMETIC BY MEANS OF RATIO AND PROPORTION.

DEFINITIONS.

I. Ratio is the relation which one quantity has to another of the same kind, as expressed by the quotient of the one divided by the other. Thus, the ratio of 4 to 2 is ₫ or 2, and the ratio of 5 to 6 is §. A ratio is sometimes written 4: 2, or 4÷2. But the fractional form, as, is the most convenient for arithmetical computations. The two numbers which constitute a ratio are called its terms.

8

II. A ratio, like a common fraction, remains unchanged in value: (1) when both its terms are multiplied by the same number; (2) when both terms are divided by the same number; (3) when the same proportional part of each is added to or subtracted from both. Thus, 12, 24, 8, 10, -, are merely different forms of the same ratio (all being equal to 2), although both terms in the second have been multiplied by 2, in the third divided by 3, in the fourth increased by 4th, in the fifth diminished by 4th.

III. Proportion is the equality of two ratios. Thus, = 12 is a proportion, both the ratios being 2. Although both terms of a ratio must relate to things of the same kind, it is not necessarily so with the two ratios of a proportion. These may relate either to things of the same kind, or to things of different kinds. Thus, the proportions

are both correct.

horses. dollars. days. days.
1=12, and 1—12

IV. Proportion is used in the resolution of arithmetical problems in cases where one of the ratios is incomplete from one of its terms being unknown. This unknown term is readily supplied by a mere change of form in the complete ratio; that is, by changing its denomination to that of the incomplete one. 4 Thus, in the proportion the unknown term is found to

16-8'

be 8 by changing the complete ratio to 16ths. Again, in the

changing to 4ths.

12

the unknown term is found to be 3, by

V. Proportion is either simple or compound. It is simple when both ratios are simple; compound when one of them is compound. Thus, the ratios, form a simple, and g of a compound proportion. The latter may be more concisely written 3:54. See p. 58, 1. 5; and 83, 1. 16.

Σ

5.9

12

VI. In every problem in proportion there is an affirmation and a question. Thus, in the problem, "Bought 9 yards of cloth for £5 12s.; what will be the cost of 72 yards?" the affirmation is that "9 yards were bought for £5 12s.;" and the question is, "what will be the cost of 72 yards?" The affirmation, however, is more commonly put in the form of a supposition, as, "If 9 yards cost £5 12s." Again, in the problem, "If a man travel 90 miles in 3 days of 12 hours long, how far will he travel in 8 days of 10 hours long?" the first clause, to the comma, is the affirmation; the last clause, to the note of interrogation, is the question.

VII. The number belonging to the imperfect ratio may always be ascertained from the words asking the question. Thus, in the first of the above problems, it is "£5 12s.,” since the question is "What will be the cost?" In the second problem it is "90 miles," the question being, "How far will he travel?" The arrangement of the terms of the perfect ratio depends on the answer to a question which should be put to every perfect ratio, and to every part of it if it be compound, namely, LESS or MORE? that is, "What effect will the numbers in the perfect ratio produce on the unknown term; increase or decrease it?" The larger term is placed above or below,