the number of leaps made by the fox, in passing over the entire distance. It might, at first, be supposed that the equation of the problem would be obtained by placing this number equal to x; but in doing so, a manifest error would be committed; for the leaps of the greyhound are greater than those of the fox, and we should thus equate numbers referred to different units. Hence, it is necessary to express the leaps of the fox by means of those of the greyhound, or reciprocally. Now, according to the enunciation, 3 leaps of the greyhound are equivalent to 7 leaps of the fox; and hence, 1 leap of the 7 greyhound is equivalent to leaps of the fox; consequently, 7x x leaps of the greyhound are equivalent to of the fox: that 3 is, had the leaps of the greyhound been no longer than those of or, by making the terms entire 14x = 360 + 9x, whence 5x 360 and x = 72. Therefore, the greyhound will make 72 leaps to overtake the fox, and during this time the fox will make 72 × 3 Verification. The 72 leaps of the greyhound are equivalent to 72 x 7 3 = 168 leaps of the fox = the whole distance. And 60+108= 168, the leaps which the fox made from the beginning. 7. A can do a piece of work alone in 10 days, and B in 13 days: in what time can they do it if they work together ? Denote the time by æ, and the work to be done by 1. Then in 1 day A could do 1 1 13 of 8. Divide $1000 between A, B, and C, so that A shall have $72 more than B, and C $100 more than A. Ans. A's share = $324, B's = $252, C's = $424. 9. A and B play together at cards. A sits down with $84 and B with $48. Each loses and wins in turn, when it appears that A has five times as much as B. How much did A win? Ans. $26. 10. A person dying leaves half of his property to his wife, one sixth to each of two daughters, one twelfth to a servant, and the remaining $600 to the poor: what was the amount of his property? Ans. $7200. 11. A father leaves his property, amounting to $2520, to four sons, A, B, C, and D. C is to have $360, B as much as C and D together, and A twice as much as B less $1000: how much does A, B, and D, receive? Ans. A $760, B $880, D $520. 12. An estate of $7500 is to be divided between a widow, two sons, and three daughters, so that each son shall receive twice as much as each daughter, and the widow herself $500 more than all the children: what was her share, and what the share of each child? Widow's share $4000. Each son $1000. Each daughter $500. Ans. { 13. A company of 180 persons consists of men, women, and children. The men are 8 more in number than the women, and the children 20 more than the men and women together: how many of each sort in the company? Ans. 44 men, 36 women, 100 children. r ( 14. A father divides $2000 among five sons, so that each elder should receive $40 more than his next younger brother: what is the share of the youngest? Ans. $320. 15. A purse of $2850 is to be divided among three persons, A, B, and C; A's share is to be to B's as 6 to 11, and C is to have $300 more than A and B together: what is each one's share? Ans. A's $450, B's $825, C's $1575. 16. Two pedestrians start from the same point; the first steps twice as far as the second, but the second makes 5 steps while the first makes but one. At the end of a certain time they are 300 feet apart. Now, allowing each of the longer paces to be 3 feet, how far will each have travelled? Ans. 1st, 200 feet; 2d, 500. 17. Two carpenters, 24 journeymen, and 8 apprentices, received at the end of a certain time $144. The carpenters received $1 per day, each journeyman half a dollar, and each apprentice 25 cents: how many days were they employed ? Ans. 9 days. 18. A capitalist receives a yearly income of $2940: four fifths of his money bears an interest of 4 per cent., and the remainder of 5 per cent.: how much has he at interest ? Ans. $70000. 19. A cistern containing 60 gallons of water has three unequal cocks for discharging it; the largest will empty it in one hour, the second in two hours, and the third in three: in what time will the cistern be emptied if they all run together? 1 Ans. 3211 min. 20. In a certain orchard are apple-trees, 4 peach-trees, plum-trees, 120 cherry-trees, and 80 pear-trees: how many trees in the orchard? Ans. 2400. 21. A farmer being asked how many sheep he had, answered that he had them in five fields; in the 1st he had 1, in the 2d, in the 3d, in the 4th, and in the 5th 450: how many had he? Ans. 1200. 22. My horse and saddle together are worth $132, and the horse is worth ten times as much as the saddle: what is the value of the horse? Ans. $120. 23. The rent of an estate is this year 8 per cent. greater than it was last. This year it is $1890: what was it last year? Ans. $1750. 24. What number is that from which, if 5 be subtracted, of the remainder will be 40? Ans. 65. G 25. A post is 1 in the mud, & in the water, and ten feet above the water: what is the whole length of the post? 26. After paying and 3 of my money, I had 66 guineas left in my purse: how many guineas were in it at first? Ans. 120. 27. A person was desirous of giving 3 pence apiece to some beggars, but found he had not money enough in his pocket by 8 pence; he therefore gave them each two pence and had 3 pence remaining: required the number of beggars. Ans. 11. 28. A person in play lost shillings; after which he lost of his money, and then won 3 of what he then had; and this done, found that he had but 12 shillings remaining: what had he at first? Ans. 20s. 29. Two persons, A and B, lay out equal sums of money in trade; A gains $126, and B loses $87, and A's money is now double of B's: what did each lay out? Ans. $300. 30. A farmer bought a basket of eggs, and offered them at 7 cents a dozen. But before he sold any, 5 dozen were broken by a careless boy, for which he was paid. He then sold the remainder at 8 cents a dozen, and received as much as he would have got for the whole at the first price. How many eggs had he in his basket? Ans. 40 dozen. 31. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern, he there spends 2 shillings also; then borrowing again as much money as was left, he went to a third tavern, where likewise he spent 2 shillings and borrowed as much as he had left; and again spending 2 shillings at a fourth tavern, he then had nothing remaining. What had he at first? Of Equations of the First Degree, involving two or more Unknown Quantities. 95. Although several of the previous questions contained in their enunciation more than one unknown quantity, we have nevertheless resolved them all by employing but one symbol. The reason of this is, that we have been able, from the conditions of the enunciation, to represent the other unknown quantities by means of this symbol and known quantities; but this cannot be done in all problems containing more than one unknown quantity. To explain the methods of resolving problems of this kind, let us take some of those which have been resolved by means of one unknown quantity. 1. Given the sum of two numbers equal to a, and their difference equal to h; it is required to find the numbers. Let x= the greater, and y the less number. - b. By subtracting (Art. 86, Ax. 2), 2y = a Each of these equations contains but one unknown quantity. 2. A person engaged a workman a number of days, denoted by n. For each day that he labored he was to receive a cents, and for each day that he was idle he was to pay b cents for his board. At the end of the n days, the account was settled, when the laborer received c cents. Required the number of working days and the number of days he was idle. |