12 Note. Since of x + 25 is 3x-35, x+25 must be twice 35. 3 x 2. Two men talking of their horses, A says to B, my horse is worth $25 more than yours, and of the value of my horse is equal to of the value of yours. What is the value of each? Let x denote the value of B's horse. Ꮖ Ans. A's $125, B's $100. Proof. The first condition is evidently answered. With regard to the second, of 125 is 75, and of 100 is 75. 394 of 3. Two men talking of their ages, one says, my age is now yours, but in twenty years from this time, if we live, it will be of yours. Required the age of each. Suppose the age of the elder x. Then the younger will be 3 x 4 In 20 years the age of the elder will be x + 20, and of the = 60= age of younger. 4. A man being asked the value of his horse and chaise, answered, that the chaise was worth $50 more than the horse, and that one half of the value of the horse was equal to one third of the value of the chaise. Required the value of each. h. 5. Two persons talking of their ages, the first says, of my age is equal to of yours; and the difference of our ages is 10 years. What are their ages? /. 25. 6. There are two towns situated at unequal distances from Boston, and on the same road. They are 30 miles apart. of the distance of the second from Boston is equal to of the distance of the first. What is the distance of each from Boston? 7. A man being asked the value of his horse and saddle, answered, that his horse was worth $ 114 more than his saddle, and that of the value of his horse was 7 times the value of his saddle. What was the value of each? 8. A hare is 40 rods before a greyhound, but she can run only as fast as the greyhound. How far will each of them run before the greyhound will overtake the hare? 9. A gentleman paid 4 laborers $ 136; to the first he paid 3 times as much as to the second wanting $4; to the third one half as much as the first, and $ 6 more; and to the fourth 4 times as much as to the third, and $5 more. How much did he pay to each? 10. A man bought some cider at $4 per barrel, and some beer at $7. There were 6 barrels more of the cider than of the beer; and of the price of the beer was equal to of the price of the cider. Required the number of barrels of each, 11. Two men commenced trade together; the first put in £40 more than the second, and the stock of the first was to that of the second as 14 to 5. What was the stock of each? 14 to 5 signifies the second is of the first. 14 12. A man's age when he was married was to that of his wife as 3 to 2; and when they had lived together 4 years, his age was to hers as 7 to 5. What were their ages when they were married? 13. A and B began trade with equal sums of money. In the first year A gained £40, and B lost £ 40; but in the second, A lost one third of what he then had, and B gained a sum less by £40 than twice the sum A had lost; when it appeared that B had twice as much money as A. What money did each begin with? Let x be the number of pounds each had at first. The x + 40 will be the sum A had at the end of the first year; and x 40 the sum B had. 13 The second year A lost of what he then had, consequently 2 x 80 he saved; his sum will then be 3 B gained twice as much as A lost wanting £40; his will be in both Note. In this example the result had the sign members, but by transposing it has the sign +. It would have been the same thing if the signs had been changed without transposing. The result would have come out right if the first member had been made the second, and the second first, in the first equation. 14. A person playing at cards, cut the pack in such a manner, that of what he cut off were equal to of the remainder. How many did he cut off? # What 15. Divide $ 183 between two men, so that of what the first receives, shall be equal to of what the second receives. will be the share of each? 16. A man sold 20 bushels of grain, rye and wheat; the rye at 5s. and the wheat at 7s. per bushel; of the rye came to as much as cf the wheat. How much was there of each? 17. What number is that from which if 5 be subtracted two thirds of the remainder will be 40? 18. A man has a lease for 99 years; and being asked how much of it was already expired, answered, that two thirds of the time past was equal to four fifths of the time to come. Required the time past, and the time to come. $4, past 19. It is required to divide the number 50 into two such parts, that three fourths of one part added to five sixths of the other may make 40. 20.30 20. Two workmen received equal sums for their work; but if one of them had received 18 dollars more, and the other 3 dollars less, then of the wages of the latter would have been equal to of the wages of the former. How much did each receive? S 14 21. A certain man, when he married, found that his age was to that of his wife as 7 to 5; if they had been married 8 years sooner, his age would have been to hers as 3 to 2. What were their ages at the time of their marriage?); VI. 1. Divide the number 68 into two such parts, that the difference between the greater and 84, may be equal to three times the excess of 40 above the less. Let the less. x= the greater. x is 68- x must be subtracted from 84. Observe that 68 not so great as 68 by x. Therefore if I subtract 68 from 84, I shall subtract too much by the quantity x, and I must add a to obtain the true result. Then we have 84-68+x for the difference between 84 and 68 x. The excess of 40 above the less is 40 is 120- 3 x. x, and 3 times this 3x = 104 84 68+x=120 4 x 68-26 x= 26 = less. 42= greater. x was subtracted from 84. In Note. In this question 68 stead of x, now put its value, 68-26. Now 68—26=42, that is, the number to be subtracted from 84 is 42, and the answer must be 42. When 68 is subtracted from 84, the result is 16, which is too small by 26, the value of x; to this it is necessary to add 26, and it makes 42, the true result, 84 68+ 26= 42. This shows that we did right in adding a after subtracting 68. This will always be found true. Therefore, when any of the quantities to be subtracted have the sign — before them, they must be changed to + in subtracting, and those which have must be changed to 2. A gentleman hired a laborer for 20 days on condition that, for every day he worked, he should receive 7s., but for every day he was idle, he should forfeit 3s. At the end of the time agreed on he received 80 shillings. How many days did he work, and how many days was he idle? Let x= the number of days he worked. Then 20-x — the number of days he was idle. x days, at 7s. a day, would come to 7 x shillings. -x, at 3s. per day, would be 60 20 must be taken out of 7 x. By the above rule 60 3 x shillings. This 3 x, subtracted from 7 x, leaves 7 x 60+3x; for 60 is too much to be subtracted by 3 x. 3. Two men, A and B, commenced trade; A had twice as much money as B ; A gained $ 50, and B lost $90, then the difference between A's and B's money was equal to three times what B then had. How much did each commence with? 4. Two men, A and B, played together; when they commenced they had $20 between them, after a certain number of games, A had won $ 6, then the excess of A's money above B's was equal to of B's money. How much had each when they commenced? 5. Divide the number 54 into two such parts that the less subtracted from the greater, shall be equal to the greater subtracted from three times the less. What are the parts? 6. It is required to divide the number 204 into two such parts, that of the less being subtracted from the greater, the remainder will be equal to of the greater subtracted from four times the less. |