171. Two mixtures are made of brandy and sherry; the quanti ties of brandy in each being as 4 to 3; and the difference of the quantities of sherry being greater by twenty-five gallons than the difference of the quantities of brandy. Also, if three times the quantity of brandy had been put into the first mixture, and twice the quantity into the second, the quantities of brandy would have been proportional to the quantities of sherry. But if the sherry in the second mixure had been mixed with the brandy in the first, and the sherry in the first with the brandy in the second, the whole mixtures would then have been in the ratio of 5 to 6. Required the quantities of brandy and sherry in each mixture. Let 47 and 3 be the quantities of brandy, .. (12x: 6x::)2: 1, the ratio of the quantities of sherry. Let .. 2y and y be the quantity of sherry, .. y=x+25. Also, 4x+y: 3x+2y::5: 6, and 24x+6y=15x+10y,.. 91-4y=4x+100, and 199-20, therefore the quantities of brandy are 80 and 60, and that of sherry 90 and 45, Ans. 172. During a winter, when fuel was scarce, 2 men, A and B, went in quest of coals and turf, which they agreed to use in common. A met with three bushels of coals, and B two, at the same price per bushel, and also seven baskets of turf. A stipulated that he should consume twice as many coals as B. B assented, but demanded of him 2s. 10d. When this stock was exhausted, B purchased one bushel of coals, and A five, together with six baskets of turf, at the same rates respectively as before; but now B consumed three times as many coals as A, and paid him 18s. 6d. What was the price of a bushel of coals, and of a basket of turf; equal quantities of turf having been consumed by each person? Let x= the value of a bushel of coals in pence, and y = the value of a basket of turf. Now in the first case A consumes of coals; ... }.5x = the value of coals consumed by him, and = the value of turf consumed by him .. 13 +2y=3x+34 In the second case B consumed of coals, ... 4·6x the value of coals consumed by him, and 4·6x+5=x+222, .. 7x+6y= 7y 2 by 2 147 2 10 7 444. Hence (7x+6y=444)—(7x+y=714) 4 and x=60. 135 and y Ans. The price of a bushel of coal was 5s. and of a basket of turf 4d. 173. Two Spanish muleteers, A and B, were seated under a tree in order to dine; and on examining, found their stock of provisions to consist of five small loaves of bread, three of which were A's property, and a bottle of wine, which was B's. A stranger, who happened to come up at the time, was invited to partake of their fare, which was just sufficient for three persons; and at parting, being pleased with their behavior, he gave them what Spanish money he had about him, which amounted to 6s. 51⁄2d., to be equitably shared between them. Now as many shillings as a loaf cost pence would, with four pence more, at the next town have bought six such loaves and four bottles of the same wine; and when the money was divided, B received 1s. 101d. more than A. What was the price of each loaf, and a bottle of wine? Ans. A loaf cost 7 pence, and a bottle of wine 11 pence, the price of a bottle of wine, .. 3x = the price of A's loaves, and 2x price of B's, and 12x+4-6x+4y, ... y = 3x+2 = the they all eat equal portions; ... each eat of 5 loaves. A, .. eat § and gave to the stranger; and B, having 2 loaves, gave to the stranger. But B had a bottle of wine, of which he gave to the stranger. 4x-y Hence is the price of the provisions A furnished to the stranger, and (x+2y) the price of what B furnished. Now A receives 55 pence, and B 50 pence, ... (4x-y) : }(x+2y) :: 55: 50, or 4x-y: x+2y::11: 20, ... 3y—3x: x+2y::9: 20, or y―x: x+2y::3: 20, .. 20y—20x=3x+6y, and 14y—23x, 23x 3x+2 and 46x =42x+28, and there14 forex=287D, therefore y= 111D. 23x Hence 14 2 174. Find numbers, which are in the proportion of 8 to 5, and whose product is equal to 360. Ans. 24, and ±15. Let 5x and 8x be the numbers, ... 40x2=360, x2 = 9, or x=3. 175. There are 2 numbers, whose sum is to their 8 to 1, and the difference of whose squares is 128. numbers? Let 8x their sum; difference as What are the Ans. 18, and ± 14. = x= their difference, whence x the greater, and the less, .. (81x-49x)-1·32x2-8x= 128, and 16, ̊... x—±4, and the numbers are 18 and 14. 176. In a court there are 2 square grass-plots; a side of one of which is ten yards longer than the side of the other; and their ares are as 25 to 9. What are the lengths of the sides? Let x= a side of the one, .. x + 10 = a side of the other, and (x + 10): x2::25 : 9, .. x + 10:x::5: 3, and x=15, and the sides are 15 and 25, Ans. 177. A person bought 2 pieces of linen, which together measured 36 yards. Each of them cost as many shillings per yard as here were yards in the piece; and their whole prices were in the proportion of 4 to 1. What were the lengths of the pieces? Let x, and 36-x be the lengths, .. x: (36-x)2::4: 1, and x=24, and the lengths are 24 and 12, Ans. 178. There are 2 numbers, whose sum is to the less as 5 to 2: and whose difference, multiplied by the difference of their squares, is 135. Required the numbers. Ans. 9, and 6. Let 2x the less, and 3x the greater, and xX5x135, and x3-27, and x3, .. the numbers are 9 and 6, Ans. 179. There are two numbers, which are in the proportion of 3 to 2; the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Required the numbers. ... the Ans. 6, and 4. Let 3x and 2x be the numbers, .. 81x1—16x1 : 27x3+8x3 :: 26 : 7, or 65x: 35:: 26: 7, or 5x : 5 2 : 1, and x=2, numbers are 6 and 4. 180. There is a field in the form of a rectangular parellelogram, whose length is to its breadth in the proportion of 6 to 5. A part of this, equal to one-sixth of the whole, being planted, there remain for ploughing 625 square yards. What are the dimensions of the field? Ans. The sides are 30, and 25 yards. Let 6x and 5x = the sides, .. the area = 30x2, and 25x2 = 625, ... 5x=25, x=5, and the sides are 30 and 25. 181. Some gentlemen made an excursion; and every one took the same sum. Each gentleman had as many servants attending him as there were gentlemen, and the number of dollars which each had was double the number of all the servants; and the whole sum of money taken out was $3456. How many gentlemen were there? Ans. 12. Let x= the number, and 2x2 be the number of servants, and of dollars each took, ... 2x2=3456, and x3—1728, .'. x=12, Ans. 182. Divide the number 49 into two such parts, that the quotient of the greater divided by the less may be to the quotient of the less divided by the greater as to 2. Ans. 28, and 21. Let x, and 49-x be the numbers, then by the question we have 49-x 4 3 49-x : : and 2 (49-x)2::16: 9, x 3 4' or x : 49-x::4: 3. Hence x: 49::4 : 7, and x= 28, Ans. 183. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished four times as many men as there were companies in the regiment; but there being found to be insufficient, each company furnished three more men; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment? Let x= the number, .. 4x2 = the number first finished, and 4x: 3x::16: 1, and x:3::4 : 1, and x=12, Ans. 184. A charitable person distributed a certain sum amongst some poor men and women, the numbers of whom were in the proportion of 4 to 5. Each man received one-third of as many shillings as there were persons relieved; and each woman received twice as many shillings as there were women more than men. Now the men received all together 18s. more than the women. How many were there of each? Ans. 12 men, and 15 women. Let 4x, and 5x be the number of men and women, and 3x, and 2x the sum each man and woman received, then 12x2 10x2, and 2x2-18, x2-9, and x=3. Ans. 12 and 15 women. 185. A gentleman who had a certain number of horses, kept part of them at livery stables, for which he paid £4 10s. per week. The rest he kept at home, and their number was to the number kept at the livery stables as 7 to 3. He found that the 18+ expense of keeping 5 at home was just equal to that of keeping 4 at the stables; and the number of shillings that one horse cost him at home was to the number of horses kept at home as 6 to 7. How many horses had he? 180, 22 the price - 4, and Let 3x and 7x be the number at the stable and at home, and 6x the number of shillings one at home cost, .. 15x = of one at the stables, and 3xXx=90, 45x2 — x = 2. Ans. 6 at the livery stables, and 14 at home. 186. A city barge, with chairs for the company and benches for the rowers, went a summer excursion, with two bargemen on every bench. The number of gentlemen on board was equal to the square of the number of bargemen, and the number of ladies was equal to the number of gentlemen, twice the number of bargemen, and one over. Among other provisions, there were a number of turtles equal to the square root of the number of ladies; and a number of bottles of wine less than the cube of the number of turtles by 361. The turtles in dressing consumed a great quantity of wine, and the party having staid out till the turtles were all eaten, and the wine all gone, it was computed, that supposing them all to have consumed an equal quantity, (viz. gentlemen, ladies, bargemen, and turtles,) each individual would have consumed as many bottles as there were benches in the barge. Required the number of turtles. Let x, x2, (x+1)2, x+1, and (x+1)3 361 the number of bargemen, gentlemen, ladies, turtles, and bottles of wine respect(x+1)3 361 x ively, and or x3+3x2+3x+1=x3+2x2+x— 361, and 2+2x+1-361, .. x+1=19, the number required. 187. From two towns, C and D, two travellers, A and B, set out to meet each other; and it appeared when they met, B had 2(x+1)2 2' gone 35 miles more than three-fifths of the distance that A had travelled; but from their rate of travelling, A expected to reach C in 20 hours and 50 minutes; and B to reach D ́in 30 hours. Required the distance of C from D. Ans. 275 miles. Let 5x, and 3x+35 be the number of miles A and B each travels, and 3x+35: 5x::20: the number of hours A has travelled 625x ; and in the same way, the number B has travelled 6(3x+35) 6(3x+35) ; x 625x 6(3x+35) x , and 6252 36(3x +35)2, ... 25x = 6(3x+35) and 7x= 6×35 and x = 30, 188. A Farmer bought two flocks of sheep, the first of which contained 18 fewer than the second. If he had given for the first flock as many pounds as there were sheep in the second, and for the second as many pounds as there were sheep in the first, then the price of six sheep of the first flock would have been to the price of 7 sheep of the second in the proportion of 7 to 6. quired the numbers in each flock. ReAns. 108, and 126. Let x and x +18 be the numbers, then xx 18 :: 6 : the 6(x+18) price of six of the first flock and the price of 7 of the 2d = 7x 6(x+18) 7x : x+18::7: 6, and 36(x+18)2 = 49x2 x+18' and 6(x+18): =7x, and x = 108, the numbers are 108, 126. 189. A Poulterer bought a number of ducks and turkeys, the number of ducks exceeding the number of turkeys by 8. For each duck he gave half as many shillings as there were turkeys, and for each turkey half as many shillings as there were ducks. He afterwards bought another small flock of turkeys containing 4 fewer than the number of turkeys he bought before; and having given for each of them as many shillings as there were turkeys in the flock, he found, that if his former purchase had cost 16 shillings more, it would have cost exactly four times as much as the present one. How many ducks and turkeys did he buy at first? Let x, and x+8 be the number of turkeys and ducks, and x(x+8) the prices of each set, .. x2+8x+16=4(x-4)2, and x+4=2(x-4), and x= =12, and the numbers were 12 and 20. 190. Two men, A and B, entered into partnership with stocks, which are in the proportion of 9 to 8; and after trading one year, A found his share of their gain to amount to one-third of his stock. They continued to trade for as many years as are equal to three-fourths of the number of dollars which B contributed to the stock, and found their whole gain amount to $1666. What did each contribute to the stock; and how many years did they |