24. A Gentleman gave away a certain sum in charity to 14 men and 15 women. Had the sum been less by 12 shillings, and only half the number of men relieved, the rest being divided amongst the women, each woman would have received two shillings more than each man did. But if there had been only 8 women, and the rest had been divided amongst the men, each man would have received twice as much as each woman. much money was given away? Ans. 24 guineas. How 25. When wheat was 5 shillings a bushel, and rye 3 shillings, a man wanted to fill his sack with a mixture of rye and wheat for the money he had in his purse. If he bought 7 bushels of rye, and laid out the rest of his money in wheat, he would want 2 bushels to fill his sack; but if he bought 6 bushels of wheat, and filled his sack with rye, he would have 6 shillings left. How must he lay out his money, and fill his sack? Ans. He must buy 9 bushels of wheat, and 12 bushels of rye. 26. A Stage Coach carries six inside, the fare outside is 138. and one-third of the sum of the outside fares exceeds one-fifth of those inside by £1.18.84d. An opposition arising, the coachman loses three outside and two inside passengers, and also reduces the inside fare by 58. and halves the outside; and then the whole loss is £7. os. 6d. Find the number of outside places, and the fare inside. Ans. 10 outside places, and 198. fare inside. 27. A Draper bought two pieces of cloth of different kinds for £37. 48.: there were 6 yards of the coarser more than there were of the finer; and had the coarser cost 2 shillings a yard more than it did, 6 yards of the coarser would have cost just as much as 5 yards of the finer. He afterwards bought 4 yards of the finer, and 12 of the coarser at the same prices per yard, and found their value less than that of the former pieces in the ratio of 20:31. How many yards did he buy the first time, and what did he give per yard for each? 28. Ans. 9 yards of the finer, and 15 of the coarser; and the prices were 36, and 28 shillings per yard. A Mercer bought two pieces of silk of different lengths for £50; the price of two yards of the shorter was 6s. 8d. more than the price of 3 yards of the longer; and each piece cost the same sum. He cut off two yards from each, and sold the rest for £53. 128. Now if he had sold the whole at that rate, he would have gained £5 by each piece. How many yards did each piece contain ? Ans. 25, and 15 yards. 29. A sets out express from C towards D, and three hours afterwards B sets out from D towards C, travelling 2 miles an hour more than A. When they meet, it appears that the distances they have travelled are in the proportion of 13 to 15; but had A travelled five hours less, and B gone 2 miles an hour more, they would have been in the proportion of 2: 5. How many miles did each go per hour, and how many hours did they travel before they met? 30. Ans. A went 4, and B 6 miles an hour, and they travelled 10 hours after B set out. The revenue of a state was increased to provide for a war in the ratio of 24: 1; and after deducting the expence of collecting, and the interest of the National Debt, the available income was augmented in the ratio of 31: 1. Now it was found upon calculation, that had circumstances on the contrary permitted the revenue to be reduced in the ratio of 17:1, the sum remaining after the specified deductions would have been diminished in the ratio of 7: 1, and would in fact have only amounted to four millions. Required the amount of the revenue, and the interest of the debt; on supposition that the expence of collecting varies as the square root of the amount collected. 31. 2:3. Ans. The revenue before the increase was 64 millions, and the interest of the National Debt 28 millions. A and B engaged to reap a field of corn in 12 days. The times in which they could severally reap an acre are as After some time, finding themselves unable to finish it in the stipulated time, they called in C to help them; whose rate of working was such, that if he had wrought with them from the beginning, it would have been finished in 9 days. Also the times in which he could severally have reaped the field with A alone, and with B alone, are in the proportion of 7 to 8. When was C called in? Ans. After 6 days. 32. Two mixtures are made of brandy and sherry; the quantities of brandy in each being as 4 to 3; and the difference of the quantities of sherry being greater by 25 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 mixture 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. Ans. The quantities of brandy are 80, and 60 gallons, and the quantities of sherry are 90, and 45 gallons. 33. During a winter, when fuel was scarce, two 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. Bassented, but demanded of him 28. 10d. When this stock was exhausted, B purchased one bushel of coals, and A five, together with 6 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? Ans. The price of a bushel of coals was 5s., and of a basket of turf 4d. 34. 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 5 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 behaviour, he gave them what Spanish money he had about him, which amounted to 68. 5d., 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 18. 10d. 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 111⁄2 pence. VIII. Problems producing Pure Equations. 1. FIND two numbers, which are in the proportion of 8 to 5, and whose product is equal to 360. Ans. 24, and ± 15. BB 2. There are two numbers, whose sum is to their difference as 8 to 1, and the difference of whose squares is 128. What are the numbers? 3. In a court there are two square grass-plots; a side of one of which is 10 yards longer than the side of the other; and their areas are as 25 to 9. What are the lengths of the sides? Ans. 25, and 15 yards. 4. A person bought two pieces of linen, which together measured 36 yards. Each of them cost as many shillings per yard, as there were yards in the piece; and their whole prices were in the proportion of 4 to 1. What were the lengths of the pieces? Ans. 24, and 12 yards. 5. There are two 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. 6. 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. Ans. 6, and 4. 7. There is a field in the form of a rectangular parallelogram, 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? 8. Ans. The sides are 30, and 25 yards. Some Gentlemen made an excursion; and every one took the same sum. Each gentleman had as many servants |