X. 1. 9×4 2. The difference is : the product 1. 3. 4. 4. 10. 5. 6. 7. The quotient is greater than the product. 8. 9. 991'r 473 11. 1. 18500 1485839 10. 130. 404 1. 213. 2. 11830. 3. 128. 4. 2143. 5. 2. 6. 149. 7. 207 XII. 1800 The operations indicated in the first four expressions offer no difficulty. 5. The expression is equivalent to 23.29 3.37 2.13 X 6. The left side of the expression can be put under the form— 1 148 110.135 3 + + } 1 3 And + = 25 11.12.13.12 = 49.50.51 11.12.13.36.37.38 4. The direct and converse. 5. £2 10s. 533d. 6. 9. 144. 10. of 1 guinea. 11. £%, and 3s. 12. 3. 13. The direct and converse. 14. of estate = £1003 17s. 6d.: £200 15s. 6d., and 17 or whole estate = £3413 3s. 6d. XIV. 720000 1. Art. 18. 2. £5 12s. 43d. 3. £7 14s. 54d. 4. £140 10s. 1114 d. 5. £54 2s. 24d. 6. 2 of 1 guinea. 7. 12 of £100. 8. £1 10s. 6d. 9. 17 of half a sovereign. 10. of £1. 11. 23 of a crown. 12. 4 of a crown. 13. Remaining share, and its value £3080. A received £2 5s. 10d., and B £2 1s. 8d. XV. 1. See on Multiplication and Division of Concrete integers. 2. 1† of £1= £1 12s. 11 d. 3. The shorter course is to add of the sum to thirteen times the The result is £89 6s. 144d. 4. 164083. 5. 1243 of £1. 6. £5 1s. 13. 7. 2633s. 8. The quotient 9143 indicates the number of times the latter sum is con sum. tained in the former, when both sums are expressed in the same units. 9. The difference is contained 443 times in the sum. 10. of 1 guinea is equal to 1s. 7d. 11. 9981744, an abstract number denoting the number of times the less sum is con tained in the greater. 12. Here 17s. 6d. is equivalent to 840 farthings, and 25 new to 24 old farthings, and 1 new is of one old farthing: 840 new=24×840=8063 old farthings. 840-8063=333 old farthings loss per week. And 33 X52=1747 farthings= £1 16s. 4 d. loss yearly. XVI. 7 of £1: £100 gives £35, or 5 per 6×2 1. £34 8s. 41d. 2. £1 gives 1s. 2d. cent. 3. First one halfpenny is the gain on three halfpence, the gain is of the capital, and the gain on £100 is £33, or 131 per cent. Secondly, one halfpenny is the gain on four halfpence, the gain is 4 of the capital, and the gain on £100 is £25, or 25 per cent. The difference is 8 per cent. 4. £100 bequeathed gives £90 to legatee, or he receives £90 for 100 bequeathed, and £1 for 1, and therefore, £1000 for 10000, or £1111 the sum to be bequeathed. 5. First prime cost 3s. 6d. or £ 15×100 7 X 20 or £10 gain, or 10 9 gives 44d. gain: £1 gives s. gain, and per cent. Secondly, 3s. 103d., or £37, gives 44d. loss, and £1 gives ffs. loss, and therefore £100 gives £100 gives 100 × 60 or £931 loss, or 931 per cent. 7. 33 of £1. 8. Neither gain nor loss. 9. 31 X 20 6. 33 per cent. With £41 gain, the prime cost is 100, and £4 is contained 3 times in £47 14s. Hence £47 14s. gives 3×100=£530, the prime cost. 10. 213s. 11. 2s. 1123d. 12. 10s. 12. Let 1 denote wholesale price to the retailer, who makes a profit of 25 per cent. Next, let 1 denote the manufacturer's price to the wholesale dealer, who makes a profit of 15 per cent, 15 Then 1+ 100 = 3 23 =1+ or 124s. therefore 1 20 20 Thirdly, let 1 denote the original cost of the article to the manufacturer, who makes a profit of 10 per cent. 1. A franc, 10d. =gs. =£24=1 of 1 guinea. 2. 1 guinea=504-6=498 halfpence and 1 franc-19 halfpence. As 498 and 19 are prime to each other. The L.C.M. 19 x 498. Hence 19 sovereigns, or 498 francs, are the least numbers of these coins that can be exchanged. 3. 25714 francs. 4. 8571 rupees. 5. 10 scudi=521 francs, and 1 scudi of 1 franc; 20 francs-16s., and 1 francof = 1s.; 41s. 12 carlini, and 1s. =3 of 1 carlini. Hence 1 scudi=2 of 4 of % of 1 carlini, and therefore 500 scudi =500×××6048 carlini. 6. 7054 carlini. 7. 6204 thalers. 8. 100 francs. 9. £6 in London is equivalent to 85 marks in Hamburgh. XVIII. 1. 17s. 6464d. in the pound, and they lose 13 of their debt. 2. He loses 18 of £5342 5s. 3. lost, whole debt £120 8s. 44d. 4. The creditors lose 283 part. The whole debt was £2440. 5. £178 1s. 6d. 6. £810, £621, £729. 7. £ ́ 137 3 £78, £58. Each receives 394 per cent. of his debt. 8. First, £664 was to gain 2 1 £166, or † of the capital employed, which is £25 in the £100, or 25 per cent. Secondly, £830 lost £166, or of the capital employed, or 20 per cent. 9. By dividing £195 into three parts,,, and . 10. £180, £320, £500. 11. Here +++&+&=}} of the whole, of which share is not assigned. The gain £125 for their share must be deducted from £1,500, which leaves £1,375 to be divided. Their three capitals are denoted by 6, 3, 2, and the first takes 7 per cent. on his 6, the second 5 per cent. on his 3, and the third 3 per cent. on his 2; and the sums of these 42+15+6=63, or 3+2+2=1, which gives the shares of the gains £9163, £327, £13039. 12. £200×16}, £200×371, £200 × 5311, £200 × 63. XIX. 1. The questions are one the reverse of the other. 2. £333 6s. 241d. 3. 5dwts. of pure gold. 4. The supposition gives 96 grains of pure gold, neglecting the alloy. 5. 8 out of 100 parts, or 83 per cent. 6. The true weight of a sovereign is 5dwts. 2grs. The loss on 100 light sovereigns is to be calculated on the supposition of a loss of 1d. on £3 17s. 101d., the value of one ounce of standard gold. 7. Neither gain nor loss. 8. 10lbs. of jewellers' gold consists of 5lbs. of pure gold, and 44lbs. of alloy, of which alloy 3 of 11b. added to 51⁄2lbs. pure gold, makes 6lbs. standard gold; the rest of the alloy, 3lbs., is neglected. The question now requires the price of 64lbs. of standard gold at £3 17s. 103d. per ounce. 9. 17 carats pure gold implies 7 carats alloy, 211 ounces of pure gold. 10. The sum of £3 17s. 101⁄2d. could have paid £4 13s. 6d. in 1815; but in 1821 the depreciation was 15s. 71⁄2d. on £4 13s. 6d., or of £4 13s. 6d. Hence the depreciation on £100 will be £163, or 163 per cent. 11. If the number of grains in one ounce Avoirdupois be divided by the number of grains in the weight of one sovereign, the quotient will give the number of sovereigns required. 12. The alloy in a sovereign is 10 grains. The difference will be the excess in value of 10 grains of silver at 5s. 6d. an ounce, and 10 grains of copper at 1s. 6d. a pound Avoirdupois of 7,000 grains. XX. 1. 12dwts. 2. 5s. 34d. 3. Sdwts. 1143 grains of pure silver, and 1311 grains alloy. 4. 5181 farthings. 5. Supposing the value of the alloy neglected. 80 per cent. of pure silver implics 20 per cent. of alloy, so that +2=1, or +=1. And if the unit be taken as one ounce, one ounce of alloyed silver is in value of an ounce of pure silver, or 4s. And 15 times the value of one ounce of alloyed silver is £3, and £3 is of the value of 1 ounce of pure gold. But three parts of pure gold and one part of copper make up the ounce of alloyed gold. of an ounce of copper, or 25 per cent., must be mixed with 75 per cent. of pure gold. 6. Estimating the coins by the pure silver in them, 1 sestertium=250×54 grains of pure silver, and 23 of 20×24 grains=37×12 grains, which are worth 5s. 2d. The sestertium is worth £7 17s. 1d. 7. 513 issued. of the sovereign 123 and 5 grains over. 8. 12cwt. of gold, 3 of silver, and 2 of copper. 9. Weight grains, of the half-crown 215 grains. 10. 80 shillings 11. 883 grains. 2. of 18cwt. or of one ton. 5. £860 7s. 2d. 7. 71141 pence. XXI. 5185 16921 3. 1520 tons of ore. 4. 20 lbs. and 171 lbs. 8. 11+100=111..+??f=1. The unit is 24cwt., of which 234cwt. is tin, and 2133cwt. copper. 9. 75 per cent. of copper and 25 per cent. of tin. 10. 55124lbs. of oxygen, 4857lbs. of carbon, and 789 lbs. of hydrogen. 11. 734 ounces of oxygen, 334 ounces of sulphur, and 434 ounces of calcium. 12. 15cwt. of nitre, 2cwt. of sulphur, and 3cwt. of 1250 charcoal. XXII. 1. 4413 lineal inches, 4413 square inches. 2. 4413 cubic inches. 3. 1 lineal inches; 1 an abstract number. 4. 1 square inches; 1 lineal inches; 1 69 an abstract number. 5. The former is double of the latter. 6. 2 lincal feet. 7. lineal foot. 8. The first quotient is 192 cubic inches, the second 192 square inches, the third 192 lineal inches, and the fourth indicates that 9 cubic inches is contained 192 times in a cubic foot. 9. The difference is 88418 square yards. 10. 306 square feet. The quotient 9 gives the number of times the latter length is contained in the former. 11. 21 inches. 12. 452 square feet. 13. Meadow and arable is +=23, the rest 13-1 acre, 3 roods, 26 poles=306 poles, and 18 poles... or =144 poles, meadow: and 15 or 3=270 poles, arable. 500+350+800+160=1810..+M+M+M=1. In this case the unit is 200 acres, ... 0 of 200 acres=55451, the allotment to the possessor of the rental of £500 a year. In a similar way the rest of the allotments may be found. 14. Here 18 XXIII. = = 1575 plethra 1575 × 124×7 =82212 yards. 5. 3+4+5=12, 1+1+=1. The unit is 492 yards. Hence,, and of 492 yards are respectively 123, 164, and 205 yards. 6. If the error be in defect, the apparent length is 502 yards, and 24 inches over. If the error be in excess, the apparent length is 499 yards and 3 inches over. 7. 5 miles is of the earth's diameter. The height of Mont Blanc above the surface of a globe of 16 inches diameter is represented by 1 of an inch, or between and of an inch. The height of Kunchingunga above the surface of a globe of 30 inches diameter, is represented by 49 of an inch, or between and of an inch. 14089 697488 XXIV. 1. 94 square yards, 847 square feet, 10174 square inches. 2. 429 yards. 3. 91 yards. 4. 104 feet wide. 5. 280 plots each 13 foot wide. 6. 266 yards. 7. 17723 square yards. 8. 28 planks. 9. 199127 yards. 10. 25 yards. 11. 1036500 stones. 12. 136 yards of paper. XXV. 2. 14s. 103d. per yard. 3. 10 feet high and 16 feet broad. £15 14s. 31d. 6. 6 feet high. 7. £5 17s. 01d. XXVI. 1. 366 cubic feet. 2. 1728 cubes. 3. 320 cubes. 6. 633600 cubic yards. 7. 33411g cubic feet. 8. 2 10. 33 feet deep. 11. 7,800 bricks. Quantity of wood used 81 cubic feet. surfaces of the chest 8 square feet. 893 4. 6 feet long. 5 16 yards. feet long. 9. 960 feet long. 12. The content of the chest is 561% cubic feet. Difference between the external and internal XXVII. 1. 1980lbs. 2. 8190lbs. 3. The weight of a cubic foot of the stone is 160lbs. The weight of the block 235,950lbs. 4. Supposing the internal dimensions given, the external dimensions will be 10ft. 11in., 8ft. 7in., and 5ft. 1in. The vessel consists of 35 cubic feet of iron, of which 4 cubic inches weigh one pound Avoirdupois. 5. The internal surface of the cistern is 14342 square feet, and 1735 cubie feet of lead are required to line it. 6. 1 yard of wire requires of 1lb. of copper, and 50 miles require 4,000lbs. The area of a section of the wire can be found by dividing the number of cubic inches of copper in the wire by its length. 7. 384 feet. 8. 7617 square yards. 9. 275625 leaves. XXVIII. 1. Since six men earn £7 6s. 3d. in 71⁄2 days, they earn 19s. 6d. in one day, and one man earns 3s. 6d. in one day. The 10 men will earn £1 15s. in one day, and in 11 days they will earn £20 11s. 3d. 2. Take an hour as the unit of time. 3. Here 3 men and 5 boys in 4 weeks earn £11: therefore 3 men and 5 boys in 1 week earn £2, and 6 men and 10 boys in 1 week earn £5. But 7 men and 10 boys in 1 week earn £6: and 3 men in 1 week earn £1. and 1 boy in 1 week earns £4. Wherefore 5 men and 4 boys earn in 1 week £31. But 5 men and 4 boys earn in some number of weeks, £14. The time required is 14-33=4 weeks. 4. 18 days. 5. Let the work of one child be taken as the unit. Then the work of 9 men, 11 women, and 9 children is equivalent to the work of 71 children. The men receive £1 14s. 44d, the women £1 5s. 24d., and the children 5s. 6d. 6. Let the wages of 1 child be taken as the unit. Wages of 1 man and one child are equivalent to the wages of 4 women. Wages of 1 man and 3 children are equivalent to the wages of 5 women. Hence the wages of 2 children are equivalent to the wages of 1 woman, and the wages of 1 man is equivalent to the wages of 34 women or 7 children. Therefore the wages of 1 man, 1 woman, and 3 children, are equivalent to the sum of the wages of 7 children, 2 children, and 3 children, or 12 children. But the amount of these wages of 12 children is 30s. The man's wages are 17s. 6d., the woman's 5s., the 3 children 2s. 6d. cach. 7. £350 for 7 weeks' work gives £50 weekly. 2 men or three women receive as much as 5 children, therefore 1 man receives as much as children, and 1 woman as children. And 20 men receive as much as 50 children, 40 women as 663 children, and consequently 20 men, 40 women, and 50 children receive together as much as 166 children in 1 week. Each child receives £50-1663=6s. weekly. 1 woman ×6=10s., and 1 man 2×6=15s. 1. IIere 1 man performs perform in one day. Hence ΧΧΙΧ. of the work in 1 day, and a certain number of men 3000-1200, the number of men. |