Miscellaneous Examples, depending on Arts. (116–128), worked out. Ex. 1. A person bought 500 yards of cloth at 15s. 9d. a-yard, and retailed it at 16s. 3d. a-yard: what was his profit? His profit on 1 yard=16s. 3d.-15s. 9d. Ex. 2. A spring of water, which yields 75 gallons an hour, supplies 600 families: how much water may each family use daily? The daily supply of water = (75 × 24) gallons; therefore each family may use daily 75 × 24 gals., or 3 gals. Ex. 3. How many revolutions will a wheel, which is 4 yards in circumference, make in 3 miles? 3 miles (3 × 1760) yards=5280 yards, = and since the wheel passes over 4 yards in one revolution; Ex. 4. The value of a mark being 13s. 4d., and that of a moidore 278., how many half-crowns are there in 30 marks + 40 moidores? 30 marks+40 moidores (13s. 4d.) × 30+27s. × 40 =592. Ex. 5. How many guineas, sovereigns, half-crowns, and shillings, and of each an equal number, are there in £1246 ? Now, 1 guinea +1 sovereign+1 half-crown + 1 shilling = (42+40 +5+2) sixpences = =89 sixpences; and £1246 (1246 × 20 × 2) sixpences=49840 sixpences; the question therefore is reduced to this: How often are 89 sixpences contained in 49840 sixpences? Ex. 6. How much water must be added to a cask containing 60 gallons of spirit at 12s. 6d. a gallon, to reduce the price to 8s. a gallon? Cost of cask=(12s. 6d.) × 60, =(150 × 60)d. 88.=(8 x 12)d.; 375 or or 932=the number of gallons which the cask must contain, in order that its contents may be sold at 8s. a gallon. Therefore (933-60), or 333=the number of gallons of water which have to be added, Ex. 7. How many yards of cloth, worth 3s. 71d. a-yard, must be given in exchange for 144 yards of cloth, worth 18s. 11⁄2d. a-yard? The value of 144 yards at 18s. 11⁄2d. a-yard, Or thus, since 18s. 1d. (3s. 7d.) × 5, it is clear that the number of yards required=144 × 5, =720. Ex. 8. A traveller walks 22 miles a-day, and after he has gone 84 miles another follows him at the rate of 34 miles a-day; in what time will the second traveller overtake the first? The second traveller has to walk over 84 miles more than the first before he can overtake him. Each day he walks (34–22) or 12 miles more than the first; therefore or 7 is the number of days required. Ex. 9. A mixture is made of 8 gallons of spirits at 12s. 10d. a gallon, 7 gallons at 10s. 6d. a gallon, and 10 gallons at 9s. 1d. a gallon; at what price per gallon must the mixture be sold, 1st, that the seller may neither gain nor lose by his bargain; 2nd, that he may gain £1. 13s. by it; 3rd, that he may lose 7 guineas; and 4th, that he may reserve 10 gallons of the mixture for himself, and sell the remainder so as to realize the money he laid out? 1st. If he is neither to gain or lose, he must sell 1 gallon for £13. 78. 25 ; which, worked out, gives 10s. 8d. as the price required. 2nd. If he is to gain £1. 13s. 25 gallons must be sold for £13. 7s. + £1. 13s., or £15.; therefore, 1 gallon must be sold for £15 ́; which, worked out, gives 12s. as the price required. 3rd. If he is to lose 7 guineas, 25 gallons must be sold for £13. 7s. — £7. 7s. or £6.; £6 therefore 1 gallon must be sold for ; which, worked out, gives 48. 9 d. g. as the price required. 25 4th. If he is to retain 10 gallons for his own use, 15 gallons must be sold for £13. 78. ; £13. 78. therefore 1 gallon must be sold for ; which, worked out, gives 15 178. 91d. q. as the price required. Ex. 10. A club, consisting of 56 persons, joined for a lottery ticket of 12 guineas value, and it came up a prize of £7000: what sum did each man contribute, and what did each man gain? Ex. 11. Divide £20 among A, B, and C, so that B may have 2 guineas more than A, and that C may have 28. less than B. A's share + B's share+ C's share= £20, or A's share + (A's share + £2. 28.) + (A's share + £2.) = £20, or 3 times A's share + £4. 2s.= £20; therefore evidently 3 times A's share= £20.- £4. 2s. Ex. 12. Divide £8. 11s. 6d. among 5 men, 6 women, and 7 boys; giving each woman twice as much as each boy, and each man thrice as much as each woman. Since each woman's share=twice each boy's share, therefore 6 women's shares = 12 boys' shares. Again, since each man's share=thrice each woman's share, but 5 men's shares + 6 women's shares +7 boys' shares = £8. 118. 6d, or 30 boys' shares +12 boys' shares +7 boys' shares = £8. 11s. 6d., or 49 boys' shares £8. 11s. 6d. =343 sixpences. 129. It may be well to notice here some of the advantages which would result from a decimal coinage of pounds, florins, cents, and mils; the pound being of the same value as the pound sterling at present; the florin being of £1.; the cent being=1th of a florin, or= £1.; the mil (m.) being=1th of a cent, or=th of a florin, or= of £1. = The Table would stand thus : =10th of 1000th 130. In such a system, much of the labour of reducing superior to inferior denominations, and the converse, would be done away with; for we could at once say, £24. 3 fl. 7 c. 2 m. =24372 m. Since by performing the operation of reduction at length, we obtain |