From the foregoing examples it appears, that the less the price of any simple differs from that of the mixture, the quantity required of that simple to form the mixture will be proportionately greater, and the greater the dif ference the less the quantity; and that the differences between the values of the simples and the given value of a mixture of those simples, mutually exchanged, express the relative quantities of those simples necessary to make a mixture of the given value. Exchanging these differences in the above examples, wo have in the first, 5 lb. at 40 cents, with 5 lb. at 60 cts., or equal quantities of each; and in the second, we have 2 lb. at 9 cts. with 1 lb. at 12. RULE. 210. Reduce the rates of all the simples to the same de nomination, and write them in a column with the rate of the required compound at the left hand. Connect each rate which is less than the rate of the compound, with one that is greater, and each that is greater with one that is less. Write the difference between each rate and that of the compound against the number with which it is connected. Then if only one difference stand against any rate, it will express the relative quantity to be taken of that rate; but if there be more than one, their sum will express the relative quantity to be taken of that rate in making up the compound. QUESTIONS FOR PRACTICE. 3. A farmer wishes to mix rye worth 4s., corn worth 3s., barley worth 2s. 6d., and oats worth 2s., so that the mixture may be worth 2s. 10d. per bushel; what proportion must he take of each sort? Alligation Alternate is the reverse of Alligation Medial, and may be proved by it Questions under this rule admit of as many different answers as there are different ways of linking. 211. When the whole composition is limited to a certain quan-` tity. RULE.-Find the differences by linking as before; then say, As the sum of the quantities or differences, thus determined is to the given quantity :: so is each of the differences: to the required quantity of that rate. 212. When one of the simples is limited to a certain quantity. RULE. Find the differences as before; then, As the difference standing against the given quantity is to the given quantity :: so are the other differences, severally, to the several quantities required. 6. A and B barter; A has 150 gallons of brandy, at $1.20 per gal. ready money, but in barter, would have $1.40; B has linen at 60 cents per yard, ready money; how ought the linen to be rated in barter, and how many yards are equal to A's brandy? Ans. barter price, 70 cents, and B must give A 300 yards. 7. C has tea at 78 cents per lb., ready money, but in barter, would have 93 cents; D has shoes at 7s. 6d. per pair, ready money; how ought they to be rated in barter, in exchange for tea? Ans. $1.49 8. C. has candles at 6s. per | dozen, ready money; but in barter he will have 6s. 6d. per dozen; D has cotton at 9d. per lb. ready money; what price must the cotton be at in barter, and how much cotton must be bartered for 100 dozen of candles? Ans. the cotton 94d. per lb. in barter, and 7cwt. Oqrs. 161b. of cotton must be given for 100 doz. candles. NOTE. The exchange of one commodity for another, is called Barter. 9. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 32 days; in what time will 12 men build a wall 100 feet long, 4 feet high, and 3 feet thick? Ans. 40 days. 10. If a family of 8 persons in 24 months spend $480; how much would they spend in 8 months, if their number were doubled? Ans. $320. ture for $48; A puts in 80 B 7.20 Ans. 12. If I have a mass of pure gold, a mass of pure copper, and a mass, which is a mixture of gold and copper, each weighing 10 lb., and by immersing them in water, find the quantities displaced by each to be 8 by the copper, 7 by the mixture, and 5 by the gold; what part of the mixture is gold, and what part copper? { 8-2 5-1 3: 10 :: And 2: 63 copper {1:3 gold. This is the celebrated problem of Archimedes, by which he detected the fraud of the artist employed by Hiero, king of Syracuse, to make 11. Three men hire a pas-him a crown of pure gold (211). ASSESSMENT OF TAXES. 1. Supposing the Legislature should grant a tax of $35000 to be assessed on the inventory of all the rateable property in the State, which amounts to $3000000, what part of it must a town pay, the inventory of which is $24600? $ inv. inv. $. 3000000 35000 :: 24600: 287 tax. Ans. 2. A certain school, consisting of 60 scholars, is supported on the polls of the scholars, and the quarterly expense of the whole school is $75; what is that on the scholar, and what does A pay per quarter, who has 3 scholars? Ans. $1.25 on the scholar, and A pays $3.75 per quarter. 213. In assessing taxes, it is generally best, first to find what each dollar pays, and the product of each man's inventory, multiplied by this sum, will be the amount of his tax. In this case, the sum on the dollar, which is to be employed as a multiplier, must be expressed as a proper decimal of a dollar, and the product must be pointed according to the rule for the multiplication of decimals (122); thus 2 cents must be written .02, 3 cents, .03, 4 cents, .04, &c. It is sometimes the practice to make a table by multiplying the value on the dollar by 1, 2, 3, 4, &c. as follows: 10 " 3.00 .30 This table is constructed cents on the dollar, as in rateable property is $276? $70 pay $2.10, and that $6 Thus $200 is 6.00 70 is 2.10 6 is 0.18 276 $8.28 100 on the supposition that the tax amounts to three example 5th. USE.-What is B's tax, whose By the table, it appears that $200 pay $6, that pay 18 cents. Proceed in the same way to find each individual's tax, then add all the taxes together, and if their amount agree with the whole sum proposed to be raised, the work is right. It is sometimes best to assess the tax a trifle larger than the amount to be raised, to compensate for the loss of fractions. REVIEW. 1. What is meant by ratio? How is ratio expressed? What is the first term called? the second term? 2. What is proportion? What general truth is stated respecting the four terms of a proportion? How is this truth shown? 3. Does changing the place of the two middle terms affect the proportion? Why not? |