QUESTIONS FOR PRACTICE. 10. A grocer would mix teas 11. How much wine at 59., at 12s., 10s., and 6s., with 20 at 5s. Cd., and 6s, per gallon, Ib. at 4s. per lb.; how much of must be mixed with 8 gallons each sort must he take to at 43. per gallon, so that the make the composition worth mixture may be worth 5s. 4d. 8s. per lb.? per gallon? 4 against the given gal. quantity. 2 at 5s. 2 Ans. 4 at 5s. 6d. per gall. 4 lh. 16 at 6s. 2 : 10 at 65.2 4:20:: 2:10 at 10s. Ans. 2 { 12 4:20 at 12s. MISCELLANEOUS. cwt. cwt. qr. 1. A has 350 yards of cloth 5. A has coffee, which he at Is. 4d. per yard, which he barters with B at 10d. per lb would exchange with B for more than it cost him, against sugar at 25s. 6d. per cwt. ; | tea, which stands B in 10s. the how much sugar will the cloth lb., but puts it at 12s. 6d. : I come to? would know how much the 350 yards at ls. 4d.466s. coffee cost at first. 3d.-5600d. and 25s.6d.=300d. Ans. 3s. 4d. d. d. Then 306 : 1 :: 5600 6. A and B barter; A has Ib. 150 gallons of brandy, at $1.20 Ans. 18 1 5nearly. per gal. ready money, but in barter, would have $1.40; B 2. A has 7 cwt. of sugar, at has linen at 60 cents per yard, 8d. per 1b., for which В gave ready money, how ought the him 12} cwt. of flour; what linen to be rated in barter, was the four per lb. ? and how many yards are equal Ans. 4 d. to A's brandy? 3. How much tea, at 9s. 6d. Ans. barter price, 70 cents, per lb., must be given in bar- and B must give A 300 yards. ter for 156 gallons of wine, at 7. C has tea at 78 cents per 12s. 3 d. per gallon ? Ans. 2011b. 13 11.02. lb., ready money, but in barter, would have 93 cents; D has 4. B delivered 3 hhds. of shoes at 7s. 6d. per pair, ready brandy, at 6s. 8d. per gallon, to money; how ought they to be C for 126 yards of cloth ; what | rated in barter, in exchange was the cloth per yard ? for tea? Ans. $1.49 Ans. 10s. 8. C. has candles at 6s. per ture for $48; A puts in 80 dozen, ready money; but in sheep for 4 months, B 60 sheep barter he will have 6s. 6d. per for 2 months, and C 72 sheep dozen; D' has cotton at 9d. for 5 months; what share of per lb. ready money; what the rent ougit each to pay? price must the cotton be at in A $19.20 barter, and how much cotton B 7.20 Ans. must be bartered for 100 dozen C 21.60 of candles ? 12. If I have a mass of pure Ans. the cotton 9 d. per lb. in barter, and 7cwt. Oqrs. 16lb. gold, a mass of pure copper, of cotton must be given for and a mass, which is a mixture 100 doz. candles. of gold and copper, each weighNote.- The exchange of one ing 10 lb., and by immersing commodity for another, is called them in water, find the quanti ties displaced by each to be 8 9. If 6 men build a wall 20 | by the copper, 7 by the mixfcet long, 6 feet high, and 4 feet ture, and 5 by the gold; what thick, in 32 days; in what time part of the mixture is gold, and will 12 men build a wall 100 what part copper? feet long, 4 feet high, and 3 2 5-1 10. If a family of 8 persons 3 : 10 :: Burter. 7 And feet thick? Ans. 40 days. 52 : 63 copper in 24 months spend $480; how 1:3} gold. much would they spend in 8 This is the celebrated problem of months, if their number were Archimedes, by which he detected doubled ? Ans. $320. 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 2. A certain school, consistshould grant a tax of $35000 ing of 60 scholars, is supportto be assessed on the inventory | ed on the polls of the scholars, of all the rateable property in and the quarterly expense of the State, which amounts to the whole school is $75; what $3000000, what part of it is that on the scholar, and must a town pay, the inventory what does A pay per quarter, of which is $24600? who has 3 scholars? $ inv. $ tax, $ inv. $. Ans. $1.25 on the scholar, 3000000 : 35000 :: 24600 : 287 and A pays $3.75 per quarter: Ans. 3. If a town, the inventory that on the dollar, and what is of which is $24600, pay $287, C's tax, whose property invenwhat will A's tax be, the in- | tories at $76.44 ? ventory of whose estate is $4325 : 86.50 :: 1:.02 cts. $525.75? Ans. 24600.00 : 287 : : 525.75 : & 76.44X.02-$1.523, C's tax, $6.133 Ans. 5. If a town, the inventory 4. The inventory of a cer- of which is $16436, pay a tax tain school district is $1325, of $493.08, what is that on the and the sum to be raised on dollar? this inventory for the support $16136 : $493.08 ::1:.03 cts, of schools, is $86.50; what is Ans. 213. In assessing taxes, it is generally best, first to find what eaclı 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: TABLE. 20 6.00 9.00 400 12.00 500 15.00 600 18.00 700 21.00 24.00 90 900 27.00 1000 30.00 This table is constructed on the supposition that the tax amounts to thi ce cents on the dollar, as in example 5th. Use.- What is B's tax, whose rateable property is $276? By the table, it appears that $200 pay $6, that $70 pay $2.10, and that $6 pay 18 cents. Thus $200 is 6.00 Proceed in the same way to find each indi70 is 2.10 vidual's tax, then add all the taxes together, 6 is 0.18 and if their amount agree with the whole sum proposed to be raised, the work is right. It is 276 88.28 sometimes best to assess the tax a trifle larger than the amount to be raised, to compensate B's tax. for the loss of fractions. 66 66 .800 REVIEW. is 1. What is meant hy ratio ? How four terms of a proportion? How is ratio expressed? What is the first this truth shown ? term called ? the second term? 3. Does changing the place of 2. What is proportion? What the two middle terms atfect the prom general truth is stated respecting the portion? Why not ! 4. What is meant by inverse pro j 7. What is Fellowship? What is portion ? meant hv capital or stock? What 5. What is meant by the Single by dividend'? What is the rule Rule of Three ? What is the gen- when the times are equal ? What, eral rule for stating questions in the when they are unequal? What is Rule of Three ? How is the an- the method of proof? swer then found ? If the first and 8. What is Alligation? What is third terms be of different denomi- į Alligation Medial !--Alligation Alnations, what is to be done? What, ternate ? What is the rule for if there are different denominations finding the proportional quantities in the second term ? Of what de- to form a mixture of a given rate ? nomination will the quotient be ? ¡ Explain by analysis of an example. What, if the quotient be not of the When the whole composition is same denomination of the required limited to a certain quantity, how answer ? What is the method of would you proceed? How, whej proof in this rule ? one of the simples is limited to a 6. What is compound proportion ? | certain quantity ? How is Alligation By what other name is it called ? proved ? What is the rule for stating questions 9. What is Barter? What is in compound proportion ?-1or per- meant by a tax? What is the comforming the operation ? mon meihod of making out taxes ? SECTION VII. Fractions. DEFINITIONS. 214. 1. Fractions are parts of a unit, or of a whole of any kind. If any number, or particular thing, be divided into two equal parts, those parts are called halres ; if into 3 equal parts, they are called thirds ; if into 4 equal parts, they are called fourths, or quarters (11); and, generally, the parts are named from the number of parts into which the thing, or whüle, is divided. If any thing be divided into 5 equal parts, the parts are called fifths ; if into they are caller! sixths ; if into 7, they are called sevenths; and so on. These broken, or divided quantities are called fractions. Now if an apple be divided into fire equal parts, the value of one of those parts would be one fifth of the apple, and the value of two parts two fifths of the apple, and so on. Thus we see that the name of the fraction shows, at the same time, the number of parts into which the thing, or whole, is divided, and how many of those parts are taken, or signified by the fraction. Suppose I wished to give a person two fifths of a dollar; I must first divide the dollar into five equal parts, and then give the person two of these parts. A dollar is 100 cents--100 cents divided into 5 equal parts, each of those parts would be 20 cents. Hence, one fifth of 100 cents, or of a dollar, is 20 cents, and two fifths, twice 20, or 40 cents. The tédiousness and inconvenience of writing fractions in words has led to the invention of an abridged method of expressing them by figures. One huif is written }, one third, }, two thirds, 3, &c. The figure below the line shows the number of parts into which the thing, or whole, is divided, and the figure above the line shows how many of those parts are signified hoy the fraction. The number below the line gives name to the fraction, and is therefore called the denominator ; thus, it the number below the line be 3, the parts signified are thirds, if 4, fourths, if 5, fifths, and so on. The number written above the line is called the ruinerator, because it enumo rates the parts of the denominator signified by the fraction. As there are no limits to the number of parts into which a thing, or whole, may be divided, it is evident that it is possible for every number to be a numerator, or a denominator of a fraction. Hence the variety of fractions must be unlimited. 2. Fractions are of two kinds, Vulgar and Decimal, which differ in the form of expression, and the modes of operation. 3. A Vulgar Fraction is expressed by two numbers, called the numerator and denominator, written the former over the latter, with a line between, as }, the former before the latter, as 3-8=. 4. A Decimol Fraction, or a Decimal, is a fraction which denotes parts of a unit which become ten times smaller by each successive division (113), and is expressed by writing down the numerator only. (See Part II. Seci. Dll). A decimal is read in the same manner as a vulgar fraction; thus 0.5 is read 5 tenths, 0.25 25 hundredths, and it is put into the form of a vulgar fraction by drawing a line under it, and writing as many ciphers under the line as there are figures in the decimal, with a l at the left hand; thus, 0.5 becomes 1'5, 0.25, 100, and 0.003, TOUT: VULGAR FRACTIONS 215. 1. A proper fraction is one whose numerator is less than its denominator; as }, 1, , &c. (23). 2. An improper fraction is one whose numerator is greater than its denominator ; as, j, k, 4, &c. (24). 3. The numerator and denominator of a fraction are called its terms (30). 4. A compound fraction is a fraction of a fraction; as, ļ of . 5. A mired number is a whole number and a fraction written together, as 12:1, and 6} (23). 6. A common divisor, or common measure, of two or more numbers, is a number which will divide each of them without a remainder. 7. The greatest common dirisor of two or more numbers, is the greatest number which will divide those numbers severally without a remainder. 8. Two or more fractions are said to have a common denominator, when the denominator of each is the same number (25). 9. A common multiple of two or more numbers is a number, which may be divided by each of those numbers without a reinainder. The least common multiple is the least number, which may be divided as above. 10. A prime number is one which can be divided without a remainder, only by itself, or a unit. 11. An aliquot part of any number, is such part of it, as being taken a certain number of times, will exactly make that number. 8* |