012. ASSESSMENT OF TAXES. 141 8. C has candles at 6s. perture for $13; A pits in 80. dozen, ready money; but in sheep for 4 months, B Go sheer barter he will have 6s. 63. 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 ought each to pay ? price must the cotton be at in A $19.20 ) barter, and how much cotton B 7.20 Ang. must be bartered for 100 do C 21.60) zen of candles ? 12. If I have a mass of pure - Ans. the cotton 9 d. per lb. i in barter, and 7cwt. Ogrs. 16ib. | gold, a mass of pure corper, of cotton must be given for and a mass,which is a mixture 100 doz. of 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 quanBarter. tities displaced by each to be 9. If 6 men build a wall 20 | 8 by the copper, 7 by the mixfeet long, 6 feet high, and 4 | ture, and 5 by the gold; that feet thick, in 32 days; in what part of the mixture is gold, time will 12 men build a wall and what part copper? 100 feet long, 4 feet high, and 2 And 3 feet thick? Ans. 40 days. 10. If a family of 8 persons | 3:10 :: 3.10..52:6copper, in 24 months spend $480; how 11:31 yoid. inuch would they spend in 8 This is the celebrated problem (1 inonths, if their number were Archimedes, by which he deiected doubled ? Ans. $32. the frauid ofine artist employed by Hiero, king of Syracuse, to make 11. Three men hire a pas- | him a crowvl of pure golit (11) And -ASSESSMENT OF TAXES. 1. Supposing the Legisla-1 2. A certain school, con ture should grant a tax of sisting of 60 scholars, is sup$35000 to be assessed on the | ported on the polls of the scheinventory of all the rateable i lars, and the quarterly exproperty in the State, which pense of the whole school is amounts to $3000000, what $75; what is that on the schopart of it must a town pay, the lar, and what does A pay per inventory of which is $24600? quarter, who has 3 scholars ? Sinv. $ tax. $ inv. $ Ans. $1 25 on the scholar. 3000000 : 35000:: 24600 : 287 | and A pays $3.75 per quarter. Ans. I 3. If a town, the inventory | that on the dollar, and what is of which is $24600, pay $287, 1 C's tax, whose property inwhat will A's tax be, the in. | ventories at $76.44 ? ventory of whose estate is $4325 : 86.50 :: 1:.02 cents, $525.75 ? Ans. 24600.00 : 287 :: 525.75 : 1876.44.02=$1.528 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 $4325, I of $493.08, what is that on the and the sum to be raised on ! dollar ? this inventory for the support | $16436 : $493.08 :: 1:.03 cts. of schools, is $86 50; what is! Ans. 20 2010 300 30 12 1.20 1111111 213. In assessing taxes, it is generally best, first to find what eacls 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 recimal 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 cenis..03, 4 cenis, .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. .60 6.00 .90 9.00 400 12.00 50 1.50 500 - 15.00 60 - 1.80 600 - 18.00 - 21 70 - 2.10 700 - 21.00 80 - 2.40 800 - 24.00 9 .27 90 - 2.70 900 -- 27.00 10 - .30 100 – 3.00 1000 - 30.00 This table is constructed on the supposition that the tax amounts to three cents on the dollar, as in example 5th. USE.- What is B's tax. shose raleable preperty is 8276? By the table it appears that $200 pay 86, that $70 pay 82. 10, and that $6 pay 18 cents. Thus $200'is 86 00 Proceed in the same way to find each indi70 - 2. 10. vidual's tax, then add all the taxes together, 6 - 0.18 and if their amount agree with the whole sum proposed to be raised, the work is right. 58.28 li is sometimes best to assess the tax a trife k's tax. Jarger than the amount to be raised, lo con pensate for the loss of fractions, 13 REVIEW 1. What is meant by ratio ? How the four terms of a proportion? is ratio expressed? What is the first How is this muth shown? ierm called the second term ? 3. Does changing the places of 2. What is proportion? What the iwo middle terms affect the pro: , general truth is stated respecting i portion? Why not? 214, 215. FRACTIONS. 143 proportion? 4. What is meant by inverse 7. What is Fellowship? What l is meant by capital or stock? 5. What is meant by the Single | What by dividend? What is the Rule of Three? What is the gen- ! rule when the times are equal ? eral rule for stating questions in the What, when they are unequal? Rule of Three? How is the an- What is the method of proof?" swer then found ? If the first and I 8. What is Alligation? What is tiird terins be of different denomi. Alligation Medial?-Alligation Al. nations, what is to be done? What, ternate ? What is the rule for is there are different denominations finding the proportional quantities in the second terin? 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 I would you proceed? How, when proof in this rule ? one of the simples is limited to a 6. What is compound propor certain quantity? How is Alliga tion? By what other name is it | tion proved ? called? What is the rule for stat- 9.“What is barter? What is ing questions in compound propor- meant by a tax? What is the comrion? - for performing the operation? | mon method of inaking out taxes ? SECTION VII, Fractions. DEFINITIONS. -214. 1. Fractions are parts of a unit, or of a whole of any kind.(21.) 2. Fractions are of two kinds, Vulgar and Decimal, which differ in the forni of expression and the modes of operation. 3. A Vulgar Fraction is expressed by two numbers, called the numerator and denominator, written the foriner over the latter, with a line between (21) 4. A Decimal Fraction, or a Decimal, is a fraction, which denotes parts of a unit wlich become ten times smaller by each successive division, is expressed by writing down the numerator only. (Sec Part II. Sect. III.) 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 1 a: the left hand; thus, 0.5 becomes to, 0,25, 25, and 0.005, TOO VULGAR FRACTIONS. 215. 1. A proper fraction is one whose namerator is Jess than its denominator; as, }, , , &c.23) 2. An improper fraction is one whose numerator is greater than its denominator; as, , , , &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, foft. 5. A mixed number is a whole number and a fraction written together, as, 125, and 64-(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 divisor of two or more numbers, is . the greatest number, which will divide those numbers several y without a remainder. 8. Two, or more fractions are said to have a common de. nominalor, when the denominator of each is the same number. (25) 9. A coinmon multiple of two or more nuinbers, is a number, which may be divided by each of these numbers without a remainder. 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 Lumber. 12. A perfeci number is one which is just equal to the sum of all its aliquot parts. The smallest perfect number is 6, whose aliquot parts are 3, 2, and 1. and 3-+2+1=6; the next is 23, the next 496, and the next 8128. Only len perfect numbers are yet known. 216. WHOLE NUMBERS CONSIDERED UNDER THE FORM OF FRACTIONS. ANALYSIS 1. Change to a whole 1. Change 25} to an imor mixed number proper fraction. 3) 76 As the denominator de- | 2563+1_76 I denotes the notes the pomber of parts 251 into which one whole, or u-1 3 division of I by 3, nit, is divided, and the nu (129); if now we multiply 25 by 3, and add the promerator shoul's how many of those duct io 1, making (25*344-=) 76, parts are contained in tlie fraction. and then write the 76 over 3, thus, (22) there are evidently as many wholes, as the number of times the , we evidently both mul. numerater contains the denomina- 1 tiply and divide 25 by 3; but as tor; or, otherwise, since every , the multiplication is actually per. fraction denotes the division of the formed, and the division only de. numerator by the denominator,(129) | noted, the expression becomes an where the numerator is greater than | improper fraction. the denominator, we have only to A whole number is changed 10 an perform the division which is de- / improper fraction, by writing 1 an. noted Ider it, with a line between 217, 218, 219. VULGAR FRACTIONS. 145 217. To change an impro- | 218. To change a whole or per fraction to an equivalent mixed number to an equivalent whole or mixed number. improper fraction. RULE.—Divide the nume. Rule.—Multiply the whole rator by the denominator, and number by the denominator the quotient will be the whole of the fraction, add the numeor mixed number required. rator to the product, and write the sum over the depominator for the required fraction. QUESTIONS FOR PRACTICE, 2. Change 26 to a mixed 1 2. Change 8p to an im Dumber. proper fraction. . 3. Change 26 to a mix | 3. Change 27 to an imed number. proper fraction. 4. In 236s. shillings, how 4. In 19 . how many many shillings? | 12ths ? 5. in 24 of a week, how 5. In 3 week, how many many weeks ? | 7ths ? 219. MULTIPLICATION AND DIVISION OF FRACTIONS BY WHOLE NUMBERS, ANALYSIS. 1. James had of a peck | 1. Henry had of a peck of plumbs, and Henry had of plumbs, which were twice twice as many ; how many | the quantity James had ; bad Henry ? | how many had James ? Here we have evidently 1 Here we have evidently to multiply by 2; but two to divide into 2 equal times is ; hence to inul. I parts; but divided into tiply if by 2, we multiply | 2 parts, one of them is }; the numerator 2 by 2, and then to divide by 2, we · write the product, 4, over | must divide the numerator 8, the denominator; or oth- | by 2 and write the quotient i erwise, if we divide 8, the over 4, the denominator ; denominator, by 2, and write or, otherwise, if we multithe quotient, 4, under 2, the ply 4, the denominator, by numerator, thus, &, the frac-\ %, and write the product, 5 tion becomes multiplied ; | under 2, the numerator, thus, for while the number of the fraction becomes die parts signified remains the vided by 2, for while the same, the division has ren number of parts remains the |