THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES ; /■■ - -•»> '- ■■ » • * m/. 'V* ^ ARITHMETIC SIMPLIFIED, PREPARED FOR THE USE OP PRIMARY SCHOOLS, FEMALE SEMINARIES, AND HIGH SCHOOLS, IN THREE PARTS; ADAPTED TO CLASSES OP DIFFERENT AGES, AND OF DIFFERENT DEGREES OF ADVANCEMENT. BY CATHARINE E. BEECHER, LATE PRINCIPAL OF THE HARTFORD FEMALE SEMINARY. HARTFORD : PUBLISHED BY Df F. ROBINSON & CO. 1832. Entered according to Act of Congress, in the year 1832, by D. F. Robinson, & Co. in the Clerk's office of the District Court of Con. necticut. P. CANCTELD, PRINTER. HARTFORD. PREFACE. \o\ — T539cu The public have this claim upon any author, who offers a new school book, that such a work shall contain some es- sential advantages, which are not to be found, in any othei work already in use. If a writer cannot sustain such a claim, the public are needlessly taxed, for an article which is not wanted. It therefore seems proper, that a statement should be made of what are supposed to be, the peculiar advantages and improvements in this work. The writer, for several years, has been engaged in in- struction, and has either used, or examined, all the most popular works on Arithmetic. The following are the de- ficiencies, which have been experienced, and which it is the aim of this work to supply. It should, however, be pre- viously remarked, that all these difficulties have not been experienced in every work of the kind, heretofore examined ; but some have existed in one, and some in another, and no one work, yet known to the writer, obviates them all. 1. The first difficulty, for which a remedy is here at- tempted, originates from the fact, that in every school, there is such a variety of age, intellect, and acquisition, that no (me book is fitted for them all. If a work is found adapted to advanced classes, it is too difficult for the younger and less advanced. If it is fitted to these last, it is too easy for the others. To remedy this, the following work is divided into Three Parts. The First Part is adapted to the comprehension of young children. The Second Part is fitted to older classes. The Third Part completes an entire system of Arithmetic, containing all that is required of students on entering col- lege. The whole work embraces every thing of any con- sequence, that can be found in the most complete and ex- tended works ever used, and yet so simplified as to occupy much less space, and to demand much less labor. There are enough subjects, of difficulty, to call forth mental in- dustry and effort, without allowing those which are plain to remain involved in needless difficulties. 2. Another difficulty which this arrangement remedies. 1* 865520 Vi PREFACE. arises from the fact, that in most works of this kind, owing to the length of the various exercises under each head, the pupils lose the general principles they gain in one part, before they reach another. Thus, before "Reduction is attained, the principles employed in Addition and Subtraction, are partially forgotten, and the pupils do not gain a clear and general view of the whole science. But in the First Part of this work, in the compass of twenty pages, the pupil gains all the fundamental principles of the science, which in the succeeding parts, are developed in more minute particulars. To aid in the same object, and to secure other advanta- ges, a new method of classification has been adopted. Tin chiefbenefit aimed at, by this new arrangement, is to sim- plify the science, by leading the pupils to understand, that all the various exercises of Arithmetic are included under the same general principles of Addition, Subtraction. Mul- tiplication, Division, and Reduction. The common method of teaching the Simple Rules first, and then of introducing Vulgar and Decimal Fractions, has a tendency to render the science much more compli- cated and perplexing. Thus the pupil is first taught the process of Simple Addition. Then follows the exercises of the other simple rules, and by the time the pupil has some- what forgotten the principles of Simple Addition, comes Compound Addition, which seems to the child as much on a new principle, as Multiplication, or Division. Then, after another interval, comes Decimal Addition, and then the Addition of Vulgar Fractions. But a child who is first taught the system of numeration, as including whole numbers and fractions, and the nature of each of these modes of expressing numbers, can immedi- ately commence the operation of Addition, in all its various particulars, and recognize the general principle, that runs through the whole, and at the same time the peculiarity which distinguishes each. The difficulties arising from the common mode of ar- rangement, are particularly felt, when the processes of multiplying and dividing by fractions, are introduced. In all previous operations, the pupil has found that multiplication increases a number, and division diminishes it. But when fractions are introduced, a new science seems to commence, in which multiplication lessens and division increases a number,' and all heretofore learned, seems to be contradicted and' undone. But if, at the commencement of the science, the pupil under- PREFACE. Vll stands the peculiar character of Fractions, and then finds them arranged with whole numbers, so as to be able to compare and distinguish the same general principles of the various exercises, and at the same time the specific differ- ence, the perplexity arising from an apparent multiplicity of operations, and their seemingly contradictory nature, is avoided. When this plan was first attempted, some difficulty was felt from the necessity of the operation of Division in the previous operation of multiplying by a fraction. But this dif- ficulty has been obviated by having the First Part precede, in which Multiplication and Division are explained, with- out entering minutely into the various processes of Frac- tional Multiplication. It will be found that pupils, by learning the Division Ta- ble, and the First Part, can perform all the exercises in Simple and Fractional Multiplication, without any other knowledge of the rule of Division. There are two or three exceptions, however, where some exercises for the slate are introduced, where the rule of Division must beemployed. These are intended for older pupils, who are supposed to understand the process of Division, and may be omitted by new beginners, until a review. The writer has herself employed this method of classifi- cation, in teaching, and it has been used by the teachers in the institution under her care, for several years ; and thus exjierience has enforced the conviction, which at first was the result of reasoning, that this mode of classification will better secure the benefits sought for, in all attempts at gen- eralization. It certainly attains advantages, and avoids difficulties, much more than the common method. 3. Another difficulty experienced in using some of the most popular works of this kind, has arisen from the fact, that the mental and ivritten exercises have been entirely separate ; in some cases being placed in different books. Thus the pupil, after completing an Arithmetic designed for mental exercise alone, will often be found repeating ex- actly the same processes in written Arithmetic, without recognizing the principles, which, in mental operations, have been constantly employed. To remedy this, both mental and written exercises are placed together under every general rule. 4. Another defect in teaching this science, has arisen from a want of some method of stating and explaining the rationale, of each arithmetical process. In many Arithme- tics, a mechanical method is presented, of performing cer- Vlll PREFACE. tain operations according to rule, without any exhibition of the reason for such operations. Thus, in Subtraction, ichy one is carried, and ten borrowed ; or, in Multiplication, why the figures are placed in a certain method ; or, in Division, why multiplication and subtraction are performed, is never explained or illustrated. To a child, they are a sort of ca- balistical process, which he finds will bring the right answer, and this is all he can know from any thing he gains from the book. To remedy this, in the following work, every rule is accompanied by a full explanation of the reason, for each process employed. In the mental operations also, a proper mode of stating each process is given. The defini- tions, rules, and explanations, will be found to be more simple and concise than in many works of this kind, and perhaps may be considered as improvements. As the writer has been in a situation, in which she has had to employ various teachers, of different qualifications, it has been one great aim to furnish a work, by which new and inexperienced teachers could avail themselves of the ex- perience of others. This work has been maturing for several years, and the results of the experience of several able and ingenious teachers employed by the writer, have been added to her own. It is believed that any teachers with common talents and industry, can, with the aid of this work, do all for their pupils in this science, which needs to be done, in order to make them thorough and expert pro- ficients. For the purpose of perfecting such a work as this, and to make a fair trial of the several improvements contem- plated, a small work on this plan was printed some years ago, for the use of the pupils of the writer. But as it was intended for an experiment, and was necessarily very im- perfect and incomplete, it was never published. Yet, as in several cases, those who have been teachers and pupils in this institution, have introduced it into their schools, it may be proper to add, that this work is very different from the former, being much easier, much more extensive and com- plete, and improved in several respects which it is unne- cessary to mention. The writer does not lay claim to any great originality, in these various particulars, but has aimed to unite in one work the various excellencies, which might be otherwise scattered among a variety. Hartford Female Seminary, Jan. 1, 1S32. TO TEACHERS. It is very desirable that new beginners should review the First Part, till it is very thoroughly understood. It will save much trouble to both teacher and pupil. It will be found advantageous, to require older pupils te study the First Past, previous to commencing the Second ; for though some of the exercises are very simple, there are some important explanations and illustrations, not found in the Second Part. It is very desirable that pupils should become thorough and expert in Numeration, especially in Decimal Numera- tion, before taking the next lessons, and one or two reviews are recommended previous to proceeding. In Compound Addition and Fractional Midtiplicalion, if the pupil has never practised Simple Division, omit those exer- cises which require this rule, till a review. The Second Part should be reviewed, before commencing the Third Part. If any teachers have a preference for the common method of classification, it is very easy to direct the pupils to learn the Simple Rules first. But every pupil will find it advantageous at least to review on the plan of arrangement adopted in this work. When young beginners take the Second Part, it is re- commended, that they take the easiest exercises, and re- serve the more difficult, till a review. ERRATA. N. B. Pupils are requested to make these corrections with a pen, before prooeeding to study the book. Page 39, line 13 from tire bottom, for ascending read descending. P. 53, line 12 from the bottom, for orders read periods. P. 82, line 14 from the bottom, change the signs from multiplication to addition. P. 92, line 10, for gr. read qrs. P. 98, line 8 from the bottom, for 2 read 8. P. 110, line 1 for 10, read 18 ; in the answer to the fifth sum, for 5 read 11 ; the answer to the seventh sum should be 256 yds. ; line 8 from the bottom, for qrs. read roods. P. 118, line 6 from the bottom, for 4 read 12. P. 169, lines 6 and 14, for 3 read 3 12 7 «' P. 173, line 5, for lb. read £. P. 200, at the end of line 3 from the bottom, insert at 6 per cent. P. 201, at line 15, insert in 1 yr. 4 mo. P. 233, lines 4 and 5 from the bottom, for seconds and thirds, read twelfths and seconds. In many places in the first and second parts of the work, an error will be per- ceived in the manner of expressing decimals, tens and hundreds being used in- stead of tenths and hundredths ; thus, tens of thousandths, instead of tenths of thousandths, or ten thousandths. INDEX. ARITHMETICAL TABLES. Addition Table, .... Page 13 Subtraction Table, ----- 13 Multiplication Table, ■ - - - - . 14 Division Table ------ 15 Table of Weights and Measures, - - - 16 Table of Foreign Coins in Federal Money, - - 19 Table of Scripture Weights, Measures, and Coins, - 19 ARITHMETIC.— PART FIRST. Addition, ---... 25 Subtraction, ------ 26 Multiplication, ...... 28 Division, ...... 31 Reduction, ...... 37 PART SECOND. Numeration. * Numeration of Whole Numbers, 42 Numeration of Vulgar Fractions, 55 Decimal Numeration, ----- 58 Addition, ...... 66 Simple Addition, ..... 67 Decimal Addition, ..... 73 Compound Addition, ..... 78 Addition of Vulgar Fractions, .... 82 Subtraction. Simple Subtraction, ..... 83 Decimal Subtraction, ..... 86 Compound Subtraction, ..... 91 Subtraction of Vulgar Fractions, .... 93 Multiplication. Simple Multiplication, ..... 94 Decimal Multiplication, .... 103 Compound Multiplication, .... 109 Multiplication of Vulgar Fractions, - - - 111 Division, ...... 123 Simple Division, ..... 124 Compound Division, ..... 132 INDEX. XI Division of Vulgar Fractions, .... 135 Decimal Division, .... - 144 Reduction. Reduction Ascending and Descending, - • • 154 Reduction of Fractions to Whole Numbers, - - 158 Reduction of Whole Numbers to Fractions, - - 159 Reduction of Vulgar to Decimal Fractions, Reduction of Fractions to a Common Denominator, - 162 Reduction of Fractions to the Lowest Terms, - - 167 Reduction of Fractions from one order to another, - 170 Reduction of Fractions of one order, to Units of another order, 171 Reduction of Units of one order to Fractions of another order, 172 Reduction of a Compound Number to a Decimal, Reduction of a Decimal to Units of Compound Orders, - 175 Reduction of Currencies, - Reduction of Different Currencies to Federal Money, - 179 Reduction of Federal Money to Different Currencies, 180 Reduction from one Currency to another, - 181 PART THIRD. Numeration. Roman Numeration, .... 184 Other methods of Numeration, .... 185 Common, Vulgar, and Decimal Numeration, - 186 Addition. Simple, Vulgar, and Decimal Addition, - - 187 Subtraction. Simple, Vulgar, and Decimal Subtraction, - - 188 Multiplication. Simple, Vulgar, Compound, and Decimal Multiplication, 189 DrvisioN. Simple, Vulgar, Compound, and Decimal Division, - 191 Exercises in Reduction ..... 193 Interest, - - . . . - 194 Simple Interest, ..... 196 To find the Interest on Sterling Money, ... 200 Various Exercises in Interest, .... 200 Endorsements, ...... 202 First Method, ...... 203 Massachusetts Rule, ..... 204 Connecticut Rule, ..... 205 Compound Interest, . . .. . . 208 Discount, ...... 210 Stock, Insurance, Commission, Loss and Gain, Duties, . 210 Equation of Payments, .... 214 Ratio, ....... 215 Xll INDEX. Proportion, ...... 216 Simple Rule of Three; or Simple Proportion, . - 218 Double Rule of Three ; or Compound Proportion, - 221 Fellowship, ...... 225 Alligation, ...... 228 Duodecimals, ...... 232 Involution, - - - - . - 235 Evolution, ...... 237 Extraction of the Square Root, .... 239 Extraction of the Cube Root, .... 243 Arithmetical Progression, .... 248 Geometrical Progression, .... 252 Annuities, ...... 254 Permutation, - - - - ' - - 259 Miscellaneous Examples, - 260 Position, Ex. 40—50. Barter, Ex. 51—58. To find the Area of a Square, Ex. 69 — 71. „ „ of a Parallelogram, Ex. 72 — 74. „ „ of a Triangle, Ex. 75, 76. To find the Solid Contents of a Cube, Ex. 77 — 79. To find the Circumference, Diameter and Area of a Circle, Ex. 80—84. To find the Area of a Globe or Ball, Ex. 85. To find the Solid Contents of a Globe or Ball, Ex. 86. „ „ „ of a Cylinder, Ex. 87. „ „ „ of a Pyramid, Ex. 88, 89- Forms of Notes, Receipts, and Orders, - - - 269 Book-Keeping, ...... 272 ARITHMETICAL TABLES. ADDITION TABLE. ■! and are 2 4 and are 4 6 and are 6 8 and are a 2 1 V 1 1 7 8 1 9 2 2 44 o 6 6 2 8 a 2 10 1 3 54 3 3 9 ■< 3 11 s 4 6 4 4 86 4 10 8 4 12 2 5 7 4 5 g s 5 11 8 5 IS S 6 8 4 6 iim; 6 ■ 12 8 6 14 3 7 9 4 7 n r. 7 13 8 7 15 2 8 10 4 8 12(1 8 14 8 8 16 o 9 11 4 9 1316 9 15 8 9 17 :< and are 3 5 and are 5,7 and are 7 9 and are 9 3 1 4 5 1 eb I 8 9 1 10 3 2 5 5 2 " ■ n 9 9 2 11 3 3 6 5 3 87 3 10 9 3 12 3 4 7 5 4 9 7 4 11 9 4 13 S 5 8 5 5 10 7 5 12 9 5 14 3 6 9 5 G 11 7 6 13 ii 6 IS 3 7 10 5 7 I -J 7 7 14 9 7 16 3 8 11 5 8 13 : 8 15 9 8 17 3 9 12 5 9 14 7 9 16 9 18 SUBTRACTION TABLE. from 1 leaves 4 from 4 leaves 7 from 7 leaves 2 1 4 5 1 7 8 1 3 2 4 G o 7 9 2 4 3 4 7 3 7 10 3 5 4 4 8 4 7 11 4 6 5 4 9 5 7 12 T: 7 6 1 10 G 7 13 6 8 7 I 11 7 7 14 7 9 8 4 12 8 7 15 8 10 9 4 13 9 7 16 g 2 from 2 leaves 5 from 5 leaves 8 from 8 leaves 8 3 1 5 6 1 8 9 i 2 4 2 5 7 2 ,-■ 10 s ! 5 3 5 8 3 8 11 3 2 6 1 5 9 4 8 12 4 2 7 5 5 10 5 8 13 5 S 8 6 5 11 6 8 14 G 2 9 7 5 12 7 8 15 7 g 10 * 8 5 13 88 16 8 <> 11 9 .5 14 9 8 17 9 a from 3 leaves 6 from G 9 fran 9 leaves 3 4 1 6 7 I 9 10 1 3 5 2 6 8 2 9 11 3 3 6 3 6 9 3 'J 12 3 3 7 4 6 10 4 9 13 4 3 8 5 6 11 5 '.) 14 5 3 9 B 6 12 6 9 15 6 3 10 7 6 13 7 9 16 7 3 11 8 6 14 89 17 a 3 12 9 6 15 9 9 18 g 14 MULTIPLICATION TABLE. MULTIPLICATION TABLE. 2 times ; ire 5 times are 3 times ire 1 1 times are 2 x 1 = 2 •"> X 1 = 5 8 X 1 = 8 11 X i = = 11 2 fit 4 5 2 10 8 2 16 11 2 22 2 3 6 5 3 15 8 3 24 11 3 33 2 4 8 5 4 20 8 4 32 u 4 44 2 5 10 5 5 25 8 5 40 11 5 55 2 6 12 5 6 30 8 6 48 11 6 66 2 7 14 5 7 35 8 7 56 11 7 77 2 8 16 5 8 40 8 8 64 11 8 88 2 9 18 5 9 45 8 9 72 11 9 99 2 10 20 5 10 50 8 10 80 11 10 110 2 11 22 5 11 55 8 11 88 11 11 121 2 12 24 5 12 60 8 12 96 11 12 132 3 times are 6 times are 9 times are 12 i imes are 3 X I = 3 6 X 1 = 6 9 X 1 = 9 12 x i = 12 3 2 6 6 2 12 9 2 18 12 2 24 3 3 9 6 3 18 9 3 27 12 3 36 3 4 12 6 4 24 9 4 36 12 4 48 3 5 15 6 5 30 9 5 45 12 5 60 3 6 18 6 6 36 ,9 6 54 12 6 72 3 7 21 6 7 42 9 7 63 12 7 84 3 8 24 6 8 48 9 8 72 12 8 96 3 9 27 6 9 54 9 9 81 12 9 108 3 10 30 6 10 60 9 10 90 12 10 120 3 11 33 6 11 66 9 11 99 12 11 132 3 12 36 6 12 72 9 12 108 12 12 144 4 times are 7 times are | tirnesO are 1 3 times are 4 X 1 = 4 7 X 1 = 7 10 X i = 10 13 X 1 = = 13 4 2 8 7 2 14 10 2 20 13 2 26 4 3 12 7 3 21 10 3 30 13 3 39 4 4 16 7 4 28 10 4 40 13 4 52 4 5 20 7 5 35 10 5 50 13 5 65 4 6 24 7 6 42 10 6 60 13 6 78 4 7 28 7 7 49 10 7 70 13 7 91 4 8 32 7 8 56 10 8 80 13 8 104 4 9 36 7 9 63 10 9 90 13 9 117 4 10 40 7 10 70 10 10 100 13 10 130 4 11 44 7 11 77 10 11 110 13 11 143 4 12 48 7 12 84 10 12 120 13 12 156 DIVISION TABLE. 15 DIVISION TABLE. 2 in 2 1-| 6 in 6 11 10 in 10 11 2 4 2 6 12 2 10 20 2 2 6 3 6 18 3 10 30 3 2 6 4 ~ G 24 4 ~ 10 40 4 *■*■ 2 10 5 ! . 3 CD 6 30 5 CD 10 50 5 . 3 2 12 6 CO 6 36 6 CO 10 60 6 CO 2 14 7 6 42 7 10 70 7 2 16 8 6 48 8 10 80 8 2 18 9j 6 54 9J 10 90 9J 3 in 3 11 7 in 7 11 11 in 11 11 3 6 2 7 14 11 22 2 3 9 3 7 21 3 11 33 3 3 12 4 -. 7 28 4 a 11 44 4 <-» 3 15 5 TO 7 35 5 ■ 3 11 55 5 CD 3 18 6 QD 7 42 6 tn 11 66 6 CO 3 21 7 7 49 7 11 77 7 3 24 8 7 56 8 11 88 8 3 27 9J 7 63 9. 11 99 9, 4 in 4 1^ 8 n 8 11 12 in 12 1^ 4 8 2 8 16 2 12 24 2 4 12 3 8 24 3 12 36 3 4 16 4 8 32 4 r* 12 48 4 ** * 4 20 5 > 5 CD 8 40 5 •I 12 60 5 > 3 CO 4 24 6 Cfl 8 48 6 QQ • 12 72 6 QQ 4 28 7 8 56 7 12 84 7 4 32 8 8 64 8 12 96 8 4 36 9. 8 72 9. 12 108 9 5 in 5 I* ! 9 in 9 1- 13 in 13 1- 5 10 2 9 18 2 13 26 2 5 15 3 9 27 3 13 89 3 5 20 4 r*- 9 36 4 13 52 4 r*' 5 25 5 J' 9 45 5 i 3 • CD 13 65 5 ► 3 f CD 5 30 6 00 9 54 6 05 13 78 6 CO • 6 35 7 9 63 7 I l3 91 7 5 40 8 9 72 8 1 13 104 8 5 45 9. 9 81 9 1 13 117 9, 16 WEIGHTS AND MEASURES. 1. Troy Weight. 24 grains (gr.) make 1 penny-weight, marked pwl~ 20 penny-weights, 1 ounce, oz. 12 ounces, 1 pound, lb. 2. Avoirdupois Weight. 16 drams {dr.) make 1 ounce, oz. 16 ounces, 1 pound, lb. 28 pounds, 1 quarter of a hundred weight, qr. 4 quarters, 1 hundred weight, cwt. 20 hundred weight, 1 ton. T. By this weight are weighed all coarse and drossy goods, grocery wares, and all metals except gold and silver. 3. Apothecaries Weight. 20 grains (gr.) make 1 scruple, 9 3 scruples, 1 dram, 3 8 drams, 1 ounce, g 12 ounces, 1 pound, ft Apothecaries use this weight in compounding their medicines. 4. Cloth Measure. 4 nails (na.) make 1 quarter of a yard, qr. 4 quarters,, 1 yard, yd. 3 quarters, 1 Ell Flemish,. E. Fl. 5 quarters, 1 Ell English, E. E. 6 quarters, 1 Ell French, E. Fr. 5. Dry Measure. 2 pints (pL) make 1 quart, qt. 8 quarts, 1 peck, pk. 4 pecks, 1 bushel, bu. This measure is applied to grain, beans, flax-seed, salt, oats, oysters, coal, &c. 6. Wine Measure. 4 gills (gi.) make 1 pint, pt. 2 pints, 1 quart, qt. 4 quarts, 1 gallon, gal. 3l£ gallons, 1 barrel, bL WEIGHTS AND MEASURES. 17 42 gallons, 1 tierce, tier. 63 gallons, 1 hogshead, hhd. 2 hogsheads, 1 pipe, p. 2 pipes, 1 tun, T. All brandies, spirits, mead, vinegar, oil, &c. are meas- ured by wine measure. Note. — 231 solid inches, make a gallon. 7. Long Measure. 3 barley corns (&. c.) make 1 inch, marked in. 12 inches, 1 foot, ft. 3 feet, 1 yard, yd. 5| yards, 1 rod, pole, or perch, rd. 40 rods, 1 furlong, fur. 8 furlongs, 1 mile, m. 3 miles, 1 league, lea. 691 statute miles, 1 degree, on the earth. % 360 degrees, the circumference of the earth. The use of long measure is to measure the distance of places, or any other thing, where length is considered, without regard to breadth. N. B. In measuring the height of horses, 4 inches make 1 hand. In measuring depths, six feet make one fathom or French toise. Distances are measured by a chain, four rods long, containing one hundred links. 8. Land, or Square Measure. 144 square inches make 1 square foot. 9 square feet, 1 square yard. 30| square yards, or ) , , o-ro e t. l ! square rod. 272£ square wet, ) ^ 40 square rods, 1 square rood. 4 square roods, 1 square acre. 640 square acres, 1 square mile. Note. — In measuring land, a chain, called Gunter's chain, 4 rods in length, is used. It is divided into 100 links. Of course, 25 links make a rod, and 25 times 25=625 square links make a square rod. In 4 rods, there are 792 inches. Of course, 1 link is 7 |-|._ 9. Solid, or Cubic Measure. 1728 solid inches make 1 solid foot. 2* 18 WEIGHTS AND MEASURES. 40 feet of round timber, or ) , , , 50 feet of hewn timber, $ 128 solid feet or 8 feet long, > rf f ^ 4 wide, and 4 high, $ All solids, or things that have length, breadth and depth, are measured by this measure. N. B. The wine gallon con- tains 231 solid or cubic inches, and the beer gallon, 282.. A bushel contains 2150,42 solid inches. 10. Time. 60 seconds (S.) make 1 minute, marked M. 60 minutes, 1 hour, h. 24 hours, 1 day, d. 7 days, 1 week, w. 4 weeks, 1 month, mo. 13 months, 1 day and 6 hours, 1 Julian year, yr. Thirty days hath September, April, June, and Novem- ber, February twenty-eight alone, all the rest have thirty- one. N. B. In bissextile or leap-year, February hath 29 days. 11. Circular Motion. 60 seconds (") make 1 minute, 60 minutes, 1 degree, 30 degrees, 1 sign, S, 12 signs, or 300 degrees, the whole great circle of the Zodiac. 12 units make A Dozen. 12 dozen A Gross. 144 dozen A Great Gross. 20 units A. Score. 24 sheets of paper A Quire. 20 quires A Ream. VALUE OF FOREIGN COINS. 19 Value of Foreign Coins in Federal Money. Shilling Sterling, «0.222 ' Crown 5s. 1.111 Sovereign, (a gold coin, - £,) 4.444 Guinea, (21s. nearly out of ; use in England,) $ Livre of France, 0. 185-j- Franc " 0.1875— Pistole* 10 livres" 1.852— Louis d'or, " 4,444-f Five franc piece, " 0.937 Real of Plate, of Spain, 0.100 RealofVellon, " 0.050 Pistole, " 3.60 Dollat; " 1.00 Re. of Portugal, $0.0012-|- Testoon, " 0.125 Milre,* " 1.250 Moidore, " 6.000 Joanese, " 8.000 Marc Banco of Hamburgh, 0.333-f- Pistole of Italy, 3.200 * Those denominations which have the asterisk, (as the Pistole of France, and the Milre of Portugal,) are merely nominal; that is, they are represented by no real coin. In this respect, they are like the Mill in Federal Money. Rix Dollar of Austria, 0.778— Rix Dollar of Denmark ^ 1.000 and Switzerland, J Rix Dollar* of Sweden, 1.037 Rix Dollar of Prussia, 0.778— Florin, " 0.259-j- Ducat of Sweden and ) Prussia, ) 2.074 Piaster of ex, of Spain, 0.80 Ducat of ex, » " 1.102— Stiver of Holland, 0.019-4- Guilder or Florin, " 0.388 Rix Dollar, " 0.970 Ducat, " 2.079 Gold Ducat, " 8.000 Ducat of Denmark, 8.833+ Ruble, of Russia, 1.000 Zervonitz, " 2.000 Tale, of China, 1.480 Pagoda, of India, 1.840 Rupee, of Bengal, 0.500 Xeriff, of Turkey, 2.222 A TABLE OF SCRIPTURE WEIGHTS, MEASURES, AND MONEY. A Cubit, A Span, half cubit, A Hand breadth, A Finger, A Fathom, Ezekiel's reed, The measuring line, Sabbath day's journey, Eastern mile, Stadium, or Furlong, Day's journey, S OF LENGTH. feet. inches. • • ■ 1 9,88 • . . 10,94 ... 3,68 ... 0,91 . • . 7 3,55 • • • 10 11,32 . • • 145 11,04 miles* furlongs. roa s. feet. 5 21 H i 3 2 3 o 1 4 3 33 1 12 6 20 SCRIPTURE WEIGHTS AND MEASURES. MEASURE OF LIQUIDS. The Homer or Cor. gall. 75 pints, sol. inch. 5 7,6 The Bath, 7 4 15,2 The Hin, 1 2 2,5 The Log, 24,3 The Firkin, . 7 4,9 MEASURE OF THINGS. The Homer, bushels, pecks. 8 pints. 1,6 The Lethech, 4 0,8 The Ephah, The Seah, 3 1 3,4 1,1 The Orner, 5,1 The Cab, 2,9 WEIGH TS. A Shekel, lb. oz. 9 pwt. 9 gr- 2,0 The Maneh, 2 3 6 10,3 A Talent, 113 10 1 10,3 MONE Y. dolls. cents. mills. A Shekel, 50 5 The Bekah, (half Sheh.) . 25 3 The Zuza, 12 5 TheGerah, 02 5 Maneh or Mina, 25 29 6 A Talent of Silver, 1,H 85 7 A Shekel of Gold, 8 09 4 A Talent of Gold, 24,2£ 71 4 Golden Daric or Drachm, 4 85 7 dolls. ■ cents. mills. Piece of Silver, (Drachm) Tribute money, (Didrachm) Piece of Silver, (Stater) Pound, (Mina) 14 14 28 57 35 3 7 4 1 Penny, (Denarius) 14 3 Farthing, (Assarium) 00 6 Farthing, (Quadrands) . Mite, 00 00 3 1 ARITHMETIC. PART FIRST. Arithmetic is the science of numbers. A unit is a whole thing of any kind. A fraction is a part of a thing. Thus a dollar is a unit ; a man is a unit; a picture is a unit ; a bushel of apples is a unit, &c. A half of an apple is a fraction ; a quarter of a dollar is a fraction ; a third of a loaf of bread is a fraction, &c. Let the pupil mention other units and fractions., If an apple is cut into two equal parts, each part is called one half of the apple. If it is cut into three equal parts, each part is called one third. If it is divided into four equal parts, each part is called one fourth. If it is divided into jive equal parts, each part is called one fifth, halves, arc how many halves ? Five sixths of an 'apple, and four sixths, and nine sixths, are how manv sixths ? J Three fifths of an orange, and four fifths, and nine fifths, and twelve fifths, are how many fifths ? Addifmi is uniting several numbers in one. When whole numbers are added, it is Simple Addition. When fracf ions are added, it is Fractional Addition. Six dimes, five dimes, and four dimes are how many ? Seven dollars, ei^ht dollars, and nine dollars are how many ? Nine cents, three cents, twelve cents, and ten cents are how many ? Four, three, and seven are how many ? Eight, five, and three are how many ? Nine, six, and two are how many ? 3 26 ARITHMETIC. FIRST PART. Seven, five, and six are how many 1 Eight, nine, and two are how many? Seven, eight, and one are how many? Eleven, five, and six are how many? Ten, seven, and three are how many? Ten twentieths, six twentieths, and five twentieths are- how many twentieths ? One thirteenth of a unit, four thirteenths, and seven thirteenths are how many thirteenths ? One fifth of a dollar, three fifths, and eight fifths are how many fifths ? One ninth of an orange, four ninths, and six ninths are how many ninths ? Seven tenths of an eagle, two tenths, and five tenths are how many tenths ? Three eighteenths, nine eighteenths, and four eight- eenths are how many eighteenths ? Ten thirtieths, six thirtieths, and five thirtieths are how many thirtieths ? Two fourths, six fourths, nine fourths, ten fourths, and five fourths, are how many fourths ? Sixteen halves, five halves, nine halves, and six halves, are how many halves ? Six eighths, four eighths, seven eighths, sixteen eighths, are how many eighths? The number made by adding several numbers together, is called he sum. What is the sum of four, six, nine and five ? What is the sum of four tenths, six tenths, and nine tenths ? SUBTRACTION. If you take two cents from three cents, how many re- main ? If you take three dollars from six dollars how many re- main ? If you take four dollars from seven dollars how many remain ? SUBTRACTION'. 27 If you take five eagles from nine eagles how many re- main ? If you take six dimes from ten dimes how many re- main ? If two tenths are taken from four tenths how many re- main ? If four ninths are taken from eight ninths how many re- main ? If two tenths are taken from seven tenths how many re- main ? Subtraction is taking one number from another, to find the remainder. When whole numbers are subtracted it is Simple Sub- traction. When fractions are subtracted, it is Fractional Subtraction. What is the remainder, when four cents are taken from nine cents ? What is the remainder, when three mills are taken from eight mills ? What is the remainder, when seven dimes are taken from twelve dimes ? What is the remainder, when five dollars are taken from ten dollars ? Five from eleven ? Seven from thirteen ? Eight from twelve ? Five from fourteen ? Nine from sixteen .' Five from twelve ? Eight from thirteen ? Ten from Twenty ? What is the remainder, when two sevenths of an apple, are taken from eight sevenths ? When four sevenths of a dollar are taken from six sevenths? Eight twelfths from ten twelfths? Three ninths from eight ninths? Ten twentieths from twelve twentieths ? Six elevenths from ten elevenths ? Seven twelfths from twelve twelfths ? Eight ninths from thirteen ninths ? Three sevenths from nine sevenths ? Four eighths from eleven eighths ? Four thirds from twelve thirds ? Five twentieths from seven twentieths ? The number which has a number subtracted from it, is called the minuend. The number which is to be subtracted from another num- ber is called the subtrahend. If eight is subtracted from twelve, what is the subtra- hend and what is the minuend ? 28 ARITHMETIC. FIRST PART. If four tenths, is .subtracted from nine tenths, what is the subtrahend and what the minuend ? If ten cents be taken from thirteen cents, what is the subtrahend, and what the minuend ? MULTIPLICATION. If you take two cents, three times, what is the amount of the whole 1 If you take three dollars, four times, what is the amount of the whole ? If you take half of an apple three times, what is the amount ? If you take two thirds of a dollar four times, what is the amount ? If 3 r ou take two fourths of an eagle, six times, what is the amount 1 Multiplication is repeating a number as often as there are units in another number. If you take five dollars four times, what is the amount 1 If you repeat four dollars five times, what is the amount 1 If you take six dollars five times, what is the amount 1 If you repeat six dollars six times, what is the amount ? Seven times ? Eight times ? If you take seven dollars three times, what is the amount ] If you repeat seven, four times, what is the amount ? Five times ? Six times ? Seven times ? If you repeat eight twice, what is the amount ? If you repeat eight three times, what is the amount ? Four times ? Five times ? Six times ? Seven times 1 Eight times ? If you repeat nine three limes, what is the amount ? &c. If you take one fifth of a dollar six times, what is the amount ? Seven times 1 Eight times ? Nine times? Tfyou repeat two sixths of a dollar three times, what is the amount ? MULTIPLICATION. M If you repeat two sixths of a thing four times, what is the amount ? Five times ? Six times ? Seven times ? Eight times ? What is the amount, if four sevenths be repeated four times ? Five times ? Six times ? Seven times ? Eight times ? What is the amount if Jive ninths be repeated eight times ? Nine times ? Ten times ? Eleven times ? What is the amount, if eight twentieths be repeated seven times ? Nine times ? Eight times ? &c. The number to he repeated, is the multiplicand. The number which shows how often the multiplicand is to bo repeated, is called the multiplier. The midtiplier and multiplicand together, are called the factors. The answer obtained is called the product. Height is repeated four times what is the product ? What is the multiplier 1 The multiplicand ? The fac- tors ? If three sixths are repeated/owr times what are the fac- tors ? The multiplier ? The multiplicand ? If you take a. fourth of twelve and repeat it three times, what is the multiplicand ? The multiplier ? The pro- duct ? If you take a sixth of eighteen and repeat it three times, what is the product ? factors 1 multiplier ? multipli- cand ? Simple Multiplication is where both factors are whole numbers. Fractional Multiplication is where one or both factors are fractions. If twelve is repeated four times, is it simple or fraction- al multiplication ? If one fourth of twelve is repeated three times, is it sim- ple or fractional multiplication ? If one sixth is repeated seven times, which kind of multiplication is it ? Exercises in Simple Multiplication. 1. If a man spends three dollars a week, how much does he spend a month ? 3* 30 ARITII31ETIC. FIRST PART. Let the pupil state the sum in this manner. As there are four weeks in a month, a man will spend four times as much in a month, as in a week ; four times three is twelve. He will spend twelve dollars. Let all the following sums be stated in the same way. Both teachers and pupils will find great advantage in be- ing particular to follow this method of stating. 2. If a man spend five dollars a month, how much does he spend in a year ? 3. If a man can make eight pens in a minute, how many can he make in ten minutes ? 4. If one orange cost six cents, what costs eight oran- ges ? 5. Eight beys have seven cents apiece, how much have all? 6. There is an orchard in which there are six rows of trees, and seven in each row, how many trees in the or- chard ? 7. The chess board has eight rows of blocks, and eight blocks in each row, how many blocks in the whole ? 8. Twelve young ladies have each five books apiece, how many have they all ? 9. If a young lady spends six cents a week, how much does she spend in a month ? 10. There are nine desks in a school room, and six scholars at each of the desks, how many are in the room? 11. There are in a window five rows of panes of glass, and seven panes in each row, how many in the whole ? 12. If one lemon cost four cents, how much will twelve lemons cost ? EXERCISES IN FRACTIONAL MULTIPLICATION. Multiplication of a fraction by whole numbers. 1. If vou repeat one half four times what is the pro- duct ? 'I. If you multiply three fourths by seven, what is the product ? 3. What is two thirds multiplied by eight ? DIVISION. 31 4. If a man spend two twelfths of a dollar a day, how many twelfths does he spend in a week ? Ans. As there are seven days in a week, a man spends seven times as much in a week as in one day. Seven times two twelfths is fourteen twelfths. He spends four- teen twelfths of a dollar in a week. 5. If a man gives two eighths of a pound of meat to six persons, how many eighths does he give away ? 6. If a boy gives two fourths of an orange to seven of his companions, how many fourths does he give away 1 7. If a man drinks three jourths of a pint of brandy a clay, how many fourths does he drink in a week? 8. What is three times three eighths ? Six times si.) sevenths ? 9. If a man lays by two eighths of a dollar a day, how much does he save in a week 1 10. If there are two thirds of a pound of meat for each one in a family of seven, how much is there in the whole ? 11. What is six times four tenths ? 12. What is nine times two thirds ? 18. What is seven times four ninths? 14. What is eight times six tenths ? 15. What is twelve times two fourths ? 16. What is nine times three tenths? 17. What is five times three sixteenths? 18. What is six times seven twentieths ? The multiplication of whole numbers by fractions, is defer- red to the Second Part, because it involves the process of Division, which must first be explained. DIVISION. How many two cents are there in four cents ? How many two cents in six cents ? How many two cents in eight ? How many two cents in ten ? How many two cents in twelve ? How manv three cents are there in six cents ? How many in nine ? How many in twelve ? 32 ARITHMETIC. FIRST PART. How many four cents are there in eight ? How many in twelve ? How many five cents are there in ten ? What part part of two cents is one cent ? What part of four cents is two ? What part of six is two ? What part of eight is two ? What part of ten is two ? What part of twelve is two ? Three cents is what part of six ? Three is what part of nine ? Of twelve 1 What part of eight is four ? What part of twelve is four ? What part of five cents is one ? What part of five is two ? What part of five is three ? Four ? Five ? Six? &c. How many two sixths are there in four sixths ? How many three fourths are there in six fourths ? How many four twelfths in eight twelfths ? What part of two twelfths is one twelfth ? What part of four twelfths is two twelfths ? What part of nine twelfths is three twelfths? Division is finding how often one number is contained in another, and thus finding ivhat part of one number is another number. How many times is six contained in twelve ? In eighteen ? What part of twelve is six ? What part of eighteen is six ? How many times is five contained in ten ? In fifteen ? Five is what part often ? Of fifteen ? How many times is seven contained in fourteen ? In twenty-one ? What part of fourteen is seven 1 What part of twen- ty-one is seven ? How many times is nine contained in eighteen ? How many times is ten contained in twenty? In thir- ty? In forty ? What part of sixteen is four 1 What part of eighteen is six ? What part of sixteen is eight ? One is what part of thirty ? Two is what part of thir- DIVISION. 33 ty ? Three is what part of thirty ? Six ? Eight ? Eleven ? Fourteen? Twenty is what part of thirty ? &c. How many two sevenths are there in ten sevenths ? ' How many three eighths arc there in nine eighths ? How many six tenths in eighteen tenths ? How many seven ninths in twenty-one ninths 1 How many five elevenths in twenty elevenths ? How many three eighteenths arc there in twelve eight- eenths ? Two sixths is what part of four sixths ? Two sevenths is what part often sevenths ? Three eighths is what part of nine eighths ? What part of eighteen tenths is six tenths? What part of fourteen ninths is seven ninths ? What part of fifteen elevenths is five elevenths? What part of twelve eighteenths is three eighteenths ? The number which is divided is called the Dividend. The number by which you divide is called the Divisor. The answer is called the Quotient. If you find how many times three there are in twelve. which is the Divisor ? The Dividend ? The Quotient ? If twelve is divided by six, which is the Dividend ? The Divisor ? The Quotient ? When whole numbers are divided by whole numbers, it is called Simple Division. When either the divisor or dividend is a fraction, it is called Fractional Division. Exercises in Simple Division. 1. If you divide 12 cents equally among three boys, how many will each one have ? Ans. Each one will have as many as there are threes in twelve ; ox four cents. 2. If there are forty-eight panes of glass in a window, and there are eight panes in each row, how many rows are there ? Ans. As many as there are eights in forty-eight ; ov six rows. 3. How much broadcloth, at six dollars a yard, can you buy for twenty-four dollars ? 34 ARITHMETIC. FIRST PART. 4. How many hours would it take you to travel twen- ty-one miles, if you travelled three miles an hour 1 5. If you divided thirty-six apples equally among four hoys, how many would you give them apiece ? 6. How many pounds of raisins, at nine cents a pound, can you buy for sixty-three cents ? 7. How many reams of paper, at seven dollars a ream, can you buy for forty-nine dollars ? 8. A man agreed to work eight months, for seventy-two dollars, how much did he receive a month ? 9. If you buy a bushel of pears for forty-eight cents, how much is it a- peck ? 10. If there are six shillings in a dollar, how many dol- lars in thirty-six shillings ? 11. Four men bought ahorse for forty-eight dollars, what did each man pay ? 12. A man gave sixty-three cents for a horse to ride nine miles, how much was that for each mile ? 13. A man agreed to pay eight cents a mile for a horse, and he paid sixty-four cents, how many miles did he go ? 14. A man had forty-two dollars, which he paid for wood, at seven dollars a cord, how many cords did he buy? 15. Two boys are running, and are forty-eight rods apart. The hindermost boy gains upon the other, three rods a minute, in how many minutes will he overtake the foremost boy 1 16. A vessel contains sixty-three gallons, and dischar- ges seven gallons an hour, in how many hours will it be emptied ? 17. If you wish to put sixty-four pounds of butter in eight boxes, how many pounds would you put in each box? EXERCISES IN FRACTIONAL DIVISION. Division of whole numbers by Fractions. 1. How many halves are there in six oranges? 2. How many thirds are there in four apples ? DIVISION. 35 Ans. One apple has three thirds, four apples haxc four times as many, or twelve thirds. 3. How many fourths are there in three oranges ? 4. How many fifths are there in four apples ? 5. How many sixths are there in two oranges ? C. How many half dollars are there in four dollars? 7. How many quarters of a dollar in five dollars 1 8. How many half eagles in eight eagles 1 9. In two dollars how many thirds of a dollar ? 10. If there are six one thirds in two dollars, how many two thirds are there ? Ans. There are only half as many too thirds as there are one thirds, or three two thirds. 11. In two dollars, how many one sixths? How main two sixths ? 12. A man divided two dollars anions his workmen, and gave them a third of a dollar apiece, how many work- men had he ? 13. A man divided four dollars equally among his chil- dren, and gave them each two thirds of a dollar, how ma- ny children had he ? Ans. As many children as there are two thirds in four dollars. In four dollars there are twelve one thirds. There are half as many two thirds, or six. He had six children. 14. If a man gave two "sevenths of a dollar to each of his servants, and gave away in the whole four dollars, how many servants had he ? 15. How many two sixths in four ? 16. How many two eighths in four ? 17. How many two thirds in eight ? 18. How many two ninths in six? 19. How many two twelfths in two ? 20. How many two twelfths in four ? Dicis ion of Fractions by whole numbers. In dividing fractions by whole numbers, we do not find how many times a whole thing is contained in a part of the same thing, for that would be absurd ; but we find what part of once, a whole number is contained in a fraction. Thus if we wish to divide one halj by one, we say, one unit is contained in one half, not once, but one half of once. 36 ARITHMETIC. FIRST PART. 1. One is contained in one fourth, what part of once ? 2. One is contained in one fifth, what part of once ? 3. One is contained in one sixth, what part of once 1 4. One is contained in one seventh, what part of once ? 5. One is contained in one eighth, what part of once ? 6. One is contained in one ninth, what part of once 1 7. One is contained in one tenth, what part of once ? 8. One is contained in one eleventh, what pail of once 1 9. One is contained in one twelfth, what part of once 1 10. If you divide one fourth, by one, which is the divi- sor ? The dividend ? What is the quotient ? 11. If you divide one sixth by one, what is the quotient ? The divisor ? The dividend ? 12. If you divide one third by one, what is the quo. tient 1 The divisor ? The dividend ? 13. If one fourth contains one, a fourth of once, what part of once does two fourths contain it? Ans. Twice as much, or two fourths of once. 14. If two sixths is divided by one, what is the answer ? Ans. Tifo sixths of once. 15. Two eighths contain one, what part of once ? Six eighths contain one, what part of once ? 16. Two twelfths contain one, what part of once ? Four twelfths contain one, what part of once ? 17. Eight twelfths contains one, what part of once ? "18. Six twelfths contains one, what part of once ? If six twelfths contains one, six twelfths of once, it would contain two, only half as often, or three twelfths of once. 19. Four eighths contains one, what part of once ? Contains two, wiiat part of once? It contains two, only half as often, or two eighths of once. 20. Six tenths contains one, what part of once 1 Con- tains two, what part of once ? 21. Eight tenths contains one, what part of once ? Contains two, what part of once ? 22. Four eighths contains one, what part of once ? — Contains two, what part of once? 23. Six elevenths contains one, what part of once 1 — Contains two, what part of once ? 24. Eight twelfths contains one, what part of once ? Contains two, what part of once ? REDUCTION. 37 REDUCTION. One dime is how many cents ? How many mills ? One unit of the order of dollars, is how many units of the order of dimes? How many of the order of cents? How many of the order of mills ? One eagle is how many dollars ? How many dimes ? Cents ? One unit of the order of dimes is how many units of the order of cents ? Reduction is changing units of one order, to those of another. A unit of the order of eagles is how many units of the order of dollars ? Of dimes? Two eagles are how many dollars ? How many dimes ? How many dollars in two hundred cents ? How many dollars in twenty dimes ? Thirty units of the order of dimes, is how many units of the order of dollars ? Two pints are one quart. Eight quarts are one peck. Four pecks are one bushel. Two units of the order of quarts, are how many units of the order of pints ? Eight pints are how many quarts ? Two bushels how many pecks ? Eight pecks how many bushels ? Three barley-corns are one inch. Twelve inches are one foot. Three feet are one yard. One inch how many barley-corns ? Two inches are how many ? Twelve barley-corns how many inches ? One foot how many inches ? Three feet how many 1 One yard is how many feet ? How many inches ? How many barley-corns ? Two yards are how many feet ? How many inches ? How many barley-corns ? Three yards are how many feet ? How many inches ? How many barley-corns ? 4 38 ARITHMETIC. FIRST PART. How many feet are there in five yards ? How many inches in five yards ? How many barley-corns 1 How many barley-corns are there in seven yards ? From the preceding exercises, you learn that a unit of one order may contain several units of another order. What do you learn from the preceding exercises ? How many units of the order of cents, are there in one unit of the order of dimes ? How many units of the order of dollars, are there in one unit of the order of eagles ? How many units of the order of mills, are there in one unit of the order of cents ? How many units of the order of pints, are there in one unit of the order of quarts ? How many units of the order of pecks, are there in one unit of the order of bushels 1 How many units of the order of barley-corns, are there in one unit of the order of inches ? N How many units of the order of feet, are there in one unit of the order of yards ? How many units of the order of da3"s, are there in one unit of the order of weeks 1 How many units of the order of weeks, in one unit of the order of months ? Change two units of the order of dimes, to units of the order of cents. Change twenty units of the order of cents, to units of the order of dimes. Change three units of the order of yards, to units of the order of feet. Change nine units of the order of feet, to units of the order of yards. Change ten units of the order of pints, to units of the or- der of quarts. Change five units of the order of quarls, to units of the order of pints. Change twenty-one units of the order of days, to units of the order of weeks, &c. When units of one order are changed to units of a high- REDUCTION. 39 er order, the process is called Reduction ascending ; and when units of one order are changed to those of a lower order, the process is called Reduction descending. If twenty cents are changed to dimes, which kind of reduction is used ? Iftwenty cents are changed to mills, which kind of re- duction is used ? If four gallons are changed to pints, which reduction is used? If eight feet are changed to inches, which kind of re- duction is used ? In changing twelve barley-corns to inches, which kind of reduction is used ? In changing fourteen days to weeks, which reduction is used? In chancins five hours to minutes, which reduction DO ' is used ? In changing one hundred and twenty minutes to hours, which reduction is used? Reduce three dimes to cents ; to mills. Which kind of reduction is it ? Reduce three hundred mills to cents ; to dimes ; and which kind of reduction is it ? Reduce ihree hundred mills to dollars, and which kind of reduction is it ? Reduce two halves to quarters, and which kind of re- duction is it ? Ans. As a half is of more value, it is a higher order than a quarter, therefore it is reduction ascending. In performing this last exercise, the pupil will find the necessity for the following distinction in regard to units. A unit has been defined as " any whole thing of a kind," and a fraction is defined as " a part of a thing." But it is very often the case, that fractions are consid- ered as units. Thus when we reduce quarters to halves, and halves to quarters, we change units of the order called quarter, to units of the order called half. When we say a whole quarter of an apple, and a half a quarter of an apple, we think of a quarter as a whole thing of its kind. The difference between the two kinds of units is this : 40 ARITHMETIC. FIRST PART. When we think of a whole quarter, we think of another thing of which the quarter is a part. We think of it as a whole thing in one respect, and as a part of a thing in another respect. But when we think of a whole apple, we do not necessarily think of another thing of which it is a part. When we think of a half of a loaf of bread, do we think of something of which the half is a part? When, we think of a biscuit, do we necessarily think of something of which it is a part ? When we think of a third of an orange, do we necessa- rily think of something of which it is a part ? When we think of a house, do we necessarily think of any thing of which it is a part ? Those units which do not require us to think of any oth- er thing of which they are parts, are called whole numbers, and those units which do require us to think of other things of which they are parts, are called fractions. What is the difference between units that are whole numbers, and units that are fractions ? Reduce two yards to quarters, and which kind of re- duction is it ? Reduce twenty-four inches to feet, and which kind of reduction is it ? Reduce three feet to inches, and which kind of reduc- tion is it ? Which is of highest value, a half or a quarter ? Reduce eight quarters to halves, and which kind of re- duction is it ? Reduce two halves to quarters, and which kind of re- duction is it ? Reduce sixteen quarters to halves, and which kind of reduction is it 1 Reduce two fifths to tenths ; six tenths to fifths ; eight tenths to fifths ; twelve tenths to fifths ; three fifths to tenths ; six fifths to tenths. Reduce one seventh to fourteenths ; four fourteenths to sevenths ; four sevenths to fourteenths ; eight four- teenths to sevenths. SUMMARY OF DEFINITIONS. 41 Reduce two sixths to twelfths ; four twelfths to sixths ; eight twelfths to sixths ; five sixths to twelfths ; four twelfths to sixths. SUMMARY OF DEFINITIONS. A unit is any whole thing of a kind. A fraction is a part of a thing. Addition is uniting several numbers in one. Subtraction is taking one number from another, to find the remainder. The largest number is the minuend, the smallest num. ber is the subtrahend. Multiplication is repeating one number as often as there are units in another number. The multiplicand is the number to be repeated ; the multiplier is the number which shows how often the multi- plicand is to be repeated ; the factors are both the multi- plier and multiplicand ; and the product is the number ob- tained by multiplying. Division is finding how often one number is contained in another number, and thus finding what part of one num. ber, is another number. The dividend is the number to be divided. The divi- sor is the number by which you divide. The quotient is the answer obtained by dividing. Reduction is changing units of one order, to units of another order. Reduction ascending, is changing units of a lower, to a higher order. Reduction descending is changing units of a higher, to a lower order. Note to Teachers. — A review of this First Part, will be found more useful than an increased number of ex- amples. 4* ARITHMETIC. SECOND PART. NUMERATION. Numeration is the art of expressing numbers by words, or by figures. Figures are sometimes called numbers, because they are used to represent numbers. Thus the figure 4, is oft- en called the number four, because it is used to represent that number. There are thirty-five words, that are commonly used in numeration ; viz : one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, six- teen, seventeen, eighteen, nineteen, twenty, thirty, forty, fif- ty, sixty, seventy, eighty, ninety, hundred, thousand, million, billion, trillion, quadrillion, quintillion, sextillion. Those words ending in tetn, are the words two, three, four, &c with teen, which signifies and ten, added to them. What is the meaning of fourteen ? Ans. Four and ten. What is the meaning of thirteen? of nineteen? of seven- teen ? Those ending in ty, are the words two, three, four, &c. with ty, which means tens, added to them. What is the meaning of sixty ? of seventy ? of eighty ? of twenty? of thirty? The words of spoken numeration would be more uni- form, it eleven and twelve, had been called oneleen and tivo- teen. The Latin and Greek numerals are so often used in the various sciences, that it is important for pupils to learn their names. They are therefore put down with the fig- ures, and the English names. The figures are called Arabic, because first introduced into Europe from Ara- bia. NUMERATION. 43 ENGLISH, LATIN, AND GREEK NUMERALS. Greek Names. Eis. Duo. Treis. Tessares. Pente. Hex. Hepta. Okto. Ennea. Dcka. Endcka. Doileka. Dekatreis. Dekatessares. Dekapente. Dekaex. Dekacpta. Dekaocto. Dekaennea. Eikosi. Triakonta. Tesserakonta. Pentakonta. Hexakonta. Hebdomekonta. Ogdoekonta. Ennenekonta. Hckaton. Chilio. Billion, Trillion, (Quadrillion, Quintillion, Sextillion, &c. are made, by adding ciphers to 1. If any higher number than sextillion is to be expressed, the names are made by the Latin numerals, with illion added to them ; as seplillion, octillion, &c. A unit has been defined as " a single thing of any kind." But a unit of one kind, maybe made up of several units of another kind. Thus the unit one dollar is made up of ten units, of the kind, ox order called dimes; and one dime is made up often units of the order called cents. A unit which is of the most value, is called a unit of a higher order. Arabic Figures. English Name, s. Latin Names. 1 One. Unus. 2 Two. Duo. 3 Three. Tres. 4 Four. Quatuor. 5 Five. Q,uinque. 6 Six. Sex. 7 Seven. Septem. 8 Eight. Octo. 9 Nine. Novem. 10 Ten. Decern. 11 Eleven. Undecim. 12 Twelve. Duodecim. 13 Thirteen. Tredecim. 14 Fourteen. Quatuordecim 15 Fifteen. Quindecim. 16 Sixteen. Sexdecim. 17 Seventeen. Septendecim. 18 Eighteen. Oetodecim. 19 Nineteen. Novemdecini. 20 Twenty. Viginti. 30 Thirty. Triginta. 40 Forty" duadraginta. 50 Fifty. Q,uinquaginti. 00 Sixty. Sexaginta. 70 Seventy. Septuaginta. HO Eighty. Octoginta. 90 Ninety. Nonaginta. 100 Hundred. Centum. 1000 Thousand. Mille. 1000000 Million. 44 ARITHMETIC. SECOND PART Which unit is of the highest order, a dollar or a cent? How many units of the order of dimes, are there in one unit of the order of dollars ? How many units of the order of mills, make one unit of the order of cents? How many units of the order of cents, make one unit of the order of dimes ? Every figure represents a certain number ; but the num- ber it represents, depends upon the order in which it is placed. If the figure (2) stands alone, it represents two units, and is said to be in the^rs^or unit order. But if it has a figure to the right of it, thus (20) it rep- resents two tens, or twenty, and is in the second order, or the order of tens. The cipher is put to the right, to make the 2 stand in the order of tens, and to show that there are no units of the unit order. If some figure was not placed there, the 2 would be in the unit order. If the figure* 2 has two figures to the right of it, thus (20G) it represents two hundreds, and stands in the third, order, or the order of hundreds. From this it appears, that in numeration, (he number ex. pressed by any figure, depends upon the order in which it stands. The number which any figure expresses when it is con- sidered alone, is called its simple value. The number it expresses when placed with other figures, is called its lo- cal value. When 2 is considered alone, what is its simple value ? When it is considered as in the order of tens, what is its local value 1 When in the order of hundreds, what is its local value '! Questions. — What does every figure represent ? What does the number which any figure represents depend up- on ? If a figure stands alone, in what order is it ? If it has one figure at the right of it, in what order is it ? If it has two figures tit the right of it, in what order is it? In this number, (234) in what order is the 2 ? the 3 ? the 4 ? Write one ten. — Why is the cipher used ? What would the number be., if the cipher were removed ? NUMERATION. 45 Write one ten and one unit. What is the name of this number? Ans. Eleven. Write one ten and two units. What is the name of this number? Write one ten and three units. What is the name 1 Write one ten and four units. What is the name ? Write one ten and Jive units. What is the name ? Write one ten and six units. What is the name ? Write one ten and seven units. What is the name ? Write one ten and eigjit units. What is the name 1 Write one ten and nine units. What is the name ? Write two tens. What is the name ? Ans. Twenty. Write three tens. What is the name ? W rite four lens ; Jive tens ; six tens; seven tens; eight tens ; nine tens ; and tell their names. Write one of the order of hundreds. Write two of the order oHiundreds ; one of the order of lens ; and Jour of the order of units. Write two of the order of hundreds ; no tens ; four units. Write 4 hundreds, no tens, no units. Write two hundreds, eight tens, and nine units. Sev- en hundreds, six tens, and three units. Two tens, and two units. Nine tens, and six units. Four hundreds, six tens, and four units. Five hundreds, five tens, and five units. Nine hundreds, seven tens, and three units. Four hundreds, eight tens, and four units. Eight hun- dreds, nine tens, and nine units. Two hundreds, six tens, and three units. One hundred, two tens, and three units. Two hundreds, five tens, and seven units. One ten, and three units. Seven tens, and three units. Nine hundreds, nine tens, and nine units. In readin« numbers, we can either mention each order separately, or simply mention the names of the numbers. Thus we can call this number, (21) either two tens, and one unit, or twenty-one. This number (305) can be read, 3 hundreds ; tens ; 5 units ; or it can be called three hundred and five. The following numbers are read both ways, thus ; 10 One ten ; no units ; or ten. 1 1 One ten ; one unit ; or eleven. 46 ARITHMETIC. SECOND PART. 208 Two hundreds ; no tens ; eight units ; or two hun- dred and eight. 40 Four tens ; no units ; or forty. Let the pupil read the following numbers both ways. 111. 203. 41. 37. 542. 1. 11. 12. 60. 300. 101. 639. 700. 305. In this number, (20U) why is the cipher put in? What would the number be if it were left out? In numeration, every unit of one order, is considered as composed of ten units of a lower order ; just as in the coins of this country, ten units of the order of cents, make one unit of the order of dimes, and ten units of the order of dimes, make one unit of the order of dollars. So in numeration, ten units of the order of units, make one ten ; ten units of the order of tens, make one unit of the order of hundreds ; ten hundreds, make one unit of the order oUhousands ; ten thousands make one of the or- der of tens of thousands; ten tens of thousands, make one of the order of hundreds of thousands ; ten hundreds of thousands, make one of the order of millions, &c. Wherever there are nine units of any order, if there is another added, the number becomes one unit of the next higher order. If we had nine cents, and should add another, instead of calling the amount ten cents, we could call it one dime ; and so when ten units are added together, we can call them one unit of the order of tens, instead of ten units of the unit order ; and when we have ten units of the order of tens, we can call them one unit of the order of hundreds. Questions. — If nine cents have one more added, in what order do they become a unit ? If nine dimes have another added, in what order do they become units 1 Ten units of the order of dollars, make one unit of what order ? Ten tens, make one unit of what order ? Ten units, make one unit of what order ? Ten hundreds make one unit of what order ? The following are the names of the orders. First order, Units. NUMERATION. 47 Second order, Third order, Fourth order, Fifth order, Sixth order, Seventh order, Eighth order, Ninth order, Tenth order, Eleventh order, Twelfth order, Thirteenth order, Fourteenth order, Fifteenth order, Sixteenth order, Seventeenth order, Eighteenth order, Nineteenth order, Twentieth order, Tens. Hundreds . Thousands. Tens of thousands. Hundreds of thousands. Millions. Tens of millions. Hundreds of millions. Billions. Tens of billions. Hundreds of billions. Trillions. Tens of trillions. Hundreds of Trillions. Quadrillions. Tens of Quadrillions. Hundreds of Quadrillions. Quiutillions. Tens of Quiutillions. Hundreds of Quintillions. Sextillions. Twenty-first order, Twenty-second order, Sextillions are as high as there is ordinarily any need of writing or reading. In all the above orders, " Ten units of one order, make one unit of the next higher order. If a figure 2 stands in thefrst order, what number does it express? What number does it express, if it stands in the fourth order? In the second order ? In the fjth or- der ? In the sixth ? seventh? eighth? Let the pupil write the following : 1. Five units. 13. One hundred and sixty. 2. Three tens ; two units. 14. One hundred, and six tens. 3. Thirty-two. 15. Two hundred, two tens. -1. Three and ten, or thirt< :n. 16. Two hundred and twenty. 5. Four and ten. 17. Two hundred and thirty. 6. Four tens, or forty. 18. Two tens and two units. 7. Six and ten. 19. Twenty-two. 8. Six tens. 20. Two hundreds and two units. 9. Sixteen. 21. Five tens and two units. 10. Sixty. 22. Five hundreds. 11. One hundred and sixteen. 23. Five tens. 12 One hundred, one ten, and six. 24. Fifty. 48 ARITHMETIC. SECOND PART. 25. Five hundred, and five units. 26. Five and ten. 27. Fifteen. 28. Fifty seven. 29. Four hundreds, six tens. 30. Four hundred and sixteen. 31. Four hundreds, one ten, and six. 32. Four hundred, and six. 33. Two hundred and sixty-six. 34. Three hundred, ten, and one. 35. Three hundred and eleven. 3G. Three hundred, ten and two. 37. Three hundred and twelve. 38. Four hundred and one. 39. One hundred and forty-two. 40. Two hundreds, two tens. Let the pupil write the following : 1. One unit of the fourth order. What number is if. ? Which orders have ciphers in them ? 2. Two units of the fourth order ; one unit of the sec- ond order, and one unit of the first order. What number is it ? What order has a cipher in it ? 3. Two thousands ; one hundred ; five tens ; six units. 4. Twenty-one hundreds ; five tens ; six units. Is there any difference between the two last numbers ? 5. Three thousands, four hundreds, six tens and three units. 0. Thirty-four hundred, and sixty-three. Is there any difference in the two last numbers ? 7. Three thousands and three units. Which orders have ciphers placed in them ? 8. Three thousands, six hundreds. Which orders have ciphers placed in them ? 9. Thirty-six hundred. What two ways of reading this last number? 10. Twenty thousand. 11. Two tens of thousands. Is there any difference between these two last num- bers? 12. Twentyifour thousand. What two ways of reading this last number? 13. One hundred thousand, two tens of thousands, five thousands, six hundreds, four tens, and three units. 14. One hundred and twenty-five thousand, six hun- dred and forty-three. Is there any difference between these two last numbers ? 15. Two tens of thousands, one thousand, four hun- dreds, six tens, five units. What two ways of reading this number ? 16. Four hundred and sixty-two thousand, five hundred and six. NUMERATION. 49 What two ways of reading this last ? 17. Forty-four thousand, four hundred and forty-four. What two ways of reading this last ? 18. Four hundreds of thousands, five thousands, six hundreds, two tens, five units. What two ways of reading this last numher ? 19. Two hundred thousand, two thousand, two units. What orders have cyphers placed in them ? 20. Twenty thousand, and two units. 21. Two hundred and six thousands, four hundred and six. 22. Sixty-four thousand and three. 23. Sixteen thousand. 24. Fourteen thousand and seven. 25. Five tens of thousands, and six units. 26. Two hundreds of thousands, two hundreds, two units. 27. Two hundred and sixty-four thousand, and six. 28. Four thousand, and five units. 29. One hundred thousand, and three. 30. Sixteen thousand, six hundred and six. 31. Twenty-four thousand and three. In order to read and write numbers more conveniently, they are divided into periods of three figures each, by means of commas, thus : 870,409,764,256,622,895,946,852. Thefirst right hand period is called lhe unit period ; and contains the orders called units, fens and hundreds The second period, is called the thousand period ; and contains the orders called thousands, tens of thousands, and hundreds of thousands. The third period is called the million period, and con- tains the orders called millions, tens of millions, and hun- dreds oj millions. The fourth period is called the billion period; and con- tains the orders called billions, tens of billions, and him- dreds of billions. The fifth period is called the trillion period; and con- tains the orders called trillions, tens of trillions, and hun- dreds of trillions. 5 50 ARITHMETIC. SECOND PART. The sixth period is called the quadrillion period ; and contains the orders called quadrillions, tens of quadrillions, and hundreds of quadrillions. The seventh period is called the quintillion period ; and contains the orders called quintillions, tens of quinlillions, and hundreds of quintillions. The eighth period is the sextillion. The following are the periods which must be learned in succession, beginning with the highest, as well as with the lowest ; thus, First Period Unit. Eighth Period, Sextillion. Second Period, Thousand. Seventh Period, Quintillion. Third Period, Million. iSixth Period, Quadrillion. Fourth Period, Billion, Fifth Period, Trillion. Sixth Period, Quadrillion. Seventh Period, Quintillion Fifth Period, Trillion. Fourth Period, Billion. Third Period, Million. Second Period, Thousand. Eighth Period, Sextillion. iFirst Period, Unit. What is the first period ? ' the third ? the fifth ? the second ? the fourth ? the seventh ? the sixth ? the eighth 1 The pupil may write the names over the periods until accustomed to reading them ; thus, Trill. Bil. Mil. Thous. Units. 32 427 983 254 693 The above may be read in the following manner : The first left hand period is read, 3 tens of trillions ; 2 units of trillions : or thirty-two trillions. The next period is read, 4 hundreds of billions ; 2 tens of billions ; 7 units of billions ; or four hundred and twen- ty-seven billions. The next period is read, 9 hundreds of millions; 8 tens of millions ; 3 units of millions, or nine hundred and eigh- ty-three millions. The next period is read, 2 hundreds of thousands ; 5 tens of thousands ; 4 units of thousands ; or two hundred and fifty-four thousand. The next period is read, 6 hundreds ; 9 tens ; 3 units; or six hundred and ninety-three. The following is a number in which several orders are omitted, having ciphers in place of numbers. NUMERATION. 51 Quin. Quad. Tril. Bil. Mil. Th. U. 33 067 004 803 064 000 400 Let the pupil first tell what periods and what orders are omitted, having ciphers instead ofnumhers. The above number may be read thus : Begin at the left and read ; 3 tens of quintillions, and 3 units of quintillions ; or thirty three quintillions. The next period is, no hundreds of quadrillions ; 6 tens of quadrillions ; and seven units of quadrillions ; or sixty- seven quadrillions. The next period is, no hundreds of trillions ; no tens of trillions; 4 units of trillions ; or four trillions. The next period is, 8 hundreds of billions; no tens of billions ; 3 units of billions ; or eight hundred and three billions. The next period is, no hundreds of millions ; 6 tens of millions ; 4 units of millions ; or sixty-jour millions. The next period, as it has no hundreds, tens, or units of thousands, may be omitted entirely, when reading. The next period is, 4 hundreds ; no tens ; no units ; or four hundred. The best and most common way of reading, is that in the italics, and then all together, it reads thus : Thirty-three quintillion ; sixty-seven quadrillion ; four trillion ; eight hundred and three billion ; sixty-four mil- lion ; four hundred. Let the pupil read the following sum in both ways : Quin. Quad. Tril. Bil. Mil. Th. Un. 607 300 000 763 490 068 002 RULE* FOR READING WHOLE NUMBERS. Point off into periods of three figures earh, beginning at the right. Read each period as if it stood alone, and then add the name of the period. Note. — When a period or order is omitted, it is not ne- cessary to mention it at all. Before reading, let the pupil tell what periods and or- ders are omitted, and represented by ciphers. 52 I ARITHMETIC SECOND PART. Let the pupil point oft', and re ad the following figures 1 2 31 304 300046 200200200 111 24 40 600 300005 2030003000 100 136 400 611 1200437 311001300 101 3024 4040 693 1200039 60009090 1011 2002 6000 4004 4960004 100100001 2002 46900 40640 103006 1430096 2071113603 3041 60021 600003 1063007 6000007 1000673 201 62003 100014 103964 86004369 101700013 2010 6040064 600436 140001 20064000 600040006 3004 46923 64003 400006 400400400 300010000 227034293 9623000062 200004900 10043259054 3690200000 43600078609 30006340200 459643723007 602030004296 612942004000040367 40000643209437 3907650060042300000 237 6000964 300600C )0 396770000543965000076 It is necessary for the pupil to understand, that the French and English arithmeticians use different methods of numeration. The English have their periods contain six orders, and the French only three. This makes no difference till we come to hundreds of millions. After that, it makes a great difference, as will be seen by the following comparison. It must be noticed, that the same figures are used in both. English Method. Trillions. 579364, Billions. 028635, Millions. 419763, Units. 215468. French Method. Sext. Quin. Qua. Trill. Bill. Mill. Th. Units. 579, 364, 028, 635, 419, 763, 215, 468. From the above it can be seen, that all the orders above hundreds of millions, in both methods, give the same name, to a very different value. Thus, the orders of thousands of millions, tens of thous- ands of millions, and hundreds of thousands of millions, in the English method, would be read as billions, tens of bill- ions and hundreds of billions, in the French method. NUMERATION. 53 Billions, tens of billions, and hundreds of billions, in the English method, are equivalent to trillions, tens of trillions, and hundreds of trillions, in the French method. Five trillion, in the French method, would be read five billion, in the English ; and five trillion, in the English method, would be read five quadrillion, in the French. Questions. — How would a billion, in the English meth- od, be read in the French? How would one hundred billion, in the English method, be read in the French ? How would one billion, in the French method, be read in the English ? How would six hundred billion in the French method, be read in the English ? The French method is adopted in this woik, because it is both the most convenient, and the most common. But the pupil needs to understand the difference be- tween the two modes, and the teacher should make the class point off and read numbers by both. Point oft' and read the following numbers, first by the French, and then by the English method. 7G543217G50U431 9870000654321765432 32698000000040000360093 436789643645964379629364 In order to write numbers correctly, the pupil must learn thoroughly, the succession of the orders beginning at the left. Thus, SextiUion, Quintillion, Quadrillion, Trillion, Billion, Thousand and Unit. Rule for Writing Whole Numbers. Begin with the highest period, and write first the hundreds, then the tens, and then the units of that period. Proceed thus, until all the periods are written. Place a comma between each period. If any period or order is omitted, place ciphers in their place. Note. — Ciphers prefixed to a whole number, have no effect upon the value. A number, therefore, should nev- er be begun with a cipher. 5* 54 ARITHMETIC. SECOND PART. Write two thousand and two. What orders are omit, ted? Write two million, two thousand, and four. What or- ders are omitted ? Write Three hundred and twenty. four. What period and orders in this number ? Write Two hundred thousand and four. What orders omitted in this last number ? Write ; Two million and six ? What period omitted ? what orders omitted ? Write, Six million ; two hundred and three. Which period and what orders are omitted ? Write, Twenty. lour million ; three hundred. Which period and what orders are omitted ? Write the following sums and mention the periods and orders which, are omitted. 1. One billion; twenty-four million; three thousand and three. 2. Four hundred and sixty-nine billion ; forty. four thousand ; and seventeen. 3. Fifty billion ; three hundred million ; four hundred and fifty thousand ; and nineteen. 4. Fifty billion, and seven. 5. Four hundred and thirteen million, and two thous- and. 6. Nineteen billion, and one million. 7. Six trillion ; nine thousand, and ten. 8. Seven trillion ; nineteen billion ; ten thousand, and four hundred. 9. Four hundred and nine trillion ; sixteen million ; eleven thousand and forty. 10. Fifteen billion ; two hundred and four million ; six thousand, and twenty-one. 11. Sixty-four million ; Tour hundred thousand ; three hundred. 12. Sixteen million ; five hundred thousand, and six. 13. Three trillion ; fourteen million ; seven thousand. 14. Two hundred and sixteen million. 15. Two billion ; sixteen million, and sixteen. 16. Three hundred and six trillion ; four thousand, and six. NUMERATION. 55 17. Two quintillion ; six quadrillion and five. 18. Three hundred and sixty-four thousand. 19. Three million and six. 20. Fourteen trillion ; three hundred. 21. Sixteen trillion, four million, two hundred and four thousand, seven hundred and one. 22. Three sextillion, one hundred quadrillion, fourteen trillion, two hundred and sixty billion, four hundred mill- ions, sixteen thousand, four hundred and one. 23. Five million, two hundred thousands, and sixty-two. 24. Two hundred and five millions, and seventy-four. 25. Twelve hundred and six billions, four millions, and six thousand. 26. Two hundred sextillions, four hundred millions, three hundred and four thousand, two hundred and six. 27. Fifteen quintillion, six quadrillions, one hundred trillions, forty-four billions, two millions, and forty-nine. 28. Fifty "quadrillions, six hundred trillions, forty-three millions, two thousands four hundred and six. 29. Two hundred and six trillions, forty-three billions, four hundred and nine millions, sixty-four thousands, four hundred and ninety-six. 30. One hundred and four billions, six millions, forty- nine thousands, four hundred and ninety-six. 31. Thirteen millions, four hundred "thousands, six hun- dred and forty-nine. 3x5. Six sextillions, five quintillions, four quadrillions, three trillions, two billions, and one million. NUMERATION OF VULGAR FRACTIONS. Figures are of two kinds, — 'Figures for a number of whole things, and figures for a number of parts of things. A unit is a whole thing of any kind. A fraction is a part of one thing ; or a part of several things. Figures may therefore be divided into fractional and unit figures. The following is the mode of showing when the num- lg 13 lin, «.vw VI ^v,„.w & 56 ARITHMETIC. SECOND PART. bers represented are several whole things, and when they are several parts of things. When there are two whole things, their number is ex- pressed thus, (2). This is called a unit figure. But if a whole thing is divided into three parts, and we wish to express two of these, by figures, we write one fig- ure, to show into how many parts the whole thing is divi- ded, and then above it, write the number of parts we wish to express ; thus, (|). This is called a fractional figure. The lower figure shows into how many parts the whole thing is divided, and the upper figure shows how many of these parts are expressed. In f , into how many parts is the whole thing divided, and how many of these parts are expressed? In f, into how many parts is the whole thing divided, and how many parts are expressed ? In 4? In a? In*] In f ? In T \? In & ? Fractional figures show into how many parts one whole thing is divided, and how many of these parts are expres- sed. Besides this, they can show what part is taken from several whole things. Thus f shows that one thing is divi- ded into four parts, and three of them are taken ; or that three whole things, have a fourth taken from each of them. For, three fourths of one whole thing, is the same quantity as one fourth of three whole things. If you have three apples, and take one fourth out of each, how much will you have, and how will you express it in figures 1 If you divide one apple into four parts, and take three of these parts, how do you express the quantity taken ? If you have two apples, and take one sixth from each, how much will you have, and how will you express it in figures ? If you divide an apple into six parts, and take two of these parts, how much will you have, and how will you express it in figures ? If an apple is divided into eight parts, and you take three of them, how much wdl you have, and how will you express it in figures 1 If the fraction is considered as showing how many parts NUMERATION OF VULGAR FRACTIONS- 57 are taken from one unit, then the lower figure shows into how many farts a unit is divided, and the upper figure shows how many of these parts are taken. But if the frac- tion is considered as showing what part is taken out of several units, then the upper figure shows the number of uni&s, and the lower figure shows what part is taken from each. Thus the fraction § may be considered as expressing, two sixths oCone thing, or as one sixth of two things. -j 3 2 is either one twelfth of three things, or three twelfths of one thing. £ is either four fifths of one thing, or one fifth of four things. | either shows that one ninth is taken out of two things ; or that two ninths are taken out of one thing. If | is considered as showing how many parts are taken out of one thing it is four sevenths of one unit. If it is con- sidered as showing what part is taken out of several things, it is one seventh of four units. If | shows how many parts are taken out of one thing, it is two thirds of one thing. If it shows what part is taken out of several things, it is one third of two things. If | is considered as showing how many parts are taken out of one unit, what does the 8 show, and what does the 7 show? If it is considered as expressing what part is taken out of several units, what does the 7 show, and what do.es the 8 show ? If * is considered as expressing how many parts are ta- ken out of one unit, what does the 6 show, and what does the 4 show 1 If it is considered as expressing what part is taken out of several units, what does the 4 show, and what does the 6 show 1 Whenever the numerator is larger than the denominator , the fraction is called an improper fraction, and always is to be considered as expressing what part is taken out of several units'. Which of the following are improper fractions ? 6. 7 _8_ 9. 1_2 1_4 6. 4. R. 3 4 14 6^2 ,5 9 13* What does an improper fraction show 1 58 ARITHMETIC. SECOND PART. RULE FOR READING VULGAR FRACTIONS. Read the number of parts expressed by the numerator, and tlien the size oftheparts expressed by the denominator ; or Read the part expressed by the denominator, and then the number of units, expressed by the numerator. Read the following fractions in both ways, thus : £ is either three fourths of one thing, or one fourth of three things. § is either three fifths, or one fifth of three. 4. 6. 2. 13. _S_ 6_ _9_ 6 9 6 13 10 12 18 RULE FOR WRITING VULGAR FRACTIONS. Write the number of parts into which a unit is divided, and draw a line above it. Over it ivrite the number of parts which are to be expressed; or Write the whole numbers which have a certain part taken from them, and draio a line wider. Beneath it write the figure which expresses the part which is to be taken out of each of the units above. Let the. pupil write the following : If a man divided an apple into eight parts, and gave away five of these parts, how do you express the quantity he gave away, and the quantity he kept? If a man had three apples, and cut out a fourth part of each, and gave it away, how do you express what he gave away I If a man had twelve oranges, and one sixth of each was decayed, how do you express the quantity of decayed or- anges he had ? If a man had five casks of wine, and a twelfth part leak- ed out of each, how do you express what he lost ? DECIMAL NUMERATION. There is another mode of writing fractions, in which the numerator only, is written. The denominator, although not written, is always understood to be 1, and a certain number of ciphers. These fractions are called Decimals. NUMERATION OF VULGAR FRACTIONS. 50 Thus in writing decimals, if we are to express two tenths, instead of writing it thus j\, the numerator only is written, and a comma, called a separatrix, is placed before it, thus ,2. The following is the rule, by which it is known what is the denominator. The, denominator of a decimal is always I , and ax many cyphers as there are figures in the numerator, or decimal. What is the denominator of this decimal, ,2 ? Ans. 1 and one cypher. How many cyphers in the denominators of these deci- mals, ,34. ,000. ,3240. ,50945. ,3694. ? If the decimal has one figure, it expresses tenths. Thus ,2 is two tenths. If it has two figures, it expresses hundredths. Thus ,02 is two hundredths. If it has three figures, it expresses thousandths. Thus ,002 is two thousandths. If it has four figures, it expresses tens of thousandths. Thus ,0002 is two tens of thousandths. If it has five figures, it expresses hundreds of thous- andths Thus ,00002 is two hundreds of thousandths. What does this decimal express, ,3? Ans. Three tenths. What does this decimal express, ,30 ? Ans. Thirty hundredths. What does this decimal express, ,003 ? Ans. Three thousandths. What does this decimal express, ,0003 ? Ans. Three tens of thousandths. What does this decimal express, ,5 ] Ans. 5 tenths. What does this decimal express, ,15 ? Ans. Fifteen hundredths. What does this decimal express, ,110? What does this decimal express, ,2000? W T hatdoes this decimal express, ,00002? Ans. Two hundredths of thousandths. A decimal must always have the number of figures in the numerator, equal to the number of ciphers in the de- nominator ; therefore it is necessary to learn how many ciphers there are in each kind of denominator. GO ARITHMETIC. SECOND PART. If the decimal is tenths, there is one cipher in the de- nominator ; if hundredths, there are two ciphers ; if thousandths, there are three ciphers ; if tens of thous- andths, there are four ciphers ; if hundreds of thous- andths, there are five ciphers, &c. Of course in writing decimals, if tenths are to be ex- pressed, there must be only one figure in the numerator, or decimal ; if hundredths, there must be two figures ; if thousandths, there must be three figures ; if tens of thous- andths, there must be four figures ; if hundreds of thous- andths, there must be Jive figures, &c. If you are to write two tenths, how many figures must there be in the numerator or decimal, and how many ci- phers are understood to be in the denominator ? Write two tenths. (,2). If you are to write two hundredths how many ciphers are understood to be in the denominator, and how many figures must there be in the numerator 1 Write, two hundredths. In writing this last, the pupil must first write the 2, and then as there must be as many figures in the numerator, as there are ciphers in the denominator, a cipher is placed before the 2, and then the separatrix is prefixed thus, ,02. If the cipher were placed after the 2, how would it read ? Ans. Twenty hundredths, instead of two hundredths. If the cipher were not placed before the 2, how would it read ? Ans. Two tenths. If another cipher is placed before the ,02 thus, ,002 how does it read 1 What does the. denominator express, when there are three figures in the decimal. Ans. Thousandths. What does it express when there are four figures in the decimal ? Let the pupil write the following. 1. Two tenths. 2. Two hundredths. 3. Two thousandths. 4. Two tens of thousandths. 5. Two hundreds of thousandths. 6. Five tenths. DECIMAL NUMERATION. 61 7. Fifteen hundredths. 8. Fifteen thousandths. 9. Fifteen tens of thousandths. 10. Fifteen hundreds of thousandths. 11. One tenth. 12. Eleven hundredths. 13. One hundred and fifteen thousandths. 14. Five tenths. 15. Fifty-five hundredths. 16. Five hundred thousandths. 17. Five hundred and five thousandths. 18. Fifteen thousandths. 19. Five thousandths. 20. Two hundred thousandths. 21. Twenty-nine thousandths. 22. Five hundredths. 23. Forty hundredths. 24. Nine tews o/" thousandths. 25. Nineteen tens of thousandths. 26. Nine hundred tens of thousandths. 27. Two thousand tens of thousandths. 28. Two thousand and two tens of thousandths. 29. Three thousand three hundred tews o/ thousandths. 30. Thirty-two hundred te»s o/* thousandths. 31. Six tens of thousandths. 32. Four hundreds of thousandths. 33. Fourteen hundreds of thousandths. 34. Four hundred hundreds of thousandths . 35. .Two thousand and six hundreds of thousandths. 36. Sixty-four thousand hundreds of thousandths. 37. Sixteen thousand and four hundreds of thousandth's. 38. Four thousand and nine hundreds of thousandths. 39. Six hundreds of thousandths. 40. Five thousand and four hundreds of thousandths. 41. Sixty-five thousand hundreds of thousandths. 42. Nine hundred and one hundreds of thousandths. 43. Twenty-nine hundred hundreds of thousandths. 44. Twelve tens of thousandths. 45. Fifteen hundredths. 46. Sixty-four thousandths. 47. Nine hundred and one tens of thousandths. 6 62 ARITHMETIC. SECOND PART. Decimals can be read in two different ways. Thus, 21 can be read, either as two tenths, and one hundredth ; or as twenty-one hundredths. This can best be illustrated, by the coin of the United States. Thus, 2 dimes, 1 cent, can be read, either as twenty-one cents, or as two dimes and one cent. Thus again, 1 dollar, 3 dimes, and 2 cents, can be called, either 132 cents ; or 13 dimes, 2 cents ; or 1 dol- lar, 3 dimes, and 2 cents. In like manner, decimals may be read in different ways. Thus, 234 can be read either as 234 thousandths ; or 2 tenths, 3 hundredths, and 4 thousandths ; or 23 hun- dredths, and 4 thousandths ; or 2 tenths, and 34 thou- sandths. Write two tenths. Write twenty hundredths. ,2 is how many hundredths ? Ans. There are ten times as many hundredths as there are tenths in a thing. Therefore ,2 is ten times as many hundredths, or 20. Is there any difference in the value of ,2 and ,20 ? What is the difference between them ? Ans. The ,20 has ten times more pieces, and each piece is ten times smaller than the ,2 ; but there is no difference in the value. ,3 is how many hundredths ? ,4 is how many hun- dredths ? ,30 is how many tenths ? ,40 is how many tenths ? Write two tenths, and four hundredths. In this sum how many hundredths ? Write thirty-four hundredths. In this sum how many tenths ? Write 2 tenths, hundredths, or twenty-six hundredths. Write 4 tenths, 9 hundredths, and read it both ways. Write 6 tenths, 7 hundredths, five thousandths, or six hundred and seventy-five thousandths. Write 6 tenths, 4 hundredths, and 5 thousandths. Write nine tenths, six hundredths, and six thousandths, and read them both ways. Write seven tenths, six hundredths, five thousandths, and nine tens of thousandths, and read them both ways. DECIMAL NUMERATION. 63 Write nine tenths, no hundredths, six thousandths, no tens of thousandths, and five hundreds of thousandths, and read it both ways. Write six tenths, no hundredths, no thousandths, and five tens of thousandths, and read it both ways. Write six thousand four hundred and thirty-six, tens of thousandths, and tell how many tenths, hundredths, and thousandths there are. Write four hundred and seventy. nine thousandths, and tell how many tenths, and hundredths there are. Write five hundred and six thousandths, and tell how many tenths there are. Write five hundred and ninety-six hundreds of thou, sandlhs, and read it both ways. From the above it appears, that in decimals, the order next to the separatrix is tenths ; the second order from the separatrix is hundredths ; the third order is thousandths ; the fourth order is tens of thousandths ; the fifth order is hundreds of thousandths, &c. Questions. — In decimals what is the first order, at the right of the separatrix ? What is the second order? What is the fourth order ? What is the third ? the fifth ? Decimals, are often written with whole numbers. Thus, 2,5. 36,349. Whole numbers and decimals together, are called mixed decimals. Write twenty-four whole numbers, and twenty-four hundredths. Two hundred whole numbers, and five tenths. What are the mixed decimals ? Rule for reading decimals. Read the numerator, as if it were whole numbers, and then add the name of the denominator ; or, Read the number of each separate order, and follow it with the name of the order in which it stands. Read the following decimals both ways. ,11. ,020. ,5005. ,32568. ,0505. ,521. ,43002. 24,690. 6,40043. 6,4000. 69,9604. 86,0092. 2,002. 16,00020. In writing decimals from the dictation of the teacher, 64 ARITHMETIC. SECOND PART. the pupil needs to understand the two methods very clearly. Thus for example, he may have this decimal, ,00205, dictated in two ways, viz. : 205 hundreds of thousandths, or 2 thousandths, and 5 hundreds of thousandths. In the first mode of dictation, he must write the 205 as if it were whole numbers, and then prefix ciphers to make the figures of the numerator equal to the ciphers of the denominator. In the second mode of dictation, he must put a cipher in each order which is not mentioned ; viz. : in the orders tenths, hundredths, and tens of thousandths, and a 2 in the order of thousandths, and a 5 in the order of hun~ dreds of thousandths. Let the pupil write the following in both methods of dic- tation. 8 hundredths, 6 tens of thousandths ; or 806 tens of thousandths. 2 tenths, 4 tens of thousandths ; or 2004 tens of. thou- sandths. 2 thousandths, 5 tens of thousandths ; or 25 tens of thou- sandths. 3 hundredths, 6 thousandths, 5 tens of thousandths ; or 365 tens of thousandths. RULE FOR WRITING DECIMALS. Write the numerator as if it were whole numbers, and tlten prefix a separatrix. If the figures of the decimal, do not equal in number the ciphers of the denominator, prefix ciphers to make them equal, before placing the separatrix ; or Write each order separately, placing ciphers in the orders omtted. Write the following : 1. Two hundred and ten thousandths. 2. Two tenths, five thousandths, six tens of thousandths. Here the order of hundredths is omitted, and has a cipher put in it. 3. Two hundred and four hundreds of thousandths. 4. Two thousandths ; four hundreds of thousandths. What orders are omitted ? 5. Sixteen tens of thousandths. DECIMAL NUMERATION. 65 6. One thousandth, six tens of thousandths. What or- ders are omitted ? 7. Four hundred and five thousandths. What orders omitted 1 8. Four tenths, five thousandths. What orders are omitted ? 9. Three hundred and sixty-five tens of thousandths. What order has a cipher placed in it ? 10. Four hundredths, five tens of thousandths. What orders are omitted ? 11. Twenty-six thousand, nine hundred and forty-six hundreds oj thousandths. 12. Two tenths, six hundredths, nine thousandths, four tens of thousandths, six hundreds of thousandths. In mixed decimals, it will be seen, that the orders are reckoned from the separatrix, both ways. Thus in 98423,40795, the first order at the right of the separatrix is tenths, and thefirst order at the left is units. What is the second order at the right, and the second or- der at the left of the separatrix ? What is the third order at the right, and at the left of the separatrix ? What is the fourth order at the right, and at the left of the separatrix ? What is the fifth order at the right, and at the left of the separatrix ? If you have the decimal ,2, and place a cypher at the right, thus ,20, what does it become ? Is the value alter- ed ? How is it altered 1 Ans. The parts are made ten times smaller, and there are ten times more of them, so that the value remains the same. If you place a cypher at the left of ,2 thus, ,02, what does it become ? How much smaller is a hundredth, than a tenth ? How much smaller does it make a decimal to prefix a cipher to it ? If you put ttco ciphers at the right of ,2, what effect is produced ? If you put them at the left of it, what effect is produced ? The following principle is exhibited above : Ciphers placed at the right of decimals, change their names but not their value. 6* 66 ARITHMETIC. SECOND PART. Ciphers placed at the left of decimals, diminish their vah tie ten times, for every cipher thus prefixed. Prefix a cipher to ,91 and read it. Annex a cipher to ,91 and read it. Prefix a cipher to ,20 and read it. Annex a cipher to ,20 and read it. Signs and Abbreviations used in Arithmetic. The following signs are used instead of the words they represent. -f- signifies plus or added to. — signifies minus or lessened by. X signifies multiplied by. -H- signifies divided by. = signifies equals. E. signifies Eagles. $ signifies Dollars. d. signifies Dimes. cts. signifies cents. m. signifies mills. ADDITION. Addition is uniting several numbers in one. There are four different processes of addition. The first is Simple Addition, in which ten units of one order make one unit of the next higher order. Thus, ten units make one of the order of tens — Ten tens make one of the order of hundreds — Ten hundreds, make one of the order of thousands, &c. The second is Decimal Addition, in which decimal frac- tions are added to each other. Thus, ,5 ,50 ,505 are ad- ded together. The third is Compound Addition, in which other num- bers besides ten, make units of higher orders. Thus, four units of the order of farthings, make one unit of the order of pence. Twelve units of the order of pence, make one of the shilling order. Twenty of the shilling SIMPLE ADDITION. 67 order, make one of the pound order, &c. The fourth is the Addition of Vulgar Fractions, in which parts of units are added to each other. Thus \ i and f are added to each other. SIMPLE ADDITION. If 8 units are added to 9 units, how many are there of the order of tens ? Write the 8 under the 9, and draw a line under. Place the units of the answer, under the figures added, and set the 1 ten before them. If 13 apples are added to 25 apples, how many are there in the whole ? Write the units under units, and tens under tens. Add the units first, and place the answer under the unit column. Then add the tens in the same way. Add 12 cents to 5 cents. Add 13 apples lo 14 apples. Add 14 dollars to 19 dollars. Add 5 and 2 and 12 together. Add 13 and 12 and 14~together. Let the pupil add small sums, which do not amount to ten of any order, till it can be done quickly and with a full understanding of the process. In the next process let the coins be used to illustrate. If 25 cents be added to 16 cents, how many cents are there ? Let 2 dimes be laid on the table, and 5 cents placed at the right of them. Under the 2 dimes place 1 dime, and under the 5 cents place G cents. Let the child then add the to the 5, and the answer will be 11 cents. Eleven cents are 1 dime and 1 cent. Let him leave 1 cent under the column of cents, and substitute 1 dime for the 10 cents. Let him place this dime with the 2 dimes, and his answer will be 3 dimes 1 cent. Ask how many cents in 3 dimes 1 cent, and the answer will be 31 cents. Thus his answer will be either 3 dimes 1 cent, or 31 cents. If the pupil thus sees the principle once illustrated, by 68 ARITHMETIC. SECOND PART. a visible process, the method will be much more readily understood and remembered. Let the following sum also, be done by the coins. Add $1,36 to $2,97. Add 2$. 6d. 8 cts. to 3$. 8d. 9 cts. Add 7 E. 2$. 5d. 6 cts. to 4 E. 8$. 6d. 4 cts. Add 5d. 6 cts. 7m. to 8d. 4 cts. 9m. Add 4 E. 0$. 6d. 5 cts. to 5 E. 0$. 4d. 6 cts. Let the teacher dictate such simple sums until the pro- cess of writing and adding is well understood, and can be done with rapidity and accuracy. Note to teachers. It is very desirable that pupils should, be required to write their figures with accuracy and neatness, and learn to place them in strait lines, both perpendicular and hori- zontal. Also that they learn to add by calculation, and not by counting, as young scholars are very apt to do. If a teacher will but be thorough, at the commencement, in these respects, much time and labor will be saved. Mary has 4 apples, James 5, and Henry 7, how many have all together 1 One boy has 6 marbles, another 4, and another 9, how many have all together ? A man gave 9 cents to one boy, 8 to another, and 11 to another, how many did he give to all 1 10 and 11 and 9 are how many? 12 and 7 and 4 are how many ? 4 and 5 and 7 are how many ? One man owns 6 horses, another 8, and another 9, how many have they all 1 In a school, 10 study history, 11 geography, and 15 grammar, how many scholars in the whole ? One house has 10 windows, another 7, and another 12, how many are there in all ? James lent one boy 8 cents, another 6, and another 17, how many did he lend them all ? If a lady pays 7 dollars for a veil, 9 dollars for a dress, and 3 dollars for a necklace, what amount does she spend ? 6 and 9 and 18 are how many 1 SIMPLE ADDITION. 69 1 and 5 and 7 are how many ? 8 and 11 and 14 are how many? Let the pupil be taught to add using the signs. the last sum. 8 + 11 + 14 = 33 Thus Rule for Simple Addition. Place units of the same order in the same column, and draw a line under-. Add each column separately, begim&ng at the right hand. Place the units of the amount, under the column tovMch they belong, and carry the tens to tlw next higher or- der. Add 2694 and 3259 and 6438. Placing units of the same order in the same column, they stand thus. 2694 3259 6438 12391 Let the pupil at first learn to add in this manner. 8 units added to 9, are 17, and 4 are 21 units, which is 1 of the unit order, to be written under that order, and 2 of the order of tens, to be carried to that order. 2 tens carried to 3 tens, are 5, and 5 are 10, and 9 are 19 tens ; which is 9 of the order of tens, to be written under that order, and 1 of the order of hundreds, to be carried to that order. Thus through all the orders. Add the following numbers. (1) 22321 (2) 23432 (3) 110331 (4) 222311 41332 42212 224212 131232 12123 13124 103123 101221 13220 21101 220320 234031 88996 99S69 657986 688795 (5) 275496 (6) 456789 (?) 369543 (8) 4976432 8732 654321 695432 4976432 70 ARITHMETIC FIRST PART. 54976 456789 567897 6325498 843215 654321 432591 5192346 7621 543219 526387 8763945 49673 345678 489549 763497 1239713 3111117 3081399 30998150 (9) (10) (11) (12) 30648 30430 764325 29367 46469 25895 70504 29367 74057 57644 98469 29367 63396 72919 57157 29367 55275 3647 46946 29367 90534 57246 3284 29367 8953 30142 363 247781 176202 1041 04ft 399474 Let the pupil now learn to place units of the same order in the same column, by the following examples. Let the teacher dictate the following. The pupils should be required previously to . attempt writing them, while studying their lesson. 1 One million, four hundred and sixty thousand, and two. Twenty- four million, six hundred and one. Three hundred and sixty thousand, four hundred and six. Ninety-four million, five hundred and seventy-eight thousand, three hundred and forty-one. Six million, seven thousand, and forty-three. 2 Two hundred and six thousand, five hundred and forty- two. One million, one thousand, and one. Nine hundred and ninety million, nine hundred and ninety-nine. Eighty-eight thousand, eight hundred and eighty-eight. Ninety-nine million, seven hundred and sixty-five thou- sand. SIMPLE ADDITION. 71 3. Two hundred and six million, five thousand, four hun- dred and one. Fifty-six million, four hundred thousand, five hundred and six. Three billion, ninety-nine thousand, and four. Five hundred million, thirty thousand, four hundred and forty. Seven million, six hundred and fifty-four thousand, three hundred and seventeen. 4. Four million, four hundred and thirty-two thousand, one hundred and seventy-six. Forty-nine thousand, and three. Nineteen million, seven hundred and sixty-five thou- sand, nine hundred and eighty-four. Five hundred and ninety-one. Seven hundred and sixty-three thousand, nine hundred and forty-three. Ninety-nine million, nine thousand and ninety. 5. Four hundred and four. Five million, six hundred and forty-three thousand, two hundred and seventeen. One million, and two. Nine thousand, and ninety-nine. Four million, five hundred and seventy-six thousand, three hundred and eighty-four. Forty-four million, three hundred and twenty-one thou- sand, seven hundred and four. 6. One hundred million, one thousand, and ten. Nine billion, eight hundred thousand, nine hundred and forty. Four hundred and eighty-eight million, nine hundred and five thousand. Eighty-eight million, seven hundred and seventy-seven thousand, and nine. Nine hundred and ninety-nine. 72 ARITHMETIC. SECOND PART. 7. Ninety-nine million, eight, thousand, and four. Five hundred and eighty-seven million, six hundred and forty-nine thousand. Twenty-eight thousand, eight hundred and ninety-nine. Four hundred thousand, eight hundred and seven. One billion, fifty-nine million, four thousand and eighty- seven. 8. Seven hundred million, ninety-nine thousand, and sev- enty-nine. Fifty-five thousand, seven hundred and forty-four. Nine million, eight hundred thousand, eight hundred. Eight thousand, eight hundred. Seven billion, and seventeen. 9. Eighty-four thousand, and nineteen. Nine million, fifty-four thousand, seven hundred. Seven hundred and sixty-eight thousand, eight hundred and four. Four billion, twenty million, ten thousand and fifty. Sixty million, two hundred thousand. Eleven hundred and forty-two. 10. Forty thousand, and twelve. Nine billion, eight thousand. Sixty million, seven hundred thousand, and ten. Nine billion, ninety million, eighty thousand, and sev- enty-eight. Sixty-five million, and four hundred. One billion, and four. 11. Nine hundred thousand. Four million, fifty-five thousand, and eighty. Three hundred and sixty-four thousand, seven hundred and thirty-eight. Forty million, four hundred and four. Six hundred and thirty thousand. 12. Ten million, four hundred. DECIMAL ADDITION. 73 Seventy-six thousand, three hundred and twenty-one. Eight million, forty-two thousand, six hundred and seventy-three. One thousand, four hundred. Sixty-four thousand, three hundred and twenty. One billion, and seventy-three. DECIMAL ADDITION. Rule for adding decimals. Place figures of the same order under each other. Add each column, as in Simple Addition, and in the answer place a separatrix between the orders of units and tenths. Example. What is the sum of 234,406. 4,6490. 13,234. 2,2. 3650,4002. 990,4699. Placing units of the same order under each other, they stand thus : — 234,406 4,6490 13,234 2,2 3650,4002 999,4699 4904,3591 Let the pupils proceed as in Simple Addition, calling the names of each order, thus : — 9 tens of thousandths added to 2, are 11 tens of thousandths ; which is 1 ten of thousandths, to be written under that order ; and 1 of the order of thousandths, to be carried to that order. 1 thousandth carried to 9, is 10, and 4 are 14, and 9 are 23, and 6 are 29 thousandths ; which is 9 thousandths, to be written under that order, and 2 hundredths, to be carried to the next order. Thus through the other orders, observing to place a separatrix between the orders of units and tenths. Arrange the following mixed decimals according to their orders, and then add them. 7 74 ARITHMETIC. SECOND PART. (1) 306,42001. 20,3391. 3246,42. .39,4695. 634,001. 84,6302, (2) 99,987. 65432,02564. 64,65. 596,32. 87632,- 51739. 36,50. 51639,2154. 63,204. 6359,42591. 8642,39. 86423,2915. 68,241. (4) 63,9876. 59432,1103. 95,02. 876,3254. 8634,251. 3426,549. Let the pupil write and add the following sums in De- cimals. 1. Four units, six tenths, four hundredths, five thou- sandths. Two tens, four units, six hundredths. Three tens, two units, two hundredths, seven thou- sandths. Six units, five tenths, seven hundredths, four thou- sandths, three tens of thousandths. One unit, three tenths. 2. Forty-two units, sixteen thousandths. Five units, sixty-three hundreds of thousandths. Seventy. four units, seven thousand five hundred and fifty-three tens of thousandths. Two units, five hundred and sixty tens of thousandths. 3. Two hundred and forty-three units, two hundred and forty-three thousandths, seventeen units, nine hundred and seventy-three tens of thousandths. Fifty units, six thousand seven hundred and forty-three hundreds of thousandths. Five units, eight thousandths. One thousand units ; one thousand tens of thousandths. 4. One thousand and one units ; one thousand and one hundreds of thousandths. DECIMAL ADDITION. 75 Nine hundred and ninety-nine units, nine thousand nine hundred and thirty hundreds of thousandths. Four units, thirty tens of thousandths. Five units, fifty-five thousand and forty-three millionths. 5. Sixteen units, seven hundred and sixty-four thousandths. Two units, forty-five hundreds of thousandths. Fifty units, forty-two millionths. Seven units, nine hundred and ninety -eight tens of thou- sands. Six units, five hundred and forty-nine millionths. 6. Four thousand units, four thousand thousandths. Forty-one units, four thousand, four hundred and nine hundreds of thousandths. Seven units, eighty-seven tens of thousandths. Four hundred and forty-one units, ninety-nine hun- dredths. Four units, four hundreds of thousandths. 7. Seventeen units, nine thousand eight hundred and sixty hundreds of thousandths. Nine units, sixteen tens of thousandths. Four units, fifty-five hundredths. Sixty-three units, ninety-nine millionths. One unit, seventy-four thousandths. 8. Five hundred and forty-four units, eight thousand seven hundred and fifty-five millionths. Ninety-nine units, four hundred hundreds of thou, sandths. Six units, eight hundred and eighty-eight thousandths. Eight thousand units, seventy -four tens of thousandths. Six units, eighty-eight hundredths. 9. Seventeen units, forty thousandths. Five units, ninety-three millionths. Forty -four units, eighty-seven hundredths. Six units, nine hundred and ninety-nine thousandths. Four hundred and twelve units, seventy-five tens of thousandths. 76 ARITHMETIC SECOND PART. 10. Seventy-eight units, four thousand and five tens of thou- sandths. Two units, five hundred hundreds of thousandths. Seven units, eighty-nine millionths. Five hundred and seventy-two units, seventy-six thou- sand, eight hundred and sixty-four hundreds of thou- sandths. Nine thousand and fifty units, nine thousand and fifty millionths. 11. Five hundred and eighty-seven units, twenty-nine hun- dred tens of thousandths. Forty units, five hundred and sixteen millionths. Eight units, four hundred and ninety-six thousand mill- ionths. Five hundred and forty-two units, two thousand hun- dreds of thousandths. Seventeen units, nine thousand nine hundred hundreds of thousandths. 12. Sixty-five units, sixty-five hundreds of thousandths. One hundred and eighty units, one hundred and eighty tens of thousandths. Twenty-four units, twenty-four millionths. Sixteen units, sixteen hundredths. Five units, five thousandths. Fifty units, fifty hundreds of thousandths. 13. One hundred and seventy-six units, one hundred and seventy-six hundreds of thousandths. Four units, two thousand four hundred and seventy-five tens of thousandths. Eighty-four units, seven hundred and sixty-three mil- lionths. Two hundred units, two thousand and forty tens of thousandths. Seventeen units, four thousand and four millionths. 14. Seventy-four units, nine hundred and eighty millionths. Four units, four hundreds of thousandths. DECIMAL ADDITION. 77 Eighty-one units, nine thousand four hundred hundreds y units of the next higher order are contained, in the svu, and carry them to that order. Set the remainder under th'' order added. EXAMPLE. £. *. d. 5 „ 6 „ 8 4 „ 9 „ 9 9 „ 9 „ 5 19„5„10 Let the pupil add thus : 5 pence added to 9 are 14, and 8 are 22 pence. This sum contains 1 of the order of shil- ling *o he carried to that order, and 10 to bo written un- der ,j 2 „ 55 * 5> 17 36 Solid Measure. tons. 116 „ 109 „ ft. 24 39 cords 72 41 . ft. „ 114 „ 120 Time. yrs. 54 43 mo. „ 11 » 11 we. „ 3 „ 3 20 „ min. 41 „ 49 „ sec. 20 19 Circular Motion. 9 „ 23 3 „ 7 ! „ 45 „40 o 29 19 „ 34 „ „ 40 „ it 54 36 SUBTRACTION OF VULGAR FRACTIONS. If a boy has 6 ninths of an apple, and gives away 4 ninths, how much remains ? If he has 8 ninths, and gives away 5 ninths, what re. mains ? 94 ARITHMETIC. SECOND PART. If he has 7 twelfths, and gives away 4 twelfths, what remains ? In doing those sums let the pupil tell first which is the minuend and which the subtrahend. A man has 9 twentieths of a dollar and loses 5 twenti- eths, how much remains ? If he has 11 twentieths and loses 7 twentieths, what re- mains ? If he has 8 sixteenths, and loses 5 sixteenths, what re- mains 1 Subtract T 3 5 from r \. Subtract -^ from if. Rule for Subtracting Vulgar Fractions. Subtract the numerator of the subtrahend, from the nu- merator of the minuend, and place the remainder over the common denominator. Let the pupil in doing the sums, use the signs in this way. Subtract f of a dollar from f . Alia. 8 8 — 8 . Subtract ^ from ¥ \. Subtract f f from f f . (( _6_4_ It ?LO.L t< 2.10. 5 U We invert a fraction when we exchange the places of the numerator and the denominator. Thus \ inverted is \, and | inverted is f and 4g in- verted is f £ &c. Now it appears, as above, that if we wish to divide 3 by | we are to multiply its numerator (3) by the denomi- nator (6) and its denominator (4) by the numerator (2). This is more easily done, if we invert the divisor f , thus #. When the divisor is thus inverted we can multip/y the numerators together for a new numerator and the denomi- nators for a new denominator and the process is the same. Thus let us divide a by |. Inverting the divisor § the two fractions would stand to- gether thus £ f . We now multiply the numerators and denominators together and the answer is }| and it is the same process, as if we had not inverted the divisor, but multiplied the numerator of the dividend by the denomi- nator of the divisor and its denominator by the numerator of the divisor. This method therefore is given as the easiest rule, but it must be remembered that in this process we always mul- tiply the dividend by the denominator of the divisor and di- vide it by the numerator, as we do in case of whole numbers. 144 ARITHMETIC. SECOND PART. Common rule for dividing one fraction by another. Invert the Divisor, and then multiply the numerators and denominators together. Examples for the slate. Divide |f by j\. Invert the divisor and the fractions stand thus ££ y . Multiply them together, and the answer is £i|. Divide f| by 44 I Divide || by |f ii 32. (( 5.6. it 'J.4. (( 1.4 4 9 3 9 1 1 1 9 1 T «( 3.8 << .32 I (I _5 6 «< _9_3_ 56 2 1 I 412 506 il 6_5_ it 3 4.X I : ' ! 6. " 9_3_ 138 302 I 49 102 DECIMAL DIVISION. In order to understand the process of Decimal Division, it is needful to recollect the method of dividing and multi- plying, by ciphers and a separatrix. If we wish to multiply a number by a sum composed of 1 with ciphers added to it, we add as many ciphers to the multiplicand, as there are ciphers in the multiplier. Thus if we wish to multiply 61 by 10, we do it by adding one cipher, 040. If we are to multiply by 100, we add two ciphers thus, 6400, &c Examples. Multiply 3 by 100 Multiply 46 by 100 « 19 " 1000 " 2 " 100000 If we wish to multiply a decimal by any number com- posed of 1 with ciphers annexed, we can do it by removing the separatrix as many orders to the right, as there are ci- phers in the multiplier. Thus if ,2694 is to be multiplied by 10, we do it thus ; 2,694. If it is to be multiplied by 100, we do it thus ; DECIMAL DIVISION. 145 26,94. If it is to be multiplied by 1000 we do it thus ; 269,4. But to multiply by a million, we must add ci- phers also, in order to be able to move the separatrix as far as required, thus ; 209400,. Examples. Multiply 2,64 « 36,9468 3,2 by 10 100 " 1000 Multiply 6,4 1,643 3,2 by 10000 « 10 « 1000000 The same method can be employed in dividing deci- mals, by any number composed of 1 and ciphers an- nexed. The rule is this. Remove the separatrix as many or- ders to the left, as there are ciphers in the divisor. Thus if we wish to divide 23,4 by 10 we do it thus ; 2,34. If we wish to divide it by 100 we do it thus, 234. But if we wish to divide it by a thousand it is necessary to pre- fix a cipher thus ,0234. If we divide it by 10,000 we do it thus ,00234. Examples. Divide 2,4 2,46 3,2 « 2,4 Multiply 2,4 Divide 328,94 Mult'y. 326,94 by (< 100 10 1000 10 10 100 100 Divide 24,3 246,9 " 2,3 34,26 Mult'y. 34,26 Divide 3,2 Multiply 3,2 by « 10 100 100000 1000 1000 10000 10000 It is needful to understand that a mixed decimal, can be changed to an improper decimal fraction. For example, if we change 3,20 to an improper decimal fraction, it becomes 320 hundredths (f§£), which is an improper fraction, because its numerator is larger than the denominator. But we cannot express the denominator of 320 hun- dredths, by a separatrix in the usual manner, for the rule requires the separatrix to stand, so that there will be as 13 I 146 ARITHMETIC. SECOND PART. many figures at the right of it, as there are ciphers in the denominator. If then we attempt to write 320 hundredths in this way, it will stand thus 3,20, which is then a mixed decimal, and must be read three units and 20 hundredths. If it is writ- ten thus, 2-f-j}, it is then a vulgar and not a decimal frac- tion. But it is convenient in explaining several processes in fractions, to have a method for expressing improper deci- mal jr actions, without writing their denominator. The fol- lowing method therefore will be used. Let the inverted separatrix be used to express an im- proper decimal fraction. Thus let the mixed decimal 2,4 which is read two and four tenths, be changed to an im- proper decimal thus, 2'4 which may be read twenty-Jour tenths . The denominator of an improper decimal, (like that of other decimals) is always 1 and as many ciphers as there are figures at the right of the separatrix. It is known to be an improper decimal, simply by having its separatrix inverted. Thus 24'69 is read, two thousand four hundred and six- ty. nine hundredths. 239'6 is read, two thousand three hundred and ninety-six tenths, &c. Examples. Change the following mixed decimals to improper deci- mals, and read them. 246,3 24,96 32,1 326,842 3,6496 49,2643 8,4692 368,491 26,3496 Rule for writing an Improper Decimal. Write as if the numerator were whole numbers, and place an inverted separatrix, so that there will be as many figures at the right, as there are ciphers in the denominator. Write the following improper decimals. Three hundred and six tenths. Four thousand and nine hundredths. DECIMAL DIVISION. 147 Two hundred and forty-six thousand, four hundred and six tenths. Three millions, five hundred and forty-nine tens of thou- sandths. Two hundred and sixty-four thousand, five hundred and six thousandths. Five hundred and ninety. six teiiths. Decimal Division when the Divisor is a whole number. The rules for Decimal Division are constructed upon this principle, that any quotient figure must always be put ^ in the same order as the lowest order of that part of the dividend taken. Thus if we divide ,25 (or two tenths", five hundredths,) by 5, the quotient figure must be put in the hundredth or- der, thus, (,05) because the lowest order of the dividend is hundredths. Again, if ,250 is divided by 50, the quotient figure must be 5 thousandths, (,005) for the same reason. Let us then divide ,«56 by 2. We proceed exactly as in the Short Division of whole numbers, except in the use of a separatrix. Let the pupil proceed thus : 2),256 ,128 2 tenths d'vided by 2, gives 1 as quotient, which is 1 tenth, and is set under that order with a separatrix before it. 5 hundredths divided by 2, gives 2 as quotient, which is 2 hundredths, and is set under that order. 1 hundredth remains, which is changed to thousandths, and added to the 6, making 10 thousandths. This, divided by 2, gives 8 thousandths as quotient, which is placed in that order. If the divisor is a whole number, and has several or- ders in it, we proceed as in Long Division, except we use a separatrix, to keep the figures in their proper order. Thus if we divide 15,12 by 36, we proceed thus : • 148 ARITHMETIC. SECOND PART 36)15,12(,42 14,4 ,72 ,72 ,00 We first take the 15,1 and divide it, remembeiing that the quotient figure is to be of the same order as the lowest order in the part of the dividend taken, of course the quo- tient 4 is 4 tenths (,4) and must be written thus in the quotient. We now subtract 30 times ,4 which is 14,4. (See rule for Decimal Multiplication page 108) from the part of the dividend taken and 7 tenths (.7) remain. To this biing down the 2 hundredths. Divide, and the quotient figure is 2 hundredths which must be set in that order in the quotient. Subtract 36 times ,02 (or ,72) from the dividend and nothing remains. Let the following sums be performed and explained as above. Divide 76,8 by 24 Divide 37,8 by 21 « 94,6 « 43 «( 85,8 " 20 Sometimes ciphers must be prefixed to the first quotient figure, to make it stand in its proper order. For example, let ,1512 be divided by 36, and we pro- ceed thus, 36),1512(,0042 ,144 ,0072 ,0072 0000 I . DECIMAL DIVISION. 149 We take ,151 first, which is 151 thousandths (for the denominator of any decimal is always of the same order as the lowest order taken). This divided by 36 gives 4 as quotient. This 4 is 4 thousandths, because the lowest order in the part of the dividend taken is thousandths. Therefore when it is put in the quotient it must have two ciphers and a separatrix prefixed thus ,004. We now subtract from the dividend 36 times, ,004 or ,144. (See rule for Decimal Multiplication.) It is desirable in such cases to place ciphers and a sep- aratrix in the remainders, to make them stand in their proper orders. To the remainder (,007) bring down the 2 tens of thousandths making 72 tens of thousandths. This divided by 36 gives 2 tens of thousandths as quo- tient which is set in that order. 36 times 2 tens of thou, sandths (or ,0072) being subtracted, nothing remains. Sometimes we must add ciphers to the dividend before we can begin to divide. For example, let ,369 be divided by 469, and we pro- ceed thus, 460),3690(,00078 ,3283 ,04070 ,03752 ,00318 z±% I We find that ,369 cannot be divided by 469, so we add a cipher to it, making it 3690 tens of thousandths. This divided by 469 gives 7 as quotient, which is 7 tens of thousandths, (,0007) because the lowest order of the dividend is of that order. We now subtract 469 times ,0007 (which is ,3283) from the dividend, and ,0407 remain. 13* 150 ARITHMETIC. SECOND PART. To this remainder we add a cipher, and change it from 407 tens of thousandths to 4070 hundreds of thousandths. This divided hy 469 gives 8 as quotient, which is 8 hun- dreds of thousandths, hecause the lowest order in the div- idend is hundreds of thousandths. We now subtract 469 times 8 hundreds of thousandths (or ,03752) from the dividend and ,00318 remain. We could continue dividing, by adding ciphers to the remainders, but it is needless. Instead of this we can set the divisor under the remainder as in common division, thus -3J.1 It is not needful to retain the separatrix and ciphers when thus writing a remainder, because when put in the quotient, it is not considered as the |£| part oi' a whole number, but as a part of the lowest order in the decimal, by which it is placed. Thus when this is put with the above quotient, we read the answer thus 78 hvndreds of thousandths, and |£| of another hundred of thousandth. Let the following sums be performed and explained as above. Divide 42869 by 95 " 3,69428 « 49 « ,269 " 482 « 481,4 " 81 28,1 " 15 Decimal Division when the Divisor is a Decimal. When the divisor is a decimal, we proceed as in divi- ding by a Vulgar Fraction, viz. We multiply by the denominator, and divide by the nu- merator. Thus if we are to divide 24 by ,4, we are to find how ma- ny 4 tenths there are in 24. We first multiply 24 by the denominator 10, to find how many one tenths there are, and then divide by the numer- ator 4, to find how many 4 tenths there are. 24 is multi- plied by ten, thus ; 24'0, and has the inverted separatrix, to show that it is not 240 whole numbers, but tenths. We now have found that in 24 there are 240 one tenths, Divide 3,694 by 84 cc ,36946 « 841 (< 3,26 " 589 (< 32,4 " 386 a 364,6 " 99 * DECIMAL DIVISION. 151 we now divide by 4, to find how many 4 tenths there are. The answer is 60, which according to the rule, must be of the same order as the lowest order in the dividend, or 60 tenths, and must be shown by the inverted separatrix thus (0'0.) This may be changed to whole numbers by revert- ing the separatrix thus (6,0.) When the dividend is a decimal, we can multiply by re- moving the separatrix. Thus let 8,64 be divided by ,36. Here we are to multiply by 100, to find how many one hundredths there are in the dividend, and then divide by 36 to find how many 36 hundredths there are. We multiply by 100, by removing the separatrix two orders toward the right, and then dividing by 36, we have "24 as answer, which is 24 units, because the dividend is units, as appears below. 36)864,(24 72 000 If the divisor is a mixed decimal, we change it to an im- proper decimal, and then proceed as before, multiplying by the denominator and divide by the numerator. Thus let 10,58 be divided by 4,6. We first change the divisor into an improper decimal thus, 4'G (46 tenths.) We now are to multiply the 10,58 by 10, to find how many one tenths there are, and then divide by 46, to find how many 46 tenths there are. We multiply by 10 by removing the separatrix thus, 105,8, and proceed as follows. 152 ARITHMETIC. SECOND PART. 46)105,8(2,3 92 13,8 13,8 000 Here we divide 105 units by 46, and the quotient fig- ure is 2 units. We then subtract 46 times 2 units from the dividend, and 13 units remain. To this bring down the 8 tenths. This is divided as if whole numbers, but the quotient 3 is 3 tenths, because the lowest order in the dividend is tenths. It is set in the quotient with the separatrix before it, and then 46 times ,3 (or 13,8) is taken from the dividend, and nothing remains. Let the following sums be performed, and explained as above. Divide a The following then is the rule for Decimal Division. Rule for Decimal Division. If the divisor is a ivhole number, divide as in common di' vision, placing each quotient figure in the same order as the lowest order of the dividend taken. If the divisor is a decimal, multiply by the denominator, and divide by the numerator, placing each quotient figure in the same order as the lowest order of the dividend taken. If the divisor is a mixed decimal, change it to an improper decimal, and then proceed to midtiply by the denominator and divide by the numerator. N. B . The ride for multiplying and dividing Federal Money, is the same as for Decimals. 46,4 by 3,6 Divide 891,6 by ,2 ,431 2,41 (< 8,964 " 8,6 4,56 " 3,64 CI 89,96 " 4,861 464,92 « 3,2649 re 8,641 » ,4169 DECIMAL DIVISION. 153 Examples. How many times is $2,04 contained in $9,40 ? Divide $2,04 by $,84 02 « 8,41 " 2,41 « 19,24 324,07 " 64,81 20,46 " ,49 As it is found to be invariably the case that the decimal orders in the divisor and quotient always equal those of the dividend, the common rule for decimal division, is formed on that principle, and may now be used. Common Rule for Decimal Division. Divide as in whole numbers. Point off in the quotient enough decimals to make the decimal orders of the divisor and quotient together equal to those of the dividend, counting every cipher annexed to the dividend, or to any remainder, as a decimal order of the dividend. If there are not enough figures in the quotient prefix ciphers. In pointing off by the above rule, let the teacher ask these questions. How many decimals in the dividend ? How many in the divisor ? How many must be pointed off in the quo- tient, to make as many in the divisor and quotient, as there are in the dividend ? Examples. At $,75 per bushel, how many bushels of oats can be bought for $14,23/ How much butter at 16 cents a pound, can be bought for $20? A half cent can be written thus, $,005 (for 5 mills is half a cent, or 5 thousandths of a dollar.) A quarter of a. cent can be written thus, $,0025 (for { of a cent is 25 tens of thousandths of a dollar.) At 12£ cents per hour, in how much time will a man earn #46" ? 154 ARITHMETIC SECOND PART. At 6^ cents per pint, how much molasses may be bought for $2 ? At $,06 an ounce, how much camphor can be bought for $3 ? At$,l2i a bushel, how much coal could be bought for $5? > V* Divide ,032 by ,005. Exercises in Decimal Multiplication and Division. Multiply ,25 by ,003. Divide ,25 by ,003. Multiply 3,4 by 2,68. Divide 3,4 by 2,68. Multiply ,005 by ,005. Divide ,004 by 16,4. If you buy 24 bushels of coal, at $,09 per bushel, what does the whole cost? If a man's wages be fifty hundredths of a dollar a day, what will it be a month ? What will be the cost of 25 thousandths of a cord of wood, at $2 a cord 1 What will be the cost of twelve hundredths of a ton of hay, at $11 a ton? If a man pays a tax of two mills on a dollar, how much must he pay if he is worth $350 ? If a man pays $,06 a year for the use of each dollar he borrows of his neighbor, how much must he pay in a year if he borrows 264 dollars ? How much in two years ? REDUCTION. Reduction is changing units of one order, to units of another order. Reduction Ascending, is changing units of a lower to a higher order. Reduction Descending, is changing units of a higher to a lower order. Examples for Mental Exercise. In 4 gallons how many quarts ? Note. Let each sum be stated thus. One gallon con. REDUCTION. 155 tains four quarts, and four gallons four times as much. 4 times 4 is 16. In 4 gallons how many pints ? In 8 yds. 3 qrs. how many quarters ? In 8 feet how many inches ? In 4 bushels how many quarts ? In 5 hours how many minutes ? Are the above sums in Reduction Ascending or De- scending ? In 32 quarts how many gallons ? Let such sums be stated thus. One gallon contains 4 quarts. In 32 quarts therefore, there are as many gal. Ions as there are 4's in 32. In 42 pints how many gallons ? In 49 quarters how many yards ? In 50 nails, how many quarters and how many yards ? In 64 inches how many feet ? In 36 barley corns how many inches 1 In 96 quarts how many bushels ? In 120 minutes how many hours ? In 48 feet how many yards 1 In 94 inches how many feet ? In 3 yards how many inches 1 In 4 gallons how many pints ? In 32 quarts how many gallons ? In 80 penny weights how many ounces ? In 24 ounces how many penny weights I In 8 pounds how many shillings ? In 40 shillings how many pence ? In £2, 9s. 6d. 3 qrs. how many farthings ? In doing this sum we proceed in the following manner : £. s. d. qr. 2 « 9 '« 6 " 3 20 49 shillings. 594 pence. 4 2379 farthings. 156 ARITHMETIC. SECOND PART. We first change the pounds to shillings, by multiplying by 20, and add the 9 shillings to them, making 49 shil- lings. We then change the 49 shillings to pence, by multiply- ing by 12, and add the 6 pence to them, making 594 pence. We then change the 594 pence to farthings, by multi- plying by 4, and add the 3 qrs. and thus we obtain the an- swer 2379 qrs This is Reduction Descending, because we have chan- ged units of a higher order to those of a lower. Why did we multiply by 20, 12, and 41 Let us now reverse the process, and change 2379 far- things to pounds. We proceed thus : £. s. d. qr. 4)2379(2 "9 "6 " 3. 12)594 20)49 2 We first change the 2379 farthings to pence, by divi- ding by 4, and the answer is 594 pence, and 3 farthings (or qr.) over, which is put in the quotient with qr. over it. We then change the 594 pence to shillings, by dividing by 12, and the answer is 49 shillings, and six pence over, which is put in the quotient with d. written over. We next change the 49 shillings to pounds, by dividing by 20, and find there is £2 and 9s. over, which are both put in the quotient with their signs written over them. Why did we divide by 4, 12, and 20 1 Let the following sums be performed and explained in the same way. Change 2486 farthings to pounds. Change £2 18s. 4d. 2qr. to farthings. Change 241 shillings to pounds. Change 249 pence to shillings and pounds. Change £21 2s. to farthings. t Change 361 pounds to pence. Change 35 shillings to pounds. REDUCTION. 157 Rule for Reduction. To reduce from a higher to a lower order. Multiply the highest order by the number required of tin next lower order, to make a unit of this order. . Add the next lower order to this product, and multiply it by the num. ber required of the next lower order, to make a unit of this order, adding as before. Thus through all the orders. To reduce from a lower to a higher order. Divide the amount given, by the number required to make, a unit of the next higher order. Divide the answer in the same way, and continue thus till the answer is in units of the order demanded. The remainders are of the same order as the dividend, and are to be put as a part of the answer. Exercises. Bought a tankard of silver weighing 5 lb. 3 oz. for which I paid $1,12 an oz. how much did it cost ? Reduce 2 lb. 8 oz. 11 pwt. to grains. In 81b. 93.43. 29. 16grs. how many grains? In 11924 grains how many pounds ? What cost 4 cwt. 3 qrs. 17 lb. of sugar, at 12a cents per lb? In 436 boxes of raisins, each containing 24 lbs. how many cwt. ? In 63469542 drams, how many tons ? In 546 yards how many nails ? In 5486 nails how many yards 1 In 118J yards, how many Ells Flemish ? How many barley corns will reach round the globe, it being 360 degrees ? How many miles in 836954621 barley corns ? In 18 acres, 3 roods, 12 rods, how many square feet 1 How many square feet in 16 square miles ? In 9269546231 square feet how many square miles ? In 37 cords of wood how many solid feet ? In 20486 solid feet how many cords ? In 4 pipes of wine how many pints ? In 9120854 pints how many pipes ? 14 158 ARITHMETIC. SECOND PART. In 464 bushels how many quarts ? In 964693 pints how many bushels ? REDUCTION OF FRACTIONS TO WHOLE NUMBERS. 1. In ten fifths, how many units ? 2. In fourteen sevenths, how many units ? 3. Change fifteen fifths to units. 4. Change thirteen fourths to units, and what is the an- swer ? 5. Change eighteen fourths to units, and what is the an- swer? 6. Change fourteen sixths to units. It will be perceived, that in answering these questions, the pupil divides the numerator by the denominator. Thus in changing twelve fourths to units, the numerator twelve, is divided by the denominator four. The above sums are to be performed mentally first, and the answers given, and then they are to be written, thus, 7. Change fourteen sixths to units. Ans. L- 4 = 14 -r- 6 = 2 § Let the pupil be required to perform all the above sums, in this manner. Rule for Reducing Fractions to Whole Numbers. Divide the numerator by the denominator ; write the re- mainder, if there be any,- over the denominator, and annex the fraction, thus formed, to the quotient. Examples. 1. Reduce 3 T 9 to a whole or mixed number. Ans. 9|. 2. Reduce V- Ans - 9 I- ¥• Ans - H- V- Ans - 151. y. Ans. 2f. REDUCTION OF FRACTIONS. 159 3. Reduce 'j 8 . Ans. 52§. ays. Ans. 565. 21 f3s. Ans. 2425. 4. Reduce 8, £*'. 62 g 7 6. 5 18_4 32. 9 15_8 7 3. 132|9 6S # 5. Reduce 08765.4321. 700070007, 600344002. 6. Reduce vii23_45499 49563J0217 33322.UU6 9 • 1 ' 6 * 59248 32 1768 REDUCTION OF WHOLE NUMBERS TO FRACTIONS. 1. In three units, how many fourths, and how is the answer expressed in figures ? 2. How many fifths is three units and two fifths, and how is the answer written ? 3. Reduce nine units to sixths. 4. Reduce seven units and two twelfths to twelfths. Rule for reducing Whole Numbers to Fractions. Multiply the whole number, by the denominator of the fraction to which it is to be reduced, and place the product over this denominator. If there is with the units, a fraction of the same denominator, add the numerator of this fraction to the product, before placing it over the denominator. Examples. 1. How many 4ths. in 1 ? How many in U f In la t In If ? ^ 4 4 • 2. How many 5ths. in 1 ? In 5? In 14 ? In 1^ 1 In 74? 55 3. How many 7ths. in 7 ? In 8 ? In 12 ? In 73 t In 4. How many 12ths. in 9 A ? In 7 T % ? In 3 A 1 In 5 - 5 - ' In 8 ' 1 18* 12" 5. How many 6ths. in 3 ? In 4 ? In 5 # ? In 7 « ? In 8? In 9 i? In 12? 160 ARITHMETIC. SECOND PART. 6. How many 27ths. in 3 ? In 2 ? In 5 ft ? Ans. |i. 5 4 I_4_4 27* 2 7. How many 19ths. in 15 ? In 13 T \ 1 In 17 }f ? Ans 2 - 8 - 5 . 2 - s -° 3-Aj REDUCTION OF VULGAR TO DECIMAL FRACTIONS. Decimal Fractions are generally used in preference to Vulgar, because it is so easy to multiply and divide by their denominators. Vulgar fractions can be changed to Decimals by a pro- cess which will now be explained. In this process, the numerator is to be considered as units divided by the denominator. Thus £ is 3 units divided by 4, for | is a fourth of 3 units. We can change these 3 units to an improper decimal ihus, 3'0 (30 tenths), and then divide by 4 ; remembering that the quotient is of the same order as the dividend. 4)3'0(,75 2'8 ,20 ,20 Thus the 30 tenths are divided by 4, and the answer is 7 tenths, which is placed in the quotient, with a separatrix prefixed. 4 times 7 tenths (or 28 tenths) are then sub- tracted, and the remainder is ,2. This in order to divide it by 4, must have a cipher annexed, making it 20 hund- redths. The quotient of this is 5 hundredths, and no re- mainder. (In performing this process, particular care must be taken in using the separatrix, both for proper and improper decimals.) Let | be reduced in the same way. The two units are first changed to an improper decimal thus : REDUCTION OF FRACTIONS. 161 8)2<0(,25 1<6 ,40 ,40 00 We proceed thus. 20 tenths divided by 8, is 2 tenths, which is placed in the quotient. 8 times ,2, or 16 tenths (1*6) is then subtracted, and ,4 remain. This is changed to 40 hundredths (,40) by adding a cipher, and then divided by 8. The quotient is 5 hund. redths, which is put in the quotient and there is no re- mainder. Note. After 3 or 4 figures are put in the quotient, if there still continues to be a remainder, it is not needful to continue the division, but merely to put the sign of addition in the quotient to show that more figures might be added. Examples. Reduce T 3 ^ to a decimal, and explain as above. Reduce $~ f $ T 2 T T 8 3 4 f £ eacn t0 a decimal of the same value. Let the pupil be required to explain sums of this kind as directed above, until perfectly familiar with the principle. When fractions of dollars and cents are expressed, their decimal value is found by the same process. For example, change \ a dollar to a decimal. Here the 1 of the numerator, is one dollar, divided by 2. By adding a cipher to this 1 and using the inverted separatrix, the dollar is changed to 10 dimes, and when this is divided by 2, the answer is 5 ; which being of the same order as the dividend is 5 dimes. The answer is to be written with the sign of the dollar before it, thus $0,5. The only difference between the answer when \ is re- duced to a decimal, and when \ a dollar is reduced to a decimal, is simply the use of the sign of a dollar ($) and a cipher in the dollar order. 162 ARITHMETIC. SECOND PART. 1. Reduce \ to a decimal. Ans. ,5. 2. Reduce | a dollar to a decimal. Ans. $0,5. 3. Change \ of a dollar to a decimal. Ans. $0,125. 4. Change t l of a dollar to a decimal. Ans. $0,0625. In this last sum there must be two ciphers added to the numerator, changing the 1 dollar to cents, instead of dimes ; and in this case a cipher is put in the order of dimes, and the quotient (being of the same order as the dividend) is placed in the order of cents. 5. Reduce \ of a dollar to a decimal. Ans. $0,2. 6. Reduce f of a dollar to a decimal. Ans. $0,625. 7. Re-duce T 3 g of a dollar to a decimal. Ans. $0,1871. 8. Reduce J^ to the decimal of a dollar. Ans. $0,01. Rule fok the reduction of Vulgar to Decimal fractions. Change the numerator to an improper decimal, by annex- ing ciphers and using an inverted separatrix. Divide by the denominator, placing each quotient figure in the same or- der as the lowest order of the part divided. 1. Reduce ^i^ to a decimal. Ans. .0016. 2. Reduce 5 | u to a decimal. Ans. .028. 3. Reduce z f 7 to a decimal. Ans. .05625. 4. Reduce i to a decimal. Ans. .3333333-f Note. We see here, that we may go on forever, and the decimal will continue to repeat 33, &c. therefore, the sign of addition + in such cases may be added, as soon as it is found that the same number continues to recur in the quotient. REDUCTION OF FRACTIONS TO A COMMON DENOMINATOR. Before explaining this process, it must be remembered that | | | | &c. or a fraction which has the numerator and denominator alike, is the same as a unit. If therefore we take a fourth of £ it is the same as taking a fourth of B REDUCTION OF FRACTIONS. 163 one. If we take a sixth of £ it is the same as taking a sixth of one. If we take § of £ it is the same as taking f of one. Whenever therefore we wish to change one fraction to another, without altering its value, we suppose a unit to be changed to a fractional form, and then take such apart of it, as is expressed by the fraction to be changed. For example, if we wish to change \ to twelfths, we change a unitto twelfths and then take i of it, and we have i of }f, which is the same as ± of one. If we wish to change \ to eightJis, we change a unit to | and then take i of it, for \ of f is the same as i of one. Change | to twelfths, thus, a unit is }§. One third of is. is T V Two thirds is twice as much, or ^, Then I are T \. Change j to twentieths. A unit is §£. One fifth of |§ is 5 4 ¥ . Four fifths is four times as much, or ^. Change the following fractions and state the process in the same way. Change f to twenty fourths. Change £ to twelfths. Reduce § to twenty sevenths. Reduce f to sixty fourths. Reduce | to twenty fifths. Reduce £ to twenty sevenths. Reduce f- to thirty sixths. Reduce £ to forty ninths. Reduce j\ to thirty sixths. Reduce £ to sixteenths. Reduce fe to fortieths. Reduce }£ to thirty thirds. Reduce # to thirty sixths. Reduce f and f each to twelfths. Reduce | and T \ each to twentieths. Reduce A f and £ each to twelfths. Reduce a T 2 g and ^ each to fortieths. Reduce i § t \ and ¥ 8 T each to sixty fourths. Reduce f £ T 8 T5- and if- each to forty eighths. In the above examples it is seen that when several frac- tions are to he reduced to a common denominator, a unit is changed first to a fractional form with the required deno- 164 ARITHMETIC. SECOND PART. minator. Then it is divided by the denominator of each fraction, to obtain one part, and multiplied by the numera- tor, to obtain the required number of parts. Thus changing | and f each to twelfths, we first change a unit to a fraction with the required denominator 12 ; thus, ||. We then divide it by the denominator of f , to obtain one fourth, and multiply the answer by 3, to obtain three fourths. In like manner with the §. We divide }§ by the denominator 6, to obtain one sixth, and multiply by the numerator to obtain two sixths. In changing fractions to common denominators then, the unit must be changed to that fractional form which will enable us to divide it by all the denominators of the frac tions (which are to be reduced) without remainder. Thus if we wish to reduce ^ and f to a common denom- iuator, we cannot reduce them to twelfths, because || can- not be divided by either the denominator 5, or 7, without remainder. We must therefore seek a number that can be thus divided, both by 7 and 5. 35 is such a number. We now take | of ff and f of f f and the two fractions are then reduced to a common denominator. One mode of reducing fractions to a common deno. MINATOR. Change a unit to a fraction whose denominator can be divided by all the denominators of the fractions to be redu- ced, without remainder. Divide this fraction by the deno- minator of each fraction to obtain one part, and multiply by the numerator to obtain the required number of parts. FURTHER examples for mental exercise. Reduce § | and J to a common denominator. Let the unit be reduced to f £ . Reduce f | f to a common denominator. Let the unit be reduced to ||. Reduce f f \ and T 4 F to a common denominator. Reduce § £ f to a common denominator. Reduce £ f £ to a common denominator. Reduce rV£ f f £ *° & common denominator. But there is another method of reducing fractions to a REDUCTION OF FRACTIONS. 165 common denominator which is more convenient for opera- tions on the slate. When a fraction has both its terms (that is its numerator and denominator) multiplied by the same number, its value remains the same. For example ; multiply both the numerator and deno- minator of | by 4, and it becomes T 8 ¥ . But § and T 8 5 are the same value, with different names. The effect, then, of multiplying both terms of a fraction by the same number is to change their name, but not their value. If therefore we have two fractions, and wish to change them so as to have both their denominators alike, we can do it by multiplication. For example ; Let §- and £ be changed, so as to have the same deno- minator. This can be done by multiplying both terms of the | by 9, and of •* by 3. The answers are i-f and £f, and the value of both fractions is unaltered. In this case both terms of each fraction were multiplied by the denominator of the other fraction. Let the following fractions be reduced to a common de- nominator in the same way. 1. Reduce f and f to a common denominator. Multiply 'the | by the denominator 7, and the f by the denomina- tor 5. . 2. Reduce § and f to a common denominator. 3. Reduce f and £ to a common denominator. 4. Reduce ' T \ and £ to a common denominator. The same course can be pursued, where there are sev- eral fractions, to be reduced to a common denominator. Thus if i | and £ are to be reduced to a common deno- minator, we can multiply both terms of the \ first by the denominator 3, and then multiply both terms of the answer by the denominator 4, and it becomes \\ and its value re- mains unaltered. For \ and \\ have the same value with a different name. Then we can multiply both terms of the § first by the denominator 2, and then by the denominator 4, and it be- comes i| and its value remains unaltered. Then f may be multiplied, first by the deneminator 2, 166 ARITHMETIC SECOND PART. and then by the denominator 3, and it becomes i| and its value is unaltered. The three fractions J § and £ are thus changed to \\ \% and \\ which have a common denominator, and yet their value is unaltered. But instead of multiplying each fraction, by each sepa- rate denominator, it is a shorter way to multiply by the product of these denominators. Thus in the above example, instead of multiplying the \, first by 3, and then the answer by 4, it is shorter to mul- tiply by 12 (the product of 3 and 4), and the answer will be the same. In like manner, if we were to reduce £ £ and | toa common denominator, we should multiply both terms of each fraction by the denominators of all the other frac- tions. But instead of each denominator separately, as multiplier, we can take the product of them for the mul- tiplier. Reduce f £ and i to a common denominator. Here both terms of the § are first multiplied by the pro- duct of the other two denominators (which is 12). Then both terms of £ are multiplied in the same way by the pro- duct of the other two denominators (15). Then both terms of \ are multiplied by the product of the other two denominators (20). Rule for reducing fractions to a common denomi- nator. Multiply both terms of each fraction by the product of all the denominators except its own. Reduce \ £ f to a common denominator. Reduce £ f\ and \\ to a common denominator. An„ difficult to reduce. Thus if we wish to reduce yWe" to ^ ts lowest terms, we could not so readily do it. In such a case as this there are two ways of doing it ; the first is as follows. Rule for reducing a fraction to its lowest terms. Divide the terms of the fraction by any number that will divide both, without a remainder. Divide the answer ob- tained in the same way. Continue thus, till no number can be found, that will divide both terms without a remainder. Thus, Reduce T 2 g 3 3 4 F to its lowest terms. N. B. The brackets at the right of the fractions show that both terms of the fraction are to be divided by the di- visor, and not (he fraction itself, as in the division effrac- tions. 234 \_i_3 _7_8_ 18 36^ • " 6 I 2 6 1 2/^^ 3 6 3 3 o 9 a)^-3= T V 3 2 Answer. In the above process, both terms of the fraction T 2 g 3 g 4 g are divided by 3 ; the answer is divided by 2 ; and this answer again is divided by 3. The last answer is T \\ which cannot have both terms divided by any number without a remainder. The other method of reducing a fraction to its lowest terms, is first to find the number which is the greatest common measure, and then to divide the fraction by this number. The following is the method of finding the greatest common measure, and reducing to the lowest terms. Reduce §4- to its lowest terms. The denominator is first placed as a dividend, and the numerator, as a divisor; (below.) After subtracting, the remainder (14) is used fnr the divisor, and the first divisor (21) is used for the dividend. This process of dividing REDUCTION OF FRACTIONS. 16 ( .) the last divisor by the last remainder is continued till nothing remains. The last divisor (7) is the greatest common measure. We then take the fraction §•]• and divide both terms by 7, the greatest common measure, and it is reduced to its lowest terms, viz. |. 21)35(1 21 14)21(1 14 7)14(2 14 00 S-M-T-7: 35/ ' Rule for finding the greatest common measure of a Fraction and reducing it to its lowest terms. Divide the greater number by the less. Divide the divi- sor by the remainder, and continue to divide the last divisor by the last remainder, till nothing remains. The last divisor is the greatest common measure, by which both terms of the fraction are to be divided, and it is reduced to its lowest terms. Reduce the following Fractions to their lowest terms. 486 • 144 • 324. 14 29. 16. 4.4. . _4J>_8_ . _4_7_4_S_ . . S_ 0_S__ . 9120 ' T72 J ' 64 8' 28S8 ' 2192' 1184' 38433' 42315' Hff ; f&H&s- Ans - 2V ; tV ; I ; \ ; 1 5 H£ ; ,¥8?, _3_ • 3 . X. RpHiipp thp fnllnwinir • 516 . 4932 . _i_2_3JL5_ . 2yj7 . JXeUUUt; Hie lOUUVVlUg . -943- , 8^64 ' 678910 ' 342954 ' 8.31^8 • 3 9_9_72_ • _9.9.8.811 • JULSJLSJL 95636' 812322' 9998881T' 7 3 28472* 15 170 ARITHMETIC. SECOND PART. REDUCTION OF FRACTIONS FROM ONE ORDER TO ANOTHER ORDER. It will be recollected that in changing whole numbers from one order to another, it was done by multiplication and division. Thus, if 40 shillings were to be changed to pounds, we divided them by the number of shillings in a pound, and if £2 were to be reduced to shillings, we midtiplied them by the number of shillings in a pound. The same process is used in changing jr actions of one order to fractions of another order. Thus, if we wish to change -^^ of a £ to a fraction of the shilling order, we multiply it by 20, making it ^Vo* For 2V0 °f a shilling is the same as ^£0 of a pound. Ifwewishto change ^W °f a shilling, to the same value in a fraction of the pound order, we divide ^Vo D y 20, making it ^^. (This could also be divided by multi- plying its denominator by 20.) If then we wish to change a fraction of a lower order to the same value in a higher order, we must divide the frac- tion, by multiplying the denominator, by that number of units (of the order to which the fraction belongs) which make a unit of the order to which it is to be changed. Thus if we wish to change | of a penny to the same value in the fraction of a shilling, we multiply its denomi- nator by 12, making it g\ of a shilling. If we wish to change this to the same value in a fraction of the pound order, we must now multiply its denominator by the num- ber of shillings which make a pound, making it T? 2 5? of a pound. It must be remembered that multiplying the deno- minator of a fraction, is dividing the fraction. If, on the contrary, we wish to change a fraction of a higher order to one of the same value in a lower order, we must multiply. Thus, to change T | T of a shilling to the penny order, we must multiply it by 12. This we do, by multiplying Us numerator by 12, and the answer is ^\ . For as there are 12 times as many whole pence in a whole shilling, so there are 12 times as many T | ? of a penny in T | T of a shilling. reduction of fractions. 171 Rule for reducing fractions of one order to an- other order. To reduce a fraetion of a higher to one of a lower order. Multiply the f'action by that number of units of the next lower order, which are required to make one unit of the order to which the fraction belongs. Continue this process till the fraction is reduced to the order required. To reduce a fraction of a lower to one of a higher order. Divide the fraction (by multiplying the denominator) by the number of units which are required to make one unit of the next higher order. Continue this process till the frac- tion is reduced to the order required. Examples. Reduce jo\i °? a guinea, (or of 28 shillings,) to the traction of a penny. Reduce a of a guinea to the fraction of a pound. Reduce ^\ of a pound Troy, to the fraction of an ounce. Reduce T 3 7 of an ounce to the fraction of a pound Troy. Reduce -^ of a pound avoirdupoise to the fraction of an ounce. A man has F | T of a hogshead of wine, what part of a pint is it ? A vine grew T / T ^ of a mile, what part of a foot was it ? Reduce | off of a pound to the traction of a shilling. Reduce § off of 3 shillings, to the fraction of a pound. REUCTION OF FRACTIONS OF ONE ORDER, TO UNITS OF A LOWER ORDER. It is often necessary to change a fraction of one order, to units of a lower order. For example, we may wish to change | of a unit of the pound order, to units of the shilling order. This | of a £ is 2 pounds divided by 3. These 2 pounds 172 ARITHMETIC. SECOND PART are changed to shillings, by multiplying by 20, and then divided by 3, and the answer is 13^ shillings. This i of a shilling maybe reduced to pence in the same way, for ^ of a shilling is 1 shilling divided by 3. This 1 shilling can be changed to pence, and then divided by 3, the answer is 4 pence. Rule for finding the Value of a Fraction in units of a lower order. Consider the numerator as so many units of the order in which it stands, and then change it to units of the order in which you wish to find the value of the fraction. Divide by the denominator, and the quotient is the answer, and is of the same order as the dividend. Examples. 1. How many ounces in f of a lb. Avoirdupoise ? 2. How many days, hours and minutes, in f of a month ? 3. What is the value of f of a yard ? 4. What is the value of -^ of a ton ? 5. How many pence in § of a lb. ? 6. How many drams in f of a lb. avoirdupoise ? 7. How many grains in f of a lb. Troy weight ? 8. How many scruples in | of a lb. Apothecaries weight ? 9. How many pints in f of a bushel ? REDUCTION OF UNITS OF ONE ORDER TO FRACTIONS OF ANOTHER ORDER. It is necessary often to reverse the preceding process, and change units, to fractions of another order. For ex- ample, to change 1 3s. 4d. to a fraction of the pound or- der. To do this we change the 13s. 4d. to units of the lowest REDUCTION OF FRACTIONS. 173 order mentioned, viz. 160 pence. This is to be the numer- ator of the fraction. We then change a unit of the pound order to pence (240) and this is the denominator of the fraction. The answer is -i££ of a pound. For if 13s. 4d. is 160 pence, and a lb. is 240 pence, then 13s. 4d. is |££ of a pound. RULE FOR REDUCING VISITS OF ONE ORDER TO FRACTIONS OF ANOTHER ORDER. Change the given sum to units of the lowest order men- tioned, and'make them the numerator. Change a unit of the order to which the sum is to he re- duced, to units of the same order as the numerator, and place it for the denominator. Examples. Reduce 6oz. 4pwt. to the fraction of a pound Troy. Reduce 3 days, 6 hours, 9 minutes to the fraction of a month. Reduce 2cwt. 2qrs. 161bs. to the fraction of a ton. Reduce 21b. 4oz. to the fraction of a cwt. REDUCTION OF A COMPOUND NUMBER TO A DECIMAL FRACTION. It is often convenient to change a compound number, to a decimal fraction. Thus we can reduce loz. lOpwt. to a decimal of the pound order. Let the figures be placed thus, and the process will be explained below. The 10 pwts. are first written, and then the 1 oz. set under. 20)10'0 pwt. 12) l'5oz. <125lb. 15* 174 ARITHMETIC. SECOND PART, We first change the lowest order (10 pwts.) to an im- proper decimal, thus lO'O. Now as 20 pwts. make an oz., there are but one twentieth as many ounces in a sum as there are penny weights. For the same reason, in any sum there are but one twen- tieth as many tenths of an ounce as there are tenths of a penny weight. As there are then 100 tenths ofapwt. in this sum, if we take one twentieth of them, we shall find how many tenths of an oz. there are. We therefore divide the lO'O pwts. by 20, and the amount is ,5. This ,5 is placed (beside the 1 oz. of the sum) under the lO'O pwts., and thus, instead of reading the sum as loz. 10 pwts., we read it as 1,5 oz., or loz. and 5 tenths of an oz. As the pwts. are thus reduced to the decimals of an oz. we now reduce the l,5oz. to the decimal of a lb. in the same way. We make the 1,5 an improper decimal, thus 1*5 (15 tenths) of an oz. Now as there are 12oz. in a lb., there are but one twelfth as many tenths of a lb. in a sum, as there are tenths of an oz. We therefore divide the 15 tenths of an oz. by 12, and the answer is ,1 of a lb. and 3 left over. This 3 is re- duced to hundredths by adding a cipher and dividing it again. The quotient is 2 hundredths. The next remain- der is changed to thousandths in the same way, and the answer is ,125 of a £. Rule for changing a compound number to a deci- mal. Change the loxcest order to an improper decimal. Divide it by the number of units of this order ; which are required, to make a unit of the next higher order, and set the answer be- side the units of the next higher order. Repeat this process till the sum is brought to the order required. Examples. Reduce 10s. 4d. to the decimal of a lb. REDUCTION OF FRACTIONS. 175 Reduce 8s. 6d. 3qrs. to the decimal of a lb. Reduce 17hrs. 16min. to the decimal of a day. Reduce 3qrs. 2na. to the decimal of a yd. Reduce 32gals. 4qts. to the decimal of a hogshead. Reduce lOd. 3qrs. to the decimal of a shilling. REDUCTION OF A DECIMAL, TO UNITS OF COMPOUND ORDERS. The preceding process can be reversed, and a decimal of one order, be changed back to units of other orders. Thus, if we have ,125 of a lb. Troy, we can change it to units of the oz. and pwt. order. In performing the process, we place the figures thus. ,1251b. 12 l,500oz. 20 I0,000pwt. We reason thus. In ,125 of a lb. there must be 12 times as many thousandths of an oz. (for 12 oz. = 1 lb. We therefore multiply by 12, and point off according to rule, and the answer is 1 oz. and 500 thousandths of an oz. Now as we have found how many oz. there are, we must find how many pwts. there are in the ,500 of an oz. There must be 20 times as many thousandths of a pwt. as there are thousandths of an oz. therefore multiply the de- cimal only, by 20, and point off according to rule, and we find there are 10 pwts. We have thus found that in ,125 of a lb. there are loz. and lOpwts. 176 arithmetic. second part. Rule for changing a decimal of one compound or- der, TO UNITS OF OTHER ORDERS. Multiply the decimal by the number of units of the next lower order which are required to make one unit of the order in which the decimal stands. Point off according to rule, and multiply the decimal part of the answer in the same way, pointing off as before. Thus till the sum is brought into the order required. The units of each answer make the final answer. In ,1257 of a £ how many shillings, pence and farth- * ings ? What is the value of ,2325 of a ton ? What is the value of ,375 of a yard ? What is the value of ,713 of a day ? What is the value of ,15834821 of a ton ? REDUCTION OF CURRENCIES. There are few exercises in Reduction, of more prac tical use than the Reduction of Currencies, by which a sum in one currency is changed to express the same val- ue in another currency. An example of this kind of reduction occurs, when the value of $1 is expressed in British currency thus, 4s. 6d. The necessity for using this process in this country, re- sults from the following facts. Before the independence of the U. States, business was transacted in the currency of Great Britain. But at vari- ous times, the governments of the different States, put bills into circulation, which constantly lessened in value, until they became very much depreciated. For example, a bill which was called a pound or twenty shillings, British currency, was reduced to be worth only fifteen shillings, in the New England states. This depreciation was greater in some states than it was in others, and the result is, that pounds, shillings, and pence have different values in different states. 12 pence make a shilling, and 20 shillings make a pound, in all cases, but the value of a penny, a shilling, or a pound, depends upon the currency to which it be- longs. REDUCTION OF CURRENCIES. 177 The following table shows the relative value of the sev- eral currencies, by showing the value oi one dollar in each of the different currencies. VALUE OF ONE DOLLAR IN EACH OF THE DIFFERENT CUR- RENCIES. equals 6*. New England currency. " 8s. New York currency. " 7s. 6d. Pennsylvania currency. " 4s. 8d. Georgia currency. " 4s. Gd. Sterling money, or Eng. currency. " 5s. Canada currency. " 4s. I0±d. Irish currency. " £2. 14s. Scotch currency. VALUE OF ONE FOUND OF EACH OF THE DIFFERENT CUR- RENCIES, EXPRESSED IN FEDERAL MONEY. £1 N. England currency equals $3,333 A- £1 N. York currency " $2,50 £1 Pennsylvania currency " $2,66(5 £1 Georgia currency " $4,2855 £1 Sterling money " $4,444f £1 Canada currency " $4,00 £1 Irish currency « $4,10| £1 Scotch currency " $0,370fe The following sums for mental exercise, will be found of much practical use, and should be practised till they can be readily answered. Examples in N. England currency for mental ex- ercise. 1. If 6 shillings equal a dollar or 100 cts. how many cents in 3 shillings ? in 2 shillings ? in 1 shilling ? in 4 shillings 1 in 5 shillings ? 2. If 1 shilling is 16| cts. how many cents in 6 pence ? in 3 pence ? in 9 pence 1 in 4 pence ? in 7 pence? in 8 pence ? in 11 pence ? 3. How many cents in Is. 6d. ? in Is. 9d. ? in Is. 3d.? in 2s. 6d ? in 2s. 9d. ? in 3s. 4d. ? in 5s. 6d. ? in 7s. 6d.? 178 ARITHMETIC. SECOND PART. in 8s. 6d. ? in 9s. 1 in 9s. 6d. ? in 10s. 6d. ? in lis. ? in Us. 6d. ? in 12s. ? 4. If 6d. is 8i cts. how many cents is 3d. ? how many- is Id. ? now many is 2d. ? 5. If you buy 8 yds. of ribbon at Is. 6d. yer yd. how much will the whole cost ? 6. If you buy 2f yds. of muslin at 2s. 6d. per yd. how much will it cost in dollars and cents ? 7. If you buy 31 yds. of ribbon at Is. 9d. per yd. how much will it cost ? 8. If you buy a brush for 2s. 3d. and a penknife for 4s. 6d. and a comb for Is. 6d. how much is given for the whole ? 9. If you pay 3s. 6d. for scissors, 2s. 4d. for a thimble, and Is. 9d. for needles, how much will the whole cost ? 10. If linen is 4s. 6d. per yd. how much will 4f yds. cost ? 11. If a piece of calico is 2s. 3d. per yd. how much will Q\ yds. cost ? 12. If muslin is 4s. 6d. per yard, what will 2f yds cost ? 13. How much is lHd. ? 10id. ? 9|d. ? 8£d. ? 7id. 1 12£d.? 16id. ? Examples in N. York ovrrency for mental exercise. 1. If a dollar in N. York currency is 8s. how many cents in 4s. ? in 2s. ? in Is. ? in 5s. i in 6s. ? in 7s. ? in 9s. ? in 10s. ? in lis. ? in 12s. ? in 13s. ? in 14s. ? in 15s. ? in 16s, ? 2. If one shilling is 12| cts. how many cents in 6d. ? in 3d. ? in Id. ? in 2d. 1 in 4d. 1 in 7d. ? in 8d. ? in 9d. ? in lOd. ? in lid.? 3. How many cents is Is. 6d. N. York currency ? is 2s. 6d. ? is 3s. 6d. ? is 5s. 3d. ? is 6s. 9d. 1 is 4s. 8d. ? Questions can be asked in the other currencies in the same manner. REDUCTION OF CURRENCIES. 179 REDUCTION OF CURRENCIES TO FEDERAL MONEY. Sums of this kind, which are too complicated to be done mentally, may be performed on the slate, by the following rules. To REDUCE BRITISH CURRENCY TO FEDERAL MONEY. Reduce the sum to a decimal of the pound order, and di- vide the answer by -fa. The. reason of this rule is that a dollar is JL. of a £ of this currency, and therefore there are as many dollars in the sum as there are ^ in it. Note. Before reducing any currency to Federal mon ey, the sum must be reduced to a decimal of the pound or' der. After this process the following rules may be used. To reduce Canada currency. As a dollar is £ of a £ in this currency, there will be as many dollars as there are } in the sum. Therefore, Reduce the sum to the decimal of a £ and divide itby\. To reduce New England Currency. As 1 dollar is ,3 of a pound in this currency, so there are as many dollars in a sum of N. England currency as there are ,3 in it. Therefore Reduce the sum to the decimal of a £ and divide it by ,3. To reduce New York Currency. As 1 dollar is ,4 of a pound in this currency, there will be as many dollars in a sum of New York currency, as there are ,4 in it. Therefore Reduce the sum to the decimal of a £ and divide it by ,4. To reduce Pennsylvania Currency. As 1 dollar is f of a £ in this currency there are as many dollars in the sum as there £ contained in it. Therefore Reduce the mm to the deciinal of a £ and divide it by %. 180 ARITHMETIC. SECOND PART. To reduce Georgia Currency. As 1 dollar is -fa of a pound in this currency there are as many dollars in the sum as there ^ contained in it. Therefore Reduce to the decimal of a £ and divide the sum by £^. 1 REDUCTION OF FEDERAL MONEY TO THE SEVERAL CURRENCIES. To change a sum in federal money to the different cur- rencies, the preceding process is reversed, and the sum is to be multiplied [instead of divided) by the several frac- tions. The answer is found in pounds and decimals of a pound. The decimal can be reduced to units of the shil- ling and pence order by a previous rule. (p. 176.) Examples. 1. Reduce Is. 6d. in the several currencies to Federal money. Answers. Of Canada Currency, it is $,30 British, N. England, N. York, Penn. Georgia, $,333i $,25 $,187i $,20 ' $,321f 2. Reduce 4id. of the several currencies to Federal money. 3. Reduce 4s. 6d. of the several currencies to Federal money. 4. Reduce 35£ 3s. 7{d of the several currencies to Federal money. 5. Reduce $118,25 to the several currencies. REDUCTION OF CURRENCIES. 181 I Ansicers to the lasi !. p £ s. d. In Canada ( British, currency, it is 29 " 11 " 3 26 " 12 " H N. Eng. N. York, M (( 35 " 9 " 6 47 « 6 " Penn. u 44 " 6 « 10i Georgia, <( 27 " 11 " 9a" 4 Reduce 2s. Dd. of N. England currency to the same value in all other currencies. Reduce 4s. 6d. N. York currency to the same value in all the other currencies. REDUCTION FROM ONE CURRENCY TO AN- OTHER. The following table will enable the pupil to reduce a sum from one currency to another, with more facility than by any other method. Each fractional figure shows the relative value of a sum in one currency to the same sum in another currency. For example, the £ in the second perpendicular and the fourth horizontal column, shows that£l sterling is £ of the number which expresses the same value in New England currency. Thus £6 sterling is £ of the number which expresses the same value in New England currency. That is, £6 is £ of the answer to be obtained when the same value is expressed in New England currency. To find the answer, we reason thus. If 6£ is three fourths, £2 is one fourth, and 8£ is the answer. Thus dividing by |. Rule for changing a sum in one currency, to the same value in another currency. To change a sum in a currency written in the upper space to one written in the right hand space, divide by the fraction that stands where both spaces meet. If there are shillings, pence and farthings in the sum, first reduce them to the decimal of a £. 16 . 182 ARITHMETIC. SECOND PART. TABLE EXHIBITING THE COMPARATIVE VALUES OF THE 3EVERAL CURRENCIES. ANY SUM EXPRESSED IN m a k it on CD it O CD O ft 1— 1 H t— • CO ft o p P • ft H ft CD 13 3 ft as ft CO o o o IS 1 27 1 1 2 7 8 I 2 5 2 7 6 9 5 54 i 9 5 36 4 27 ,1 £ Scot. o •} K K > 3 K Of e S a X » B w M O z 1 4 9 I 6 7 1 2 112 5 184 6 5 8 3 4 1 6 1 5 27 4 £N.Y. 8 3 3 5 2 8 4 5 6 9 2 3 2 3 4 5 1 6 1 5 36 5 £ Pen. 1 3 3 i 4 9 75 92 3 5 6 5 4 4 3 9 limes £N.E. 4 times 9 1 I 4 1 5 9 9 2 3 6 5 3 2 8 5 5 4 5 £ Can. 92.3 22 5 92 3 i o c n 64 6 1 6 7 5 9 2 3 9 92 3 7 5 92 3 6 18 4 6 112 5 2 7 6 9 2 5 £ Irish. 3 7 2 7 2 8 6 7 5 6 4 6 1 15 1 4 9 7 4 5 2 8 7 8 1 7 £ Geo. 4 9 I 2 8 1 27 10 9 2 3 1 9 4 3 5 3 1 6 ! ' 2 -a 1 times 1 £ Ster. 9 7 4 1 3 22 5 9 2 3 1 4 3 1 3 8 4 2 7 10 10 $F.M. EXAMPLES FOR PRACTICE. 1. Reduce £4 N. E. to F. M. 2. Reduce 2£ 3s. 9d. N. E. to F. M 8. Reduce £6 N. Y. to F. M. 4. Reduce £8 ; 4 ; 9 N. Y. to F. M. 5. Reduce £3 ; 2 ; 3 Penn. to F. M. 6. Reduce $152.60 to N. E. 7. Reduce $196.00 to N. E. 8. Reduce $629.00 to N. Y. Ans. $13.3331. Ans. $7,2913. Ans. $15.00. Ans. $20.593f . Ans. $8.30. Ans. £45 ; 15 ; 7.2. Ans. $58 ; 16. Ans. 251 ; 12. 9! Reduce £35 ; 6 ; 8 sterling to N. E Ans. £47 ; 2 ; 2 ; 2| 10. Reduce £120 N. E. to Can. Ans. £100 11. Reduce £155 ; 13 N. E. to Sterling. Ans. £116; 14; 9 REDUCTION OF CITRKENCIK3. 183 12. Reduce £104 ; 10 Can. to N. Y. Ans. £167 ; 4. 13. Reduce £300 ; 10 ; 4 ; 2 Can. to Penn. Ans. £450 ; 15 ; 6 ; 3. 14. Reduce £937 ; 18; 11 ; 1 N. E. to Geo. Ans. £721; 14; 8; 3. 15. Reduce $224 ; 60 to Can. Ans. £56 ; 3. 16. Reduce £225 ; 6 N. E. to F. M. Ans. $752.00. 17. Reduce £880 15 ; 11 ; 1 Perm, to Sterling. Ans. 528 9 ; 6 ; 3. 18. Reduce £6,750 Irish to Geor. Ans. £6,461. 19. Reduce £1,846 Ster. to Irish. Ans. £2,000. 20. Reduce £1,722 ; 18 ; 9 ; 3 N. E. to N. Y. Ans. £2,298 ; 5 ; 1. 21. Reduce £2,114 ; 1 ; 3 Can. to F. M. Ans. $8,456.25. 22. Change £784 ; 5 ; 6 ; 2 Penn. to Geor. Ans. £487; 19; 10 ; 2ff. 23. Change £923 Sterling to Irish. 24. Change £,4000 Irish to Sterling. 25. Change £157 ; 8 ; 3 ; 3 N. Y. to N. E. 26. Change £1,654 ; 3 ; 8 ; 1 Penn. to N. E. 27. Change £ 947 ; 9 ; 4 ; 2 N. E. to F. M. 28. Change $1,444.66 to N. E. To N. Y. To Penn. 29. Change $945.22 to N. Y. To Geor. To Can. 30. Change £1,846 ; 15 ; 4 N. E. to F. M. To Penn To Georgia. 31. Change $4,444,444f to Sterling. 32. Reduce £1,000,000 Sterling to F. M. ARITHMETIC. THIRD PART. NUMERATION. In the following, Third Part, there will be a reveiw of the preceding subjects, embracing the more difficult ope- rations. The rules and explanations will not be repeated, as the pupils can refer to them in the former part. ROMAN NUMERATION. Before the introduction of the Arabic figures, a method of expressing numbers by Roman Letters was employed. As this method has not entirely gone out of use, it is im- portant that it should be learned. The following letters are employed to express numbers. I. One. X. Ten. II. Two. L. Fifty. III. Three. C. One Hundred. IIII. or IV. Four. D. Five Hundred. V. Five. M. One Thousand. The above letters, by various combinations, are made to express all the numbers ever employed in Roman Nu- meration. RULE FOR WRITING AND READING ROMAN NUMBERS. As often as a letter is repeated, its value is repeated. When a less number is put before a greater, the less number is subtracted. But when the less number is put after the great- er, it is added to the greater. Examples. In IV. the less number I. is put before the greater number V. and is to be subtracted, making the number four. In VI. the less number is put after the greater, and it is to be added, making the number six. NUMERATION. J8. r i In XL. the ten is subtracted from the fifty. In LX. the ten is added to the fifty. The following is a table of Roman Numeration : TABLE. LXXXX. orXC C. CC. CCC. CCCC. D. or Io* DC. DCC. DCCC. DCCCC. M. or Cly.t IQO. or v.t_ CtJIOO- or X. iooo- ICCCIooO-orC. M MM \q. is used instead of D. to represent five hundred, and for every additional Q. annexed at the right hand, the number is increased ten times. t C13. is used to represent one thousand, and for every C. and 3. put at each end, the number is increased ten times, t A line over any number increases its value one thousand timts. One I. Ninety Two II. One hundred Three in. Two hundred Four IIII. or IV. Three hundred Five V. Four hundred Six VI. Five hundred Seven VII. Six hundred Eight VIII. Seven hundred Nine Villi, or IX. Eight hundred Ten X. Nine hundred Twenty XX. One thousand Thirty XXX. Five thousand Forty XXXX.orXL. Ten thousand Fifty L. Fifty thousand Sixty LX. Hundred thousa Seventy LXX. One million Eighty LXXX. Two million Write the following numbers in Roman letters : 5. 7. 3. 9. 8. 10. 4. 14. 5. 1*5. G. 16. 26. 36. 30G. 1. 11. 111. 7. 17. 77. 777. 1800. 1832. 1789. Read the following Roman numbers : VI. XIX. XXIV. XXXVI. XXIX. LV. XLI. LXIV. LXXXVIII. XCIX. MDCCCXVIII. OF OTHER METHODS OF NUMERATION. By the common method of numeration, ten units of out order, make one unit of the next higher order. But it is equally practicable, to have any other number than ten. to constitute a unit of a higher order. Thus we might have six units of one order make one unit of the next higher order. Or twelve units of one order might make one of the next higher order. The number which is selected to constitute units of the higher orders, is called the radix of that system of nu- meration. 16. 186 ARITHMETIC. THIRD PART. The radix of the common system is ten, and this num- ber it is supposed was selected, because men have ten fingers on their hands, and probably used them in express- ing numbers. Before the introduction of the Arabic figures, Ptolemy introduced a method of numeration, in which sixty was the radix. The Chinese and East Indians use it to this But in Ptolemy's system there were not sixty different characters employed. Instead of this, the Roman method of numeration was used for all numbers as far as sixty, and then for the next higher orders the same letters were used over again, with an accent (') placed at the right. For the third order two accents (") were used, and for the fourth order three accents ("'). To illustrate this method by Arabic figures, 31' 23 signifies 31 sixties and 23. We have some remnants of this method in the division of time into 60 seconds for a minute, and 60 minutes for an hour, and also the division of the degrees of a circle, into 60 seconds to a minute, and 60 minutes to a degree. EXERCISES IN NUMERATION, COMMON, VULGAR, AND DECIMAL. (See rides on pages 53, 58, and 64.) 1. Two million, four thousand, one hundred and six. 2. Two hundred thousand, and six tenths. 3 Twenty six billion, six thousand, and fifteen thou- sandths. 4. Two hundred and sixty thousand millionths. 5. One sixth of two apples how much and how written ? 6. One ninth of twenty o?'anges, how much, and how written ? Is it a proper or improper fraction ? 7. One sixth of four bushels how much? how written ? is it a proper, or improper fraction ? 8. One tenth of forty bushels, how much ? how written ? is it a proper or improper fraction ? 9. One tenth of three oranges, how much ? how express- ed ? ADDITION. 187 10. Three tenths of three oranges, how much 1 how ex- pressed. 11. Four sixths of twelve apples, how much ? how ex- pressed ? 12. Three thousand tenths of thousandths. 13. Four billions, six thousand, and five ten thousandths. 14. Sixteen billions, three hundred and six millions, five hundred thousand, and six tenths of millionths. 15. Five trillion, five million, five units, and three hundred and sixty five millionths. It!. Sixteen hundred and twenty four, and four tenths of billionths. ADDITION. Let the pupil add the following numbers : 1 Two hundred and six million ; twenty four thousand, five hundred and six. Thirty seven billion, twenty six thousand and three. Four hundred and seventy nine billion, six hundred and sixty seven million, nine hundred and eighty four thou- sand, six hundred and ninety nine. Fifteen million, seventy seven thousand, nine hundred. Thirty six trillion, four hundred million, and six. Four quadrillion, seventeen million, three hundred and six. Six quadrillion, fourteen trillion, seventeen million, four- teen thousand, three hundred and nine. Twenty four sextillion, five hundred million and nine. 2 Sixteen thousand, four hundred and sixty four, and nine tenths. Two hundred and sixty nine million, fourteen hundred and three, and thirteen hundredths. Forty four million three thousand and six, and twenty thousandths. Five hundred million, nine hundred and ninety nine thousand, eight hundred and seventy nine, and two hun- dred and sixty four tenths of thousandths. 188 ARITHMETIC. PART THIRD. Six hundred and seventeen thousand, four hundred and sixty eight, and five hundred and seventy nine hundredths of thousandths. Forty six million, nine thousand, and seventy millionths. 3 Add two twelfths, three fourths, and four sixths. (See page 166.) Add twenty four fiftieths, sixteen tenths, and twenty halves. 5 Add forty nine eightieths, seventy nine fortieths, and two hundred thousandths. 6 Add nine twenty sevenths, thirteen forty fourths, and twenty nine seventieths . SUBTRACTION. 1 From, Three hundred and sixty nine million, four hun- dred twenty seven thousand, three hundred seventy six, Subtract, Two hundred and ninety three million, four hun- dred and eighty three thousand, nine hundred and eighty seven. From, Twenty four billion six hundred and thirteen mil- lion, four hundred and forty four thousand, eight hundred and eighty six, and twenty nine hundredths, Subtract, Sixteen billions, twenty four thousand and sixteen, and four hundred and six thousandths. 3 From, Sixty four sextillion, ninety trillion, seven billion, twen- ty nine million, forty thousand three hundred and six, and twenty nine tenths of millionths, MULTIPLICATION. 189 Subtract, Fourteen quintillions, nine quadrillions, seven trillions, fourteen thousand and eighty, and seven hundredths of milliontJis. 4 From nine twelfths, subtract two fifths. (See page , 166.) 5 From thirteen twenty sevenths, subtract three twenty fourths. 6 From threejijths, subtract twenty nine seventy sevenths. 7 From, Twelve hundred and six, four hundred and twentieths, Subtract, Four hundred and nine, nine hundred and ninetieths. MULTIPLICATION. 1. Multiply 32694302 by 365. 2. Multiply 24,2 by 27 (See page 108.) 3. Multiply 321,92 by 236. 4. Multiply 236,49 by 2,4. 5. Multiply 47,2935 by 2,08432. 6. Multiply 870,24 by 32,94. 7. Multiply 14 yds. 3 qrs. 2 na. by 28. 8. Multiply 8 le. 2 m. 6 fur. 22 po. by 362. 9. Multiply 2 bu. 3 pk. 1 qr. 1 pt. by 172. 10. Multiply | by 3 (Seepage 112.) 11. Multiply ^ bv 48. 12. Multiply i| by 32. 13. Multiply 12 by § (See page 116.) 14. Multiply 24 by f. 15. Multiply 324 by T V 16. Multiply 2342 by £&. 17. Multiply f by | (Se'e page 123.) 18. Multiply f by f . 19. Multiply f by f 20. Multiply T liole time. Find the amount of each payment to the time of settlement. Add the amounts of the payments, and subtract them from ike amount of the principal. Example. On April 1st, 1825, 1 gave a note to A. B. promising to pay him $300 for value rec'd. and interest on the same at - 6 per cent, till settlement. Oct. 1, 1825, I paid #100. April 16, 1826, paid $50. Dec. 1, 1827, paid $120. What do I owe on April 1st, 1828 ? $ cts. m. 300,00,0 principal dated April 1, 1825. yrs. mo. da. 54,00,0 interest up to April 1st, 1828. 3. 0. 0. 354,00,0 amount of principal. 100,00,0 1st payment, Oct. 1, 1825. 15,00,0 interest up to April 1st, 1828. 2. 6. 0. 1 15,00,0 amount of 1st payment. 50,00,0 2nd payment, April J 6th, 1820. 5,87,5 interest up to April 1st, 1828. 1. 11. 15. 55,87,5 amount of second payment. 120,00,0 3rd payment, Dec. 1st, 1827. 2,40,0 interest up to April 1st, 1828. 0. 4. 0. 122,40,0 amount of 3rd payment. 55,87,5 " 2nd payment. 115,00,0 " 1st payment. 293,27,5 total amount of payments. 204 ARITHMETIC. THIRD PART. 354,00,0 amount of principal. 293,27,5 total amount of payments subtracted. A. 60,72,5»remains due April 1st, 1828. Rule in Massachusetts. Find the Amount of the Principal to the time when one payment, or several payments together, exceed the interest due. From this subtract the payments and the remainder will be a new Principal. Proceed thus till the tune of settlement. Examples. For value received I promise to pay James Lawrence $116,666 with interest. May 1st, 1822. $116,666. John Smith. On this note were the following endorsements. Dec. 25, 1822, received $16,666. July 10,1623, " $ 1,666. Sept. 1, 1824, " $ 5,000. June 14, 1825, " $33,333. April 15, 1826, " $62,000. What was due August 3, 1827 ? Ans. $23,775. The first principal on interest from May 1, 1822, $116,666 Interest to Dec. 25, 1822, time of the first payment (7 months 24 days), 4,549 Amount, $121,215 Payment, Dec. 25, exceeding interest then due, 16,666 Remainder for a new principal, 104,549 Interest from Dec. 25, 1822, to June 14, 1825 (29 months, 19 days), 15,490 Amount, $120,039 INTEREST. 205 Payment. July 10, 1823, less than interest then due, $ 1,606 Payment, Sept. 1, 1824, less than interest then due, 5,000 Payment June 14, 1825, exceed, ing interest then due, 33,333 $39,909 Remainder for a new principal (June 14, 1825), 80,040 Interest from June 14, 1825, to April 15, 1826 (10 months 1 day), 4,015 Amount, $84,055 Payment, April 15, 1825, exceeding inte- rest then due, 62,000 Remainder for a new principal (April 15, 1826), $22,055 Interest due Aug. 3, 1827, from April 15, 1826 (15 months 18 days), 1,720 Balance due Aug. 3, 1827, $23,775 The Rule now adopted in Connecticut, is founded on the principle that interest is to be paid by the year, so that if a man pays before a year is ended, he receives interest on all he pays, from the time he pays it, to the end of the year when the interest is due. Rule in Connecticut. If the payment be made at the end of a year or more, add the interest due on the whole sum, at this time, to the princi- pal, and subtract the payment. Whenever other payments are made, proceed in the same manner, calculating interest on the principal from the time of the last payment. If payment is made before a year has elapsed (from the date of the note, or jrom the last payment), find the amount of the principal for one year. Find also the amount of the payment from the time of payment to the end of the year when 18 206 ARITHMETIC. PART THIRD. the interest would he due, and subtract the latter from the former. If hotoever a year extends beyond the time of settle- ment, find the amount up to that time, instead of for a year. If any remainder after subtraction, be greater than the preceding principal, then the preceding principal is to be con- tinued as the principal for the succeeding time instead of the remainder, and the difference to be regarded as so much unpaid interest. Let interest on the following note be calculated by the three different rules. A note for 820,000 is given July 1st, 1825. 1st payment, January 1st, 1826, $1400 2d do. : January 1st, 1827, 2000 3d fajM. September 1st, 1827, 5000 Settlement. January 1st, 1829. What is due on the note ? Ansu'ers. By the common rule, $14,90^,00 By the Massachusetts rule, 15,212,96 By the Connecticut rule, 15,209,47 Let the following be calculated by the Connecticut rule. $1000,00 Hartford, Jan. 4, 1826. On demand I promise to pay James Lowell, or order, one thousand dollars with interest ; value received. Hiram Simpson. On this note were the following endorsements. Feb. 19, 1827, received June 29, 1828, " Nov. 14, 1828, Dec. 29, 1831, $200.00 500.00 260.00 25.00 What is the balance, June 14, 1832 ? Answer $204.49 Find the balance due on the following note by the Mas- sachusetts rule. v $500.00. Hartford, Feb. 1, 1820. Value received 1 promise to pay A. B. or order five hundred dollars with interest. Samuel Jones. INTER Ei IT. 2 Endorsements. May 1, 1820, received, $40.00 Nov. 14, 1820, " 8.00 April 1, 1821, 12.00 May 1,1821, 30.00 How much remains Sept. 16, 1821 ? Am. $445.57 207 Find the balance due on the following note by the Con- necticut rule. For value received I promise to pay (i. B. or order, eight hundred and seventy-rive dollars, with interest. $875.00. S \MUEL J^MiS. Hartford, Jan. 10, 1821. Endorsements. Aug. 10, 1824, received S260.00 Dec. 16, 1825, 300.00 March 1, 1826, " 50.00 July 1, 1827, « 150.00 What was due Sept. 1, 1828 ? Ans. $474.95. The three rules used above, are all considered as objec- tionable. By the first rule, when a man pays a part of his debt, his payments are not applied to discharging the interest, but entirely to lessening the principal. By this rule, if a man should borrow a sum and promise to pay it, with the interest, in twenty-five years, if he should simply pay what would be the yearly interest, and have it endorsed, at the end of 25 years the debt would be entirely extinguished. Whereas if he should wait till the end of the time agreed upon, he would have to pay the original sum borrowed, and the yearly interest upon it also. The objection to the other two rules is, that the man who makes payments before the time of settlement, actu- ally is obliged to pay more than one who pays nothing be- fore that time. Thus the most punctual man is obliged to pay more than the negligent. Compound Interest is the only method, which will do exact justice to both creditor and debtor. For a man who 208 ARITHMETIC. PART THIRD. lends money is fairly entitled to receive interest at the end of each year ; and then by investing the interest in other stock, he can obtain compound interest. The borrower, therefore, who detains this yearly interest, ought, in jus- tice, to pay what the creditor could gain, if the debtor were punctual. COMPOUND INTEREST. Compound Interest is an allowance made for the use of the sum lent, and also for the use of the interest when it is not paid. Rule. Calculate the Interest, and add it to the principal at tlie eyul of a year. Make the Amount a new principal for the next year, with which proceed as before, till the time of set- tlement. 1. What is the compound interest of $256 for 3 years, at 6 per cent. ? $256 given sum, or first principal. ,6 15,36 interest ) be added t ther . 256,00 principal, $ to 271,36 amount, or principal for 2d year. ,00 16,2816 compound interest, 2d year, > added 271,36 principal, do. $ together. 287,6416 amount, or principal for 3d vear. ,06 17,25846 compound interest, 3d year, ) added 287,641 principal, do. $ together. 304,899 amount. 256 first principal subtracted. A $4^,899 compound interest for 3 years. INTEREST. 209 3. At 6 per cent, what will be the compound interest, and what the amount, of $ 1 for 2 years ? what the amount for 3 years ? for 4 years ? for 5 years ? for 6 years ? for seven years ? for 8 Ans. to the last, $1,593+ years ? It is plain that the amount of $2 for any given time, will be 2 times as much as the amount of $ 1 ; the amount of $3 will be 3 times as much, &c. Hence, we may form the amounts of $1, for several years, into a table of multipliers for finding the amount of any sum for the same time. TABLE, Showing the amount of $1, or 1£, &c. for any number of years, not exceeding 24, at the rates of 5 and 6 per cent, compound interest. Y'rs. 1 2 3 4 5 5 per cent. 1,05 1,1025 1,15762 + 1,21550+ 1,27628+ 6| 1,34009+ 1,40710 + 1,47745 + 1,55132 + 1,62889+ 1,71033+ 12 1,79585 + 7 8 9 10 11 6 per cent. 1,06 1,1236 1,19101 + 1,26247 + 1,33822 + 1,41851 + 1,50363 + 1,59384 + 1,68947+ 1,79084 + 1,89829 + 2,01219 + Y'rs. | 5 per cent. 13 1,88564+ 14|1,97993 + 152,07892 + 162,18287 + 172,29201 + 132,40661+ 192,52695 202,65329+ 2112,78596 + 222,92526 + 233,07152 + 6 per cent. 2,13292+ 2,26090+ 2,39655+ 2,54035+ 2,69277+ 2,85433+ 3,02559+ 3,20713-|- 3,39956 + 3,60353 + 3,81974 + 24 3,22509+14,04893+ will be Note 1. Four decimals in the above numbers sufficiently accurate for most operations. Note 2. When there are months and days, you may first find the amount for the years, and on that amount cast the interest for the months and days ; this, added to the amount, will give the answer. 3. What is the amount of $000,50 for 20 years, at 5 per cent, compound interest ? at 6 per cent. ? $1 at 5 per cent., by the table, is $2,65329 ; therefore, 2,65329 X600,50=$1593,30+ Ans. at 5 per cent. ; and 3,20713x600,50=^1925,881+ Ans. at 6 per cent. 18* 4 210 ARITHMETIC. PART THIRD. 4. What is the amount of $40,20 at 6 per cent, com- pound interest, for 4 years ? for 10 years ? for 18 years ? for 12 years ? for 3 years and 4 months 1 for 24 years, 6 months, and 18 days ? Ans. to last, $168,137 DISCOUNT. Discount is a deduction made from a debt, for paying it before it is due. If, for example, I owe a man $300 two years hence, and am willing to pay him now, I ought to pay only that sum, which, with its interest, would in two years, amount to $300. The question then is, what sum, together with its inte- rest at 6 per cent., would, in two years, amount to $300 ? Such operations are performed by the rule for finding the principal, when the time, rate, and amount are given (see page 201). The sum which, in the time mentioned, would, by the addition of its interest, amount to the sum which is due, is called the present worth. What is the present worth of $834, payable in 1 yr. 7 mo. 6 days, discounting at the rate of 7 per cent. 1 Ans. $750 What is the discount on $321,63, due 4 years hence, at 6 per cent. 1 Ans. $62,26 What principal, at 8 per cent., in 1 yr. 6 mo. will amount to $85,12 ? Ans. $76 What principal, at 6 per cent, in 1 1 mo. 9 d. will amount to $99,311 ? Ans. $94 How much ready money must be paid for a note of $18, due 15 months hence, discounting at the rate of 6 per cent. ? Ans. $16,744 STOCK, INSURANCE, COMMISSION, LOSS AND GAIN, DUTIES. Stock is a name for money invested in banks, in trade, in insurance companies, or loaned to a national govern- ment, for the purpose of receiving interest. STOCK, INSURANCE, &C. 211 Persons who invest money thus, are called stockholders. When stockholders can sell their right to stock, for more than they paid, it is said that stock has risen, and when they cannot sell it for as much as they paid, it is said that stock has fallen. Stock is bought and sold in shares, of from $50 to $100 a share. The nominal value of a share is the amount paid, when the stock was first created. The real value is the sum for which a share will sell. When stock sells for its nominal value, it is said to be at par. When it sells for more than its nominal value, it is said to be above par, and when for less it is below par. When stock is above par it is said to be so much per cent, advance. An Insurance Company, is a body of men, who in re- turn for a certain compensation, promise to pay for the loss of property insured. The written engagement they give, is called a Policy. The sum paid to them for insurance, is called Premium. Commission, is a certain sum paid to a person called a correspondent, agent, Jactor, or broker, for assisting in trans- acting business. Loss and Gain refer to what is made or lost, by mer- chants, in their business. The calculations relating to stock, insurance, commis- sion, loss and gain, and duties, are performed by the rule for calculating interest, when the time is one year ? Rule. Multiply the sum given, by the rate per cent, as a deci- mal. (See page 199.) Examples. Stock.— 1. What is the value of $350.00 of stock at 105 per cent, that is, at 5 per cent, advance ? Ans. $367.50 The rate here is 105 per cent —105 hundredths. The question, then, is, what is 105 hundredths of 350 ; or, mul- tiply 350 by 1.05. '212 ARITHMETIC. PART THIRD. 2. What is the value of 35 hundred dollar shares of stock, at | per cent, advance? Rate 1.0075 Ans $3,526.25 3. At I12i per cent, what must I pay for $7,564.00 of stock? Rate"l. 125. Ans. 8,509.50 4. What is the value of $615.75 of stock, at 30 per cent, advance ? Ans. $800,475 5. What is the value of $7,650.00 of stock at 119i per cent. ? Ans. $9,141.75 6. What is the value of $1,500.00 of stock at 110 per cent. ? Ans. $1,650.00 7. What is the value of $3700 bank stock at 95i per cent., that is at 4| per cent, below par? Ans. $3,533.50 Insurance. — 1. What premium must be paid for the insurance of a vessel and cargo, valued at $123,425.00, at 15i per cent. ? 15i per cent. =.155, and the question is, what is .155 of 123,425. Ans. $19,130,875 2. What must I pay annually for the insurance of a house worth $3,500.00, at If per cent. ? Ans. $61.25 3. What must be paid for the insurance of property, at 6 per cent., to the amount of $2,500.00 ? Ans. $150.00 4. What insurance must be paid on $375,000-00, at 5 per cent. ? . Ans. $18,750.00 5. What premium must be annually paid for the insur- ance of a house worth $10,050.00, at 3 per cent.; and a store worth $15,875.00, at 4 per cent. ; and out houses worth $3,846 00, at 5 per cent. ? 6. What premium must be annually paid for the insur- ance of a Factory worth $30,946.00, at 10 per cent. ; and 7 duelling houses, worth 875.00 each, at 8 percent. ; and 3 grist mills, worth $1,930.00, apiece, at 7 per cent. ; and one storeing house, worth $9,859.00, at 6 per cent. ? Also, what is the average rate of insurance on the whole ? 7. If I pay $930.00 annually for insurance, at 5 per cent., what is the value of the property insured? Here 930 is .05 of the answer ; 930 ~. 05=$ 18,600 An. Profit and Loss. — 1. Sold a bale of goods at $735.00, by which I gain at the rate of 6 per cent. What sum do I gain? Ans. $44.10 2. In selling 50 hhds. of molasses, at 38 dollars a hhd., I gain 10 per cent. What is my gain ? Ans. $190.00 DUTIES. 213 3. In selling 25 bales of cloth, each containing 27 pieces, and each piece 50 yards, a merchant gained 20 per cent, on the cost, which was 10 dollars a yard. What did he gain, and what did he sell the whole for ? Ans. Gain $07,500.00. Whole $405,000.00 4. A merchant gained at the rate of 15 per cent, in selling the following articles : 6 hhds. of brandy for which he paid $1.50 per gal. ; 7 barrels of flour, cost 11 dollars a barrel ; 2 quintals of fish, cost 4 cents a pound; 16 hhds. of molasses, cost 56 cents per gal. and 25 bis. of sugar, containing each 175 lbs., cost 9 cents per lb. What was his gain on the whole, and what did he receive in all ? Commission. — 1. If my agent sells goods to the amount of $2,317.46, what is his commission at 3i per cent. ? Ans. $75.31745 2. What commission must be allowed for a purchase of goods to the amount of $1,286.00, at 2~ per cent. Ans. $32.15 3. What commission shall I allow my correspondent for buying and selling on my account, to the amount of $2,836.23, at 3 per cent. ? 4. A merchant paid his correspondent $25.00 commis- sion on sales to the amount of $1,250.00. At what per cent, was the commission ? He paid him T ||^= J '-^ T |^=.02=2 per cent. Ans. Duties. — Duty is a certain sum paid to government for articles imported. When duty is at a certain rate on the value, it is said to be ad valorem, in distinction from duties imposed on the quantity. An Invoice is a written account of articles sent to a pur- chaser, factor, or consignee. In computing duties, ad valorem, (or ad vol. as it is commonly written,) it is usual in custom houses to add one tenth to the invoice value, before casting the duty. This makes the real duty one tenth greater than the nominal du- ty. It will be equally well to make the rate one tenth greater, instead of increasing the invoice. 1. Find the duty on a quantity of tea, of which the in- voice is $215.17, at 50 per cent. Ans. $118.3435=$! 18.343X 214 ARITHMETIC. PART THIRD. In this example we may add, as directed above, one tenth of 215.17, to 215.17. Thus, 215.17+21.517= 236 687. Then 236.687X50=$ 118.3435. Or we may add to the rate .50, one tenth of itself=.05 : thus, .50-j- .05=55. Then, 215.17 x-55=$l 18.3435, as before. 2. Find the duty on a quantity of hemp at 131 per cent., of which the invoice is $654.59. The second of the above modes is recommended. Another might be used, viv. : to find, first, the duty on the invoice at the given rate, and add to it one-tenth of itself. Thus, 654.59 X 13-' =$88. 36965. Ans. 897.2066 15 3. What is the duty on a quantity of books, of which the invoice is $ 1,670.33," at 20 per cent. ? Ans. $367.4726 EQUATION OF PAYMENTS. Equation of payments is a method of finding a time for paying several debts due at different times, all at once ; and in such a way that both creditor and debtor will have the same value, as if the debts were paid at the several times promised. For if a man pays a debt before it is due, the creditor gains ; if he pays it after it is due, the debtor gains. In how many months will $1 gain as much at interest as $8 will gain in 4 months ? Now as the $1 is 8 times less than 8, it will require 8 times more time, or 8x4=32 months. In how many months will the interest on $9 equal the interest on $1 for 40 months ? Supposing a man owes me $12 in 3 months, $18 in 4 months, and $20 in 9 months. He wishes to pay the whole at once ; in what time ought he to pay ? $12 for 3 months=$l for 36 months. $18 for 4 months=$l for 72 months. $20 for 9 months=$l for 180 months. $50 288 months. Now it appears that it will be the same to him to have $1 for 36, for 72, and for 180 months, as it would to have the 12, the 18, and the 20 dollars for the number of months specified. RATIO. 215 He might therefore keep $1 just 288 months, and it would be the same as keeping the $50 for the number of months specified. But as the whole sum of money lent was -$50, he may keep this only one fiftieth (j\) of the time he might keep $1. Therefore divide the 288 months by the 50, and the answer is 5|| months. Rule for finding the mean time of several payments. Multiply each sum by the time of its payment. Divide the sum of these products by the sum of the payments, and the quotient is the mean time. A man is to receive $500 in 2mo. ; $100 in 5 mo. ; $300 in 4 mo. If it is paid all at once, at what time should the payment be made 1 A man owes me $300, to be paid as follows ; i in 3 months ; } in 4 months ; and the rest in G months ; what is the mean time for payment ? Ans. 4| months. RATIO. The word ratio means relation ; and when it is asked what ratio one number has to another, it means in what relation does one number stand to another. Thus, when we say the ratio of 1 to 2 is ^, we mean that the relation in which 1 stands to 2 is that of one half to & whole. Again, the ratio of 3 to 4 is £ , that is, 3 is J of 4, or stands in the relation of £ to the 4. So also the ratio of 4 to 3 is | ; for the 4 is 4 thirds of 3, and stands to it therefore in the relation of |. What is the relation of 11 to 12 1 of 12 to 11 ? When therefore we find the ratio of one number to an- other, we find what part of one number another is. Then the ratio of 4 to 6 is £ ; that is, 4 is 4 sixths of 0. The ratio of one number to another, then may always be expressed by a fraction in which the first number (called the antecedent) is put for numerator, and the second number (called the consequent) is put for denominator. Thus the ratio of 8 to 4 is f. This is an improper fraction, and, changed to whole numbers, is 2 units. The ratio of 8 to 216 ARITHMETIC. PART THIRD. 4, then, is 2. That is, 8 is twice 4, or stands to 4 in the relation of a duplicate or double* PROPORTION. When quantities have the same ratio, they are said to be proportional to each other. Thus the ratio of 2 to 4 is £, and the ratio of 4 to 8 is i ; that is, 1 has the same relation to 2, that 4 has to 8, and therefore these numbers are called proportionals. Again, 4 is the same portion or part of 8, as 10 is of 20, and therefore these numbers are called proportionals. A proportion, then, is a combination of equal ratios. Points are used to indicate that there is a proportion between numbers. Thus 4:8::9:18 is read thus ; 4 has the same ratio to 8, that 9 has to 18. Or more briefly, 4 is to 8, as 9 to 18. It will always be found to be the case in proportionals, that multiplying the two antecedents into the two consequents, produce the same product. Thus, 2 : 4 : : 6 : 12 Here let the consequent 4 be multiplied into the ante- cedent 6, and the product is 24 ; and let the atecedent 2 be multiplied into the consequent 12, and the product also is 24. If then we have only three terms in a proportion, it is * The pupil needs to be forewarned that there is a difference be- tween French and English mathematicians in expressing Ratio. The French place the antecedent as denominator, and the conse- quent as numerator. The English, on the contrary, place the ante- cedent as numerator, and the consequent as denominator. It seems desirable that there should be an agreement on this subject, in school books at least. Two of the most popular Arithmetics now in use, have adopted the French method, viz. Colburn and Adams. It seems needful to mention this, that pupils may not be needlessly perplexed, if called upon to use different books. The method used here, is the English ; as the most common, and as most consonant with perspicuity of language. For there seems to be no propriety in saying that the relation of 2 to 4 is A. The ratio between these two numbers maybe either | or |, but the rela- tion of 2 to 4, to use language strictly can be nothing but a. PROPORTION. 217 easy to find the fourth. For when we have multiplied one antecedent into one consequent, we know that the term left out, is a number that, multiplied into the remaining term, would produce the same product. Thus let one term be left out of this proportion. 8 : 4 : : 12 : Here the consequent is gone from the last ratio. We multiply the antecedent 12 into the consequent 4, and the answer is 48. We now know that the term left out, is a number which, multiplied into the 8, would produce 48. This number is found by dividing 48 by 8, the answer is 6. Whenever, therefore, a term is wanting to any propor- tion, it can be found by multiplying one of the antecedents by one of the consequents, and dividing the product by the remaining number. What is the number left out in this proportion? 3 : 12 : : 24 : What is the number left out in this proportion ? 9 : 8 : : 27 : In a proportion, the two middle terms are called the means, and the first and last terms are called the extremes. Rule for finding a fourth term in a Proportion. Multiply the means together, and divide the product by the remaining number. It is on this principle, that what is commonly called the " Rule of Three," is constructed. By this process, we find a fourth term when three terms of a proportion are given. Such sums as the following are done by this rule. If 4 yards of broadcloth cost $12, what cost 9 yards ? Now the cost is in proportion to the number of yards ; that is, the same ratio exists between the number of yards, as exists between the cost of each. Thus, — as 4 yards is to 9 yards, so is the cost of 4 yards to the cost of 9 yards. The proportion, then, is expressed thus : yds. yds. $ 4 : 9 : : 12 : Here the term wanting, is the cost of 9 yards ; and if we multiply the means together, and divide by the 4, the 19 218 ARITHMETIC. PART THIRD, answer is 27 ; which is the other term of the proportion; and is the cost of 9 yards. Again, if a family of 10 persons spend 3 bushels of malt a week, how many bushels will serve at the same rate when the family consists of 30 ? Now there is the same ratio between the number of bushels eaten, as between the numbers in the family. That is, as is the ratio of 10 to 30, so is the ratio of 3 to the number of bushels sought. Thus, 10 : 30 : : 3 : Rule of Proportion ; or Rule of Three. When three numbers are given, place thai one as third term, which is of the same kind as the answer sought. If the an- swer is to be greater than this third term, place the greatest of the remaining numbers as the second term, and the less num- ber as first term. But if the answer is to be less, place the less number as second term, and the greater as first. In either case, multiply the middle and third terms toge- ther, and divide the product by the first. The quotient is thf answer, and is always of the same order as the third term. Note. This rule may be used both for common, com- pound, and decimal numbers. If the terms are compound, they must be reduced to units of the lowest order men- tioned. Many of the sums which follow will be better understood if performed by the mode of analysis, which has been explained and illustrated in a former part. For example, we will take the first sum done by the rule of proportion. If 4 yards of broadcloth cost $12, what cost 9 yards ! We reason thus, — If 4 yards cost $12, one yard must cost a fourth of $12. Therefore, divide $12 by 4, and we have the cost of one yard. Multiply this by 9, and we have the cost of 9 yards. (It is usually best to multiply first and then divide, and it has been shown that this is more convenient, and does not alter the answer.) PROPORTION. 219 Lot the following sums be done by the Rule of Propor- tion, and then explained by analysis. 1 . If the wages of 15 weeks come to 64 dols. 19 cts. what is a year's wages at that rate ? Ans. $222, 52 cts. 5m. 2. A man bought sheep at 81.11 per head, to the amount of $51.6 ; how many sheep did he buy ? Ans. 46 3. Bought 4 pieces of cloth, each piece containing 31 yds. at 1 6s. 6d. per yard, (New England currency,) what does the whole amount to in federal money ? Ans. 8341 When a tun of wine cost 8140, what cost a quart ? Ans. 13 cts. S^m. 4. A merchant agreed with his debtor, that if he would pay him down 65 cents on a dollar, he would give him up a note of hand of 2 19 dollars 88 cts. I demand what the debtor must pay for his note ? Ans. $162.42 cts. 2m. 5. If 12 horses eat 30 bushels of oats in a week, how many bushels will serve 45 horses the same time ? Ans. 112^ bushels. 6. Bought a piece of cloth for 848.27 cts. at 8 1.19 cts. per yard ; how many yards did it contain ? Ans. 40 yds. 2 qrs. ^ 7. Bought 3 hhds. of sugar, each weighing 8cwt. lqr. 12 lb. at $7.26 cts. per cwt. ; what come they to ? Ans. $182.1 ct. 8 in. 8. What is the price of 4 pieces of cloth, the first piece containing 21, the second 23, the third 24, and the fourth 27 yards, at 81.43 cts. a yard ? Ans. $135.85 cts. 21+23+24+27=95 yds. 9. Bought 3 hhds. of brandy, containing 61, 62, 62£ gals, at 81 38 cts. per gallon. I demand how much they a- mount to ? Ans. $255.99 cts. 10. Suppose a gentleman's income is $1836 a year, and he spends $3.49 cts. a day, one day with another, how much will he have saved at the year's end? Ans. $562.15 cts. 11. A mereh't bought 14 pipes of wine, and is allowed 6 months credit, but for ready money gets it 8 cents a gallon cheaper ; how much did he save by paving ready money ? " Ans. $141.12 cts 12. Sold a ship for 537Z. and I owned f of her ; what was my part of the money? Ans. £201.7s. 6d, 220 ARITHMETIC. PART THIRD. 13. If T 5 ? of a ship cost $718. 25 cents, what is the whole worth ? 5 : 781,25 : : 16 : $2500 Ans. 14. If I buy 54 yards of cloth for £31. 10s. what did it cost per Ell English 1 Ans. 14s. 7d. 15. Bought of Mr. Grocer 11 cwt. 3 qrs. of sugar, at $8,12 per cwt. and gave him James Pay well's note for £ 19.7s. (New England currency) the rest I pay in cash ; tell me how many dollars will make up the balance. Ans. $30,91 16. If a staff 5 feet long casts a shade on level ground 8 feet, what is the height of that steeple whose shade at the same time measures 181 feet 1 Ans. I13i ft. 17. If a gentleman has an income of 300 English guineas a year, how much may he spend, one day with another, to lay up 500 dollars at the year's end ? Ans. $2,46cts. 5m. 18. Bought 50 pieces of kerseys, each 34 Ells Flemish, at 8s. 4d. per Ell English ; what did the whole cost ? £425 19. Bought 200 yards of cambric for £90, but being damaged, I am willing to lose £7. 10s. by the sale of it ; what must I demand per Ell English ? Ans. 10s. 3f d. 20. How many pieces of Holland, each 20 Ells Flemish, may I have for £23.8s. at 6s. 6d. per Ell English ? Ans. 6 pieces. 21. A merchant bought a bale of cloth containing 240 yds. at the rate of $7i for 5 yards, and sold it again at the rate of $11| for 7 yards ; did he gain or lose by the bargain, and how much ? Ans. He gained $25,71 cts. 4m. + 22. Bought a pipe of wine for 84 dollars, and found it had leaked out 12 gallons ; I sold the remainder at 12i cents a pint ; what did I gain or lose ? Ans. I gained $30 23. A gentleman bought 18 pipes of wine at 12s. 6d. (N. Jersey currency) per gallon ; how many dollars will pay the purchase ? Ans. $3780 24. Bought a quantity of plate, weighing 15 lb. 11 oz. 13 pwt. 17 gr. how many dollars will pay for it, at the rate of 12s. 7d. (New York currency,) per ounce ? Ans. $301,50 cts. 2 T \ m. 25 A factor bought a certain quantity of broadcloth and drugget, which together cost £81 per yard, the quantity of broadcloth was 50 yards, at 18s. per yard, and for every 5 yards of broadcloth he had 9 yards of drugget ; 1 de. PROPORTION. 221 mand how many yards of drugget he had, and what it cost him per yard ? Ans. 90 yards at 8s. per yard. 26. If I give 1 eagle, 2 dollars, 8 dimes, 2 cents and 5 mills, for 675 tops, how many tops will 19 mills buy ? Ans. 1 top. 27. If 100 dollars gain 6 dollars interest in a year, how much will 49 dollars gain in the same time 1 Ans. $2,94 cts. 28. If 60 gallons of water, in one hour, fall into a cistern containing 300 gallons, and by a pipe in the cistern, 35 gal- lons run out in an hour ; in what time will it be filled ? Ans. in 12 hours. 29. A and B depart from the same place and travel the same road ; but A goes 5 days before B, at the rate of 15 miles a day ; B follows at the rate of 20 miles a day ; what distance must he travel to overtake A ? Ans. 300 miles. COMPOUND PROPORTION. Compound proportion, is a method of performing such operations in proportion, as require two or more Stat- ings. It is sometimes called Double Rule of Three, be- cause its operations can be performed by two operations of the Rule of Three. For example : If 56 lbs. of bread are sufficient for 7 men 14 days, how much bread will serve 21 men 3 days? Here the amount of bread consumed depends upon two circumstances, the number of days, and the number of men. We will first consider the quantity of bread as depend- ing upon the number of men, supposing the number of days to be the same. The proportion would then be this ; 7 men : 21 men : : 56 lbs. to the number of lbs. re- quired. Here we multiply the means together, and divide the answer by 7, and the answer is 168. That is, if the time was the same, viz. 14 days, the 21 men would eat 168 lbs, in that time. We now make a second statement thus : 19* . 222 ARITHMETIC. PART THIRD. 14 days : 3 days : : 168 lbs. : number of lbs. requir- ed. The result of this statement is 36 lbs. which is the an- swer. In performing this operation, let the pupil notice that in the first statement, the 56 was multiplied by the 21 and the answer divided by 7. This gives the same answer as would be given, did we divide first, and then multiply. That is, 56 multiplied by 21, and the product divided by 7, is the same as 56 divided by 7 and the quotient multi- plied by 21. We divide by 7, to find how much one man would eat in the same time, or 14 days, and multiply by 21, to find how much 21 men would eat. When we make the second statement, as we have found how much 21 men would eat in 14 days, we divided the quantity (168 lbs.) by 14, to find how much they would eat in one day, and then multiply by 3, to find how much they would eat in 3 days. But in this case also, the mul- tiplication is done first. Let the pupil also notice that the 56 lbs. was multiplied by 21 and divided by 7, and then that the answer to this (168 lbs.) was multiplied by 3 and divided by 14. Here 21 and 3 are used as multipliers, and 14 and 7 are used as divisors. The answer will be the same (as may be found by trial) if 56 is multiplied by the product of these multipliers, and the answer divided by the product of the divisors. It is on this principle that the common rule in compound proportion is constructed, which is as follows. Rule of Compound Proportion. Make the number which is of the same kind as the answer required, the third term. Take any two numbers of the same kind, and arrange them in regard to this third term, according to the rule of proportion. Then take any other two numbers of the same kind, and arrange them in like manner, and so on till all the numbers are used. Then multiply the third term, by the product of the second PROPORTION. 228 terms, and divide the answer by the product of the first terms. The quotient is the answer. Examples. 1. If a man travel 273 miles in 13 days, travelling only 7 hours a day, how many miles will he travel in 12 days at the rate of 10 hours a day ? Here the number, which is of the same kind as the answer required, is the 273 miles, and this is put as third term. We now take two numbers of the same kind, viz. 13 days and 12 days, and placing them according to the rule of simple proportion, the question would stand thus. 13 : 12 : : 273 : We next take two other numbers of the same kind, viz. 10 hours, and 7 hours, and arrange them under the former proportion according to the same rule, thus : 7 : 10$ ' * * We now multiply the 273 by the product of 12 and 10, and divide by the product of 13 and 7 and the quotient is the answer. We can explain this process analytically, thus. We divide by 13, to find how much the man would tra- vel in one day, at the rate of 7 hours per day. We multiply by the 12, to find how much he would travel in 12 days, at the same rate. We divide by 7 to find how much he would travel in one hour, and multiply by 10 to find how much he would trave 1 in 10 hours. Let the pupils explain the following in the same man- ner. Examples. 2. If £100 in one year gain £5 interest, what will be the interest of £750 for 7 years ? Ans. £262. 10s. 3. What principal will gain £262. 10s. in 7 years, at 5 per cent, per annum ? Ans. £750. 4. If a footman travel 130 miles in 3 days, when the days are 12 hours long ; in how many days, of 10 hours each, may he travel 360 miles ? Ans. 9|f days. 224 ARITHMETIC. PART THIRD. 5. If 120 bushels of corn can serve 14 horses 56 days, how manv days will 94 bushels serve 6 horses ? Ans. 102if days. 6. If 7 oz. 5 pwts. of bread be bought at 4fd. when corn is at 4s. 2d. per bushel, what weight of it may be bought for Is. 2d. when the price of the bushel is 5s. 6d. ? Ans. 1 lb. 4 oz. 3££§ pwts. 7. If the carriage of 13 cwt. 1 qr. for 72 miles be £2. 10s. Gd. what will be the carriage of 7 cwt. 3 qrs. for 112 miles ? Ans. £2.5s. lid. l^q. 8. A wall, to be built to the height of 27 feet, was raised to the height of 9 ft. by 12 men in 6 days ; how many men must be employed to finish the wall in 4 days at the same rate of working ? Ans. 36 men. 9. If a regiment of soldiers, consisting of 939 men, can eat 351 quarters of wheat in 7 months ; how many soldiers will eat 1464 quarters in 5 months, at that rate 1 Ans. 5483fl&. 10. If 248 men, in 5 days of 11 hours each, dig a trench 230 yards long, 3 wide and 2 deep ; in how many days of 9 hours each, will 24 men dig a trench of 420 yards long, 5 wide and 3 deep ? Ans. 2883 S eV 11. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft, thick, in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick ? Ans. 80 days. 12. If the freight of 9 hhds. of sugar, each weighing 12 cwt., 20 leagues, cost £16, what must be paid for the freight of 50 tierces, each weighing 2| cwt., 100 leagues 1 Ans. £921. Is. 10|d. 13. If 4 reapers receive $11.04 for 3 days' work, how many men may be hired 16 days for $103.04 ? Ans. 7 men. 14. If 7 oz. 5 pwt. of bread be bought for 4fd. when corn is 4s. 2d. per bushel,what weight of it may be bought for Is. 2d. when the price per bushel is 5s. 6d. ? Ans. 1 lb. 4 oz. 3f|f pwts. 15. If 8100 gain $6 in 1 year, what will 400 gain in 9 months ? 16. If $100 gain $6 in 1 year, in what time will $400 gain $18 ? 17. If $400 gain $18 in 9 months, what is the rate per cent, per annum 1 FELLOWSHIP. 225 18. What principal, at 6 per cent, per arm., will gain $18 in 9 months ? 19. A usurer put out $75 at interest, and, at the end of 8 months, received, for principal and interest, $79 ; I demand at what rate per cent, he received interest. Ans. 8 per ct. 20. If 3 men receive £8 T \ for 19^ days work, how much must 20 men receive for 1001 days' ? Ans. £305 0s. 8d. 21. If 40 men in 10 days, can reap 200 acres of grain, how many acres can 14 men reap in 24 days ? Ans. 168 acres. 22. If 14 men in 24 days, can reap 168 acres of grain ; how many acres can 40 men reap in 10 days ? Ans. 200 acres. 23. If 16 men in 32 days, can mow 256 acres of grass ; in how many days will 8 men mow 96 acres ? Ans. 24 days. 24. If 4 men mow 96 acres in 12 days ; how many acres can 8 men mow in 16 days ? Ans. 256 25. if a family of 16 persons spend $320 in 8 months ; how much would 8 of the same family spend in 24 months ? Ans. 8480 26. If a family of 8 persons in 24 months spend $480 ; how much would they spend, if their number were doub- led, in 8 months ? Ans. $320 27. If 12 men build a wall 100 feet long, 4 ft. high, and 3 ft. thick, in 40 days ; in what time will 6 men build one, 20 ft. long, 6 ft. high, and 4 ft. thick ? FELLOWSHIP. The Rule of Fellowship, is a method of ascertaining the respective gains or losses of individuals engaged in joint trade. Let the pupils perform the following sums as a mental exercise. 1. Two men own a ticket; the first owns i, and the second owns f of it ; the ticket draws a prize of 40 dollars ; what is each man's share of the money ? 2. Two men purchase a ticket for 4 dollars, of which one man pays 1 dollar, and the other 3 dollars ; the ticket draws 40 dollars ; what is each man's share of the money ? 226 ARITHMETIC. PART THIRD. 3. A and B bought a quantity of cotton ; A paid $100, and B $200 ; they sold it so as to gain $30 ; what were theirrespective shares of the gain ? The value of what is employed in trade is called the Capital, or Stock. The gain or loss to be shared is called the Dividend. Each man's gain or loss is always in proportion to his share of the stock, and on this principle the rule is made. Rule. As the whole stock is to each man's share of the stock, so is the whole gain or loss, to his share of the gain or loss. 4. Two persons have a joint stock in trade ; A put in $250, and B $350 ; they gain $400 ; what is each man's share of the profit ? Operation. A's stock, $250 1 Then, B's stock, $350 j^ 60() . 250 . . 400 . $ 166#666 i A ' s gam . Whole stock $000 j 600 : 350 : : 400 : $233,333^ B's gain. The pupil will perceive that the process may be con- tracted by cutting off an equal number of ciphers from the first and second, or first and third terms ; thus, 6 : 250 : : 4 : 166.666|, &c. It is obvious the correctness of the work may be ascer- tained by finding whether the sums of the shares of the gains are equal to the whole gain ; thus, $166.666§-f- $233.333i=$400, whole gain. 5. A, B, and C, trade in company : A's capital was $175, B's 200, and C's $500 ; by misfortune they lose $250 ; what loss must eaeh sustain ? ( $ 50., A's loss. Ans. 1$ 57.1421, B's loss. ($142.857i, C's loss. 6. Divide 600 among 3 persons, so that their shares may be to each other as 1, 2, 3, respectively. Ans. $100, $200, and $300 In assessing taxes, it is customary to obtain* an inven- tory of every man's property, in the whole town, and also a list of the number of polls. Each poll is rated at a tax FELLOWSHIP. ^1 of a certain value. From the whole tax to be raised is taken out what the tax on polls amounts to, and the re- mainder of the tax is to be assessed on the property in the town. We may then find the tax upon 1 dollar, and make a table containing the taxes on 1,2, 3, dzc. to 10 dollars ; then on 20, 30, &c. to 100 dollars ; and then on 100, 200, &c. to 1000 dollars. Then, knowing the inventory of any individual, it is easy to find the tax upon his proper- ty- I. A certain town, valued at $64530, raises a tax ot $2259.90; there are 540 polls, which are taxed $,60 each ; what is the tax on a dollar, and what will be A's tax, whose realestate is valued at $1340, his personal pro- perty at 8874, and who pays for 2 polls ? 540 x ,60 = $324, amount of the poll taxes, and $2259,90,— $324=1935,90, to be assessed on property. $645301: $1935,90 : : $1 : ,03 ; or,»|£g tV=> 03 > tax on 81 TABLE. dolls. ,30 ,60 30 dolls, dolls Tax on 1 is ,03 « 2 " ,06 « 3 " ,09 « 4 " ,12 <« ,15 6 « ,18 7 « ,21 8 " ,24 9 « ,27 dolls. Tax on 10 is « 20 " dolls. dolls. Tax on 100 is 3, " 200 « 0, " 300 " 9, " 400 " 12, " 500 « 15, " 600 " 18, " 700 " 21, « 800 « 24, " 900 « 27, '« 1000 " 30, Now, to find A's tax, his real estate being $1340, I find by the table, that The tax on $1000 - is - 30, The tax on 300 . 9, The tax on 40 1,20 40 50 60 70 80 90 ,90 1,20 1,50 1,80 2,10 2,40 2,70 Tax on his real estate - - - $40,20 In like manner I find the tax on his personal > ^g oo property to be $ ' 2 polls, at ,60 each, are 1,20 Amount, $67,62 228 ARITHMETIC. PART THIRD. 2. What will B's tax amount to, whose inventory is 874 dollars real, and 210 dollars personal property, and who pays for 3 polls ? Ans. $34.32 3. What will be the tax of a man, paying for 1 poll, whose property is valued at $3482 ? at $768 ? Ans. to the last, $140.31 Two men paid 10 dollars for the use of a pasture 1 month ; A kept in 24 cows, and B 16 cows ; how much should each pay ? 4. Two men hired a pasture for $10 ; A put in 8 cows 3 months, and B put in 4 cows 4 months; how much should each pay ? The pasturage of 8 cows for 3 months is the same as oi 24 cows for 1 month, and the pasturage of 4 cows for 4 months is the same as of 16 cows for 1 month. The shares of A and B, therefore, are 24 to 16, as in the former ques- tion. Hence, when lime is regarded in fellowship, — Multiply each one's stock by the time he continues it in trade, and use the product for his share. This is called Double Fellowship. Ans. A 6 dollars, and B 4 dollars. 5. A and B enter into partnership ; A puts in $100 6 months, and then puts in $50 more ; B puts in $200 4 months, and then takes out $80 ; at the close of the year they find that they have gained $95 ; what is the profit of each ? . ( $43,711, A's share. Ans ' I $51,288, B's share. 6. A, with a capital of $500, began trade, Jan. 1, 1826, and, meeting with success, took in B as a partner, with a capital of 600, on the first of March following ; four months after, they admit C as a partner, who brought $800 stock ; at the close of the year, they find the gain to be $700 ; how must it be divided among the partners ? $250, A's share. Ans. ^ $250, B's share. $200, C's share. ALLIGATION. The rule of Alligation teaches how to gain the mean value of a mixture that is made by uniting several articles of different values. ALLIGATION. 229 Alligation Medial, teaches how to obtain the value, (or mean price,) of a mixture, when the quantities and prices of the several articles are given. Rule. As the whole mixture is to the whole value, so is any par' of the composition, to its mean price. Examples. 1. A farmer mixed 15 bushels of r) r e, at 64 cents a bushel, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel ; I demand what a bushel of this mixture is worth ? bu. cts. $ cts. bu. $ cts. bu 15 at 64=9,60 As 54 : 25,38 : : 1 18 55=9,90 1 21 28=5,88 cts. — 54)25,38(.47 Ans. 54 25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel, be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth ? Ans. 91,20 cts. 3. A tobacconist mixed 36 lb. of tobacco, at Is. 6d. per lb., 12 lb. at 2s. a pound, with 12 lb. at Is. JOd. per lb. ; what is the price of a pound of this mixture ? Ans. Is. 8d. 4. A grocer mixed 2 C. of sugar at 56s. per C. and 1 C. at 43s. per C. and 2 C. at 50s. per C. together ; I de- mand the price of 3 cwt. of this mixture? Ans. £7. 13s. 5. A wine merchant mixes 15 gallons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons at 6s. 3d. ; what is a gallon of this composition worth ? Ans. 5s. lOd. 24| qrs. Alligation Alternate, teaches how to find the quantity of each article, when the mean price of the whole mixture, and also the prices of each separate article are known. Rule. Reduce the mean price and the prices of each separate article to the same order. 20 230 ARITHMETIC. PART THIRD. Connect with a line each price that is less than the mean price, with one or more that is greater ; and each price greater than the mean price, with one or more that is less. W?ite the difference between the mean price, and the price of each separate article, opposite the price with which it is connected ; then the sum of the differences, standing against any price, will express the relative quantity to be taken oj that price. Examples. 1. A merchant has several kinds of tea ; some at ti shillings, some at 9 shillings, some at 11 shillings, and some at 12 shillings per pound ; what proportions of each must he mix, that he may sell the compound at 10 s. per pound ? The pupil will perceive, that there may be as many different ways of mixing the simples, and, consequently as many different answers, as there are different ways of linking the several prices. Operations. lbs. 8s. ,-2 \ Or, lit Here the prices of the simples, are set one directly under another, in order, from least to greatest, and the mean rate, (10s.) written at the left hand. In the first way of linking, we take in the proportion of 2 pounds of the teas at 8 and 12s. to 1 pound at 9 and lis. In the second way, we find for the answer, 3 pounds at 8 and lis. to 1 pound at 9 and 12s. 2. What proportions of sugar, at 8 cents, 10 cents, and 14 cents per pound, will compose a mixture worth 12 cents per pound ? Ans. In the proportion of 2 lbs. at 8 and 10 cents, to f> lbs. at 14 cents. Note. As these quantities only express the proportions of each kind, it is plain, that a compound of the same mean price will be formed by taking 3 times, 4 times, one half, or any proportion, of each quantity. Hence, ALLIGATION. 231 When the quantity of one simple is given, after finding the proportional quantities, by the above rule, we may say, As the proportional quantity ; is to the given quantity : . >«> is each of the other proportional quantities : to the re- quired quantities of each. 3. If a man wishes to mix 1 gallon of brandy worth 16s. with rum at 9s. per gallon, so that the mixture may be worth lis. per gallon, how much rum must he use ? Taking the differences as above, we find the propor- tions to be 2 of brandy to 5 of rum ; consequently, 1 gal- lon'of brandy will require 21 gallons of rum. Ans. 2i gallons. 4. A grocer has sugars worth 7 cents, 9 cents, and 12 cents per pound, which he would mix so as to form a com- pound, worth 10 cents per pound ; what must be the pro. portions of each kind ? Ans. 2 lbs. of the first and second, to 4 lbs. of the 3d kind. 5. If he use 1 lb. of the first kind, how much must he take of the others? if 4 lbs., what ? if 6 lbs., what ? if 10 lbs., what ? if 20 lbs., what ? Ans. to the last, 20 lbs. of the 2d, and 40 of the 3d. 6. A merchant has spices at 16d. 20d. and 32d. per pound ; he would mix 5 pounds of the first sort with the others, so as to form a compound worth 24d. per pound ; how much of each sort must he use ? Ans. 51bs. of the second, and 1\ lbs. of the third. 7. How many gallons of water, of no value, must be mixed with 60 gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon ? Ans. 8± galls. 8. A man would mix 4 bushels of wheat, at $1,50 per bushel, rye at $1,16, corn at 8,75, and barley at 8,50, so as to sell the mixture at 8,84 per bushel ; how much of each may he use ? When the quantity of the compound is given, we may say, As the sum of the proportional quantities, found by the above rule, is to the quantity required, so is each pro- portional quantity, found by the rule, to the required quantity of each. 9. A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cents, and some at 14 cents per pound, 232 ARITHMETIC. PART THIRD. so that the compound may be worth 12 cents per pound ; how much of each kind must he use ? We find the proportions to be, 2, 2, and 6. Then, 2-f-2 +6= 10, and C 2 : 20 lbs. at 8 cts. ) 10 : 100 : : 1 2 : 20 lbs. at 10 cts. V Ans. (6 : 60 lbs. at 14 cts. ) 10. How many gallons of water, of no value, must be mixed with brandy at $1,20 per gallon, so as to fill a ves- sel of 75 gallons, which may be worth 92 cents per gal. ? Ans. 17i gallons of water to 57i gallons of brandy. 11. A grocer bas currants at 4d., 6d., 9d., and lid. per lb. ; and he would make a mixture of 240 lbs., so that the mixture may be sold at 8d. per lb. ; how many pounds of each sort may he take 1 Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &q. Note. This question may have five different answers. DUODECIMALS. Duodecimal is derived from the Latin word duodecim, signifying twelve. They are fractions of a foot, which is supposed to be divided into twelve equal parts called primes, marked thus, ('). Each prime is supposed to be subdivided into 12 equal parts called seconds, marked thus, ("). Each second is also supposed to be divided into twelve equal parts called thirds, marked thus ('"), and so on to any extent. It thus appears that 1 an inch or prime is t l of a foot. 1" a second is T \ of T ' 5 or T ^ of a foot. V" a third is T \ of T » 5 of t l, or T ^j of a foot, &c. Whenever therefore any number of seconds (as 5") are mentioned, it is to be understood as so many y i T of a foot, and so of the thirds, fourths, &c. Duodecimals are added and subtracted like other com- pound numbers, 12 of a less order making 1 of the next higher, thus, 12"" fourths make 1 third 1"'. 12"' thirds make 1 second 1". 12" seconds make 1 prime or inch 1'. DUODECIMALS. 233 12' inches or primes, make 1 foot. The addition and subtraction of Duodecimals is the same as other compound numbers. These marks ' " '" "" are called indices. Multiplication of Duodecimals. Duodecimals are chiefly used in measuring surfaces and solids. How many scpiare feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide ? Note. The square contents of any thing are found by multiplying the length into the breadth. The following example is explained above. Examples. 16 r 1 3' 16 7' 4 1' 9 20 8' 9" It is generally more convenient to multiply by the higher orders of the multiplier first. Thus we begin and multiply the multiplicand first by the 1 foot, and set down the answers as above. We then multiply by the 3' or T 3 2 of a foot. 16 is chan- ged to a fraction, thus y, and this multiplied by T 3 ^ is f£, or 48', which is 4 feet, (for there are 12' in every foot,) and is set under that order. We now multiply 7' (or T \) by 3' (or T 3 ^) and the answer is T V T or21". This is 1' to set under the order of seconds, and 9" (pf T ) to be set under the order of thirds. The two products are then added together, and the answer is obtained, which is 20 feet 8 primes 9 seconds. Another example will be given in which the cubic con- 20* 234 ARITHMETIC. PART THIRD. tents of a block are found by multiplying the length, breadth and thickness together. How many solid feet in a block 15 ft. 8' long, 1 ft. 5' wide, and 1 ft. 4' thick ? Length, Breadth, i 15 1 Operation. 8' 5' 4" 15 6 8' 6' Thickness 22 1 2' 4' 4" 22 7 2' 4' 4" 9" 4'" Ans. 29 r 1" 4'" Let this example be studied and understood before the rule is learned. If any difficulty is found, let both multi- plier and multiplicand be expressed as Vulgar Fractions, and then multiply. In duodecimals it is always the case that the 'product of two orders, will belong to that order which is made by ad- ding the indices of the factors. Rule. Write the figures as in the addition of compound numbers. Multiply by the higher orders of the multiplier first, remem- bering that the product of two orders belongs to the order de- noted by the sum of their indices. If any product is large enough to contain units of a higher order, change them to a higher order, and place them where they belong. Examples. How many square feet in a pile of boards 12 ft. 8' long, and 13' wide ? INVOLUTION. 235 What is the product of 371 ft. 2' 6" multiplied by lttl a. 1' 9" ? Ans. 07242 ft. 10' 1" 4'" 6"". If a floor be 10 ft. 4' 5" long, and 7 ft. 8' 6" wide, what is its surface ? Ans. 79 ft. 11' 0" 6'" 6"". What is the solidity of a wall 53 ft. 6' long, 10 ft. 3' high, and 2 ft. thick ? Ans. 1096| ft. INVOLUTION. When a number is multiplied into itself, it is said to be involved, and the process is called Involution, . Thus, 2X2x2 is 8. Here the number 2 is multiplied into itself twice. The product which is obtained by multiplying a number into itself, is called a Power. Thus, when 2 is multiplied into itself once, it is 4, and this is called the second power of 2. If it is multiplied into \tse\t' twice (2x2x2=8) the answer is 8, and this is called the third power. The number which is involved, is called the Root, or first power. Thus, 2 is the root of its second power 4, and the root of its third power 8. A power is named, or numbered, according to the number of times its root is used as a factor. Thus the number 4 is called the second power of its root 2, because the root is twice used as a factor ; thus, 2x2=4. The number 8 is called the third power of its root 2 ; because the root is used three times as a factor ; thus, 2X2x2=8. The method of expressing a power, is by writing its root, and then above it placing a small figure, to show the number of times that the root is used as a factor. Thus the second power of 2 is 4, but instead of writing the product 4, we write it thus, 2 2 . The third power of 2 is written thus, 2 3 . The fourth power of 2 is 10, and is written thus, 2 4 . The small figure that indicates the number of times that the root is used as a factor, is called the Index, or Expo- nent. 236 ARITHMETIC. PART THIRD. The different powers have other names beside their numbers. Thus, the second i>ower is called the Square. The third power is called the Cube. The fourth power is called the Biquadrate. The fifth power is called the Sursolid. The sixth power is called the Square-cubed. Powers are indicated by exponents. When a power is actually found by multiplication, involution is said to be performed, and the number or root is involved. Rule of Involution. To involve a number, multiply it into itself, as often as there are units in the exponent, save once. Note. — The reason why it is multiplied once less than there are units in the exponent, is, that the first time the number is multiplied, the root is used twice as a factor ; and the exponent shows, not how many times we are to multiply, but how many times the root is used as a factor. 1. What is the cube of 5 ? Ans. 5x5X5=125 2. What is the 4th power of 4 ? Ans. 256 3. What is the square of 14? Ans. 196 4. What is the cube of 6 ? Ans. 216 5. What is the 5th power of 2 ? Ans. 32 6. What is the 7th power of 2 ? Ans. 128 7. What is the square of \ 1 Ans. J 8. What is the cube of § ? Ans. JL A Fraction is involved, by involving both numerator and denominator. 9. What is the fourth power off ? Ans. g%L 10. What is the square of 51 ? Ans. 30£ 11. What is the square of 30i ? Ans. 915 T ' F 12. Perform the involution of 8s. Ans. 32,768 13. Involve ^ }i, and f to the third power each. AnLES. 265 54. A man sold 342 pounds of beef, at 6 cents per pound, and received his pay in molasses, at 37^ cents per gallon ; how many gallons did he receive ? Ans. 54,72 gallons. 55. A man exchanged 70 bushels of rye, at -$,92 per bushel, for 40 bushels of wheat, at $1,37} per bushel, and received the balance in oats, at $,40 per bushel ; how many bushels of oats did he receive ? Ans. 23^ 56. How many bushels of potatoes, at 1 s. 6 d. per bushel, must be given for 32 bushels of barley, at 2 s. 6 d. per bushel ? Ans 53^ bushels. 57. How much salt, at $1,50 per bushel, must be given in exchange for 15 bushels of oats, at 2 s. 3 d. per bushel ? Note. It will be recollected that, when the price and cost are given, to find the quantity, they must both be re- duced to the same denomination before dividing. Ans. 3£ bushels. 58. How much wine, at $2,75 per gallon, must be given in exchange for 40 yards of cloth, at 7 s. 6 d. per yard ? Ans. 1 8 T 2 T gallons. 59. There is a fish, whose head is 4 feet long ; his tail is as long as his head and \ the length of his body, and his body is as long as his head and tail ; what is the length of the fish ? The pupil will perceive that the length of the body is i the length of the fish. Ans. 32 feet. 60. A gentleman had 7 £. 17 s. 6 d. to pay among his laborers ; to every boy he gave 6 d., to every woman 8d., and to every man 16 d. ; and there were for every boy three women, and for every woman two men ; I demand the number of each. Ans. 15 boys, 45 women, and 90 men. 61. A farmer bought a sheep, a cow, and a yoke of oxen for $82,50 ; he gave for the cow 8 times as much as for the sheep, and for the oxen 3 times as much as for the cow ; how much did he give for each ? Ans. For the sheep $2,50, the cow $20, and the oxen $60. 62. There was a farm, of which A owned |, and B if ; the farm was sold for $1764; what was each one's share of the money 1 Ans. A's $504, and B's $1260 23 266 ARITHMETIC. PART THIRD. 63. Four men traded together on a capital of $ 3000, of which A put in \, B \, C }, and D t l ; at the end of 3 yrs., thev had gained $2364 ; what was each one's share of the gain? rA's$1182 A I B's $ 591 Ans> ) C's $ 394 ' D's $ 197 64. Bought a book, the price of which was marked $4,50, but for cash the bookseller would sell it at 33^ per cent, discount ; what is the cash price ? Ans. $3,00 65. A merchant bought a cask of molasses, containing 120 gallons, for $42 ; for how much must he sell it to gain 15 per cent. ? How much per gallon ? Ans. to last, $,40i Q6. A merchant bought a cask of sugar, containing 740 pounds, for $59,20 ; how must he sell it per pound to gam 25 per cent ? Ans. $,10 67. What is the interest, at 6 per cent., of $71,02 for 17 months 12 days ? Ans. $6,178+ 08. What is the interest of $487,008 for 18 months ? Ans. $43,83+ It has been shown that the length of one side of a square multiplied into itself, will give the square con- tents. Hence to find the area, or superficial contents of a square when one side is given, Multiply the side of the square into itself. 69. There is a room 18 feet square ; how many yards of carpeting 1 vard wide will cover it ? Ans. 182=324 ft.=36 yards. 70. The length of one side of a square room is 31 feet; how many square feet in the whole room 1 Ans. 961 71. If the floor of a square room contain 36 square yards, how many feet does it measure on each side ? Ans. 18 feet. Note. This answer is obtained by finding the square root of the area 36 feet. A parallelogram, or oblong, is a four sided figure, ha- ving its opposite sides equal and parallel. To find the area of a parallelogram, Multiply the length by the breadth. MISCELLANEOUS EXAMPLES. 267 72. A garden in the form of a parallelogram is 96 feet long and 54 wide ; how many square feet of ground are contained in it 1 Ans. 5184 sq. ft. 73. What is the area of a parallelogram 120 rods long and 60 wide ? Ans. 7200 sq. rods. 74. If a board be 21 feet long, and 18 inches broad, how many square feet are contained in it ? Ans. 31} sq. feet. A triangle is a figure bounded by three lines. If a line be drawn from one corner of a parallelogram to its opposite, (as in the Fig. A B,) it will divide it into two B equal parts of the same length and breadth as the parallelogram, but containing only half its surface. These two parts are triangles. — Now supposing the length of this parallelo- gram to be 6 feet, and its breadth* 2, the area would be 12 feet. But the triangle will contain only half the sur- face, or 6 feet. Hence to find the area of a triangle, Multiply the length by half the breadth, or tJie breadth by half the length. 75. In a triangle 32 inches by 10, how many square inches ? Ans. 160 sq. inches. 76. What is the area of a triangle whose base is 30 rods and the perpendicular 6 rods ? Ans. 90 rods. It has been shown that the length of one side of a cube raised to its third power will give the solid contents of the cube. Hence to find the solid contents of a cube, when one side is given, Multiply the given side into itself twice, or raise it to ita, third power. 77. The side of a cubic block is 12 inches ; how manv solid inches does the block contain ? Ans. 12 3 =1728 78. One side of a cube is 59 feet ; what are its solid contents ? Ans. 205379 268 ARITHMETIC. PART THIRD. 79. If a cube contains 614,125 cubic yards, what is the length of one side ? Ans. 85 yards. Note. This answer is obtained bv finding the cube root of 614125. A circle is a figure contained by one line called the cir. eumference, every part of which is equally distant from a point within called the centre. The diameter of a circle, is a line drawn through the centre, dividing it into two equal parts. It is found by calculation, that the circumference of a circle measures about 3| times as much as its diameter, or more accurately in decimals, 3,4159 times. Hence to find the circumference of a circle when the diameter is known, Multiply the diameter by 3j. To find the diameter when the circumference is known, Divide the circumference by 3{. To find the area of a circle, Multiply i the diameter into \ the circumference. 80. If the diameter of a wheel is 4 feet, what is its cir- circumference 1 Ans. 12^ feet. 81. What is the circumference of a circle, whose di- ameter is 147 feet ? Ans. 462 feet. 82. What is the diameter of a circle, whose circum- ference is 462 feet ? Ans. 147 feet. 83. What is the area of a circle, whose diameter is 7 feet, and its circumference 22 feet ? Ans. 38i sq. feet. 84. What is the area of a circle, whose circumference is 176 rods ? Ans. 2464 rods. The area of a globe, or ball, is 4 times as much as the area of a circle of the same diameter. Hence, to find the area of a globe, Multiply the wliole circumference into the whole diameter. 85. What is the number of square miles on the surface of the earth, supposing its diameter 7911 miles ? Ans. 7911x24853=196,612,083. To find the solid contents of a globe, or ball, Multiply its area by J- part of its diameter. 86. How many solid inches in a ball 7 inches in diame- ter ? Ans. 179$. FORMS OF NOTES, RECEIPTS, &C. 269 A cylinder is a round body, whose ends are circles, and which is of equal size from end to end. To find the solid contents of a cylinder, Multiply the area of one end by the length. 87. There is a cylinder 10 feet long, the area of whose ends is 3 square feet ; how many solid feet does it con- tain ? Ana. 30. Solids which decrease gradually from the base till they come to a point, are called pyramids. The point at the top of a pyramid is called the vertex. A line drawn from the vertex perpendicular to the base, is called the perpen- dicular height of the pyramid. To find the solid contents of a pyramid, Multiply the area of the base by i of the perpendicular height. 88. There is a pyramid whose height is 9 feet, and whose base is 4 feet square ; what are its contents ? Ans. 48 feet. 89. There is a pyramid, whose height is 27 feet, and whose base is 7 feet in diameter ; what are its solid con- tents ? Ans. 346^ feet. FORMS OF NOTES, RECEIPTS, AND ORDERS. When a man wishes to borrow money, after receiving it, he gives his promise to repay it, in such forms as those below. NoTE- No. 1. Hartford, Jan. 1, 1832. For value received, I promise to pay D. F. Robinson, or order, two hundred sixty four dollars, twenty-five cents, on demand, with interest. John Smith. No. 2. New York, Jan. 15, 1332. For value received, I promise to pay William Dennis, or bearer, twenty dollars, sixteen cents, three months after date. George Ellis. 23* 270 ARITHMETIC. PART THIRD. No. 3. Philadelphia, July 6, 1831. For value received, we, jointly, and severally, promise to pay to Henry Reddy, or order, one hundred dollars, thirteen cents, on demand, with interest. James Barnes. Attest. James Cook. William Hedge. Remarks. 1. The sum lent, or borrowed, should be written out in words, instead of using figures. 2. When a note has the words " or order," or " or bearer," it is called negociable ; that is, it may be given or sold to another man, and he can collect it. If the note be written, to pay him " or order," (see No. I,) then D. F. Robinson can endorse the note, that is, write his name on the back of it, and then sell it to any one he chooses. Whoever buys the note, demands pay from the signer, John Smith. 3. If the note be written, " or bearer," (see note 2,) then whoever holds the note can collect it of the signer. 4. When no rate of interest is mentioned, it is to be understood at the legal rate in the state where the note is given. 5. All notes are payable on demand, unless some par- ticular time is specified. 6. All notes draw interest after the time of promised payment has elapsed, even if there is no promise of inte- rest in the note. 7. Notes that are to be paid on demand, draw interest after a demand is made. 8. If a man promises to pay in certain other articles, in- stead of money, after the time of promised payment has- elapsed, the creditor can claim payment in money. Receipts. Hartford, June 16, 1831. Received of Mr. Julius Peck, twelve dollars, in full of all accounts. John Osgood. FORMS OF NOTES, RECEIPTS, &C 271 Receipt for money on a note. Hartford, June 18, 1831. Received of John Goodman, (by the hand of Willliam Smith,) twenty dollars, sixteen cents, which is endorsed on his note of July 6, 1829. John Reed. Receipt j or money on account. Hartford, April 6, 183L Received of Albert Jones, forty dollars, on account. Peter Trusty. Receipt of Money for another Person. Hartford, June 1st, 1831. Received of A. B. one hundred and six dollars, for I. C. Samuel Wilson. Receipt for Interest due on a Note. Hartford, Aug. 1, 1832. Received of W. B. thirty dollars in full of one year's interest of $500, due to me on the day of last, on note from the said W. B. William Gray. Receipt for Money paid before it is due. Newport, June 1, 1829. Received of A. F. sixty dollars advanced, in full for one year's rent of my house, leased to said A. F. ending the first day of September next, 1829. John Graves. Note. — If a receipt is given in full of all accounts, it cuts off only the claims of accounts. But " in full of all demands" cuts off all claims of every kind. Orders. New York, June 9, 1830. Mr. John Ayers. For value received, pay to N. S. or order, fifty dollars, and place the same to my account. Solomon Green. New York, July 9, 1831. Mr. William Redfield,— Please to deliver Mr. L. D. such goods as he may call for, not exceeding the sum of one hundred dollars, and place the same to the account of vour humble servant. Stephen Birch. 272 ARITHMETIC. PART THIRD* o z> a - CJ cs 03 E ^= 3 a £ "a . c 0" * o o. . CO M ~ O — 3 <~ b *s 2 o O u *, ~ o r _ •- «n a ~ * C C eo ej ~ — '- » -2 ^ e £ ^h ~ - M « £ « o s- o ». o . 33 r- ,*» — Ce 3/ 33 farm th w e rig if <§"?- * a *o E O c u •3 00 C ^ ~- r *- C S 33 Jj oj O — fi fa5 >-,'— re C3 — e3 c s 2 aT °-o 5r ® t_ a * 1 o o .2 « ce 33 c a ^ C C3 ^E — M 03 *"" GO — i— %* ^ 3;' a;. = CO ?r .S c = "2 •° 2 DO c_y a, c o- cc "lo . a — £ 5 ■^3 r~ ~. = -2 "* £ 3 r* a 33 >^ CO S 33 ,33 easy as b the If — g a; 01 «? 5 iJ cS .^ 09 CC p E_, =. 2 J" tc - - O £ c CO b£ * 5 3! — The followiii Take a book count, at the *^ 33 s ^- 0) * -»-» T3 cS m m — c? -*• 33 « i. r^ a «s | c ^ s* 0-0 Mi "S T* *• ^ « "i y one y rep y Cas CO CC 02 . .-; ?* <-~ GC w . i-h — April May 1 Sept. m O 1 m - i> :-. O |* X l-H CO CO ■):■ 1 r-< 1 ,75, and cts. r to 33 • 3 » 3 ° t Q « S f Wood, s work, of Rye, sy your 5 V ce w 03 'O ^ 4 cord ne d 33 bush livere Q X rf 03 • G^ O HH h • l> 6 CO U T' irs « P CJL> Jan . May 3 BOOK-KEEPING. 273 to to cc © © iO © i m cs — C> © OJ C5 1 *> be <*. 5 u 1 c *« w CS «s >> hi 3 .0. o w '3* Cm O Cm O — id > >. CC £C 0) o o o ** ft, C bC o c ©" >> CC s © yi" © o m o 1 »« © o CN © 1 I- c -T3 0) * lO C5 P3 © 1 *■* -* C n i— ■ SO 'O d <2 "2 • *D o C V. ^ CO rtO • u CD CO CD # « 3 >» ** S 8* G 03 = 5 cS >> 2 JS CS fa s . • • ^2 •<-^ ^ H c o — DO s o T3 -o -a .£ — C 2> >> a c _w Q g fa ee 13 "o CS CD § W> n 6 6 O • — a • — T3 T3 13 -^ S -5 — CC _c — C c = b CS O >-, 1 s c s. _2 =- 9 O O O 01 o .22 I-* CO — 3 HI g 'J- a 3D cS CD C (S 3 fH £ 2 This book is DUE on the last date stamped below. UC SOUTHERN REGIONAL LIBRARY FA A 000 257 693 2 STACK ANN6K