yfejitek Van Antwerp, Pragg & Co. Cincinnati & New York. T l-^ 'V WHITE'S GRADED-SCHOOL SERIES. AN INTERMEDIATE ARITHMETIC, MENTAL AND WRITTEN EXERCISES NATURAL SYSTEM OF INSTRUCTION. By E. E. white, M.A. VAN ANTWERP, BRAGG A CO., CINCINNATI. NEW YORK. Entered according to Act of Congress, in the year 1870, by WILSON, HINKLE & CO., In the Clerk's Office of the District Court of the United States for the Southern District of Ohio. Entered according to Act of Congress, in the year 1873, by WILSON, HINKLE & CO., In the Office of the Librarian of Congress, at Washington, D. C. Copyright, 1876, by Wilson, Hinkle & Co. EDUCATION DEPT, ECLECTIC press: VAN ANTWERP, BRAGG k CO., CINCINNATI. l^i^ PREFACE. It is claimed for this treatise that it possesses three very important characteristics, to wit: 1. It is specially adapted to the grade of pupils for which it is designed. It presents only those operations and principles which can be mastered by intermediate classes, and each subject is treated as thoroughly as the advancement of such pupils will permit. It is also believed that the subjects are introduced in the best possible order. There are reasons in favor of placing United States Money before Fractions, but stronger reasons favor the arrangement in this work. 2. It combines mental and ivritten arithmetic in a practical and philosophical manner. This is done by making every mental ex- ercise preparatory to a written one ; and thus these two classes of exercises, which have been so unnaturally divorced, are united as the essential complements of each other. This union is. natural and complete; and, as a consequence, the several subjects are treated in much less space than is possible when mental and written exercises are presented in separate books. 3. It faithfully embodies the Inductive Method. Instead of at- tempting to deduce a principle or rule from a single example, as is usually done, each process is developed inductively, and the succcvssive steps are thoroughly mastered and clearly stated by the pupil before he is confronted with the author^s rule. This method not only places "processes before rules," but it teaches " rules through processes," thus observing two important inductive maxims. .— ^-^ ji ^a ^- r-' .^ IV PREFACE. Attention is also called to the use of visible illustrations (objects or pictures) in developing new ideas and processes. In the funda- mental rules, this illustrative or perceptive step is omitted, since it is fully presented in the Pkimary Arithmetic. The engraved cuts in Fractions, United States Money, and Denom- inate Numbers, are specially designed to be used as a means of developing and illustrating the subjects considered. Two other features, worthy of special notice, are the great variety of exercises^ and their preeminently progi^essive character. Generally, each lesson contains both concrete and abstract ex- amples, and every new process or combination is at once used in the solution of problems involving mental analysis. This arrangement avoids the mechanical monotony which character- izes long drills on a single class of exercises. The problems, all of which are original, are so graded that they present but one difficulty at a time, and all difficulties in their natural order. The pupil's progress is thus made easy and thorough. It is hoped that these and other features may commend this work to all progressive teachers, and that it may prove as suc- cessful in the school-room as its plan is natural and simple. Columbus, Ohio, May^ 1870. The Last Edition. The recent addition of the more useful processes in Per- centage and Mensuration adapts the work to those pupils who do not attend school long enough to master an arithmetic de- signed for advanced classes. It now presents a short course IN arithmetic, with thorough drills in all elementary processes, and with a brief and simple treatment of those practical appli- cations which are most frequently used in business. Lafayette, Ind., May, 1876. SUGGESTIONS TO TEAOHEES. In the preparation of this work two facts were kept in view, viz.: (1) that it is to be studied by pupils who must largely depend upon the living teacher for explanations; and (2) that those methods wiiich are most natural and simple are most suc- cessful in practice. Hence, its paojes are not cumbered with long verbal explanations and peculiar methods, of little prac- tical use to pupil or teacher. The author^ has left something for the teacher to do; and tliat this may be done wisely, he offers the following hints and suggestions: 1. Mental Exercises. — These exercises should be made a thor- ough intellectual drill. They should be recited mentally, that is, without writing the results; and, since the reasoning faculty is not trained by logical verbiage, the solutions should be con- cise and simple. See pages 23, 88, 89. They should also be made introductory to the Written Exercises, of which they are often a complete elucidation. The corresponding examples in the two classes of exercises should be recited together as well as separately. Many of the written problems may also be solved mentally. 2. Written Exercises. — The pupils should be required to solve every problem of the assigned lesson on tlie slate or paper, and the solutions should be brought to the recitation for the teacher's inspection and criticism. From three to five minutes will suffice to test the accuracy and neatness of each pupil's work. The mental problems should also be solved on the slate or paper in preparing the lesson, and then recited, not only mentally as above described, but also as a written exercise. This will in- crease the number of written problems, and, at the Fame time, it will secure a careful preparation of the entire lesson. 3. Definitions and Principles. — These should be deduced and stated by the pupils under the guidance of the teacher, and usually in connection with the solution of problems. Take for illustration the definition of multiplication. The pupil mul- tiplies 304 by 5. The teacher asks, What have you done? "I have multiplied 304 by 5." T. Do not use the word "multi- plied." (If necessary,. the teacher shows what is meant by taking a thing one or more times.) " I have taken 304 five times." (V) VI SUGGESTIONS TO TEACHERS. T. By what process have you taken 304 five times? "By mul- tiplying it.^' T. What, then, is multiplication? ''Multiplication is the process of taking one number ." T. How many times is the number taken in the above example? "It is taken five times, or as many times as there are units in the multiplier." T. Now complete your definition. ^^Multiplication is the process of taking one number cts many times as there are units in another. ^^ These steps should be repeated with other examples, until the definition is clearly reached and accurately stated. It should then be written and compared with the author's definition, which should be thoroughly memorized. 4. Eules. — These should also be deduced and stated by the pupils. The true order is this: 1. A mastery of the process without reference to the rule. 2. The recognition of the suc- cessive steps in order, and the statement of each. 3. The com- bination of these several statements into a general statement. 4. A comparison of this generalization with the author's rule. 6. The memorizing af the latter. Take for illustration the rule for adding fractions. T. What is the first step? "Write the fractions, separating them by the plus sign." (Pupils write an example.) T. What is the second step? "Reduce the fractions to a common denominator." 2\ What is the third step? "Add the numerators of the new fractions." T. The fourth step? "Under their sum write the common denominator." These ques- tions should be repeated until the answers are promptly and ac- curately given, and then they should be united in a general statement. The first step may be omitted in the rule. 5. Questions for Review. — These are designed as a final test of the pupil's knowledge. Before they are reached, the defini- tions, principles, and rules should be thoroughly mastered, and the pupils should be able to make a topical analysis of them and recite each in order. 6. Fractions. — This section presents only the elements of Frac- tions, and these in the simplest manner. The subject is more ex- haustively treated in the Complete Arithmetic. Tlie reduction of compound fractions is made introductory to the multiplication of fractions, as the two processes are best taught together. 7. Oine Method. — Elementary instruction in arithmetic should aim to make the pupil ready and accurate in the use of one method for each operation. This may not be the shortest method in every case; but, as a rule, the pupil will reach the result sooner by it than by a method that is less familiar. The attempt of the young pupil to use several methods, results in hesitation and confusion. For other suggestions see the Manual of Arithmetic. coNTE:^rTS. SECTION I.— Notation and Numeration. PAGE Oral and Written Exercises 9 Definitions, Principles, and Rules 16 Koraan Notation 20 SECTION II.— Addition. Mental and Written Exercises 23 Definitions, Principles, and Rule 35 SECTION III.— Subtraction. Mental and Written Exercises ...... 37 Definitions, Princii)les, and Rule . . . . . .45 SECTION IV.— Multiplication. Mental and Written Exercises 48 Definitions, Principles, and Rule 59 SECTION v.— Division. Mental and Written Exercises 63 Definitions, Principles, and Rules 74 SECTION VI. — Properties of Numbers. Greatest Common Divisor 80 Multiple and Least Common Multiple .... 83 SECTION VII.— Common Fractions. The Idea of a Fraction 86 Reduction of Integers and Mixed Numbers to Fractions . 88 Reductions of Fractions 90 Addition of Fractions 96 Subtraction of Fractions 98 Multiplication of Fractions 101 (vii) VlU CONTENTS. PAGE Division of Fractions 107 Fractional Parts 110 SECTION VIII. -Decimal Fractions. Numeration and Notation 114 Reduction of Decimals 119 Addition of Decimals 122 Subtraction of Decimals 123 Multiplication of Decimals 124 Division of Decimals 126 SECTION IX.—United States Money. Notation and Definitions ....... 129 Reduction of United States Money 132 Addition and Subtraction of United States Money . . 133 Multiplication and Division of United States Money . 135 Bills '.'.... 139 SECTION X. — Denominate Numbers. Reduction, Mental and Written 144 Definitions, Principles, and Rules . . . . . 171 SECTION XI.— Compound Numbers. Compound Addition . . 177 Compound Subtraction . . 180 Compound Multiplication 183 Compound Division 185 SECTION XII.— Percentage. Notation and Definitions .190 The Three Cases 191 Profit and Loss 196 Commission, Insurance, Taxes, etc 198 Simple Interest 199 Discount 203 Notes, Drafts, and Bonds 207 SECTION XIII.— Mensuration. Surfaces 209 Solids 211 INTERMEDIATE ARITHMETIC. SECTION I. %%^%%%%%%% LESSON I. ORAL EXERCISES. Article 1. — 1. Here are one hundred balls in ten rows. How many balls are there in one row? How many balls in two rows? In three rows? In five rows? In eight rows? In ten rows? 2. How many ones in ten? How many ones in two tens? In five tens? Eight tens? Ten tens? (9) 10 TNTERMEDIi^TE ARITHMETIC. 3. How * many* tens ih ten? How many tens in twenty? In thirty? Forty? Sixty? Seventy? Eighty? One hundred? Art. 2. When a number is expressed by two figures, the first or right-hand figure denotes Units^ and the second or left-hand figure denotes Tens. 4. Which figure in 25 denotes units? Which de- notes tens? 5. How many tens and units are there in 37? In 57? 46? 33? 50? 45? 64? 88? 94? 99? Art. 3. In reading numbers, the tens and units are read together as so many units. Thus, 45 is read forty-five units, or, more briefly, forty -five. Eead the following numbers, and give the number of tens and units in each : (6) (7) (8) (9) (10) (11) 1 11 21 20 14 67 3 13 23 40 34 83 5 15 25 60 55 75 9 19 29 80 95 72 Art. 4. When a number is expressed by three fig- ures, the third or left-hand figure denotes Hundreds. 12. Which figure in 245 denotes hundreds? Which figure denotes tens? Which denotes units? 13. How many hundreds, tens, and units in 426? 708? 340? 235? 406? 560? 666? Eead the following numbers, and give the number of hundreds, tens, and units in each : (14) (15) (16) (17) (18) 200 240 302 349 560 500 550 805 424 703 700 770 807 825 909 900 990 804 448 836 NOTATION AND NUMERATION. 11 19. What is the greatest number that can be ex- pressed by one figure? By two figures? By three figures ? 20. When numbers are expressed by figures, in which place or order is the figure denoting units written? The figure denoting tens? The figure de- noting hundreds? Art. 5. The first three figures, viz. : units, tens, and hundreds, constitute the first or Units' Period. 'WRITTEN EXERCISES. 1. Write in words, 4, 6, 8, 13, 14, 18, 20, 24, 30, 34. 2. Write in words, 40, 46, 60, 67, 70, 78, 80, 83, 87, 90, 95, 99. Express in figures the fi^llowing numbers : (3) (4) (5) Twelve; Twenty-one; Twenty-three; Sixteen ; Thirty-two ; Twenty-four ; Eighteen; Forty-two; Forty-seven; Twenty; Sixty-five; Sixty-five; Sixty ; Eighty-five ; Seventy-nine ; Eighty. Ninety-four. Ninety-six. 6. Express in figures the numbers composed of three tens and four units ; six tens and seven units ; seven tens and six units; seven tens. 7. Express in figures the numbers composed of six tens and eight units; three tens and nine units; nine tens and no units ; seven units. 8. Write in words, 100, 150, 200, 280, 300; 350, 390, 560, 607, 803, 340, and 908. 12 INTERMEDIATE ARITHMETIC. Express in figures the following numbers : (9) (10) Two hundred ; Four hundred and five ; Five hundred ; Five hundred and six ; Seven hundred ; Six hundred and four ; Three hundred and forty; Four hundred and forty-five; Six hundred and seventy ; Eight hundred and thirty-seven; Nine hundred and thirty. Nine hundred and twenty-seven. 11. Express in figures the numbers composed of three hundreds, five tens, and four units; six hun- dreds, four tens, and three units ; five hundreds, seven tens, and no units. 12. Express in figures the numbers composed of eight hundreds and six tens; five hundreds and four tens ; seven hundreds and five units ; two hundreds and six units; six tens. 13. What number is composed of 3 hundreds, tens, and 6 units? 2 hundreds and 3 tens? 4 hun- dreds and 6 units? 5 hundreds and 8 tens? 14. What number is composed of 5 tens and 8 units? 6 hundreds and 5 units? 7 hundreds and 6 tens? LESSON II. 2'hoiisa)ids* "Period — ^'hoiisands, 2en-ihousa7ids, Mund^^ed- thousands, ORAIi EXERCISES. Art. 6. When a number is expressed by four figures, the fourth or left-hand figure denotes Thousands. 1. How many thousands in 4,635? 3,045? 6,309? 7,554? 5,384? 8,054? 5,006? 2. Read the units' period in 6,325; 5,080; 7,009; 3,406; 5,800; 6,370; 7,590; 8,008. NOTATION AND NUMERATION. 13 Eead the following numbers: (3) (4) (5) (6) (7) 1,000 2,200 1,020 2,007 3,432 3,000 4,400 3,040 4,001 4,568 5,000 6,600 5,060 5,003 5,608 7,000 8,800 7,090 6,005 7,893 9,000 / 9,900 9,070 8,009 9,890 Art. 7. When a number is expressed by five fig- ures, the fifth or left-hand figure denotes tens of thousands, or Ten-thousands. 8. How many ten-thousands in 45,684? 50,480? 38,305? 15,056? 80,650? 9. How many ten-thousands and thousands in 36,308? 48,500? 60,070? 85,350? 90,308? Art. 8. In reading a number expressed by five fig- ures, the fifth and fourth figures are read together as so many thousands. Thus, 45,000 is read forty-five thousand. Eead the following numbers : (10) (11) (12) (13) 10,000 21,000 34,400 53,333 30,000 44,000 53,440 16,089 50,000 63,000 67,444 99,008 70,000 84,000 48,307 28,045 90,000 99,000 39,600 67,909 Art. 9. When a number is expressed by six figures, the sixth or left-hand figure denotes hundreds of thou- sands, or Hundred-thov sands. 14. How many hundred-thousands in 534,000? 308,000? 650,430? 508,080? 15. How many hundred-thousands, ten-thousands, and thousands in 354,000? 607,800? 350,307? 193,240 ? 470,386 ? 14 INTERMEDIATE ARITHMETIC. Art. 10. In reading a number expressed by six fig- ures, the sixth, fifth, and fourth figures are read together as thousands. Thus, 452,000 is read four hundred and fifty-two thousand. Eead the following numbers : (16) (17) (18) (19) 200,000 250,000 845,630 603,408 400,000 360,000 803,084 490,732 600,000 580,000 760,432 308,400 800,000 730,000 900,425 600,550 900,000 960,000 807,708 707,700 Art. 11. The fourth, fifth, and sixth figures of a number constitute the Thousands' Period, 20. Eead the thousands' period in the 16th, 17th, 18th, and 19th examples. 21. How many orders in units' period? In thou- sands' period ? 22. What are the names of the three orders in units' period? In thousands' period? 23. How may the two periods be separated? Ans. By a comma, WKITTEN EXERCISES. 1. Write in words, 3000 ; 4060; 3580; 7086; 6606; and 8080. 2. Write in words, 4400 ; 5008 ; 6070 ; 8506 ; 5087 ; 7600; and 3003. 3. Express in figures, three thousand ; seven thou- sand ; nine thousand ; four thousand five hundred ; eight thousand nine hundred. 4. Express in figures, two thousand four hundred and forty ; four thousand six hundred and sixty ; five thousand eight hundred ; six thousand five hundred and twenty -five. NOTATION AND NUMERATION. 15 5. Express in figures, seventy-five; two hundred and forty ; three hundred and six ; five hundred and forty- five; four thousand. 6. Express in figures, four hundred and forty ; five hundred and ninety ; seven thousand eight hundred ; eight thousand and fi^j. 7. Write in words, 10000; 25000; 40500; 36000; 44000; 30400; 45080; 64008; 89800. 8. Express in figures, forty-five thousand five hun- dred and four; sixty thousand seven hundred and ninety ; thirty-eight thousand and twenty ; ninety-six thousand and eighty-four. 9. Express in figures, four hundred and twenty; seven hundred and eighty-nine ; four thousand and fifty-seven ; seventy-five thousand ; sixteen thousand and ninety-eight. 10. Express in figures, as one number, 87 thousand 327 units; 60 thousand 405 units; 70 thousand 346 units ; 4 thousand 40 units ; 5 thousand 5 units ; 95 thousand 406 units. 11. Express in figures, as one number, 88 thousand 88 units ; 8 thousand 80 units ; 65 thousand 60 units ; 6 thousand 600 units ; 60 thousand. 12. Write in words, 300000; 440000; 334000; 245500; 304800; 450340. 13. Express in figures, four hundred thousand ; six hundred thousand; eight hundred and forty thousand ; seven hundred and sixty thousand. 14. Express in figures, nine hundred and fifty thou- sand four hundred ; four hundred and fifty-five thou- sand two hundred and eighty. 15. Separate the following numbers into periods : 3080 ; 44004 ; 400080 ; 20066 ; 109038 ; 160006 ; 809090 ; 706030; 40004; 30030. 16 INTERMEDIATI5 ARITHMETIC. LESSON III. DEFINITIONS, PEINOIPLES, AND EXILES. Art. 12. Arithmetic is the science of numbers, and the art of numerical computation. A Number is a unit or a collection of units. A Unit is one thing of any kind. An Integer is a whole number. Art. 13. There are three methods of expressing numbers : 1. B}^ words; as, five, ^fty^ etc. 2. By letters^ called the Roman method. (Art. 23.) 3. By figures^ called the Arabic method. Art, 14. Notation is the art of expressing numbers by figures or letters. Numeration is the art of reading numbers ex- pressed by figures or letters. The word Notation is commonly used to denote the Arabic method, which expresses numbers by figures. Art. 15. In expressing numbers by figures, ten char- acters are used, viz. : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these characters, 0, is called JVaught, or Cipher, It denotes nothing^ or the absence of mimher. The other nine characters are called Significant Figures, They each express one or more units. They are also called Digits. Art. 16. The successive figures which express a number, denote successive Orders of Units. These orders are numbered from the right ; as, first, second, third, fourth, fifth, and so on. NOTATION AND NUMERATION. 17 A figure in units' phice denotes units of the first order; in tens' place, U7iits of the second order; in hundreds' place, units of the third order, and so on — the term units being used to express ones of any order. Art. 17. Ten units make one ten, ten tens make one hundred, ten hundreds make one thousand ; and, generally, ten units of any order 7nake one unit of the next higher order. Note. — The teacher can make this principle plain by means of the illustration given on page 9. It is easily shown that 10 ones or units equal 1 ten, and that 10 tens equal 1 hundred. Art. 18. Figures have two values, called Simple and Local. The Simjjle Value of a figure is its value when standing in units' place. The Local Value of a figure is its value arising from the order in which it stands. When 3, for example, stands alone, or in the first order, it denotes 3 units; when it stands in the second order, as in 34, it denotes 3 tens; when it stands in the third order, as in 354, it denotes 3 hundreds. Hence, the local value of figures increases from right to left in a tenfold ratio. The local value of each of the successive figures which express a number, is called a Term. The terms of 325 are 3 hundredths^ 2 tens, and 5 units. Art. 19. The figures denoting the successive orders of units, are divided into groups of three figures each, called Periods. The first or right-hand period is called Units; the second. Thousands; the third. Millions; the fourth, Billions; the fifth, Trillions; the sixth, Quadrillions; the seventh, Quintillions ; etc. I. A.— 2. 18 INTERMEDIATE ARITHMETIC. Art. 20. The three orders of any period, counting from the right, denote, respectively, Units ^ Tens^ and Hundreds, as shown in the table : o o e K H P 5 5 5 1 1 1 5th Period, 4th Period, 3d Period, 2d Period, 1st Period, Trillions. Billions. Millions, Thousands, Units. The several orders may be named more briefly by calling the first order of each period by the 7ianie of the period, and omitting the word "of" after tens and hundreds, thus : .2 V O ■73 S o 1 1 1 i '6 i :3 'd e a ^ '§ ^ O '2 1 .2 2i 13 1 .2 22 o c«- B "C H 3 1 § 3 i 1 B o H O 5 5 5 od. , 4 4 4 3 3 2 2 2 , 1 1st 1 1 5th Perj 4th Period. 3d Period. 2d Period. Period. Art. 21. KuLE FOR Notation. — Begin at the left, and write the figures of each period in their proper orders, filling all vacant orders and periods with ciphers. Art. 22. EuLE for Numeration — 1. Begin at the right, and separate the number into periods of three figures each. NOTATION AND NUMERATION. 19 2. Begin at the left^ and read each period containing one or more significant figures as if it stood alone, adding its name. Note. — The name of the units' period is usually omitted. WBITTEN EXEKCISES. 1. Write in words, 20080406. Suggestion.— Separate the number into periods, thus : 20,080,406. Then write each period, thus : Twenty million eighty thousand four hundred and six. 2. Write in words, 50038456. 3. Write in words, 300607008. 4. Write in words, 40000300400. Suggestion.— Omit the third period, since it contains no sig- nificant figures, thus: Forty billion three hundred thousand four hundred. 5. Write in words, 3450000067. 6. Eead 3000080040; 10080603400. 7. Eead 15000407030; 5075803004. 8. Eead 400440300500; 130030003003. 9. Express in figures, twelve billion forty-six mill- ion and nine. Process. — First, write 12, with a comma after it, to form the fourth or billions' period, thus: 12, ; then write 46 in the next period, filling the vacant order with a cipher, thus: 12,046,; then, as there are no thousands, fill the next three orders with ciphers, thus: 12,046,000,; and, finally, write 9 in the units' period, filling the vacant orders with ciphers, thus: 12,046,000,009. 10. Express in figures, fifty million thirty-two thou- sand six hundred and forty. 11. Three hundred million nine thousand two hun- dred and six. 20 INTERMEDIATE ARITHMETIC. 12. Forty-eight billion seventeen thousand and sixty-four. 13. Five million five thousand and five. 14. One million one hundred thousand and ten. 15. Three trillion three hundred million three hun- dred and three. 16. Sixty-two million three hundred thousand and fortj^-nine. 17. Five hundred million five thousand. 18. Four hundred and six thousand five hundred and seven. 19. Two million ten thousand and eighty. 20. Ninety million seven thousand four hundred and ninety. 21. Four hundred million forty thousand four hun- dred and four. 22. Thirty billion seventy-five thousand. 23. Nine billion nine thousand and nine. 24. Fifty-four million eighty-seven thousand and eighty-six. 25. Two hundred and two thousand five hundred and eighty. 26. Fifty billion fifty million ^ve hundred thousand and seven, 27. Seventeen billion seven hundred thousand three hundred and six. 28. Ninety million ten thousand and fifty-five. LESSON IV. "ROMAJV J\rOT^TIOJ\r. Art. 23. In the Roman Notation, numbers are ex- pressed by means of seven capital letters^ viz. : I, Y, X, L, C, D, M. NOTATION AND NUMERATION. 21 I stands for one ; V for five ; X for ten ; L for fifty ; C for one hundred ; 1) for five hundred ; M for one thousand. Art. 24. All other numbers are expressed by re- peating or combining these letters. 1. When a letter is repeated, its value is repeated ; thus : II represent 2 ; XX, 20 ; CCC, 300, etc. 2. When a letter is placed before one of greater value, the less value is taken from the greater; thus: IV stands for 4 ; IX for 9 ; XC for 90. 3. W^hen a letter is placed after one of greater value, the less value is added to the greater; thus: VI stands for 6; XI for 11; CX for 110. Art. 25. In the following table, numbers are ex- pressed by letters and figures : I, 1 VIII, 8 XV, 15 XL, 40 II, 2 IX, 9 XVI, 16 L, 60 III, 3 X, 10 XVII, 17 LX, GO IV, 4 XI, 11 XVIII, 18 LXX, 70 V, 5 XII, 12 XIX, 19 LXXX 80 VI, 6 XIII, 13 XX, 20 XC, 90 VII, 7 XIV, 14 XXX, 30 C, 100. WRITTEN EXERCISES. Express the following numbers in figures: (1) (2) (3) XIV CCL MDCL XXIV DCXC MDLX XXXIX ccxc MDLIX XCVI DCCL MDCCC CXI DCLIX MDCCCLX CIX MCCL MDCCCLXIX 22 INTERMEDIATE ARITHMETIC. Express the following numbers by letters : (4) (5) (6) (7) 45 156 210 1500 76 184 550 1650 90 345 700 1850 93 433 750 1868 99 555 880 1940 Express the following numbers by letters: (8) (9) (10) (11) 204 1200 1685 2000 409 1350 1944 2050 540 1408 1865 2550 675 1590 1909 3010 Questions for Eeview. What is arithmetic ? What is a number? What is a unit? What is an integer? In how many ways may numbers be expressed ? How are numbers expressed in the Arabic method? In the Eoman method ? What is notation ? What is numeration ? How many figures are used to express numbers? Which are called significant figures? Which figure has no numerical value ? What is meant by orders of units? How are the orders numbered ? How many units of any order make one unit of the next higher order? What is meant by the simple value of a figure ? On what does the local value of a figure depend? What is the law of increase from right to left? How many orders make a period? What are the names of these orders? Give the names of the first six periods. Give the rule for notation. Give the rule for numeration. How are numbers expressed in the Koman notation? Name the letters used, and give the value of each. How are numbers expressed by these letters? SECTIOlSr II. AD'DITIOJy. LESSON I. Add/Jire JVumhers, /, 2, and 3, 1. Four and 2 are how many? 8 and 2? 6 and 2? 7 and 2? 3 and 2? 9 and 2? 2. Two and 3 are how many? 5 and 3? 6 and 3? 7 and 3? 9 and 3? 11 and 3? 3. How many are 8 and 3? 18 and 3? 38 and 3? 47 and 3? 67 and 3? 87 and 3? 4. How many are 9 and 2? 39 and 2? 59 and 2? 48 and 3? 38 and 3?' 88 and 3? 5. Frank has 5 marbles in one hand and 2 mar- bles in the other: how many has he in both hands? Solution. — 5 marbles and 2 marbles are 7 marbles : Frank has 7 marbles in both hands. 6. A drover bought 9 sheep of one farmer arid 2 sheep of another: how many sheep did he buy? 7. Jane spelled 17 words correctly and mis-spelled 3: how many words did she try to spell? 8. A grocer sold 8 pounds of sugar to one cus- tomer, 3 pounds to another, and 2 pounds to another : how many pounds of sugar did he sell? 9. A man walked 4 miles the first hour, 3 miles the second, and 2 miles the third : how many miles did he walk in the three hours? 10. Eegin with 1 and count to 45 by adding 2 suc- cessively, thus: 1, 3, 5, 7, 9, 11, 13, etc. 11. Begin with 2 and count to 50 by adding 2 suc- cessively; by adding 3 successively. (23) 24 INTERMEDIATE ARITHMETIC. WRITTEN EXERCISES. Add the following numbers: (1) (2) (3) 4) (5) 2 10 112 2112 12102 2 21 211 1201 21210 1 12 122 1122 10222 2 20 111 2021 11121 1 22 222 1212 21212 L 11 1?L 2221 12111 (6) Write the numbers so that the units shall PROCESS. ^*^^"^ ^^® ^^'^^ column ; the tens, the second column ; and the hundreds, the third column. 121 Begin with the units' column, and add, nam- 2'^'^ ing results only, thus: 3, 5, 8, 11, 12, 14, 12^ 17, 20, 21,-21 units equal 2 tens and 1 unit. ^•^-' AYrite the 1 unit under the units' column, and 2^1 add the 2 tens with the tens' column, thus: 5, ^23 8, 10, 12, 15, 18, 20, 23, 25,-25 tens equal 2 1"^^ hundreds and 5 tens. Write the 5 tens under the tens' column, and add the 2 hundreds '^^ with the hundreds' column, thus: 5, 7, 8, 1 1 2051, Sum. 13, 16, 17, 19, 20,-20 hundreds equal 2 thou- sands and hundreds. Write the hundreds under the hundreds' column, and write the 2 thousands in thousands' place. The sum is 2051. To test the accuracy of the work, add the columns downward. (7) (8) (9) (10) 13 232 1323 3232 22 123 2112 2323 20 212 2131 23213 81 131 3213 13221 12 120 1301 32233 21 102 2222 232111 23 223 nil 323212 32 121 1323 232021 ADDITION. 25 11. Add 213, 322, 203, 312, 222, 321, 231, 123, 303, 232, 311, 132. 12. What is the sum of 2132, 3113, 2323, 1313, 2132, and 3320 ? 13. 2021 + 12333 + 22031 + 332231 + 231323 = how many? 14. 3231 + 2302 + 2330 + 12332 = how many ? 15. A grocer sold 12 pounds of sugar to one cus- tomer, 21 pounds to another, 32 pounds to another, and 30 pounds to another: how many pounds did he sell? 16. July has 31 days; August, 31; September, 30; October, 31 ;" November, 30 ; and December, 31 : how many days in the last six months of the year? 17. A farm contains 120 acres, another 212 acres, another 133 acres, and another 322 acres: how many acres do the four farms contain? 18. A man bought four loads of hay, the first weigh- ing 2130 pounds, the second 2312 pounds, the third 2232 pounds, and the fourth 2322 pounds : how many pounds of hay in the four loads? LESSON II. JVeff Additive J\^ umbers, 4 aiid 5. MENTAL EXERCISES. 1. Three and 4 are how many? 5 and 4? 6 and 4? 8 and 4? 7 and 4? 9 and 4? 2. Two and 5 are how many? 4 and 5? 6 and 5? 8 and 5? 7 and 5? 9 and 5? 3. How many are 18 and 4? 28 and 4? 48 and 4? IG and 4? 36 and 4? 56 and 4? 26 INTERMEDIATE ARITHMETIC. 4. How many are 17 and 5? 27 and 5? 47 and 5? 29 and 57 49 and 5? 69 and 5? 5. There are 17 birds on one tree and 4 on another: how many birds on both trees? 6. A man gave 26 dollars for a coat and 5 dollars for a hat : how many dollars did he give for both ? 7. A drover bought 19 cows of one man and 4 of another: how many cows did he buy? 8. James picked 27 peaches from one limb and 5 peaches from another : how many peaches did he pick from both limbs? 9. Mary has written 16 lines: if she write 5 lines more, how many lines will she then have written? 10. George gave 15 cents for a slate and 5 cents for a pencil: how many cents did he give for both? 11. Begin with 2 and count to 50, or more, by adding 4 successively; by adding 5 successively. 12. Begin with 3 and count to 48 by adding 5 successively. AT^RITTEN EXERCISES. (1) (2) (3) (4) (5) 15 251 15215 23512 52134 25 153 14343 30425 34445 35 354 45046 41341 53054 45 452 50350 23301 44052 55 355 33432 41545 25253 45 254 43543 43453 34545 35 555 23343 25445 41534 25 444 45452 41505 22335 6. What is the sum of four hundred and four; four thousand and forty; forty thousand four hundred; and four million four hundred thousand? ADDITION. 27 7. A grain dealer bought 2350 bushels of wheat on Monday, 4215 bushels on Tuesday, 3245 bushels on Wednesday, 1500 bushels on Thursday, 2424 bushels on Friday, and 1350 bushels on Saturday : how many bushels did he buy? 8. In a city containing ^ve wards, there are 345 voters in the first ward, 443 in the second, 213 in the third, 523 in the fourth, and 425 in the fifth: how many voters in the city? * 9. A father gave to his eldest son 225 acres of land, to the second 155 acres, to the third 145 acres, and to the youngest 124 acres: how many acres did he give to all? 10. The first three cars of a freight train contain 35240 pounds each ; the next four cars, 25345 pounds each ; the next two cars, 31540 pounds each ; and the last car, 25432 pounds: how many pounds of freight in the ten cars? LESSON III. JV^^ A Mi/lye J^iimher, 6. MENTAL EXERCISES. 1. Two and 6 are how many? 4 and 6? 3 and 6? 5 and 6? 7 and 6? 9 and 6? 8 and 6? 2. How many are 17 and 6? 28 and 6? 48 and 6? 68 and 6? 58 and 6? 78 and 6? 3. How many are 19 and 6? 29 and 6? 59 and 6? 39 and G? 69 and 6? 49 and 6? 4. Begin with 3 and count to 63 by adding 6 successively. 5. Mary's father gave her 5 peaches and her mollier gave her 6: how many peaches did both give her? 28 INTERMEDIATE ARITHMETIC. 6. John solved 18 problems before school and 6 problems in school: how many problems did he solve? 7. A farmer bought a cow for 27 dollars and a calf for 6 dollars: how many dollars did he pay for both? 8. The head of a fish is 5 inches long, its body 16 inches, and its tail 6 inches: how long is the fish? 9. In a certain orchard there are 29 apple trees, 5 pear trees, and 6 peach trees : how many trees in the orchard? 10. William gave a blind boy 19 cents, John gave him 15 cents, and Charles 6 cents : how many cents did they all give him? WRITTEN EXERCISES. (1) (2) (3) (4) (5) 3640 24137 43260 35260 305129 2566 16126 32345 16165 224603 1654 20050 16606 32542 350164 2366 16654 46060 36344 255234 3456 33456 50050 24030 145344 5634 44162 16566 33246 242456 4565 23206 24656 21438 145346 5656 36562 32562 44546 200500 6. Add thirty-six thousand three hundred and twenty-five; fourteen thousand and forty-six; twenty- three thousand four hundred and five ; fifteen thou- sand and sixteen; and three hundred and six thou- sand three hundred and four. 7. What is the sum of three million one thousand and fifty-six; six hundred thousand six hundred and twenty -five; four million forty-two thousand and four; forty -five million six hundred and fifty thousand? ADDITION. 29 LESSON IV. JV*eH^ Additive JViember, 7. MENTAL EXERCISES. 1. Two and 7 are how many? 5 and 7? 3 and 7? 6 and 7? 8 and 7 ? 7 and 7? 9 and 7? 2. How many are 18 and 7? 48 and 7? 68 and 7? 88 and 7? 28 and 7? 3. Fifteen and 7 arc how many? 35 and 7? 65 and 7? 45 and 7? 75 and 7? 4. Begin with 4 and count to 53 by adding 7 successively. 5. Charles had 6 marbles and his father gave him 7 : how many marbles had he then ? 6. A garden contains 19 pear trees and 7 peach trees: how many trees in the garden? 7. A man bought a set of harness for 37 dollars and a saddle for 7 dollars: how much did he \)ixj for both? 8. Mr. Jones gave 8 plums to John, 6 to Henry, and 7 to George : how many plums did he give to the three boys? 9. Frank gave 10 cents for a lead -pencil, 5 cents for a piece of rubber, and 7 cents for paper: how much did the three articles cost? 10. A gentleman gave 36 dollars for a suit of clothes, 7 dollars for a pair of boots, and 5 dollars for a hat : how much did he pay for all? 11. Count by 7's from 2 to 72; from 5 to 89; from .4 to 95 ; from 6 to 97. 12. Count by 6's from 3 to 75; from 4 to 88; from 5 to 95; from 7 to 97. 30 INTERMEDIATE ARITHMETIC. AVKITTEN EXERCISES. (1) (2) (3) (4) 10640 24045 32620 7121365 14075 14036 75437 2171634 26507 25507 50743 1237773 16021 46364 64017 7143656 34412 54563 32516 2674467 53452 16057 18416 6734765 26123 72027 13673 6574636 16021 47735 31654 7147347 5. What is the sum of sixteen million four thou- sand and sixty-five ; three hundred thousand two hun- dred and fifty-six; seven thousand and forty; and five million five thousand and seven? 6. What is the sum of forty-five million seven thou- sand and seventy; six million sixty-five thousand two hundred and six ; and seventy-five thousand and forty- four? 7. January has 31 days; February (except in leap year), 28; March, 31; April, 30; May, 31; and June, 30: how many days in the first six months of the year ? 8. A gentleman owns five farms, containing, re- spectively, 285 acres, 345 acres, 146 acres, 438 acres, and 248 acres: how many acres of land does he own ? 9. A newsboy sold 327 papers in April, 465 in May, 318 in June, and 278 in July: how many papers did he sell in the four months? 10. The first ward of a city contains 1675 youth of school age ; the second, 2357 youth ; the third, 2347 ; the fourth, 3270; and the fifth, 2677: how many youth of school age in the city? ADDITION. 31 LESSON V. JVeH' AddU/'ye JVumber, 8. MENTAL EXERCISES. 1. Two and 8 are bow many? 5 and 8? 3 and 8? 6 and 8? 4 and 8? 9 and 8? 2. How many are 16 plus 8? 36 plus 8? 56 plus 8? 25 plus 8? 45 plus 8? 65 plus 8? 3.13 + 8? 33 + 8? 53+8? 29 + 8? 49 + 8? 4. Count by 8^8 from 3 to 54; from 5 to 93. 5. Jane solved 17 problems in tbe morning and 8 in the evening: how many problems did she solve? 6. A farmer raised 16 loads of wheat in one field and 8 loads in another: how much wheat did he raise? 7. Kate spelled 38 words correctly and mis-spelled 8: how many words did she try to spell? 8. Charles gave 25 cents for a speller and 8 cents for a pencil: how much did he give for both? 9. A lady paid 27 dollars for a shawl, 8 dollars for a bonnet, and 3 dollars for a pair of shoes: how much did she pay for all? 10. A merchant sold 18 yards of muslin to one cus- tomer, 7 yards to another, and 8 yards to another: how many yards did he sell? WRITTEN EXERCISES. (1) 308 (2) 2617 (3) 19864 (4) 42764 (5) 5868 280 4565 34687 38768 4384 667 6387 46768 34187 5065 444 7836 65837 63506 6008 555 bm^ 80040 24483 4873 371 4084 18608 43832 8345 736 644 8168 7846 36084 45687 41608 37860 6654 5636 32 INTERMEDIATE ARITHMETIC. 6. Add thirty thousand six hundred and fifty; fifty thousand and eighty-five; four hundred thou- sand six hundred and seven ; and three hundred and forty thousand and seventy. 7. Add eight million eight thousand and eight; eighteen million eighteen thousand and eighteen ; and eight hundred million eight hundred thousand eight hundred. 8. The distance by railroad from Philadelphia to Harrisburg is 106 miles; from Harrisburg to Pitts- burgh, 249 miles; from Pittsburgh to Crestline, 188 miles; from Crestline to Fort Wayne, 132 miles; from Fort Wayne to Chicago, 148 miles: how far is it from Philadelphia to Chicago? 9. One of the wards of a certain city contains 1384 houses; another, 2868 houses; another, 857 houses; and another, 1486 houses: how many houses in the city? 10. A steam-ship sailed 217 miles the first day; 265 miles the second ; 227 miles the third ; 187 miles the fourth ; and 168 miles the fifth ; how many miles did it sail in the five days? LESSON VI. JVen^ A.dditlre JSTianbery 9, MENTAL EXERCISES. 1. Three and 9 are how many? 7 and 9? 9 and 7? 8 and 9? 9 and 8? 5 and 9? 2. How many are 14 + 9? 24 + 9? 44 + 9? 16 + 9? 36 + 9? 56 + 9? 3. How many are 17 + 9? 37 + 9? 57 + 9? 23 + 9? 43 + 9? 63 + 9? ADDITION. 33 4. Begin with 3 and count to 57 by adding \). 5. A farmer sold 6 hogs to his neighbor and 9 to a drover: how many hogs did he sell? 6. Andrew sold 8 bunches of grapes and had 9 bunches left: how many bunches had he at first? 7. There are 17 cows in one field and 9 cows in another: how many cows in both fields? 8. A pole is 7 feet in the water and 9 feet in the air : how long is the pole ? 9. A man paid 23 dollars for a coat, 9 dollars for a pair of pants, and 8 dollars for a vest: how much did he pay for the suit? 10. A boy paid 45 cents for a ball, 8 cents for marbles, and 7 cents for an orange : how much did he pay for all? WRITTEN EXERCISES. (1) 57384 (2) 4369 (3) 45566 48 (5) 4868 5834 13846 806 76 3769 691 3482 9376 287 1804 2637 691 2038 80 786 13484 5873 4056 409 5863 596 578 8705 96 4836 43486 509 6508 378 3988 G. What is the sum of nine billion nine million and nine; nine hundred million nine hundred thou- sand nine hundred ; and ninety million nine hundred thousand and ninety? 7. The State of Maine contains 31766 square miles ; New Hampshire, 9280 square miles; Vermont, 10212; Massachusetts, 7800; Connecticut, 4674; and Rhode Island, 1306. How many square miles in all of the New England States? I. A.— 3. 34 INTERMEDIATE ARITHMETIC. 8. The distance by riiilroud from Boston to Spring- field is 98 miles ; from Springfield to Albany 103 miles; from Albany to Buffalo, 298 miles; from Buf- falo to Cleveland, 183 miles ; from Cleveland to Chi- cago, 355 miles. How far from Boston to Chicago? LESSON VII. 1. An orchard contains 25 apple trees and 8 peach trees : how many trees in the orchard ? 2. A gardener sold 17 quarts of strawberries in market and 9 quarts to a grocer : how many quarts did he sell? 3. A lady gave 15 cents for tiiread, 8 cents for needles, and 7 cents for pins: how many cents did she spend? 4. James gave 8 cherries to George, 7 to William, 6 to Thomas,- 9 to Harry, and kept 5: how many cherries had he at first? 5. A gentleman gave 95 dollars for a horse, 15 dollars for a saddle, and 5 dollars for a bridle : how much did he pay for all? 6. Begin with 2 and add to 72 by 7'8, thus: 9, 16, 23, 30, 37, etc. 7. Begin with 5 and. add to 61 by 8'8. 8. Begin with 3 and add to 69 by 6'8. 9. Begin with 4 and add to 67 by 9'8. WKITTEN EXERCISES. 1. 32545 + 8607 + 11709 -f 50063 r^ how many? ADDITION. 35 2. A man paid $3575 for a lot, $5450 for a house, $875 for a stable, and $675 for other improvements : what did the property cost him ? Note. — This character ($) denotes dollars, and is called the dol- lar sign: $35 is read 35 dollars; $1 is read 1 dollar. 3. The first book of a series contains 328 pages; the second, 392 pages; the third, 400 pages; and the fourth, 432 pages: how many pages in the series? 4. Ohio contains 39964 square miles; Michigan, 56243 square miles; Indiana, 33809 square miles; and Illinois, 55409 square miles: what is the area of these four States? 5. The distance by railroad from Pittsburgh to Co- lumbus is 193 miles; from Columbus to Cincinnati, 120 miles; from Cincinnati to St. Louis, 340 miles: how far is it from Pittsburgh to St. Louis? 6. A father divided his estate between two sons and three daughters, giving to each son $3250, and to each daughter $2750 : what was the value of the estate ? 7. A farmer raised in one year 380 bushels of wheat, 245 bushels of oats, 87 bushels of rye, and as many bushels of corn as of wheat, oats, and rye together: how many bushels of grain did he raise? DEFINITIONS, PEINOIPLES, AND EULE. Art. 26. Addition is the process of finding the sum of two or more numbers. The number obtained by adding two or more num- bers is called the Sum or Amount. The Slim contains as many units as all the num- bers added, taken together. 36 INTKIIAIKDIATE ARITHMETIC. Numbers are either Concrete or Abstract. A Concrete Number m applied to a particular thing or quantity; as, 4 pears, 7 hours, 30 stei)s. An Abstract Number is not applied to any par- ticular thing or quantity ; as, 4, 7, 30. Fourteen balls and 13 balls are numbers of the same kind; and 6 tens and 3 tens are numbers of the same order. Num- bers of the same kind or order are ealled Like Numbers. Only like numbers can be added. Art. 27. The Sign of Addition is -f . It is called plus^ meaning more. When placed between two num- bers, it shows that they are to be added. Thus, 8 -)- 5 is read 8 plus 5, and it shows that 5 is to be added to 8. The Sign of Equality is n= . It is read equals or is equal to. Thus, 7 -f 8 = 1» is read 7 ^)^w.s 8 equals 15. Art. 28. EuLE for Addition. — 1. Write the numbers to be added so that figures denoting units of the same order shall be in the same column, and draw a line underneath. 2. Beginning with units, add each column, and write the sum, when less than ten, underneath. 3. When the sum of any column exceeds nine, write the right-hand figure under the column added, and add the number denoted by the left-hand figure or figures with the next column. 4. Write the entire surn of the left-hand column. Proof. — Add the columns downward. 8T]CTTOX III. s msTHA cTiojr. LESSON I. SubtraJi 671(2 I^igiires^ /_, 2, S, 1. How many is 4 less 3? 6 less 3? 8 less 3? 10 less 3? 12 less 3? 11 less 3? 2. How many is 11 less 2? 21 less 2? 41 less 2? 19 less 2? 29 less 2? 49 less 2? 3. Three from 12 leaves how many? 3 from 22? 3 from 42? 3 from 52? 3 from 32? 3 from 20? 3 from 40? 3 from 50? 4. Begin with 50 and count back to by sub- tracting 2 successively, thus: 50, 48, 4G, 44, 42, etc. 5. Begin with 40 and count back to 1 by sub- tracting 3 successively. G. Charles bought 12 sticks of candy and ate 3 of them : how many sticks were left? Solution.— 12 sticks less 3 sticks are 9 sticks: Charles had 9 sticks left. 7. Henry wrote 21 words, but misspelled 2 of them : liow many words did he spell correctly? 8. Charles's lesson consists of 15 examples, and he has solved all but 3 of them : how many has he solved? 9. James is 11 years old, and his brother Henry is 3 years younger: how old is Henry? (37) ^^8 INTERMEDIATE ARITHMETIC. WRITTEN EXERCISES. 1. From 345 take 123. PROCESS. Write 123 under 345, placing units un- Minuendj 345 ^^^ units, tens under tens, and hundreds Subtrahend 123 under hundreds. Subtract 3 units from Difference 222 ^ units, and write 2 units, the difference, below; subtract 2 tens from 4 tens, and write 2 tens, the difference, below; sub- tract 1 hundred from 3 hundreds, and write 2 hundreds, the difference, below. The difference, or remainder, is 222. (2) (13) (4) (5) (6) (7) (8) 57 46 88 75 685 409 967 43 24 65 53 343 307 645 (9) (10) (11) (12) (13) (14) (15) 246 487 507 718 563 485 560 132 231 302 312 330 212 320 16. From four thousand and sixty-five take two thousand and thirty-one. 17. A grocer bought 585 pounds of sugar and sold 231 pounds: how many pounds had he left? 18. In a graded school, there are 345 boys and 321 girls: how many more boys than girls in the school? LESSON II. JVen' Subtrahend li*tgte7^es, Z- and 5. MENTAL EXERCISES. 1. How many is 7 less 4? 6 less 4? 9 less 4? 8 less 4? 10 less 4? 11 less 4? 2. How many is 7 less 5? 9 less 5? 8 less 5? 10 less 5? 12 less 5? 15 less 5? SUBTRACTION. 39 3. How many is 13 less 4? 23 less 4? 43 less 4? 63 less 4? 83 less 4? 53 less 4? 93 less 4? 4. How many is 14 less 5 ? 44 less 5 ? 34 less 5? 54 less 5? 64 less 5? 74 less 5 ? 94 less 5? 5. Begin with 60 and count back to by sub- tracting 4 successively. 6. Begin with 53 and count back to 1 by sub- tracting 4 successively. 7. A man gave $12 for a saddle and $4 for a bridle: how much did the saddle cost more than the bridle? 8. Charles earned 21 cents by selling papers, and gave 4 cents for a comb : how many cents had ho Mt? 9. Kate is 15 years old and her sister is 4 years younger: what is her sister's age? 10. There are 21 passengers in a car: if 5 of them leave at a station, how many will reraain? 11. There are 13 men in one coach and 5 men in another: how many men in the first coach more than in the second? WRITTEN EXERCISES. (1) (2) (3) (4) (5) (6) 335 2036 308 1565 3683 7863 214 1034 205. 1433 2542 4552 7. From five thousand and seventy-six take threo thousand and fifty. 8. A farm contains 358 acres of land : if 155 acres should be sold, how many would be left? 9. A man bought a house for $4320 and sold it for $6450: how much did he gain? 10. A man bought 3487 bushels of wheat and sold 1425 bushels: how many bushels had he left?- 40 INTERMEDIATE ARITHMETIC. 11. A ship-builder sold a vessel for $24350: if tlie vessel cost him $27585, how much did he lose? 12. The number of school-houses in Ohio, in 1867, was 11353; in Pennsylvania, 11453: how many more school-houses in Pennsylvania than in Ohio ? 13. A wool-dealer having bought 23437 fleeces of wool, shipped 12322 fleeces to Boston : how many fleeces had he left? LESSON III. jy^^'ft^ Subh^ahend J^lgtij^es, 6 and 7. MENTAL EXERCISES. 1. How many is 8 less 6? 10 less 0? 12 less G? 13 less 6? 15 less 0? 14 less (>? 2. How many is 12 less 7? 13 less 7? 15 less 7? 16 less 7? 18 less 7? 22 less 7? 3. How many is 14 less 6? 24 less 6? 44 less 6? 64 less 6? 34 less 6? 74 less 6? 4. How many is 14 less 7? 24 less 7? 44 less 7? 16 less 7? 36 less 7? 46 less 7? 5. Begin with 56 and count back to by 7's. 6. Begin with 60 and count back to by 6's. 7. Ella was absent from school 7 days in a term of 75 days: how many days was she present? 8. John earned 25 cents by selling oranges, and gave 6 cents for a pencil: how many cents had he left? 9. A boy was carrying home 21 eggs ; he fell, and broke 7 of them: how many were left? 10. A man, having 23 dollars, gave 6 dollars for a hat: how many dollars had he left? 11. A teacher pronounced 25 words to an idle pupil, who mis-spelled 7 of them : how many words did he spell correctly ? SUBTPv ACTION. 41 WKITTEN EXERCISES. 1. From 5334 take 2726. PROCESS. Since 6 units can not be taken from 4 units, Mm. 5334 ^^^ 1^ units to the 4 units, making 14 units; Sub. 2726 then subtract 6 units from 14 units, and write Dif 2608 ^ units, the difference, below. To balance the 10 units (equal 1 ten) added to the minuend, add 1 ten to the 2 tens of the subtrahend; then subtract 3 tens from 8 tens, and write tens, the differ- ence, below. Add 10 hundreds to the 8 hundreds of the minuend, mak-'^ ing 13 hundreds; subtract 7 hundreds from 13 hundreds, and write 6 hundreds, the difference, below. To balance the 10 hundreds (equal 1 thousand) added to the minuend, add 1 thousand to the 2 thousands of the subtrahend ; subtract 3 thousands from 5 thousands, and write 2 thousands, the dif- ference, below. The difference is 2608. This process may be shortened, thus : 6 units from 4 units plus 10 units, or 14 units, leave 8 units; 2 tens and 1 ten are 3 tens, and 3 tens from 3 tens leave ten ; 7 hundreds from 3 hundreds plus 10 hundreds, or 13 hundreds, leave 6 hun- dreds; 1 thousand and 2 thousands are 8 thousands, and 8 thousands from 5 thousands leave 2 thousands. The differ- ence is 2608. NoTp]. — The teacher should show that the adding of 10 to a term of the minuend and 1 to the next higher term of the subtrahend increases both minuend and subtrahend equally, and does not affect the difference. C^) (3) (4) (5) (6) 44 63 272 1385 5754 26 46 147 1276 3457 (7) (8) (9) (10) (11) 3416 3041 14406 20670 30401 2507 2637 7345 17856 20576 42 INTERMEDIATE APvITHMETIC. 12. From fourteen thousand and forty-four take six thousand and sixteen. 13. A man whose income is $1850 expends annually $1365: how much does he lay up? 14. The number of youth of school age in a certain city is 1234, and only 756 pupils are enrolled in the schools : how many youth do not attend school ? 15. The number of pupils enrolled in the public schools of Ohio, in 1867, was 704767; in Pennsyl- vania, 789389: how many more pupils were enrolled in Pennsylvania than in Ohio? LESSON IV. JVen^ S/ibirahe7id J^igiires, 8 and 9, MENTAL EXEKCISES. 1. How many is 9 less 8? 11 less 8? 13 less 8? 10 less 8? 14 less 8? 12 less 8? 2. How many is 16 less 8? 26 less 8? 56 less 8? 17 less 8? 27 less 8? 67 less 8? 3. How many is 11 less 9? 13 less 9? 15 less 9? 12 less 9? 16 less 9? 17 less 9? 4. How many is 16 less 9? 26 less 9? 36 less 9? 15 less 9? 25 less 9? 45 less 9? 5. Begin with 50 and count back to 2 by subtract- ing 8 successively. 6. Begin with 57 and count back to 3 by subtract- ing 9 successively. 7. A school has enrolled 65 pupils, and 8 are ab- sent: how m^ny are present? 8. Mr. Smith is 44 years of age and his ^"oungest son is 8 years of age: what is the ditference in their ages? 9. A school contains 9 more girls than boys : if there are 56 girls, what is the number of boys? SUBTRACTION. 43 WBITTEN EXERCISES. 1. From 800000 take 238. 2. From forty million take eighty thousand. 3. A nursery contains 705 peach trees and 428 plum trees: how many more peach trees than plum trees in it? 4. The Pilgrims landed at Plymouth in 1620, and our National Independence was declared in 1776 : how many years between the two events? 5. The first steam-boat was made in 1807, and the Atlantic Cable was laid in 1866 : how many j^ears between the two events? 6. Mont Blanc in Europe is 15668 feet high, and Mount Sorata in South America is 21286 feet high : what is the difference in the height of these two mountains? 7. A man who owned 3408 sheep, sold 1897 of them: how many sheep had he left? 8. Mt. Etna is 10874 feet high, and Mt. Yesuvius 3948 feet: how much higher is Etna than Vesuvius? 9. America was discovered in 1492, and the Pil- grims landed at Plymouth in 1620: how many years intervened? 10. The population of the State of New York in 1860 was 3881000, and that of Ohio, 2340000: how many more people in New York than in Ohio? LESSON V. 1. From a cask containing 45 gallons of molasses, 39 gallons were sold : how many gallons remained unsold ? 2. An orchard contains 56 apple trees and 48 peach 44 INTERMEDIATE ARITHMETIC. trees: how many more i\])])\e trees than peach trees in the orchard ? 3 A grocer sold 57 pounds of butter from a firkin containing G5 pounds: how many pounds remained in the firkin? 4. In a school, 63 pupils are enrolled and 54 are present: how many pupils are absent? 5. If a man earn $45 a month, and spend $36, how much does he lay up? 6. A man gave $75 for a watch and $22 for a chain : how much did the watch cost more than the chain ? 7. Charles has 17 marbles and John 8 : how many more marbles has Charles than John ? 8. A teacher asked his class 52 questions, and 8 were answered incorrectly : how many were answered correctly ? 9 In a term of 64 days, Charles attended school 55 days: how many days was he absent? 10. Subtract by 4's from 62 back to 2. 11. Subtract by 6's from 75 back to 3. 12. Subtract by 9's from 68 back to 5. 13. Subtract by 7's from 59 back to 3. 14. Subtract by 8's from 48 back to 0. \/V^BITTEN EXERCISES. 1. From 202380 take 165436. 2. 4308560 — 1674805 = how many? 3. Illinois contains 55409 square miles, and Missouri 67380: how much more area has Missouri than Illinois? 4. By the census of 1850 the entire population of the United States was 23191876, and by the census of 1860 it was 31224885: what was the increase in 10 years? SUBTRACTION. 45 5. In 1862 there were 10869 miles of railroad in Great Britain, and 33222 miles in the United States: how many more miles in the United States than in Great Britain ? 6. An army of 30340 men lost 7568 in battle: how many men did it then contain? 7. In 1862 Ohio produced 35442858 pounds of butter and 20637235 pounds of cheese: how much more butter than cheese was produced? 8. A merchant having S11315 in bank, drew out $978: how much remained in the bank? DEFINITIONS, PEINCIPLES, AND EULE. Art. 29. Subtraction is the process of finding the difference between two numbers. The Difference or lieniainder is the number found by subtracting one number from another. The Minuend is the number diminished. The SubtrahenU is the number subtracted. Art. 30. Only Li he JVinnhers can be subtracted. Three pencils can not be subtracted from 7 books, nor 3 units from 7 tens. Art. 31. The Sign of Subtraction is — . It is read minus or less. It shows that the number after it is to be subtracted from the number before it. Art. 32. EuLE for Subtraction. — 1. Write the sub- trahend under the minuend., placing units under units, tens under tens, hundreds under hundreds, etc. 2. Begin at the right, and subtract each term of the subtrahend from the term above it, and icrite the differ- ence underneath. 46 INTERMEDIATE ARITHMETIC. 3. When any term of the subtrahend is greater than the term above it^ add 10 ^o the upper term, and then subtract, and write the difference as before. 4. When 10 has been added to the upper term, add 1 to the next higher term of the subtrahend before subtracting. Proof. — Add the remainder and subtrahend ; if their sum is equal to the minuend, the work is correct. Note. — Instead of adding 1 to the next term of the subtrahend, 1 may be subtracted from the next term of the minuend. LESSON VI. 'Problems combining AddUlon and Subhead ion, 1. Eobert picked 21 peaches, and gave 7 to his Bister and 8 to bis brother: how many peaches had he left? 2. A garden contains 17 pear trees, 8 plum trees, and a certain number of peach trees : if there arc 33 trees in the garden, what is the number of peach trees ? 3. A grocer bought 35 bushels of apples, and sold 17 bushels to A, 9 bushels to B, and the rest to C : how many bushels did he sell to C? 4. Jane is 8 years old and Lucy 13, and the sum of Jane's and Lucy's ages, less 7 years, is the age of Mary: how old is Mary? 5. A man bought a firkin of butter for $17, a crock of lard for $8, and a barrel of flour for $9; but he had not money enough by $7 to pay for them : how much money had he? 6. A man earned $45, and paid $15 for house rent, $8 for flour, $7 for shoes, and $10 for groceries : how much had he left? SUBTRACTION. 47 7. A man sees 15 pigeons on one branch of a tree, and 9 pigeons on another branch : if 7 should fly away, how many would be left on the tree? 8. A farmer had 23 chickens, but 7 of them were stolen and 5 were carried off by a hawk: how many chickens had he left? 9. A drover bought 17 sheep of one farmer, 9 sheep of another, and 8 of another, and then sold 7 of them to a butcher: how many sheep had he left? 10. A man gave a watch and $9 in money for a horse valued at $75: what did he get for his watch? WRITTEN EXERCISES. 1. From a piece of carpeting containing 150 yards, a merchant sold 3 carpets, containing 27, 39, and 42 yards, respectively : how many yards were left ? 2. Ehode Island contains an 'area of 1306 square miles; Delaware, 2120; Connecticut, 4674 ; New Jer- sey, 8320; Maryland, 9356; and New York, 47000: how many more square miles has New York than the other five States named? 3. A regiment entered the service with 1088 men ; 150 were killed in battle, 65 died from disease, 24 deserted, and 250 were discharged : how many re-* mained ? 4. A grain dealer bought 1250 bushels of wheat on Monday, 2145 bushels on Tuesday, and 3240 bushels on Wednesday, and on Thursday, fearing a decline in price, he sold 5450 bushels : how much wheat had he left? 5. A man deposited $175, $141, $75, $304, and $250 in a bank, and then drew out $480 and $225: how many dollars remained in the bank? 6. A railroad train left Cincinnati for St. Louis 48 INTERMEDIATE ARITHMETIC. with 336 passengers, and during the trip 145 passen- gers came aboard, and 208 passengers left: how many were in the cars when the train reached St. Lonis? 7. In 1860 the popuhition of Maine was 628279; New Hampshire, 326073; Vermont, 315098; Massa- chusetts, 1231066; Connecticut, 460147; Khode Ishind, 174620; and New York, 3880735: how many more inhabitants in New York than in the six New Eng- hmd States? 8. A man gave to his eldest son $2380; to the second, $245 less than to the eldest; and to the youngest, $450 less than to the second : how much did he give to all ? 9. From the sum of 2348 and 1864 subtract their difference. 10. From the sum of 506703 and 340067 take their difference. SECTION IV. MUZ TI^JPZ ICA TIOJV'. LESSON I. M'leHf'pllcajfd I^fgttres, 1 , 2, and S, 1. Twice 2 arc how many? 4 times 2? 6 times 2? 5 times 2? 8 times 2? 9 times 2? 2. Twice 3 are how many? 3 times 3? 5 times 3? 4 times 3? 7 times 3? 9 times 3? MULTirLICATION. 49 3. How many are 5 times 1 ? 5 times 3 ? 7 times 1? 7 times 2? 8 times 1? 8 times 3? 4. A boy has 2 hands: how many hands have 6 boys? 8 boys? 10 boys? 5. There are 3 feet in a yard : how many feet in 2 yards? 4 yards? 5 yards? 7 yards? 6. If a" man earn 3 dollars a day, how many dollars will he earn in 6 days? Solution. — If a man earn 3 dollars in one day, in 6 days ho will earn 6 times 3 dollars, which is 18 dollars. 7. If a boy walk 3 miles a day in attending school, how many miles will he walk in 10 days? WKITTEN EXERCISES. 1. Multiply 232 by 3. PROCESS. Write the multiplier 3 under the Multiplicand, 2^2 ^^its' figure of the multiplicand, and Multiplier, 3 multiply, thus: 8 times 2 units are P- <7 / /^Q/? 6 units; 3 times 3 tens are 9 tens; ' 3 times 2 hundreds are 6 hundreds. The product is 696. (2) (3) (4) (5) (G) 3212 10202 23321 202122 303203 3 3 3 4 3 7. Multiply 230321 by 2. By 3. 8. Multiply 320201 by 3. By 4. By 2. 9. If a gold watch is worth $220, what is the worth of 4 such gold watches ? 10. A drover bought 3 horses at $133 apiece : what did they cost? 11. There are 320 rods in a mile: how many rods are there in 3 miles? In 4 miles? I. A.— 4. 50 INTERMEDIATE ARITHMETIC. LESSON II. J^ew Multiplicand JFlgures, J^ and 5, MENTAL EXERCISES. 1. Twice 4 are how many? 3 times 4? 5 times 4? 6 times 4? 8 times 4? 7 times 4? 9 times 4? 2. Twice 5 are how many? 5 times 5? 6 times 5? 8 times 5? 7 times 5? 9 times 5? 3. How many are 7 times 4? 7 times 5? 9 times 4? 9 times 5? 10 times 5? 4. If a lemon cost 4 cents, what will 6 lemons cost? 5. How much will a man earn in 7 days at $4 a day? In 8 days? In 9 days? 6. There are 5 cents in a half-dime: how many cents in 3 half-dimes? 5 half-dimes? 7. If you write 5 lines a day, how many lines will you write in 4 days? In 7 days? 8. If 5 boys can sit on 1 bench, how many boys can sit on 8 benches? 9. What will 10 oranges cost at 5 cents apiece? 10. If there are 5 school -days each week, how many are there in 6 weeks? In 8 weeks? In 10 weeks? WRITTEN EXERCISES. . 1. Multiply 434 by 6. PROCESS. Multiply the number denoted by each Multiplicand, 434 ^g^re of the multiplicand by 6. Thus: Multiplier 6 ^ times 4 units are 24 units, which P rl f or 04 Gq^^al 2 tens and 4 units; write the 4 units in units' place in the product, and reserve the 2 tens. Six times 3 tens are 18 tens, and 18 tens plus the 2 tens reserved are 20 tens, MULTIPLICATION. 51 which equals 2 hundreds and tens; write the tens in tens* place in the product, and reserve the 2 hundreds. Six times 4 liundreds are 24 hundreds, and 24 hundreds plus the 2 hundreds reserved are 26 hundreds, which equals 2 thou- sands and 6 hundreds; write the 6 hundreds in hundreds' place in the product, and the 2 thousands in thousands' place. The product is 2604. (2) (3) (4) (5) (6) (7) ^ 453 2524 4545 3545 13545 25245 8 6 7 8 4 6 8. If there are 324 pins on a paper, how many- pins are there on 3 papers? 5 papers? 9. If a train of ears run 425 miles a day, bow far will it run in 8 days? 10. If 135 tons of iron rails will make one mile of railroad, how many tons will make 7 miles? 11. What will 6 horses cost at $152 apiece? 12. A father divided his estate between four sons, giviiig to each $3545 : what was the value of the estate ? 13. There are 1440 minutes in a day : how many minutes in 7 days, or a week? 14. If it take 15520 shingles to cover a house, how many shingles will it take to cover 8 houses? LESSON III. JVe>t^ MuUipUcarid I^i^ure, 6, MENTAL EXERCISES. 1. Twice 6 are how many? 4 times 6? 3 times 6? 5 times 6? 7 times 6? 6 times 6? 8 times 6? 9 times 6? 10 times 6? 52 INTERMEDIATE ARITHMETIC. 2. There are 8 rows of trees in an orchard, and G trees in each row: how many trees in the orchard? 3. What will 7 lead-pencils cost at 6 cents apiece ? 4. What will 6 oranges cost at 8 cents apiece? 5. There are 6 days for labor in each week : how many days for labor in 6 weeks? 9 weeks? 6. John caught 6 fishes, and Harry 7 times as nrany as John : how many did Harry catch ? 7. If a horse travel 6 miles an hour, how far will it travel in 5 hours? In 10 hours? 8. There are 6 feet in a fathom: how many feet in 7 fathoms? 9 fathoms? WRITTEN EXERCISES. 1. Multiply 456 by 43. PROCESS. Write the multiplier under the mul- Multiplicand, 456 tiplicand, placing units under units and Multiplier 43 ^^^^ under tens. First multiply by the P f 1 ( 1 Qr8 ^ units, as in the preceding lesson, J J i -loo^ which sfives 1368 for the first partial products, [1824 i ^ xr ^ i^- i i. \n a ^ product. Next multiply by the 4 tens, Product, 19608 o]^)gej,ving that units multiplied by tens (or tens by units) produce tens, that tens by tens produce hundreds, and that hundreds by tens produce thousands, etc. This gives, for the second partial product, 4 te7is, 2 hundreds, 8 thousands, and 1 ten-thousand, which are to be written in their proper orders, since unlike orders can not be added. Then add the two partial products, and their sum, which is 19608, is the product required. Note. — The teacher should show that units multiplied hy tens produce tens; tens by tens, hundreds, etc. This may be done, in the above example, by changing the 4 tens into 40 units. 40 times 6 units = 240 units, or 24 tens; and 40 times 5 tens = 200 tens, or 20 hundreds, etc. The first figure of each partial product is written under the multiplier which produces it. MULTIPLICATION. 53 (2) <3) (4) (5) (6) (7) (8) 606 562 653 1446 2306 4636 40563 54 67 86 234 726 67 143 9. If a ship sail 216 miles a day, how far will it sail in 38 days? 10. What will 27 carriages cost at $165 apiece? 11. If a web of flannel contain 46 yards, how manj^ yards in 397 webs? LESSON IV. JVen^ Multiplicand I^lgure^ 7* MENTAL EXERCISES. 1. Three times 7 are how many? 5 times 7? 7 times 7? 9 times 7? 8 times 7? 2. There are 7 days in a week: liow many days in 2 weeks ? 4 weeks ? 3. How many hills of potatoes in 6 rows if there are 7 hills in each row? 4. If Charles earn 7 dollars a week, liow much will ho earn in 5 weeks? 5. If a horse travel 7 miles in an hour, how far will he travel in 8 hours? 6. If 5 men can build a wall in 7 days, how long will it take 1 man to build it? 7. If a box of crackers will last 8 men 7 days, how long will it last 1 man ? 8. An orchard contains 10 rows of trees, and there are 7 trees in each row : how many trees in the orchard ? 54 INTERMEDIATE ARITHMETIC. WRITTEN EXERCISES. 1. Multiply 2745 by 306. PROCESS. Multiply successively by the first 2745 ^^^^ third figures of the multiplier, 3Q5 observing that units multiplied by -r» ,.7 r TTTT^ hundreds produce hundreds, and hence Partial (164/0 ^/ . ^ ^ ^ ' , , 7 ^ ^ o r» o r writmo; the first figure or the second 2)roducfs, (8235 .f j ^ - u a a ^ ^ V partial product in hundreds order. Product, 8399 7 in 306 there are no tens to be used as a multiplier. (2) (3) (4) (5) (6) (7) 4086 32607 7908 8099 60772 86507 4008 4009 909 1088 1019 9003 8. Enos lived 905 years : how many days did he live, allowing 365 days to the year? 9. A planter raised 208 bales of cotton, each bale weighing 475 pounds: how many pounds of cotton did he raise? 10. If a garrison of soldiers consume 4865 pounds of bread a day, how many pounds will supply the garrison 408 days? 606 days? 11. What will 508 horses cost at $125 apiece? 12. What will it cost to build 705 miles of railroad at $7525 a mile? LESSON V. JVe^f^ MulttpUcand J^lffiirey 8, MENTAL EXERCISES. 1. Three times 8 are how many? 5 times 8? 7 times 8? 9 times 8? 8 times 8? 2. How many are 5 times 8? 8 times 5? 6 times 8? 8 times 6? 7 times 8? 8 times 7? MULTIPLICATION. 55 3. There are 8 quarts in 1 peck : how many quarts in 3 pecks? 5 pecks? 7 pecks? 4. There are 8 pints in a gallon : how many pints in 4 gallons? 6 gallons? 8 gallons? 5. If 5 men can mow a field of grass in 8 days, how long would it take 1 man to do it? 6. If a quantity of provisions will last 7 men 8 days, how long will it last 1 man? 7. If 4 equal pipes will empty a cistern in 8 hours, how long will it take 1 pipe to empty it? 8. If a tnan earn 8 dollars a week, how much will he earn in 9 weeks? 11 weeks? 9. A railroad car has 8 w^heels: how many wheels has a train of 7 cars? 9 cars? . 10. If a horse eat 8 quarts of oats each day, how many will he eat in 6 days? 10 days? 11. If a pint of oil cost 8 cents, what will 8 pints cost? 12. James has 8 marbles, and John has 6 times as many : how many marbles has John ? 13. What will 8 pounds of beef cost at 10 cents a pound? J^rtfUtpllcand 6>r Jlfnltiplle?* endhig with Ciphers. WRITTEN EXERCISES. 1. Multiply 148000 by 47. PROCESS. rpo shorten the process, write the 14 8000 multiplier under the significant fig- 47 ures of the multiplicand, and, omit- Partial (1036 ting the ciphers in forming the par- products, I 592 ^^^^ products, annex them to the Product] 7956000 P"^^"^^ obtained. The result will be the true product. 56 INTERMEDIATE ARITHMETIC. Note. — The teacher should sliow that the use of the ciphers in forming the partial products would produce the same result. (2) (3) (4) (5) 48000 308000 295 4306 36 405 43000 245000 6. There are 5280 feet in a mile : how many feet in 805 miles? 7. The earth moyes in its orbit at an average rate of 68000 miles in an hour: how far does it move in 24 hours? In 48 hours? ^ 8. If a carriage-wheel revolve 280 times in running a mile, how many times will it revolve in running 68 miles? 75 miles? 9. A canal -boat was loaded with 245 bales of hay, weighing 280 pounds each: what was the weight of* the cargo? 10. There are 480 sheets of paper in a ream : how many sheets are there in 604 reams? 11. If an acre of land produce 380 pounds of cot- ton, how many pounds will 248 acres produce? 12. A steam-boat makes 145 trips in a season, and carries, on an average, 280 passengers each trip : how many passengers does she carry during the season? LESSON VI. Ji^cH^ MiiUipl'ica7id I^l(5iire, 9. MENTAL EXERCISES. 1. Three times 9 are how many? 4 times 9? 6 times 9? 8 times 9? 7 times 9? 9 times 9? 2. How many are 5 times 9? 9 times 5? 7 times 9? 9 times 7? 10 times 9? 9 times 10? MULTIPLICATION. 57 3. How many are 5 times 10? 10 times 5? 7 times 10? 10 times 7? 9 times 10? 10 times 9? 4. A man gave 7 boys 9 rabbits each : how many rabbits did he give them all? 5. If a man earn 10 dollars a week, how much will he earn in 8 weeks? 6. Jane writes 9 lines each day at school : how many lines does she write in 8 days? 7. Charles receives 9 dollars a month as errand- boy: how much will he earn in 10 months? 8. If 7 Inen can do a piece of work in 9 days, how many men will it take to do the same work in one day? 9. If a quantity of provisions will supply 10 men 9 days, how long will it supply one man? 10. What will 6 barrels of flour cost, at $9 a barrel? ^oth Multfj^licand cmd Mnlfipller ending ^WRITTEN EXERCISES. 1. Multiply 198000 by 8900. PROCESS. Write the significant figures of 198000 t^^^ multiplier under the significant 8900 figures of the multiplicand, and ^rjoiy multiply, omitting the ciphers in ^^o4 forming the partial products, but annexina: them to the product oh- Product, 1762200000 ,^.^^^ ^^^ ,^^ ,^^^^ p,,^^,, (2) (3) (4) (5) 94000 90800 470000 950000 1600 370000 1900 360000 58 INTERMEDIATE ARITHMETIC. 6. There are 3600 seconds in one hour : how many seconds are there in 630 hours? 7. Light moves 192000 miles in a second: how far does it move in one hour? 8. A ship has provisions enough to allow the crew 130 pounds a day for 90 days : how many pounds of provisions are aboard? 9. What will 1700 tons of railroad iron cost at $250 a ton? 10. An army is composed of 54 regiments, contain- ing, on an average, 670 men each : how many men in the army? 11. If a steamer can run 260 miles a day, how far can it run in 10 days? In 100 days? 12. In a field of corn there are 70 rows, and each row contains 280 hills, and each hill 3 stalks: how many stalks of corn in the field? LESSON VII. 1. What will 4 bananas cost at 5 cents apiece? 2. What will 5 barrels of flour cost at $9 a barrel ? 3. If an orange is worth 5 apples, how many apples are 7 oranges worth? 4. If there are 8 pints in a gallon, how many pints are there in 6 gallons? 5. Two men start from the same place, and travel in opposite directions, one at the rate of 3 miles an hour and the other 4 miles an hour: how far will they be apart in 8 hours? 6. If an orange is worth 2 lemons and a lemon is worth 5 plums, how many plums are worth 6 oranges? MULTIPLICATION. 59 7. If 7 men can do a piece of work in 5 days, how long would it take 1 man to do it? 8. If 6 men can cut a field of grass in 8 days, liow many men will it take to cut it in 1 day? 9. If 3 pipes fill a cistern in 10 hours, in how many hours will 1 pipe fill it? WRITTEN EXERCISES. 1. What is the product of 4894 X 37? 2. What is the product of 5680 X 340? 3. 6084 X 3008 = how many ? 4. 704000 X 4800 = how many? 5. Multiply forty-eight thousand by sixty -five thou- sand. 6. In a train of 37 cars, each car contains 9850 pounds of freight : how much freight in the train ? 7. If 980 pounds of bread will supply the inmates of the State Prison one day, how many pounds will supply them one year, or 365 days? 8. If a sack of salt contain 168 pounds, what will be the weight of 1600 sacks? 9. A merchant bought 18 firkins of butter, each weighing 32 pounds, at 37 cents a pound : what did it cost? 10. A train of 27 cars is loaded with iron; each car contains 48 bars, and each bar weighs 365 pounds: what is the weiirht of the car^o? DEPINITIONS, PRINCIPLES, AND EULE. Art. 33. MiilUpllcation is the process of taking one number as many times as there are units in another. (See Multiplication Table, p. 213.) 60 INTERMEDIATE ARITHMETIC. The Multiplicand is the number taken or mul- tiplied. The MaltipUer is the number denoting how many times the multiplicand is taken. The Product is the number obtained by multi- plying. The multiplicand and multiplier are called the Factors of the product. Art. 34. The Sign of Multiplication is X , and is read multiplied by. When placed between two num- bers, it shows that the number before it is to bo multiplied by the number after it. Thus : 6 X 3 is read 6 multiplied by 3. Note. — Since a change in the order of the factors does not change tlie product, 6X3 may also be read 6 times 3. Art. 35. Multiplication is a short method of addition. The sum of 5 + 5 -[- ^ -|- 5 is the same as 4 times 5. Art. 36. Rule for Multiplication. — 1. Write the multiplier under the mult iplic and ^ jplachig units iinder units^ tens under tens, etc. 2. When the multiplier consists of but one term, begin at the right and multiply successively each term of the multiplicand, ivriting the right-hand term of each residt in the product and adding the left-hand term to the next result. 3. When the multiplier consists of more than one term, multiply the multiplicand successively by each significant term of the multiplier, writing the first term of each partial product under the term of the multiplier which produces it. 4. Add the particd products thus obtained^ and the sum will be the true product. MULTIPLICATION 61 Art. 37. 1. When the multiplier or multiplicand, or both, end with one or more ciphers, omit the ciphers in the partial products and annex them to the product obtained. 2. Any number may be multiplied by 10, 100, 1000, etc., by annexing to it as mayiy ciphers as there are ciphers in the multiplier. LESSON VIII. Problems co?nbhim^ cidditlo7i, Stibtractiojr, and Mulitplication . MENTAL EXERCISES 1. 6x7 + 4 + 5 + 8 + 7 — 6== how many? 2. 8X4 +6 — 3 + 2 — 5 + 6== how many? 3. A grocer bought 10 barrels of apples, at %\ a barrel, and sold them so as to gain $15 : for how much did he sell them? 4. John has 6 marbles, and Willis has 4 times as many less 9, and Charles has as many as both John and W^illis: how many marbles has Charles? 5. A lady teacher receives $9 a week, and spends S6 for board and washing : how much can she save in 8 weeks? 6. Two men start from the same place and travel in opposite directions, one at 7 miles an hour and the other at 5 miles an hour : how far will they be ap>art in 8 hours? 7. Two stages start from the same place and go in the same direction, one at 9 miles an hour and the other at 6 miles an hour: how far w^ill they be apart in 5 hours? 8. When oranges are sold at 7 cents apiece and C2 intp:rmediate arithmetic, lemons at 5 cents apiece, how many cents will buy 6 oranges and 8 lemons? 9. If a man earn $8 a week and a boy $3, how much will they both earn in 7 weeks? 10. A pedestrian left a city and walked 9 hours at the rate of 4 miles an hour ; he then returned at the rate of 3 miles an hour, but in 4 hours stopped to rest: how far was he from the city? 11. If a man earn $12 a week and 82)end $7, how much will he save in 9 weeks? WRITTEN EXERCISESo 1. From 4080 X 26 take 2024 X 16. 2. A grocer bought 275 barrels of flour for $2475, and sold it at $12 a barrel: what did he gain? . 3. A clerk receives $125 a month, and spends $68 a month: how much does he lay up each year? 4. An agent sold 48 sets of outline maps, at $16 a set ; the maps cost him $10 a set : how much did he make ? 5. If a steamer carry, on an average, 75 passengers each trip, how many passengers will it carry in 12 weeks, making 3 trips a week ? 6. A book contains 288 pages, each page contains 42 lines, and each line 13 words: how many words in the book? 7. A man bought a farm for $4780; he sold 80 acres at $33 an acre, and the remaining portion for $2560 : how much did he make by the transaction ? 8. A regiment contains 960 men, exclusive of the commissioned officers ; the men receive $16 a month, and the aggregate salary of the officers is $2800 a month: what is the monthly pay of the regiment? 9. A drover bought 480 head of cattle in Ohio, at DIVISION. 63 $45 a head, shipped them to New York, at an ex- pense of $6 a head, and then sold them at $56 a head: how much did he make? 10. A miller manufactured 560 barrels of flour, and sold it at $9 a barrel ; the wheat cost $2750, and the expense of running the mill was $960: how much did he make? 11. A man sold 5 horses at $87 apiece, and received $350 in cash and a note for the balance : what was the value of the note ? 12. The President's salary is $50000 a year: if his expenses are $2500 a month, how much can he save during his term of 4 years? 13. If a quantity of provisions will supply 960 soldiers 27 days, how many soldiers will it supply one day? SECTION Y. Divisiojy. LESSON I. 1. How many times is 2 contained in 6? 2 in 12? •2 in 16? 2 in 18? 2 in 20? 2. How many times is 3 contained in 9? 3 in 12? 3 in 15? 3 in 18? 3 in 21 ? 3 in 27? 3. How many times is 2 contained in 12? 3 in 12? 2 in 18? 3 in 18? 2 in 24? 3 in 24? 4. Two boys sit at 1 desk : how many desks will seat 8 boys? 16 boys? 64 INTERMEDIATE ARITHMETIC. 5. If a man walk 3 miles an hour, how long will it take him to walk 15 miles? 21 miles? Solution. — At 3 miles an hour, it will take as many hours to walk 15 miles as 3 miles are contained times in 15 miles, which are 5: it will take 5 hours. ^WRITTEN EXERCISES. I. Divide 848 by 2. PROCESS. Write the divisor at the left T^. . ^^ r. . r. ^. . y , ^f* thc dividcnd, and draw a JjLVisor, 2)848, Dividends j t i. 4. ^1, ^ a^Ta n curved hne between them, and ^ ^ * a straight line under the divi- dend. Begin at the left, and divide successively each term of the dividend by the divisor. The quotient is 424. (2) (3) (4) (5) (6) 2)482 2)8642 3)6936 3)9369 3)3696 7. Divide 3609 by 3. 8084 by 2. 8. Divide 4684 by 2. 6309 by 3. 9. At $3 a bushel, how many bushels of wheat can be bought for $963? For $639? 10. In how many hours can a man walk 396 miles, if he w^alk at the rate of 3 miles an hour? II. If a man earn $2 a day, how long will it take him to earn $360? LESSON II. Jl^cH' DIpIso7^ I^i^u7^es, ^ and S. MENTAL EXERCISES. 1. How many times is 4 contained in 8? 4 in 12? 4 in 20? 4 in 28? 4 in 36? DIVISION. 65 2. How many 5's in 15? 30? 40? 25? 50? 35? 45? 20? 3. Ill an orchard there arc 16 trees, in rows of 4 trees each : how many rows in the orchard ? 4. How many ranks of 4 soldiers each will 24 sol- diers make? 32 soldiers? 40 soldiers? 5. A man planted 30 peach trees in rows, setting 5 trees in each row : how many rows did they make ? 6. A school -room contains 35 desks, arranged with 5 desks in each row: how many rows of desks in the room ? 7. How many chairs, at $4 apiece, can be bought for $36? 8. How many pairs of boots, at $5 a pair, can be bought for $35? 0. Mary is reading 5 chapters a day : how long will it take her to read 45 chapters? 10, A boy had 50 peach -stones, which he jilanted in rows of 5 each: how many rows did he plant? A^TKITTEN EXEKCISES. 1. Divide 784 by 4. PROCESS. Write the divisor at the left r.. . ,s^r> T^ . T T of the dividend. Begin at the Devisor, 4 ) 784, Dwidend, i<,ft.,,,^„,, ^^^ „f the dividend, 196, Quotient. ^nd divide, thus; 4 is contained in 7 hundreds 1 hundred times, with 3 hundreds remaining. Write the 1 hundred in hun- dreds' place in the quotient, and reduce the 3 hundreds re- maining to 30 tens, which, with the 8 tens added, make 38 tens. Four is contained in 88 tens 9 ten times, with 2 tens remaining. Write the 9 tens in tens' place in the quotient, and reduce the 2 tens remaining to 20 units, which, with. the 4 units added, make 24 units. Four is contained in 24 units I. A.— 5. 66 INTERMEDIATE ARITHMETIC. 6 times. Write the 6 units in units' place in the quotient. The quotient is 196. (2) (3) (4) (5) (G) 4)764 4)936 5)640 5)870 5)765 7. Divide 1128 by 2 ; by 3 ; by 4. 8. Divide 8740 by 2 ; by 4 ; by 5. 9. Divide 18480 by 2; by 3; by 4; by 5. 10. A mMnufacturer packed 372 clocks in boxes, placing 4 clocks in each box: how many boxes were required ? 11. If 4 bushels of wheat will make a barrel of flour, how many barrels will 972 bushels make? 12. If a man earn $4 a day, how many days will it take him to earn $1584? LESSON III, A^cw divisor J^/'^urcs^ 6 and 7. MENTAL EXEKCISES. 1. Six is contained in 12 how many times? 6 in 24? 6 in 36? 6 in 48? 6 in 54? 2. How many times 7 in 7? 7 in 21? 7 in 35? 7 in 49? 7 in 63? 7 in 42? 7 in 56? 3. How many 6's in 42? 7's in 42? 6's in 30? 5's in 30? 7's in 28? 4's in 28? 7's in 35? 5's in 35? 7's in 56? 4. If 6 chairs make a set, how many sets will 36 chairs make? 48 chairs? 60 chairs? 5. There are 7 da^^s in a week : how many weeks in 49 days? In 56 days? 63 days? 6. There are 6 feet in a fathom : how many fathoms in 54 feet? In 60 feet? DIVISION. 67 7. An orchard contains 56 trees in rows of 7 trees each: how many rows of trees in the orchard? 8. How many plows, at $6 each, can be bought for $48? For $54? WRITTEN EXERCISES. 4)784(190 Solve each of the written exercises in the 4 preceding lesson (II), writing the quotient at r^ the right of the dividend, and the partial o/> dividends and products below the dividend, as at the right: 2 4 24 1. Divide 4876 by 23, PROCESS. Divide 48 hundreds by 23, and 23)4876(212, quotient. write the result, 2 hundreds, at 46 the right of the dividend for the ~^ first or hundreds' term of the quotient. Multiply the divisor (23) by this quotient term, and '*" subtract the product, 46 huur ^^ dreds, from 48 hundreds. To the remainder, 2 hundreds, an- nex the 7 tens of the dividend, giving 27 tens for the second partial dividend. Divide 27 tens by 23, and write the result, 1 ten, for the tens' term of the quotient. Multiply 23 by this 1 ten, and subtract the product, 23 tens, from 27 tens. To the remainder, 4 tens, annex the 6 units of the dividend, giving 46 units for the third partial dividend. Divide 46 units by 23, and write the result, 2, for the units' term of the quotient. Multiply 23 by 2, and subtract the result from 46. The quotient is 212. Note. — In tliis and the next eight problems, each term of the quotient may be determined hy dividing the left-hand term of the partial dividend bi/ the left-hand term of the divisor. In the last five problems (6 to 10, inchisive) each term of the quotient may also be determined by dividing; the two left-hand terms of the partial dividend by the two left-hand terms of the divisor. 68 INTERMEDIATE ARITHMETIC. 2. Divide 4686 by 22. 68952 by 221. 3. Divide 3813 by 123. 63336 by 203. 4. Divide 446886 by 2013. 5. Divide 678273 by 2113. 6. Divide 549661 by 1043. 7. Divide 69898188 by 10678. 8. Divide 4890375 by 1035. 9. Divide 35854660 by 10435. 10. Divide 45691212 by 10562. 11. Divide 1608 by 67. PROCESS. 67)1608(24, Quotient 1 34 268 268 Since 67 is not contained in the number denoted by the first two left-hand terms of the dividend, take 160 tens for the first partial dividend. Note. — It is sometimes difficult to tell how many times the divisor is contained in a partial dividend. Wlien this is the case, take the first left-hand term, or first two left-hand terms, of the divisor for a trial divisor, and the proper number of left-hand terms of the partial dividend for a trial dividend. If the divisor, multiplied by the quotient term thus found, gives a product greater than the partial dividend, tlie quotient term is too large, and should be reduced. Tiie trial divisor in the above ex- ample is 6, the first trial dividend is 16, and the second 26. The teacher should make this process plain to tlie pupil. 12. Divide 312 by 24. 374 by 17. 13. Divide 792 by 36. 1625 by 65. 14. Divide 2520 by 36. 3024 by 63. 15. Divide 64347 by 267. 49179 by 507. 16. There are 36 inches in a yard : how many yards are there in 792 inches? 17. A bushel of corn weighs 56 pounds : how many bushels of corn in 24416 pounds? DIVISION. 69 18. A hogshead of molasses contains 63 gallons : how many hogsheads in 4788 gallons? 19. If 72 books can be packed in a box, how many boxes will it take to hold 17496 books? 20. How many farms of 156 acres each can be sold from a tract of land containing 7332 acres? 21. If a vessel sail, on an average, 47 miles a day, how long will it take it to sail 2303 miles? 22. There are 365 days in a common year : how many years are there in 90155 days? LESSON IV. JVeH^ divisor I^igiire, 8, MENTAL EXERCISES. 1. How many times is 8 contained in 8? 8 in 24? 8 in 40? 8 in 56? 8 in 72? 2. How many 8's in 56? 7's in 56? S's in 48? 6's in 48? 8'8 in 72? 9's in 72? 3. 32 -- 8 == how many? 49 -r- 7? 54 -f- 6? 64 ~ 8? 56-4-7? 56 -- 8? 4. There are 8 quarts in a peck : how many pecks in 72 quarts? 5. If a steamer run 8 miles an hour, in how many hours will it run 80 miles? 6. There are 8 furlongs in a mile: how many miles in 56 furlongs? 7. At $8 a barrel, how many barrels of flour can be bought for $64? 8. If a man work 8 hours a day, in how many days will he work 96 hours? 70 INTERMEDIATE APaXHMETIC. The Quotient contahi'mg One or more Cip/iers, WRITTEN EXERCISES. 1. Divide 341:^7 by 84. PROCESS. Since the divisor is not con- 84)34137(406 tained in the second partial div- 33g idend (53 tens), Avrite in the yz^ tens^ phice in the quotient, and _ ^ . annex the 7 units for a third par- ») 04 tial dividend. As there is no hg- 3 3, Remainder. ^^^ ^^f ^j^^ dividend left to annex to 33 to form a new partial divi- dend, 33 remains undivided, and is called the remainder. 2. Divide 24399 by 48. 4G7034 by 806. 3. Divide 2845007 by 5728. 215607 by 18036. 4. Divide 1423685 by 6785. 1604083 by 2088. 5. In 1 week there are 168 hours: how many weeks in 85008 hours? 6. A drover went West with $23490 to buy cattle: how many cattle could he buy at $58 a head ? 7. If a garrison consume 648 pounds of bread in a day, how long will 134136 j)ounds last it? 8. If the average daily receipts of a ferry-boat are $275, in how many days will the receipts amount to $165825? 9. How long will it take a pipe, discharging 158 gallons of water in an hour, to empty a cistern con- taining 7584 gallons? 10. A cord of wood contains 128 solid feet: how many cords in a pile containing 5280 solid feet? 11. Divide 543513392 by 6704. DIVISION. 71 ^ LESSON V. MENTAL EXERCISES. 1. How many times 9 in 18? 9 in 27? 9 in 36? 9 in 54? 9 in 72? 9 in 90? 2. How many 9's in 45? 5's in 45? 9's in 63? 7's in 63? 9'8 in 72? 8's in 72? 3. How long will it take a steamer to make a trip of 81 miles if it run 9 miles an hour? 4. If 9 words fill a line, how many lines will 72 words fill? 81 words? 5. If a man can do a piece of work in 90 days, how many men can do it in 9 dnys? 6. If a quantity of provisions will last 72 men one day, how long will it last 9 men? 7. How many sheep, at $9 a head, can be bought for $54? For $63? 8. A copy-book contains 100 lines, with 10 lines on each page: how many pages in the book? 9. If a man earn $10 a week, how long will it take him to earn $100? 10. How many tons of hay, at $10 a ton, can be bought for $90? 11. 8x6^8 + 9 — 7-- what? 12. 9x4---6 + 7-;|-6 — 8== what? 13. 72 -- 9 X 6 -f- 8'x 3 -- 9 = what? 14. 81-f-9--3x8--Gx8 — 9 = what? 15. 7x8-^7x9 — 9 + 4 — 8::= what? To THE Teacher. — For additional examples see Manual OF Arithmetic, page 39. 72 INTERMEDIATE ARITHMETIC. The divisor ending hi One 07* more Cfp/ie7^s» WRITTEN EXERCISES. 1. Divide 350 by 10. FIRST PROCESS. By comparing these two pro- cesses, it is seen that 350 is divided by 10, by cutting off the right-hand figure. Tlie reason is obvious. Tlie cut- ting off of the right-hand fig- ure removes each of the other figures one place to the right, and thus decreases their value ten-fold. In like manner, it may be shown that cutting off the two right-hand figures divides a number by 100; cutting off three right-hand 65, Remainder, figures, by 1000, etc. 10)350(35 30 50 50 SECOND PROCESS. 1 1 )35|0 3 5, Quotient. 2. Divide 2865 by 100 1|00 )28|65 2 8, Quotient. 3. Divide 45600 by 10. By 100. 4. Divide 187000 by 1000. By 100. 5. Divide 384050 by 100. By 1000. 6. Divide 230045 by 1000 By 10000. 7. Divide 450860 by 10000. By 1000. 8. Divide 196800 by 4800. PROCESS. 48100)1968100(41, Quotient. 192 48 48 First divide both divisor and dividend by 100, which is done by cutting off the two right-hand figures. Then divide 1968, the new dividend, by 48, the new divisor. The quotient is 41. Note.— The teacher can show that both divisor and dividend may he divided by any number withont affecting the value of the quotient. PROCESS. 45100)588164(13 45 138 135 3 64, Remainder. DIVISION. 73 9. Divide 63200 by 7900. 10. Divide 116087000 by 2900. 11. Divide 70338000 by 75000. 12. Divide 58864 by 4500. First divide both divisor and dividend by 100, wliicli, in the case of the dividend, leaves a remainder of 64. Next divide 588 by 45, leaving a remainder of 3 hundreds, to wliich annex the first remainder (64), obtain- ing 364 for the true remainder. Note. — Tlie trne remainder is found by annexing the first remain- der to the second. The reason for this can be easily given by the teacher. 13. Divide 466384 by 3900. 220345 by 940. 14. Divide 99990 by 5400. 172800 by 14400. 15. A barrel of beef contains 200 pounds: how many barrels will contain 12800 pounds? 16. There are 480 sheets of paper in a ream : how many reams will 129600 sheets make? 17. There are 3600 seconds in an hour: how many hours in 172800 seconds? 18. How many city lots, at $1600 each, can bo bought for $25600? 19. How many cars, each carrying 1800 pounds, will transport 79200 pounds of hay? 20. How many barrels, each holding 196 pounds, will hold 9016 pounds of flour? 21. How many regiments, averaging 750 men each, will make an army of 30000 men ? 22. A peach orchard contains 6758 trees, and there are, on an average, 62 trees on each acre : how many acres in the orchard? T4 INTERMEDIATE ARITHMETIC. 23. A pipe discharges 94 gallons in an hour: in how many iioiirs will it empty a cistern holding 3384 gallons of water? 24. What number multiplied by 98 will produce 15288? 25. The dividend is 5292 and the divisor is (]3 : what is the quotient? 26. The divisor is $1500 and the dividend $564000: what is the quotient? DEFINITIONS, PEINOIPLES, AND EULES. Art. 38. Division is the process of finding how many times one number is contained in another. The Dividend is the number divided. The Divisor is the number by which the dividend is divided. The Quotient is the number of times the divisor is contained in the dividend. The lieniainder is the part of the dividend which is left undivided. When the dividend contains the divisor an exact number of times, there is no re- mainder. Art. 39. The Sign ^ of Division is ^, and is read divided by. When placed between two numbers, it shows that the number before it is to be divided by the number after it. Thus : 16 -f- 4 = 4 is read 16 divided by 4 equals 4. Division is also expressed by writing the dividend above and the divisor below a short horizontal line. Thus: \f- is read 18 divided 'by '6. Art. 40. One number is contained in another num- ber as many times as it can be taken from it. Hence DIVISION. 75 ' division is a short method of finding hgw many times one number may be subtracted from another. A number is contained in another as many times as it must bo taken to produce it. Hence division may be regarded as the reverse of multiplication. The divisor and quotient are factors of the dividend. Art. 41. There are two methods of division, called Short Division and Long Division. In Short Division, the partial products and par- tial dividends are not written, but are formed men- tally. This method is generally used when the divisor does not exceed 12. In Long Division, the j^artial products and partial dividends are written. Art. 42. EuLE for Short Division. — 1. Write the divisor at the left of the dividend^ draw a curved line between them^ and a straight line under the dividend. 2. Find how many times the divisor is contained in the left-hand term or terms of the dividend^ taken as a partial dividend^ and write the quotient under the last figure of the dividend used. 3. Midtiply the divisor by the quotient term found, and subtract the product from the partial dividend used, performing each process mentally. 4. Prefix the remainder^ if there be one, to the next term of the dividend for a second partial dividend^ and divide, multiply, and subtract, as before. 5. Proceed in this manner until all the terms of the dividend have been used. Proof. — Multiply the divisor by the quotient, to the product add the remainder, if there be any, and if the result equals the dividend, the work is correct. 76 INTERMEDIATE ARITHMETIC. Art. 43. EuLE for Long Division. — 1. Write the divisor at the left of the dividend, draw a curved line beticeen them, and also at the right of the dividend, to separate it from the quotient. 2. Take as many of the left-hand terms of the dividend as will contain the divisor, for a partial dividend ; find how many times this will contain the divisor, and write the quotient at the right of the dividend for the first left-hand term of the quotient, 3. Multiply the divisor by the quotient term found, write the product under the partial dividend used, and subtract. 4. To the remainder annex the next term of the divi- dend for a second partial dividend, and divide, multiply^ and subtract, as before. 5. Proceed in this manner until all the terms of the dividend have been used. Note.— When any partial dividend does not contain the divisor, write a ciplier in the quotient, and annex another term of the divi- dend to form a new partial dividend. Art. 44. When one or more of the rigbt-haxid fig- ures of the divisor are ciphers — 1. Cut off the ciphers from the right of the divisor, and an equal number of figures from the right of the dividend. 2. Divide the yiew dividend thus formed by the new divisor, and the result will be the quotient. 3. Prefix the remainder, if there be one, to the figures cut off from the dividend, and the result will be the true remainder. DIVISION. 77 Art. 45. To divide an}^ number by 10, 100, 1000, etc., — Cut off as many figures from the right as there are ciphers in the divisor. The figures cut off are the true remainder. LESSON VI. MISCBLLA^J^BO US "RBriBW T^CSLBMS. 1. The Bum of two numbers is 15 and one of the numbers is 6: what is the other? 2. The difference between two numbers is 8 and the smaller number is 9: what is the larger? 3. The product of two numbers is 56 and one of the numbers is 7: what is the other? 4. The quotient of two numbers is 6 and the di- visor is 8: what is the dividend? 5. How many barrels of flour, at $8 a barrel, will pay for 24 yards of carpeting, at $2 a yard ? 6. How many tons of coal, at $9 a ton, will pay for 15 cords of wood, at $6 a cord? 7. A grocer bought 7 barrels of flour at $6 a baiTel : for how much a barrel must he sell it to gain $14 on the lot? 8. If 1 man can build a wall in 36 days, how many men can build it in 4 days? 9. If 6 men can do a piece of work in 8 days, how many men can do it in 12 days? 10. Two vessels start from the same port and sail in the same direction, one sailing 12 miles an hour and the other 9 miles an hour: how far apart will they be in 10 hours? 78 INTERMEDIATE ARITHMETIC. WKITTEN EXERCISES. 1. The greater of two numbers is 4056 and their difference is 3650 : what is the less number ? 2. The subtrahend is 34203 and tlie remainder is 8706: what is the minuend? 3. The divisor is 534 and the quotient 43: what is the dividend? 4. The product of two numbers is 5328 and one of the numbers is 148: what is the other? 5. Multiply 486 + 392 by their difference. 6. Divide the product of 48 and 24 by their differ- ence. 7. A merchant bought 35 yards of cloth at $56, and sold it at $2 a yard: how much did he gain? 8. A drover bought 240 sheep at $8 a head, and then sold 90 of them at $12 a head, 75 at $9 a head, and the rest at $6 a head: how much did he gain? 9. A farmer exchanges 65 bushels of wheat at $2 a bushel, and 35 sheep at $6 a head, for cows at $34 a head: how many cows did he receive? 10. A man's income is $3500 a year; he pays $450 a year for house-rent, $150 for taxes, $350 for hired help, and $45 a month for other expenses: how much has he left at the close of the year? 11. A man bought 80 acres of land at $35 an acre, paid $325 for improvements, and then sold it for $3750: how much did he gain? 12. A grain merchant having 3500 bushels of oats, sold 1650 bushels, and then bought twice as much as he had left: how many bushels did he buy? 13. A man left an estate to his wife and three children ; the wife received $4500 ; the youngest child, $1500; the second child, $1850; and the eldest DIVISION. 79 child, as much as both of the others less $1350: what was the value of the estate? 14. A and B start together on a journey, A trav- eling 28 miles a day and B 33 miles : how far apart will they be in 12 days? 15. A and B start together and travel in opposite dii-ections, A going 28 miles a day and B 33 miles: how far apart will they be in 12 days? Questions for Keview. What is addition? What is meant by sum or amount? What does it contain? What is meant by like numbers? What numbers can be added ? What is the sign of addi- tion ? What does it show ? Give the rule for addition. What is the method of proof? What is subtraction ? The difference, or remainder ? The minuend? Tlie subtrahend? What numbers can be sul)- tracted? What does the sum of the remainder and subtra- hend equal ? What is the sign of subtraction ? What does it show? Give the rule for subtraction. What is a method of proof? What is multiplication ? The multiplicand? The multi- plier? The product? Of what are the multiplicand and multiplier factors? What is the sign of multiplication? What does it show ? How may the product be obtained by addition ? Give the rule for multiplication. How may you multiply when either the multiplicand or multiplier, or both, end in ciphers? How may any number be multiplied by 10, 100, 1000, etc.? What is division? The dividend? The divisor? The quotient? The remainder? What is the sign of division? What does it show? In what other way may division be expressed ? How many times may the divisor be subtracted from the dividend? Of what is division the reverse? 80 INTERMEDIATE ARITHMETIC. What is short division ? When is it used ? Give the rule. What is long division ? Give the rule. How do you proceed when a partial dividend will not contain the divisor? How may you divide when the divisor ends in ciphers ? How may any number be divided by 10, 100, 1000, etc. ? SECTIOl^ VI. LESSON I. Divisor, Greatest Common Dlyisor, and I^actor, Note. — The term number used in this section, denotes an integer. 1. What numbers besides itself and 1 will exactly divide 15? 21? 25? 30? 56? 63? 2. What numbers besides itself and 1 will exactly divide 7? 11? 13? 17? 23? 37? 41? 3. What numbers will exactly divide 4? 5? 16? 19? 24? 29? 33? 31? 42? 4. What are the divisors of 10? 28? 31? 33? 43? 49? 53? 55? 70? 90? 99? Note. — Since every number is exactly divisible by itself and 1, these divisors need not be given. 5. What number is a divisor of both 9 and 12? 15 and 20? 24 and 27? 42 and 56? 6. W^hat divisor is common to 28 and 35? 27 and 36? 42 and 54? 63 and 81? 7. What is a common divisor of 15 and 30? 45 and 60? 50 and 75? 60 and 84? PROPERTIES OF NUMBERS. 81 8. What is the greatest divisor common to 48 and 72? 27 and 54? 50 and 75? 9. What is the greatest common divisor of 24 and 36? 32 and 48? 56 and 84? 10. What is a common divisor of 16, 32, and 48? 15, 30, and 45? 36, 54, and 72? 11. What is the greatest common divisor of 32 and 48? 15, 30, and 45? 36, 54, and 72? 18, 45, and 81? Art. 46. A number that has no divisor except itself and 1, is called a Prime J\^U7riber. A number that has other divisors besides itself and 1, is called a Composite Jfumher, 12. Name all the prime numbers between and 20. Between 20 and 30. 40 and 50. 13. Name all the composite numbers between 20 and 30. 40 and 50. 60 and 70. 14. What are the prime divisors of 15? 18? 22? 28? 33? 36? 37? 40? 43? 15. What are the prime divisors of 16? 20? 25? 27? 35? 44? 55? 60? Art. 47. The divisors of a number are called its Factors; and prime divisors are called Prime Factors, 16. What are the prime factors of 21? 24? 35? 39? 42? 49? 54? 56? 63? mi 72? 17. Of what number are 2, 3, and 5 prime factors? 2, 5, and 7? 2, 2, 3, and 5? Note. — Tlie product of the prime factors of a number equals the number. 18. Of what number are 2, 2, 3, and 3 prime factors? 2, 3, 5, and 7? 3, 5, 2, and 7? I. A.— 6. 82 INTERMEDIATE ARITHMETIC. WRITTEN EXERCISES. 1. What are the prime factors of 126? PROCESS. Divide 126 by 2, a prime 2)126 divisor; next divide the quo- g \ gg tient 63 by 3, a prime divisor ; oT^T and then divide the quotient 21 by 3, a prime divisor. The prime factors are 2, 3, 3, and 7. 7 126 = 2X3X3X7. "What are the prime factors of 2. 63? 6. 175? 10. 264? 14. 440? 3. 72? 7. 147? 11. 200? 15. 500? 4. 84? 8. 275? 12. 256? 16, 648? 5. 96? 9. 325? 13. 250? 17. 900? 18. What is the greatest common divisor of 126 and 210? PROCESS. Eesolve 126 and 210 into their 126 = ^X$X3Xyf prime factors. The product of 2io_:,^\/*\/5\/r^ the factors common to both will 2 X "^ X 7 == 42, Alls. be the greatest common divisor required. Note. — Tliis process and tliat for finding the least common multiple (Art. 54) may be easily explained by means of objects. What is the greatest common divisor of 19. 54 and 90? 23. 81 and 135? 20. 72 and 96? 24. 63, 84, and 126? 21. 75 and 90? 25. 96, 144, and 192? 22. 84 and 108? 26. 128, 224, and 320? Art. 48. A Divisor of a number is a number that will exactly divide it. A Common Divisor of two or more numbers is a number that will exactly divide each of them. PROPERTIES OP NUMBERS. 83 The Greatest Common Divisor of two or more numbers is the greatest number that will exactly divide each of them. Art. 49. A Prime Number is one that has no divisor except itself and 1. A Composite Number is one that has other di- visors besides itself and 1. Art. 50. An Ei^en Number is exactly divisible by 2; as, 2, 4, 6, 8, 10, 12, etc. An Odil Number is not exactly divisible by 2; as, 1, 3, 5, 7, 9, 11, 13, etc. All even numbers except 2 are composite. Art. 51. EuLES. — 1. To resolve a composite num- ber into its prime factors. Divide it by any "prime divisor^ and the quotient by any prime divisor, and so continue until a quotient is obtained which is a prime number. The several divisors and the last quotient are the prime factors. 2. To find the greatest common divisor of two or more numbers. Resolve the given numbers into their prime factors^ and select the factors which are common. The product of the common factors will be the greatest common divisor. LESSON II. Multiple and Zeast Conimo7i Multiple, Art. 52. When a number is multiplied by an in- teger, the product is called a Multiple, Thus, 3G, or 12 X 3, is a multiple of 12. 1. What number is a multiple of 3? 4? 5? 7? 8? 10? 15? 20? 25? 30? 84 INTERMEDIATE ARITHMETIC. 2. What number is a multiple of 18? 24? 35? 45? 44? 60? 100? 250? Note. — The teacher should show that a number has any num- ber of multiples. 3. What number is a common multiple of 3 and 4? 4 and 5? 6 and 8? 5 and 6? 4. What number is a common multiple of 7 and 5? 6 and 9? 3, 4, and G? 4, 8, and 12? Note. — The teacher should show that two or more numbers have any number of common multiples. 5. What is the least common multiple of 3 and 4? 5 and 6? 3, 6, and 12? 2, 4, and 8? 6. What is the least common multiple of 3, 5, and 10? 2, 5, and 10? 2, 3, 5, and 10? WRITTEN EXEBCISES. 1. What is the least common multiple of 12, 18, and 30? PROCESS. ^ Resolve the num- 12 = ^ V ^ y 3 bers into their prime |g__2y*yj^ factors, and select 30 = 2X3X0 all the different fac- — tors, repeating each 2x2x3x3x5=r 180, L. G, M. as many times as it is found in any num- ber. The factor 2 is found twice in' 12 ; the factor 3, twice in 18 ; and the factor 5, once in 30. The product of 2 X 2 X 3 X 3 X 5 is the least common multiple required. What is the least common multiple of 2. 12, 15, and 20? 7. 24,72,18,48? 3. 21, 24, and 42? 8. 15, 24, 18, 32? 4. 32, 48, and 80? 9. 75,150,300? 5. 27, 54, and 108? 10. 125, 250, 500? 6. 24, 80, and 120? 11. $48, $7 2, $144? PROPERTIES OF NUMBERS. 85 Art. 53. A Multiple of a number is any number which it will exactly divide. Note. — Every number is an exact divisor of its product by an integer. A Coinmon Multiple of two or more numbers is any number which each of them will exactly divide. The Least Common Multiple of two or more numbers is the least number which each of them will exactly divide. Art. 54. EuLE.— To find the least common multiple of two or more numbers, Resolve the numbers into their prime factors, and then select all the different fac- tors^ taking each the highest number of times it is found in any number. Multiply the factors thus selected ; their product will be the least common multiple. Questions for Eeview. What is meant by the divisor of a number ? When is a divisor a common divisor ? Define a common divisor. What is the greatest common divisor of two or more numbers? How is it found ? By what may every number be divided ? What is a prime number ? A composite number ? What is an even number ? An odd number? What is meant by the factor of a number ? A prime fac- tor? A composite factor? How may a composite number be resolved into prime factors? What is a multiple of a number? When is a multiple a common multiple? Define a common multiple. What is the least common multiple of two or more numbers? Give the rule for finding the least common multiple. What is the differeuce between a divisor and a multiple^ SECTIOIS^ VII. I'HA CTIOJTS. LESSON I. The Idea of a JRractloji developed. 1. If a melon be cut into two equal pieces, what part of the melon will one piece be? 2. How many halves in a melon? How many halves in any thing? 3. If a melon be cut into four equal pieces, what part of the melon will one piece be? Two pieces? Three pieces? 4. How many fourths in an apple? How many fourths in any thing? 5. Which is the greater, one half or one fourth of an apple? How many fourths equal one half? 6. If a cake be cut into three equal pieces, what part of the cake will one piece be? 7. How many thirds in a cake? How many thirds in an}^ thing? 8. If a cake be cut into six equal pieces, what part (86) P^RACTIONS. 87 of the cake will one piece be? Two pieces? Three pieces? Four pieces? Five pieces? 9. How many sixths in any thing? 10. Which is the greater, one third or one sixth of a cake? How many sixths equal one third? 11. A single thing is a unit. How many halves in a unit? How many thirds? How many fourths? How many sixths? 12. What is meant by one third? Ans. One third is one of the three equal parts of a unit. 13. What is meant b}^ two thirds? One fourth? Three fourths? One sixth? Three sixths? 14. Which is the greater, two thirds or a unit? Five thirds or a unit? Three fourths or a unit? Four fourtlis or a unit? Art. 55. Such parts of a unit as two thirds, three fourths, five sixths, etc., are called Fractions. A fraction may be expressed by two numbers, one written under the other, with a horizontal line be- tween them; as, |, f, f. The number below the line denotes the number of equal parts into which the unit is divided. It is called the Denominator. The number above the line denotes the number of equal parts taken. It is called the Numerator, Eead the following fractions. How is the unit di- vided, and how many parts are taken in each case? (15) 16) (17) (18) (19) (20) f f A A A H 1 A t\ H \% M 1 % 5 TT U M M 88 INTERMEDIATE - ARITHMETIC. Write the following fractions in figures : (21) (22) (23) Two fifths; Seven twelfths; Twenty-four fortieths; Seven ninths; Ten thirteenths; Thirty-five fiftieths ; Nine tenths ; Forty fiftieths ; Twenty -two twelfths ; Ten ninths. Twenty seventeenths. Forty fifty-fifths. DEHNITIONS. Art. 56. A Fraction is one or more of the equal parts of a unit. Art. 57. A fraction is expressed by two numbers, called the Numerator and the Denominator. The Uenoininator of a fraction denotes the num- ber of equal parts into which the unit is divided. The Nmnerator of a fraction denotes the number of equal parts taken. The numerator and denominator are called the Terms of a fraction. LESSON II. Integers mid Mixed JVumbers reduced to I'r act ions. 1. How many thirds in an apple? How many thirds in 2 apples? 2. How many fourths in 3 pears ? Solution. — In 1 pear there are 4 fourths, and in 3 pears there are three times 4 fourths, which is 12 fourths. There are 12 fourths in 3 pears? 3. How many sixths in 3 oranges? In 5 oranges? 6 oranges? 8 oranges? FRACTIONS. 89 4. How many fifths in 3? 5? 8? 10? 5. How many eighths in 4? 6? 8? 10? 6. How many halves in 2 and 1 half oranges? Solution. — In 2 oranges there are twice 2 halves, which is 4 halves, and 4 halves and 1 half are 5 halves. There are 5 halves in 2 and 1 half oranges. 7. How many fourths in 2 and 3 fourths? * 8. How many thirds in 5 and 2 thirds? 8 and 1 third? 7 and 2 thirds? 9. How many sixths in 5 and 2 sixths? 8 and 3 sixths? 12 and 5 sixths? 10. How many tenths in 6 and 3 tenths? 7 and 5 tenths? 8 and 7 tenths? 11. Eead6|; 33^; 45i ; 25|1; 50,^; 66^%. 12. How many fifths in 6f ? 8|? 12|? WRITTEN EXERCISES. 13. Eeduce 157 to ninths. 157-Z- to ninths. 157 9 1571 9 PROCESS : 1413 9 , Am. PROCESS : 1413 7 1420 , Ans, 14. Eeduce 96 to eighths. 96| to eighths. 15. Eeduce 35ii^ to twelfths. 46|- to ninths. 90 INTERMEDIATE ARITHMETIC. 16. Reduce 73^^ to elevenths. 63^ to sevenths. 17. Eeduce 53|-J to a fraction. Suggestion. — Reduce the mixed number to twentieths. 18. Eeduce 33y^ to a fraction. Reduce to a fraction: 19. 85^\ 22. 236| 25. 48H 20. 361^ 23. 49,^^ 26. 6VoV 21. 4Sji 2-1. ^h'^ 27. ^hh To Teachers. — See Manual of Arithmetic for additional problems in this and the following lessons in Fractions. Art. 58. A Mixed Number is an integer and a fraction united; as, 51, 16|, 33^. Art, 59. Rules. — 1. To reduce an integer to a frac- tion, Multiply the integer by the given denominator, and write the denominator under the product, 2. To reduce a mixed number to a fraction, Multiply the integer by the denominator of the fraction, to the product add the numerator, and write the denominator under the result. LESSON III. I^ractions reduced to Integers or Mixed JVumbers. 1. How many pears in 6 half-pears? In 7 half-pears ? 2. How many days in 10 half-days? In 11 half-days? Solution. — In 11 half-days there are as many days as 2 half-days are contained times in 11 half-dayS; which is 5 J times. There are 5J days in 11 half-days. FRACTIONS. 91 3. How many pints in 14 half-pints? In 17 half- pints? In 21 half-pints? 4. How many yards in 18 thirds of a yard? In 19 thirds of a yard? In 22 thirds of a yard? 5. How many weeks in 28 sevenths of a w^eek? 30 sevenths of a week? 6. A mason was 17 half-days in building a wall : how many days did he work? 7. A boy earned 25 fourths of a dollar by selling papers: how many dollars did he earn? 8. A man walked 25 eighths of a mile in an hour: how many miles did he walk? 9. How many ones in -%5-? ^-? \^-? -«/? 10. How many ones in f^? "ff? i^? |3? WKITTEN EXERCISES. 11. Eeduce ^^^- to a mixed number. Process : W- = ^77 -:~ 15 ^^ iif|, Am. 12. Eeduce ^^- to a mixed number. Eeduce to an integer or mixed number: 13. n- 17. M 21. -w 14. w 18. -w 22. w 15. w 19. W 23. w 16. ¥/ 20. -w 24. w 25. li^ 26. V/ 27. -V_o 28. ^^ Art. 60. A Proper Fraction is one whose nu- merator is less than its denominator; as, |, |^, |-. An Improper Fraction is one whose numerator 92 INTERMEDIATE ARITHMETIC. is equal to or greater than the denominator; as, h h f- The value of a proper fraction is less than one ; and the value of an improper fraction is equal to or greater than one. Art. 61. EuLE. — To reduce an improper fraction to an integer or mixed number, Divide the numerator of the fraction by the denominator. LESSON IV. JFh^actions reduced to Zojper Terms. 1. How many half-inches in 2 fourths of an inch? In 4 fourths of an inch? 2. How many thirds of an inch in 2 sixths? In 4 sixths? In 6 sixths? 3. How many fourths in 6 eighths? Solution. — In 2 eighths there is 1 fourth, and in 6 eighths there are as many fourths as 2 eighths are contained times in 6 eighths, which is 3. There are 3 fourths in 6 eighths. 4. How many fifths in 2 tenths? In 4 tenths? 6 tenths? 8 tenths? 12 tenths? 5. How many fifths in 6 fifteenths? 9 fifteenths? 12 fifteenths? 18 fifteenths? 6. How many sevenths in ^? -i-l? ^? 7. How many eighths in ^? |f ? -|4? 8. Eeduce ^, ^, and ^ each to fourths. 9. Eeduce If, f|, and -^ each to sevenths. 10. Eeduce -^-l, -||-, and ^ each to eighths. FRACTIONS. 93 , and II each to sixths. 12. Keduce |f , ^, and |-| each to tenths. 11. Eeduce |-|, f^, and || each to sixths. WRITTEN EXERCISES. 13. Reduce ff to its lowest terms. PROCESS. Reduce f f to |J by dividing 63^3 21-r-7 3 . both terms by 3; next reduce 34_^3~~28^ 'T" 4 ' * f i ^^ f ^y dividing both terms bv 7 ; J can not be reduced to 6 3-^21 3 * Qj.. ^ — _ lower terms, and, hence, is in 8 4 -f- 2 1 4 its lowest terms. Or, reduce f j to f by dividing both terms by 21, the greatest number which will exactly divide each term. Note. — The teaclier should show that the value of a fraction is not ciianged by dividing both of its terms by the same number. Reduce to lowest terms: 14. n 18. T% 22. m 26. m 15. m 19. 2oG 23. t'A 27. ill 16. 96 144 20. tV% 24. m 28. m 17. ■rh 21. A\ 25. m 29. m Art. 62. When a fraction is reduced to an equiv- alent fraction with smaller terms, it is reduced to lower terms. A fraction is in its Lowest Terms when no integer except 1 will exactly divide both numerator and de- nominator. Art. 63. Principle. — The division of both terms of a fraction by the same number does not change its value. 94 INTERMEDIATE ARITHMETIC. Art. 64. EuLE. — To reduce a fraction to its lowest terms, Divide both tet^ms of the fraction by any common divisor; then divide both terms of the residting fraction by any common divisor ; and so on, until the terms of the resulting fraction have no common divisor except 1. Or : Divide both terms of the fraction by their great- est common divisor. LESSON V. Israel Ions reduced to Illghe?* 2*ertns» 1. How many- fourths of an or- ange in 1 half? In 2 halves? 2. ITow many eighths of an or- ange in 1 fourth ? In 3 fourths ? Solution. — In 1 fourth there are 2 eighths, and in 8 fourths there are 3 times 2 eighths, which is 6 eighths. There are 6 eighths in 3 fourths. 3. How many ninths in 1 third? In 2 thirds? 3 thirds? 4 thirds? 4. How many tenths in |? f? |? 5. How many twelfths in f ? f ? |? 6. Change \ and ^ each to twelfths. 7. Change f , |, and |- each to eighteenths. 8. Change -|, \, and ii each to twenty-fourths. 9. Change |-, -j7^, and -^ each to thirtieths. 10. Change -f-, -J-i, and \ each to twenty-eighths. FRACTIONS. 95 WRITTEN EXERCISES. 11. Change H to seventieths. PROCESS. One thirty-fifth is as many seven- 70^35 = 2. tieths as 35 is contained times in 70, 17V9 ^4 which is 2 times, and 17 thirty-fifths = — J ^^^' are 17 times 2 seventieths, which is 34 ^ seventieths. This is the same as mul- tiplying both terms by the quotient of 70 divided by 35. 12. Change if to ninety-sixths. 13. Change ^]- and || each to eighty-fourths. 14. Change -^y ^, and |-| each to seventy-seconds. 15. Eeduce f , |^, and \^ to equivalent fractions having a common denominator. PROCESS. ^, Change the fraction to twenty-fourths^ Heduce to a common denominator: IC. 11^ 19. f I t!^ 22. I VV A tt 17. I t t 20. I f Jj 23. I H \l If ^^' Z lU To ^^- H T2- 2 4 ^^' TXT 20 25^ FO Art. 65. When a fraction is changed to an equiva- lent fraction with greater terms, it is reduced to Higher Terms. Several fractions having the same denominator, are said to have a Common Denominator. Art. 66. Principle. — TTie rmdtiplication of both terms of a fraction by the same number does not change its value. 96 INTERMEDIATE ARITHMETIC. Art. 67. EuLES. — 1. To reduce a fraction to higher terms, Divide the given denominator by the denominator of the fraction, and multiply both terms by the quotient. 2. To reduce fractions to a common denominator, Divide the least common multiple of the denominators by the denominator of each fraction, and multiply both terms by the quotient, LESSON VI. Addllion of JFractlo7is, 1. A boy gave 1 fourth of a pine-apple to his brother, 1 fourth to his sister, and 1 fourth to a playmate: what part of it did he give away? How many fourths are ^ -f ^ + |-? 2. A grocer sold 1 eighth of a cheese to one cus- tomer, 2 eighths to another, and 3 eighths to another: what part of it did he sell? How much is I + I + I? 3. How many sixths in \, |-, and f ? f , f , and A? I-, f, and f ? 4. A boy gave 1 half of his money for a knife, and 1 third of it for a ball: what part of his money did he spend? Suggestion.— Change \ and \ to sixths. 5. How many tenths in ^ and |? \ and -j^? 6. How many twelfths in \ and i? \ and |? 7. How many eighths in J and i^? | ynd |? 8. How many fifteenths in \ and i? | and |? 1- and f ? I and f ? FRACTIONS. 97 9. How many twentieths in i and |? | and |? f and I? I and f ? 10. How many twenty -fourths in i, ^, \, and J? In i 2. 3. and 4 '<* WRITTEN EXERCISES. 11. What is the sum of ^3, y^^, ^, -j^? PROCESS : A + A + A + A ^ T J ^ 2^ , ^^«. 12. What is the sum of |f, |f , -5^, and -if? 13. What is the sum of |^, if, ff^ and f|? 14. What is the sum of f , |^, and j^? PROCESS. Change the fractions to twenty- 5 _|_ 7 _|_ 5 -_ fourths ; add the numerators of 20 1 21 I 10 51 the new fractions; and reduce the resulting improper traction to a ii'=^Ti = ^y ^^^' mixed number. 15. What is the sum of f , |, and ^? 16. Add I, I, and I . 17. Add i, ^, and i^. 18. Add I, ^, and ||. 19. Add i, f, andij. 20. Add I, ^, f, and if. 21. Add f , I, j\, and i|. 22. Add f, ^,-, 1^, and |i. 23. Add^, H, /^, and ||. 24. What is the sum of 16|, 18f , and 37|? First add the fractions and then the in- tegers. | = A, i=--ft, i^A- A + « PROCESS. 16f A 371 ? 4- A = ft = IH • Write the ^ under the - — '- fractions and add the 1 with the intecrers. I. A.— 7. 98 INTERMEDIATE ARITHMETIC. 25. Add 45^-, 67|, and 62f . 26. Add 37|, 18f , 33|, and 25-^, 27. Add 30^, 66|, 84f , and 133i. 28. Add 75|, 108, 160f , and 207. Art. 68. EuLES. — 1. To add fractions, Reduce the fractions to a common denominator^ add, the numerators of the new fractions^ and under the sum write the com- mon denominator. 2. To add mixed numbers, Add the fractions and the integers separately^ and combine the results. LESSON VII. Subtraction of JFVactions, 1. Albert had 2 thirds of an orange, and he gave 1 third to his sister: how many thirds had he left? How much is | less i? | less |? 2. Charles bought 3 fourths of a pound of raisins, and then gave 1 fourth of a pound to his playmate: what part of a pound had he left? How much is f less i? f less |? 3". A farmer bought | of a bushel of flax-seed, and sold 1^ of a bushel to a neighbor: what part of a bushel had he left? SuoGESTioN.— Change \ and \ to sixths. 4. How much is f less i? f less 1? 5. How much is \ less |? \ less 1? 6. How much is -^ less \'l -h l^ss 1? 7. How much is -^ less f ? A less 4? 8. How much is if less f? H less f ? 9. How much is \ less -J-? A less 3^? FRACTIONS. 99 WRITTEN EXERCISES. 10. From ii- take ^^. PROCESS : J^ — ^ — ^4^ =: 1 , Arts. 11. From If take \\ . 12. From ^\ take ^\\. 13. From i| take |. PROCESS : 1 1 — f ^ 1 2 _ 1 5 ^ ^7^ ^ ^^5^ 14. Take f from y7_; | from f . 15. Take -^-^ from ^; -J-^ from f|. 16. Subtract if from ff ; |f from |f . 17. Subtract y^^ from -j^; -j^ from |-. 18. Subtract y\ from ^\ ; ^ from J-|. 19. Subtract ^ from -J-^] \\ from ii. 20. From 33^ take 18f . First subtract the fractions and then the PROCESS. integers. Since x% is greater than ^^, add 33 J ^ ^ to j%, making ^f , and then take the ^ 18| f^ from ^, writing/^ under the fractions, and 14 7 j^g adding 1 to the 8 units before subtracting the integers. 21. Take 30^ from 66|; 45f from 66|. 22. Take 112| from 145^; 90^ from 108|-. 23. Subtract 250f from 300; 105| from 261i. 24. Subtract 13o| from 241f ; 166| from 233|. Art. 69. EuLES. — 1. To subtract fractions, Beduce the fractions to a common denominator^ subtract the numerator of the subtrahend from the numerator of the minuend^ and under the difference write the common denominator. 2. To subtract mixed numbers, First subtract the fractions and then the integers. 100 INTERMEDIATE ARITHMETIC. LESSON VIII. Problems involving the Addition aiid Subtractio7i of I^ractions, 1. A boy spent \ of bis money for a sled, and | of it for a pair of skates: wbat part had be left? 2. John bought a knife for f of a dollar and a ball for 1^ of a dollar, and then sold both of them for |- of a dollar : what part of a dollar did he gain ? 3. Jane having |^ of a quart of plums, gave -|^ of a quart to her brother and |^ of a quart to her sister: how much had she left? 4. A farmer bought 11 bushels of clover-seed, and then sold |- of a bushel to one neighbor and f of a bushel to another: what part of a bushel had he left? 5. A student spends \ of his time in study, J^ of it in labor, and \ of it in sleep: what part has he left? 6. One sixth of a pole is in the ground, |- of it in water, and the rest in the air: what part is in the air? 7. .A man bequeathed \ of his estate to his wife, \ of it to each of his two sons, and the rest to liis daughter: Avhat part did the daughter receive? 8. A man did \ of a piece of work the first day, \ of it the second day, and then completed it the third: what part did he do on the third day? WRITTEisr EXERCISES. 9. From the sum of |, f , and | take i^. 10. From the sum of | and \ take their difference. 11. A man owning -y- of a vessel, sold \ and \ of the vessel: what part had he left? 12. A farmer bought 240-| acres of land, and sold 90|- acres and 75^ acres : how many had he left? FRACTI01S13. 101 13. From a piece of broadcloth Containing 20| yards, a merchant sold 5|- yards, 4^ yards, and 8^ yards: how many yards were left? 14. A man earned $56|- one month and $70| the next, and then gave $85^ for a horse : how much money had he left? 15. From 47f + 33^ take their difference. 16. A pedestrian walked -f-^ of his journey the first day, ^ of it the next day, and completed it the third day: what part of the journey did he travel the third day? LESSON IX. J^ractio7is JVfult /plied by Integers. 1. What j^art of a cake is twice 2 eighths of it? 3 times 2 eighths of it? 2. A father gave 3 fourths of an orange to each of 4 children : how many fourths did they all receive? 3. How much is 4 times |? 6 times f ? 4. If a boy earn 2 thirds of a dollar in a day, how much will he earn in 3 days? 5. How much is 3 times |? 5 times |? 6. IIow much is 6 times f ? 9 times f ? 7. How much is 7 times |? 8 times |? 8. How much is 5 times 6|? 7 times 8-i-? Suggestion. — Multiply the integer and the fraction separately, and add the products. 9. How much is 3 times 6|? 8 times 7|? 10. How much is 6 times 4f ? 9 times Sf? 11. How much is 8 times 6|? 8 times 7f ? 102 INTERMEDIATE ARITHMETIC. Multiply : AATRITTEN EXERCISES. 12. T-V by 8. 15. li by 25. •18. if by 16, 13. ^, by 12. 16. ii by 16. -19. 16| by 4. 14. If by 24. 17. « by 33. •20. 18| by 12. Art. 70. Principle. — A fraction is multiplied by mul- tiply ing its numerator or dividing its denominator. Art. 71. EuLES. — 1. To multiply a fraction by an integer, Multiply the numerator or divide the denom- inator. 2. To multiply a mixed number by an integer, Mul- tiply the fraction and the integer separately ^ and add the products. LESSON X. JRr actional !Pa?^is of Integers. 1. If 6 pears be , divided equally between 2 boys, what part of the whole will each receive? What is \ of 6 pears? \ of 10 pears ? 2. A father di- vided 5 melons equally between 2 children : what part of the whole did each receive? What is I of 5 melons? Suggestion.— Take 1 half of 4 melons and then 1 half of 1 melon. FRACTIONS. 103 3. Charles divided 12 plums equally between 3 boys: what part of the whole did each receive? 4. What is ^ of 9? ^ of 12? i of 16? 5. What is ^ of 20? \ of 28? 1^ of 30? 6. What is 1 of 25? 1 of 26? I of 37? 7. What is J- of 24? f of 24? Solution. — J of 24 is 4, and f of 24 is 5 times 4, which is 20. f of 24 is 20. 8. What is -^ of 40? ^ of 40? ^ of 40? 9. What is f of 63? f of 64? f of 65? 10. What is I of 45? f of 37? | of 58? 11. What is I of 33? I of 58? ^ of 50? WRITTEN EXERCISES. 12. What is I of 659^ 8)659 659 PROCESS : o Z f 824 Or: ^ 3 8)1977 247i,Ans. 247^ 13. What is | of 191? | of 367? 14. What is f of 508? ^ of 243? 15. What is f of 466? ^ of 4648? 16. What is ^ of 906? f| of 6070? Integers tnuWpUed by I^ractfons, 17. Multiply 256 by f . f is 3 times J, and hence 4 )256 256 3 times 256 is 3 times \ of 64 ^ 256. Or, f is 4 of 3, and 3 ^^ • 4 )768 hence f times 256 is J of 3 192, Arts. 192 times 256. 104 INTERMEDIATE ARITHMETIC. 18. 48 by yV 22. 163 by -^ , " 26. 248 by if . 19. 65 by I-. 23. 300 by if. '27. 406 by ^. 20. 59 by f . ^ 24. 257 by fi . 28. 856 by ^f . -^21. 87 by |. ^25, 305 by |^. 29. 794 by |f . 30. Multiply 324 by 16f . 324 1^1 First multiply by the 1944 integer and then by the process: 3 24 fraction, and add the 2 16 products. 5400, Ans. /31. 48 by 16|. 34. 246 by 12|. 37. 108 by 56f. ' 32. 72 by 18f . 35. 324 by 17i. 38. 524 by 72f . 33. 96 by 23^ . 36. 406 by S3i . 39. 684 by 66| . Art. 72. EuLES. — 1. To find the fractional part of an integer, or to multiply an integer by a fraction, Divide the integer by the denominator and multiply the quotient by the numerator. Or: Multiply the integer by the numerator and divide the product by the denominator. 2. To multiply an integer by a mixed number. Mul- tiply by the integer and the fraction separately , and add the products, LESSON XI. Coinpoii7id I^racUons reduced to Simple J^7*acHo7is, 1. A boy having i of a melon, gave -^ of it to a playmate: what part did the playmate receive? What is i of ^? i off? FRACTIONS. 105 2. If each third of a pine -apple bo cut into 2 equal pieces, what part of the pine-apple will 1 piece be? What is i of i? 3. What is i of i? ^ of ^? ^ of i? 4. What is i of i? ^ of i? i- of i? 5. What is ^ of i? i of |? -^ of -i? 6. A girl having f of an orange, divided it equally between her 2 brothers : what part of the orange did each receive? Suggestion. — Divide each fourth into 2 equal pieces, and then give 3 pieces to each. 7. What is ^ of ^? ^ off? 8. What is i of -i? i of 3.? 8 • ioff? 9. What is I of I? i of f ? I of I? Solution. — J of f is ^^, and j of f is 3 times ^%, which lb ^ry. :f OI ^ — ^^. 10. What is I of f ? I of I? f of I? 11. What is f of -I? I of f ? I of 3^? 12. What is i of 12^? i of 131? Solution.— J of 13^ == ^ of 12 + J of li or f . J of 12 is 4, and J of f is f or | . Hence, J of 13| is 4^ . 13. What is I of 18|? i of 21|? | of 31|? 14. What is | of 22|? i.of 42^? i of 46f ? 15. What is ^ of 334? I'of 644? ^^^ of 62|? 106 INTERMEDIATE ARITHMETIC. AVKITTEN EXERCISES. 16. Reduce f of | to a simple fraction. 2 3_2><^__6__2 ^^^ process: gOt 5-3^5 ~15~5' 2 . 3 2X$ Note. — The examples should be solved by both methods. The teacher should explain the process of cancellation. Eeduce to a simple fraction : 17. f of i. 21. 1 of A. 18. fof f- 22. f of|f- 19. f of,^. 23. Aofj^. 20. |of A- 24. Aofif. 25. f of f off. 26. I of f of 2I-. 27. f of 2| of I . 28. 4 of f of If . JRracti07is niulilpUed by JFractlons* 29. Multiply | by f. PROCESS. Since J is } of one, f 4X3 12 3 \Am^% 4 ^| of once f, or J of f , which equals — -^ — 4X5 ^ 4^3 4X3 3 5X^"5' ^^"^^'5>^4^.^r^ = 5' 30. I by |. 34. if by 3^. 38. 2i by 2i. 31. f by ^. 35. I^by f4. 39. 3^ by 3^. 32. ,3^ by |. 36. I by ^-. 40. 61 by 2^. 33. I by 3^. 37. i^ by V-. 41. 6^ by 12|. FRACTIONS. 107 Art. 73. A Simple Fraction is a fraction not united with an integer or another fraction ; as, f . A Compound Fraction is a fraction of a fraction ; as, I off; |of 3i. Art. 74. Rule. — To reduce a compound fraction to a simple fraction, or to multiply one fraction by an- other. Multiply the numerators together, and also the denominators. Note.— The process may often be shortened by canceling com-%k mon factors before multiplying. LESSON XII. J^rac lions divided by l7itegers, 1. A father divided J of a melon equally between 3 boys : what part of the melon did each receive ? Solution. — If 3 boys receive f of a melon, each will re- ceive J of f , which is \, 2. A woman divided f of a pound of crackers equally between 3 poor children : what part of a pound did each receive? 3. If 5 pounds of cheese cost f of a dollar, what will 1 pound cost? 4. How much is f -- 5? f -f- 5? | -f- 5? 5. If 6 men can build |^ of a wall in a day, what part of the wall can 1 man build? 6. How much is |- -- 6 ? |^ -^ 5 ? | -'r- 9 ? 7. A grocer put 161 pounds of sugar into 5 equal parcels: how much sugar was put into each parcel? 8. Divide 16^ by 5. ^ 12^ by 5. 181 by 5. 9. Divide 20| by 3. 301 by 4. 31f by 6. 108 INTERMEDIATE ARITHMETIC. A^V^KITTEnsr EXERCISES. 10. Divide ^ by 3. PROCESS : ^^~Z = \oi ^^ = /^ 12 12X3 36 11. xVby 7. 12. 6 by 12. 13. 14. 6 tV V 10. T^by 6. 15. -If by 7. 16. if by 8. 17. If- by 5, IS.fiby 3. 19. 241 by 6. 20. 29f by 7. 21. 46f by 5. 22. 66| by 8. Art. 75. Principle. — A fraction is divided by dividing the numerator or multiplying the denominator. Art. 76. EuLES. — 1. To divide a fraction by an in- teger, Divide the numerator or multiply the denominator. 2. To divide a mixed number by an integer, Divide the integral part and then the fraction. LESSON XIII. I7itegers divided by JFr actions. 1. How many times is \ of an apple contained in 2 ap- ples? \ of an apple in 2 ap- ples? 2. How many times is \ of a yard contained in 3 yards? Solution. — 3 yards = 12 fourths, and 3 fourths is con- tained in 12 fourths 4 times. r FRACTIONS. 109 3. If a basket hold |^ of a bushel, how many baskets will hold 4 bushels? 4. How many times is | contained in 4? | in 4? f in 4? 5. If I of a yard of silk will trim a hat, how many hats will 6 j^ards trim? G. How many times is f contained in 3? f in 6? f in 6? f in 3? 7. Divide 12 by f ; 15 by f ; 20 by f . 8. Divide 8 by A; lObyf; 12 by ^\. WRITTEN EXERCISES. 9. Divide 14 by f . PROCESS : 14 = 7^0-. 7^0 ^ 3 ^ 70 ^ 3 = 23 J, Ans, Note. — Since 14 is reduced to fifths by multiplying it by 5, tlie process may be shortened by omitting the denominators, thus: 14^ f = 14X5-f-3==23i. 10. 16 by |. 13. 60 by f . 16. 30 by 2| . 11. 20 by I-. 14. 21 by f. 17. 40 by 3^. 12. 45 by f . 15. 42 by f . 18. 16 by 5i. Art. 77. EuLE. — To divide an integer by a fraction, Multiply the integer by the denominator of the fraction, and divide the product by the numerator. LESSON XIV. I^ractlons divided by J^ractions, 1. If 1^ of a barrel of flour will supply a family 1 month, how many months will f of a barrel last ? 2. How many times is \ contained in |-? -J- in |? 110 INTERMEDIATE ARITHMETIC. 3. If f of a 3'ard of cloth will make a vest, how many vests will |^ of a yard make? 4. How many times f in |^? |^ in |^? 5. If a pound of butter cost ^ of a dollar, how many pounds can be bought for f of a dollar? Suggestion.— Change one fourth to eighths. 6. How many times ^ in |^? i in |^? 7. How many times ^in|^? iini? |^ini? 8. How many times iin|? |^in|? fin|? WKITTEN EXEKCISES. 1. Divide f by |. process: } = a • f = 1%^ • A -^ A = f = IJ • 5. f by |. 8. 2| by |. 6. I by |. 9. 3^ by f . 7. I by f . 10. I by 2|. Art. 78. EuLE. — To divide a fraction by a fraction, Beduce the fractions to a common denominator^ and di- vide the numerator of the dividend by the numerator of the divisor. Note. — When pupils are familiar with this method, they may be taught to divide by inverting the terms of tlie divisor and mul- tiplying. This method is fully explained in the author's Com- plete Aeithmetic. LESSON XV. JViimbers J^ractlonal 'Parts of Other JVumbers. 1. 5 is -i of what number? SoLUTioi^'. — 5 is I of 3 times 5, or 15. 2. 1 byf- 3. 1 by i- 4. f\ by |. 7. 12 is 1 of what 8. 25 is 1 of what 9, 30 is f of what 10. 33 is f of what 11. 44 is * of what FRACTIONS. Ill 2. 7 is ^ of what, number? 3. 12 is j\ of what number? 4. 12^ is ^ of what number? 5. 16| is |- of what number? 6. 10 is I of what number? Solution. — If 10 is f of a number, -J of the number is J of 10, which is 5 ; if 5 is J of a number, J of it will be 3 times 5, which is 15. number? number? number? number? number? 12. A man spent f of his money and had $21 left: how much money had he at first? 13. A boy gave 24 cents for a slate, which was ^ of his money: how much money did he have? 14. A man pays $25 a month for house-rent, which is -^-^ of his monthly salary: what is his salary? 15. A farmer sold a cow for $45, w^hich was \ more than he paid for her : what was the cost of the cow ? 16. A man sold f of a farm for $1500: at this rate, what was the value of the farm? LESSON XVI. Miscellaneous 'Problems, 1. Eeduce 18f to an improper fraction. 2. Eeduce ^^- to a mixed number. 3. Eeduce f , f , and ^ to a common denominator. 4. Add i, i, I, and ^. 5. Add 28f , 40 1-, 63|, and 19^3^. 112 INTERMEDIATE ARITHMETIC. 6. From f take f . From 28| take 16|. 7. Multiply I by 7; 13 by |; f by f. 8. Multiply 137| by 15; 256 by 21|. 9. Divide 12 by | ; | by 12; f by f. 10. Divide 243| by 11 ; 256 by ^, 11. | + | = what? f-l? |x|? i^i? 12. There are 5280 feet in a mile : how many feet in ^ of a mile ? 13. A vessel is worth $6000, and the cargo is worth I as much as the vessel: what is the value of the cargo ? 14. A man sold | of his farm to one neighbor and I of it to another: what part of the farm has he left? 15. A man owning -| of a factory sold | of his share : what part of the factory did he sell ? What part does he still own? 16. There are 161 feet in a rt)d : how many feet in 66 rods? 17. There are 5^ yards in a rod: how many rods in 66 yards? 18. At $6^ a barrel, how many barrels of flour can be bought for $150? 19. If f of a ship is worth $12000, what is the whole ship worth ? 20. A man owning ^ of an estate sells t of his tshare for $2400: at this rate, what would be the value of the estate ? 21. A farmer had 2 fields of wheat; the first yielded 840 bushels, which was ^ of the amount yielded by the second field : how many bushels did the second field yield? 22. A man owning |- of a ship sells |- of his share for $4400 : at this rate, what is the value of the ship? FRACTIONS. 113 28. The value of a certain shii^ is $9760, and | of the value of the ship is |^ of the value of the cargo : what is the value of the cargo ? Questions for Eeview. What is a fraction ? What does the denominator denote ? The numerator? What are the terms of a fraction? What is a mixed numher? What is meant by 18|? Ans. 18 -f- J- When is a fraction called proper? When improper? When is the value of an improper fraction equal to 1 ? How is an integer reduced to a fraction ? How is a mixed number reduced to a fraction ? What kind of a fraction is the result? Give examples. How is an improper fraction reduced to a whole or mixed number ? How is a fraction reduced to lower terms? On what principle does the process depend? How may a fraction be reduced to its lowest terms by one division? How are fractions having a common denominator added or subtracted? When fractions have different denominators, how are they added or subtracted ? How may mixed num- bers be added? How may they be subtracted? In what two ways may a fraction be multiplied by an integer? How may an integer be multiplied by a fraction? Give the rule for multiplying a fraction by a fraction. What is a compound fraction ? How is a compound frac- tion reduced to a simple fraction ? In what two ways may a fraction be divided by an in- teger? How may an integer be divided by a fraction? How may a fraction be divided by a fraction ? N. B. — If it is thought best to teach United States Money before Decimal Fractions^ introduce Section IX at this point. 1. A.— 8, SECTION VIII. LESSON I. JVumeratJon and JVotation, 1. If a unit be divided into ten equal parts, what is one part called? 2. If a tenth of a unit be divided into ten equal parts, what is one part? What is ^ of y^^? 3. If a hundredth of a unit be divided into ten equal parts, what is one part? What is J^ of y^? 4. What part of one tenth is one hundredth? What part of one hundredth is one thousandth? 5. How do one tenth and one hundredth compare with each other in value? ^V ^^^ rlo^ 6. How do one hundredth and one thousandth compare with each other in value? y|^ and y (jV(r ^ 7. How, then, do the fractional units, tenths, hun- dredths, and thousandths, compare in value? Art. 79. The fractional units, tenths, hundredths, and thousandths may be expressed on a scale of tens, by writing the tenths in the first order at the right of units, the hundredths in the second order, and the thousandths in the third order, and placing a period between the orders of units and tenths, to distinguish the fractional orders from the integral orders. Thus, 2.5 denotes two units and five tenths; 4.06 denotes four units and six hundredths; .05 denotes five hundredths; and .004 denotes four thousandths. (114) - DECIMAL FRACTIONS. 115 8. How many tenths in .3? In .6? In .7? 9. How many hundredths in .02? In .04? .07? 10. How many thousandths in .005? In .008? 11. How many tenths and hundredths in .24? .06? 12. How many tenths, hundredths, and thousandths in .356? In .523? In .603? In .041? Art. 80. Two tenths and five hundredths (25) de- note twenty-five hundredths; and two hundredths and five thousandths (.025) denote twenty-five thousandths. 13. How many hundredths in .34? In .45? In .06? 14. How many thousandths in .246? In .048? In .605? In .007? In .403? In .075? Art. 81. Such fractional numbers as .24 and .208 are called Decimal Fractions, or Decimals, and the orders of which they arc composed are called Decimal Orders. The first decimal order is tenths; the second, hun- dredths; and the third, thousandths. Copy and read the following decimals: (15) (16) (17) (18) (19) .12 .014 .324 .53 .004 .3 4 .06 3 .406 .6 .803 .0 6 .008 .065 .009 .550 .50 .030 .704 .057 .400 Art. 82. When fractions denoting tenths, hun- dredths, thousandths, etc., are expressed on a scale of ten, they are said to be expressed decimally. The right-hand figure is written in the order indicated by the name of the decimal. 20. Express decimally 25 hundredths. 21. Express decimally 205 thousandths. 116 INTERMEDIATE ARITHMETIC. 22. Express decimally 26^ thousandths. 23. Express decimally 6J- hundredths. 24. Express decimally forty-five hundredths. 25. Four hundred and twelve thousandths. 26. Seven hundred and eight thousandths. 27. Sixty-five thousandths. 28. Eight units and seven tenths. 29. Fifteen units and thh^ty-six hundredths. Art. 83. The fourth decimal order is called ten- thousandths; the fifth, hundred-thousandths; and the sixth, millionths. Copy and read : (30) (31) (32) (33) .445 .0304 .3256 .00267 .0445 .00304 .4048 .000267 .706 .475 .03256 .004324 .0706 .00475 .04048 .046 3 75 34. Express decimally 3205 ten-thousandths. 35. Express decimally 6008 hundred-thousandths. 36. Express decimally 40532 millionths. 37. Two hundred and seventeen ten -thousandths. 38. Four hundred and twenty -two millionths. 39. Seven hundred and twelve ten-thousandtlis. 40. Fifteen millionths. 41. Four hundred and one hundred-thousandths. Art. 84. An integer and a decimal, written together as one number, are connected by and when expressed in words. Thus, 45.14 is read forty -five units and fourteen hundredths. 42. Head 27.305. 44. Eead 7.06005. 43. Eead 463.3028. 45. Eead 4000.004. DECIMAL FRACTIONS. 117 46. ExpresB decimally forty-five units and fifty -two hundredths. 47. Forty units and forty-five thousandths. 48. Two hundred units and seventy-nine millionths. DEFINITIONS, PEINOIPLES, AND EULES. Art. 85. A Decimal Fraction is a fraction whose denominator is ten or a product of tens. The decimal denominators are 10, 100, 1000, 10000, etc. Art. 86. Decimal fractions may be expressed in three ways: 1. By words; as, three tenths, twelve hundredths. 2. B}" writing the denominator under the numer- ator ; as, y%, yV%. 3. By omitting the denominator and writing the numerator decimally; as, .3 and .12. Three tenths, -f^^ and .3 express the same decimal fraction, but the term decimal is usually applied to a decimal fraction when expressed by the third method. Decimal fractions may be read or dictated, and hence may be expressed iyi words. Art. 87. The Decimal Boiiit is a period placed at the left of the order of tenths, to designate the decimal orders. Art. 88. A Mixed Decimal Number is an integer and a decimal written together as one number; as, 2.45. It is also called a Mixed Kuiriber. The orders on the left of the decimal point are integral^ and those on the right are decimal. The decimal orders are called Decimal Places. 118- INTERMEDIATE ARITHMETIC. Art. 89. The following table gives the names of six inte«:ral and six decimal orders: n^ n3 fl a ^ OD ^ s oc *fco 13 ^ o -^ .g {» ^ o ri:5 S s j» ^ G ,£5 O CO 1 02 CO •1 •5 1 O 4^ CO .2 K O i p s ^ 5 o 2 w 1 Iniegral Orders. Decimal Orders. Art. 90. Principles.— 1. The denominator of a deci- mal fraction is 1 ivith as many ciphers annexed as there are decimal 'places in the fraction. 2. The successive decimal orders decrease in value from left to rights and increase from right to left in the same manner as integral orders. 3. The name of a decimal is the same as the name of its right-hand order. Art. 91. EuLES. — 1. To read a decimal, Bead it as though it were an integer^ and add the name of the right- hand order. Notes. — 1. A decimal is read precisely as it would be were the denominator expressed. 2. In reading a mixed decimal number, the word ''units" may be omitted when this does not change the mixed number to a pure decimal. 2. To write a decimal, Write it as an integer^ and so place the decimal point that the right-hand figure DECIMAL FRACTIONS. 119 shall stand in the order denoted by the name of the decimal. Note.— When tlie number does not fill all the decimal places, supply the deficiency by j^relixing decimal ciphers. W^RITTEN EXERCISES. 49. Write in words 4045.03007. 50. Write in words .040085. 51. Wiite in words 405.40071. 52. Write in words 35000.0094. 53. Express decimally five thousand and sixty-six milliontbs. 54. Eight hundred and forty-two ten-thousandths. 55. Seventy-five and four hundred and three hun- dred -thousandths. To Teachers. — See Manual of Arithmetic for addi- tional problems in Decimal Fractious. LESSON II. ^educfio7i of decimals. Case I. — Decimals Reduced to Higher or Lower Orders. 1. How many hundredths in 1 tenth? In 3 tenths? In 5 tenths? In .8? 2. How many thousandths in 1 hundredth? In 5 hundredths? In .12? In .25? 3. How many tenths in 10 hundredths? In 40 hundredths? In .50? In .60? In .80? 4. How many hundredths in 10 thousandths? In 30 thousandths'? In .060? In .120? In .340? 120 INTERMEDIATE ARITHMETIC. WRITTEN EXERCISES. 5. Eeduce .325 to hundred-thousandths. PROCESS. According to Art. 66, .32*5 or .3 2 5 = .3 2 5 A¥o = t%%Vo or .32500. 6. Reduce .45 to ten-thousandths. 7. Reduce 6.5 to thousandths. 8. Reduce 23 to hundredths. 9. Reduce 62.5 to ten-thousandths. 10. Reduce .048 to hundred-thousandths. 11. Reduce 406.062 to millionths. 12. Reduce .4500 to hundredths. PROCESS : .4 5 = .4 5, Ans. 13. Reduce .5000 to tenths. 14. Reduce 2.4000 to hundredths. Art. 92. Principles. — 1. Annexing ciphers to a deci- mal^ or decimal ciphers to an ititeger, does not change its value. (Art. GS.) 2. Removing ciphers from the right of a decimal^ or decimal ciphers from the right of an integer^ does not change its value. (Art. 63.) * Case II.— Decimals Reduced to Common Fractions. 1. How many fiftlis in y\? In ^? In .6? 2. How many fourths in ^25^? In y%\? In .75? WRITTEN EXERCISES. 3. Reduce .225 to a common fraction in its lowest terms. PROCESS : .225 = ^^ --= 4% J -4n6'. (Art. 64.) DECIMAL FRACTIONS. 121 4. Eediice .75 to a common fraction in its lowest terms. 5. .625. 8. .075. 11. .0096. 14. .0032. G. .0625. 9. .024. 12. 3.25. 15. 12.375. 7. .125. 10. .512. 13. 21.075. 16. 25.032. Art. 93. EuLE. — To reduce a decimal to a common fraction in its lowest terms, Omit the decwial point and supply the denominator^ and then reduce the fraction to its lowest terms. Case III. — Common Fractions Reduced to Decimals. 1. How many tenths in i? In ^? |? |? 2. How many hundredths in 2^^ ^^^ ^\ ^ A^ WKITTEN EXERCISES. 3. Reduce ^^ to a decimal. 25)3.00(.12, Arts. 2 5 Since ^\ = ^^ ^^ ^> ^"d process: -— t 3 = 3.00 (Art. 92), ^\ = ^^ ^V of 3.00 =.12. 4. Reduce | to a decimal. 6.^. 9.^. 12. 3,-V- 15- 12A. 7.1. 10.3%. 13. 21|. 16.25^-. 17. Reduce .621 to thousandths. 18. Reduce .012| to ten-thousandths. 19. Reduce 12.06L to millionths. Art. 94. Rule. — To reduce a common fraction to a decimal, Annex decimal ciphers to the numerator and divide by the denominator, and point off as many deci- mal places in the quotient as there are annexed ciphers. 122 INTERMEDIATE ARITHMETIC. LESSON III. ^dddilion of decimals, 1. What is the sum of 5 tenths and 4 tenths? 6 tenths and 9 tenths? Solution. — 6 tenths and 9 tenths are 15 tenths, which is equal to 1 unit and 5 tenths. 2. What is the sum of 8 hundredths and 7 hun- dredths? 18 hundredths and 7 hundredths? 3. Wliat is the sum of 28 thousandtlis and 9 thou- sandths? 56 tliousandths and 22 thousandths? WRITTEN EXERCISES. 4. What is the sum of 24.6, 307.08, 93.609, .456, 400.06, 37.027. PROCESS. Since only like orders can be added, 2 4.6 write the figures of the same order in 307.08 the same column. Since ten units of 9 3.6 09 any order make 1 unit of the next .4 56 left-hand order, begin at the right, and 400.06 add as in simple numbers. Place the 3 7.0 2 7 decimal point between units and tenths 86 2.8 3 2, Alls, in the amount. 5. What is the sum of .4506, .709, and 27.0508? 6. Add 15.34, 6.078, 60.804, and 99.875. 7. Add $21.94, $87,075, $9,858, and $807,621. 8. Add thirty-nine hundredths, six hundred and eight ten-thousandths, and eighty-seven thousandths. 9. What is the sum of 47.6 miles, 19.48 miles, 34.75 miles, and 76.625 miles? Art. 95. Rule. — To add decimals, 1. Write the num- bers so that figures of the same order shall be in the same column. DECIMAL FRACTIONS. 123 2. Beginning at the rights add as in the addition of integers^ and place the decimal point at the left of the tenths' order in the amount. LESSON IV. Suhtracti07i q/' Decimals. 1. From 8 tenths take 5 tenths. 2. From 5 tenths take 5 hundredths. Solution. — 5 tenths are equal to 50 hundredths, and 50 hundredtlis less 5 hundredths are 45 hundredths. 3. From 7 hundredths take 4 thousandths. WRITTEN EXERCISES. 4. From 56.403 take 18.6. Since only like units can 5 6^4 3 be subtracted, write the process: 1 8.6 numbers so that figures of 3 7.803, Ans. tlie same order shall be in the same column. Since ten units of any decimal order 5. From 56.6 take 18.403. make one unit of the next left-hand order, subtract as "^"•"^^ in integers, and place the process:. 18.403 decimal point at the left of 3 8.197, Ans. the tenths' order in the re- mainder. 6. From 45.3 take 28.756. 7. From .0407 take .008075. 8. From twelve thousandths take eight millionths. 9. From eight tenths take eight ten-thousandths. 10. From 47.065 + 36.87 take 9.08 + 43.375. 11. From the sum of twenty-five thousandths and forty-six ten-thousandths take their difference. 124 INTERMEDIATE ARITHMETIC. Art. 96. EuLE. — To subtract decimals, 1. Write the numbers so that figures of the same order shall he in the same column. 2. Subtract as in the subtraction of integers, and place the decimal point at the left of the tentlis' order in the re- mainder, LESSON V. Mnltiplicatio7i of Decimals. 1. How much is 3 times 2 tenths? 3 times ^? 2. How much is 4 times 3 tenths? 4 times .4? 3. How much is 5 times jf^? 5 times .05? 4. How much is ^2^ x i^? .2 X -3? .4 X -7? 5. How much is yV X yw? .1 X .05? .4 X -06? 6. How much is ^^ x yf^? .02 X .05. 7. How much is y|^ X y^W? -06 X -008? 8. What is the denominator of the product when tenths are multij^lied by units? Tenths by tenths? 9. What decimal order is produced when tenths are multiplied by hundredths? Hundredths by hun- dredths? Hundredths by thousandths? 10. What is the number of decimal orders in the jDroduct of any two decimals? W^RITTEN EXERCISES. 11. Multiply .435 by .65. PROCESS. 435 Since thousandths multiplied by 55 hundredths produce hundred- thou' ^TWT~ sandthsy the product of .435 and i)(*^f) -65 contsLUiQ five decimal orders, or as many as both of the factors. .2 82 7 5, Ans, DECIMAL FRACTIONS. 125 12. Multiply .347 by .73. 19. 30.3 by .044. 13. Multiply .48 by .36. 20. .008 by .007. 14. Multiply .067 by 6.5. 21. .075 by .48f 15. 3.42 by .054. 22. 2.42 by 50. 16. 47.5 by 3.4." 23. .0075 by 2.8. 17. 492 by 3.06. 24. 43.6 by .073. 18. 650 by .24. 25. .024 by .06i 26. Multiply 4.35 by 10. By 100. PROCESS. Since the value of decimal or- ders increases from right to left ten- 4.35 X 10 ==43.5, Ans. ^^^^ ^j^^^ ^^^ p^. 2)^ the removal 4.35 X 100 — 435,^^5. of the decimal point one place to the right removes each figure one order to the left, and hence multiplies 4.35 by 10. The re- moval of the decimal point two places to the right multiplies 4.35 by 100. 27. Multiply 4.085 by 100. By 1000. 28. Multiply 3.0048 by 1000. By 100000. PRINCIPLES AND EULES. Art. 97. Principles. — 1. The number of decimd pJmes in the product equals the number of decimal places in both factors, 2. Each removal of the decimal point one place to the right multiplies a decimal by 10. Art. 98. Rules. — 1. To multiply one decimal by another, Multiply as in the multiplication of integers^ and point off as many decimal places in the product as there are decimal places in both multij)licand and midtiplier. Note. — If there be not enough decimal figures in the product, supply the deficiency by prefixing decimal ciphers. 2. To multiply a decimal by 10, 100, 1000, etc., 126 INTERMEDIATE ARITHMETIC. Remove the deciinal point as many places to tlie rigid as tJwre are dplwrs in the multiplier. Note. — If there be not enough decimal places in the product, supply the deficiency by annexing ciphers. LESSON VI. Dlvisio7i of decimals, 1. How many times are 2 tenths contained in 8 tenths? 3 tenths in 9 tenths? 2. How many times are 4 hundredths contained in 12 hundredths? 6 hundredths in 42 hundredths? 3. How much is ^^ -^ t%- ^ tenths ~ 3 tenths? 4. How much is .06 -^ .02? .56 ^ .07? 5. Of what order is the quotient when tenths are divided by tenths? Hundredths by hundredths? 6. Of what order is the quotient when any number is divided by a lii<:e number? WRITTEN EXERCISES. 7. Divide 6.25 by .25. PROCESS. .2 5)62 5(25 Ans. Since 25 hundredths are con- f^Q tained in 625 hundredths (a like number) 25 times, the quotient is an integer. 125 125 8. Divide .625 by .25. PROCESS. Since the divisor (.25) and the .2 5 ).6 2 5(2.5,^1/15. first partial dividend (.62) are 50 both hundredths (like numbers), 125 the first quotient figure is units^ 12 5 and hence the second is tenths. DECIMAL FRACTIONS. 127 9. Divide 17.28 by .48. By 4.8. 10. Divide 3.528 by .042. By 12.6. 11. Divide .9408 by 8.4. By .084. 12. Divide .06241 by 79. By .079. 13. Divide 18.816 by 1.68. By 168. 14. Divide $17,595 by $.85. By $2.07. 15. Divide .0768 by 9.6. By .096. 16. Divide 62.5 by .025. PROCESS. .025)62.500(2500,^/15. % annexing two decimal 50 ciphers to 62.5, the dividend T^T and divisor are made like ^ rt c numbers, and hence their quo- tient is an integer. 00 17. Divide 25.6 by .032. By .16. 18. Divide 2.5 by 1.25. By .0125. 19. Divide 45.3 by 3.02. By .0302. 20. Divide 80.5 by .35. By .00035. 21. Divide 402.5 by 1.75. By .0175. 22. Divide 34.5 by 10. By 100. PROCESS. The removal of the decimal 8 4.5^-10=3.4 5. point one order to the left, re- 345-^100= 345 moves each figure in 34.5 one order to the right, and hence divides its value by 10; and the removal of the decimal point two places to the left divides it by 100. 23. Divide 436.7 by 100. By 1000. 24. Divide 234.6 by 1000. By 100000. PRINCIPLES AND RULES. Art. 99. Principles. — 1. The dividend contaim as many decimal places as both divisor and qnoti£nt. 128 INTERMEDIATE ARITHMETIC 2. Tlie quotient contains as many decimal places as the number of decimal places in the divideml exceeds tlie number in Hie divisor. 3. Each removal of tJw decimal point one place to ilie left divides a decimal by 10. Art. 100. EuLES. — 1. To divide one decimal by an- other, Divide as in the division of integers j and point off as many deCESS. If 9 (,Qrds cost $32,625, 1 cord will 9 ) $3 2.6 2 5 cost \ of $32,625, which is $3,625, or $3,6 2 5, ^n^. $3.62^. 10. If 12 pounds of sugar cost $2.16, what will 1 pound cost? 11. A man paid $1687.50 for 45 acres of land: what was the price an acre? 136 INTERMEDIATE ARITHMETIC. 12. A grocer paid $135 for 18 barrels of flour: what was the cost a barrel? 13. A man earned $91 in 8 weeks: how much did he earn a week ? 14. At $.12^ a dozen, how many dozen of eggs can be bought for $5? PROCESS. $5 ==5000 mills, and $.12} 125w.)5000m.(40,^ln5. ==125 mills. Hence, $o ~- 500 $.12} = 5000 mills -- 125 mills, which is 40. Note. — When both divisor and dividend are denominate num- bers, they must be reduced to the same denomination before di- viding. 15. At $1.25 a bushel, how many bushels of corn can be bought for $75? 16. At 31 cents apiece, how many lemons can be bought for $7? 17. If a boy earn 75 cents a day, in how many days will he earn $24? 18. At 37^ cents a bushel, how many bushels of oats can be bought for $57.75? 19. A farmer sold 35 pounds of butter at 20 cents a pound, and received in payment muslin at 12^ cents a yard: how many yards of muslin did ho receive? 20. A farmer exchanged 16 cows, at $27.50 a head, for sheep, at $5.50 a head: how many sheep did he receive ? 21. How many lemons, at 2^ cents each, can be bought for 20 oranges, at 5 cents each? 22. Multiply $12.62^ by 15, and divide the product by $2,525. 23. Multiply $1.25 by 18, and divide the product by $.62f TNITED STATES MONEY. 137 24. Multiply $5.75 by 25, and divide the product by $.571 Art. 105. EuLES. — 1. To multiply or divide sums of money by an abstriict immber, Multiply or divide as in diiiple numbers, separate dollars ami cents in the result by a j)eriody and prefix the dollar sign. 2. To divide one sum of money by another, Reduce both numbers to the same denomination, and divide as in simple numbers, LESSON V. M iscellaneoiis Written ^JProblenis, 1. What is the sum of $13.45, $9.87, $100, $.87, $1.40, and $14? 2. From $10 take 5 mills. 3. From $500 take 500 cents. 4. Multiply $15,331- by 33. 5. Divide $50 by 50 cents. 6. A man's income tax in 1868 was $55.75, his State and city tax $68.35, and his other taxes $7.50 : what was the amount of his taxes? 7. A man bought a house and lot for $5400, and, after expending $1500 for improvements, sold the property for $7500 : how much did he gain ? 8. What will 60 ];)Ounds of butter cost at 331^ cents a pound? 9. What is the cost of 35 reams of paper, weighing 44 pounds each, at 18 cents a pound? 10. How many yards of carpeting, at $1.75 a yard, can be bought for $350? 11. A fruit dealer makes a net profit of 20 cents on each bushel of apples he sells : how many bushels must he sell to make $80? 138 INTERMEDIATE ARITHMETIC. 12. A widow is to receive one third of an estate of $12000, and the remainder is to be divided equally between 5 children : what is tlie share of each child ? ±3. A fruit dealer sold 144 baskets of peaches for S252: what was the price per basket? 14. If 40 acres of land cost $1400, how many acres can be bought for $1750? 15. A man sold 15 cords of wood at $4.50 a cord, and received in payment 10 barrels of flour: what did the flour cost him a barrel? 16. If 8 barrels of salt cost $36, what will 13 barrels cost? 17. A grain dealer bought 15000 bushels of wheat at $1.35 a bushel, and sold it the next week for $1.48 a bushel: what was his gain? 18. A workman receives $1.50 a day, and his living costs him $.75 a day: how much can he lay up in a year, if he work 310 days? 19. A drover bought 240 sheep at $4.50 a head, drove them to market at an expense of $75, and then sold them at $6.50 a head: how much did he make ? 20. A farmer exchanged 40 pounds of butter at 22 cents a pound, and 8 dozen of eggs at 12^ cents a dozen, for cotton cloth at 10 cents a yard : how many yards of cloth did he receive? 21. A farmer exchanged 8 cows valued at $37.50 a head, for sheep valued at $7.50 a head: how many sheep did he receive? 22. If a boy pays $2.50 a hundred for papers, and sells them at 5 cents apiece, how much does he make on 100 papers? 23. A farmer sold, one year, 200 bushels of wheat, at $1.80 a bushel; 500 bushels of corn, at $1.15 a UNITED STATES MONEY. 139 bushel ; 65 bushels of potutoep., at 80 cents ii bushel ; 12 tons of hay, at $16.50 a ton; and 225 pounds of butter, at 20 cents a pound: what was the amount of his annual product? 24. A man bought 250 bushels of coal, at 15 cents a buslicl; 7 cords of wood, at $5.50 a cord; 18 bushels of potatoes, at $.90 a bushel ; and 9 barrels of apples, at $2.75 a barrel : how much did he pay for all? 25. A bookseller sold 12 geographies, at $1.75; 20 readers, at $.85; 30 arithmetics, at $.65; and 45 sj^ellers, at $.30 : what was the amount of the bill ? 26. The annual expenses of a man's family are as follows: provisions, $350; clothing, $400; fuel, $95; books and periodicals, $50; house-rent, $240; and all other expenses, $150: if he receive an annual salary of $1500, how much can ho lay up each year? To Teachers. — See Manual of Arithmetic for addi- tional review problems for dictation. LESSON VI. :bizzs. L Columbus, 0., June, 10, 1869. Mr. Charles Wilson Bought of James Cooper & Co. : Vi lbs. Coffee, @ 30c $3.90 4 lbs. Butter, @ 35c 1.40 10 lbs. B'k't Flour, @ 6c 60 12 lbs. Dried Beef, @ 24c 2.88 25 lbs. Sugar, @ 18c. 4.50 3 lbs. Starch, @ 20c ^ ^ $13.88 Received payment^ James Cooper & Co. 140 INTERMEDIATE ARITHMETIC. 2. Chicago, Jan. 3, 1869. Joseph Masox Bought of Peter & Brothers: 27 yds. Brussels carpeting, @ $2.60 23 yds. Ingrain ^* @ 1.75 8| yds. Oil Cloth, @ 1.20 32 yds. Curtains, @ .60 $ Received payment^ Peter & Brothers, Per Smith. What is the amount of the above bill? 3. Nashville, Tenn., Oct. 8, 1868. Samuel Mills To Jones, Smith & Co., Dr. To 7 yds. Broadcloth, @ $6.50 ...... *' 3^ yds. Doeskin, @ 2.75 *' 7J yds. Linen, @ .90 *^ 2-j doz. Handkerchiefs, @ 1.50 *' 12^ yds. Muslin, @ .18 . .... . " 9 yds. " bleached, @ .33 " 12 yds. Silk, @ 1.60 " 19 yds. Binding. @ .08 Becelved imymenty Jones, Smith & Co. "What is the amount of the above bill? 4. UNITED STATES MONEY. HI St. Louis, May 23, 18(39. Henry Williams IS(}9. Bought of Isaac Clarke: Mch. 10, 5 Pair Calf Boots, @ $5.75 Ladies' Gaiters, @ 3.10 . Children's Shoes, @ 1.75 . Coarse Boots, @ 2.75 . Calf Shoes, @ 3.25 . Ladies' Slippers, @ 1.20 . Calf Boots, @ 5.75 . Received payment ^ Isaac Clarke. What is the amount of the above bill ? ii. 11 8 " a ic 7 " Apr. 4, 8 '^ a a 6 " << n 7 *^ May 23, 3 ^' Pittsburgh, Pa., Dec. 15, 1868. Andrew Wilson 1868. ^oi/^A^ o/ Smith & Waring : July 5, 7 gross Shirt Buttons, @ $4.50 '' " 10 doz. Linen Napkins, @ 2.75 Aug. 12, 8 " Pair Kid Gloves, @ 12.50 '' 3^ '• Linen Handk'fs, @ 6.75 '' 4§ '' Shirt Bosoms, @ 6.00 Dec. 15, 3| '' Silk Gloves, @ 9.00 '' 8 '' Pair Socks, @ 5.50 Received payment. Smith & Waring. What is the amount of the above bill? 142 INTERMEDIATE ARITHMETIC. 6. Indianapolis, Ind., Aug. 18, 1876. Thos. M. Cochrane Bought of Jones, Dunlap & Co. : 12 doz. Scythes, @ $15 12i " Scy. Snaths, @ 16.50 6 '' Eakes, @ 2.25 o '' Hoes, @ 5.75 8| '' AVhetstones, @ 1.50 $ Cr, June 20, By Cash $75.00 Aug. 1, By 2h doz. Scythes returned . . . 37.50 $112.50 Received pmjmejit, $ Jones, Dunlap & Co. 7. Columbus, O., July 1, 1876. Smith & Bell ;i^gg^ In account with George Stationer. Feb. 1, To 2J M. Envelopes, ® $5.75 li (( " IJ reams Cap Paper, @ 8.00 ti (( '' 3 Blank Books, @ 1.25 ]\Ich. 9, *' 5 doz. Pencils, @ 1.25 (( *' 60 lbs. Wrapping Paper , @ .10 ti 11 " 6 vols. Dickens, Cr. @ 1.75 $ Juno 20, By printing 1500 Circulars $5.50 (i u By printing Letter Heads . 3.75 11 25, By 33 tokens Press- work - 16.50 $ $ UNITED STATES MONEY. 143 DEFINITIONS. Art. 106. A Bill of Goods is a written etatement of goods sold, with the price of each article and the entire cost. It also gives the date and place of the sale, and the names of the buyer and seller. A bill is drawn against the buyer, or Debtor, and in favor of the seller, or Creditor. A bill is receipted by writing the words ^^ Received payment^'' at the bottom, and affixing the seller's name. A bill may be receipted by a clerk, agent, or an}^ other authorized person, as in Bill 2. Art. 107. When sales are made at different times, the dates may bo written at the left, as in Bills 4, 5, and 7. A bill presenting a debit and credit account between the parties, may be written and receipted as in Bill 6. Questions for Eeview. What is United States money? What is it also called? Of what does United States money consist? AVhat are the principal gold coins? Silver coins? What are the lesser coins? Of what metals are the lesser coins made? Name the two kinds of paper money. What are the principal denominations of United States money? Kepeat the table. In what denominations are ac- counts kept? What use is made of the dollar sign? How are dollars and cents separated ? Is the separatrix a period or a comma? Where is the figure denoting mills written? How are dollars reduced to cents? Dollars to mills? Cents to mills? How are dollars and cents reduced to c^nts? Dollars, cents, and mills to mills? How are cents reduced to dollars? Mills to dollars? 144 INTERMEDIATE ARITHMETIC. Give the rule for adding or subtracting sums of money. Give the rule for multiplying or dividing a sum of money by an abstract number. By another sum of money. What is a bill of goods? What does it contain? Against whom is it drawn ? How is a bill receipted ? By whom ? Where are the dates of sales written ? Items of credit ? SECTIOlSr X. LESSON I. Art. 108. Dry Measure is used in measuring grain, fruit, most vegetables, coal, and many other dry articles. The denominations are piiitSf quarts, pecks, and 'bushels. DENOMINATE NUMBERS. 145 Table. 2 pints (pt) , . are 1 quart , . . . qt. 8 quarts .... are 1 peck . , , , pk. 4 pecks .... are 1 bushel .... 6m. 1 bu. -= 4 pk. = 32 qt. = 64 pt. Notes. — 1. The standard bushel is ISJ inches in diameter and 8 inches deep. It contains 2150| cubic inches. 2. In measuring grain, seeds, and small fruits, the measure nuist be even full ; but in measuring corn in the ear, potatoes, apples, and other large articles, the measure must be heaping full. 1. How many pints in 3 quarts? Solution. — In 3 quarts there are 3 times 2 pints, which is 6 pints. 2. How many pints in 5 quarts? In 8 quarts? In 10 quarts? 3. How many quarts in 10 pints? Solution. — In 10 pints there are as many quarts as 2 pints are contained times in 10 pints, which is 5 times. 4. How many quarts in 8 pints? In 14 pints? In 16 pints? In 20 pints? 5. How many quarts in 3 pecks? In 5i pecks? In 1\ pecks? In lOf pecks? 6. How many pecks in 16 quarts? In 20 quarts? In 32 quarts? In 56 quarts? 7. How many pecks in 5 bushels? In 7^ bushels? In 9f bushels? In 11 bushels? - 8. How many bushels in 12 peeks? In 20 pecks? In 32 pecks? In 40 pecks? 9. How many quarts in 8 pecks? In 12 pecks? 10. How many pints in 8 quarts? In 12 quarts? 11. What part of a quart is 1 pint? 2 pints? 12. What part of a peck is 1 quart? 3 quarts? 13. What part of a bushel is 1 peck? 2 2:)ecks? 3 pecks? 4 pecks? 5 pecks? I. A.— 10. 146 INTERMEDIATE ARITHMETIC. 14. How many pecks in 17 quarts? In 27 quarts? In 33 quarts? 15. How many bushels in 13 pecks? In 23 pecks? In 33 pecks ? 16. Wiiat will 51^ quarts of plums cost at 4 cents a pint? 17. A man carried 3| pecks of cherries to market, and sold them at 10 cents a quart: how much did he receive? 18. If beans are worth $1.60 a bushel, how much are they worth a quart? 19. When apples sell at 20 cents a peck, what are they worth a bushel? 20. A boy bought half a bushel of chestnuts for $1, and sold them at 8 cents a quart: how much did he make ? WRITTEN EXERCISES. 21. How many pecks in 12 bushels? How many quarts? How many pints? 22. Eeduce 12 bu. 3 pk. 1 pt. to pints. 1st; PROCESS. 2d PROCESS. bu. pk. qt. pt bu. pk. qt. pt. 12-^3 + + 1. 12 + 3+0 + 1. 4 4 4 8, pk. 51, pk. 3- 8 51, pk. 4 8 , qt. 8 2 408, qt. 817, pt., Ans. 2 816, pt. 1 817, pt. Ans. DENOMINATE NUMBERS. 147 23. Reduce 5 bu. 2 pk. 7 qt. to pints. 24. Eeduce 15 bu. 5 qt. 1 pt. to pints. 25. Keduce 8 bu. 3 pk. to quarts. 26. How many pints in 3 pk. 5 qt. 1 pt. ? 27. How many quarts in 3 pk. 7 qt. ? 28. How many pints in 1 bu. 1 qt. ? 29. How many bushels in 768 pints? In 817 pints? PROCESS. PROCESS. 2 )768, pt. 2)8J.7, pt. 8 )384, qt. 8 )408, qt. + 1 pt. 4)«^,pk. 4)51, pk. 12,bu. 12, bu. + 3 pk. Ans. 12 bu. Ans. 1 2 bu. 3 pk. 1 pt. 30. Reduce 168 qt. to bushels. 31. Reduce 342 pt. to bushels. 32. Reduce 51 pt. to pecks. 33. How many pecks in 37 pints? 34. How many bushels in 151 quarts? 35. What will 3 pk. 5 qt. of cherries cost at 5 cents a pint? 36. A man sold 1 bu. 3 pk. 5 qt. of clover-seed at 8 cents a quart: what did he receive? 37. A fruit dealer paid $7 for 3 bu. 3 pk. of peaches, and sold them at 75 cents a peck: what was his gain? 38. How many bushels of chestnuts can be bought for $15.50, at 5 cents a quart? 39. A fruit dealer put 3 bu. 2 pk. of strawberries into quart baskets: how many baskets were filled? 40. A boy bought 5 pecks of cherries at 60 cents a peck, and sold them at 10 cents a quart: how much did he gain? 148 INTERMEDIATE ARITHMETIC. LESSON II. Art. 109. Liquid Measure is used in measuring liquids; as, oil, milk, alcohol, etc. The denominations are gills, pints, quarts, and gallons. 4 gills {gl) . , 2 pints . . . 4 quarts . . . 1 gal. = 4 qt, Table. are 1 pint . . . are 1 quart . . . are 1 gallon . , = 8 pt. = 32 gi. pt. qt gal Notes. — 1. The standard liquid gallon contains 281 cubic inches. 2. The size of casks for liquids is variable. The capacity of vats, cisterns, etc., is usually measured in barrels of 31 J gallons. 1. How many gills in 3 pints? In 10 pints? In 20 pints? In 32 pints? DENOMINATE NUMBERS. 149 2. How many pints in 5 quarts? In 81 quarts? In 12 quarts? In lOi quarts? 3. How many quarts in 5 gallons ? In 7| gallons ? In 11 gallons? 4. How many pints in 16 gills? In 24 gills? In 32 gills? In 36 gills? In 40 gills? 5. How many quarts in 12 pints? In 16 pints? In 22 pints? 6. How many gallons in 20 quarts? 32 quarts? 28 quarts? 36 quarts? 40 quarts? 7. How many quarts in 8 pints? 15 pints? 19 pints? 13 pints? 21 pints? 8. How many pints in 8 quarts? 11 quarts? 16 quarts? 15^ quarts? 20 quarts? 9. How many gallons in 8 quarts? 13 quarts? 21 quarts? 24 quarts? 29 quarts? 10. How many quarts in 6 gallons? 9| gallons? 11 gallons? 11. What part of a gallon is 1 quart? 2 quarts? 3 quarts? 4 quarts? 12. How many quarts in | of a gallon ? 13. What will 10 quarts of milk cost at 5^ cents a pint? 14. If a gallon of wine cost $6, what will 1 pint cost? 15. If maple syrup cost $1.60 a gallon, what will 1 quart cost? 16. At 4 cents a pint, what will 5 gallons of milk cost? WRITTEN EXERCISES. 17. How many pints in 21 gallons? 18. How many gills in 7 gal. 3 qt. 1 gi.? 19. How many pints in 34 gal. 1 pt. ? 150 INTERMEDIATE ARITHMETIC. 20. Ecdiice 9 gal. 2 qt. 1 j)t to pints. 21. Eediice 38 pints to gallons. 22. Eeduce 245 gills to gallons. 23. Eeduee 130 gills to quarts. 24. Eeduce 547 gills to gallons. 25. Eeduce 45|- gallons to gills. 26. Eeduce 56 gal. 1 pt. to pints. 27. Eeduce 305 pints to gallons. 28. What will 256 pints of maple syrup cost at $1.30 a gallon? 29. How many vials, holding 2 gills each, can be filled from a gallon of alcohol ? 30. How many jugs, each containing 1 gal. 2 qt., can be filled from a barrel of vinegar containing 31i- gallons? 31. A grocer bought 25 gallons of maple syrup at S1.20 a gallon, and sold it at 40 cents a quart: how much did he gain? 32. A grocer bought 6 barrels of vinegar, contain- ing 31i gallons each, at $6.50 a barrel, and sold it at 10 cents a quart: how much did he make? 33. A merchant bought a hogshead of molasses, containing 63 gallons, and sold | of it at 75 cents a gallon, and the rest at 20 cents a quart: what did he receive for it? 34. A man bought 5 hogsheads of molasses, each containing 63 gallons, at $31.50 a hogshead, and sold 3 hogsheads at 70 cents a gallon, and 2 hogsheads at 65 cents a gallon : how much did he gain ? To Teachers. — See Manual of Arithmetic for addi- tional problems in Denominate Numbers. DENOMINATE NUMBERS. 151 LESSON III. ZOJSTG MBASUHB. Art. 110. Long Measure is used in measuring lines or distances. It is also called Linear Measure. The denominations are inches, feet, yards, rods, furlongs, and miles. Table. 12 inches {in.) . are 1 foot . . . . . ft- 3 feet . . . , are 1 yard . . , , . yd. 51 yards . . . are 1 rod . . . . rd. 40 rods . . . . are 1 furlong . . . fur. 8 furlongs . . are 1 mile . . . . mi. 1 mi. = 8 fur. -320 rd.r=1760 yd. -5280 ft. -63360 in. 152 INTERMEDIATE AKITHMETIC. The following denominations are also used : 4 inches are 1 hand, [^^^%^' measuring the height of 3 feet are 1 pace. 6 feet are 1 fathom, { ^^^f^j.''^ measuring the depth of 3 miles are 1 league, j ^^"^ '" measuring distances at 60 geographic miles, or, ] Vare 1 degree at the equator. 69^ statute miles (nearly), J 360 degrees ( ° ) make the circumference of the earth. 4 yards? In 9 yards? 4. How many yards in 15 feet? In 21 feet? 5. How man}^ yards in 2 rods? In 6 rods? fi. How many yards in 5 rods? In 9 rods? 7. How many rods in 2 furlongs? In 5 furlongs? In 8 furlongs? 8; How many furlongs in 80 rods? In 120 rods? 9. How many furlongs in 6 miles? In 9 miles? 10. How many miles in 32 furlongs? In 56 fur- longs? In 72 furlongs? DENOMINATE NUMBERS. 153 11. How many rods in 66 paces? 12. A ditch is 28 furlongs long : how many miles long is it? 13. A vessel sank in water 9 fathoms deep: what was the depth of water in feet? 14. A steamer sails 3 leagues an hour : how many hours will it take it to sail 90 miles? 15. A horse is 15 hands high: what is its height in feet? 16. How many feet in a rod? 17. How many rods in a mile? 18. What part of a foot is 9 inches? 19. What part of a yard is 2 feet? 20. What part of a mile is 5 furlongs? WKITTEN EXERCISES. 21. How many feet in 16 yards? How many inches ? 22. How many inches in 3 fur. 20 rd. 3 j^d.? 23. How many feet in 2 fur. 30 rd. 4ft.? 24. Reduce 3 mi. 5 fur. 20 rd. to yards. 25. Eeduce 1650 rods to miles. 26. Reduce 32274 inches to higher denominations. 27. Reduce 4 mi. 27 rd. 2 ft. 10 in. to inches. '28. How many steps of 2 ft. 6 in. each will a man take in walking 2 miles? 29. How many times will a wheel 6 feet in cir- cumference turn round in going 2^ miles? 30. Sound travels 1090 feet a second: how many miles will it travel in 60 seconds? 31. How many rods of fence will be required to inclose a farm which is -^ of a mile long and ^ of a mile wide? 154 INTERMEDIATE ARITHMETIC. LESSON IV. Z^J\r:D Oil SQUAOIB M£:ASU'EB. Art. 111. Land or Square Measure is used ill measuring surfaces. It is also called Superficial Measure. The denomina- tions are square inches, square fee t, square yards, square rods or perches, roods, acres, and square miles. "iOUAR^ Art. 112. A Square Inch is a square, each side of which is an inch in length. The figure at the left represents a square inch of real size. A Square Yard is a square, each side of which is a yard, or three feet, in length. It contains 9 square feet. Note. — The teacher should explain and define a right angle, a square, a rectangle, etc. J rrr. DENOMINATE NUMBERS. 155 Table. 144 square inches (sq. 9 square feet 30 1 square yards 40 perches . . 4 roods . . 640 acres . . in.) are 1 square foot . . . sq.ft. . are 1 square yard . . . sq. yd. . are 1 square rod or perch P. . are 1 rood E. . are 1 acre A. . are 1 square mile , , , sq. mi. Notes. — 1. Land Surveyors use Gunter's Chain, which is 4 rods or 66 feet long, and consists of 100 links, each link being Ty^^^^ inches long. A square chain is 16 square rods, and 10 square chains are 1 acre. 2. Glazing and stone-cutting are estimated by the square foot ; painting, plastering, paper-hanging, ceiling, and paving, by the square yard ; and flooring, roofing, tiling, and brick-laying, by the square of 100 feet. Brick-laying is also estimated by the square yard, and by the 1000 bricks. For directions for measuring lum- ber, see Art. 157, Rule 6. 1. How many square feet in 5 square yards? In 7 square yards? 2. How many square yards in 36 square feet? In 72 square feet? In 90 square feet? 3. How many perches in 2 roods? In 5 roods? 4. How many roods in 80 perches? In 120 perches ? 5. How many roods in 8 acres? In 12 acres? 6. How many acres in 16 roods? In 40 roods? 7. How many square chains in 32 square rods? In 64 square rods? In 80 square rods? 8. How many acres in 20 square chains? In 40 square chains? In 80 square chains? 9. How many square yards in a pavement 10 yards long and 4 yards wide? Solution. — In a pavement 10 yards long and 1 yard wide there are 10 square yards, and in a pavement 10 yards long and 4 yards wide there are 4 times 10 square yards, which is 40 square yards. There are 40 square yards in the pavement. 150 INTERMEDIATE ARITHMETIC. 10. How many square yards in a ceiling 8 yards long and 6 yards wide ? 11. How many square feet in a board 16 feet long and 1^ feet wide? 12. How many perches in a field 30 rods long and 10 rods wide? How many roods? 13. How many square inches in a piece of tin 15 inches long and 4 inches wide? 14. How many square yards in a floor 15 feet long and 12 feet wide? ^WRITTEN EXERCISES. 15. How many square yards in 16 perches? How manj^ square inches? 16. How many perches in 5 A. 2 E. ? 17. Eeduce 1 A. 2 R 20 P. 10 sq. yd. 7 sq. ft. to square feet. 18. Eeduce 70882 sq. fb. to higher denominations. 19. Eeduce 5280 perches to higher denominations. 20. Eeduce 5184 square inches to square yards. 21. How many acres in a field 56 rods long and 40 rods wide? 22. How many acres in a street 21 miles long and 4 rods wide? 23. How many square yards in a ceiling 72 feet long and 40^ feet wide? 24. What will it cost to pave a walk 60 feet long and 15 feet wide, at $1.25 a square yard? 25. How many trees can be planted on 3 acres of ground, if a tree be planted on each square rod? 26. If 1000 shingles will cover 100 square feet, how many shingles will cover a roof 40 feet long and 25 feet wide? DENOMINATE NUMBERS. 157 27. How many acres of land in a township 5 miles square ? 28. How many acres in a township 7 miles long and 6 miles wide? 29. How many yards of carpeting, a yard wide, will carpet a room 20| feet long and 18 feet wide? /'HO. How many bricks, 8 in. long and 4 in. wide, will pave a walk 60 feet long and 12| feet wide? ^31. What will it cost to plaster the walls and ceil- ing of a room 15 feet long, 12 feet wide, and 9 feet high, at 50 cents a square yard? LESSON V. ?»'^Nl50v . Art. 113. Cubic Measure is used in measuring solids. It is also called Solid Measure. The denominations are cubic inches, cubic feet, and cubic yards. Art. 114. A cubic inch is a cube whose edges are each one inch long. A cubic yard is a cube whose edges are each one yard long. Note.— The teacher should exx^lain and define a cube ; also its faces and edges. 158 INTERMEDIATE ARITHMETIC. Table. 1728 cubic inches (cu. in.) are 1 cubic foot . . cu.ft. 27 cubic feet .... are 1 cubic yard . . cu. ijd. 1 cu. yd. = 27 cu. ft. ^ 46656 cu. in. Note, — A cubic yard of earth is called a load, and 24| cubic feet of stone or of masonry make a perch. WKITTEN EXEKCISES. 1. How many cubic inches in 5 cubic feet? In 12 cubic feet? 32 cubic feet? 2. How many cubic feet in 15552 cubic inches? 3. How many cubic feet in 120 cubic yards? 4. How many cubic yards in 405 cubic feet? 5. Reduce 15 cu. yd. 16 cu. ft. and 1305 cu. in. to cubic inches. 6. Eeduce 1473462 cubic inches to higher denom- inations. 7. How many cubic feet in a block of marble 15 feet long, 12 feet wide, and 5 feet thick? A block 15 ft. long, 1 ft. wide, and 1 ft. thick contains 15 cu. ft.; a block 15 ft. long, 12 ft. wide, and 1 ft. thick contains 12 times 15 cu. ft., or 180 cu. ft. A block 15 ft. long, 12 ft. wide, and 5 ft. thick con- tains 5 times 180 cu. ft., or 900 cu. ft. PROCESS. 15 cu. ft. 12 180 cu. ft. 5 900 cu. ft. 8. How many cubic feet in a rock 18 feet long. 13 feet wide, and 8 feet high? 9. How many cubic feet in a pile of wood 24 feet long, 3 feet wide, and 8 feet high ? 10. How many cubic yards in a bin 9i feet long, 6 feet wide, and 4| feet deep ? 11. How many cubic feet of earth must be re- DENOMINATE NUMBERS. 159 moved to make a cellar 44 feet long, 27 feet wide, and 5 feet deep? How many cubic yards? .12. How many cubic yards of earth must be re- moved to make a reservoir 120 feet long, 54 feet wide, and 9 feet deep below the surface? -' 13. What will it cost to dig a cellar 36 feet long, 18| feet wide, and ^\ feet deep, at $2.50 a cubic yard? LESSON VI. WOO ID mjs;a.su'R£:. Art. 115. Wood Measure is used in measuring wood and rough stone. The denominations are cubic feety cord feet, and cords. Table. 16 cubic feet . 8 cord feet, or 128 cubic feet 1 cd. = 8 cd. ft. are 1 cord foot . are 1 cord . . . 128 cu. ft. cd. ft. 160 INTERMEDIATE ARITHMETIC. Note. — A pile of wood 8 feet long, 4 feet wide, and 4 feet high, contains 1 cord ; and 1 foot in length of such a pile contains 1 cord foot. See cut on page 159. Wo(k1 four feet, or nearly four feet, in length, is measured by multiplying the length of the pile by the height and dividing the product by 32. WKITTEN EXERCISES. 1. How many cord feet in a pile of wood 4 feet long, 4 feet wide, and 5 feet high? 2. How many cubic feet in 6 cord feet? 3. How many cord feet in 5^^ cords of wood? 4. How many cords of wood in 128 cord feet? 5. How many cords of wood in a pile containing 1536 cubic feet? 6. How many cords of wood in a pile 20 feet long, 4 feet w^ide, and 6 feet high ? 7. How many cords of wood in a pile 48 feet long, 21 feet wide, and 5^ feet high ? 8. A man bought a pile of wood 36 feet long, 4 feet wide, and 8 feet high, and paid $5.50 a cord. What did the wood cost him? / 9. How many cords of stone in a wall 40 rods long, 2 feet thick, and 4 feet high ? 10. At $4.50 a cord, what is the value of a pile of wood 40 feet long, 3| feet wide, and 6^ feet high ? LESSON VII. Art. 116. Circular Measure is used in measuring arcs of circles, and angles, and in estimating latitude and longitude. It is also called Angular Measure. The denominations are seconds, minutes, degrees, signs, and circumferences. DENOMINATE NUMBERS. 161 Table. 60 seconds (^^) . are 1 minute ....'' 60 minutes . . are 1 degree . . . . ° 30 degrees . . are 1 sign S. 12 signs, or ] i • ^ n - ^^^ , > . are 1 circumference . . (7. or cir. 360 degrees j 1 cir. - 12 S. - 360 ° = 21600^ = 129600^^. Notes. — 1. Circular Measure is used by surveyors in surveying land; by navigators in determining latitude and longitude at sea; and by astronomers in measuring the motion of tbe lieavenly bodies, and in computing dilference in time. 2. The portion of surface represented by the annexed figure is a circle. The curved line which bounds the circle is its circumference. Any portion of a circumference is an arc, 3. One half of a circumference is called a seiiii-clrcamference. One fourth of a circumference is called a e quadrant. One third of a quadrant is called a sign. A semi -circumference contains 180°; a quadrant, 90° ; and a sign, 30°. 4. Every circumference is divided into 360 equal parts, called degrees, and, hence, the length of a degree depends upon the size of the circle. A degree of the earth's surface at the equator contains 69^ statute miles, or CO geographical miles— a minute of space being a geographical or nautical mile. 1. How many minutes in 5 degrees? 2. How many signs in 3 quadrants? 3. How many degrees in i of a quadrant? 4. How many degrees in 3| signs? 5. How many signs in ^ of a cireumference ? WKITTEN EXERCISES. 6. How many seconds in 15° 30'? 7. Reduce 15° 33' to minutes. T. A.— 11. 162 INTERMEDIATE ARITHMETIC. 8. Eeduce 5^ signs to minutes. 9. Eeduce 10800'' to degrees. 10. The sun aj^pears to revolve around the earth once a day : how many degrees does it appear to pass over in an hour? In 6 hours? LESSON VIII. TIMB MBA.SU'RB. Art. 117. Ti m e Measure is used in measuring time or duration. The denominations are seconds, jnin- iites, hours, days, years, and centu- ries. Table. 60 seconds (sec) are 1 minute . . . . mm. 60 minutes . . are 1 hour h. 24 hours . . . are 1 day d. 365 days . . . are 1 common year . . c. yr. 366 days . . . are 1 leap year . . . I. yr. 100 years (365] d.) are 1 century .... 0. 1 d =- 24 h. ^ 1440 min. ^ 86400 sec. The following denominations are also used: 7 days .... are 1 week . . . . w. 4 weeks .... are 1 lunar month . Ir. m. 13 Ir. m. 1 d. 6 hr., ) , ^ ,. r ^^^, , \ are 1 Julian year . . J. yr. or 365} days j 12 calendar months are 1 civil year . . e, yr. DENOMINATE NUMBERS. 163 Notes. — 1. The exact length of a solar year is 365 d. 5 h. 48 min. 48 sec, which is nearly 6 hours, or i of a day, longer than the common year. Since the common year lacks I of a day of the true time, an additional day is added to every fourth 3^ear, mak- ing lea}) year. This additional day is given to February, and hence this month in leap year contains 29 days. The leap years are exactly divisible by 4; as, 1860, 1864, 1868, 1872, etc. 2. The names and order of the calendar months and the num- ber of days in each are as follows: January, 1st month, 31 days. February, 2d " 28 or 29. March, 3d *' 31 days. April, 4th '* 30 " May, 5th '' 31 '' June, 6th " 30 " July, 7ih month, 31 days. August, 8th '' 31 " September, 9th " 30 '' October, 10th " 31 ^' November, 11th " 30 '' December, 12th '' 31 '* 3. The following couplet will assist in remembering the months which have 30 days each : Thirty days hath September, April, June, and November. 4. In most business transactions 30 days are considered a month, and 360 days a year. 5. The year is divided into four seasons of three months each, as follows: {March, April, May. {June, July, August. . f September, ^^™f October, ' or Fall, [ November. {December, January, February. 1. How many seconds in 5 minutes? In 10 min- utes? In 20 minutes? 2. How many minutes in 4 hours? In 8 hours? 3. How many hours in 120 minutes? In 240 min- utes? In 300 minutes? 4. How many hours in 3 days? In 5 days? 5. How many days in 48 hours? In 72 hours? In 240 hours? In 480 hours? 164 INTERMEDIATE APJTHMETIC. 6. How many days in 6 weeks? In 8 weeks? In 10 weeks? In 15 weeks? 7. How many weeks in 35 days? In 49 days? 8. How many weeks in 5 lunar months? In 12 lunar months? 9. How many lunar months in IG weeks? In 32 weeks? 44 weeks? 60 weeks? 10. How many calendar months in 5 years? In 7 years? 10 years? 12 years? 'WRITTEN EXERCISES. 11. How many seconds in 15 hours? 12. How many hours in 28800 seconds? 13. Reduce 5 d. 13 h. 40 min. to seconds. 14. Eeduce 31 d. 30 min. 45 sec. to seconds. 15. Reduce 30600 minutes to higher denominations. 16. Reduce 52560 hours to common years. 17. How many minutes in a leap year? 18. How many seconds in the solar year, which contains 365 d. 5 h. 48 min. 48 sec? 19. How many seconds in a common year? 20. The age of a certain man is 64 yr. 45 d. 12 h.: how many hours has he lived, allowing 365^ days to the year? 21. How many hours in the three Spring months? In the three Summer months? 22. How many minutes will there be in the month of February, 1880? In February, 1882? 23. If your pulse beat 75 times a minute, how many times will it beat in 5 weeks? /24. How many days will it take a steamship to sail 3744 miles, if it sail at the rate of 12 miles an hour? DENOMINATE NUMBERS. 165 LESSON IX. ArOI'R^UTOIS WBIGIIT. Art. 118. Avoirdupois Weight is iLsed in weighing all articles except gold, silver, and the precious stones. The denominations are drams, ounces, pounds, hundred-weights, and tons. Table. 16 drams {dr.) . . are 1 ounce . , , . oz. ^ 16 ounces . . . are 1 pound . ... lb. 100 pounds . . . are 1 hundred-weight . civt 20 hundred-weights are 1 ton T. 1 T. ^ 20 cwt. = 2000 Ih. = 32000 oz. -= 512000 dr. 196 pounds of flour, ) , , , ^^^ „ , , ^ > . . . . are 1 barrel. ^ 200 lb. pork or beef, j 100 lb. of fish are 1 quintal. 14 lb. of lead or iron are 1 stone. 56 lb. of corn, rye, or flax-seed, ") 60 lb. of wheat or clover-seed, V . are 1 bushel. 32 lb. of oats, j 166 INTERMEDIATE ARITHMETIC. Notes. — 1. In wholesaling and freighting coal and in invoicing Englisli goods at the United States custom-houses, the hundred- weight is divided into 4 quarters, of 28 pounds each, and tlie ton contains 2240 pounds. This is called the long or gross ton, while the ton of 2000 pounds is called the short or net ton. 2. The dram is seldom used in business transactions, and the quarter, of 25 pounds, is never used. 1. How many drams in 2 ounces? In 5|^ ounces? 10 ounces? 15 ounces? 2. How many ounces in 48 drams? In 64 drams? 96 drams? 160 drams? 3. How many ounces in 4 pounds? In 6| pounds? lOf pounds? 12^ pounds? 4. How many pounds in 80 ounces? 5. How many pounds in 5 hundred-weight? In 8 cwt? 12f cwt.? 25 cwt? 6. How many hundred-weight in 4 tons? In 6 tons? 8f tons? 12| tons? 7. What will | of a pound of candy cost at 2 cents an ounce? 8. What will | of a hundred-weight of flour cost at 5 cents a pound? WRITTEN EXERCISES. 9. Reduce 5 tons to ounces. 10. Reduce 3 T. 14 cwt. 56 lb. to pounds. 11. Reduce 5 cwt. 77 \h. 13 oz. to ounces. 12. Reduce 34920 pounds to tons. 13. Reduce 4560 ounces to higher denominations. 14. Reduce 11 T. 38 lb. 15 oz. to drams. 15. What will a barrel of flour cost at 6 cents a pound ? 16. What will 3 barrels of pork cost at 15 cents a pound? DENOMINATE NUMBERS. 167 17. How many barrels will 3920 pounds of flour make ? 18. A farmer sold 3600 pounds of wheat at $1.75 a bushel : how much did he receive ? 19. A hay-stack contains 9000 pounds of hay: what is it worth at $12 a ton? ^ 20. What will it cost to transport 50 T. 15 cwt. 75 lb. of freight, at ^ cent a pound ? 21. A farmer exchanged 45| pounds of butter, at 20 cents a pound, for sugar, at 15 cents a pound: how much sugar did he receive? LESSON X. Art. 119. Troy Weight is used in weighing gold, silver, and precious stones, and also in philosophical exj)eriments. The denominations are grains, pennyweights, ounces, and -pounds. Table. 24 grains (^r.) . are 1 pennyweight . . pwt. 20 pennyweights are 1 ounce .... 02. 12 ounces . . . are 1 pound . . . . ?6. 1 lb. = 12 oz. = 240 pwt. =-- 5760 gr. 168 INTERxMEDIATE ARITHMETIC. Notes. — 1. Diamonds are weighed by carats and fractions of carats. A carat is 4 Troy grains. 2. The purity of gold is also expressed in carats, a carat mean- ing ^1^ part. Gold that is 22 carats fine contains 22 parts of pure gold, and 2 parts o^ alloy. 1. How many grains in 5 penny weigbtB? In 3 pwt.? 8 pwt.? 10^ pwt.? 2. How many pennyweights in 3 ounces? In 6 ounces? 9 ounces? 10 ounces? 3. How many ounces in 40 pennyweights? In 80 pwt.? 100 pwt.? 120 pwt.? 4. How many ounces in 4 pounds? In 7^ pounds? 12| pounds? 20 pounds? 5. How manj^ pounds in 36 ounces? In 60 ounces? 84 ounces? 96 ounces? 6. What part of a pound is 1 ounce? 6 ounces? 8 ounces? 9 ounces? -WRITTEN EXERCISES. 7. Eeduce 44 lb. 3 oz. 13 pwt. to pennyweights. 8. Eeduce 7 oz. 15 pwt. to grains. 9. Eeduce 56 lb. 13 pwt. to grains. 10. Eeduce 13486 pwt. to higher denominations. 11. Eeduce 40408 grains to higher denominations. 12. Eeduce 5680 ounces to pounds. 13. Eeduce 5280 grains to ounces. 14. A lady bought a pearl necklace, weighing 8 oz. 15 pwt, at 75 cents a grain: what did it cost? 15. What will be the cost of a gold chain, weighing 31 oz., at 75 cents a pwt.? 46. How many gold rings, each weighing 3 pwt., can be made from a bar of gold weighing | of a ? pound ? DENOMINATE NUMBERS. 169 LESSON XI. Art. 120. Apothecaries Weight is used by physi- cians in prescribing and by apothecaries in mixing medicines. I The denominations are grains, scruples, drams, .-^^ ounces, and pounds. Table. 20 grains {gr.) . are 1 scruple . . 9. 3 scruples . are 1 dram . . • . 3- 8 drams . . are 1 ounce . . . . I. 12 ounces . . are 1 pound . . . . lb. Note. — Medicines are bought and sold in quantities by avoir- dupois weight. 1. How many grains in 2 scruples? 2. How many scruples in 5 drams? In 7 drams? 9 drams? 12 drams? 20 drams? 3. How many drams in 21 scruples? In 27 scru- ples? 33 scruples? 40 scruples? 4. How many drams in 5 ounces? In 8 ounces? 10 ounces? 12 ounces? 5. How many pounds in 36 ounces? In 72 ounces? 96 ounces? 120 ounces? 6. How many ounces in 5 pounds? In 8 pounds? 10^ pounds? 12 pounds? 170 INTERMEDIATE ARITHMETIC. \?VrRITTEN EXERCISES. 7. Eeduce 16 lb. 11 g 5 5 2 9 10 gr. to grains. 8. Keduce 10 g 83 to grains. 9. Keduce 356 5 to pounds. 10. Eeduce 26484 gr. to higher denominations. 11. How many pounds in 5760 9? 12. How many doses, of 18 gr. each, in 5 3 2 9 of tartar emetic? 13. How many pills, of 5 gr. each, can be made from 1 § 2 3 2 9 of calomel ? 14. How many ounces of calomel will make 480 pills, each weighing 6 grains? LESSON XII. MISCBZLAJVJBJOUS T;A^LB. PAPER. 24 sheets are 1 quire. 20 quires are 1 ream. 2 reams are 1 bundle. 5 bundles are 1 bale. 12 things are 1 dozen. 12 dozen are 1 gross. 1 2 gross are 1 great gross. 20 things are 1 score. Note.— A sheet of paper folded in 2 leaves is called a folio; in 4 leaves, a quarto, or 4to; in 8 leaves, an octavo, or 8vo; in 12 leaves, a duodecimo, or 12mo ; in 18 leaves, an ISmo. 1. How many sheets of paper in 5|- quires? 2. How many quires of paper in 4 reams? In 8 reams? 12| reams? 15 reams? DENOMINATE NUMBERS. 171 3. How many bundles of paper in 6 reams? In 12 reams? 18 reams? 32 reams? 4. How many eggs in 5 dozen? In 7f dozen? 8^- dozen? 12 dozen? 20 dozen? 5. How many years are 4 score years? 3 scoro years and 10? WRITTEN EXERCISES. 6. How many sheets of paper in 12i reams? 7. Eeduce 6 rm. 15 qu. 12 sheets to sheets. 8. What will 7200 sheets of paper cost at $8.50 a ream? 9. How many crayons are there in 36 boxes, if each box contains 1 gross? 10. If a shirt require 6 buttons, how many shirts will 12 gross of buttons trim? 11. What will 44 gross of lead-pencils cost at 75 cents a dozen? 12. A stationer bought 15 reams of letter-paper at $3.50 a ream, and sold it at 25 cents a quire: how much did he gain? LESSON XIII. DEFINITIONS, PEINOIPLES, AND RULES. Art. 121. A Denominate Number is a number composed of concrete units of one or several denom- inations. Art. 122. Denominate Numbers are either Simple or Compound, A Simple Denominate Number is composed of units of the same denomination ; as, 7 quarts. 172 INTERMEDIATE ARITHMETIC. A Compound Denominate Num^her is composed of units of several denominations; as, 5 bu. 3 pk. 7 qt. Note. — Compound Denominate Numbers are properly called Compound Numbers, since every compound number is necessarily denominate. Art. 123. Denominate Numbers express Ciirrenc[j , Measure, and Weight. Carreiicy is the circulating medium used in trade and commerce as a representative of value. Measure is the representation of extent, capacity, or amount. Weight is a measure of the 'force called gravity, by which bodies are drawn toward the earth. Art. 124. The following diagram represents the three general classes of denominate numbers, their subdivisions, and the tables included under each : (1. Coin, ) 1. Currency, \ \ United States Money. (2. Paper Money, j 1. Lines, P- ^^'^ or arcs, 1 2 ci 2. Measure, 1. Of extension, 1 . Long Measure. Circular Measure. 2. Surfaces : Square Measure. ri. Cubic Measure, o n r.x4-,. 2. Wood Measure. 3. Capacity, 3 j^^^ Measure. [4. Liquid Measure. .2. Of duration: Time Measure, fl. Avoirdupois Weight. 3. Weight, -j 2. Troy Weight. [3. Apothecaries Weight. Art. 125. The Heduction of a denominate number DENOMINATE NUMBERS. 173 is the process of changing it from one denomination to anotlier without altering its value. Art. 126. Eeduction is of two kinds: Eeducfion Descending and Eeduction Ascending, Reduction Descendhtg is the process of changing a denominate number from a higher to a lower de- nomination. It is performed by multij)lication. Reduction Ascending is the process of changing a denominate number from a lower to a higher de- nomination. It is performed by division. Art. 127. EuLE for Eeduction Descending. — 1. Midtiply the number of the highest denomination by the number of units of die next loiver ivhich equals a unit of the higher, and to the product add the number of the loiver denomination, if any. 2. Proceed in like manne)* ivith this and each successive result thus obtained until the number is reduced to the re- quired denomination. Note. — Tlie successive denominations of the compound num- ber sliould be written in their proper order, and the vacant de- nominations, if any, filled with ciphers. Art. 128. EuLE for Eeduction Ascending. — 1. Divide tlie given denominate number by the number of units of its oivn denomination which equals one unit of the next higher, and place the remainder, if any, at the right. 2. Proceed in like manner tvith this and each successive quotient thus obtained until the number is reduced to the re- qidred denomination. 3. The last quotient, with the several remainders annexed in proper orda% will be the answer required. 174 INTERMEDIATE ARITHMETIC. Questions for Keview. ' What is a number? What is an abstract number? What is a denominate number ? Into what two classes are denom- inate numbers divided? Define each. By what other names are compound denominate numbers usually called ? Why may the word " denominate " be omitted? What is currency ? Of how many kinds of money is United States currency composed ? What is meant by measure ? Name the two kinds of measures. How are the measures of extension divided? What tables are used in measuring lines? Surfaces?- Con- tents? Capacity? What table is used in measuring duration? What ones are used in measuring the weight of bodies? What is reduction ? Name the two kinds of reduction. Define each kind. Repeat the rule for each. For what is Dry Measure used? Name the denominations. Repeat the table. For what is Liquid Measure used ? Name the denominations. Repeat the table. For what is Long Measure used? Name the denominations. Repeat the table. For what is Square Measure used ? Name the denomina- tions. Repeat the table. What is a square inch ? A square yard ? For what is Cubic Measure used ? Name the denom- inations. Repeat the table. What is a cubic inch ? A cubic yard ? For what is Wood Measure used ? Name the de- nominations. Repeat the table. For what is Circular Measure used ? Name the denomina- tions. Repeat the table. For what is Time Measure used? Name the denominations. Repeat the table. Name the calendar months in their order, and give the number of days in each. How many days has February in leap years? Name the four seasons of the year, and the months of each. For what are the three weights respectively used? Give the denominations and repeat the table of each. Repeat the miscellaneous table. DENOMINATE NUMBERS. 175 LESSON XIV. Miscellaneous ^erte}p ^7*oblems, 1. How many quarts in | of a bushel? 2. How many pints in 3| gallons? 3. How many hours in ^ of a week ? 4. How many ounces in 2|- pounds of sugar? 5. What will -I of a cwt. of sugar cost at 15 cents a pound? 6. What will I of a gallon of oil cost at 25 cents a pint? 7. What will f of a ream of paper cost at 20 cents a quire? 8. What costs -| of a ton of hay at 75 cents a cwt. ? 9. A boy picked 3 pecks of cherries, and sold them at 10 cents a pint; how much did he receive? 10. If a ship sail 3 leagues an hour, in how many hours will it sail 63 miles? 11. How many half-pint bottles will a gallon of sweet oil fill? 12. How many quart baskets will 3 pk. 5 qt. of strawberries fill ? 13. How many leap years in every century? 14. How many calendar months in 20 years? WRITTEN EXERCISES. 15. A fruit dealer bought 24 barrels of apples, con- taining 2| bushels each, at $2.50 a barrel, and sold them at $1.25 a bushel; what was his gain? 16. What will 20 yd. 2 ft. of iron railing cost at $1.25 a foot? 17. What will 40 miles of telegraph wire cost at 25 cents a yard? 176 INTERMEDIATE ARITHMETIC. 18. How many times will a carriage- wheel 11 feet in circumference turn round in running 2 miles? 19. How many times will a car- wheel 5 feet in cir- cumference turn round in running from Columbus to Cincinnati, the distance being 120 miles? 20. How many acres in a township 6 miles square? 21. What will a piece of land 40 rods long and 32 rods wide cost at $75 an acre? 22. How many hills of corn can be planted on 5 acres, allowing 1 hill to every square yard? 23. How many people can stand on a terrace 250 feet long and 120 feet wide, allowing 4 persons to each square yard ? 24. What will it cost to gravel a street 129 rods long and 60 feet wide at 75 cents a square yard? ' 25. How many square yards in the walls and ceiling of a room 21 ft. long, 18 ft. wide, and 9 ft. high? ' 26. How man}^ yards of carpeting, a yard wide, will carpet a room 18^ feet long and 15 feet wide? ' 27. If 1000 shingles will cover 100 square feet, how many shingles will cover a roof each side of which is 48 feet long and 15 feet wide? 28. A park containing 40 acres is 50 rods wide: how long is it? 29. How many cubic feet in a bin 12 feet long, 8 feet wide, and 3|- feet deep? 30. How many perches of stone in a wall 99 feet long, 8 feet high, and \\ feet thick? 31. What will it cost to dig a ditch 80 rods long, \\ feet wide, and 2 feet deep, at 15 cents a cubic yard? 32. At $4.50 a cord, what will be the cost of a pile of wood 48 feet long, 6 feet high, and 4 feet wide? 33. Ho\v many times will a clock that ticks seconds tick in the month of June? COMPOUND NUMBERS. 177 34. If a person read a half hour each day, how many hours will he read in 40 years, of 365^ days each? 35. How many gold rings, each weighing 4 pwt., can be made from a bar of gold w^eighing 1 lb. 4 oz. ? 36. A car contains 80 barrels of pork, and another 80 barrels of flour: what is the difference in the freight of the tw^o cars? 37. How many gross of pens will supply 4320 pupils one year, if each pupil require 4 pens? 38. If 10 sheets of paper will make a 16mo. book of 320 pages, how many reams will it take to publish an edition of 2000 copies? SECTION XI. LESSON I. Addition of Compound A^ambers, 1. What is the sum of 5 bu. 3 pk. 6 qt. 1 pt. ; 8 bu. 2 pk. 1 pt. ; 10 bu. 1 pk. 3 qt. ; and 3 pk. 5 qt. 1 pt.? Write the compound numbers so that terms of the same denom- ination shall stand in the same column. Add first the column of pints. The snm is 3 pints, which equals 1 qt. 1 pt. Write 25 bu. 2 pk. 7 qt. 1 pt. the 1 pt. under the pints, and add the 1 qt. with the column of quarts. The sum of the quarts is 15 quarts, which equals I. A. 12. PROCESS 5. bu. pk. qt. pt. 5 3 6 1 8 2 1 10 1 3 8 5 1 178 INTERMEDIATE ARITHMETIC. 1 pk. 7 qt. Write the 7 qt. under the quarts, and add the 1 pk. with the column of pecks. The sum of the pecks is 10 pecks, which equals 2 bu. 2 pk. Write the 2 pk. un- der the pecks, and add the 2 bu. with the column of bushels. The sum of the bushels is 25 bushels. The sum of the four compound numbers added is 25 bu. 2 pk. 7 qt. 1 pt. (2) (3) (4) bu. pk. qt. pt. gal. qt. pt. gi. mi. fur. rd. yd, ft. in. 16 2 6 1 21 3 1 3 19 7 39 5 2 10 23 1 4 16 1 2 27 3 24 3 1 6 40 3 1 48 2 45 4 33 7 9 2 35 1 3 (6) - 6 17 2 (7) 1 (5) cwt . lb. oz. dr. lb. OZ. pvvt. gr. lb. S. 5. 9. gr. 15 63 11 13 9 11 19 23 44 11 7 2 19 18 85 10 18 6 13 20 23 9 6 8 6 15 15 7 10 8 11 10 5 2 16 75 8 7 9 15 16 27 7 6 1 14 19 36 14 15 23 10 9 16 3 18 8. What is the sum of 15 w. 5 d. 22 h. 45 min. 34 sec. ; 8 w. 6 d. 13 h. ; 3 w. 20 h. 52 min. ; 4 d. 22 h. 33 min. 55 sec; 1 w. 2 d. 3 h. 30 min.? 9. Add 14° 30' 46'^; 53° 16' 49"; 26° 34' 15"; 18° 44' 33"; 62° 36'; and 43° 45". 10. Add 5 sq. mi. 625 A. 3 E. 35 P. ; 14 sq. mi. 546 A. 2 K. 28 P. ; 486 A. 1 R. 27 P. ; 94 A. 24 P. ; and 14 sq. mi. 300 A. 3 R. 36 P. 11. A wood dealer bought 5 piles of wood, the first containing 21 cd. 5 cd. ft. 15 cu. ft. ; the second, 45 cd, 12 cu. ft. ; the third, 18 cd. 7 cd. ft. ; the fourth, 50 cd. 6 cd. ft. 14 cu. ft; and the fifth, 16 cd. 5 cd. ft: how much wood did he purchase? 12. A printer used 3 bundles 1 ream 16 quires of paper, Monday ; 2 bundles 1 ream, Tuesday ; 4 bundles r COMPOUND NUMBERS. 179 16 quires, Wednesday; 3 bundles 1 ream 18 quires, Thursday; 5 bundles, Friday; and 3 bundles 1 ream, Saturday: how much paper did he use? 13. The four quarters of an ox weighed respectively 2 cwt. 84 lb. 10 oz. ; 3 cwt. 1 lb. 14 oz. ; 2 cwt. 76 lb. 4 oz. ; and 2 cwt. 98 lb. 14 oz : what was the weight of the four quarters ? 14. A garden has four unequal sides; the first is 4 rd. 3 yd. 2 ft. 8 in.; the second, 5 rd. 1 ft. 10 in.; the third, 4 rd. 5 yd. 4 in. ; and the fourth, 3 rd. 4 yd. 2 ft. 9 in.: what is the distance round the garden? 15. A cistern full of water was emptied by 3 pipes ; the first discharged 45 gal. 3 qt. : the second, 54 gal. 1 pt. ; and the third, 61 gal. 2 qt, 1 pt. : how much water did the cistern contain. DEFINITIONS, PEINCIPLE, AND RULE. Art. 129. A Cofnpoitfid Number is a number com- posed of units of several denominations. Art. 130. The numbers expressing the successive de- nominations of a compound number are called its Tejvns. Compound numbers are of the sajne hind when their corresponding terms express units of the same denomination; as, 3 bu. 2 pk., and 6 bu. 3 pk. 5 qt. Art. 131. Compound Addition is the process of finding the sum of two or more compound numbers of the same kind. Art. 132. Principle. — In both simple and compound addition, the sum of each column is divided by the num- ber of units of that denomination which equals one of the 180 INTERMEDIATE ARITHMETIC. next higher denomination. In simple addition this di- visor is 10; in compound addition it is a varying number, since the several denominations are expressed on a varying scale. Art. 133. Rule. — 1. Write the compound numbers to to he added so that units of the same denomination shall stand in the same column. 2. Add first the column of the lowest denomination^ and divide the sum by the number of units of that de- nomination which equals a unit of the next higher de- nomination ; write the remainder under the column added, and add the quotient with the next column. 3. In like manner add the remaining columns^ writing the sum of the highest column under it. LESSON II. SuhtractloH of Co?7zpou?id JVumbers, 1. From 13 lb. 5 oz. 16 pwt. 21 gr. take 9 lb. 4 oz. 18 pwt. 15 gr. Write the subtrahend under the minuend, placing terms of the same denomination in the same column. Subtract 15 gr. 4 lb. oz. 18 pwt. 6 gr. fi'««i 21 gr., and write 6 gr., the difFeretice, under the grains. Since 18 pwt. are greater than 16 pwt., add 20 pwt. to 16 pwt., making 36 pwt. Subtract 18 pwt. from 36 pwt., and write 18 pwt., the difference, under the pennyweights. Since 20 pwt. were added to the minuend, add 1 oz. (which equals 20 pwt.) to the 4 oz. of the subtrahend, making 5 oz. Sub- tract 5 oz, from 5 oz., and write oz., the difference, under PROCESS. lb. oz. pwt. gr- 13 5 16 21 9 4 18 15 COMPOUND NUMBERS. 181 the ounces. Subtract 9 lb. from 13 lb., and write 4 lb., the difference, under pounds. The difference is 4 lb. 18 pwt. 6 gr. (2) (3) (4) cwt. lb. oz. dr. lb. g. 5. 9. gr. w. d. h. min. From 48 73 10 15 7 10 7 1 14 13 1 13 45 Take 29 47 14 9 3 11 5 2 16 8 6 17 33 (5) (6) (7) mi. fur. rd. yd. rd. yd. ft. in. gal. qt. pt. gi. From 405 5 25 4 35 5 2 10 44 3 1 2 Take 384 6 37 5 27 4 1 11 26 3 1 3 8. A farmer raised 7 bu. 1 pk. 4 qt. of clover-seed, and sold 5 bu. 6 qt. 1 pt. : how much had he left ? 9. A man bought a farm containing 356 A. 2 R. 25 P., and sold 148 A. 3 R 36 P. : how much land had he left? 10. Washington is 77° 2' 48" W. longitude, and San Francisco 122° 26' 15" W. longitude: how much farther west is San Francisco than Washington? 11. From a stack of hay containing 5^ tons, a farmer sold 3 T. 12 cwt. 65 lb. : how much hay re- mained unsold? 12. From a hogshead of molasses containing 63 gallons, a grocer sold 38 gal. 3 qt. 1 pt. : how much molasses remained in the hogshead? 13. A silversmith bought a bar of gold weighing 1 lb. 5 oz. 12 pwt., and a bar of silver weighing 3 lb. 8 oz. 16 pwt. 10 gr. : how much more silver than gold did he buy? 14. A company contracted to build 65 mi. 4 fur. of railroad, and completed the first year 27 mi. 7 fur. 20 rd. : how much remained to be built? PROCESS. y- mo. d. 1869, 9 15 1863, 7 23 6y] r. 1 mo. , 22d. 182 INTERMEDIATE ARITHMETIC. 15. A note was given July 23, 1863, and paid Sept. 15, 1869: how long did it run? Write the earlier date under the later, writing the number of the year, month, and day in proper order, and subtract, allowing 30 days to a month, and 12 months to a year. 16. What is the difference of time between Oct. 23, 1856, and June 15, 1866? 17. How long from April 12, 1861, to May 22, 1865? 18. Abraham Lincoln was born Feb. 12, 1809, and died April 15, 1865: what was his age? 19. The American Eevolution began April 19, 1775, and ended Jan. 20, 1783: how long did it continue? 20. America was discovered Oct. 14, 1492, and the Declaration of Independence was signed July 4, 1776 : how much time elapsed between these two events? 21. The laying of the Atlantic Cable was consum- mated July 28, 1866, and the Pacific Railroad was completed May 10, 1869: how much earlier was the first event than the second? 22. Andrew Jackson died at Nashville, Tenn., June 8, 1845, aged 78 yr. 2 mo. 23 days: what was the date of his birth? DEFIITITION AO EULE. Art. 134. Compound Subtraction is the process of finding the difference between two compound num- bers of the same kind. Art. 135. EuLE. — 1. Write the subtrahend under the minuend, placing terms of the same denomination in the same column. COMPOUND NUMBERS. 183 2. Beginning at the rights subtract each successive term of the subtrahend from the corresponding term of the minuend^ and write the difference beneath. 3. If any term of the subtrahend he greater than the corresponding term of the minuend^ add to the term of the minuend as many units of that denomination as equal one of the next higher^ and from the sum subtract the term of the subtrahend^ writing the difference beneath. 4. Add one to the next term of the subtrahend^ and proceed as before. Note, — Instead of adding one to the next term of the subtra- hend, one may be subtracted from the next term of the minuend. LESSON III. Multiplicatio7i of Compound A^umbers. 1. Multiply 34 gal. 3 qt. 1 pt. by 9. PROCESS. Write the multiplier under the lowest denomination of the multipli- cand. 9 times 1 pt. are 9 pt., equal to 4 qt. 1 pt. Write the 1 pt. under 313 gal. 3 qt. 1 pt. pints, and reserve the 4 qt. to add to the product of quarts. 9 times 3 qt. are 27 qt., and 4 qt. added are 31 qt., equal to 7 gal. 3 qt. Write the 3 qt. under quarts, and reserve the 7 gal. to add to the product of gallons. 9 times 34 gal. are 306 gal., and 7 gal. added are 313 gal. Hence, 9 times 34 gal. 3 qt. 1 pt. = 313 gal. 3 qt. 1 pt. gal. qt. pt. 34 3 1 9 (2) (3) (4) lb. oz. pwt. yd. ft. in. mi. fur. rd. 15 6 13 8 7 2 11 12 15 3 22 6 184 INTERMEDIATE ARITHMETIC. (5) (6 ) (7) lb. 5. 9. bu. pk. qt. w. d. h. 8 10 4 2 4 27 3 5 9 4 6 13 7 8. If a barrel of sugar weigh 2 cwt. 45 lb. 8 oz., how much will 6 barrels weigh ? 9. How much gold will make a dozen rings, each weighing 7 pwt. 15 gr. ? 10. If a pupil study 4 h. 30 min. each day, how many hours will he study in 12 school weeks of 5 days each? 11. If a ship sail 3^ 25' 33^' in 1 day, how far will it sail in 15 days? 12. If 1 man can build 5 rd. 4 yd. 2 ft. of fence in a day, how many yards can 8 men build? 13. How much wheat in 12 bins, if each bin con- tain 50 bu. 2 pk. 5 qt.? 14. John's age is 7 yr. 9 mo. 16 d., which is one- fifth of the age of his father: how old is his father? 15. If a load of wood contain 6 cd. ft. 12 cu. ft., how much wood will 15 loads make? 16. If a family use 2 gal. 3 qt. 1 pt. of milk a week, how much will it use in a year? 17. How much hay is there in 6 stacks, each con- taining 4 T. 16 cwt. 70 lb.? /18. What is the distance round a square field, each side' of which is 24 rd. 3 yd. 2 ft.? 19. If a printer use 3 reams 15 quires 12 sheets of paper each day, how much paper will he use in 4 weeks of 6 days each ? 20. If a man walk 2 mi. 7 fur. 32 rd. an hour, how far will he walk in 12 hours? 21. A field contains 25 rows of corn : if each row yield 5 bu. 3 pk., how much corn will the field yield? COMPOUND NUMBERS. 185 DEFINITION AND EULE. Art. 136. Comjyoiind Multiplication is the process of taking a compound number a given number of times. Art. 137. EuLE. — 1. Write the multiplier under the lowest denomination of the multiplicand. 2. Beginning at the rights multiply each term of the multiplicand in order, and reduce each product to the next higher denomination, writing the remainder under the term multiplied, and adding the quotient to the next product. Note. — In both simple and compound multiplication the suc- cessive products are each divided by the numhei' of units of their denom- ination which equals one of the next higher denomination. LESSON IV. division of Cofnpotmd JV*?imhers. 1. Divide 15 w. G d. 13 h. 12 min. by 12. PROCESS. Write the divisor at the 12 )15 w. 6 d. 13 h. 12 m in. left of the dividend, as in 1 w. 2 d. 7 h. 6 min. simple division. -^^ of 15 w. — 1 w.. with 3 w. remain- ing. Write the 1 w. under weeks. The 3 w. remaining equal 21 d., and 21 d. and 6 d. equal 27 d. ^V ^f 27 d. — 2 d., with 3 d. remaining. Write the 2 days under days. The 3 d. remaining equal 72 h., and 72 h. and 13 h. equal 85 h. ^ of 85 h. = 7 h., with 1 h. remaining. Write the 7 h. under hours. The 1 h. remaining equals 60 min., and 60 min. and 12 min. equal 72 min. -^ of 72 min. := 6 min. Write the 6 min. under minutes. The quotient is 1 w. 2 d. 7 h. 6 min. 186 INTERMEDIATE ARITHMETIC. (2) (3) (4) 8 )14 lb. 12 oz. 15 dr . 10 )53 yd. 2 ft. Sin. 5 )52 A. 3 R. 30 P. (5) (6) (7) 6 )9 cwt. 73 lb. ;2 oz. 11 )65 w. I d. 1 h. 58 min. 7 )Ulb. 5g. 6.^. 8. If a inan sleep 52 h. 30 min. in a week, how long does he sleejD, on an average, each day? 9. A man bought a stack of hay containing T. 19 cwt. 86 lb., and drew it home in 7 equal loads: how much hay did he draw at each load? 10. If a dozen silver spoons weigh 8 oz. 15 pwt., what is the weight of each spoon? 11. A farm of 3-15 A. 3 R 24 P. was divided equally between 6 heirs: how much did each receive? 12. Five equal casks of vinegar contain 218 gal. 2 qt. : how much vinegar in each cask? 13. If a man can dig a ditch 36 rd. 4 yd. 2 ft. long in 8 days, how much can he dig in 1 day? 14. If 9 men can pave 22 sq. rd. 25 sq. yd. in a day, how much can 1 man pave in a day? 15. A ship sailed 48° 24' 45'' in 15 days: how flir did it sail each day? 16. How many goblets can be made of 5 lb. 6 oz. 12 pwt. of silver, if each goblet weighs 7 oz. 8 pwt. ? PROCESS. Reduce both dividend 5 lb. 6 oz. 12 pwt. = 1332 pwt. and divisor to penny- 7 oz. 8 pwt. -- 148 pwt. weights, and divide as 1332 pwt. ^ 148 pwt. = 9, Ans, "^ '^^^P^^ ^^^^'^^"• 17. How many bottles, holding 3 qt. 1 pt. each, can be filled from a cask containing 45^ gallons? 18. How many baskets of peaches, containing 3 pk. 4 qt. each, will make 3| bushels? COMPOUND NUMBERS. 187 19. How many lengths of fence, each 10 ft. 4 in., will make 28 rd. 3 ft. of fence? 20. How many castings, weighing 12 lb. 8 oz. each, can be made from 5 cwt. 50 lb. of iron ? 21. How many times will a wheel, 11 ft. 8 in. in circumference, revolve in going 2 mi. 4 fur. ? 22. How many rings, weighing 5 pwt. 16 gr. each, can be made from a bar of gold weighing 1 lb. 8 oz. ? 23. How many stej^s, of 2 ft. 4 in. each, will a man take in walking | of a mile? 24. K a man can walk 2 mi. 6 fur. in an hour, how long will it take him to walk 22 miles? DEHNITIOIT AND EULES. Art. 138. Compound Division is the process of dividing a compound number into equal parts. Art. 139. EuLE I. — 1. Write the divisor at the left of the dividend, as in simple division. 2. Beginning at the left, divide each term of the divi- dend in order, and write the quotient under the term divided. 3. If the division of any term give a remainder, reduce it to the next lower denomination, to the result add the number of that denomination in the dividend, and then divide as above. Note. —When the divisor is a large number, the successive terms of the quotient may be written at the right of the dividend, as in long division. Art. 140. EuLE II. — 1. To divide a compound num- ber by another of the same kind, Beduce both com- pound numbers to the same denomination, and then divide as in simple division. 188 INTERMEDIATE ARITHMETIC. LESSON V. Miscellaneotis Problems. 1. If 5 sheets of copper contain 28 lb. 10 oz. 8 dr., how much copper is there in each sheet? 2. How much silver will it take to make 4 dozen spoons, each spoon weigiiing 15 pwt. 12 gr. ? 3. If a milk dealer sell daily 7 cans of milk, each holding 12 gal. 2 qt., how much milk does he sell in 4 weeks? 4. From the sum of 15 lb. 8 oz. 15 pwt. and 9 lb. 10 oz. 18 pwt. take their difference. 5. John Jones was born Aug. 8, 1856, and on Jan. 1, 1862, his age was just i of the age of his father: how old was his father? 6. Two small casks, each holding 21 gal. 3 qt., were filled from a cask of cider containing 56 gal. 2 qt. : how much cider remained in the large cask? 7. A father owning a ftirm of 256 A., 3 R 24 P., gave 100 A. to his son, and then divided what re- mained equally between his two daughters : what was each daughter's share? 8. A farmer having cut 12 T. 15 cwt. of hay from a meadow, sold 6 loads of 1 T. 3 cwt. 75 lb. each, and then put the rest in a stack: how much hay was in the stack? 9. A merchant bought 3 chests of tea, each weigh- ing 2 cwt. 45 lb., and in one month sold 4 cwt. 80 lb. 12 oz. : how much tea had he left? 10. A publisher bought 20 bundles of paper, and used daily 3 reams 15 quires 12 sheets: how much paper had he left at the close of 10 days? 11. A railroad company bought 145 cords of wood, i COMPOUND NUMBERS. 189 piled in 3 ranks; the first rank contained 36 cd. 5 cd. ft., and the second, 64 cd. 6 cd. ft. 12 cu. ft. : how much wood was in the third rank? ^.^ 12. A man bought 3 loads of hay, which, with the wagon, weighed respectively 1 T. 8 cwt. 40 lb. ; 1 T. 11 cwt. 80 lb.; and 1 T. 9 cwt. 60 lb.; and the wagon alone weighed 10 cwt. 90 lb. : how much hay did he buy? 13. If 4 horses eat 15 bu. 3 pk. 4 qt. of oats in 12 days, how much will they eat in one day? 14. If 5 horses eat 21 bu. 1 pk. 6 qt. of oats in 4 weeks, how much will 3 horses eat in the same time ? 15. How many steps, of 2 ft. 6 in. each, will a man take in walking 4 fur. 20 rd.? 16. How many times will a carriage-wheel, 11 ft. 4 in. in circumference, turn round in running 10 miles? 17. How many yards of carjieting, a yard wide, will carpet a room 18 ft. by 21 ft.? 18. How many yards of Brussels carpeting, | of a yard wide, will carpet a room 20 ft. by 28 ft.? 19. Three men, A, B, and C, bought a hogshead of sugar, weighing 13 cwt. 60 lb. ; A received ^ of it, B I of the remainder, and C what was left: bow much- sugar did each receive? 20. A company graded 25 mi. 5 fur. 36 rd. of road ; ^ of the job was completed the first month, ^ of it the second month, | of it the third month, and the rest the fourth month: how many miles of road were graded each month? To Teachers. — See Manual of Arithmetic for addi- tional review problems. SECTION XII. TU'M C UJVTA G U. LESSON I. JSTo tat 1071 and Definitions. Art. 141. The term Fer Cent means hy the hundred. One per cent of a number is one hundredth of it, two per cent is two hundredths, three per cent is three hundredths, etc. 1. How many hundredths of a number is 5 per cent of it? 7 percent of it? 2. How many hundredths of a number is 75 per cent of it? 125 per cent of it? 3. How many hundredths of a number is 6^ per cent of it? 16| per cent of it? 4. How many hundredths of a number is 2|- per cent of it? f of one per cent of it? 5. What per cent of a number is y^^ of it? -^-^ of it? 125 of it? 6. What per cent of a number is .06^ of it? .24 of it? .331 of it? 1.12 of it? 1.45 of it? 7. What per cent of a number is 12^ hundredths of it? /^%ofit? .23fofit? 8. How many hundredths of a number is 22^ of it? 21% of it? 63% of it? 142% of it? 9. What per cent of a number is ^ of one hundredth of it? f of one hundredth of it? Note.— The character % is used instead of the term "per cent." 22 fo denotes 22 per cent; 21 fo denotes 2\ per cent, etc. (190) : PERCENTAGE. 191 WRITTEN EXERCISES. 10. Exi)res8 decimally G j^er cent; 12i per cent; | of one per cent. process: 6fo = .06 ; U^fo = .12i; |^ -^ .OOf. 11. Express decimally 125%; 150%; 200%. process: 1257^ = 1.25; 150% = 1.50; 200% = 2.00. 12. Express decimally 5%; 7%; 16%; 24%; 40%. 13. Express decimally i%; |%; f%; ^%; ^\%. 14. Express decimally .12^%; 16|%; 62^%; 17^^%. Suggestion.— 12i% = .12i, or .125; 62i% = .62^, or .6225. 15. Express decimally i%; f%; i%; f%; ^V%- 16. Express decimally 7j\%i 20^%; 112i%; 1331%. DEFINITIONS. Art. 142. Any Per Cent of a number or quantity is 80 many hundredths of it. The term per cent is a contraction of the Latin per centwiiy which means by the hundred. Art. 143. The Rate Per Cent is the number of hundredths. The character fo is^the per cent sign, and is read per cent. Art. 144. Percentage embraces all numerical opera- tions in which one hundred is the basis. LESSON II. Case L— To Find a Given Per Cent of any Number. 1. How much is 5 per cent of 200? Solution. — b% of 200 is ^^ of 200 : -^ of 200 is 2, and r§TT of 200 is 5 times 2, which is 10: 5% of 200 is 10. Or, 5^ of 200 is ^ of 200, which is 10. 192 INTERMEDIATE ARITHMETIC. 2. Wluit is 3 per cent of 400? 8% of 500? 3. What is 6 per cent of 150? 4% of 250? 4. What is 10 per cent of $900? 15% of $600? 5. What is 8% of 2000? 12% of 4000? 3% of 2500? To Teachers.— Show that Sfo of 2500 may also be found by multiplying 2500 by .03, and then require the pupils to solve the above problems in this manner. WRITTEN EXERCISES. 6. What is 16% of 324? "51% of $724.50? 324 .16 process: r 1944 324 51.84, Am. 7. What is 8% of $3250? 8. 6 % of 245? 9. 9 % of 1200? 10.- 15 % of 644? 11. 5 % of 1540? 12. 10 % of 1050 ft.? 13. 33 % of 560 lb.? 14. 31% of 321 oz. ? 15. 12|% of 960 men? 24. A man has an income of $2540, and his expenses are 62i per cent of his income: how much are his ex- penses ? 25. A man having 285 acres of land gave 334^% of it to his daughter: how many acres had he left? 26. A drover bought 245 sheep of A, and 60% as many sheep of B : how many sheep did he buy? 27. A ship is valued at $15800, and the cargo is $724.50 .05i SSS I 24150 362250 $38.6400, Ans. 16. i% of $450? 17. f % of $525? 18. 1 % of 365 days? 19. ^% of $9650? 20. 6% of $.621? 21. ^ of 6.45? 22. 3% of 40.5 ft.? 23. 6-1 % of 96.6 miles? PERCENTAGE.' 193 worth 15^ less than the ship: what is the value of the cargo? 28. An army of 8450 men lost 22% of its men in battle: how many men did it lose? 29. A school enrolled 320 pupils in a term, and the average number in daily attendance was 82^% of the number enrolled: what was the average number of pupils in daily attendance ? 30. A farmer raised 2450 bushels of grain, and 42% of the grain was wheat, 24% oats, and the rest corn: how many bushels of each kind of grain did he raise? Art. 145. EuLE. — To find a given per cent of any number, Multiply the number by the given rate per centy expressed decimally. Note. — When the rate is an aliquot part of 100, the per cent may be found by taking the same aliquot part of tlie number. Thus, 33i% of $48 is i of $48; 12hfo of 320 is h of 320; 25% of 84 is i of 84. LESSON III. Case IL— To Find what Per Cent one Number is OF Another. 1. What per cent of 12 is 3? Solution.— 1 is yV of 12, and 3 is x\, or ^ of 12; J is .25, or 25%. 3 is 25% of 12. 2. What per cent of 75 is 15? 3. 120 is (^0? 5. 90 is 15? 7. 320 is 32? 4. 125 is 25? 6. 72 is 36? 8. 128 is 16? To Teachers. — Show that the rate per cent in each of the above problems may also be found by dividing the number which is the percentage by the other number. Thus: (1) j\ = 3 ^ 12 := .25 =. 2hcJo\ (8) ^2-8 =- 16 -^ 128 ^ .125 == 12^%. T. A.— 13. 19^ INTERMEDIATE ARITHMETIC. WRITTEJJ EXERCISES. 9. What per cent of $520 is $23.40? $520)$23.40(.04r) = 4^;^, Ans, 2080 PKOCESS: -j^^ 2600 10. What per cent of 75 is 13.5? 11. 640 is 48? 18. $324 is $356.40? 12. $650 is $32.50? 19. $324 is $32.40? 13. 38 lb. is 5.32 lb.? 20. $3.20 is $3.36? 14. 900 yd. is 112.5 yd.? 21. $2.40 is $2.04? 15. $128 is $5.76? 22. 12|- cts. is 5 cts.? 16. $7.20 is $1.08? 23. 12| cts. is 10 cts.? 17. $392.50 is $3.92|? 24. 4^ is |? 25. A drover bought 45 horses, and sold 18 of them : what per cent of the drove did he sell? 26. A merchant bought 432 yards of silk, and sold 288 yards : what per cent of the silk did he sell? 27. A school enrolled 225 pupils in a term, and the average number in daily attendance was 198 : what was the per cent of attendance? 28. An army of 15450 men lost 1236 men in battle : what per cent of the army was lost? 29. If 3740 pounds of ore contain 2618 pounds of iron, what per cent of the ore is iron? 30. A man having an income of $2750 pays $440 rent, and $1650 for other expenses: what per cent of his income has he left? Art. 146. EuLE.— To find what per cent one num- ber is of another, Divide the number which is the per- centage by the other number, and the quotient, expressed as hundredths, will be the rate per cent. PERCENTAGE. 195 LESSON IV. Case III. — To Find a Number when a Per Cent of IT IS Given. 1. 60 is 20% of what number? Solution. — If 60 is 20^ of a number, 1% of it is ^V of 60, which is 3, and 100^, or the number, is 100 times 3, which is 300 : 60 is 20 fo of 300. 2. 50 is 25% of what number? 3. 75 is 15% of what number? 4. $36 is 3% of how many dollars? 5. $70 is 10% of how many dollars? 6. 240 acres are 12% of how many acres? 7. 230 miles are 115% of how many miles? 8. 500 rods are 125% of how many rods? To Teachers. — Show that the result obtained above by- analysis may also be obtained by dividing the given number by the rate per cent expressed decimally. Thus: (1) 60 -^ 20 X 100 = 60 -f- .20 == 300 ; (8) 500 rd. ^ 125 X 100 = 500 rd. -4- 1.25 = 400 rd. WRITTEN EXERCISES. 9. A man owes $3175, which is 12|^% of his estate : what is the value of his estate? process: $3175 -V- .125 = $25400, Am, 10. 75 is 37|% of what number? 11. $23.10 is 5% of what sum of money? 12. $61.60 is 110% of what sum of money? 13. $8.16 Js 121% of what sum of money? 14. $180 is 831% of what sum of money? 15. A lady paid $31.50 for a chain, which was 45% of what she paid for a watch : what was the cost of the watch? 196 INTERMEDIATE ARITHMETIC. 16. A farmer sold 420 pounds of wool, which was | 15% of his clip: how much wool did he shear? 17. A man invested $4050, which was 7^% of his property: what was tiie value of his property? 1 18. A man paid $375 a year for a house, $1275 for other expenses, and laid up 25% of his income: what was his income? Art. 147. EuLE. — To find a number when a per cent of it is given, Divide the number which is the percentage by the rate per cent, expressed decimally, LESSON V. Profit and Loss, 1. A man paid $80 for a horse, and sold it for 10% more than it cost him : for how much did he sell it ? Solution.— 10/^ of $80 is $8, and $80 plus $8 is $88: he sold it for $88. 2. A man bought a horse for $80, and sold it for $88: Avhat per cent did he gain ? Solution.— If he gave $80, and sold for $88, he gained $88 less $80, which is $8 ; $8 is f^, or ^^ of the cost, and ^V equals .10, or 10^ : he gained 10 per cent. 3. A man sold a horse for $88 and gained 10%: what was the cost of the horse? Solution.— Since he gained 10%, $88 is 110% of the cost; if $88 is 110%, 1% is $.80, and 100% is $80: the horse cost $80. 4. A dealer paid $5 for a hat, and sold it at 20% profit: what was the selling price? 5. A dealer paid $5 for a hat, and sold it for $G: what per cent did he gain? PERCENTAGE. 197 G. A dealer sold a hat for $G and gained 20^ : what was the cost of the hat? 7. For how much must silk, that cost $1.20 a yard, be sold to gain 25%? 8. Silk that cost $1.20 a yard was sold for $1.50: what was the gain per cent? 9. Silk was sold for $1.50 a yard, at a gain of 25% : what was the cost? 10. For how much must butter, that cost 20 cents a pound, be sold to gain 10%? 11. A carriage that cost $150 was sold for $120: w^hat was the per cent of loss? 12. A merchant bought velvet at $4 a yard, and sold it at $5 a yard: what per cent did he gain? 13. A merchant bought velvet at $5 a yard, and sold it at $4: what was the per cent of loss? 14. A merchant sold velvet at $4 a yard, and lost 20%: what was the cost? "WKITTEN EXEKCISES. 15. A man paid $8750 for a farm, and sold it at 40% profit; what was the selling price? 16. A man sold a farm for $12250, and gained 40%: what was the cost? PROCESS: $12250 -- 1.40 == $8750, the cost. 17. A cargo of wheat, which cost $24650, was sold at a loss of 12%: for how much was it sold? 18. A drover paid $135 a head for horses, and sold them at 30% profit: what was the selling price? 19. A house that cost $3840 was sold for $4128: what was the gain per cent? 20. For how much must teas, that cost $.90, $1.05, and $1.10 a pound, be sold to gain 20%? 198 INTERMEDIATE APaTHMETIC. LESSON VI. Comtmssiony Disurance^ Taxes, etc. TATRITTEN EXERCISES. 1. An agent sold a farm for $3500, and received a commission of 5^: how much did he receive? Note. — The teacher should explain the business terms used in these problems, and add such information as will be of value to the pupil. 2. An attorney collected a debt of $324.50, and charged 10% for his services: what was his com- mission? 3. An architect furnished plans and superintended the erection of a building for 2^% of its cost, which was $25400: how much did he receive? 4. A commission merchant sold 320 barrels of flour at $6.50 a barrel, and charged \\% commission: how much did he receive? 5. A broker bought $12300 worth of cotton, and charged i%: what was his commission? 6. A store worth $25600 is insured for f of its value, at \%\ what is the premium? ff of $25600 = $19200, amount insured. process: { ($19200 X .005 -.$96, premium. 7. A house worth $7500 is insured for | of its value, at f%: what is the premium? 8. A man has his life insured for $4000, at $23.50 per $1000: what annual premium does he pay? 9. A man's property is listed at $10450, and the tax levy is 15 mills on the dollar: what is his tax? process: $10450 X .015 = $156.75, tax. PERCENTAGE. 199 10. A man j^ays a tax of 12|^ mills on property listed at $4960 : what is his tax ? 11. The taxable property in a certain village is listed at $316000: if a tax of $4740 is assessed to build a school-house, what will be the rate in mills? 12. A merchant imported a lot of silk invoiced at $32600: what was the duty, at 37^%? LESSON VII. Simple Tnte7^est. Art. 148. Interest is money paid for the use of money. The Vrinripal is the sum of money for the use of which interest is paid. The Amount is the sum of the principal and the interest. Art. 149. Simple Interest is interest on the princi- pal only. Note.— See Complete Arithmetic for methods of computing Annual Interest and Compound Interest. The Six Per Cent Method. Art. 150. When money is loaned at 6 per cent per annum, the interest of $1 for 1 year is .06 of $1, or 6 cents. 1. What is the interest of $1 for 2 years, at 6%? Solution. — Since the interest of $1 for 1 year, at 6%, is 6 cents, the interest of $1 for 2 years is 2 times 6 cents, which is 12 cents. 2. What is the interest of $1 for 4 years, at 6%? 200 INTERMEDIATE ARITHMETIC. 3. What is the interest of $1 for 5 years, at 6% ? 7 years? 9 years? 10 ^^ears? 4. What is the interest of $1 for 1 month, at 6% ? Solution. — Since the interest of $1 for 12 months is 6 cents, the interest for 1 month is ^^ of 6 cents, which is J cent, or 5 mills. 5. What is the interest of $1 for 3 months, at G% ? Solution. — Since the interest of $1 for one month, at 6%, is 5 mills, the interest for 3 months is 3 times 5 mills, which is 15 mills. 6. What is the interest of $1 for 4 months, at 6% ? 7. What is the interest of $1 for 5 months, at 6% ? 7 months? 9 months? 11 months? 8. What is the interest of $1 for 2 years 4 months, at G%? Solution. — The interest for 2 years is 12 cents, and for 4 months 2 cents; 12 cents and 2 cents are 14 cents: the interest of $1 for 2 yr. 4 mo. is 14 cents. 9. What is the interest of $1 for 3 yr. 6 mo.? For 1 yr. 10 mo.? 2 yr, 8 mo.? 10. What is the interest of $1 for 2 yr. 5 mo.? 3 yr. 7 mo.? 4 yr. 3 mo.? 1 yr. 1 mo.? 11. What is the interest of $1 for 6 days, at 6%? Solution.— Since the interest of $1 for 1 month, or 30 days, is 5 mills, the interest for 1 day is ^^o ^^ ^ mills, which is J mill, and the interest for G days is 6 times J mill, which is 1 mill. 12. What is the interest of $1 for 12 days, at G%? Solution. — Since the interest of $1 for 6 days is 1 mill, the interest for 12 days, or 2 times 6 days, is twice 1 mill, which is 2 mills. PERCENTAGE. 201 13. What is the interest of $1 for 24 days, at 0%? 18 days? 9 dajs? 21 days? 14. What is the interest of $1 for 21 daj^s at 6%? 8 days? 14 days? 20 days? 15. What is the interest of $1 for 2 mo. 18 da.? 4 mo. 24 da.? 3 mo. 6 da.? 16. What is the interest of $1 for 2 mo. 9 da.? 5 mo. 12 da.? 5 mo. 8 da.? 1 mo. 1 da.? WRITTEN EXERCISES. 17. What is the interest of $1 for 2 yr. 3 mo. 14 da., at 6%? Int. of $1 for 2 yr. = $.12 '' " '' '' 3 ino.-^ .015 " " '' " 14 da. ^ .0021 $.1371, A71S. What is the interest of $1, at 6%, for 18. 1 yr. 9 mo. 6 da.? 22. 2 yr. 2 mo. 2 da.? 19. 2 yr. 1 mo. 25 da.? 23. 1 yr. 1 mo. 1 da.? 20. 3 yr. 3 mo. 20 da.? 24. 7 mo. 7 da.? 21. 1 yr. 1 mo. 24 da.? 25. 5 jr. 5 da.? 26. What is the interest of $425 for 2 yr. 3 mo. 12 da., at 6%? Since the intere.st of $1 for 2 yr. 3 mo. 12 da., at 6fo, is $.137, or .137 of $1, the interest of $425 is .137 of $425, which is $58,225. 27. What is the interest of $145.60 for 1 yr. 5 mo. 24 da., at 6% ? For 2 yr. 2 mo. 9 da.? Int. of $1 = $.137. $425 .137 DESS: 2 97 5 1275 425 $5 8,2 2 5, Int. 202 INTERMEDIATE ARITHMETIC. 28. What is the interest of $64.20 for 1 yr. 3 mo., at 6%? For 5 mo. 15 da.? 29. What is the amount of $85.50 for 1 yr. 1 mo. 1 da., at 6%? For 4 yr. 10 da.? Note. — The amount is the sum of principal and interest. 30. What is the amount of $184.80 for 9 mo. 27 da., at 6%? 2 yr. 25 da.? 31. Wliat is the interest of $31.20 from Oct. 23, 1855, $31.20 .208 ^ 520 24960 6240 I to Apr. 12^ 1859, at 8%? 1859 4 12 1855 10 23 Syr. 5 mo. 1 9 da. In t. of$lat6% = $.208;. 8)$6.49480, Int. at 6%. 2.16493, '' '' 1%, $8.65973, " " 8^. 32. What is the interest of $540.50 from May 10, 1873, to March 4, 1874, at 7% ? At 9% ? Suggestion.— The interest at 7% is \ more than the interest at 6%, and the interest at 9% is J more than the interest at 6^^ . 33. Wliat is the amount of $121.60 from Feb. 12, 1874, to May 22, 1876, at 8% ? At 12%? ' 34. A man borrowed $460, July 28, 1866, and paid it May 16, 1869, with interest at 5%: what was the amount? 35. A note of $243.75, dated June 8, 1873, was paid Nov. 14, 1875, with interest at 10% : what was the amount ? 36. A note of $600, dated Sept. 9, 1872, was paid Jan. 21, 1874, with interest at 7.3% : what was the amount? PERCENTAGE. 203 Art. 151. EuLES. — 1. To comijute interest at 6 per cent, Find the interest of ?1 for the given time^ and then multiply the given principal by the abstract deci- 7nal which corresponds to the interest of %\. Note. — The interest of $1 may be found by taking six times as many cents as there are years, one- half as many ceiits as there are months, and one-sixth as many mills as there are days. 2. To compute interest at any rate per cent, Find the interest at 6 per cent^ and then increase or diminish this interest by such a part of itself as will give the in- terest at the given rate. Notes. — 1. The following table may be found convenient: 7 % = 6fo + i of (yfo. 12 ^0 = 6fo X 2. n% -- (jfc + i of Qfc, 15 fc ^ 6fo X 2J. 8 % -= Qfo + 1 of 6fc. 10 % = 6% -^- 6 X 10. 9 % = G% + i of 6%. 11 fo = Qfo -^- 6 X 11. 5 /. = 6fc — i of Gfo. 7.3 fo = Qfo -f- 6 X 7.3 4 fo = Gfo — 1 of Qfo, bifo = Qfo - 6 X 5i. ^fo = 6fo — i of efo. X fo = 6fo -- 6 X ^. 2. The interest at 10% is found by dividing the interest at 6% by 6 and removing the decimal point one place to the right. 3. The interest of $1 at 7.3% is 1 mill for 5 days, and hence interest for any number of days, at 7.3%, (for 365 days,) may be computed by multiplying the principal by | as many thousandths as there are days in the time. The answer to problem 36, p. 202, is $659.88, if the interest be computed for the actual number of days. LESSON VIII. !2)fscou7it. Art. 152. jyiscount is a deduction from a debt for its payment before it is due. A note due at a future date without interest is usually discounted in business by deducting the interest for the 204 INTERMEDIATE ARITHMETIC. time, with or without grace, as per agreement. The rate of interest allowed is usually greater than the current rate. Bills are often discounted by deducting a certain per cent of their face without regard to time. These deductions are ^called Business Discount. Deductions from the nominal price of articles sold are often computed as a certain per cent off. This deduction is called Trade Discount. Note. — The deduction known as *' True Discount^* is seldom made in business, and hence is not presented in this book. It is fully treated in the Complete Arithmetic {p. 192). Art. 153. Sank Discount is the interest on a note paid in advance. When a bank loans money, the borrower gives his note, payable at a specified time without iriterest. The interest on this note for the time, plus three days, is subtracted from its face, and the remainder, called the proceeds^ is paid to the borrower. The three days added to the time are called Days of Grace. WRITTEN EXERCISES. 1. A bank discounted a note of $250, payable in 60 days, at 8% : what were the proceeds? PROCESS. $250 .0105 60 days + 3 days == 63 days. 12 5 Int. of $1 at 6^ = $.0105. ^^^ 3)$2.6250, Discount at 6^. 8750 $3.50, Discount at 8%. $250 — $3.50 ==$246.50, Proceeds. 2. What are the proceeds of a note of $240.60, pay- able in 90 days, discounted by a bank at 10% ? PERCENTAGE. 205 What are the bank proceeds of a note of 3. $22.50, payable in 60 days, discounted at 9% ? 4. $720, payable in 30 days, discounted at 8% ? 5. $62.40, payable in 45 days, discounted at 10%? 6. $125, payable in 90 days, discounted at 7|% ? 7. What are the proceeds of a note of $90.60, dated March 10, 1876, and payable June 5, 1876, discounted by a bank at 9% ? PROCESS. In March, 21 days. $90.6 " April, 30 - .015 "May, 31 " "J^^^ " June, 5 " 9060 ^'''!\-^ 2)11.35900, 6 ) 90 days. ^^^r^ Int. of $1 at 6fe = 15 mills. "^oAoor t^» . ^^ ^ $2.0 3 8 5, Dis. at d%, $90.60 — $2.0385 = $87,561 5, Proceeds. Note. — When the time of a note is short, it is the general cus- tom of bankers to compute interest for the actual number of days in the time, inchiding grace, each day being considered as ^J^ of a year. This mode of finding the time of interest, with or without grace, is called the Method by Days. 8. What are the proceeds of a note of $142, dated Aug. 15, 1875, and payable Nov. 4, 1875, discounted at 10% ? 9. A note of $360, dated Sept. 15, 1875, and pay- able Nov. 15, 1875, was discounted by a bank at 8% : what were the proceeds? 10. A note of $225, dated May 11, 1875, and payable July 31, 1875, with interest at 6%, was discounted June 4, 1875, at 10%: what were the proceeds? Suggestion. — Compute the interest for 81 days -[- 3 days, or 84 days, and discount the amount thus found for 60 days. 206 INTERMEDIATE ARITHMETIC. 11. A merchant discounted a bill of $750. due in 3 months, by deducting the interest for the time, without grace, at 8% : what were the cash proceeds? PROCESS. Int. of $1 for 3 mo. at 6^ = : $.015 $750 .015 8)$11.250, Dis. at 6/.. 3 75 $15.00, Dis. at 8^. $750 — $15 = $735, Cash proceedf?. 12. A man sold a note of $150, due in 6 months, at a discount of 10^ for the time, without grace: how much did he receive for the note? 13. A merchant deducted for cash 5% from a bill of goods amounting to $540: how much did he re- ceive for the goods? process: $5.40 X .05 -= $27. $540 — $27 =^ 51 3, Ans. 14. A man asked $12500 for a farm, but sold it for 6% off for cash down: what did he receive for the farm? 15. A man bought a bill of goods amounting to $1500 on 60 days' credit, but was offered a discount of 5% for cash: how much did he pay for the goods? Art. 151. EuLES. — 1. To compute bank discount. Find the interest of the sum discounted for the number of days in the time plus three days. 2. To find the proceeds, Subtract the discount from the sum discounted. PERCENTAGE. 207 LESSON IX. JVbteSy Drafts, and !Bonds, Art. 155. A Proniissorif Note is a written agree- ment by one person to pay another a specified sum of money at a specified time. The sum of money specified is called the Face of the note. The person who signs a note is its Maker ; the person to whom it is payable is the Payee; and the owner is the Holder. An Indorser is a person who signs his name on the back of a note as security for its payment. Art. 156. A Draft is an order by one person upon another to pay a specified sum to a third person named. It is also called a Bill of Exchange, The process of making payments at distant places by the remittance of drafts is called Exchange. Art. 157. Bonds are interest bearing notes issued by nations, states, cities, railroad companies, and other corporations, as a means of borrowing money. They are issued under seal. The market value of bonds is quoted at a certain per cent of their face, or par value. Bonds quoted at 109 are worth, in currency, 109^ of their face. To Teachers. — For fuller information respecting notes, drafts, and bonds, see Complete Arithmetic, pp. 198-202, and 311. 208 INTERMEDIATE ARITHMETIC. A^TBITTEN EXERCISES. 1. What will be the cost of a draft on New York for $640, when exchange is f % premium? $640 X .00J=-$2.40, Cost of exchange. process: , $640 + $2.40 = $642.40, Cost of draft. 2. A merchant in St. Louis wishes to remit $2450 to a creditor by draft on New York: what will be the cost of the draft, at f% premium? 3. A merchant in New Orleans bought a sight draft on New York for $5600, at |% discount: . what was the cost of the drafl? 4. When gold was quoted at 112 J, what was the value in currency of $5440 in gold ? PROCESS : $5 440X1-125 = $6120, Value in currency. Note. — When gold is quoted at 112K, it is worth 112K^, or 12>^ more than currency; that is, $1 in gold is worth $1.12^ in currency, and $100 in gold is worth $112.50 in currency. Gold has not been at a premium and thus quoted since 1878. 5. When gold was quoted at 112^, what was the value in gold of $6120 in currency? process: $6 1 20^1.1 25:=:=$5440, Value in gold. 6. When gold was quoted at 112, what was the value in currency of $1752.50 in gold? 7. When gold was quoted at 112, how large a gold draft could be bought for $196280 in currency? 8. When gold was quoted at 110, what was the gold value of a one dollar bank note? 9. When gold was quoted at 112^, what was the value in currency of a five dollar gold piece? 10. What was the cost in currency of a gold draft for $500, when gold was quoted at $110? SECTIOK XIII. j. — I — i I 1 4 LESSON I. Su7''faces* 1. How many square feet in a piece of zinc, 5 feet long and 3 feet wide? PROCESS : 5 X 3 = 1 5, sq. ft. In a piece 5 feet long and 1 foot wide there are 5 sq. ft., and in a piece 3 feet wide there are 3 times 5 sq. ft., which is 15 sq. ft. 2. How many acres in a field 45 rods long and 32 rods wide? 3. How many square inches in a triangle whose base is 15 inches and altitude 10 inches. PROCESS. 15 X 10 — 150, sq. in. in rectangle. 150 -r- 2 = 75, sq. in. in triangle. The area of a triangle is one half of the area of a rectangle with the same base and altitude. Base, 15 inches. 4. How many square yards in a triangular surface with a base of 18 feet and an altitude of 12 feet? 5. How many square feet in a board 14 feet long and 8 inches wide at one end and 4 inches wide at the other? I. A.— 14. (209) 210 INTERMEDIATE ARITHMETIC. Note. — The average width of the board is i of the sum of 8 inches and 4 inches, which is 6 inches, or i foot. In j)ractice, this would be found by measuring the width of the board at its middle. 6. How many square feet of lumber in 15 boards, each 12 ft. long and 7 in. wide, and in 21 boards, each 14 ft. long and 8 in. wide? 7. How many square feet of timber in 12 joists, 2 in. by 8 in., and 15 ft. long, and 20 scantling, 3 in. by 4 in., and 12 ft. long? 8. The diameter of a circle is 5 inches : how many inches in its circumference? PROCESS. y^^^^ \ ^^X,^^ 5 in. X 3.1416 = 15.708 in., circum- / I \ ference. [ '^ - It may be shown by geometry that \ / the circumference of a circle is 3.1416 \^^ / (nearly 3f) times the diameter. 9. What is the circumference of a wheel whose diameter is 15 feet? 10. A circular room is 40 feet in diameter: how many square feet in the floor? PROCESS. 4 ft. X 3.1416^ 125.664 ft, circumference. 125.664X40-4-4 = 1256.64, sq. ft. in the floor. It may be proven by geometry that the area of a circle is equal to the circumference multiplied by one-fourth of the diameter, or One-half of the radius. 11. A horse is tied to a stake by a rope 40 feet long: on how much surface can the horse graze? Suggestion. — The surface is a circle 80 feet in diameter. MENSURATION. 211 12, How many square inches in the surface of a ball 3 inches in diameter? PROCESS : 3 X 3 X 3.1 4 1 6 = 2 8.2 7 4 4 , sq. in. in surface. Art, 158. EuLES.— 1. To find the area of a rectangle, Multiply the length by the width. 2. To find the area of a triangle, Multiply the base by one half of the altitude. 3. To find the circumference of a circle, Multiply the diameter by 3.1416. 4. To find the area of a circle, Multiply the circum- ference by one fourth of the diameter. 5. To find the number of square feet in a board not exceeding one inch in thickness, Multiply the length in feet by the width in inches , and divide the product by 12. Notes. — 1. If tlie board is H inches thick, add i of this surface measure; if li inches thick, add i of surface measure. 2. Planks, joists, sills, and other timber are measured by multi- plying the nmiiber of square feet in one surface by the thickness in incites. LESSON II. Solids, 1. How many cubic feet in a block of stone 4 ft. long, 3 ft,, wide, and 2 ft. thick? PROCESS. ^' y ,..- ..- ^ 4 X 3 = 12, cu. ft. in 1 ft. thick. (fj im i ^'^^^ Note. — For fuller explanation see iMliliilliliiililillilift^^ Complete Arithmetic, p. 105. iiillMiilM^^ 2. How many cords of wood in a pile 40 ft. long, 12 ft. high, 5 ft. thick? 212 INTERMEDIATE ARITHMETIC. 3. How many cords of four- foot wood in a pile 64 ft. long and 5^ ft. high? (Art. 115, Note 1.) 4. How many cords of three-foot wood in a pile 28 ft. long and 6 ft. high? 5. How many bushels of wheat will fill a bin 6 ft. long, 5 ft. wide, and 3 ft. deep? Note. — A bushel contains 2150J cubic inches. 6. How many cubic feet in a round block of timber 5 ft. long and 2 ft. in diameter? PROCESS. 2 ft. X 3.1 4 1 6 -- 6.2 8 3 2 ft., cir. of base. 6.2 8 3 2 X 2 -- 4 = 3.1 4 1 6, 8q. ft. in base. 3.1 4 1 6 X 5 = 1 5.7 08, cu. ft. in block. Note. — The block of timber is a cylinder. 7. How many cubic feet in a circular well 30 ft. deep and 3 ft. in diameter? 8. A liquid gallon contains 231 cubic inches: how many gallons in a barrel 27 inches long and 18 inches in diameter (average) ? 9. How many cubic inches in a globe 10 inches in diameter? flOX 10 X 8.1416 = 314.16, sq. in. in surface. process: < 1314.16X1 = 528.6, cu. in. in globe. Art. 159. Rules. — 1. To find the solid contents of a rectangular solid, Multiply the lengthy width, and thickness together. 2. To find the solid contents of a cylinder. Multiply the area of the base by the altitude. 3. To find the solid contents of a sphere. Multiply the surface by one third of the radius. MULTIPLICATION TABLE. 213 MULTIPLICATION TABLE. IX 2X 3X 4X 5X «Xl 7X1 8X1 9X1 10X1 11 XI 12X1 1 2 8 4 5 6 7 8 9 10 11 12 1 X 2X 3X 4X 5X 6X 7X 8X 9X 10 X 11 X 12 X 2 = 2 = 2 = 2 = 2:= 2:z:r 2 = 2 =: 2=r 2 = 2 = IX 2X 3X 4X 5X 6X 7X 8X 9X 10 X 11 X 12 X 3 = ^^ 3 = 3 = 3=r= 3=: 3=r 3 = 3 =: 3 6 9 12 15 18 21 24 27 30 33 36 IX 2X 3X 4X 5X 6X 7X 8X 9X 10 X 11 X 12 X 12 16 20 24 28 32 36 40 44 48 1X5: 2X5: 3X5t: 4X5: 5X5: 6X5: 7X5: "X5: X5: 10 X 11 X 12 X- 5 10 15 20 25 30 35 40 45 50 55 60 IX 2X 3X 4X 5X GX 7X 8X 9X 10 X 11 X 12 X 6 12 18 24 30 86 42 48 54 60 66 72 IX 2X 3X 4X 5X 6X 7X 8X 9X 10 X 11 X 12 X 7 14 21 28 35 42 49 56 63 70 77 84 IX 2X 3X 4X 5X 6X 7X 8X 9X 10 X 11 X 12 X 16 24 32 40 48 56 64 72 80 88 96 X9: X9 X9: X9: X9: X9. X9: 8X9: 9X9: 10X9: 11X9: 12X9: r 9 -: 18 -- 27 z 36 = 45 3 54 = 63 r 72 r 81 z: 90 r 99 rl08 1X10: 2X10: 3X10: 4X 10: 5X10: 6X10: 7X10: 8X10: 9XJ0: 10 X 10: U XIO: 12 X 10: 10 1 Xl^ [ ^ 11 20 2X 1^ I := 22 80 3X11 I =: 33 40 4X11 L .= 44 50 5X1] '--=. 55 60 6X11 I = 66 70 7X11 \= 77 80 8X11 L — 88 90 9 X 11 I =r 99 00 10 X 11 =110 10 11 Xli [ =121 20 12X11 L =132 1 X 12 : 2X12^ 3X12: 4X12: 5X12: 6X12: 7X12: 8X12: 9X12: 10 X 12: 11X12: 12 X 12 : = 12 ^ 24 r 36 r 48 r 60 = 72 = 84 = 96 =108 rl20 nl32 =144 ANSWERS TO THE WRITTEN PROBLEMS. N. B.— The last answer is given when a problem has several answers, and also when several problems are united. NOTATION. Pagre 19. 15. 3,000,300,000,303.22. 30,000,075,000. 10. 50,032,640. 16. 62,300,049. 23. 9,000,009,009. IL 300,009,206. 17. 500,005,000. 24. 54,087,086. Page 20. 18. 406,507. 25. 202,580. 12. 48,000,017,064. 19. 2,010,080. 26. 50,050,500,007. 13. 5,005,005. 20. 90,007,490. 27. 17,000,700,306. 14. 1,100,010. 21. 400,040,404. ADDITION. 28. 90,010,055. Page 24. 18. 8,996 pounds. 6. 395,096. 1. 10. Page 26. 7. 53,293,685. 2. 96. 1. 280. Page 30. 3. 899. 2. 2,^18. 1. 197,251. 4. 9,889. 3. 270,724. 2. 300,334. 5. 87,978. 4. 270,527. 3. 319,076. 6. 2,051. 5. 307,352. 4. 40,805,643. 7. 174. 6. 4,444,844. 5. 21,316,368. 8. 1,264. Page 27. 6. 51,147,320. 9. 14,736. 7. 15,084 bushels. 7. 181 days. 10. 861,566. 8. 1,949 voters. 8. 1,462 acres. Page 25. 9. 649 acres. 9. 1,388 papers. 11. 2,925. 10. 295,612 pounds. 10. 12,326 youth. 12. 14,333. Page 28. Page 31. 13. 599,939. 1. 29,537. 1. 4,005. 14. 20,195. 2. 214,353. 2. 47,169. 16. 95 pounds. 3. 262,105. 3. 347,575. 16. 184 days. 4. 243,571. 4. 327;008. 17. 787 acres. (214) 5. 1,868,776. 5. 46,833. ANSWERS. Page 32. 2. 29,348. 1. 102,924. 6. 821,412. 3. 77,055. Page 35. 7. 826,826,826. 4. 1,374. 2. $10,575. 8. 823 miles. 5. 25,914. 3. 1,552 pages. 9. 6,595 houses. 6. 10,000,800,999. 4. 185,425 sq. miles. 10. 1,064 miles. 7. 65,038 sq. miles. 5. 653 miles. Page 33. Page 34. 6. $14,750. 1. 124,112. 8. 1,037 miles. 7. 1,424 bushels. SUBTRACTION. Page 38. 7. 909. Page 44. 16. 2,034. 8. 404. 1. 36,944. 17. 354 pounds. 9. 7,061. 2. 2,633,755. 18. 24. 10. 3,314. 3. 11,971 sq. miles. Page 39. 11. 9,825. 4. 8,033,009. 1. 2. 3. 121. 1,002. 103. 12. 13. 14. 15. Page 42. 8,028. $485. 478 youth. 84,622 pupils. 5. 6. P^ge 45. 22,853 miles. 22,772 men. 4. 5. 132. 1,141. 7. 8. 14,805,623 lbs. $10,337. 6. 3,311. Page 47. 7. 2,026. Page 43. 1. 42 yards. 8. 203 acres. 1. 799,762. 2. 21,224 sq. miles. 9. $2,130. 2. 39,920,000. 3. 599 men. 10. 2,062 bushels. 3. 277. 4. 1,185 bushels. 11. 12. Page 40. $3,235. 100 sch'l-houses, 4. 5. 6. 7. 156 years. 59 years. 5,618 feet. 1,511 sheep. 6,926 feet. 5. 6. 7. $240. 273. Page 48. 745,452. 13. 11,115 fleeces. 8. 8. $6,200. Page 41. 9. 128 years. 9. 3,728. 6. 2,297. 10. 1,541,000. 10. 680,134. MULTIPLIOATION Page 49. 6. 909,609. 11. 1,280 rods. 2. 9,636. 7. 690,963. Page 51. 3. 30,606. 8. 640,402. 2. 3,624. 4. 69,963. 9. $880. 3. 15,144. 5. 808,488. 10. $399. 215 4. 31,815. ANSWERS. 5. 28,360. 6. 54,180. 7. 151,470. 8. 1,620 pins. 9. 3,400 miles. 10. 945 tons. 11. $912. 12. $14,180. 13. 10,080 min. 14. 124,160. Pagre 53. 2. 32,724. 3. 37,654. 4. 56,158. 5. 338,364. 6. 1,674,156. 7. 310,612. 8. 5,800,509. 9. 8,208 miles. 10. $4,455. 11. 18,262 yards. Pasr^ 54. 2. 16,376,688. 3. 130,721,463. 4. 7,188,372. 5. 8,811,712. 6. 61,926,668. 7. 778,822,521. 8. 330,325 days. 9. 98,800 pounds. 10. 2,948,190 lbs. 11. $63,500. 12. $5,305,125. Paire 56. 2. 1,728,000. 3. 124,740,000. 4. 12,685,000. 5. 1,054,970,000. 6. 4,250,400 feet. 7. 3,264,000 miles. 8. 21,000 times. 9. 68,600 pounds. 10. 289,920 sheets. 11. 94,240 pounds. 12. 40,600. Pagre 57. 2. 150,400,000. 3. 33,596,000,000. 4. 893,000,000. 5. 342,000,000,000. Paee 58. 6. 2,268,000 sec. 7. 691,200,000. 8. 11,700 lbs. 9. $425,000. 10. 36,180 men. 11. 26,000 miles. 12. 58,800 stalks. PiHpe 59. 1. 181,078. 2. 1,931,200. 3. 18,300,672. 4. 3,379,200,000. 5. 3,120,000,000. 6. 364,450 lbs. 7. 357,700 lbs. 8. 268,800 lbs. 9. 21,312 cents, or $213.12 10. 473,040. Page 63. 1. 73,696. 2. $825. 3. $684. 4. $288. 5. 2,700. 6. 157,248 words. 7. $420. 8. $18,160. 9. $2,400. Page 63. 10. $1,330. 11. $85. 12. $80,000. 13. 25,920 soldiers. Pase 64. 2. 241. 3. 4,321. 4. 2,312. 5. 3,123. 6. 1,232. 7. 4,042. 8. 2,103. DIVISION. 9. 213. 10. 132. 11. 180. Paee 66. 2. 191. 3. 234. 4. 128. 216 5. 174. 6. 153. 7. 282. 8. 1,748. 9. 3,696. 10. 93 boxes. 11. 243 barrels. 12. 390 days. ANSWERS. «8-M I Paee 68. 2. 312. 3. 312. 4. 222. 5. 321. 6. 527. 7. 6,546. 8. 4,725. 9. 3,436. 10. 4,326. 12. 22. 13. 25. 14. 48. 15. 97. 16. 22 yards. 17. 436 bushels. Page 69. 18. 76 hogsheads. 19. 243 boxes. 20. 47 farms. 21. 49 days. 22. 247 years. Page 70. 2. 579, with 360 R. 3. 11, with 17,211 R. 4. 768, with 499 R. 5. 506 weeks. 6. 405 cattle. 7. 207 days. 8. 603 days. 9. 48 hours. 10. 41cd.32sq. ft.R. 11. 81,073. Page 72. 3. 456. 4. 1,870. 5. 384, witli 50 R. 6. 23, with 45 R. 7. 450, with 860 R. Page 73. 9. 8. 10. 40,030. 11. 937,w'h63,000R. 13. 234, with 385 R. 14. 12. 15. 64 barrels. 16. 270 reams. 17. 48 hours. 18. 16 h)ts. 19. 44 cars. 20. 46 barrels. 21. 40 regiments. 22. 109 acres. Page 74. 23. 36 hours. 24. 156. 25. 84. 26. 376. Page 78. 1. 406. 2. 42,909. 3. 22,962. 4. 36. 5. 82,532. 6. 48. 7. $14. 8. $285. 9. 10 cows. 10. $2,010. 11. $625. 12. 3,700 bushels. 13. $9,850. Page 79. 14. 60 miles. 15. 732 miles. PROPERTIES OF NUMBERS, Page 82. 13. 2, 5, 5, 5. 26. 32. 2. 3, 3, 7. 14. 2, 2, 2, 5, 11. Page 84. 3. 2, 2, 2, 3, 3. 15. 2, 2, 5, 5, 5. 2. 60. 4. 2, 2, 3, 7. 16. 2, 2, 2, 3, 3, 3, 3. 3. 168. 5. 2, 2, 2, 2, 2, 3. 17. 2, 2, 3, 3, 5, 5. 4. 480. 6. 5, 5, 7. 19. 18. 5. 108. 7. 3, 7, 7. 20. 24. 6. 240. 8. 5, 5, 11. 21. 15. 7. 144. 9. 5, 5, 13. 22. 12. 8. 1,440. 10. 2, 2, 2, 3, 11. 23. 27. 9. 300. 11. 2, 2, 2, 5, 5. 24. 21. 10, 500. 12. 2,2,2,2,2,2,2,2. 25. 48. 11. 144. 217 89-103 Page 89. 14. ip. 15. HK Page 90. 16. *F. 17. m^. 18. -W- 19. HiK 20. W- 21. -W- 22. -t-¥-*. 23. W- 24. mK 25. HF- 26. Wi- 97 1860£ <»«• ~2 0(y • Page 91. 12. 17xV 13. 6xV 14. 8tV 15. 15,V 16. 6. 17. 3fr 18. lOM. 19. 2. 20. 11. 21. 20,V 22. 20Jt. 23. 10. 24. mi 25. 2I3V 26. 9. 27. 28ii. 28. I7ii Page 93. 14. |. 16. J. ANSWERS. FEAOTIONS. 16. f. 22. 21!- 17. A- 23. m- 18. |. 19. i. 20. If 21. f. Paare98. 25. 175|. 26. 27. 115|. 315. 22. f. 23. tV 28. SolA- 24. f. Pace 9». 25. I5V 11. i. 26. t't- 12. h 27. h 14. A- 28. i 15. h 29. f. 16. i«- 17. A- Page 95. 18. if- 12. If. 19. Ih- 13. II, If. 21. 21A- 14. H, ff, n- 16. t\, f?, t's- 17. if, if, if. 18. U, fJ, il- 22. 23. 24. 155A. 66|. 19. if, il. A- Page 100. 20. !f. A, A- 9. n- 21. A, M, A- 10. f- 22. if. A, A, U- 11. A- 23. if, 11, fl, If. 12. 74iJ. 24. A^A^./A.AV Pace 101. Paee»7. 13. 2f. 12. 21 14. 41J. *"»• 2* 13. 2ii. 15. lA- 15. 16. 66|. 16. 2f. Page lOS. 17. lA- 12. 31. 18. m. 13. 8f. 19. lA- 14. 15|. 20. 2A. 15. 21. 21. 2 A- 16. 3^. 218 ANSWERS, II 17. in. 22. |. 16. 32. 18. n. 23. /s- 17. 12. 19. 66|. 24. i 18.3. 20- 225. 25. f. . Pai^ .,«. raee 103. 26. f. 2. IJ. 13. 305|. 11 \\ 3. IJ. 14. loij. ;°- '• 4. f. 15. 2,112^. 30.,. 5.1^. 16. l,972i. il- f 6. I. 17. 192. ^i- ^; 7. 2|. <><*• Ti- 8. 3|. Page 104. 34. 1. g g| 18. 20. 35. f. lo! A. 19. 56|. 36. J. 20. 50f. 37. If . 21. 69|. 38. 6}. ^ 22. USA. 39. IH. J 5- ,„,, 23. 260. 40. 15|. ^- f "»' T^s- 24. 176H. 41.781. *-;|„, 25.2561. r^,^ ^'''^"■ 26. 119,V. 1, f"*****- rareiw. 27. 96f. "• V 6. llH. 28. 247if. f ;• Tf 7. Pace 111. T¥- 29. 553H. ; J \^^- 8. 5,536. 31. 792. ;:• \^- 9. A- 32. 1,350. !^- V 10. 46i-V 33. 2,292. }°- Y- 11. ItV 34. 3,136|. , ^- Y' 12. 1,188 feet. 35. 5,670. iQ f; 13. $3,750. 36. 13,5331. g^- 7/- 14. A. 37. 6,094f. i^Z^f 15. Soldi; owns f. 38. 38,042|. i^- gf 16. 1,089 feet. 39. 45,600. **• ^" 17. 12 rods. Pace 100. 18. 24 barrels. Page 106. jQ_ 24. 19. $21,000. 17. f. 11. 20. 20. $8,400. 18. h 12. 75. 21. 2,016 bushels. 19. h. 13, 68f 22. $10,560. 20. /j. 14. 49. Paite 113. 21' A- 15. 70. 23. $6,832. 219 ANSWERS. Paice 120. 6. .4500 7. 6.500 8. 23.00 9. 62.5000 10. .04800 11. 406.062000 13. .5 14. 2.40 Page 121. 1 J' 3 4. }. 5. |. 6. 7. 8. 9. 10. 11. 12. 3 J. 13. 21^3^. 14. ^1,. 15. 12|. 16. 25^1^. 4. .75 5. .625 6. .0625 7. .125 8. .075 9. .024 10. .256 11. .0075 12. 3.08i DECIMAL rRAOTIONS. 13. 21.75 14. .0032 15. 12.0375 16. 25.032 17. .625 18. .0124 19. 12.062500 Page 122. 5. 28.2104 6. 182.097 7. $926,498 8. 5378 9. 178.455 miles. Page 123. 6. 16.544 7. .032625 8. .011992 9. .7992 10. 31.48 11. .0092 Page 125. 12. .25331 13. .1728 14. .4355 15. .18468 16. 161.5 17. 1505.52 18. 156. 19. 1.3332 20. .000056 21. .03625 22. 121. 23. .021 24. 3.1828 25. .00156 27. 4085. 28. 300480. Page 127. 9. 3.6 10. .28 11. 11.2 12. .79 13. 112 14. 8.5 15. .8 17. 160. 18. 200. 19. 1500. 20. 230000. 21. 23000. 23. .4367 24. .002346 Page 12!^. 1. .001 2. .0064 3. iin?. 4. 8.7 5. 1020. 6. 1120. 7. .1743 8. .0072 9. 176000. 10. $721,446. UNITED STATES MONEY. Page 132. 8. 560 c. 11. 3125 m. 6. 10800 c. 9. 801 c. 12. 105 m. 7. 23000 c. 10. 40000 ra. 220 13. 3030 m. ANSWERS. 14. 50 m. 15. 1000 c. 16. 45000 m. 17. ic. 18. $75. 19. $75.50 20. $3,125 21. $4. 22. $15.07 23. $10.01 24. $1.50 25. $10.25 26. $50. 27. $5. 28. $3.75 29. $.375 Page 133. 30. 4500 m. 31. $4.50 32. 1010 c. 33. $1.01 2. $17,825 Page 134. 3. $97,285 4. $308,365 6. $.49| 7. $98,744 8. $9.90 9. $327.75 10. $1,375 11. $2.50 12. $4,875 13. $493.25 14. $2.75 15. $281.73 Page 133. 2. $120. 3. $45. 4. $144.90 5. $2.25. 6. $87.50 7. $1200. 8. $146.25 10. $.18 11. $37.50 Page 136. 12. $7.50 13. $11,375 15. 60 bushels. 16. 200 lemons. 17. 32 days. 18. 154 bushels. 19. 56 yards. 20. 80 sheep. 21. 40 lemons. 22. 75. 23. 36. Page 137. 24. 250. 1. $139.59 2. $9,995 3. $495. 4. $506. 5. 100. 6. $131.60 7. $600. 8. $20. 9. $277.20 10. 200 yards. 11. 400 bushels. Page 138. 12. $1600. 13. $1.75 14. 50 acres. 15. $6.75 16. $58.50 17. $1950. 18. $191.25 (365 days in year). 19. $405. 20. 98 yards. 21. 40 sheep. 22. $2.50 23. $1230. Page 139. 24. $116.95 25. $71. 26. $215. Page 140. 2. $140.15 3. $91.79 Page 141. 4. $132.95 5. $288,375 Page 143. 6. $329. 7. $26,125 DENOMINATE NUMBEES. Page 146. 21. 768 pt. Page 147. 23. 366 pt. 24. 971 pt. 25. 280 qt. 26. 59 pt. 27. 31 qt. 221 28. 66 pt. 30. 5 bu. 1 pk. 31. 5bu. 1 pk. 3qt. 32. 3 pk. 1 qt. 1 pt. 147-166 ANSWERS. 33. 2 pk. 2 qt. 1 pt. 34. 4 bu. 2 pk. 7 qt. 35. $2.90 36. 14.88 37. $4.25 38. 9bu. 2pk. 6qt. 39. 112 baskets. 40. $1.00 Page 149. 17. 168 pt. 18. 249 gi. 19. 273 pt. Paere 150. 20. 77 pt. 21. 4gal. 3qt. 22. 7 gal. 2 qt. 1 pt. 23. 16qt. 2gi. 24. 17 gal. 3 gi. 25. 1456 gi. 26. 449 pt. 27. 38 gal. 1 pt. 28. $41.60 29. 16 vials. 30. 21 jugs. 31. $10. 32. $36.60 33. $48.30 34. $56.70 Page 153. 21. 576 in. 22. 27828 in. 23. 1819 ft. 24. 6490 yd. 25. 5mi.lfur.10rd. 26. 4fur. 2rd. 5 yd. 1 ft. 6 in. 27. 258820 in. 28. 4224 steps. 29. 2200. 30. 12 mi. 3 fur. 3rd. 3 yd. 1ft. 6 in. 31. 240 rd. Page 156. 15. 627264 sq. in. 16. 880 P. 17. 70882 sq. ft. 18. 1 A. 2 R. 20 P. 10 sq. yd. 7 sq. ft. 19. 33 A. 20. 4 sq. yd. 21. 14 A. 22. 20 A. 23. 324 sq. yd. 24. $125. 25. 480 trees. 26. 10000 shingles. Page 157. 16000 A. 26880 A. 41 J yd. 3442i bricks. 27. 28. 29. 30. 31. $37. Page 158. 1. 55296 cu. in. 2. 9 cu. ft. 3. 3240 cu. ft. 4. 15 cu. yd. 5. 728793 cu. in. 6. 31 cu. yd. 15 cu. ft. 1206 cu. in. 8. 1872 cu. ft. 9. 576 cu. ft. 10. IO3L cu. yd. 11. 220cu. yd. Page 159. 12. 2160 cu. yd. 13. $406.25 21i2 Page 160. 1. 5 cd. ft. 2. 96 cu. ft. 3. 44 cd. ft. 4. 16 cd. 6. 12 cd. 6. 3 cd. 96 cu. ft. 7. 5 cd. 20 cu. ft. 8. $49.50 9. 41 cd. 32 cu. ft. 10. $31,993^ Page 161. 6. 55800"^ 7. 933^ Page 162. 8. 9900'. 9. 3° or iV S. 10. 90°. Page 164. 11. 54000 sec. 12. 8h. 13. 481200 sec. 14. 2680245 sec. 15. 21 d. 6 h., or 3 w. 6 h. 16. 6 c. yr. 17. 527040 min. 18. 31556928 sec. 19. 31536000 sec. 20. 562116 h. 21. 2208 h. 22. 40320 min. 23. 3780000. 24. 13 days. Page 166. 9. 160000 oz. 10. 7456 lb. 11. 9245 oz. 12. 17T.9cwt.20Ib. ANSWERS. 166-181 13. 2cwt. 851b. 14. 5641968 dr. 15. $11.76 16. $90. Page 167. 17. 20bbl. 18. $105. 19. $54. 20. $507,875 21. 61 lb. Page 168. 7. 10633 pwt. 8. 3720 gr. 9. 322872 gr. 10. 56 lb. 2 oz. 6 pwt. 11. 7 lb. 3 pwt. 16 gr. 12. 473Jlb. 13. 11 oz. 14. $3150. 15. $52.50 16. 60 rings. Page 170. 7. 97790 gr. 8. 4980 gr. 9. 3 lb. 8 g 4 5. 10. 41b.7gl3l9 4gr. 11. 20 1b. 12. 18 doses, with 16 gr. R. 13. 128 pills. 14. 6g. Page 171. 6. 6000 sheets. 7. 3252 sheets. 8. $127.50 9. 5184 crayons. 10. 288 shirts. 11. $396. 12. $22.50 Page 175. 15. $22.50 16. $77.50 17. $17600. Page 176. 18. 960 times. 19. 126720 times. 20. 23040 acres. 21. $600. 22. 24200 pills. 23. 13333 p'ple, with tV sq. yd. R. 24. $10642.50 25. 120 sq. yd. 26. 30| yd. 27. 14400 shingles. 28. 128 rods. 29. 336 cu. ft. 30. 48 P. 31. $QQ. 32. $40.50 33. 2592000 times. Page 177. 34. 7305 hours. 35. 80 rings. 36. 320 1b. 37. 120 gross. 38. 41f reams. COMPOUND NTJMBEES. Page 1*8. 2. 89 bn. 3 pk. 5 qt. 3. 121 gal. 3 qt. 1 pt. 4. 99 mi. 35 rd. 2 ft. 11 in. 5. 60 cwt. 77 lb. 2 oz. 13 dr. 6. 55 lb. 3 oz. 8 pwt. 7 gr. 7. 113 1b.73 25 29l5gr. 8. 29 w. 6 d. 10 h. 41 min. 29 sec. 9. 7 S. 8° 43^ S'^ 10. 36 sq. mi. 134 A. 30 P. 11. 153 cd. 1 cd. ft 9cu. ft. 12. 23 bundles, 10 quires. Page 179. 13. 11 cwt. 61 lb. 10 oz. 14. 15. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 223 18rd. 3 yd. 1 ft. 7 in. 161 gal. 2 qt. Page 181. 19 cwt. 25 1b. 12 oz. 6 dr. 3 1b. 11 g 1 .5 1 9 18 gr. 4 w. 1 d. 20 h. 12 m. 20 mi. 6 fur. 27 rd. 4 yd. 1 ft. 6 in. 8 rd. 1 yd. 11 in. 17 gal. 3 qt. 1 pt. 3 gi. 2 bu. 5 qt. 1 pt. 207 A. 2 R. 29 P. 45° 23^ 27"^ 1 T. 12 cwt. 35 lb. 24 gal. 1 pt. 181-18(1 ANSWERS. 13. 2 lb. 3 oz. 4 pwt. 10 gr. 9, 19 cwt. 98 lb. 14. 37 mi. 4 fur. 20 rd. 10. 14 pwt. 14 gr. Page 182. 1 1. 57 A. 2 E. 24 p. 16. 9 yr. 7 mo. 22 d. 12. 43 gal. 2 qt. If pt. 17. 4 yr. 1 mo. 10 d. 13. 4 rd. 3 yd. 1 ft. 18. 56 yr. 2 mo. 3 d. 14. 2 sq. rd. 16 sq. yd. 2 sq. ft. 19. 7 yr. 9 mo. Id. 15. 3° 13^ 39^^ 20. 283 yr. 8 mo. 20 d. 17. 52 bottles. 18. 4 baskets. 21. 2 yr. 9 mo. 12 d. Page 187. 22. March 15, 1767. 19. 45 lengths. Page 183. 20. 44 castings. 2. 124 lb. 5 oz. 4 pwt. 21. 1131 f times. 3. 95 yd. 2 ft. 22. 70 rings, with 80 gr. R. 4. 92 mi. 5 fur. 12 rd. 23. 1697^ steps. 24. 8 hours. Page 184. Page 188. 5. 35 lb. 6 g 2 3 2 9. 1. 5 lb. 11 oz. IH dr. 6. 251 bii. 5 qt. 2. 3 lb. 1 oz. 4 pwt. 7. 34w. 3d. 19 h. 3. 2450 gal. 8. 14 cwt. 73 lb. 4. 19 lb. 9 oz. 16 pwt. 9. 4 oz. 11 pwt. 12 gr. 5. 32 yr. 4 mo. 18 d. 10. 270 h. 6. 13 gal. 1 1. 51° 23^ 15^^ 7. 78 A. 1 R. 32 P. 12. 2571 yd. 8. 5 T. 12 cwt. 50 lb. 13. 607 bu. 3 pk. 4 qt. 9. 2 cwt. 54 lb. 4 oz. 14. 38 yr. 11 mo. 20 d. 10. 2 reams, 5 quires. 15. 12 cd. 5 cd. ft. 4 cu. ft. 11. 43 cd. 4 cd. ft. 4 cu. ft. 16. 149 gal. 2 qt. Page 189. 17. 29 T. 20 lb. 12. 2 T. 17 cwt. 10 lb. 18. 2 fur. 18 rd. 3 yd. 2 ft. 13. 1 bu. 1 pk. 2i qt. 19. 90 reams, 12 quires. 14. 12 bu. 3 pk. 3f qt. 20. 35 mi. 5 fur. 24 rd. 15. 1188 steps. 21. 143bu. 3pk. 16. 4658}^ times. Page 186. 17. 42 yd. 18. 82|f yd. 2. 1 lb. 13 oz. 9 J dr. r A. 6 cwt. 80 lb. 3. 5 yd. 1 ft. 2 in. 19. < B. 4 cwt. 53i lb. 4. 10 A. 2 R. 14 P. i C. 2 cwt. 26f lb. 5. 1 cwt. 62 lb. 4 oz. lOf dr. r 1st mo., 8 m. 4 fur. 25| r. 6. 5w.6d.llh.5min.l6T\sec. q^ j 2d " 6m.3fur. 19r. 7. 2 lb. 6 5 1 9 14f gr. ** • I 3d " 9 m. 5 fur. 8i r. 8. 7 h. 30 min. i 4th " 1 m. fur. 23^ r. 224 ANSWERS. 192-21S PEEOENTAGE. Page 192. 7. $260. 8. 14.7 9. 108. 10. 96.6 11. 77. 12. 105 ft. 13. 184.8 1b. 14. 10.7 oz. 15. 120 men. 16. $2.25 17. $3.15 18. 2.43ida. 19. $28.95 20. $.03| 21. $.516 22. 1.215 ft. 23. 6.118 mi. 24. $1587.50 25. 190 acres. 26. 392 sheep. 27. $13430. Pagre 193. 28. 1859 men. 29. 264 pupils. 30. 1029 b.wH. 588 b. oats. 833 b. corn. Pagre 194. 10, 18/,. 11. 7if,. 12. 5f,. 13. 14/,. 14. 12J/,. 15. 4iy,. 16. 15/,. 17. 1/.. 18. no/,. 19. 10/,. 20. 105/, . 21. 85/,. 22. 40/, . 23. 80/,. 24. 16f /, . 25. 40/,. 26. 66f/,. 27. 88/,. 28. 8/,. 29. 70/, . 30. 24/,. Page 195. 10. 200. 11. $462. 12. $56. 13. $65.28 14. $216. 15. $70. Page 196. 16. 28001b. 17. $54000. 18. $2200. Page 197. 15. $12250. 17. $21692. 18. $175.50 19. 71-/,. 20. $1.08 $1.26 $1.32 Page 198. 1. $175. 2. $32.45 3. $635. 4. $93.60 6. $41. 7. $18.75 8. $94. Page 199. 10. $62. 11. 15 mills. 12. $12225. Page 201. 18. $.106 19. $.129i 20. $.198i 21. $.069 22. $.130 J 23. $.065 J 24. $.036i 25. $.300f 27. $19,146 + Page 202. 28. $1,765 + 29. $106,162 4- 30. $207,746 32. $39.726| 33. $154,837 + 34. $524.40 35. $303,062 36. Page 204. 2. $234,384 + Page 205. 3. $22,145 + 4. $714.72 5. $61,568 6. $122,578 + 8. $138,686 + 9. $354.88 10. $224,347 + Page 206. 12. $142.50 14. $11750. 15. $1425. Page 208. 2. $2468.375 3. $5565. 6. $1962.80 7. ^175250. 9. $5.62i 10. $550. Page 200. 2. 9 acres. 4. 12 sq. yd. 5. 7 sq. ft. MENSUEATION. Page 210. 6. 301 sq. ft. 7. 480 sq. ft. 9. 47.124 ft. 11. 5026.56 s. ft. Pp. 211, 212. 2. 18| cords. 3. 11 cords. 225 72.32 + bu. 212.058 c.f. 8. 29.743 + g. "^ .b'O L .SSj 3 ^ ,af / .Z6~ . ^i / / ^ ■— V ^..- ' / ^ / 'VMM. *» 1 Z " .Zo . 16% 1 ' .. / M - '^^i iz 10 ^ \oL-^ m 35853 ivi24:9554 THE UNIVERSITY OF CALIFORNIA LIBRARY ECLECTIC EDUCATIONAL SERIES. Tew Eclectic Penmanship. n^HE simplest, 7?iost leg^ible, and business-like style of Capitals and Small Letters is adopted. In the Copy-Books each letter is given separately at Jlrst, and then in combination ; the spacing is open ; analysis simple, and indicated in every letter when first presented ; explanations clear, concise, and complete are given on the covers of the books, and not over and around the copies. NEW ECLECTIC COPY-BOOKS.— Revised and Re-engraved. No's I, 2, 3, 4, 5 Boys, 5 Girls, 6 Boys, 6 Girls, 6 3^, 7, 8 Boys, 8 Girls, and No. 9. Girls' Copy-Books identical, word for word, with the Boys' , but in smaller hand-writing. First-class paper, engraving, and ruling. ECLECTIC ELEMENTARY COURSE The Elementary course comprises three books, smaller than the Copy Books, but the same in form. No's i and 2 are Tracing-Books. ECLECTIC PRIMARY COPY-BOOK. A complete Primaiy Penmanship, designed for use during the second year of school life. It contains all the small letters, figures and capitals, each given separately and of large size, the object being to teach the form of the letter. It is designed to be written with the lead-pencil. Furnished either in white or manilla paper. ECLECTIC EXERCISE-BOOK. Contains a variety of exercises especially designed to develop the differ- ent movements, and so arranged as to give as much or as little practice on each exercise as may be desired. It is a little larger than the Copy- Books, and has a strong cover, so that the latter may be placed within it, thus making it convenient to keep the two together. THE ECLECTIC PRACTICE-BOOK Is made of the same size and weight of paper as the Copy-Books, ruled with -'ouble Hues for No's i, 2,3, 4, and with ?^ngle lines for the higher number-.. NEW HAND-BOOK OF ECLECTIC PENMANSHIP. A Key to the Eclectic r ^steni oi Penmanship. A complete description and analysis of movement ana the letters, and a brief summary of what is required in teaching penmauonip, ECLECTIC WRITING-CARDS. 72 No's on 36 Cards. Oue T.etter or Principle on each Card: Capital Letter on one side. Small Letter on the reverse. Each illustration accom- panied with appropriate explanations and instructions. Size of Cards, 9x13 inches ; loop attached for suspending on the walls. SAMPLE BOOK OF ECLECTIC PENMANSHIP. Containing nearly 200 copies selected from all the Copy-Books in the Series. Will be sent by mail for 15 c. to any teacher or school officer desir- ing to examine it with a view to introducing the Eclectic Penmanship. ECLECTIC PENS. School Pen, No. 100, 90 c. per gross; small box (2 doz.), 25 c. Com- mercial Pen, No. 200, 90 c. per gross. Ladies' Pen, No. 300, 90 c, per gross. Free Writing* Pen, No. 400, 90 c per gross. Sample Card of Eclectic Pens, 10 c. VAN ANTWERP, BRAGG & CO., Publishers, Cincinnati.