OF- THK University of California. GIKT OP^ Accession H5970 ^^^^^ HEATH'S COMPLETE PRACTICAL AEITHMETIC BY CHARLES E. WHITE SYRACUSE, N.Y. BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1901 Copyright, igoi, By D. C. Heath & Co. 6 PREFACE. In preparing this book, the aim has been to make it complete, so that it might cover with sufficient fulness the topics usually required in common and grammar schools. The aim also has been to make the work practical, i.e. to use only such rules, definitions, solutions, and problems as would accustom the learner to reason, compute, and esti- mate with the same facility and economy of thought with which business men reason, compute, and estimate in prac- tical life. Great care has been exercised to omit the merely theo- retical and the unpractical, as well as problems that are too hard for the grades for which the book is intended. The relations of the inductive work, the definition, the solution, the rule, are such that the learner, with a mini- mum of help, should easily master the difficulties through the logical progression of the steps from the known to the unknown. While care has been exercised in selecting a great variety of practical business problems and in arranging them pro- gressively, the development of mental power has been kept constantly in view. The arrangement of this book is topical, but subjects previously studied are kept fresh in the minds of the pupils by frequent carefully prepared reviews. IV PREFACE. Questions of relation, as treated in connection with divi- sion of fractions, will be found helpful in overcoming a group of difficulties which, in the author's experience, are trying to all children. When these are mastered by the learner, he will later have little or no difficulty with Per- centage and its applications. ^ The practice of referring percentage problems back to the original questions of relation has proved highly suc- cessful in the experience of many teachers. A wide variety as well as a great number of topical review and miscellaneous problems are given in the last part of the book. Thanks are due to the various superintendents of city schools who have kindly furnished copies of recent exami- nation questions, which largely constitute the topical review of this book. The author has also received invaluable aid from many leading educators, all of whom he desires to thank most cordially. C. E. W. Syracuse, N.Y., January 24, 1901. CONTENTS. TAQZ Notation and Numeration 1 Arabic Notation 2 llomau Notation 9 Notation of United States Money . . . . . .10 Addition 12 Subtraction 22 Multiplication 31 Division 42 Short Division 48 Long Division 51 Indicated Operations 54 Principles of Division 55 Factors 61 Cancellation 63 Greatest Common Divisor 65 Least Common Multiple 68 Common Fractions 73 Ked action of Fractions 75 Addition 83 Subtraction 86 Multiplication 89 Division .... 93 The Three Questions of Relation 98 Review 101 Decimal Fractions 108 To read and write a Decimal 109 Reduction of Decimals 110 V si VI CONTENTS. PAGE Addition of Decimals . . 112 Subtraction of Decimals 113 Multiplication of Decimals 114 Division of Decimals 116 To divide by 10, 100, 1000, etc 117 Parts of 100 and 1000 118 Aliquot Parts of $1.00 119 Review of Decimals 121 Accounts and Bills . . . . . . . . . 125 Compound Numbers 135 Linear Measure 135 Surveyor's Measure 136 Square Measure 136 Cubic Measure 136 Liquid Measure 137 Apothecaries' Fluid Measure 137 Dry Measure . . • 137 Avoirdupois Weight 137 Troy Weight 138 Apothecaries' Weight 138 Comparison of Weights * . . 138 Measure of Time 138 Circular Measure 139 Federal Money 140 English Money . . 140 Counting Table 141 Paper Table 141 Reduction Descending 141 Reduction Ascending 143 Reduction of Denominate Fractions to Integers of Lower Denominations 147 eduction of Denominate Numbers to Fractions of Higher Denominations 148 /-^ CONTENTS. Vii PAGE To find what Part One Denominate Number is of Another . 150 Addition of Compound Numbers 151 Subtraction of Compound Numbers 153 Difference between Dates 155 Multiplication of Compound Numbers 156 Division of Compound Numbers 157 Miscellaneous Problems 161 Measurements, Surfaces 164 Carpeting Eooms 169 Plastering and Painting 171 Papering Walls ......... 173 Board Measure 175 Miscellaneous Problems 176 Measurements, Volumes 178 Wood Measure 181 Capacity of Bins and Cisterns 182 Longitude and Time ......... 183 Standard or Railroad Time 186 The Metric System 191 Linear Measure 192 Surface Measure 194 Volume Measure 19'/ Capacity Measure 198 Measures of Weight . . 200 Review Questions 201 General Review 202 Percentage 210 Profit and Loss 220 Commission 223 Insurance 227 Trade Discount 233 Taxes 235 Duties 237 Vm CONTENTS. ' PAGE Review Questions 238 Miscellaneous Review of Percentage '..... 239 Simple Interest 246 The Six Per Cent Method 247 Exact Interest 251 Problems in Interest . 252 Promissory Notes 255 Partial Payments . . . 258 Compound Interest 262 Review of Interest 263 Discount • 267 True Discount 267 Bank Discount 269 To find the Face of a Note when Proceeds, Time, and Rate are known 275 Review of Discount 276 Stocks and Bonds 278 Bonds 281 Average of Payments 286 Ratio and Proportion 292 Ratio . . . 292 Proportion 294 Simple Proportion . . . . . . . . 295 sQompound Proportion 298 j ~T ( Partnership 301 / Involution 307 t--^VOLUTION 309 Evolution and Involution . . . . . . . 310 Square Root 310 Right-angled Triangles . . . ... . . 315 Similar Surfaces 317 Cube Root 318 Similar Solids . . .325 -'I Pi CONTENTS. IX PAGB Questions ....*. 326 General Review 327 Topical Review 340 Common Fractions 346 Decimals 349 Denominate Numbers 363 Percentage . . , 360 Interest and Discount 36§ Proportion and Partnership 376 Involution and Evolution 379 Miscellaneous Problems ........ 381 Mensuration . . . . . . . . . . 391 Surfaces 392 Solids 393 Pyramids and Cones 394 APPENDIX. Mariners' Measures 397 Surveyors' Linear Measure 397 Surveyors' Square Measure ....... 398 Government Lands ......... 398 Miscellaneous Measures of Weight 399 Apothecaries' Fluid -Me a sure 401 Business Forms .......... 402 Computing Taxes 404 Table of Legal Rates of Interest . . . . . 405 Exchange 406 Domestic 406 Foreign 409 \: \ B R A rf^ UNX/EHSITY COMPLETE PEACTICAL AEITHMETIC. 3>© z z O z z 3 U I 3 UJ 27. For convenience in reading and writing numbers they are separated into groups of three figures each, q^Wq^ periods. Each group takes the name of its right hand order of units ; thus, the first group is the group of units, the second of thousands, the third of millions, the fourth of billions. The comma is used to separate the groups. Thus, in the number 624,503,275,320, the first group is 320 units, the second group is 275 thousands, the third group is 503 millions, the fourth group is 624 billions. The above number is read as follows : 624 billion, 503 million, 275 thousand, 320. Note. — In reading numbers the last group-name is always omitted t) NOTATION AND NUMERATION. PRINCIPLES OP NOTATION. 28. 1. Ten units of any order are equal to one unit of the next higher order. 2. The value of a figure is increased teyifold by removing it one place to the left, and decreased tenfold by removing it one place to the right. The following table shows the grouping of the orders into periods : =! 2 5 z z < go S o o 5 oil f2 5g ii ^^ -§^ 2gi 25- S5^ So 't/vQL "-1— "-;^o "-«a u-.co u- Q?Q Q^H Q"?^ qS2 Qh3 Qco£ zz< zz=i zz^ zzH zzo zzt 3li]3 Zi \ii a. DuJ= DuJ— 3LiiZ 3lJZ Il-Cr Ihh- IhCQ IHS XHI- XI-3 32, 406, 398, 040, 324, 763 Groups, Gtli 5th. AtYi 3d 2d 1st Name, Quadrillions Trillions Billions Millions Thousands Units The above number is read 32 quadrillion, 406 trillion, 398 billion, 40 million, 324 thousand, 763. Note 1. — The names of groups above quadrillions are quintillions, sextillions, septillions, octillions, nonillions, decillions, etc. Note 2. — Each group except the one at the left must contain three figures. TO BEAD NUMBERS. 29. Rule. — Begin at the right, and separate the numbers into groups of three figures each, using the comma. Begin at the left, and read the number in each group, giving to it the name of that group. No name is given to the number in the last group. ARABIC NOTATION. 30. Copy, separate into groups, and read : 1. 3896 2. 26432 3. 897063 4. 20396 5. 390403 6. 704503 7. 2987652 8. 356293603 9. 290030052 10. 387523729842 14. 500004 15. 329000101 16. 3424300000 17. 800003000 18. 29856323155824 19. 1487603035006201 20. 35601600 21. 18008 22. 180506 23. 1658838 11. 5030473694026 24. 200800 12. 500320 25. 32004060 13. 3400093 26. 13087 27. 716042 28. 2730010 29. 126003184 30. 47250627 . 31. 1002970 32. 17042 33. 14390023 34. 11935079 35. 4000030 36. 29307070 37. 14280643 31. Write in figures : 1. Thirty-six million, twenty-four thousand, two hundred seventy-two. 36,024,272. Solution, — Write 35 for the millions' group, following with a comma ; then write the 24 in the thousands' group, prefixing naught to make the group complete, and follow it with a comma. This is followed by 272 in the units' group. TO WRITE NUMBERS. 32. Rule. — Beginning at the left, write the figures of each group in their proper order, filling vacant places with ciphers. Place a comma after each group before writing the following group. 8 NOTATION AND NUMERATION. Write in figures : 1. Twenty-seven thousand, three hundred sixteen. 2. Eighty-four thousand, seven hundred twenty-six. 3. One hundred twenty-two thousand, one hundred forty-five. 4. Two hundred thousand, sixteen. 5. Eleven thousand, two. 6. Four million, six hundred eight thousand, three hun- dred seventy-five. 7. Twenty-five thousand, three hundred eighty-seven 8. Nineteen thousand, seventeen. 9. Twenty-seven million, six hundred fifty-two. 10. Eighty million, six hundred nine thousand, four hundred twenty-eight. 11. Four hundred thirty-six thousand, forty-one. 12. Six hundred twenty million, seventeen thousand, four hundred seventy-seven. 13. One hundred fifty-seven million, six hundred eight thousand, four hundred seventy-seven. 14. Write a number containing four groups. 15. Six hundred four million, seventy-eight thousand, nine hundred two. 16. Three hundred twenty-four billion, two thousand, six hundred forty. 17. Six hundred thousand, fifty-five. 18. Four million, three hundred six thousand, one hun- dred eight. Forty thousand, ten. 19. 75 million, 136 thousand, 265. 20. 356 billion, 208 million, 708 thousand, 16. 21. Five billion, five million, five thousand, five. Four thousand, four. ROMAN NOTATION. 9 22. 306 million, 20 thousand, 12. Seventeen million, 2 thousand, 406. 23. Ninety-four trillion, sixteen billion, four hundred six million, fifteen thousand, seven hundred. 24. 20 million, 20 thousand, 20. Four hundred twenty- five million, seven hundred two thousand, one hundred eighty-one. 25. 60 billion, 40 thousand. 1 billion, 1 thousand, 1. ROMAN NOTATION. 33. The Eoman system of notation uses seven capital letters to express numbers, viz. : I, V, X, L, C, D, M, Values, 1, 5, 10, 50, 100, 500, 1000. All other numbers are formed by repeating or combining these letters. 34. The following principles are used in expressing !koman numbers : Principles. — 1. Eepeating a letter repeats its value. Thus, I stands for one ; II for two ; III for three ; X for ten ; XX for twenty ; XXX for thirty, etc. Note. — Only I, X, C, and M, are thus repeated. 2. When a letter is placed before another of greater value, its value is taken from that of the greater. Thus, IV stands for four ; IX for nine ; XIX for nineteen ; XL for forty ; XC for ninety. 3. When a letter is placed after another of greater value, their values are united. Thus VI stands for six ; XII for twelve ; XV for fifteen ; XXXV for thirty-five ; LV for fifty-five. 10 NOTATION AND NUMERATION. value a thousand-fold. L for fifty thousand ; 4. A dash over a letter increases its Thus, V stands for five thousand ; M for one million. The following table illustrates the use of Roman letters in forming numbers : I . . 1 XI . . 11 XXIV . . 24 C . 100 II . . 2 XII . . 12 XXIX . . 29 cc . 200 Ill . . 3 XIII . . 13 XXX . . 30 cccc . 400 IV . . 4 XIV . . 14 XL . . 40 CD . 400 V . . 5 XV . . 15 L . . 50 D . 500 VI . . 6 XVI . . 16 LX . . 60 DCC . 700 V^II . . 7 XIX . . 19 LXX . . 70 DC . 600 IX . . 9 XX . . 20 LXXX . . 80 M . 1000 X . . 10 XXI . . 21 xc . . 90 MD . . 1500 35. Read the following : XXIX; XXXV; LXX; XXXIX; XLIV; CXV ; XCV; LXXXIX; CXIV; XCIV ; XLIX; CCCIX; CDXIV; XDLXII; MDCCCC. 36. Write in Roman : 39, 63, 98, 161, 515, 654, 1560, 1899, 1902, 1040, 3762. UNITED STATES MONEY. 37. The Dollar Sign is $, and is placed before the number. Thus, $26 is read 26 dollars. • Dollars are written at the left of a period (.) called the Decimal Point. Cents are written at the right of the deci- mal point, and always occupy two places. Mills, or tenths of a cent, occupy the third place at the right of the decimal point. Twenty-four dollars sixty-five cents three mills is written $24,653. UNITED STATES MONEY. 11 Any number of cents, less than 10, requires a naught be- tween it and the decimal point. Thus, one dollar and 8 cents is written ^1.08. Copy and read : 1. $25.34 6. .flj.03 11. $204,865 2. $74.98 7. $2,843 12. $384,467 3. $25.09 8. $200,504 13. $293,062 4. $87.15 9. $192,003 14. $398,405 5. $58.16 10. $715.38 15. $294,066 Write the following : 1. Sixteen dollars, fifteen cents. Nineteen dollars, seventy-five cents. Sixty-one dollars, twenty-eight cents. 2. Twenty-five dollars, twenty cents. Two hundred dol- lars, eight cents. 24 dollars, 5 cents. 3. 1864 dollars, 11 cents. 87 dollars, 9 cents. 28 dol- lars, 28 cents. 19 dollars, 1 cent. 4. 8 dollars, 5 cents, 5 mills. 12 dollars, 16 cents, 6 mills. 4 dollars, 4 cents, 4 mills. 30 dollars, 30 cents, 3 mills. 5. 306 dollars, 10 cents, 4 mills. 806 dollars, 20 cents, 2 mills. 1349 dollars, 9 cents, 9 mills. 6. 85,600 dollars, 20 cents, 1 mill. 28 dollars, 7 cents, 7 mills. 28 dollars, 7 mills. 7. 1840 dollars, 4 mills. 268 dollars, 15 cents, 9 mills. 84 dollars, 84 cents, 8 mills. ADDITION. 38. 1. How many apples are 4 apples and 2 apples ? 2. How many pencils are 3 pencils and 4 pencils ? 3. Willie had 5 cents and his uncle gave him 4 cents. How many had he then ? 4. 5 books and 3 books are how many books ? 5. How many oranges are 5 oranges and 2 oranges ? 6. How many are 6 and 3 ? 7 and 4 ? 3, 2, and 4 ? 7. How many books are 2 books, 4 books, and 5 books ? 39. Addition is the process of uniting two or more num- bers into one sum. 40. The result obtained by adding is called the Sum or Amount. 41. The Sign of Addition is an upright cross, +. It is called plus and is sometimes placed between numbers to be added. Thus, 5 + 4 is read 5 plus 4, and means that 5 and 4 are to be added. 42. The Sign of Equality is two short horizontal lines, and means equals, or equal to. Thus, 5 + 3 = 8 is read 5 plus 3 equals 8, or 5 and 3 are 8. 12 DRILL IK ADDITION. _v 13 DRILL IN ADDITION. 43. The following are all the combinations of two num- bers from 1 to 9. 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 2 2 2 2 2 2 2 2 3 4 5 6 7 8 9 3 3 3 3 3 3 3 4 5 6 7 8 9 4 4 4 4 4 4 5 6 7 8 9 5 5 5 5 5 6 7 8 9 6 6 6 6 7 8 9 7 7 7 8 9 8 8 9 9 Oral. 44. 1. John found 4 eggs in one nest and 6 in another. H^w many eggs did he find ? 2. James had 7 cents and found 4 more. How many- cents did he then have ? 14 ADDITION. 3. If Mary paid 10 cents for a tablet, 5 cents for a pen- cil, and 3 cents for pens, how much did she pay for all ? 5 + 3 = ? 4. In a pasture there are 6 black horses, 6 bay horses, and 3 white ones. How many horses are in the field? 6+5+3=? 5. There are 9 yellow apples in one basket and 5 red ones in another basket. How many apples in both baskets ? 6. I spent 4 dollars for a chair, 6 dollars for a table, and 3 dollars for a lamp. How much money did I spend ? 7. Edward had 9 cents in the bank and put in 9 cents more. How many cents did he then have in the bank ? 8. 3 + 4 + 6 = ? 4 + 5 + 3 = ? 9. John caught 5 trout on Monday, 6 on Tuesday, and 4 on Wednesday. How many trout did he catch ? 45. Add by twos from to 28. Thus, 2, 4, 6, 8, 10, 12, 14, etc. 1. Add by 2's from 1 to 31. 2. Add by 3's from to 39. From 1 to 40. 3. Add by 4's from 2 to 34. From to 30. 4. Add by 4's from 1 to 37. From 3 to 38. 5. Add by 5's from to 40. From 1 to 36. 6. Add by 5's from 3 to 38. From 4 to 39. 7. Add by 6's from to 30. From 1 to 49. 8.- Add by 6's from 4 to 64. From 5 to 53. 9. Add by 7's from to 70. From 1 to 71. 10. Add by 7's from 2 to 72. From 5 to 89. 11. Add by 8's from to 80. From 5 to 93. 12. Add by 9's from to 90. From 2 to 92. ORAL EXERCISES. 15 46. Add. 1. 4 2. 2 3. 7 4. 7 5. 6 6. 5 7. 6 8. 4 8 6 1 4 9 9 7 9 7 5 3 5 4 8 3 6 3 4 4 8 3 6 9 5 2 3 9 3 5 9 4 3 51 6 2 4 4 5 8 9. 7 + 5 + 8 + 3 + 9 + 8+4 = ? 10. 9 + 6 + 8 + 4 + 7 + 3 + 5 + 3 = ? Oral. 11. What is the sum of 43 and 24. 12. A railway train ran 35 miles the first hour, and 40 miles the second hour. How many miles did it run in the two hours ? 13. In a certain class there are 26 boys and 35 girls. How many pupils are in the class ? 14. A newsboy made 31 cents on Monday, 15 cents on Tuesday, and 52 cents on Wednesd^ay. How many cents did he make in the three days ? 15. Lucy bought a pineapple for 21 cents, and two pounds of sugar for 11 cents. How much did she pay for both ? 16. A farmer had three flocks of sheep. The first flock contained 26 sheep, the second 37, and the third 41. How many sheep had he ? 17. A father gave 25 cents to his son, and 20 cents to each of his two daughters. How much money did he give to his three children? 18. A farmer sold four jars of butter. The first con- tained 24 pounds, the second 26 pounds, and the third and fourth 25 pounds each. How many pounds of butter did he sell ? 16 ADDITION. 19. In a certain grove there are 45 maple trees, 34 oak trees, and 28 beech, trees. How many trees in the grove ? 20. There are 45 cattle in each of three pastures. How many cattle in the three pastures ? Written. 21. What is the sum of 635, 726, and 893 ? Solution. — Write the numbers so that units of the same order shall stand in the same column. The sum of the units' column is 34-6+5=14. 14 units are equal to 1 ten and 4 units. Place the 4 units under ' -^6 the units' column, and add the 1 ten to the column of 893 tens. noK4 1 + 9 + 2 + 3 = 15, the sum of the tens. 15 tens are equal to 5 tens and 1 hundred. Place the 5 tens under the tens' column, and add the 1 hundred to the hundreds' column, 1+8 + 7+6 = 22, the sum of the hundreds. 22 hundreds are equal to 2 thousands and 2 hundreds, which are placed under the thousands' and hundreds' columns. Hence the sum is 2254, Note 1. — The columns should be added a second time, beginning with the top, and, if the sums are not alike, this should be repeated till the sums agree. Note 2. — When the sum of any column is 10, 20, or any number ending with naught, the naught is placed under the column added, and the other number added to the next column. Note 3. — In adding do not name the numbers after the first one. Say 3, 9, 14, instead of 3 and 6 are 9 and 5 are 14. Note 4, — In adding United States money, write the numbers in columns, with the decimal points standing in a vertical line, and add as above. Place the decimal point in the sum directly under the points above, and prefix the dollar sign. Rule. — Write the numbers so that units of the same order shall be in the same column. Beginning at the right, add the columns, placing the units of the sum under the column added, and add the tens, if any, to the next column. Write the entire sum of the last column. WRITTEN EXERCISES. 17 47. Add and prove : 1. 2673 2. 837 3 . 628 4. 8063 846 2964 4307 259 1025 418 526 8264 92 3825 8279 1287 837 842 428 428 642 29 4273 3064 5. $26.43 6. $715.30 7. $ 165.00 8, . $ 20.863 18.75 21.86 8.429 129.40 2.93 9.246 113.82 5208.00 4.10 10.163 6.804 .926 128.06 7.208 39.625 128.753 663.13 516.00 11.31 37.15 28.00 8,096 476.203 192.097 9. 4286 10. 3184 11. 2306 3804 2929 1242 9273 3641 8936 6518 8207 6084 8274 9243 8346 2936 6041 2920 8142 2938 1289 9370 8465 9016 6425 3721 2914 3184 4936 8563 6293 8749 8472 12. 178469 13. 729476 14. 428756 738527 835694 937524 592946 209731 129305 846953 569420 747925 362431 182134 165213 234234 234123 243121 18 ADDITION. 16. $15,684 17.326 28.11 39.487 24.16 29.312 43.291 16. $11,164 21.178 31.486 23.12 34.196 29.394 84.684 17. 16.98 13.043 1.29 31.751 48.006 28.775 8.44 18 19. 20. 21. 22. $19,723, $5.80, Add 2468, 9416, 7843, 6974, 1306. Add 395, 25682, 50600, 39, 48732. Add 639, 746, 892, 948, 769, 498. Add 2493, 79621, 98725, 16053, 972341, 28739. Add $21.98, $17,543, $2.17, $1554, $.155, $3.82, $ 1.756. 23. Add $39,412, $17,694, $34,006, $6.94, $2.97, $4.62. Add and prove : 24. 2786 25. 1947 1838 5287 6496 8035 2956 8375 1698 3263 28. The sum of 369, 298, and 492, added to the sum of 1628, 38, and 297, equals what ? 29. Add seventeen thousand, nine hundred six; four thousand, two hundred eighty-nine; eight hundred twelve thousand, seven hundred eight; six hundred two; forty- two thousand, nine hundred two ; twelve thousand. 5278 26. 43562 27. 22879 4492 84601 43012 8913 92873 92874 6472 26461 69154 7258 30725 82738 5603 92837 56425 8450 68154 78997 7921 15210 37684 2864 2835 1872 1342 5832 5832 WRITTEN EXERCISES. 19 30. Find the sum of eleven thousand, six hundred seven- teen; sixty-eight; four thousand, twenty-five; two thou- sand, three hundred nine; eighty-five thousand. 31. A merchant's sales were $2963.84 in January, $ 1463.27 in February, $ 3846.25 in March, and $ 2016.92 in April. What did his sales amount to in the four months ? 32. Find the sum of all the numbers between and including 167 and 174. 33. A man had $ 170 in his pocket, which was $ 70.75 less than he had in his safe. How much money had he in the safe? 34. Add nine dollars six cents ; fifteen dollars seventy- two cents; sixty dollars eighty-seven cents; fifty-nine cents ; four dollars five cents ; two hundred dollars thirteen cents. 35. Find the cost of the following articles : coal, $10.50; sugar, $4.38; flour, $13.72; wood, $5.28; pork, $12.93; beef, $16.05; potatoes, $15.97; apples, $9,875; clothing, $46,195. 36. The expenses for one year for a family of four persons were as follows : table, $ 375 ; fuel and light, $ 125 ; physician, $48 ; car fare, $25 ; clothing for man, $ 95 ; for wife, $ 125 ; for two children, $ 75 ; church expenses, $ 57 ; newspapers and magazines, $ 28 ; servant, $ 175 ; all other expenses, $ 148. What were the entire expenses for the year ? 37. A man bought a lot for $ 2125, erected a house upon it at a cost of $ 5486, paid taxes amounting to $ 58, and insurance amounting to $ 75. He .desires to sell his prop- erty at an advance of $ 1200. What shall he ask for the house and lot ? 20 ADDITION. 38. Find the total population of the five largest cities in the world. 39. A speculator bought potatoes for $ 2680, corn for $ 5870, apples for $ 1596, wheat for $ 7942, oats for ^ 6793, barley for $ 3978, and sold his entire purchase at a profit of $ 2984. What did he receive ? 40. A, B, C, and D are partners in the dry goods business. A has put in $ 5825 ; B, $ 3246 more than A ; C, as much as A and B together ; and D, as much as all the others. What is the entire capital of the firm ? 41. How far will a man ride a bicycle in a week if he travels 75 miles on Monday, 84 miles on Tuesday, 86 miles on Wednesday, 100 miles on Thursday, 95 miles on Friday, and 101 miles on Saturday ? 42. The landing of the Pilgrims occurred 128 years after the discovery of America by Columbus ; the Declaration of Independence followed in 156 years ; Washington was made President 13 years later and 72 years before the Civil War broke out. In what year did the Civil War begin ? 43. The population of Minnesota in 1890 was 1,301,826, of Iowa 610,070 more than Minnesota, and of Missouri 767,288 more than Iowa. What was the total population of the three states ? 44. Por how much must I sell a horse, a cow, and a pig, that cost me $ 125, $ 40, and $ 4 respectively, if I gain $ 35 on the horse, $ 16 on the cow, and $ 1.50 on the pig ? 45. Five barrels of sugar weighed respectively 348, 327, 354, 335, and 329 pounds. What was the weight of the whole ? 46. In an election, A received 27,423 votes, B, 19,804, and C, 5366 votes more than both A and B. How many votes did all three together receive ? WRITTEN EXERCISES. 21 47. I have $ 12,450 invested in bonds, $ 15,850 in busi- ness, $32,745 in real estate, and $4395 in bank. How much do I have in all ? 48. The area of the United States is 3,556,300 square miles, of Canada 3,470,000 square miles, and of Mexico 767,000 square miles. What is their total area ? 49. If it takes 1254 feet of lumber to build a sidewalk, 2248 feet to repair a htmse, 25,235 feet for a barn, and 5160 feet for a fence, how many feet will it take for all ? 50. Six loads of hay weighed respectively 1942, 2126, 2049, 1807, 1645, and 2214 pounds. How many pounds of hay in all ? 51. 816 52. 7016 53. $16.75 54. $24,076 391 5609 29.48 18.92 175 2854 45.22 45.805 762 6327 19.80 7.19 549 4005 32.15 19.843 433 1963 14.05 3.457 257 842 11.63 80.76 624 396 20.49 22.40 988 1901 10.72 64.068 8450 8.14 30.12 55. 18324 56. 88632 57. $ 29.845 16309 79045 107.16 92745 129836 95.423 47528 95207 16.95 9216 47658 75.09 16127 21030 40.206 4556 19019 148.75 65901 410803 862.60 18104 36521 104.494 90027 17084 53.51 51042 SUBTRACTION. 48. 1. A farmer had 9 sheep and sold 4 of them. How many sheep had he left ? 2. Annie is now 8 years old. How old was she 5 years ago? 3. John had 11 marbles and gave away 6 of them. How many had he left ? 4. Henry lives 10 miles north of the city, and James 7 miles north. How far is it from James's house to Henry's? 5. There are 12 pnpils in a class. Five of them are boys. How many are girls ? 6. A boy picked 9 quarts of cherries, and sold 6 quarts. How many quarts had he left ? 49. Subtraction is the process of finding the difference between two like numbers. 50. The Minuend is the number from which we subtract. 51. The Subtrahend is the number subtracted. 52. The result in subtraction is called the Difference or Remainder. 53. The Sign of Subtraction is a short horizontal line — . It is called rainus, and when placed between two numbers signifies that the second is to be subtracted from the first. Minus means less. 12 — 8 = 4 is read twelve minus (or less) 8 are 4. 22 ORAL EXERCISES. 23 Oral. 7. Tom is 20 years old, and Jack 12. How much younger than Tom is Jack ? 8. There were 14 eggs in a nest, but 6 have been taken away. How many eggs are left ? 9. I bought a picture for $ 20, and sold it at a loss of $ 5. What did I sell it for ? 10. What number taken from 16 leaves 9 ? 11. A girl bought a bill of goods amounting to f 11. She gave the merchant a ten-dollar bill and a five-dollar bill. How much change should she receive ? 10-3 13 - 5 9-4 14-7 12-5 16-7 15-12 17-7 20-12 18-12 22. 10-f? = 15 15-10 = ? 9 + ? 23. 11 + ? = 1T 17-11 = ? 8 + ? = lT 17-8 = ? 24. 12 + ? = 18 18-12 = ? 7 + ? = 20 20-7 = ? 25. 6 + ? = 15 15-6=? 5-f? = 17 17-5 = ? 26. Subtract by 2's from 40 back to 0. 27. Subtract by 3's from 27 back to 9. 28. Subtract by 4's from 30 back to 10. 12. 12-8 15-5 13. 15 - 10 8-2 14. 7-2 8-3 15. 16-9 15-8 16. 10-3 11-4 17. 18-5 17-6 18. 17-10 16-11 19. 19-5 18-6 20. 20-10 20-11 21. 18-10 18-11 16-10 9-4 12-6 16-6 10-6 11-5 13-6 12-7 13-6 14-8 15-8 14-6 14-- 2 13-3 16-8 15-9 20-13 20-14 18-13 18-14 ? = 14 14-9 = ? 24 SUBTRACTION. 29. Subtract by o's from 68 back to 18. 30. Subtract by 6's from 90 back to 30. 31. Subtract by 7's from 35 back to 0. 32. Subtract by 8's from 49 back to 1. 33. Subtract by 9's from 84 back to 12. 54. Principles. — 1. Only like numbers can be sub- tracted. 2. The sum of the subtrahend and remainder must equal the minuend. 34. Find the difference between 987 and 563. Minuend, 987 Solution. — We write the less number under Subtrahend 563 ^^^ greater, with units of the same order in the 15 . -I 7 m ^ > -^ li ^ ^1 95 A prime number used as a factor is a Prime Factor. 6. 3 and 5 are prime factors of 15, but 4 and 6 are not prime factors of 24. Why not ? Note. — Every number may be multiplied by 1 to produce itself. While both are factors, according to the definition, they are not so used in practice. 61 62 FACTORS. 7. Name the prime factors of 20, 18, 24, 25, 27, 38, 34, 48, 49, 63. 96. A number that is divisible by 2 is an Even Number. Thus, 8, 6, 10, 14, are even numbers. 8. Name all the even numbers from 1 to 40. 97. A number not divisible by 2 is an Odd Number. Thus, 3, 5, 9, 11, are odd numbers. 9. Name the odd numbers from 1 to 40. 98. Every prime number except 2 and 5 ends with 1, 3, 7, or 9. 10. Name a prime number ending with 1, with 3, with 7, with 9. 11. Name a composite number ending with 1, with 3, with 7, with 9. 2 is an exact divisor of any even number, or of any num- ber ending with an even number, or with 0. 3 is an exact divisor of any number the sum of whose digits is divisible by 3. 4 is an exact divisor of any num- ber when the number expressed by its two right-hand figures is divisible by 4. 5 is an exact divisor of any number ending with or 5. Separating a number into its factors is called Factoring. 99. Written. 12. Find the prime factors of 60. ^ Lr2: Solution. — We first divide by tlie prime number 2. 2 30 The quotient, 30, being even, we also divide by 2. The 3 15 quotient is 15, which we divide by 3, giving a quotient 5 5 of 5, which we divide by 5. The last quotient is 1. -j Hence the prime factors of 60 are 2, 2, 8, 5. Rule. — Divide the given niimher by cpiy prime number that will exactly divide it. Divide this quotient by any prime number, and so continue until the quotient is 1. The several divisors are the prime factors. WRITTEN EXERCISES. 63 Find the prime factors : 13. 63 18. 720 23. 2431 28. 13104 14. 84 19. 1572 24. 2310 29. 11550 15. 250 20. 1872 25. 7007 30. 17325 16. 210 21. 2800 26. 3150 31. 64384 17. 636 22. 2310 27. 3465 32. 10323 CANCELLATION 33. What is the quotient of 42 -^ 21 ? 34. What is the quotient of 3 x 2 x 7 divided by 3 x 7 ? 35. How many times is 3 times 5 contained in 6 times 5 ? 3 times 8 in 12 times 8 ? 36. Divide 12 x 16 by 4 x 16. 9 x 11 by 3 x 11. 37. Divide 18 x 7 by 9 x 7. 38. In the last example what factor is found in both dividend and divisor ? 39. Would the quotient be the same if the factor 7 were rejected from both dividend and divisor ? 100. Principles. — 1. Eejecting a factor from a number divides the number by that factor. 2. Eejecting the same factors from both dividend and divisor does not affect the quotient. (See Principles of Division.) 101. Rejecting the same factors from both dividend and divisor is called Cancellation. 102. 1. Divide 8 X 18 X 15 X 7 by 4 X 6 X 11 X 9. Solution. — We first indicate the division by writing the dividend over the divisor with a line between. Since 4 2^5 and 6 are factors of 8 and 18, respectively, they p X ]Lp X Jp X i may be omitted, or cancelled, from both divi- ^ X ^ X 11 X f^ dend and divisor. Since 8 in the dividend is 15 a factor of 9 in the divisor, it is cancelled from 64 CANCELLATION. both, leaving 3 in the divisor. 3 in the divisor, being a factor of 15 in the dividend, is cancelled from both. The product of the uncancelled factors in the dividend is 70, and in the divisor 11. The quotient is therefore |^, which equals 6^. Indicate, and find quotients by cancellation. 2. Divide 36 x 27 x 49 x 38 x 50 by 70 x 18 x 15. 3. (28 X 38 X 48) - (14 X 19 X 24 X 2 X 2) = ? 4. (26 X 5 X 54) - (13 X 5 X 6) = ? 5. What is the quotient of 36 x 48 x 16 divided by 27 X 24 X 8 ? 6. Divide 5 x 45 x 7 x 20 by 49 x 5 x 4 x 9. 7. Divide 5 x 51 x 7 x 9 x 4 by 17 x 20 x 12 x 7 x 2. 8. Divide 25 x 2 x 72 x 14 by 6 x 9 x 120. 9. How many bushels of potatoes at 50 cents a bushel must be given in exchange for 15 pounds of tea at 40 cents a pound ? 10. If 60 yards of cloth cost $120, how many yards can be bought for $40? 11. 15 oranges cost 45 cents. How much will 7 oranges cost? 12. A dairyman sells 100 quarts of milk daily at 5 cents a quart. How many bushels of corn at 45 cents a bushel can he buy with 10 days' milk receipts ? 13. A farmer sold a grocer 45 bushels of apples at 50 cents a bushel, taking his pay in flour at 90 cents a sack. How many sacks did he receive ? 14. In what time can a boy, at 60 cents a day, earn as much as a man can earn in 40 days at $ 3 a day. Note. — Change the $ 3 to cents. 15. If 32 quarts of chestnuts cost $ 2.50, what will 800 quarts cost ? WRITTEN EXERCISES. 65 16. There are 16 ounces in a pound. 30 pounds of steel will make how many horseshoes, each weighing 6 ounces ? 17. A man sold 15 cords of wood at $ 6 a cord and received payment in wheat at 90 cents a bushel. How many bushels of wheat did he receive? 18. If I buy 10 yards of cloth at f 2 a yard, and pay for it in wool at 50 cents a pound, how many pounds of wool will it require ? 19. If wood is worth ^4 a cord, and coal $5 a ton, how many cords of wood will pay for 20 tons of coal ? 20. Divide the product of 18, 6, 9, and 4 by the product of 10, 7, 6, and 2. 21. Divide the product of 10, 75, 9, and 96 by the product of 5, 12, 15, and 9. 22. Find the quotient of 51 times the product of 54 and 12 divided by 36 times the product of 17 and 3. 23. How many jars of butter each containing 10 pounds at 20 cents a pound must be given for 10 sacks of granulated sugar each containing 5 pounds at 5 cents a pound ? 24. A grocer sold 20 boxes of soap, each containing 100 packages at 4 cents a package, and took in payment hay at ^ 16 a ton. How many tons did he receive ? GREATEST COMMON DIVISOR. 103. 1. What number will exactly divide 12 and 15? 12 and 36 ? 15 and 20 ? Note. — When two or more numbers have the same factor, it is called a common factor of those numbers. 2. Name a common factor of 9, 12, and 15. 3. What factor is common to 15, 20, and 25 ? What factor is common to 14, 21, and 28? 4. Name two common divisors of 10 and 20. 5. Name the greatest factor that is common to both 18 and 30. 66 GREATEST COMMON DIVISOll. 104. A number that is a factor of two or more numbers is called a Common Divisor of them. Thus, 5 is a common divisor of 10 and 15. 105. The greatest factor of each of two or more numbers is called the Greatest Common Divisor of them. Thus, 6 is the greatest common divisor of 18 and 24. 106. When two or more numbers have no common factor or divisor, they are Prime to each other. Thus, 8 and 15 are prime to each other. 107. Principle. — The greatest common divisor of two or more numbers is the product of all their common prime factors. V,. Written. 1. What is the greatest common divisor of 90 and 150. 90 = 3x3x5x2 Solution. — The prime factors common to both 60 and 150 are 2, 3, and 5. And since the greatest common divisor of two or more numbers is tlie product of their common prime factors, 30 is the greatest common divisor of 90 and 150. 150 = 2x5x5x3 2 X 3 X 5 = 30, Ans. OR 2 90 150 5 45 75 3 9 15 3 5 2 X 3 > : 5 = 30, Ans. T'ind the greatest common divisor : 2. 84, 132 7. 40, 60, 80 12. 45, 60, 90 3. 63, 42 8. 64, 144, 560 13. 36, 72, 81 4. 90, 105 9. 36, 48, 24 14. 44, 121, 132 5. 112, 168 10. 40, 56, 72 15. 63, 126, 189 6. 132, 156 11. 18, 54, 32 16. 36, 81, 135 WRITTEN EXERCISES. 67 108. To find the greatest common divisor when the numbers cannot be readily factored. 17. What is the greatest common divisor of 510 and 935 ? Solution. — The greatest common divisor must be a factor of both these numbers. It cannot be the larger. It is not the smaller, for we find a remainder of 425 610)935(1 after dividing the larger by gj^Q the smaller. If the remainder 425 is a 425)510(1 factor of 510, it will be the 425 greatest common divisor of _ ^ ^ -^ ,.. . ~77\.or/i? 425 and 510, and therefore of Greatest Common Divisor 85)425(5 ^-.^ -, nor tj ^ v • / ^ 610 and 935. But it is not, ^^'^ for we find a remainder of 85 after dividing 610 by 426. If the remainder 85 is a factor of 425, it will be the greatest common divisor of itself and 425, also of 426 and 610 ; also of 510 and 986. We find that 85 is a divisor of 425. It is therefore the greatest common divisor of 510 and 935. Note 1. — An exact divisor of a number is an exact divisor of any number of times that number. Note 2. — An exact divisor of each of two numbers is an exact divisor of their sum and of their difference. Rule. — Divide the greater number by the smaller, and the last divisor by the last remainder until there is no remainder. Tlie last divisor will be the greatest common divisor. If more than two numbers are given, find the greatest common divisor of two of them, then of this divisor and a third number, and so on. The last divisor will be the greatest common divisor. Find the greatest common divisor : 18. 270,810-" 22. 504, 560 f ^ 26. 120, 180, 240 19. 360, 420 23. 646,950 ^^ 27. 140,280,420 20. 294, 567 . ' 24. 216, 324 1^ < 28. 288,432,1152 21. 264, 312j4 25. 300, 480 fpO 29. 225, 360, 405 30. Find the greatest common divisor of 72, 153, 315, 2187. (\ 68 LEAST COMMON MULTIPLE. 31. I have 32 bushels of wheat, 48 of barley, and 128 of oats. I desire to put all this grain into boxes of the largest possible size, so that no box shall contain more than one kind of grain. How many bushels must each box con- tain? \i. There will be how many boxes of wheat? Of barley? Of oats ? LEAST COMMON MULTIPLE. 109. 1. Name a number of which 3 is a factor. Of which 5 is a factor. 2. Name several numbers that are exactly divisible by 2. By 7. By 5. 3. Name a number that is exactly divisible by both 3 and 2. Name another. Another. Another. What is the least number that is exactly divisible by 3 and 2 ? 4. What is the smallest number that will exactly contain 5 and 6 ? 110. A Multiple of a number is a number that exactly contains it. Thus, 5, 10, and 15 are multiples of 5. Note. — Pupils sometimes mistake multiples for factors. A multiple is a product. A factor is a divisor. 111. A Common Multiple of two or more numbers is any number that exactly contains each of them. Thus, 60 is a common multiple of 4, 5, and 6. 112. The Least Common Multiple of two or more numbers is the smallest number that exactly contains each of them. Thus, 30 is the least common multiple of 3, 5, and 6. DBB^INITIONS. 69 113. Principle. — The least common multiple of two or more numbers is the product of all the prime factors *in the largest number multiplied by the product of such prime factors of the other numbers as are not found in the largest. 114. Written. 1. What is the least common multiple of 21, 28, and 30 ? Solution. — Separating the numbers into their prime factors, and multiplying the product of the prime factors of the largest number, 2 X 3 X 5 = 30, by the product of the 21 = 3 X 7 prime factors of the other numbers not Qo _ 2 y 2 y 7 found in the largest, we have 2x7 = 14. QA ~ o Q K Therefore 14 x 30 = 420, the least com- 30 = 2 X 3 X O jjjQjj multiple. The prime factors that 2x3x5x2x7= 420. enter into this least common multiple are 2, 3, 5, 2, 7. The least common multiple must contain all the factors of 30 (2 X 3 X 5) or it would not contain 30. It must contain the prime factors of 21 (3 x 7). 3 is also a prime factor of 30, and is not again included, but 7, not being a factor of 30, must be included in the least common multiple, or it will not contain 21. Of the prime factors of 28 (2x2 x7), there are two 2's. Since the largest number has but one factor 2, the factor 2 must be again included in the least common multiple, or it would not contain 28. The factor 7 is excl\ided because it is also a factor of 21. We now find that all the factors of the three numbers are found among the factors of the least common multiple 2x3x5x2x7. The practical method of finding the least common multiple is as follows : Solution. — We divide^ by any prime number that is contained in two or more of them, and the quo- tients and undivided numbers again in like manner, until the remaining quotients are prime to each other. "J 2 5 The product of all the divisors and the last quotients will be the least 2x3x7x2x5 = 420. common multiple. 2 21 28 30 3 21 14 15 7 7 14 5 70 LEAST COMMON MULTIPLE. Find the least common multiple : 2. 18, 27, 30 6. 36, 40, 48 10. 24, 42, 54, 360 3. 9, 12, 18 7. 18, 24, 36 11. 25, 20, 35, 40 4. 16, 48, 60 8. 15, 30, 21, 28 12. 14, 21, 35, 45 6. 21, 27, 36 9. 15, 60, 140, 210 13. 24, 48, 96, 192 14. Find the contents of the smallest box that may be filled with wheat by using a 4-quart, a 5-quart, or a 6-quart measure. How many 4-quart measures will fill it ? How many 5-quart measures ? 6-quart ? 15. Three boys ride around a circular track. A goes around once in 5 minutes, B once in 8 minutes, C once in 10 minutes. If they start together, how many minutes must elapse before they all come together at the starting- point ? How many times will each have gone around the circle ? 115. Review of Factors, Multiples, Divisors, and Cancel- -iation. 1. Define factor, composite number, prime number, and prime factor. 2. Find the prime factors of 5075; of 9576; of 3150; of 6006. 3.' Find the sum of the prime factors of 34650. 4. Find the prime factors of 2310 ; of 17199 ; of 6840. 5. 81158 is the product of what prime factors ? 6. Find the largest prime factor of 12600. 7. What is a common divisor of two or more numbers ? 8. What is the greatest common divisor of two or more numbers ? 9. When are numbers prime to each other ? REVIEW. 71 Find the greatest common divisor of : 10. 672 and 960. 13. 1650 and 1920. 11. 616 and 1012. 14. 696, 1218, and 1160. 12. 272 and 428. 15. 450, 720, and 810. 16. What is the greatest prime factor common to 4242 and 2626 ? 17. A grocer had 84 bananas and 126 lemons, which he wished to put into bags, each bag containing the largest number possible, and each containing the same number. How many could be put into each bag ? 18. A man Has three fields containing respectively 14, 18, and 22 acres. He wishes to cut them into the largest pos- sible lots of equal size. How much land will each lot con- tain ? How many lots will each field contain ? 19. What is a multiple of a number ? A common mul- tiple ? The least common multiple ? Find the least common multiple of : 20. 96, 196, 42, and 54. 23. 252, 462, and 1092. 21. 45, 36, 70, and 90. 24. 120, 280, and 308. 22. 36, 40, 42, and 48. 25. 36, 110, 98, and 66. 26. Find the least common multiple of the even numbers to and including 20. 27. What is the least sum with which I can buy an exact number of chairs at $ 6, $ 8, or $ 5 each ? 28. What is the smallest sum of money that may be expended by using an exact number of nickels, dimes, quar- ters, or 3-cent pieces ? How many pieces of each kind will the sum contain ? 29. John can run around a block in 6 minutes, James in 8 minutes, and Henry in 9 minutes. If they start together, how long before they will all be together again at the starting-point ? 28 x56 x30 14x3 x5 28 x32 x7 u x35 x2 34 X 9 x5 72 EEVIEW OF FACTORS. 30. What is the shortest piece of rope that can be cut into pieces 32, 36, and 44 feet long ? 31. What is cancellation ? 32. Of what use is cancellation ? Find results of the following by cancellation : 33. ^^ X '^6 X 48 24 X 6 X 12 34. lO^xAxi 37 4x6 35. ^^X^X^^X^ ' 38. 6x8x2 25x17x3 39. 240 X 48 X 70 X 18 --42 x 15 x 54 X 7 = ? 40. Divide the product of 25, 14, and 11 by the product of 15, 7, and 22. 41. How many bushels of wheat at $1.10 a bushel must be given for 6 pieces of cloth each containing 33 yards at 50 cents a yard ? 42. How many cords of wood at $3 a cord will pay for 30 lb. of sugar at 5 cents a pound ? 43. If 8 men can do a piece of work in 6 days, in how many days can 12 men do it ? 44. How many pounds of sugar can be bought for $7 if 21 lb. cost $1.05? 45. How many pounds of maple sugar at 12 cents a pound must a farmer exchange for 15 pounds of coffee at 24 cents a pound ? 46. A milkman exchanges 8 cans of milk, 30 quarts in a can, at 4 cents a quart, for 3 pieces of sheeting, 40 yards in a piece. What is the price of the sheeting per yard ? COMMON FRACTIONS. 116. 1. When any whole thing, as an apple, is divided into two equal pieces, what part of the whole will each piece be ? 2. If anything is divided into 3 equal parts, what is each part called ? Into 4 equal parts ? Into 5 equal parts ? Into 8 equal parts ? 3. One of the two equal parts of an apple is one half of it. One half is written ^. 4. One of the two equal parts of a number is \ of the number. 5. How many are ^ ^f 4 oranges ? -I- of 20 cents ? -J of 10 ? 6. One of the three equal parts of anything is one-third of it. One-third is written \. How many are ^ of 6 men ? \ of 12 dollars ? i of 18 days ? i of 24 ? 117. One or more of the equal parts of a unit is called a Fraction. The unit of which the fraction is a part is called the Unit of the Fraction. One of the equal parts is called a Fractional Unit. Two or more fractions having the same fractional unit are Like Fractions. 73 I f 74 COMMON FRACTIONS. 118. A Fraction is written with two numbers, one above the other, with a line between them ; as \. The number below the line is the Denominator, and it shows into how many equal parts the unit is divided. Thus, in the fraction -|, 8 is the denominator. The number above the line is the Numerator, and it shows how many of the parts are taken. '- Thus, in the fraction f , 5 is the numerator. xy^^tJ^^n^ ^ r '^ ^ V '^f li6 numerator and denominator are called the Terms of a Fraction. Thus, 3 and 4 are the terms of f. ^ 119. A Common Fraction is a fraction written with its numerator above its denominator with* a line between. •' 120. A Proper Fraction is a fraction whose value is less than 1. Its numerator is less than its denominator; as 4 3 7 7? ¥> 8- * 121. An Improper Fraction is a fraction whose value is 1. or more than 1. Its numerator is equal to or greater than its denominator ; as |, |, -y-. 122. An integer may be written in fractional form by giving it 1 for a denominator, when it becomes an improper fraction. Thus, 5 = 1, 4 = f •^ 123. A number composed of an integer and a fraction is a Mixed Number. Thus, 31 12|. 124. A fraction is indicated division, the numerator being the dividend, and the denominator the divisor. Thus, I means 3 h- 4. -i^ means 12 -r- 3. REDUCTION OF FKACTIONS. _ 75 125. The Value of a Fraction is the quotient of the numerator divided by the denominator. Write in figures : 7. Four sevenths. 10. Seventeen eighteenths. 8. Five eighths. 11. One twenty -fourth. 9. Nine sixteenths. 12. Eight forty-seconds. 13. Nine and seven tenths. 14. Twenty-five and eight elevenths. 15. Fourteen and seven ninths. Eead the following : 16. \ 19. \\ 22. ¥ 25. f 28. 17. fV 20. A 23. \\ 26. If 29. 18. 3%V 21. 1 24. 1 6 2'3 27. 1 5 9""0 30. 120 TTTOT 3 _2_ 1"(5"1 REDUCTION OF FRACTIONS. 1. In one apple there are how many halves ? How many fourths ? How many eighths f 2. In |- of an apple, how many fourths? How many eighths f 3. How many eighths in J of an apple ? 4. In one apple there are how many thirds f How many sixths ? 5. In 1^ there are how many sixths f How many imiths 9 In J how many sixths f How many ninths f 6. Name a fraction that is equal to i. Name a fraction equal to \. To \. 7. Change | to halves. Change | to thirds. 8. Change |^ to fourths. To eighths. 76 COMMON FRACTIONS. 9. Change ^ to sixths. To ninths. 10. Change | to sixths. To ninths. 11. Express f in larger terms. What operations did you perform ? Express | in smaller terms. What did you do ? Reduction of fractions is the process of changing their form without changing their value. 126. To higher terms. 12. Eeduce f to sixteenths. Solution. — Since we must change 4ths to 16ths, the new denomi- nator must be 4 times the given denominator. Since 3x4 12 tlie new denominator will be 4 times the given 4x4 16 denominator, the new numerator must be 4 times the given numerator. Therefore, multiplying both terms of the fraction | by 4 gives i|. J Principle. — Multiplying both terms of a fraction by the same number does not change the value of the fraction. Change the following : 13. ftolOths 19. fto27ths 14. I to 9ths 15. ito30ths 16. f to 18ths 17. |tol2ths 18. I to 24ths 20. I to 56ths 21. -i-to48ths 22. fto21sts 23. ^ to 50ths 24. i5_to22ds f to 84ths 25. 4 to 25ths 26. 27. ii to eOths 28. I to 72ds 29. 1 to 63ds 30. |to40ths To reduce a fraction to higher terms. Rule. — Multiply both terms of the fraction by the same * number. Note. — To find the multiplier, divide the required denominator by the given denominator. REDUCTION OF FRACTIONS. 77 Written. Change the following: 31. iJto96ths 35. ||tol40ths 39. |f to 500ths 32. I to 64ths 36. ^ to 150ths 40. |-J to 168ths 33. ^% to 75ths 37. If to 144ths 41. ij to 522ds 34. i|tol20ths 38. fi-tol28ths 42. iff to 9375ths 127. To lowest terms. A fraction is expressed in its lowest terms when the terms are prime to each other. 1. Change || to lowest terms. Solution. — Since we must change 16ths to 4ths, the new denomi- nator must be i of the given denominator. Since the 1_ -j- 4 _ 3 new denominator will be I of the given denominator, 16-5-4 4 the new numerator must be I the given numerator. Therefore, dividing both terms of ^f by 4 gives |. ^ Principle. — Dividing both terms of a fraction by the same number does not change the value of the fraction. Change to lowest terms : 2. f 6. 3. H 7. 4. 1 8. 5. A 9. 11 1 8 56 6T 42 10. 11. 12. 13. .3 6 T2 24 H 14. 15. 16. 17. 1 5 2T 18 ■2¥ 45 ¥¥ if 18. u 19. u 20. M 21. u y Written. ^^^Jl^^ Reduce |^ to lowest terms. — "^ =- Solution. — Dividing both terms of |-f by 5 we 4o -5- o y have f . Dividing both terms of f by 3 we have |. 6-5-3 2 Since the terms of f are prime to each other, they Q _s_ 3 3 are the lowest terms of f^. . a /. ^ ./ . ^ . 78 COMMON FRACTIONS. To reduce a fraction to its lowest terms. Rule. — Divide both terms by any common factor, and divide the result in the same way until the terms are prime to each other. If the terms are large numbers, divide by their greatest common divisor. Eeduce to lowest terms : 23. II 30. t\\ 37. tW^ 44. t¥A 24. tVs 31. iH 38. -Hi 45. m 25. a 32. 2% 39. m 46. AV 26. n 33. m 40. tW 47. t¥t 27. If 34. m 41. »¥t 48. iff 28. 5 5 35. 2¥^ 42. m 49. iff 29. aV^ 36. m 43. m\ 50. m 51. How many thirds in ffl? 52. Express in simplest form 98 divided by 392. 53. Change -^y-^j to a fraction whose denominator is 5. 54. Express in simplest form the quotient of 288 divided by 504. 55. What is the simplest form of -J-|| ? 128. Integers and mixed numbers to improper fractions. 1. In 2 apples how many halves? How many fourths? How msLTij fifths? 2. In 5 apples how many halves? How many fourths? fifths? eighths? 3. How many 4ths in 2 ? in 8 ? in 10 ? in 15 ? in 20 ? 4. How many fifths in 1 ? in 2 ? in 2i ? in 3| ? in 4f ? 5. Keduce 5 J to halves. 7^ to eighths. 4|- to sixths. 4^ to sevenths. 6. Change 4i to 9ths. 3| to 3ds. 5^% to lOths. 8f to 5ths. 7t\ to llths. REDUCTION OF FRACTIONS. 79 Reduce to improper fractions : 7. ^ 14. n 21. 8A 28. m 8. H 15. H 22. H 29. m 9. 3f 16. H 23. lOf 30. 111 10. H 17. n 24. iH 31. n 11. H 18. H 25. 12f 32. H 12. n 19. H 26. lOA 33. 12| 13. 4f 20. ^A 27. ii« 34. 10| 35. How many 9ths in 13 ? 36. Change 26 to a fractio^ial form. $7. How many fifths in* 32 ? in 24 ? in 16 ? in 50 ? 38. In 5^ weeks how many 7ths of a week ? 39. What improper fraction is equal to 11|^? Written. 40. Reduce 25 to fifths. ^^ . ^ J Solution. — In 1 there are ^. In 25 there Zo times ^ = -^- jj^^g^ ^^ 25 times f , or i|^. 41. Change 28 to sevenths. 42. Reduce 16^ to an improper fraction. 16^ 7 sevenths Solution. — Since in 1 there are ^, in 112 16 there are 16 times ^j or ip. 4 sevenths 116 sevenths, = i^ 6 To reduce an integer or mixed number to an improper fraction. / Rule. — Multiply the integer by the denominator, add the nu- merator of the fraction, if any, and write the residt over the denominator. 80 • COMMON FRACTIONS. Reduce to improper fractions : 43. 25i 50. 35tV 57. 59f 64. 238| 44. i^fi 51. m 58. 67f 65. 359A 45. 35A 52. 251 59. 4|? 66. iio^VV 46. 49tV 53. I^tV 60. 5if 67. 483^^ 47. 270f 54. 27A 61. 29A 68. 846ff 48. 19J 55. 4H 62. i^H 69. 359ff 49. 28A 56. 253V 63. l^A 70. 482if 71. In 560 how many otlis ? 72. Reduce 250 to IGtlis. 349 to 15ths. 73. Change 12| to 16ths. ^4| to 18ths. 74. In $ 730 how many fourths of a dollar ? 75. Change 156 to a fraction whose denominator shall be 12. 129. Improper fractions to integers or mixed numbers. 1 . How many dollars in 4 quarter-dollars ? In 8 quarter- dollars ? In 16 quarter-dollars ? In 20 quarter-dollars ? 2. How many dollars in $ -2/ ? In ^ Y_ ? In f 2_8 9 3. 12 fourths of a bushel are equal to how many bushels ? 36 fourths ? 40 fourths ? 4. 4 fourths of a dollar are equal to how many dollars ? 5. To how many dollars are 8 fourths of a dollar equal? 9 fourths ? 11 fourths ? Reduce to integers or mixed numbers : 6. f 11. i 16. ¥ 21. ¥ 26. H 7. 1 12. ¥ 17. ¥ 22. 124 27. ff 8. f 13. ¥ 18. ¥ 23. a 28. M 9. i 14. ¥ 19. 80 1 1 24. 46. 1 1 29 W 0. -¥ 15. ¥ 20. ¥ 25. a 30. In -3^ of a pound how many pounds ? 31. In -2^1^ of a dollar how many dollars? / REDUCTION OF FRACTIONS. 81 Written. 32. Reduce -^^^ to an integer or mixed number. 16 385'*'' ■'ort S0T.UT10N, — Since ^f equal 1, W^- will equal as — -— many times 1 as 16 is c'bntained in 385, or 24^ times. 65 54 ^^i^s. 24tV. ~1 To reduce an improper fraction to an integer or mixed number. Rule. — Divide the numerator by the denominator. Reduce to integers or mixed numbers : 33. ¥ 40. -w- 47. Hi^ 54. 3M^63 34. ^ • > 41. w 48. 4973 55. i_9_^69 35. W 42. w 49. X2J4 56. 8_8_^2 36. H 43. -w- 50. • Hr- 57. 38_0_00 37. If 44. w 51. 3|.24 58. 25001, 38. H 45. \¥ 52. 1806 59. -W/^ 39. a 46. -w 53. 3^|2 60. 8_y^6^ 61. How many bushels in ^^^ bushels ? 130. Fractions to fractions having a common denominator. 1. How many eighths in 1 ? In \ ? In J ? 2. How many sixths in 1 ? In ^ ? In J ? 3. How many twelfths in i ? | ? i ? 4. Write 1 and ^ as twelfths. As sixths. 5. Write \ and ^ as eighths. As twelfths. 6. Change \ and ^ so both may have 20 for a denominator. 7. Change -J-, i, and ^ each to 12ths. 8. Change | and J each to 24ths. v When fractions have the same denominator they are Like Fractions, and their denominator is called a Common Denominator. Thus, ^J, ^J, and |^} have a common denominator. 82 COMMON FRACTIONS. 131. The smallest common denominator of two or more fractions is their Least Common Denominator. Thus, If, ^|, and if become y^^, y%, and ■^-^, when changed to their least common denominator. The common denominator of two or more fractions is a common multiple "of their denominators. The least common denominator of two or more fractions is the least common multiple of their denominators. Written. 1. Reduce J and |- to fractions having a common denomi- nator. 3x6 18 Solution. — The common denominator must be a 4 x^ g 24 common multiple of the denominators 4 and 6, and since 24 is the product of the denominators it is a com- _ '*^ '* _. zH mon multiple of them. Therefore 24 is a common 6x4 24 denominator of | and |. f=|f,andf = ff. Beduce to fractions having a common denominator : q5 11 ifil234 2. hi 3. hi 4. hi 5. hi 6. hi 7. hi 8. hi 1 1 1 3"? 4? Z 3_ 5 12? 9' 10. I, f, f 17. 11. h h I- 18- 19 5 2 1 1 q 4 i _8_ _7_ 13. j\, 1 i 20. i, i, ^^, -/_ 14. h if 21. y%, f, I, I 1 K 7 5 4 22 i _2_ _3_ i ■'■°- ¥? 6? 9" '^''' 2? 11? 13' 3 23. Eeduce |, f, and -^-^ to fractions having the least common denominator. ^? ^> 1^ Solution. — The least common denominator 3, 3, 6 must be the least common multiple of the denomi- 1, 1, 2 nators 3, 6, and 12, which is 12. 2x3x2 = 12 j = ^ l = IS ^, = A 30. h h h 1 31. h tV rf tt? f 32. h IZ' 2"T' H REDUCTION OF FRACTIONS. 83 To reduce fractions to fractions having a common denominator. Rule. — Reduce each fraction to its lowest terms, divide the least common midtiple of the denoininators by the denominator of each fraction, and midtiply both terms by the quotient. Note. — If the denominators are prime to each other, their product will be their common denominator. Reduce to fractions having the least common denominator : 25. I, h fV 26. h h tV 97 4 5 3 28. T% 2h f 33. f ^\, A, 4 34. Which fraction is larger, ^ or J ? 132. Oral. 1. Add land i |., |, and f |, |, and f . |-, f, and f . 2. rind the sum of f, J, and f Qf f, ^, and f . Of |, |, and |. 3. Find the sum of f , |, |, and -i-. Of fV? tV A? and ^^g-. ^f tVj tVj tV. and |i. 4. In ^ how many fourths ? Add ^ and i. 5. ^ equals how many tenths? ^ equal how many tenths ? Add | and f . 6. ^ equals how many sixths ? |- ? Add ^ and ^. 7. Find the sum of ^ and \. Of i and ^. Of | and f 8. Add i I, and i. Add i -|, and i. Add |, |, and f 9. If I pay J of a dollar for breakfast and |- of a dollar for dinner, what will both meals cost me ? 84 COMMON FRACTIONS. 10. A boy paid f of a dollar for a book and :^ of a dollar for paper. How much did he pay ? 11. A owns ^ of a store, and B f . How many eighths do both own ? 12. John saves i a dollar a week, and Charles f of a dol- lar. How many fourths do both save ? 13. Henry gave -\ of his marbles to one boy and J of them to another. How many twelfths do both receive ? 14. A clerk sold i a pound of tea to one customer, J to another, and |- to another. How many eighths did he sell ? 15. A man pays ^ of his salary for rent, ^ for table expenses, and -f^ for clothing. What part of his money was expended for rent, table, and clothing ? 16. A farmer planted | of his seed in one field and f of it in another. What fraction represents the seed planted in both fields ? 17. A man paid $2^ for a hat and f 4i for shoes. What is the cost of both ? 18. Jack deposits $ 3\ in the bank, Elsie $ li, and Susie $ |. How much do all deposit ? Principle. — Only like fractions can be added. Written. 19. What is the sum of f, |, and -^^ ? Solution. — The least common multiple of the denominators is 48. l)ividing this by the denominator of each fraction and multiplying both terms of the quotient, f + f + T'6=if + M + H = -W- we have ff, If, and ||. i_03 _ 2 7. Ans. '^^^^ fractions are now like fractions, and are added by adding their numerators and placing the sum over the common de- nominator. Hence the sum is -L"^/, or 2^^^. ADDITIOK. 85 20. What is the sum of of, Ty^^, 6^^ ? ^f ~ •-* 3Tr Solution. — Since there are both integers and f rac- T-j^ = 7|^ tions to be added, we find first the sum of the fractions, 6rj^ = 6^-^ which is fl, or Iff. This is added to the sum of the 192 3 integers, 18. 18 + Iff = 19|f . Ans. 3 Rule. — If the fractions are not like fractions, reduce them to a common denominator, add their numerators, ayid place the sum over the common denominator. Reduce the result to lowest terms. If the result Is an improper fraction, reduce it to an integer or mixed number. When there are integers, or mixed numbers, add the frac- tions and integers separately, and unite their results. Find the sum of : 32. 1,1,8,1 33. i 3f,f,6 34. i,hn>T% 35. 4|-, y, 3, -^ 36. |,i,TV6 37. 7f,8f,f I OQ 4. 2 5 1 39. 7|,9i,6i,4| 40. lOi, 74,11,11 41. ^,^,l0i,48 42. 19f, 18f, 15i, 12tV 43. A man walked 21J miles on Monday, 27| miles on Tuesday, and 28^*^ miles on Wednesday. How far did he walk in the three days ? 44. A farmer owned three fields, containing respectively 42^^ acres, 36f acres, and 322^^^ acres. How many acres were there in all ? 21. h A, i 22. h h ^T 23. h h A 24. hhl 25. A, h f 26. iii.A 27. h A» TdJ i 28. 'S'? ■§■? 12 > g"!" 29. h h h i\ 30. 1% h h T5 31. ¥? T2"? 2"?? T¥ 86 COMMON FRACTIONS. 46. How many yards of cloth in four pieces containing 13|- yards, 21i yards, 31| yards, and 45| yards ? 46. John lives 241 rods from school, Harry 6/g- rods farther than John, and Thomas lOjf rods farther than Harry. How far does Thomas live from school ? 47. A farmer sold hay for $ 45J, oats for $ 15y%, corn for f 20|, and potatoes for $ 35|. What was the amount of his sales ? 48. I paid $ 2^ for car-fare, $ 4^ for cotton cloth, $ 5f for shoes, $ 25^ for a suit of clothes, and had $ 1 ^^^ left. How much money did I have at first ? 49. A boy was absent from school the first week of the term 12i hours, the second week 16f hours, the third week 8j7^ hours, and the fourth week 7^% hours. How many hours was he absent during the four weeks ? 50. I bought 31 tons of coal in January for $ 19^, 2J tons in February for $ 15|, and 5j\ tons in March for $ 30 j?^. How much coal did I buy in all, and what was the cost ? 61. James weighs 5S^^ pounds, William 65| pounds, Charles 67-J pounds, and their father as much as all three of them. How much does their father weigh ? 62. I bought of A 102% *^^s of hay, of B 18|-% tons, and of C 16| tons. How many tons did I buy in all ? 63. Four piles of wood contain respectively 24| cords, 18f cords, 27|- cords, and 30i- cords. How many cords in all? SUBTRACTION. Oral. 133. 1. From f take i- How much is 1 less f ? i less 5 ? ii-less^^? 2. From | take i- How much is ^ minus f ? | minus i ?|-iV? SUBTRACTION. -. 87 3. A boy had $ ^^ and spent $ ^. What part of a dollar had he left ? 4. A owns I of a store, and B |. How much of the store does A own more than B ? 5. John runs ^ of a mile, and Jerry | of a mile. Which runs farther, and what part of a mile ? 6. Mr. Ames owned | of a farm, and sold i of it. What part remained ? 7. What is the difference between | of anything and 4 of it ? Which is greater ? 8. Lucy has $ ^, and Alice $ |. Which has the more, and how much ? Subtraction. 10. f - i 15. f - J 20. 4 - I 11. i-i 16. A-i 21. 2-l| 12. i-i 17. J -i 22. 5-3| 13. i-i 18. J^-i 23. 4i-2i Principle. — Only like fractions can be subtracted. Written. 24. From f subtract |. Solution. — We find the least com- ^ ~" "g — T8 "■ Tt — TS' mon denominator of | and | to be 18. _3 ^ 1 jij^s 6 = t!' and I = tI- Their difference ' ^ ' is r\ or 1. 25. From 11^ subtract 4|. Solution. — We subtract the fractions and inte- 111 __ 10 s gers separately. After changing the fractions to 45 _ 45 their least common denominator, we have 11§ — 4|. — ~ I cannot be subtracted from |, hence we take 1 6^ of the 11 units, change it to sixths, and add the f , making 10|. lOf - 4| = 6| = 6^. Ans. 88 COMMON FRACTIONS. Rnle. — If the fractions are not like fractions, reduce them to a common denominator, and write the difference of their numerators over the common denominator. When there are integers, or mixed numbers^ subtract the fractions and integers separately. Note. — Mixed numbers may be changed to improper fractions, and subtracted as fractions. Subtraction. 26. i-l 34. 18i-4i 42. h-i. 27. l-i 35. 16 -If 43. ii-U 28. T^^-i 36. 16^-23 44. M-H 29. A-f 37. «-| 45. M-A 30. 16 -J 38. M-l 46. H-A 31. 31 -i 39. tV-I 47. M-i 32. 3i-i 40. If-oV 48. ii-^% 33. 8-21 41. \i-^\ 49. M-M 50. From 3846J take 2944f. Subtract : 51. 58627 52. 3169 f 53. 98701 54. 1000^2,. im^\ 30502^ 4963| 358A 55. From a cask of oil containing 42 gallons I sell 8i gallons, 17f gallons, and 5| gallons. How much oil remains in the cask ? 56. A rod is 16J feet. Take 12J feet from a rod. 57. From a cask of vinegar containing 43| gallons 17| gallons were drawn. How many gallons remained ? 58. A farmer having 217 bushels of wheat, sold 95JJ bushels. How many bushels had he left ? 69. The minuend is 123 J^ J, and the remainder 381^. What is the subtrahend? MULTIPLICATION. 89 60. From a piece of cloth containing 54| yards, 15^^ yards were sold at one time and 21J yards at another. How many yards were left? 61. If a grocer gained $ 1| by selling a barrel of flour for $ 6 j, what did it cost him ? 62. A lady bought a hat for $ 4|, shoes for f 5|, and some cotton cloth for $ 3^-^, and gave in payment a twenty- dollar bill. How much change should she receive ? M UL TI PLICA TION. OraL 134. 1. 4 times 1 apple are how many apples ? 7 times 1 apple ? 4 times 6 apples ? 2. 5 times 1-ninth are how many ninths? 6 times one- ninth ? 3 times | ? 3. 4 times J equals how many 6ths ? 5 times |? 8 times f ? 4. 5 times y\ = how many sixteenths. 5 times ^j — how many elevenths ? 5. 6 times 2% = ? 8 times f ? 10 times | ? 6. 3xJ=? 8xf = ? 4xi = ? 3xf = ? 7. How much is ^ of 4 ? i of 6 ? ^ of 10 ? i- of 12 ? 8. How much is i of 1 ? ^ofi? lofi? 9. How much is i of 1 ? ^ of ^ ?, ^ of 1. i of ^ ? 10. How much is | of |^ of an orange ? ^^ of |- of an orange? 11. At $ f each what will 8 books cost ? 12. If a horse eat |^ of a bushel of grain in a week, how much will 5 horses eat ? 90 COMMON FRACTIONS. 13. If a pound of tea costs 48 cents, what will J of a pound cost ? What will | of a pound cost ? 14. A owned -| of a ship, and sold ^ of his share. What part of the ship did he sell ? What part was left ? 15. What will i yard of ribbon cost at 16 cents a yard? ^ of a yard ? f of a yard ? 16. 8 boys earn $ f each. What do all earn ? 17. John earns $f, Henry ^ as much, and Edward ^ as much. What do Henry and Edward earn ? 18. John earns $ J, and Henry | as much. How much does Henry earn ? I of I = ? Since -^ of f = i | will be 2 times i or | = ^. A71S. Written. 19. Multixjly f by f. Solution. — i multiplied by | is the same as f § X - = — = -; of |. ^ of f = ^, and I is 3 times ^, or f. This o 4 20 5 solution, in effect, is the same as multiplying the J 3 3 numerators together for a new numerator, and ^^f K ^ J ~ K* the denominators for a new denominator. Cancellation shortens the process. -»/ Rule. — Reduce integers and mixed numbers to improper fractions. Multiply the numerators together for the numerator of the product, and the denominators for the denominator of the product. Cancel when possible. Two or more fractions joined by of form a Compound Fraction. The word of between two fractions is equivalent to the sign of multiplication. To change a compound to a simple fraction, multiply the fractions together. Thus,! of |of | = |x|x| = |. MULTIPLICATION. 9l Find the products : 20. |x J 29. i Of i of 1 38. 5| X 24 X 20 21. |x| 30. f of ,% of t 39. 7.} X 5f X f 22. Axf 31. ¥ X A X f 40. 9i X 3V X 2i 23. J of A 32. fV X 1 X 1 41. H X ii X A 24. t of U 33. Ax^xf 42. 3V X 4 X 51. 25. tV X T 4 34. \^ X 34 X 1 43. i\ X 80 X 5i 26. if xf 35. It X 90 X i 44. 3^ X 51 X T^ 27. ifxA 36. 8xf xf 45. f X 16 X If 28. iVx^^ 37. 16 X 1 X 1 46. 45 X 41 X t\ Find the value : 47. 1 of 40 51. 1 of 328 55. ij of 342 48. f of 42 52. f of 721 56. If of 800 49. 1 of 16 53. 4 of 90 57. j\ of 2222 50. A of 17 54. f of 131 58. 1 of 1632 59. Find the product of 124| by 5. 124f 5 Solution. — Multiplying the fraction and inte- •^t ger separately by 5, we have 5 times |= J^ = 3f, ^^Q and 5 times 124 = 620. 620 + 3| = 623|. Ans. 623f 5 X f = -V- = 3f Find the products : 60. 13i X 4 65. 16f X 30 70. 95| x 45 61. 184 X 10 66. 21-1 X 29 71. 64| x 81 62. 16f X 5 67. 451J X 15 72. 84f X 16 63. 28f X 9 68. 48J X 63 73. 34/y x 63 64. 48f x^8 69. 24| X 25 74. 4} x 18 92 COMMOJT FRACTIONS. 75. Find the product of 127 x 4J. 127 43 — ^ Solution. — Multiplying by the fraction and integer separately, we have f of 127 = 95^. 4 times 127 = 608. 508 + 95^ = GOSJ. Ans. 95} 508 603} of 127 = 95} Multiply : 76. 65 by 7| 77. 45 by 34 78. 83 by 5f 79. 72 X 16| 80. 84xlli^ 81. 26 by 9f 86. 3156 by ^ 82. 64 by 5J 87. 8165 by 7f 83. 89 by 5| 88. 4950 by 9| 84. 56 by 4f 89. 2835 by 16| 85. 92 by lOf 90. 5872 by 25f 91. ^offoff of |oft = ? ^.^^of^of-Vof A = ? 92. T^off|of,-«^ofifof| = ? |ofifof|of|| = ? 93. II X J of -f of ii of 42 = ? 51 X f of ^\ of I = ? 94. 4f of if of fl of if of I of if of 5j\ = ? 95. Mr. Brown earns $40 J a month, and his son | as much. How much does the son earn ? 96. At $ 12| a ton, how much will 9^-^ tons of hay cost? 97. What will be the cost of 48f yards of cloth at $ f a yard ? 98. A man gave 124Y^g- acres of land to his two sons, giv- ing f of it to the elder and f to the younger. How many acres did eaxih receive ? 99. If it requires 21| days for a man to dig a ditch, in what time can he dig ^ of it ? 100. A man owning Jg- of a cotton mill sold -fl- of his share. What part of the mill did he sell ? MULTIPLICATION. ' 93 101. In a school containing 945 pupils, j- of the number were boys. How many boys in the school ? 102. What is the cost of 15^ acres of land at $45| an acre? 103. Mr. Clark, owning f of a farm of 128 acres, sold his share at $ 45^ an acre. How much should he receive from the sale ? 104. At ^ J a bushel, what will | of f of a bushel of wheat cost ? 105. A lady purchased lOf yards of silk at $1|- a yard. What was the cost ? 106. I paid $150 for a horse, and ^J- as much for a carriage. What did the carriage cost ? 107. What will be the cost of a side of beef, containing 252 pounds, at 9^ cents a pound ? 108. The divisor is 15f and the quotient 21|. What is the dividend ? DIVISION, 135. 1. How many times is J contained in 1 ? Solution. — 1 equals f . Therefore ^ is contained in f , 4 times. t-^i = 4. 2. How many times is J contained in 1 ? Solution. — Since ^ is contained in 1, 4 times, f is contained in It I as many times. ^ of 4 times is 2 times, f -^ f = 2. 3. How many times is f contained in 1 ? Solution. — Since \ is contained in 1, 4 times, | is contained in It I as many times. ^ of 4 times equals f times. 4. How many times is ^ contained in 1 ? | in 1 ? ^ in 1? finl? finl? 94 COMMON FRACTIONS. 5. ^ js contained in 2 how many times ? Solution. — 2 equals |. | -r- 1- = 6. 6. I is contained in 2 how many times ? 7. 2-r-i = ? 4--i = ? 3--| = ? 3--f = ? 8. If you earn f of a dollar in a day, how long will it take to earn 7 dollars ? 9. At ^ I apiece how many books can be bought for $ 6 ? 10. If I save $ f a day, in how many days can I save $8? 11. How much is 4 divided by 2? Solution. — f -^ 2 is the same as ^ of |. ^ of | = |. Therefore fH-2 = |. 12. If 3 balls are worth -^^ of a dollar, what will 1 ball be worth ? 13. What will one book cost if 3 books cost | of a dollar ? 14. I divided -^j of my money equally among 4 boys. What part did each boy receive ? 15. Divide f by 3. i^ by 5. if by 5. 16. I is contained in | how many times ? Solution. — f is contained in 1, f times. Therefore it is contained in f , I of f times, or f times. Written. 17. I is contained in f how many times ? Solution. — | is contained in 1, | times. Since f is contained | times in 1, in I it is contained f of f times, or | times. Thus, we see that the divisor | has become inverted, and multipUcation performed. ^ Rule. — Multiply the dividend by the divisor inverted. Cancel when possible. Note. — Change integers and mixed numbers to improper fractions. DIVISION. 95 Find the quotients : 18. JH-I 23. S^^i-i 19. H^l 24. 6i^/^ Oft 5 _s_ 3 25. -^^5f y^^^ 21. if -I 26. 10 22. H^ft 27. 11^51 38. 4 X 3^5 of 3 = ? 28. 29. 8 30. 10 31. i 32. ii 5. 6 14 8 33. 2|-5-5^ 34. 7i-^li 35. 23H-H 36. 2J--3i 37. 8|H-9| Solution. — Inverting the divisor, indicating the operations, and cancelling, we have -• A71S. 5 ^ ^ 3 5 4 3 - X - X ^ X - 39. f of 9 -J- f of 6- 40. 4ofi|^|of4 41. i7^x|^Aof22 42. TT -tVx X* 43. 3^-1 X J of 2 44. 4x5x3^7| 45. Hx||xH-H 46. 8ix5i^|of| 47. Divide 31563 by 5. 5)31561(6312^ 30 ~T5 lf = J 6 5 If Solution. — When the integer of a mixed number is large, it may be divided as folio v^^s : 5 is contained in 3156|, 631 times, with a remainder of 1|. This remainder being divided by 5, gives -^^, which we place at the right of the quotient. Find the quotients : 48. 47f--7 49. 384f-!-5 50. 287^9j^~8 61. 3854-5-5 52. 139871 -I- 9 53. 897243 H- 15 54. 69834^-- 24 55. 969851^25 96 COMMON FRACTIONS. 56. Divide 3682 by 51 Solution. — When the dividend contains several figures and the Ki \ Qgco divisor is a mixed number it is often more convenient 2 2 ^^ divide as above. TT YT^OA ^^ multiply both dividend and divisor by 2, when i the divisor becomes 11 (halves), and the dividend 7364 boy j-y (halves). Dividing, the quotient is 669 j\. Principle. — Multiplying both dividend and divisor by the same number does not change the quotient. 3f ind the quotients : Si 57. 356 --4^ 61. 39846 58. 728 --8i 62. 44077 71- 59. 397 --5J 63. 76582 60. 296-- 101- 64. 28769 65. If 16 bushels of apples cost $ 8|, vrhat will 1 bushel cost? 66. Five heirs shared equally in the division of a legacy of $ 35,862|. What was the share of each ? 67. When 15 bushels of wheat sell for $ 17f, what is the price per bushel ? 68. The product of two numbers is 326|. One of the numbers is 5. What is the other ? 69. There are 5^ yards in a rod. How many rods in 3158 yards ? 70. If a man walks 15|^ miles a day, in how many days can he walk 155 miles ? 71. What is the price of coal per ton when 16 tons cost $73f? 72. How much does a man earn in a day if he earns $ 84J in a month of 26 days ? DIVISION. 97 73. When flour is $ 6| per barrel, how many barrels can be bought for $ 297 ? 74. At 6^ cents apiece how many tablets can be bought for $ 5 ? 75. If coffee is 37^ cents a pound, how many pounds can be bought for f 60 ? 76. If a boy can read 17J pages of a book in an hour, in how many hours can he read 175 pages ? 77. If IJ bushels of corn cost $1.^*^, what will 1 bushel cost? 78. How many books can be bought for $31^, if 1 book [costs $ 3^ ? >J A fraction having a fraction for one or both of its terms is called a Complex Fraction. Note. — Like all fractions, it is an expression of division. 42 79. Eeduce -^ to a simple fraction. 75 ^ 4| -M Solution. — 7! = A- Since all fractions indicate the division of the numerator by the denominator, JI means 1^ -^ ^. Dividing, we have ^-^ x 5^7 = ff . Ans. Therefore to simplify a complex fraction, divide the tiumerator by the denominator. Change to simple fractions : 80. li 82. 13 84. ii 86. 5i 88. i^LI « 4 16f J^ I of I 81. m 83. « 85. i 87. ^ 89. i^L3 i , 16 A A J 90. If f of an acre of land is worth $ 72^, what is the value of an acre at the same rate ? 9B COMMON FRACTIONS. 91. There are 5 J yards in a rod. How many rods in 70|- yards ? 92. At ^ 5 J a ton, how many tons of coal can be bought for I 73^ ? 93. If I of a yard of silk costs $ |, what will 1 yard cost? 94. How many bags will be needed to hold 92^- bushels of wheat, if 1 bag holds 2i bushels ? 95. At $ f per bushel, how many bushels of corn can be bought for $ 62^ ? 96. The product of three fractions is ^^g, and two of them are f and ii. What is the third. 97. What will 12|- yards of broadcloth cost if | of a yard costs $ 41 ? THE THREE QUESTIONS OF RELATION, 136. 1. 3 times 4 equals what ? A71S. 12. 2. 12 is how many times 4 ? Ans. 3. 3. 12 is 3 times what ? An,s. 4. In question 1, we have two factors, to find their product. In questions 2 and 3, we have the product and one factor, to find the other. 1. Form questions like 2 and 3, from the following state- ment: 5 X 6 = 30. a. |- of 8 equals what ? Multiplying 8 by |-, we have 4. Ans. h. 4 is ^ of what ? Since ^x8 = 4, 4-5-^ = 8. Ans. c. 4 is what part of 8 ? Since |x8 = 4, 4-8 = ^. Ans. QUESTIONS OF RELATION. 99 Principle. — The product of two numbers divided by one of them gives the other. To THE Teacher. — In such examples as question a, after the product is found, it may be used with each of the two numbers to form, successively, question b and question c. Drill upon these three ques- tions of relation should be so thorough that each question will suggest its own solution instantly. 2. i of 24 = what ? (Question a.) 3. After finding the product in example 2, form question b. Question c. 4. 8 is 1^ of what ? (Question b.) Solution. — From the question it is evident that 8 is the product of two numbers, and that ^ is one of them. Therefore, 8 -^ ^ = 24. 8 is I of 24. 5. What part of 24 is 8 ? (Question c.) Solution. — It is evident that 8 is the product of two numbers, and 24 is one of them. Therefore, 8 -4- 24 = -^j or f 8 is | of 24. Question a. Find result, and form questions b and c: 6. How much is f of 12 ? 9. f of 15 = ? 7. How much is f of 16 ? 10. ^ of 21 = ? 8. How much is I of 20 ? 11. | of 40 = ? Question b. Find result, and form questions a and c : 12. 15 is f of what? 15. 18 is -f- of what ? 13. 4 is I of what ? 16. 24 is | of what ? 14. 9 is f of what ? 17. 25 is | of what ? 100 COMMON FRACTIONS. Question c. Find result, and form questions a and h \ 18. What part of 24 is 8 ? 21. 21 is what part of 35 ? 19. What part of 18 is 12 ? 22. 28 is what part of 63 ? 20. What part of 9 is 2 ? 23. 15 is what part of 25 ? Find result, form the other two questions, and solve each : 24. I of bQ equals what? 28. How much is -^^ of 96 ? 25. What part of 49 is 14 ? 29. 38 is y2_ of what number ? 26. 26 is I of what ? 30. 16 is what part of 80 ? 27. 64 is what part of 120? 31. 18 is y% of what number ? Remark. — Each of the following problems contains one or more of the three questions of relation. Before attempting to solve any of them, the pupil should state the question in each of them. 32. James had 56 marbles, and John | as many. How many had John ? The question is. How much is f of 56 ? — a. 33. John had 42 marbles, which was f as many as James had. How many had James ? The question is, 42 is j of what ? — h. 34. James had 56 marbles, and John 42. John's marbles are what part of James's ? The question is. What part of 56 is 42 ? — c. 35. A man sold 50 acres of land, which was ^ of all he had. How many acres had he at first ? 36. A boy had 20 cents *and spent 15 cents. What part of his money did he spend ? What part was left ? 37. Mr. A has 640 sheep, and Mr. B -^^ as many. How many has Mr. B ? 38. f of a ton of hay cost $ 12. What was the cost of a ton? QUESTIONS OF RELATION. 101 39. I of a basket of eggs were sold for f 6. What was the value of the entire basket ? 40. At $ 45 an acre, how much land will $ 25 buy ? 41. If I of a factory is worth $ 6300, what is the value of the factory ? 42. A man. who owed $ 7825 failed, and could pay only | of his debts. How much could he pay ? 43. A man lost ^ of his money and had $ 210 left. How much had he at first ? 44. If -^-^ of a merchant's capital is $ 35,000, what is his entire capital ? 45. If a man can do a piece of work in 24 days, what part of it can he do in 18 days ? 46. In my pasture are 75 sheep, which is f of all my sheep. How many sheep have I ? 47. Henry runs 540 yards, which is ^ as far as Frank runs. How far does Frank run ? 48. A barrel can be filled by a pipe in 40 minutes. What part of a barrel can be filled in 25 minutes ? 49. A bushel contains 32 quarts, and a peck 8 quarts. What part of a bushel is a peck ? 50. I bought a house and lot, and made a payment of $ 4500, which was f of the cost. What was the cost of the property ? MISCELLANEOUS REVIEW OF COMMON FRACTIONS. 137. Oral. 1. Add { and ^ ; -^ and ^ ; | and | ; -f and | ; f and | ; i and If 102 COMMON FRACTIONS. 3- Reduce to improper fractions 3^, 7|, 8|, 5|-, 7^, 8|, 16|, 15i. 4. Reduce to integers or mixed numbers -\^-, -y-, J^, -U-, ¥, ¥, f i ¥tS -VV- 5. Multiply 16 by |; 45 by f ; 18 by |; 45 by j\', I by 9; iby 32; || by 16; J by 27. 6. Find product of: f x ^; A X y^^ 5 ^J X ^5 f X if; ^x-V-;4Jx6|. 7. Find | of 24; f of 12; | of 30; f of 27 ; f of 45; I of 40. 8. Find i of i; i of |; ^ of f; ^-^ of f ; | of If; i of 21. 9. Divide f by 3; f by 4; -^ by 12; ^e, by 11; 4^ by 3; ^by 6; 4|by6; 10 by f ; 8 by f ; If by 8; Aby22; fby9. 10. Divide 4 by i; 8 by 4; 9 by f ; 16 by f ; 24 by |; 13 by I; llbyt; 12 by If. 11. Divide 1 by J; f by |; i by i; | by f ; j\ by |; fbyf; 51. by 21 12. Divide 1 by : 111111 132 3> "§"? 6' I'S"? T> Y:5"> TIT' T' "S"* Note. — 1 divided by a fraction equals that fraction inverted. 13. I of 12=? 9 = what part of 12 ? 9 is J of what? 14 Qy?_2. A -^9 — L' 3y?_l. 5._:_9— 1- 3_s_9_l. 5_:^9 — 8 ■3" • ^ — "5" 5 ¥ • • — "9 • 15. i-? = i; i + ? = i; !-? = !; l + ? = ii; 16. What part of 6 is 4 ? i is i ? f is I ? 5^ is 2^ ? 1 ? 11 is 5? i is i? lis J? |is ^ 17. 9 is I of what ? 5 is | of what ? 6 is f of what ? REVIEW OF ERACTIONS. 103 18. Change f to 24ths ; | to loths; |- to 32ds; f to 20ths. 19. A man owning J of a farm, sold ^ of his share. What part did he sell ? How much remains ? 20. At 121^ a dozen, how many dozen of eggs can I buy for $3? 21. At 6J^ a box, how much will 8 boxes of berries cost ? 22. John has 56 cents, and James |- as much. How much have both? 23. A can do a piece of work in 4 days ; B can do the same piece of work in 2 days. What part of the work can each do in a day ? 24. A can mow a field in 3 days, and B in 4 days. What part of the field can they mow in a day if both work together ? A can mow \ of it in I'day, and B can mow ^ of it in 1 day. Both working together can mow the sum of ^ and ^ = -^^ of it in 1 day. 25. C can do a piece of work in 2 days, and D can do it in 4 days. In what time can they both do it, working together ? C does I of it in 1 day, and D ^ of it in 1 day. Therefore both can do ^ + ^ = f of it in 1 day. Since both can do | of it in 1 day, it will take as many days to do |, or the whole of it, as | is contained times in |, or 1^ days. A7is. Note. — | divided by f gives the same result as 4 divided by 3. 26. ^ of my money is gold, and ^ as much is silver. What part of my money is silver ? 27. If a boy can earn $ 2i in 1 week, how much can 3 boys earn in 4 weeks ? 28. James sold a book for 28 cents, which was |- of what it cost him. What did it cost him ? 104 COMMON FRACTIONS. 29. The difference between ^ of a number and \ of it is 6. What is the number ? Solution. — The difference between | and { is {. Now the ques- tion is, 6 is ^ of what ? 30. A boy 12 years of age is \ as old as his father. How old is his father ? Written. 31. A farmer having 1200 bushels of potatoes sold i of them at one time, ^ at another, and 350 bushels at another. How many bushels had he left ? 32. A mechanic whose wages are $5 per day uses -^ of his weekly earnings for board, and | for clothing and other expenses. How many dollars does he save weekly ? 33. Which is greater and how much, yf (3r -^ ? 34. If it takes 27 days to do a piece of work, how long will it take to do -| of it ? 35. If a horse is worth $100, and a cow is worth |- as much as the horse, what is the cow worth ? 36. John has in the bank $45 and draws out f of it. How much remains in the bank ? 37. What will 16 pair of shoes cost at $ 3|^ a pair ? 38. If a farmer has 23 sheep and sells them at $3/^ apiece, how much does he receive for the sheep ? 39. What is the cost of -^-^ of a pound of cheese at 10 ^ a pound ? 40. What is the cost of | of a yard of cloth at $ IJ a yard? 41. f xf x6ixy\x3xlf = ? 42. What is the value of 3 1 of 8| of | of J| ? 43. A man sold 3f tons of hay at one time, 7|- at another, and enough the third time to make 20 tons. How many tons did he sell the third time ? EEVIEW OF FKACTIONS. 105 44. jij plus ^ plus J plus ^ and how many more will make 3 ? 45. A man having a farm of 96 acres sold ^ of an acre to one man, ^ of an acre to another, ^ of an acre to another, and -^j of an acre to another. How many acres had he left? 46. If two men were 90 miles apart, and each should travel 23^ miles toward the other, how many miles would they then be apart ? 47. If George has ^ of a dollar and -^ of a dollar, and Henry has J of a dollar and ^^ of a dollar, which has the greater amount, and how much ? 48. A man bought 3 loads of wood containing respectively IJ cords, 1| cords, and 1| cords. How many cords of wood did he buy ? 49. I paid $101 for hay, f lof for coal, and $6i for wood. What did I pay for ail ? 50. Mr. Jones paid $ 525J for a span of horses, and sold them for $ 6251-. How much did he gain ? 51. L. W. and J. E. Connell paid f 4500f for a store and its contents. They sold it for $5025f. How much did they gain by the operation ? 52. A, B, C, D, and E own respectively ^, |, |, -fi^, and ii acres of land. How much do they all own ? 53. A gentleman having f 1700 paid $825J for horses, $230| for cows, $1.50 J for oxen, and $407|- for sheep. How much money had he left ? 54. Mr. Blanchard paid $ 8^^ for shovelling his walk, $ 5|- for trimming his grape-vines, and $ 6| for sifting his ashes. He gave the man a 20-dollar bill and a dollar bill. How much money should Mr. B. receive in return? 55. If I add 2 to each term of the fraction ^, will its value be increased or diuimished; and how much ? 106 COMMOl^ FRACTIONS. 56. Mr. Homer has lOi acres of wheat, 6| acres of corn, 20| acres of barley, and 16| acres of rye. How many acres of grain has he ? 57. What is the quotient of 389 divided by 1556, ex- pressed in its simplest form ? 58. I of if of ^9_ off 69. 816 is f of what number ? 60. From f of |f take f 61. The product of two factors is 10^; one factor is 3f. What is the other ? 62. 3+624-9i + ^6_ + _7_ = 9 63. (j\ of 21 of j\) X (I of 3i of 8 X J) = ? 64. The sum of two numbers is 19||. One of the num- bers is 12f . What is the other ? 43 65. Reduce -1 to its lowest terms. 28 66. a-i)xa+i)=? 67. Change to simple fractions : if H tBii n fofi If L5LI. 9' 5i' 16 ' fof2i' J of J' jj' J of I 68. A can do a piece of work in 4 days, B can do the same work in 5 days, and C in 6 days. In what time can all do it together ? 69. A tank has 3 supply pipes. It can be filled in 6 hours by the first pipe, in 7 hours by the second, and in 8 hours by the third. In how many hours can the tank be filled by the three pipes together ? 70. A and B can do a piece of work in 3 days. A can do it alone in 54 days. In what time can B do it alone ? REVIEW OF FKACTIONS. 107 Solution. — Both can do ^ of it in 1 day. A, alone, can do j^ of it in 1 day. ^ — j\ = -^■^, the part A can do in 1 day. Since he can do ^5 of it in a day, he can do f|, or the whole of it, in as many days as j\ is contained times in f f, or 33 -=- 5 = 6f days. 71. J of my property is invested in land, f of the re- mainder in business, and | of the remainder, which is $2400, is in the bank. How much property have I? 72. What is the value of (^1 J + 6 - f of f + i^ - 3i ? 73. A farmer sold 11 doz. eggs at 14^^ a dozen, and took his pay in sugar at 5| ^ a pound. How much did he receive ? 74. Find the value of — |— -f- i of i -- ^. I off 2 3 3 75. A boy having spent ^ of | of his money for a knife, had $ 2.25 left. How much did he pay for the knife ? 76. A father left $ 39,000 to his two children, dividing it so that the daughter received | as much as the son. What was the share of each ? 77. A person owning f of a steamboat, sold f of his share for $ 17360» What was the value of the boat ? 78. After spending i of my money and J of the remain- der I had $300 left. How much had I at first ? 79. If i of I of a bushel of apples cost f of y^^ of a dol- lar, what will J of j of a bushel cost ? 80. How many pounds of honey at |^ of f of a dollar a pound can be bought for | of 2| dollars ? 81. Simplify ri-y^-^^- 12 • 2 X 3J • DECIMAL FRACTIONS. 138. A Power is the product of equal factors, as 5 x 5 = 25, 5x5x5 = 125. 25 is the second power of 5. 125 is the third power of 5. 10 x 10 = 100. 10 x 10 x 10 = 1000. 100 is the second power of 10. 1000 is the third power of 10. > 139. A Decimal Fraction or Decimal is a fraction whose denominator is 10 or a power of 10. y Note. — The denominator of a common fraction may be any number, but the denominator of a decimal fraction must be 10,- 100, or 1000, etc. 140. A decimal is written at the right of a period (.) called the Decimal Point. Note. — It is not customary to write the denominator of a decimal. It is determined by the position of the decimal point. 141. A figure at the right of a decimal point is called a Decimal Figure. Tenths are written like dimes with one decimal figure ; thus, ^^ = .5. Hundredths are written like cents, with two decimal figures ; thus, jW = .25, -j^ = .07. Thousands are written like mills, with three decimal figures ; thus, Jo%V = -125, TitTr = -016, tAi7 = -004. Ten-thou- sandths require four decimal figures ; hundred-thousandths, five ; millionths, six ; etc. 142. Name the denominators in the following : .36 ; .08 ; .294; .1406: .0001; .263402. PViQ-ncTA fr» rlppimnlQ* 25. 125. 10063. 3 6 . 108 TiAFr- -^ DECIMAL FRACTIONS. 109 ^ 143. A Mixed Decimal is an integer and a decimal; as, 16.04. 144. To read a decimal. Rule. — Read the decimal as an integer, and give it the denomination of the right-hand figure. Read the following numbers : 1. .7 8. .0000054 15. 235.850062 2. .07 9. 35.18006 16. 100.000104 3. .007 10. .0005 17. 9.1632002 4. .700 11. .500 18. 3543.4536982 5. .03065 12. 4.98625 19. 30.3303303 6. .16984 13. 38694.06 20. 303.303303 7. .10016 14. 9.98463004 • 21. 9.999999 145. To write a decimal. \l Rule.— Write the numerator, prefixing ciphers when necessary to express the denominator, and place the point at the left. Note. — There must be as many decimal places in the decimal as there are ciphers in the denominator. Express decimally : • 22. Four tenths. Seventeen hundredths. Five hun- dredths. Three hundred twenty-five thousandths. Five thousandths. Fifteen thousandths. Nineteen and seven hundred twenty-four thousandths. 23. Seven thousand five hundred four ten-thousandths. Sixteen, and 125 ten-thousandths. Six ten-thousandths. Five thousand ten-thousandths. 24. Seventeen thousand two hundred eleven hundred- thousandths. Four hundred-thousandths. Fifteen hun- dred-thousandths. Eighteen, and two hundred sixteen hundred-thousandths. One hundred twelve hundred-thou- sandths. 110 DECIMAL FRACTIONS. 25. Twenty-nine hundredths. Twenty-nine thousandths. Twenty-nine ten-thousandths. Twenty-nine hundred-thou- sandths. One and one tenth. One and one hundredth. One and one thousandth. One and one ten-thousandth. One and one hundred-thousandth. 26. 324 and one hundred twenty-six millionths. 4582 and 36242 hundred-thousandths. Seventeen millionths. Five hundred-thousandths. Twenty-four, and three thou- sand four hundred six ten-millionths. 27. 10 millionths. 824 ten-thousandths. 31 hundredths. 216 hundred-thousandths. 7846 hundred-million ths. Four and 15 hundred-thousandths. 28. A 32. ToVAiy 36. t\ 40. iTFOOOUiy 29. 1 5 100 33, ylttfio 37. riTT 41. nn 30. T%\\ 34. To'G^^'SU'U 38. 500A 42. TTTO^OT 31. tWA 35. I^tAtt 39. TWOTT 43. TOOlJTJ REDUCTION OF DECIMALS. ^ 146. Principles. — Ciphers annexed to decimals do not change their value. V For each cipher prefixed to a decimal, the value is dimin- ished tenfold. The denominator of a decimal when expressed is always 1 with as many ciphers as there are decimal places in the decimal. ^ 147. To reduce two or more decimals to a Common Denomi- nator. V Rule. — Aymex ciphers so that each decimal will have the same number of decimal figures. BEDUCTION OF DECIMALS. Ill 148. Reduce to a eommon denominator: 44. .5, .017, .1256, .000155, 29.803. 45. .80062, 305.24, 70.5, 3.85263. 46. .1, .0001, 1000.001, 1.0100385. 47. .26, .13682, 9.4, 25, 8.63521. 149. Reduce .375 to a common fraction. V .375 as a common fraction is xVoV ^^^^ ^^ lowest terms _ 3 — 8- s/ Rule. — Write the numerator, omitting the point. Supply the denominator, and reduce to lowest terms. Reduce to common fractions : 48. 1.24 53. .32 58. 16.144 49. .16 54. .113 59. 28.3695 50. .325 55. .7282 60. 34.000010 51. .098 56. 2.25 61. 25.0000100 52. .875 57. .2425 62. 1084.0025 150. 63. Red uce 37i to a common fraction. Solution. - 100 -fo=--^- Ans. 64. .12-1 67. .161 70. .87| 65. .06J 68. .33i 71. .661 66. .621 69. .831 72. .367 151. To reduce a common fraction to a Decimal. Reduce f to a ( iecimal. 1 = 3 times \. 3 = (3.0), 30 tenths. \ of 3.0 = (.7), 7 tenths, and 2 tenths remainder. 2 tenths = 20 hun- dredths, i of .20 = .05. Hence | = .7 + .05 = .75. 112 DECIMAL FRACTIONS. \J Rule. — Annex decimal ciphers to the numerator, and divide by the denominator. Point off from the right of the quotient as many places as there are ciphers annexed. Notes. — A decimal cipher is a cipher at the right of the decimal point. If there are not enough figures in the quotient, prefix ciphers. The division will not always be exact. In such cases write the remainder over the divisor as a common, fraction, or place the sign + after the decimal to show that the result is incomplete. Thus, |=.142f or .142+. 162. Reduce to decimals : 73. t 77. A 81. f 85. t\ 89. 66| 74. f 78. 1 82. ^4 86. « 90. 25.121 75. 1 79. f 83. i 87. 12i 91. 161 76. 1 80. i 84. t\ 88. 331 92. 16.251 ADDITION. 153. Add .35, 4.375, 28.3065. .35 Rule. — Write the numbers so that decimal points 4.375 stand in a column. Add, as in integers, and 28.3065 place the point in the sum directly under the 33.0315 points above. Find the sum : 93. 24.36 94. 38,28006 95. 1.186 1.358 1.005 .285 .004 2.16 .003 1632.1 1873.148^ 203. 96. .175 4-1.75 + 17.5 + 175. +1750. 97. 145. + 14.5 + 1.45 + .145 + .0145. 98. 32.58 + 28963.1 + 287.531 + 76398.9341. 99. 1. + .1 + .01 + .001 + 100 + 10. + 10.1 + 100.001. SUBTRACTION. 113 100. 1.923 + .008 ^ 251.47 + 1.961 + 0.0543 + .006 + 18.7. 101. Add 750.3521, 698.42001, .005321, 3.5, 749.006984, 36950.06, 875.942, 286.753. 102. Add 5 tenths; 8063 millionths; 25 hundred-thou- sandths; 48 thousandths; 17 millionths; 95 ten-millionths ; 5, and 5 hundred-thousandths ; 17 ten-thousandths. 103. Add 24|, 17i, .0058, 7^, 9J^. SUBTRACTION. 154. Rule. — Wi'ite the riumbers so that the decimal point of the subtrahend stajids directly under the decimal point in the minuend. Subtract as in integers, and place the point directly under the points above. Note. — It is sometimes convenient to give the decimals the same denominator by annexing ciphers. 104. From 6.008 105. 38. 106. 26.34 107. 16.2600 Take 3.154 .356 1.28983 1.0001 108. 32.90596 -.75 114. .00011 - .000011 109. 9.5-3.35006 115. 10 -.1 + .0001 110. 856.2-8.562 116. 8.75 -h .95 + .125 111. .1 -.00001 117. 16-.00001 + 27.69852 112. 1000 -.001 118. 2.5 - .09 + 1.85 - 1.283 113. 20 -.00205 119. 83.1 - 8.31 4- .831 120. From one thousand take five thousandths. 121. Take 17 hundred-thousandths from 1.2. 122. From 8.5 take eighty-four hundredths. 114 DECIMAL FRACTIONS. 123. Find the sum of 500 thousandths and 5 hundred- thousandths and from it subtract ^^. 124. From 17.371 take 14.16i 125. Find the difference between f-^-^-^ and yfft^. 126. From 10 take J^ ; j^^-, 4.98; 1.05. 127. From one million and one millionth take one tenth. 128. From 1 tenth take 1 millionth. 129. Which is the greater and how much, one tenth or 100 thousandths ? 130. Prove that i and .500 are equal. M UL TIP Lie A TI ON. 155. Every decimal equals a corresponding common fraction, and for each cipher in its denominator there is a decimal figure in the decimal fraction. TTO" ^ i^ — tMu- (Three ciphers in the denominator.) .05 X .3 = .015. (Three decimal places in the decimal.) Rule. — Multiply as in iritegers, and give to the product as many decimal figures as there are in both multiplier and multiplicand. Note. — If there are not figures enough, prefix ciphers. Ciphers at the right of a, decimal have no value, and may be omitted. Find the products : 1. .38 X 1.6 7. .296 x 124 2. .015 X .05 8. 1.001 X 1.01 3. 1\ X 3.4 9. 13.33 X 1.3 4. 50 X .304 10. 25.863 x 4J 5. 2.65 X .104 11. 1.04 x ^ 6. 257 X. 354 12. 327f x 4| MULTIPLICATION. 115 13. 58.42 X 20.06 17. .001542 x .0052 14. .0001 X 1000 18. 26 X 36.82 15. .325 xl2|- 19. 2.84 x 3J 16. .333 X .333 20. 11.11 x 100 156. To multiply by lo, lOO, looo, etc. 21. Multiply 1.265 by 100. 1.625 Remove the point one place to the right for 100 each cipher in the multiplier. 126.500 Do not write the multiplier. Oral. 22. 3689.25 x 10 27. .5 X 100 23. 38.6422 x 100 28. .5 X 1000^ 24. 269.8342 x 1000 29. 384.2 X 10 25. 100 X 23.85 30. .3659 X 100 26. 1000 X 1.52 31. .1000 X .01 157. To multiply by 200, remove the point to the right and multiply by 2. Oral. 32. 86.44 X 200 35. 750.5 x 5000 33. 3.894 X 3000 36. 1.892 x 2000 34. 88.42 X 20 37. 156.2 x 200 158. Written. 38. Find the product of 1 thousand by one thousandth. 1 million by one millionth. 39. Multiply 700 thousandths by 7 hundred-thousandths. 40. Multiply the sum of 2 millionths and 10 thousandths by their difference. 41. Multiply together .35, 18.5, 28.004. 116 DECIMAL FKACTIONS. DIVISION. 159. Since in multiplication there are as many decimal places in the product as in both multiplier and multiplicand, in division the quotient must have as many places as the number of places in the dividend exceeds those in the divisor. 1. Divide 12.685 by .5. K\-j2fjcK Solution. — Since there are three decimal places or'orr " i" the dividend and one in the divisor, there must be two m the quotient. iZizie. »— 1. In all cases divide as in integers, then place the decifial point. 2. Divide 399.552 by 192. 1Q9\QQQ^W ■^w-'^' — 2- When the divisor is an integer, ^^ooj^' place the point in the quotient directly TbbE ^^^^ ^^^ point in the dividend in long 1536 division (directly under in short divi- 192 sion). Prove by multiplying divisor by 192 V quotient. Principle. — Multiplying both dividend and divisor by the same number does not change the quotient. 3." Divide 28.78884 by 1.25. 23.031 + 1.25')28.78'884 Rule. — 3. When the divisor contains 250 decimal figures, move the point in 378 both divisor and dividend as many ^^^ places to the right as there are deci- mal places in the divisor (this, in 388 §Z5_ Ex. 3, multiplies both by 100), then 134 125 place the point in the quotient as if ' the divisor were an integer. DIVISION. 117 Note 1. — The new points may be placed on a line with the tops of the figures, and the original points may stand to preserve the reading of the decimals. Note 2. — If the quotient does not have a sufficient number of figures, prefix ciphers. Note 3. — Before commencing to divide, see that there are at least as many decimal places in the dividend as in the divisor. Note 4. — If there is a remainder after all the figures of the divi- dend are used, annex decimal ciphers and continue the division. Note 5. — It is not usually necessary to have more than four deci- mal figures in the quotient. Find the quotients : 1. .288 -.64 11. 315.432 - .132 2. .36-600 12. 1.5906 - 241 3. 144 -.12 13. 36.25 - 1.25 4. .25 -.2500 14. 75 - .0125 5. .12-30 15. 125 - .12^ 6. .96H-.08 16. 25 - .25 7. 384.526-1.16 17. .25 - 25 8. 1440 -.0018 18. 1000 - .001 9. 1.225-4.9 19. .001 - 1000 10. 9.156-12 20. 18.65 - 100 160. To divide by lo, loo, looo, etc., remove the point one place to the left for each cipher in the divisor. , Oral. 21. 38.64-10 25. 3.91 -H- 1000 22. .5-^10 26. 1.155 h- 100 23. 558-^100 27. 398.42-1000 24. 1684.32-1000 28. 2.46-200 Note. — To divide by 200, remove the point to the left, and divide by 2. 29. 386.54-2000 31. 865.45-5000 30. 38.28-5-400 32. 2.5-500 118 ' DECIMAL FRACTIONS. PARTS OF lOO OB lOOO. 161. 1. What part of 100 is 121 ? 25? 33i? 2. What part of 1000 is 125 ? 250 ? 333| ? 3. How much is J of 100 ? Of 1000 ? 4. How much is i of 100 ? Of 1000 ? 5. Find J of 100. Of 1000. » 6. How much is 25 times 24 ? Solution. — 100 times 24 = 2400. 25 times 24 = ^ as much as 100 times 24, = 600. 162. To multiply by 25, annex two ciphers, and take J of the result. 7. Tell how to multiply by 33i ; by 12i; by 250; by 125 ; by 3331 Oral. 8. 36 X 25 11. 444 x 25 14. 3331 x 30 9. 48 X 121 12. 320 X 33J 15. 168 x 250 10. 24 X 33i 13. 125 X 80 16. 12J x 48 17. What cost 650 oysters at 50 cents a hundred ? Solution. — 650 h- 100 = 6.50 hundred. $.50x6.50=? 18. What will be the cost of 3850 laths at 40 cents a hundred ? 19. What is the freight on 685 pounds of baggage at $ 1.10 per 100 lb. Note. — C. means 100 ; M., 1000. 20. What is the cost of 4862 ft. of pine lumber at $ 30 per M. ? 21. Find the cost of 38,586 bricks at $ 8.25 a thousand. 22. What will 583 heads of cabbage cost at $ 3.50 a hun- dred ? WRITTEN EXEKCISES^ .. 119 23. At $3.50 a thousand, what will be the cost of 7800 shingles ? 24. At $8.25 per C, what will be the cost of 2864 lb. of dried fish ? 25. At $ 50 per M., what will be the cost of 3865 feet of cherry lumber ? 26. What is the cost of laying 5890 bricks at $ 9.00 a thousand ? To find the cost of merchandise sold by the ton, divide the price by 2 and proceed as above. 27. Three loads of hay weigh 7894 lb. What will the hay bring at $ 12 a ton ? Note. — 1000 lb. will cost ^ of $ 12 = $ 6. $ 6 x 7.894 = ? 28. What cost 48986 lb. of railroad iron at $ 35 a ton ? 29. Four loads of coal weigh respectively 3896 lb., 3524 lb., 4106 lb., and 3123 lb. What is the cost of the coal at $ 4.82 a ton. ALIQUOT PARTS OF $1.00. 163. The Aliquot Parts of a number are the numbers which are exactly contained in it. The aliquot parts of 100 are 5, 20, 12-i-, 16|, 33|, etc. 164. The aliquot parts of $ 1, commonly used, are as follows : 61 cents = $J^ 25 cents = $ J 8i cents = $^ 33^ cents = $ J 12| cents = $ |- 50 cents = $ J 16| cents = f J- 1. What is the cost of 69 books at 16|^ each ? Solution. —69 books will cost 69 times 16|j?, or69x$J = $^ = $11.60. Ans. 120 DECIMAL FRACTIONS. 165. Oral. Multiply : 2. 33 J cents by 36 5. 25 cents by 40 3. 121 cents by 24 6. 75 cents by 4 4. ej cents by 32 Note. — ^ means cents, lb., pounds, and yd., yards. 7. What is the cost of : 48 lb. of bacon at 121^ a pound ? 80 hand balls at 50^ each ? 36 yd. of ribbon at 33^^ a yard ? 80 lb. of candy at 25^ a pound ? 166. Written. 8. Find the cost of the following : 66 lb. of pork at 121^, 148 lb. of veal at 16f^, 48 boxes of strawberries at 25^, 48 lb. of honey at 25^, 64 bars of soap at 6^^, 60 doz. of eggs at 16|^. Find the cost of : 9. 1580 lb. of sugar at 6J^ a pound. 10. 500 books at 25^ each. 11. 16 yd. of dress-goods at 33^^ a yard. 12. At 25^ a pound, how many pounds of butter can be bought for $ 8.00 ? Solution. — As many pounds as 25j? or $ | is contained times in 167. Oral. Divide : 13. $ 5 by 331^ 16. $ 3 by 8J^ 14. $6 by 6i^ 17. ^4 by 25^ 15. $ 9 by 12^^ 18. f 4 by 66|^ REVIEW OF DECIMALS. 121 19. At 25^ each, how many hats can be bought for $ 6 ? 20. At $ J a pound, how many pounds of cheese can be bought for $ 6 ? 21. At 33 J^ a yard, how many yards of linen can be bought for i 10 ? 168. Written. 22. At 75^ a bushel, how many bushels of barley can be bought for $ 125 ? 23. When butter is 25P a pound, how many pounds can I buy for $50? 24. How many dozen eggs at 16J cents a dozen can be bought for $ 38 ? 25. At 12|^ cents a quart, how many quarts of nuts can be bought for $ 10 ? REVIEW OF DECIMALS, 169. 1. Tell how to locate the decimal point in any sum. In any remainder. In any product. In any quotient. 2. In the number 777, what is the local value of the 7 at the right ? The second 7 ? The left-hand 7 ? 3. Upon what does the value of any figure depend ? 4. In the decimal .777, what is the value of the first 7 at the right ? The second 7 ? The third 7 ? 5. What is the effect of removing an integral figure one place to the right ? A decimal figure ? 6. What is the effect of removing an integral figure one place to the left ? A decimal figure ? 122 DECIMAL FRACTIONS. Eead: 7. .0001 .00196 4.3 .0006 .02789 71.86 .0014 .52000 329.400 .0282 .050798 ' 1.001 .5897 .725386 200.3278 .00001 .500001 579000.00005 .00027 .000829 437.050609 Copy and write decimally : 8. 1 tenth; 24 hundredths ; 379 thousandths ; 1000 ten- thousandths ; 85 hundred-thousandths ; 20079 millionths. 9. One thousand six, and five hundred two millionths. 10. Three hundred fifteen thousand one, and eleven ten- thousandths ; thirty-eight, and seven thousandths ; 8 mil- lion 270 thousand 942, and 5 thousandths ; seventeen tenths. 11. Four hundred 21, and 5 ten-thousandths ; 1 thousand 27, and 27 hundredths; ninety -nine, and ninety-nine ten- millionths. Write without the denominator : 42t%, 78t%Vt5, 2003-VA^^. 13. Change to common fractions in lowest terms : .028, .0015, .2175, .000048, .00075, .45, .8, .75, 8.9375, 91.16, 4001.645, 9.156575. Change to equivalent decimals : 14- h h -h, H, A. I. 20H, 8ji^, 4J5, losm. Change to common fractions, then to simple decimals : 15. .1^, .07i, .18f, .mi, .121, .08i, .221, .045^%, ,37J, ,381, .541, .000051 j8|, .38J REVIEW OF DECIMALS. 123 Reduce to a common denominator and add : 16. 50.06, 367.41, 200.200, .12304, 40.0056, 7.5620, .096071. 17. 1301.6, 904.02, .547, .0009, .00001, 218.94, 203.410, 1000, .01. 18. 100.101, 82.4, 401.009, .00038, 60702, 10.10, 574.68139. 19. 5.628, 850.002, 9.00256, 37.0005, 724.6811, 3759, 7000.0036, 2.25. 20. $11.78, $347, $5.06,' $218, $20.07, $42.0244, $7,104, $37,625. 21. 4.76, .390, .0915, .00207, 841, 63.2, .00234, 1.43, .00536. 22. .00908, .0371, 24.5, 7.03, .0127, 354, .000781, .0436, 20.7354. Subtraction : 23. 5.74-3.23 = ? 26. 367-1.52 = ? 24. .876 -.343 = ? 27. 200 - .02 = ? 25. 67.5-41.5 = ? 28. Which is greater, | or 4 tenths ? . 29. How much more is $ 20 than $ 17.84 ? 30. Erom two million take two millionths. 31. I bought 4 farms: one contained 19.368 acres; one, 27.96 acres; one, 473.0008 acres; and one, 73.7561 acres. I sold 300.25 acres. How much land had I left ? 32. From 1 inch take one ten-thousandth of an inch. Multiply : 33. 7.945 by .3 37. 7.853 by 23.16 34. 350 by .42 38. 1.36 x 20.04 = ? 35. One tenth by one hundredth. 39. 27.27 x 4.0004 = ? 36. 25 units by 25 tenths. 124 DECIMAL FRACTIONS. 40. If wheat is worth $ .38 a bushel, what will 117.75 bushels cost? 41. Apples sell for $1.28 a bushel. How much money will 24 barrels bring, each containing 2i bu. ? 42. Find the cost of 3.325 lb. of butter at 18.75 cents a pound. 43. What will 6| yd. of broadcloth cost at $ 1.375 a yd. ? 44. A boy paid $ .125 a dozen for 1.75 dozen eggs. What did they cost him ? 45. 3.64 X .0002 X 1.756 x 4.004 = ? Divide : 46. 1738.89 by .00417. 52. 42.475681 by .29. 47. 1237.6 by 26. 53. 40.20 by .000012. 48. 36.11 by .021. 54. $ 302.03 by 200. 49. 2.38 by .17. 55. 64.64006 by .002. 50. 36.82 by .0003. 56. 12.9643 by 18.4. 51. 437.96 by 2.8. 57. 759.806 by 90.3. 58. 16| + 3.06 - 1 + .002 - 2.1 + .03 - 1 + .OOi = ? 59. ^ + 3 _ .65 + .5 + J - i + 3.14 = ? 60. Find the product of .003 multiplied by .06, and divide it .by 3. 61. A certain decimal divided by 1000 is 35.002. What is one fifteenth of the decimal ? 62. The sum of two numbers is 306.52 ; one of them is 100. What is the other ? 63. A man spent $450, which was .125 of his money. How much money had he ? 64. Mr. A. bought a cow for $ 45, which was .375 of what he paid for a horse. How much did he pay for the horse ? 65. John spent .75 of his money for a book and had $ .50 left. How much had he at first ? ACCOUNTS AND BILLS. 170. An Account is a record of indebtedness for articles bought or sold, cash paid or received, or services rendered. 171. A Debtor is a person who owes a debt. 172. A Creditor is a person to whom a debt is owed. 173. A Bill is a written statement of a debtor's account, made by the creditor. 174. A Receipt is a creditor's written acknowledgment that he has received payment of part or all of a debt. 175. A bill is receipted when its payment is acknowl- edged in writing, by the creditor, or by some authorized person. Note. — The sign @ is for at. Dr. is for debtor, and Cr. for creditor. 1. BILL FORMS. James P. Barnes, Chicago, July 1, 1902. Bought of Dey Bros. & Co. 50 yd. Brussels Carpet @ 24 " Oil Cloth " 4 doz. pair Merino Hose " 2 Willow Chairs " $i 15 35 f : 8 50 4 50 1$ 125 126 ACCOUNTS AND BILLS. RECEIPTED BHili WITH CREDITS. Rochester, jST. Y., Jan. S, 1896, Mrs. Johx F. White, 1895 To Burke & White, Dr. Nov. 6 11 6 li 18 Dec. 11 « 15 ii 19 Nov. 18 Dec. 28 4 lb. Coffee 28 lb. Sugar 5 gal. Molasses 18 lb. Rice 2 bbl. Potatoes 28 lb. Butter @ Cash Balance due, 27 5% 60 7% 80 21 50 75 $ Received payment, Jan. 15, 1896, BuKKE & White, By John R. Pierce. FORM OV A RECEIPTED BIIX. New York, June SO, 1896. Jerome A. Phelps, In account with D. 0. Potter & Co. May H 12 bbl. Flour @ $6 50 $ K U 6 tubs Butter, 684 l^- ii 24 June 10 5 bbl. Beef u 25 28 u 25 450 lb. Ham u 9% Received payment, D. O. Potter & Co. WRITTEN EXERCISES. 127 4. Mr. John Q. Adams buys of D. McCarthy & Co. : 14 pounds of coffee at 27 cents a pound, 28 pounds of sugar at 5^ cents a pound, 15 gallons of molasses at 60 cents a gallon, 16 pounds of rice at SJ cents a pound. Make out the bill. 5. James Smith, farmer, sold Kichard Dunn, grocer, the following : 16 barrels of potatoes at $1.80 a bbl., 12 tons of hay at $ 16 a ton, 13 cords of wood at $ 4 a cord, 360 pounds of butter at 24A ^ a pound. Make a receipted bill. , 6. Chicago, Dec. 5, 1900. Edward Smith sold B. M. Watson 65 yd. Brussels carpet at $ 1.25 ; 24 yd. oil cloth @ 35^; one dozen pair of merino hose @ $3.50; 2 willow chairs @ $ 4.50.# Make bill, find the footing, and properly receipt it. 7. Make out a bill of groceries. Foot it, and receipt it, with F. H. Mead as creditor and Wm. H. Scott as debtor. To THE Teacher. — See that the prevailing prices are used, and that the quantities are consistent. 8. J. H. Acker bought of H. A. Strong of San Francisco, the following articles : 15 bbl. flour @ $ 8.00 ; 6 tubs of butter, 120 pounds in a tub, at 24 cents a pound ; 5 barrels of beef, 200 lb. to the barrel, at 5^; 25 sacks of flour @ 95^; 450 pounds of ham at 11^^. Make out bill, and receipt it. 9. Make out a bill of hardware, another of groceries, and another of dry goods. , 10. Make out a bill of goods bought at a shoe store ; at a 128 MISCELLANEOUS. MISCELLANEOUS. 176. M. I have four pieces of broadcloth. The first con- tains 13.7642 yd.; the second, 22.008 yd.; the third, 15.027 y(L; and the fourth, 19.255 yd. How many yards in all ? "• 2. From a piece of ribbon containing 103f yd., 73| yd. were sold. How many yards were left ? 3. How many yards of muslin at $ .121 a yard will it take for 4 pair of curtains, if each curtain contains 3.375 yd. ? 4. I have 14.735 yd. of lace, and desire to cut it into seven equal strips. How much will there be in each strip ? 5. What will be the cost of a hat at $ 7.50, a pair of gloves at $ 1.13, a veil at $ 1.25, and a parasol at $ 3.375 ? 6. Arrange the following articles in the form of a bill : 7 qt. of molasses at $.15 a qt., | bu. of apples at $1.28 a bushel, 30 lb. of sugar at $.08^ a pound, and 12 bu. of potatoes at $ .29 a bushel. 7. A grocer bought three bunches of bananas at $ 1.54 a bunch. The first bunch contained 73 bananas, the second 54, and the third 97. He sold them all at 30^ a dozen. Did he gain or lose, and how much ? 8. The first year in business a grocer made $ 2374.68, the second $ 1529.47, and in the third year he lost $ 300. His expense each year averaged $928.45; how much money had he gained at the end of three years ? 9. What will 9 barrels of flour cost, if 28 barrels cost $173.60? 10. I bought 437 heads of lettuce at $ 5 a hundred, and sold them at $ .08 a head. What was my gain ? Find the cost of ; 11. 6824 1b. of coal at $4.68 a ton. 12. 2384 lb. of coal at $ 5.67 a ton. 13. 8972 ft. of lumber at $ 35.40 a thousand. 17. What part of 4.50 is 3.33J ? MISCELLANEOUS. 129 14. 6854 lb. of hay at $ 16.50 a ton. 15. 4836 bricks at I? 9.45 per M. 16. 895 ft. of lumber at $ 19.75 per M. 9 18. What part of 3.625 is 1.5 ? 19. What part of 6.2 is 3.25 ? 20. 1.1 is what part of 7.4 ? 21. A father left his son $24,000, which was .375 of his estate. What was the value of the estate ? 22. Divide 26 by 2^, and multiply the result by 17.345. 23. Divide | of .375 by f of | of .298. 24. The product of three numbers is 167.7. Two of the numbers are 3.25 and 5.16. What is the other ? 25. What number divided by 2.86 equals .34 ? 26. What number diminished by 38.64 leaves .356 ? 27. A man bought 8.5 yd. of cloth at $3.33J a yard, 12.4 yd. at $ 2.75, 18^ yd. at $4,375, and 24f yd. at $ 2.875. How many bushels of corn at 43| cents a bushel will pay for the cloth ? 28. .5 of a number exceeds .45 of it by 20. What is the number ? Solution. — .5 — .45 = .05. Now the question is, 20 is .05 of what ? 20 -f- .05 = 400. 29. At 85 j^ a yard, how many yards of cloth can be pur- chased for $29.75? 30. Divide $ 785 among A, B, and C, so that C will have $ 185 more than each of the others. 31 1_ .0045 " .05 .4 X .005 + .002 x .125 130 MISCELLANEOUS. 32. What part of .876 is »31536 ? 33. If .375 of a ton of coal cost $ 1.25, what will 7.125 tons cost ? 34. What is .3 of a number when .8 of it is 80 ? 35. How many thousandths in 3 units ? 36. How many thousandths in .1 ? 37. Express \ of one hundredth as a decimal. 38. The salary of the President of the United States is $ 50,000 a year. How much does he receive per day ? 33 i of 4 5| 39. Divide the product of 5 times j| plus ^ •'^ by ^• 40. Divide 2 of il by J of ^. Find the cost of the following : 41. 3151 lb. of tea at $ .37i a pound, 34f lb. of coffee at $ .18f a pound, 3105| lb. of pork at $ .121 a pound, 30691 bu. of wheat at f 1.121 a bushel, 36| doz. of eggs at $ .121 a dozen, 26| yd. of sheeting at $ .07| a yard. 42. A owns f of a farm and B owns the remainder; J of the difference of their shares is worth $ 2400. What is the value of the farm ? 43. Divide $ 3J among some poor children, giving each J of a dollar. W^hat will be the number of children ? 44. Two men hire a pasture for $ 25. A puts in 8 horses and B 12 horses. How much should each pay ? Note. — Both have put in 20 horses. A must pay ^ and B i^ of $ 25. 45. Add 8 to both terms of the fraction ■^, and find how much you have increased or diminished it. QUESTIONS. 131 46. Subtract 4 from each, term of the fraction ^, and find how much it has been increased or diminished. 47. Find the least common multiple of 28, 34, 42, and 56. 48. Divide the least common multiple of 240 and 600 by their greatest common divisor. 49. Name all the prime numbers between 75 and 100. The odd numbers. 50. Divide the product of 21 x 11 x 6 x 26 x 10 by the product of 5 X 13 X 3 X 14 X 2. Use cancellation. 51. A merchant bought 10 casks of vinegar, each contain- ing 42 gallons, at 20 cents a gajlon, and paid for them in pieces of cloth, each containing 35 yards, at 10 cents a yard. How many pieces of cloth did he give ? QUESTIONS. / 177. l.^What is a decimal ? mow are decimals written ? ' Why are they called decimals ? 2. How many decimal places are needed to write ten- thousandths ? • Millionths ? * Hundredths ? 3. ''What is the first place at the right of the decimal point? "^'What is the first period called ?* The second place ? ^ The second period ? 4. ' What is a mixed decimal ? . 5. What must the denominator of a decimal be ? 6. What is the effect of removing the decimal point one place to the right ? ' To the left ?^ Two places to the right ? ^ Three places to the left ? 7."^ What is the effect of annexing a cipher to an integer ? To a decimal? ' Of prefixing a cipher to an integer ? ^ To a decimal ? 132 MISCELLANEOUS. 8. How do we reduce decimals to common fractions? Common fractions to decimals ? 9. Give rules for adding, subtracting, multiplying, and dividing decimals. 10. How do we locate the decimal point in the sum ? In the remainder ? In the product ? In the quotient ? 11. What are coins ? 12. What are the gold, silver, bronze, and nickel coins used in the United States ? 13. What are the aliquot parts of a number ? What are, the aliquot parts of ^ 1 ? Of 100 ? Of 1,000 ? 14. What is a bill? An account? A creditor? A debtor? Tell how to receipt a bill. 178. 1. Define unit, number, the unit of a number, ab- stract number, concrete number, li-ke numbers. 2. Define notation, numeration, Arabic notation. 3. What is the value of the unit figure of a number? The tens ? The hundreds ? 4. What is the largest number which can be expressed by four figures ? 5. What is the simple value of a figure ? The local value ? 6. What name is given to the first period to the right of the decimal point? The second? The third? 7. What is addition? What kind of numbers can be added? 8. Define subtraction, minuend, subtrahend, remainder. What is a proof of subtraction? What is the sign of sub- traction, and where placed? QUESTIONS. . 133 9. What is a parenthesis? A vinculum? For what are they used? 10. What is multiplication ? The multiplier? The mul- tiplicand? The product? 11. The multiplier and the multiplicand are what of the product ? 12. What is the sign of multiplication and how is it used ? Define division, divisor, dividend, quotient, re- mainder. 13. What is the sign of division, and how is it used? 14. Express the division of 12 by 8 in as many ways as you can. 15. To what terms in multiplication do the divisor, quo- tient, and dividend correspond? 16. How do you find the dividend when the divisor, quo- tient, and remainder are given? 17. When is the quotient an abstract number? 18. When the quotient and dividend are like numbers, what kind of a number is the divisor? 19. How can we divide when the divisor is 10? 100? 1000? When the divisor is 20? 50? 300? 20. Multiplying both dividend and divisor by the same number affects the quotient how? 21. Dividing both divisor and dividend by the same number affects the quotient how? 22. Multiplying the dividend affects the quotient how? The divisor? Dividing the dividend? The divisor? 23. Define exact divisor, factor, prime factor, factoring. 24. How can you find the prime factors of a number ? 134 . Mli^rELTvAKEOITS. 26. Dej&ne divisor. Common divisor. The greatest com- mon divisor. Give the riile to find the greatest common divisor. 26. Define multiple, common multiple, least common multiple. Give the rule for finding the least common multiple. 179. Define fraction, fractional unit, unit of a fraction, denominator, numerator, terms of a fraction, common frac- tion, integer, proper fraction, improper fraction, mixed num- ber, simple fraction, compound fraction, complex fraction. What is the value of a fraction? State the principles of fractious. ■ What is it to reduce a fraction ? How are fractions reduced to lowest terms? To higher terms ? How can an improper fraction be reduced to a whole or a mixed number? A whole or a mixed number to an improper fraction? What are like fractions? Unlike fractions? How can fractions be reduced to others having a common denominator? A least common denominator? How can two or more fractions be added? How can the sum of fractions be found? Mixed num- bers? How can the difference of fractions be found? Mixed numbers ? How can a fraction be multiplied by a fraction ? A frac- tion by an integer? How can an integer be multiplied by a fraction? By a mixed number? How can a fraction be divided by a fraction? How do you reduce a complex fraction to a simple frac- tion? COMPOUND NUMBERS. 180. A number composed of only one kind of unit is a Simple Number ; as, 5 pk., 4 apples, 6. 181. A Denomination is a name given to a unit of measure or of weight. 182. A number composed of different kinds of units is a Compound Number ; as, 3 bu. 2 pk. 1 qt. A number having one or more denominations is also called a Denominate Number. 183. Reduction is the process of changing a number from one denomination to another without changing its value. 184. Changing to a lower denomination is called Reduction Descending; as, 2 bu. 3 pk. = 88 qt. 185. Changing to a higher denomination is called Reduc- tion Ascending ; as, 88 qt. = 2 bu. 3 pk. 186. Linear Measure is used in measuring lines or distances. TABLE. 12 inches (in.) = 1 foot, ft. 3 feet = 1 yard, yd. 5|- yards, or 16^ feet = 1 rod, rd. 40 rods = 1 furlong, fur. 8 furlongs = 1 mile, mi. 320 rods, or 5280 feet = 1 mile. 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63360 in. 136 136 COMPOUND NUMBERS. 187. Surveyors' Measure is used in measuring land. TABLE. 7.92 inches = 1 link, li. 100 links = 1 chain, ch. 80 chains = 1 mile, mi. Note. — A surveyors' chain is 4 rods long, and contains 100 links. A chain, or steel measuring tape, 100 feet long, is sometimes used by engineers. 188. Square Measure is used in measuring surfaces. J / TABLE. y "^ y 14:4: square inches = 1 square foot, M aM "^ / "^ ^ square feet = 1 square yard, sq. ft. sq. yd. 304; square yards ) . , o'roi i 4. r = 1 square rod, sq. rd. 272J square feet ^ ^ ? ^ f y#^< ^vilGO square rods = 1 acre, A / 640 acres = 1 square mile, sq. mi. 1 sq. mi. = 640 A. = 102400 sq. rd. = 3097600 sq. yd. 189. A square mile of land is called a Section. A square rod is sometimes called a perch (P.). A rood (R.) is 40 sq. rods. Note, — 1 acre = 43560 sq. ft. There are 10 square chains in an acre. Eoofing, paving, etc., are often estimated by the Square, which is 100 square feet. 190. Cubic Measure is used in measuring volumes or solids. TABLE. 1728 cubic inches = 1 cubic foot, cu. ft. 27 cubic feet = 1 cubic yard, cu. yd. 16 cubic feet = 1 cord foot, cd. ft. 8 cord feet, or 128 cu. ft. = 1 cord, cd. 1 cu. yd. = 27 cu. ft. = 46656 cu. in. TABLES. 137 191. Liquid Measure is used in measuring liquids. TABLE. 4 gills (gi.) = 1 pint, pt. 2 pints = 1 quart, qt. 4 quarts = 1 gallon, gal. 1 gal. = 4 qt. = 8 pt. = 32 gi. A gallon contains 231 cubic inches. The standard barrel is 31 J gal., and the hogshead 63 gal. 192. Apothecaries' Fluid Measure is used in mixing medi- cines in liquid form. table 60 minims (ni) — 1 fluid dram, f. 3. 8 fluid drams = 1 fluid ounce, f. S« 16 fluid ounces = 1 pint (0). 193. Dry Measure is used in measuring roots, grain, vege- tables, etc. ^^3^^ 2 pints = 1 quart, qt. 8 quarts = 1 peck, pk. 4 pecks = 1 bushel, bu. 1 bu. = 4 pk. = 32 qt. = 64 pt. The bushel contains 2150.42 cubic inches. 194. Avoirdupois Weight is used in weighing all common articles ; as, coal, groceries, hay, etc. TABLE, 16 ounces = 1 pound, lb. . ^„ , (1 hundred- weight, cwt. 100 pounds = ■{ ^ 1 ., ^ (or cental, ctl. 20 cwt., or 2000 lb. = 1 ton, T. 1 T. = 20 cwt. = 2000 lb. = 32000 oz. The Long Ton of 2240 pounds is used at the U. S. Cus- tom-House and in weighing coal at the mines. 138 COMPOTTND NUMBERS. The ounce is considered as 16 drams. The Avoirdupois pound contains 7000 grains. A hundred-weight is sometimes called a Cental. 195. Troy Weight is used in weighing gold, silver, and jewels. TABLE. 24 grains (gr.) = 1 pennyweight, pwt. 20 pennyweights = 1 ounce, oz. 12 ounces = 1 pound, lb. 1 lb. = 12 oz. = 240 pwt. = 5760 grains. 196. Apothecaries' Weight is used by druggists and physi- cians in weighing medicines that are not liquid. TABLE. 20 grains (gr.) = 1 scruple, sc. or 3. 3 scruples = 1 dram, dr. or 3. 8 drams = 1 ounce, oz. or 5 . 12 ounces = 1 pound, lb. or lb. 1 lb. = 12 oz. = 96 dr. = 288 sc. = 5760 gr. Dry medicines are bought and sold in large quantities by Avoirdupois weight. Comparison of Weights. 1 lb. Avoirdupois = 7000 gr 1 oz. Avoirdupois = 437^ gr. 1 lb. Troy or Apothecary = 5760 gr. 1 oz. Troy or Apothecary = 480 gr. 197. Measure of Time. TABLE. 60 seconds (sec .) = 1 minute, min. 60 minutes = 1 hour, hr. 24 hours = 1 day, da. 7 days = 1 week, wk. 365 days = 1 year, yr. 366 days = 1 leap year. 100 years = 1 century. TABLES. 139 The Civil Day begins and ends at midnight. ■ The exact time in which the earth makes one revolution of the sun is 365 da. 5 hr. 48 min. 49.7 sec, or 365J days, nearly. For convenience the common year is regarded as 365 days; the fraction being disregarded until it amounts to a full day, which is in four years, nearly. Accordingly every fourth year contains 366 days. This day is added to the shortest month, February, and the year in which it is added is called Leap Year. But 365J days is a little more than the exact year, and we have added a little too much when we. have added 1 day to every fourth year, therefore only every fourth cen- tennial year is considered as leap year. This nearly cor- rects the excess, so that the error is less than 1 day in about 3600 years. Every year divisible by 4, and every centennial year divisible by 400, is a leap year. CIRCULAR OR ANGULAR MEASURE. 198. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from the centre. 199. The bounding line of a circle is the Circumference. 200. Any part of a circumfer- ence is an Arc. ^ to J3 and C to D are arcs of a circle. 201. A straight line through the centre of a circle terminat- ing at the circumference is the ^ circle. Diameter. 202. A straight line from the centre to the circumference is the radius ; as, E to D, or E to C. 140 COMPOUND NUMBERS. 203. The circumference of every circle is divided into 360 equal parts called Degrees, each degree into 60 parts called Minutes, and each minute into 60 parts called Seconds. 204. An Angle is the difference in direction between two straight lines. The point of meeting is the Vertex. The vertex is at the centre of a circle, and the angle is measured in degrees by the arc between its sides. Thus BD is the measure of the angle BED. TABLE OF CIRCULAR MEASURE. o 60 seconds (") = 1 minute, 60 minutes = 1 degree, 360 degrees = 1 circumference, Cir. Note. — An arc of 90 degrees or J of a circumference is called a quadrant. A degree upon a great circle of the earth is 69. 16 statute miles, or 60 geographical miles. A sign is an arc of 30 degrees. 205. Federal Money is the currency of the United States. TABLE. 10 mills = 1 cent, ct. 10 dimes = 1 dollar, $. 10 cents = 1 dime, d. 10 dollars = 1 eagle, E. The gold coins of the United States are the double-eagle, eagle, half-eagle, quarter-eagle, and one-dollar piece. The silver coins are the dollar, half-dollar, quarter-dollar, and the ten-cent piece. The five-cent piece is nickel, and the one-cent piece bronze. 206. English or Sterling Money. TABLE. 4 farthings = 1 penny, d. 12 pence = 1 shilling, s. 20 shillings = 1 pound, £, or 1 sovereign. The coin which represents the Pound Sterling is the Sovereign, equal in value to $4.8665. REDUCTION. 141 207. Counting. TABLE. 12 things = 1 dozen, doz. 12 dozen = 1 gross, gr. 12 gross = 1 great gross, G. gr. 208. Paper. TABLE. 24 sheets = 1 quire. 2 reams = 1 bundle. 20 quires = 1 ream. 5 bundles = 1 bale. REDUCTION DESCENDING. 209. 1. Keduce 5 lb. 6 oz. 12 pwt. 6 gr. to grains, 5 lb. 6 oz. 12 pwt. 6 gr. 12 — Solution. — Since there are 12 oz. m 1 lb., in 5 lb. ^^ there are 5 times 12 oz. = 60 oz. (add 6 oz.) = 66 oz. 6 Since there are 20 pwt. in 1 oz., in 66 oz. there QQ oz. are m times 20 pwt. = 1320 pwt. "'(add 12 pwt.) = 20 1332 pwt. -1Q2Q Since there are 24 gr. in 1 pwt., in 1332 pwt. ^ o there are 1332 times 24 gr. = 31968 gr. (add 6 gr. ) = 31974 gr. 1332 pwt. ^ 24 E-educe to lower denominations : 5328 2664 ^/2. 17 yd. 2 ft 9 in. to inches. ^3. 46 rd. 4 yd. 2 ft. to feet. -4. 3 mi. 75 rd. 4 ft. to inches. 31968 6 31974 ffr. *^* 16 -^- 140 sq. rd. 2Q> sq. yd. to square yards. »^6. 4 A. 15 sq. rd. 4 sq. ft. to square inches. .^. 50 ch. 45 li. to links. ^. 16 cu. yd. 25 cu. ft. 900 cu. in. to cubic inches. /9. 8 cd. 12 cu. ft. to cubic feet. -10. 15 gal. 3 qt. 1 pt. to pints. 142 COMPOUND NUMBERS. ^11. 4 0. 6 f. 5 3 f. 3 25 ni to minims. ^2. 7 bu. 3 pk. 5 qt. 1 pt. to pints. ^13. 16| bu. to quarts. '^ 14. 25 lb. 5 oz. 16 pwt. 10 gr. to grains. ^16. 2 T. 6 cwt. 10 lb. 14 oz. to ounces. \/l6. 16 tb. 5 § 4 3 2 3 11 gr. to grains. ^7. 28° 14' 18" to seconds. ^ 18. £18 15s. Sd. 3 far. to farthings. "19. 27 da. 18 h. 49 min. to seconds. "20. 3 wk. 48 min. 52 sec. to seconds. -^21. How many quires in a bundle of paper? '^%2. How many pints in a cask of molasses holding 84 gallons ? 23. How many articles in 7 G. gr. 5 gr. ? 24. How many hours in 10 years, allowing for two leap years ? 25. How many inches in 4 J rods ? 26. What is the cost of 10 miles of telephone wire at 28 cents a pound, if a pound measures 75 ft. ? 27. Find the number of square inches in a square yard ; square feet in a square chain ; cubic inches in a cubic yard. 28. How many hours in the month of February, 1896 ? 29. How many cubic inches in 5 gallons? 30. How many square yards in 4 sq. miles ? 31. How many square feet in 2^- acres ? 32. How many ounces in 3 lb. of silver ? 3 lb. of iron ? '33. If I buy 3 bu. of nuts at $ 4 a bushel, and sell them at 5^ a pint, how much shall I lose ? 34. How many ounces in a long ton ? BEDUCTION. 143 35. At $ 12 a ton, what will f of a ton of hay cost ? 36. In 1800 years how many centuries ? 37. If you can count sixty a minute, how long will it take to count 180000 ? 38. Through how many degrees does the hour-hand of a clock pass in 6 hours ? 39. Through how many degrees does the minute-hand pass in 6 hours ? 40. What will 3 reams of paper cost at 40^ a quire ? 41. Reduce 3 mi. 4 fur. 20 rd. 5 yd. 2 ft. 8 in. to inches. 42. Reduce 6 mi. 240 rd. to feet. 43. Reduce 3 A. 8 sq. rd. 5 sq. yd. 3 sq. ft. to sq. inches. 44. Reduce 16 cu. yd. 9 cu. ft. 3 cu. in. to cu. inches. 45. Reduce 58 cd. to cu. feet. 46. Reduce 2 T. 3 ctl. 16 lb. to ounces. 47. Reduce 3 lb. 9 oz. 15 pwt. 12 gr. to grains. 48. Reduce 60 gal. 3 qt. 3 gi. to gills. 49. How many sheets in 5 bales of paper? 50. Reduce 3 wk. 6 da. 5 hr. to minutes. REDUCTION ASCENDING. 210. 1. Reduce 1306 gills to higher denominations. 1306 gi. Solution. — Since in 1 pt. there are 4 gi., 326 pt 4- 2 si. '^"^ -^'^^^ »^* t^6^^ ^^6 ^s many pints as 4 gi. "TTTT; 7 is contained times in 1306 gi. , or 326 pt. and ^ * 2 gi. remainder. 40 gal. -t-3 qt. since in 1 qt. there are 2 pt., in 326 pt. 40 gal. 3 qt. 2 gi. there are as many quarts as 2 pt. is con- tained times in 326 pt. , or 163 qt. Since in 1 gal. there are 4 qt. , in 163 qt. there are as many gal- lons as 4 qt. is contained times in 163 qt., or 40 gal., and 3 qt. remainder. Therefore, in 1306 gills there are 40 gal. 3 qt. pt. 2 gi. 144 COMPOUND NUMBERS. 2. How many rods in 334 yd. ? 5^ yd. 334 yd. Solution. — Since in 1 rd. 2" 2 there are 5^ yd., in 334 yd. 11 half yd. | 668 half yd. ^^^^re are as many rods as 60 rd +8 half vd ^^ Y^- is contained times in •^ * 334 yd., or 60 rd., and 4 yd. remainder. 334 yd. -^ 5| yd. = 668 iialf yd. -=-11 lialf yd. 8 half yd. = 4 yd. 334 yd. = 60 rd. 4 yd. 3. Reduce 225932 in. to miles, etc. 4. How many miles and rods are there in 35640 ft. ? 5. Reduce 19922544 sq. in. to higher denominations. 6. Reduce 762051 cu. in. to cu. yards, etc. 7. How many cords in 7424 cu. ft. ? 8. Reduce 69056 oz. to tons, etc. 9. Reduce 21076 gr. to higher denominations. 10. Reduce 1947 gi. to gallons, etc. 11. How many bales in 24000 sheets of paper? 12. Reduce 39180 min. to weeks, etc. 13. Reduce 5762 far. to higher denominations. 14. Reduce 84623" to higher denominations. 15. Reduce 62341 ni to higher denominations. 16. How many chains, etc., in 13025 li. ? 17. How many bushels, etc., in 35842 pints? 18. How many pounds, etc. (Troy) in 32563 gr. ? 19. Reduce 39632 gr. to lb., etc. (Apoth.). 20. How many tons, etc., in 35682 lb. ? 21. A box contains 75832 pens. How many Gr. gross, etc., in the box ? 22. Change 1384 dry pints to higher denominations. 23. In 139843 sq. in. how many square miles, rods, etc. ? 24. How many cords of wood in 3692 cu. feet ? REVIEW. 145 REVIEW PROBLEMS. 211. 1. Bought 2 gal. 8 oz. of fluid extract at 20^ an ounce, and sold it at 15^ an ounce. What was lost ? 2. How many minims are there in 10 fluid ounces (f. S ), 7 fluid drachms (f. 3) ? 3. Find the difference in value b^ween 4 gal. of ammo- nia water at 10 cents a pint and 8 ounces of cinnamon water at 5 cents an ounce. 4. The pendulum of a certain clock beats seconds. How many times will it tick in 1 day, 9 hours, 25 minutes ? 5. How many degrees in 3492.58 statute miles, measured on the equator, a degree being equal to 69.16 statute miles ? 6. How many degrees of longitude will a steamship pass through, sailing due west on the equator, at the rate of 15 knots an hour for 5 days ? Note. — A knot = 1 geographic mile or minute. 7. Find cost of each of the following: (a) 5 gallons, 3 qt. 1 pt. of molasses at 20^ a gallon ; (b) 10 acres, 50 sq. rd. of land at $ 50 an A. 8. What will it cost to build 112 rd. 3 yd. of fence at 48^ a yard ? 9. If a man steps 2^ ft. at each step, how many mile's will he travel in stepping 4820 times ? 10. If 17 ft. is f of the height of a tree, how high is the tree? 11. Eeduce 6| mi. 317 rd. 4 yd. 2 ft. to feet. 12. Change 16571 ft. to miles. 13. At $3.20 a bu. how many quarts of nuts can be bought for $ 4.80 ? 14. How many pint bottles of camphor may be filled from 96 fluid ounces (f . 3 ) ? 146 COMPOUND NFMBEES. 15. Find the cost of the following: 4 oz. iodine at 10^, 8 oz. spts. camphor at 5^, 10 oz. aqua ammonia at 10^, 14 oz. cinnamon water at 5^. 16. Reduce 4 bu. 3 pk. 7 qt. 1 pt. to pints. 17. How many quart boxes will hold 2 bu. 3 pk. 5 qt. of berries ? 18. If 4 bu. of bef ries are bought for $ .70 per bushel and sold for $ .05 per quart, what is the gain ox loss ? 19. Both sides of a railroad track are fenced with wire for 40 yards. What is the cost of the fence at 4^ a foot. ? 20. What will 8 lb. 6 oz. of sugar cost at 8^ a pound ? 21. When pens are bought at 75^ a gross, and sold at 2 for 3^, what is the gain ? 22. If a man can walk 10 miles in 2 hours, how far can he walk in 6 hours ? In 30 minutes ? In 50 minutes ? 23. What will ^ bu. berries bring at 8^ a quart ? 24. A silver chain weighs 18 pwt. What is its value, when silver is worth $ .65 an ounce ? 25. What will 24 qt. of milk cost at 20^ a gallon ? 26. If I buy peanuts at 5^ a quart, and retail them so as to gain $ 6.40 on 4 bushels, what do I sell them for ? '27. At 4 pens for 3 cents what will 1 great gross cost ? 28. How many table-forks, each weighing 2J oz., can be made from 4 lb. 4 oz. 10 pwt. of silver ? 29. In f of a gallon how many pints ? 30. How many rods of fence will enclose a mile square of land ? 31. What is the cost of 1 yd. and 27 inches of fringe at 60 cents a yard ? 32. How many rods of fence are required to enclose a lot that is 20 rods wide and three times as long ? BEDUCTION. 147 33. Required the distance around a room that is 13 feet long and 15 feet wide. 34. A shoe-box is 4 in. deep, 6 in. wide, and 12 in. long. How much twine will it take to wind twice around the box each way to hold on the cover, allowing 6 inches for tying.'/ 35. I have a lawn that is 30 ft. by 70 ft., and wish to lay a board walk around it that is 3 ft. 6 in. in width. What is the distance around the walk, outside measurement ? 212. A Denominate Fraction is a fraction having a de- nomination. 213. To reduce denominate fractions to Integers of Lower Denominations. 1. Reduce f of a mile to rods, yards, feet, etc. SoLDTioK. — f of 320 rd. = i-«^ rd. = 228f rd. I of -\i yd. = ff yd. = 3f yd. f of 3 ft. = Of ft. f of 12 in. = -3^ in. = 5} in. f of a mile = 228 rd. 3 yd. ft. 5^ in. Note. — The same process applies to denominate decimals. 2. Reduce .87 bu. to pecks, quarts, etc. .87 bu. .87 of 4 pk. = 3.48 pk. 4 3.48 g .48 of 8 qt. = 3.84 qt. 3.84 2 '.84 of 2pt. =1.68pt. 1.68 .87 bu. z= 3 pk. 3 qt. 1.68 pt. Rule. — Change the given fraction (or decimal) to the next lower denomination. Treat the fractional (or decimal) part of the product in the same way, and so proceed to the required denominatioii. 148 COMPOUND NUMBERS. Eeduce to integers of lower denominations. 3. I of a mile. 9. .375 of a month. 4. f of an acre. 10. .3125 of a gallon. 5. I" of a pound (Troy). 11. .4267 of an acre. 6. f of a ton. 12. .2364 of a ton. 7. f of a gallon. 13. .363 of a sign. 8. I of a mile. 14. .51625 of a mile. 15. Reduce ^| mi. to lower denominations. 16. Change f of a year to months and days. 17. In -^ gal. how many qt. and pt. ? 18. Reduce -^^ lb. to oz. and dr. 19. -f^ acre are equal to how many sq. rods, etc. ? 20. Reduce f^ bu. to integers of lower denominations. 21. What is the value of ^ of ^ of a hhd. in integers of lower denominations ? 22. What is the value of ^7^ of an acre in integers of lower denominations ? 23. Reduce £ J to integers of lower denominations. 24. W^hat is the value of ^ of 1^ of a mile ? 214. To reduce denominate numbers to Fractions of Higher Denominations. 1. Reduce 2 qt. 1 pt. 2 gi. to the fraction of a gallon. Solution. — 2 gi. -^ 4 = | pt. = ^ pt. Upt. = f pt. . f pt. -2 = |qt. 2f qt. = J^ qt. -^ 4 = \\ gal. Ans. 2. Reduce 2 qt. 1 pt. 2 gi. to the decimal of a gallon. BEDUCTION. 149 gi. Rule. — Change the number of the lowest 1.5 pt. denomination to a fraction (or deci- 2.75 qt. mal) of the next higher, write this 6875 eal fraction (or decimal) as a part of the number of that higher denominor Hon, and reduce this number in like manner, and so proceed to the required denomination. 3. Reduce 213 rd. 1 yd. 2 ft. 6 in. to a fraction of a mile. 4. What fraction of an acre is 3 E,. 13 sq. rd. 10 sq. yd. 108 sq. in. ? 5. What part of a year is 273 da. 18 hr. ? 6. Reduce to a fraction of a pound 8 oz. 11 pwt. 10|- gr. 7. What part of a ton is 857 lb. 2f oz. ? 8. Change 3 fur. 19 rd. 5 yd. 1 ft. 4.7328 in. to the deci- mal of a mile. 9. Reduce 1 da. 14 hr. 24 min. to the decimal of a month. 10. What decimal of a gallon is 1 qt. 2 gi. ? 11. Reduce 68 sq. rd. 8 sq. yd. 2 sq. ft. 7.488 sq. in. to the decimal of an acre. 12. What decimal of a pound Troy is 6 oz. 3 pwt. 21.6 gr.? 13. Reduce 131 da. 18 hr. 21 min. 36 sec. to the decimal of a year. 14. Reduce 2 qt. If gi. to the fraction of a gallon. 15. What fraction of a mile is 71 rd. 1 ft. 10 in. ? 16. Reduce 12 da. 34 min. 17^^ sec. to the fraction of a month. 17. What decimal of a ton is 4 cwt. 72 lb. 128 oz. ? 18. Reduce 48 cu. ft. 1636.7616 cu. in. to the decimal of a cord. 150 COMPOUND NUMBERS. 19. What decimal of a circle is 10° 53' 24" ? 20. Reduce 4 fur. 5 rd. 1 yd. 3.6 in. to the decimal of a mile. 21. Reduce 6 pwt. to a fraction of a pound. 22. 3 qt. 1 pt. 2 gi. are what part of a gallon ? 23. Change 6 rd. 4 yd. 1 ft. to the fraction of a mile. 24. What part of a cord of wood are 8 cu. ft. ? 25. Reduce 5 gross 7 doz. to the fraction of a great gross. To find what part one denominate number is of another. I. What part of 2 gal. 1 qt. 1 pt. is 3 qt. 1 pt. 1 gi. ? 3 qt. 1 pt. 1 gi. = 29 gi. 2 gal. 1 qt. 1 pt. = 76 gi. The question now is, 29 gi. is what part of 76 gi. ? 29 gi. is ^ of 76 gi. Ans. Note. — To find the decimal part, divide numerator by denominator. 2. What part of 5 lb. 9 oz. 3 pwt. is 2 lb. 8 oz. 6 pwt. 10 gr. ? 3. What part of 3 mi. 24 rd. 5 yd. is 2 mi. 34 rd. 4 yd. ? 4. What part of 3 da. 5 hr. 22 min. is 1 da. 10 hr. 3 min. 12 sec. ? 5. What decimal of 3 gal. 2 qt. 1 pt. is 2 gal. 2 qt. 2pt.? 6. What decimal of 4 T. 5 cwt. 10 lb. is 2 T. 6 cwt. 13 lb. ? 7. What part of a rod is 4 yd. 2 ft. 7 in. ?' 8. What part of 6 rods is f of 7 feet ? 9. What part of 3| mi. is 160 rd. 5 yd. ? 10. ^ pint is what part of a bushel ? II. What decimal of 8 bu. 3 pk. 4 qt. is 4 bu. 1 pk. 5 qt. ? ADDITION. 151 ADDITION OF COMPOUND NUMBERS. 215. 1. Add 14 lb. 5 oz. 17 pwt. 12 gi., 18 lb. 10 oz. 14 gr., 6 lb. 4 oz. 8 pwt. 16 gr. lb. oz. pwt. gr- 14 5 17 12 18 10 14 6 4 8 16 39 8 6 18 Solution. — The sum of the grains = 42 gr. = 1 pwt. 18 gr. We place the 18 gr. under the column of grains, and add the 1 pwt. to the col- umn of pennyweights. Add the other columns in like manner. rd. yd. ft. rd. ft, 2. 17 4 1 3. 6 12 12 4 2 4 14 11 6 5 n 17 15 9 8 3 2 6 12 8 46 H n 36 5^ 10 u= .^yd. 6 = ^ft. 46 2 m tons. cwt. lb. oz. 4. 14 13 Q>^ 15 13 17 88 11 46 16 86 13 14 15 57 6 11 17 85 15 deg. min. sec. 6. 29 59 59 15 45 42 18 11 40 13 19 17 sq. yd. sq. ft. sq. in. 6. 45 8 113 45 3 112 75 8 139 49 115 589 8 90 yr. da. hr. min. see. 18 345 13 37 15 87 169 12 16 28 316 144 20 53 18 13 360 21 57 15 152 COMPOUND NUMBERS. bu. pk. qt. pt. cd. cd. ft. ctu ft 8. 40 2 6 1 9. 5 7 89 1 3 2 2 12 75 2 1 1 6 15 69 2 3 '^1 49 1 3 2 1 1 1 3 2 65 10. Find the sum of 3 T. 15 ewt. 25 lb. 9 oz., 4 T. 17 cwt. 30 lb. 10 oz., 6 T. 18 cwt. 15 lb. 12 oz., 2 T. 12 cwt. 20 lb. 16 oz. 11. Find the sum of 7 hr. 30 min. 45 sec, 12 hr. 25 min.. 30 sec, 20 hr. 15 min. 33 sec, 10 hr. 27 min. 46 sec 12. Add 10 mi. 101 rd. 3 yd. 2 ft. 11 in., 16 mi. 4 yd. 6 in., 3 mi. 560 rd. 3 ft., 175 rd. 4 ft. 7 in. 13. Add 3 A. 50 sq. rd. 25 sq yd. 10 sq. ft. 102 sq. in., 5 A. 110 sq. rd. 30 sq. yd. 8 sq. ft. 34 sq. in., 6 A. 75 sq. rd. 14 sq. yd. 7 sq. ft. 82 sq. in., 7 A. 215 sq. rd. 17 sq. yd. 16 sq. ft. 53 sq. in. 14. Find the sum of 18 cd. 6 cd. ft. 12 cu. ft., 19 cd. 4 cd. ft. 6 cu. ft., 24 cd. 2 cd. ft. 1 cu ft. 15. Find the sum of 18 T. 18 lb. 12 oz., 16 cwt. 21 lb., 14 cwt. 75 lb. 10 oz. 16. What is the entire length of a railway consisting of 5 different lines measuring respectively 160 mi. 185 rd. 2 yd., 97 mi. 63 rd. 4 yd., 126 mi. 272 rd. 3 yd., 67 mi. 199 rd. 5 yd., and 48 mi. 266 rd. 5 yd. ? 17. A merchant sold 48 gal. 3 qt. 1 pt. of coal oil and had 15 gal. 1 qt. 1 pt. left. What quantity had he at first ? 18. A starts from a point in Lat. 21° 25' 35" N. and travels north 24° 36' 45". At what latitude does he arrive ? 19. Find the difference in longitude between a point 46° 15' 30" E. and a point 21° 18' 16" W. SUBTRACTION. 153 Note, — When one place is in east and the other in west longi- tude, their difference in longitude is the sum of their longitudes. 20. Charles walks 5 mi. 15 rd. 2 ft. north of the school- house, and James 6 mi. 28 rd. 5 yd. south. How far are they apart? 21. Find the sum of f mi. 35 rd. 4| rd. Note. — Reduce each to integers of lower denominations, then add. 22. Add f bu. 17| pk. 4f pt., 6 bu. 3| pk. 2 qt., J bu. J pk. 5 qt. 23. What is the sum of f T. f cwt. and f lb. ? SUBTRACTION OF COMPOUND NUMBERS. lb. oz. pwt. gr. Solution. — 15 gr. - 12 216. 1. From 6 2 14 15 S^- = '^ ^'- ^« ^^ ^^^"^^ Take 4 10 18 12 ^^^^, !f T'' 'T l^ ^"\ we take 1 oz., which equals 1 3 16 3 20 pwt., and add to the 14 pwt. = 34 pwt. ; 34 pwt. — 18 pwt. =16 pwt. We have taken 1 oz. from the 2 oz., leaving 1 oz. ; but as we cannot take 10 oz. from 1 oz., we take 1 lb. = 12 oz., and add it to 1 oz. = 13 oz., from which take 10 oz. = 3 oz. Since we took 1 of the 6 lb., we have 5 left ; from which take 4 lb. = 1 lb. 2. 3. A. sq. rd. i sq. ft. hr. min. sec. From 10 50 7 5 54 30 Take 4 106 5 1 17 50 4. '> 5. gal. qt. pt. gi. A. R. sq. rd. sq. yd. sq. ft From 39 2 2 1 5 1 39 15 7 Take 16 2 3 3 6. 2 2 26 • 21 7. 8 da. hr. miu. sec. T. cwt. lb. oz. 200 17 54 36 20 15 75 10 135 20 24 48 5 16 25 12 154 COMPOUND NUMBERS. 8. From 260 mi. take 23 mi. 7 fur. 25 rd. 5 yd. 2 ft. 10 in. 9. A man having i an acre of ground sold 25 sq. rd. 11 sq. yd. 8 sq. ft. to one man, and 50 sq. rd. 9 sq. yd. 4 sq. ft. to another. How much land had he left ? 10. From 12 cwt. subtract 9 cwt. 14 lb. 12 oz. 11. From a hogshead of molasses 25 gal. 3 qt. 2 pt. were drawn at one time, and at another time 10 gal. 1 pt. How many gallons remained ? . 12. From 2 bu. 3 pk'., 1 bu. 2 pk. 6 qt. were sold. How much remained ? 13. An apothecary bought 2 lb. of quinine, and sold 1 lb 3 oz. 5 dr. 2 sc. 11 gr. How much had he left ? 14. What is the difference in longitude between. New York 74° 0' 3" W. and San Francisco 122° 25' 40" W. ? Note. — When both places are in east or in west longitude, their difference of longitude is found by subtraction. 15. Charles walks 22 mi. 4 rd. 2 yd. south of the school, and Henry 16 mi. 160 rd. 3 yd. in the same direction. How far are they apart ? 16. From J of a mile take 16\ rd. Note. — Change both to integers of lower denominations, then subtract. 17. Rome is in longitude 12° 28' 40" E., and Paris in longitude 2° 20' 14" E. What is their difference in longi- tude ? 18. From | of a pound Avoir, take 3f oz. 19. From 16| bu. take 71 pk. 20. Take .325 T. from 6.54| cwt. 21. Take .7 of a rod from 4 yd. 2 ft. 8 in. 22. From 22 da. 16 hr. 20 min. take 2^ weeks. SUBTRACTION. 155 DIFFERENCE BETWEEN DATES. 217. 1. Find the time from Jan. 25, 1842, to July 4, 1896. 1896 7 4 Solution. — It is customary to con- . „ . ,. . „^ sicler 80 days to a month. July 4, 1896, is ^ ^ 1 wO ^^^ 1896th yr. 7th mo. 4th da., and Jan. 54 yr. 5 mo. 9 da. 25, 1842, is the 1842d yr. 1st mo. 25th da. Subtract, taking 30 da. for a month. 2. What is the exact number of days between Dec. 16, 1895, and March 12, 1896 ? j-/c«^. Ltj Solution. — Do not count the first day Jan. 31 mentioned. There are 15 days in December, Feb. 29 after the 16th. January has 31 days, Feb- Mar. 12 ruary 29 (leap year), and 12 days in March ; ~ - making 87 days. Ans. ■ 3. How much time elapsed from the landing of the Pil- grims, Dec. 11, 1620, to the Declaration of Independence, July 4, 1776 ? 4. How much time elapsed from the beginning of the Civil War, April 14, 1861, to the close of the war, April 9, 1865? 5. Washington was born Feb. 22, 1732, and died Dec. 14, 1799. How long did he live ? 6. Washington was first inaugurated April 30, 1789. How long ago was his inauguration ? 7. How much time will have elapsed since Columbus discovered America, Oct. 12, 1492, to your next birthday? 8. Mr. Smith gave a note dated Feb. 25, 1896, and paid it July 12, 1896. Find the exact number of days between its date and time of payment. 9. A carpenter earning $ 2.50 per day, commenced Wednesday morning, April 1, 1896, and continued work- ing every week day until June 6. How much did he earn ? 156 COMPOUND NUMBERS. 10. Fred was born Dec. 20, 1875 ; how old is he now ? 11. How much time has elapsed since George Washing- ton was 15J years old ? 12. General Grant was born April 27, 1822. How old would he be if he were alive to-day ? 13. How long since Lee surrendered to General Grant? 14. Find the exact number of days between Jan. 10, 1896, and May 5, 1896. 15. When can a boy who was born May 5, 1882, cele- brate his 25th birthday ? 16. John goes to bed at 9.15 p.m. and gets up at 7.10 a.m. How many minutes does he spend in bed ? MULTIPLICATION OF COMPOUND NUMBERS. 218. 1. Multiply 4 yd. 2 ft. 8 in. by 8. Solution. — 8 times 8 in. = 64 in. = 5 ft. 4 in. ^ ■ ' ' Place the 4 in. under the inches' column, and reserve the 5 ft. to be added to the product of 2 ft. by 8, 5 which equals 16 ft. (add 5 ft.)= 21 ft. 21 ft. -- 3 39 4 =7 yd., with no remainder. Add 7 yd. to the product of 4 yd. by 8 = 32 yd. (add 7 yd.) = .39 yd. 2. gal. qt. pt. gi. bu. pk. qt. pt. 31 3 2 3 12 3 2 1 5 8 /-^/ / / A 3. If a man travel at the rate of 60 mi. 240 rd. 16 ft. in one day, how far will he travel in ten days ? 4. A man owns 6 farms, each containing 75 A. 49 sq. rd. 25 sq. yd. of land. How much land in all the farms ? 5. If 6 loads of hay weigh 6 T. 18 cwt. 75 lb., how much will 48 loads weigh ? Note. — 48 loads will weigh 8 times as much as 6 loads. MULTIPLICATION. 157 6. If 12 spoons weigh 3 lb. 8 oz. 15 pwt., how much will one gross of spoons weigh? 7. How much oil will 7 barrels hold if each barrel con- tains 35 gal. 2 qt. ? 8. What is the value at ^ 4 per cord of 10 piles of wood, each containing 5 cd. 5 cd. ft. 12 cu. ft. ? 9. What is the weight of 15 packages, each weighing 1 lb. 4 oz. (Avoir.) ? 10. If a bicyclist travels 75 mi. 140 rd. in one day, how far can he travel in ten days ? 11. In a watch-chain there are 2 oz. 12 pwt. 15 gr. of gold. How much gold is required for 25 such chains ? 12. A farmer has six bins, each containing 60 bu. 2 pk. of wheat. How much wheat has he ? 13. If a train is run for 8 hours at the average rate of 50 mi. 30 rd. 10 ft. per hour, how great is the distance covered ? 14. It takes John Smith 5 hr. 20 min. 11 sec. to plough one acre of ground. At the same rate, how long will it take him to plough 4 acres ? 15. 4 gal. 3 qt. 1 pt. X 11 = ? 16. 2 A. 40 sq. rd. 16 sq. yd. x 20 = ? DIVISION OF COMPOUND NUMBERS. 219. 1. Divide 16 lb. 9 oz. 17 pwt. 8 gr. by 10. Solution. — J^ of 16 lb. = 1, and 6 lb. remaining. 6 lb. = 72 oz. IK ^r, «^f „. 72 oz. + 9 oz. = 81 oz. ^ of 81 oz. = 8 oz., lb. oz. pwt. gr. 1" ' 10 "116 9 17 8 ^'^^^ '^ °^* ^^maining, = 20 pwt., to which ^—^ — • add 17 pwt., = 37 pwt. J^ of 37 pwt. = 3 TIF pwt., with 7 pwt. remaining, = 168 gr., to which add 8 gr. ; and taking ^-^ of the sum, we have 17j% gr, "\Mien the divisor is large, employ long division. 158 COMPOUND NUMBERS. 2. Find ^ of 42 rd. 4 yd. 2 ft. 8 in. Solution. — ^^ of 42 rd. = 1 rd. ; re- mainder, 7 rd. = 38^ yd. ; add 4 yd. = 421 yd. ^1^ of 42-1 yd. = 1 yd. ; remainder, 7-1 yd., = 22ift. ; add 2 ft. = 24^ ft. j\ of 24^ ft. = ft. 24^ ft. = 294 in. ; add 8 m. = 302 in. ^^ of 302 in. = 8|f in. Note. — When both dividend and di- visor are compound, reduce them to tbe same denomination, and divide. The quotient will be abstract. 3. Divide 169 bu. 3 pk. 5 qt. by 7. 4. If a man travelled 607 mi.. 169 rd. 11 ft. 6 in. in 10 days, what average distance did he travel in 1 day ? 5. If one gross of spools weighs 44 lb. 9 oz., how much will one dozen weigh ? 6. If one bottle holds 1 pt. 3 gi., how many dozen bottles will be required to hold 65 gal. 2 qt. 1 pt. ? 7. A man has 451 A. 138 sq. rd. 29 sq. yd. of land, which he wishes to divide equally among his six children. How much land will each child receive ? 8. If 12 persons share equally in the contents of a bin containing 20 bu. 2 pk. 4 qt. of apples, what is the share of each ? ,9. If the entire area of 24 equal fields is 242 A. 20 sq. rd. 15 sq. yd., what is the size of each field ? rd. yd. ft. in. 35)42 4 2 8(1 rd. 35 7 ^ H 35 38i + 4 35)42J-yd. (1 yd. 35 H 3 221 ft. + 2 35)241. ft. (Oft. 12 294 + 8 35)302(8^2 ii, I. 280 22 1 rd. 1 yd. m in. Ans. DIVISION. 159 10. A man walked 50 mi. 71 rd. 2 yd. in 15 hours. What was his rate per hour? 11. If it takes a man 12 hr. 35 min. 15 sec. to walk 45 miles, what is the average time taken for each mile? (Divide by the factors of 45.) 12. AVhen .f 12 will buy 11 gal. 2 qt. 1 pt. of maple syrup, how much will $1 buy? 13. A man travelled 100 miles in 9 hours. What was the average rate per hour? 14. If a horse eats 12 qt. of oats per day, how long will 10 bu. 1 pk. 4 qt. last him ? 15. If a package weighs 4 cwt. 15 lb., how many such packages will it take to weigh 3 T. 2 cwt. 25 lb. ? 16. A man had 5 acres of land which he divided into 12 equal parts. How much land did each part contain ? 17. Divide 102 T. 15 cwt. 27 lb. 9 oz. by 8. 18. Divide 16 bu. 3 pk. 6 qt. by 2 bu. 1 pk. 19. I have 84 lb. 14 oz. of salt which I wish to put into packages of 2 lb. 6 oz. each. How many packages will there be? 20. If a horse eats 1 pk. 2 qt. of oats a day, how many days will 16 bu. 3 pk. 6 qt. last him ? 21. How many sacks, each containing 2 bu. 3 pk. 2 qt., will be needed to hold 165 bu. 2 pk. of meal ? 22. 16 cwt. 75 lb. 9 oz. of butter are to be jjut into jars each containing 9| lb. How many jars will be needed ? To multiply or divide a compound number by a fraction. Note. — To multiply by a fraction, multiply by the numerator, and divide the product by the denominator. To divide by a fraction, multiply by the denominator, and divide the product by the numerator. 160 COMPOUND NUMBERS. 23. How much is f of 16 hr. 17 min. 14 sec. ? 24. How much is J of 30 S ^ 3 1 3 8 gr. 25. Divide 120 cd. 50 cd. ft. 34 cu. in. by f . 26. How many times is ^|^ contained in 840 T. 15 cwt. 98 lb. 3 oz. ? . 27. A man sold 4 bu. 3 pk. 2 qt. of potatoes, which was ■J of what he raised. How much did he raise ? 28. A butcher sells 120 tons, 9 cwt. 75 lb. of beef every month. How much does he sell in | of a month ? 29. If 6 bottles hold 5 gal. 2 qt. of milk, how much milk will 3 such bottles hold ? 30. A field contains 10 acres 12 sq. rd. of land, which is f the size of the whole farm. Find the size of the farm. • 31. 'A railroad track extends 144 miles, 40 rd. 3 yd. How far has a train of cars gone which has travelled -^^ of this distance ? 32. If a pipe discharges 25 gal. 3 qt. 1 pt. of water in 1 hr., how much will it discharge in 5j hr., if the water flows with the same velocity ? Note. — When the multiplier or divisor is a mixed number, reduce to an improper fraction, and proceed as above. 33. Divide 8 lb. 11 oz. 15 pwt. 18 gr. by 2|. 34. If a railroad train runs 60 mi. 35 rd. 16 ft. in one hour, how far will it run in 12| hr. at the same rate of ' speed ? 35. Divide 14 bu. 3 pk. 6 qt. 1 pt. by |. 36. Divide 5 yr. 1 mo. 1 wk. 1 da. 1 hr. 1 min. 1 sec. by 3|. MISCELLANEOUS PROBLEMS. 161 MISCELLANEOUS PROBLEMS. 220. 1« Name two numbers which multiplied together make 14. 2. Write three sets of factors for 24. 3. Find the prime factors of 2205. 14x6x3x2x8 ^^ •5x6x2x9x24"' 5. How many yards of silk, 24 inches wide, will it take to line a skirt containing six yards of cloth 28 inches wide ? 6. Find the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9. 7. Find the greatest common divisor of 285, 465. 8. Find the smallest number that will exactly contain 9, 15, 18, 20. 9. Find the length of the longest stick that will exactly measure the sides of a room 216 yd. by 111 yd. 10. What is the smallest sum of money with which you can buy pears at 75^ a basket, peaches at 90^, and grapes at 50^, using the same amount of money for each kind ? 11. How many times is 1 contained in i ? 12. How many times is \ contained in 1 ? 13. A man bought a horse for $ 240, which is f of what he sold it for. What did it sell for ? 14. A man bought a horse for f 240, and sold it for 4 of what he paid for it. What did it sell for ? 15. A pole stands \ in the mud, \ in the water, and the remaining 10 feet are above the water. How long is the pole? 16. A man owns 4 farms containing 3651 375f, 284|, and 254-1 acres respectively. How many acres in all ? 162 MISCELLANEOUS PROBLEMS. 17. The man owning the above farms sells A 234/^ acres, and B 366/^ acres. How many acres has he left ? 18. What is the value of 3| x y^^ X f X 14 x Sj^ x ^ xAxfx^xi? 19. Find the least common multiple of 273, 462, 1785, and 399. 20. A man owns a farm containing 400 acres. He sells J of the farm, and divides the remainder among his six children. How many acres does each child receive ? 21. Find the sum of 3 bu. 6 pk. 2 qt. 1 pt, 3 pk. 1 qt 1 pt., 7 bu. 3 pk., 4 bu. 7 qt. 1 pt., and 19 bu. 2 pk. 2 qt. 1 pt. 22. Find the sum of 4 lb. 6 oz. 21 pwt. 9 gr., 5 oz. 11 gr., 3 lb. 9 oz. 18 pwt., 11 oz. 17 pwt. 5 gr., 16 lb. 4 oz. 11 pwt., 18 lb. 17 gr., 21 lb. 15 pwt. 11 gr., 23 lb. 10 oz. 21 pwt. 23 gr. 23. What part of a mile is 214 rd. 2 yd. 2 ft. 3 in. ? 24. Keduce 1 pk. 4 qt. 1| pt. to the fraction of a bushel? 25. What is the quotient of 184 bales, 4 bundles, 1 ream, 13 quires, 20 sheets, divided by | ? 26. A farm is 60 ch. 25 1. long. How many rods long is it? 27. A surveyor measured my farm, and found that it is 80 ch. long and 60 ch. broad. How many acres does it contain ? 28. In 133128 in. how many miles? 29. How many times will a wheel 12 ft. 3 in, in circum- ference turn round in going 15 mi. 20 rd. 12 ft. 2 in. ? 30. How many steel rails 30 ft. long are needed in the construction of 7 mi. 305 rd. 7 ft. 6 in. of double-track rail- road? MrSCELLANEOUS PROBLEMS. 163 31. What fraction of the year is contained in the months of July and August, 1896 ? 32. The greatest depth of the Atlantic telegraph cable is 2 mi. 250 rd. 5 yd. 1 ft. How many feet deep is it ? 33. How many statute miles in 45° 22' 30", measured on the equator ? 34. On an average, when walking, Isaac steps 24 inches twice every second. How many minutes will it take him to walk 1^ miles ? 35. Reduce -^-^ of an acre to square rods and decimals of a square rod. 36. A silversmith in making spoons uses 2 lb. 3 oz. 19 pwt. of silver in one day, 3 lb. 18 pwt. 20 grs. on the sec- ond day, and 11 oz. 19 pwt. 23 gr. on the third day. How much silver does he use altogether ? 37. What decimal of a pound Troy are 4 oz. 14 pwt. ? 38. Reduce 18 pwt. 164 gr. to the fraction of a pound. 39. Find the difference between | lb. and 4 lb. 7.84 oz. Troy. 40. Find the cost of 21 doz. spoons, each weighing 9 oz. 8.76 pwt., at ^.045 a pennyweight. 41. What decimal part of a grain is j-^-^-^ of a pound ? 42. AVhat part of an acre is f sq. rd. ? 43. Multiply 16 bu. 9 pk. 8 qt. by |. 44. What is the product of 20 lb. 9 oz. 11 pwt. 15 gr. multiplied by | ? 164 MEASUREMENTS. An Angle. ME A S U RE ME NTS, 221. An Angle is the difference in direction of two lines. 222. A Right Angle is the angle of a square. 223. Anything that has length and breadth, but not thickness, is a Surface. 224. A surface that does not change its direction is a Plane Surface. 225. A figure having four straight sides and four right angles is a Rect- angle. 226. A Square is a rectangle having equal sides. A Right Angle. A Rectangle. 227. A Square Foot is a square 1 foot long and 1 foot wide. 228. A Square Yard is a square 1 yd. long and 1 yd. wide. 229. The Area of a surface is the number of square units that it con- tains. 6 IN. z A Square. Note, — There are 6 sq. in. in a row, and in 4 rows there are 4 times 6 sq. in. =24 sq. in. The multiplier is abstract, and the unit of the product must be the same as the unit of the multiplicand. MEASUREMENTS. 165 230. The lengtli and breadth of a rectangle are called its Dimensions. Length x Breadth = Area. Area -r- Length = Breadth. Area -i- Breadth = Length. 231. A figure having three straight sides and three angles is a Triangle. The Base of a triangle is the line upon which it stands, and the Altitude is its height above the base, or the base extended. Thus, AC is the base, and BD the altitude, of the triangle shown below. ( It is evident from the accompanying figure that the Y area of a triangle is equal to one-half the area of a rectangle \of the same base and altitude. 232. Every circle may be regarded as composed of many equal triangles, the radius of the circle forming the alti- tudes, and the circumference forming the sum of the bases. Therefore, the area of a circle is equal to i the prod- uct of the circumference and radius. Principle. — The circumference of a circle is 3.1416 times the diameter, or about 3^. 166 MEASUREMENTS. 233. Circumference -7- 3.1416 = Diameter. Diameter x 3.1416 = Circumference. Oral. Find the areas of rectangles as follows : 1. 10 ft. by 8 ft. 4. 50 ft. by 20 ft. 2. 16 ft. by 4|- ft. 5. 6 ch. by 8 ch. 3. 14 rd. by 10 rd. 6. 9 yd. by 6 yd. Find the other dimension : 7. Area 24 sq. ft., length 8 ft. 8. Area 72 sq. yd., length 8 yd. 9. Area 100 sq. in., breadth 5 in. 10. Length 16 yd., area 64 sq. yd. 11. Breadth 4 ft., area 84 sq. ft. 12. Area 56 sq. ch., length 8 ch. Find the areas of the following triangles : 13. Base 10 ft., alt. 12 ft. 16. Base 5 ft., alt. 10 ft. 14. Base 9 yd., alt. 6 yd. 17. Base 12 in., alt. 8 in. 15. Base 15 in., alt. 6 in. 18. Base 10 rd., alt. 5^ rd. Find the circumferences of circles having the following diameters : Note. — Indicate the operation only. 19. 12 ft. 21. 16 in. 23. 16 rd. 25. 62 yd. 20. 18 ft. 22. 14 yd. 24. 25 ch. 26. 84 ft. Find the diameters having the following circumferences : Note. — Indicate only. 27. 78 ft. 28. 19 ft. 29. 316.14 rd. 30. 189.68 ch. Fi^DiisG arp:as. 167 Written. Find areas : 31. Circumference 37.6992 ft., radius 6 ft. 32. Circumference 47.124 ft., diameter 15 ft. 33. Circumference 62.832 ft., diameter 20 ft. 34. Circumference 94.248 ft., diameter 30 ft. 35. Diameter 24 ft. 38. Diameter 160 ft. 36. Radius 16 ft. 39. Radius 62 ft. 37. Circumference 50 ft. 40. Circumference 314.16 ft. 41. How many square yards are there in a floor 24 ft. long and 15 ft. wide ? 42. The base of a triangle is 20 ft. and the altitude 18 ft. What is the area ? 43. The circumference of a circle is 31.416 ft. and its radius is 5 ft. What is its area? 44. When the diameter of a circle is 50 ft., what is the circumference ? 45. When the radius is 6 ft., what is the circumference ? 46. When the circumference is 78.54 ft., what is the radius ? 47. Mr. Clark's farm is 35 ch. long and 25 ch. wide. How many acres does it contain ? 48. A certain field is 70 rd. long and 65 rd. wide. How many acres are there in the field ? 49. A piece of land is 65 rd. wide. How long must it be to contain 56^ acres ? 50. How many sods 10 inches square will be required to turf a lawn 100 ft. long and 50 ft. 6 in. wide ? 51. A building lot measures 60 ft. in front. What must be its depth to contain \ of an acre? \^ \''i»^X'^ • 168 MEASUREMENTS. ^^^ 52. How many tiles, each 8 in. square, will be required for the floor of a room 24 ft. by 30 ft. ? 53. How many shingles will be required for a roof 45 ft. long, and each of its two sides 20 ft. wide, allowing 8 shingles to the square foot ? > / 54. How many square yards of oil-cloth are needed to L--^ cover a floor 18 ft. by 24 ft. 6 in. ? 55. A owns a city lot 168 ft. long and 42 ft. wide. He ->/^ uses f of it for a lawn. How many square yards does the ■ * lawn contain? 56. How long will it take a man to mow the above lot,, if it takes him a minute to run a 2-foot lawn-mower length- wise of the lot ? 57. A pony can reach 40 feet in any direction from the stake to which he is picketed. Over how many square rods of surface can he graze ? 58. What is the diameter of a tree that is 10 ft. in circumference ? 59. A basin measures 9 in. across the top and 6 in. across the bottom. How much farther around the top than around the bottom? 60. How many acres of land are enclosed by a circular mile track ? 61. A landscape gardener lays off a circular grass-plot whose radius is one rod, and near it a semicircular plot having a radius two rods in length. Compare their areas. . 62. If the area of a triangle is 9 acres and the base is \/ 80 rods, what is the altitude ? 63. Find the area of one gable end of a building 40 ft. wide, the ridge being 15 ft. above the eaves. 64. Find the area of a triangle whose altitude is 14 ft. and its base 12 ft. 65. Find the area of a triangle whose altitude is 4 in. and base 12 in. CABPETING KOOMS*. 169 CARPETING ROOMS. 234. In making a carpet, the carpeting is cut from a roll into strips which are usually laid from end to end of the floor, or lengthwise. Sometimes the strips are laid across the room. 1. How much carpeting must I purchase to cover a room 6 yd. long and 4f yd. wide, strips running lengthwise ? Solution. — It will be necessary to purchase as much carpeting as if the room were 5 yd. wide, the excess of \ yd. being turned under in the last strip. 1 strip contains 6 yd. 5 strips = 5 times 6 yd. = 30 yd. Ans. 2. How many yards must I purchase, if the strips are laid across the room ? Solution. — 1 strip contains 4| yd. 6 strips = 6 times 4f yd. = 28-1- yd, ^j^s. Carpeting is commonly 1 yd. or J yd. wide. Note. — It is often necessary to 6 yd. purchase more than enough carpet- ing to cover a room, on account of the waste in matching patterns. This diagram represents the &oor in Example 1, in which the strips are laid lengthwise. Pupils should draw a similar diagram for each floor. Note. — Carpeting is sold by lineal yards or metres, not by square measure. 3. A merchant bought a roll of carpet containing 74 yd. at 85^ a yard, and sold it at $ 1.15 a yard. What was his profit ? 4. A roll of carpet f yd. wide contains 60 yards. How many square yards of surface will it cover ? 170 MEASUREMENTS. 5. How many strips of carpet 3 ft. wide will cover a floor 15 ft. wide ? 17 ft. wide ? 18 ft. wide ? 6. How many strips 27 in. wide are required for a floor 12 ft. wide ? 14 ft. wide ? 16 ft. wide ? 7. If, in Example 5, the room is 16 ft. long, how many linear yards of carpet will be needed to cover the floor ? 8. If, in Example 6, the room is 19 ft. long, how many yards will be required to cover the floor ? 9. How many yards of ingrain carpet J yd. wide will be required for a floor 17 ft. wide and 20 ft. long, strips run- ning across the room ? 10. How much will a carpet cost at $ .90 a yard to cover a floor 22 ft. long and 15 ft. wide, if the strips run crosswise, and no allowance is made for matching ? 11. How many yards of carpet f yd. wide will be re- quired for a floor 20 ft. long and 15 ft. wide, if the strips run across the room ? 12. How many yards of carpet 1 yd. wide will be re- quired for a floor 18 ft. long and 14 ft. wide, strips running across the room ? 13. How many yards of carpeting are needed to cover a floor 24 ft. long and 17 ft. wide, strips running lengthwise and I yd. wide ? 14. If my room is 16^ ft. long and 12 ft. wide, how many yards of carpeting 24 inches wide must I buy, if in cutting 6 inches is allowed on each strip for matching ? 15. I wish to have a carpet woven. My room is 21 ft. long and 17 ft. wide. How much carpeting, 34 inches wide, must I order to exactly cover the room, no allowance being made for matching ? PLASTEKINCi AND PAINTING. 171 16. How many yards of carpet 2^ ft. wide will cover a floor 7^ yd. long and 14 ft. wide, if strips run length- wise, and it requires ^ yd. for matching ? 17. What will it cost to carpet a room 15 ft. by 17-^ ft., with carpet 30 in. wide, at $1.20 per lineal yard, if the strips run lengthwise, and an allowance of 9 in. to each strip be made for matching? 18. How much less would be the cost with no loss for matching? (Ex. 17.) 19. A room 31 ft. by 17 ft. is to be covered with carpet- ing 30 in. wide. How many yards must be purchased, and how wide a strip must be turned under ? 20. At $ 2.50 a yard, what will be the cost of a carpet to cover a parlor floor 6 yd. long and 5^ yd. wide, if | yd. is wasted in matching ? 21. How many yards of matting li yd. wide will be required for an assembly room 85 ft. 8 in. long and 64 ft. 6 in. wide, strips running across the room ? 22. A room 17 ft. 6 in. long, 14 ft. wide, is to be car- peted with carpet J yd. wide. A border -| yd. wide goes around the outside. How many yards of border, and how many yards of carpet, if strips run lengthwise, and there is a waste of one foot on each strip for matching? PLASTERING AND PAINTING, ETC. 235. Plastering and painting are usually done by the square yard. Allowance is sometimes made for doors and windows, which are called openings. Allowance is also sometimes made for base-boards and wainscoting? 1. A room is 18 ft. long, 12 ft. wide, and 10 ft. high. How many square yards in the walls and ceiling, making no allowance for openings ? 172 MEASUREMENTS. Note. — Let the pupils draw a diagram for each room, represent- ing the four walls in a line. The entire length of the walls will be 2 x (18 ft. + ^o ft 12 ft.) = 60 ft. The area of the four walls, ; """j 60 ft. by 10 ft. = 600 sq. ft. I | Area of ceiling 18 ft. by 12 ft. =216 sq. ft. -'1 • 600 sq. ft. +216 sq. ft. = 816 sq. ft. = 90| ^1 \ , area of ^ svalls and ceilmg. ; i j 1 . : ; 1 ««-i • 1 ' I o ' 1 1 18 a. .12.11. .18 ft. 12 ft. 2. Find the cost of plastering the walls and ceiling of a room 35 ft. long, 26 ft. 6 in. wide, and 15 ft. high, at $ .45 a square yard, allowing 1024 sq. ft. for doors, windows, and base-board ? 3. How many square yards of plaster in the sides and ceiling of a room 30 ft. long, 24 ft. wide, and 10 ft. high, allowing for a base-board 1 ft. high, 2 doors 3 ft. by 8 ft., and 4 windows 3 ft. by 6 ft. ? 4. Find the cost of plastering the ceiling of a room 18 ft. by 20 ft., at 10 cents a square yard. 5. A room 15 ft. by 18 ft., and 10 ft. high, has 4 doors each 3 ft. by 7 ft., and 3 windows each 3 ft. by 6 ft. Find the cost of plastering the walls and ceiling of the room at 30 cents a square yard. 6. How many square yards of plastering in the ceiling of a room 20 ft. long, 9 ft. high, and 15 ft. wide, no allowance for openings ? 7. At $ .30 a square yard, how much will it cost to plaster a room 21 ft. 6 in. long, 16 ft. wide, and 9 ft. high, the base-board being 8 in. wide, and allowing for 3 windows PAPERING WALLS. 173 7 ft. by 2^ ft., and 6 doors of the same dimensions as the windows ? 8. Find the cost of plastering the walls and ceiling of a room which is 36 ft. long, 27 ft. wide, and 9 ft. high, at 25i^ per square yard. 9. My study is 18 ft. long, 16 ft. wide, 8i ft. high, and contains 1 door 3 ft. by 7 ft., and 2 windows, each 3 ft. by 6 ft. The base-board is 9 in. high. What will it cost, at 36^ per square yard, to plaster it, making full deduction for openings ? 10. At 35 cents a square yard, what will be the cost of plastering the walls and ceiling of a room 6 yd. long, 5 yd. wide, and 3 yd. high, an allowance of 20 sq. yd. being made for openings, etc. ? 11. Find the cost of plastering a room 18 ft. square and 10 ft. high, at 25 cents a square yard, ^ being deducted for openings ? 12. A close fence 6i ft. high surrounds a vacant lot 450 ft. by 380 ft At 7 cents a square yard, what will be the cost of painting both sides of the fence ? 13. Find the cost at 18j^ per square yard to plaster the sides and bottom of a cistern 8 ft. 6 in. square, and 9 ft, deep. 14. Find the square yards of plastering on a room 20 ft, long, 17 ft. 6 in. wide, 9 ft. high. Allow for 6 windows, each 7 ft. 6 in. high, 3 ft. wide, and 4 doors, each 7 ft, high and 3 ft. 9 in. wide. PAPERING WALLS. 236. Wall-paper is sold by the roll. A Single Roll is 8 yd. long, a Double Roll, 16 yd. long. Borders are sold by the lineal yard. 174 MEASUREMENTS. The number of rolls needed for a room is found by dividing the area of the space to be papered by the area of one roll. The width of wall-paper' is commonly 18 in. Notes. — Unless otherwise stated, a roll is considered as 8 yd. long and 18 in. wide. Dealers in wall-paper do not sell a part of a roll. If a part of a roll is needed, a whole roll must be purchased. 1. What would it cost to paper the walls of a room 18 ft. long, 12 ft. wide, and 9 ft. high, with paper 8 yd. to the roll, and i yd. wide, at 45^ a roll ? 2. How many strips of paper, and how many double rolls, will paper the sides of a room 15 ft. long, 12 ft. wide, and 8 ft. high, each roll being 1^ ft. wide and 16 yd. long, no allowance being made for matching ? 3. How many double rolls of paper 16 yd. to the roll, ■1- yd. wide, will be required to paper the walls and ceiling of a room 25 ft. long, 10 ft. wide, and 10 ft. high, 110 sq. feet being deducted for doors, windows, etc. ? 4. How many rolls of paper will be required for the walls of a room 16 ft. by 20 ft., and 9 ft. high above the base-board, allowing for 3 doors, each 3 ft. by 7 ft., and 3 windows, each 3 ft. by 6 ft. ? 5. What will be the cost of the paper and border for the above room at 30 cents a roll for the paper, and 15 cents a yard for the border ? 6. How many rolls of paper must be purchased to paper the walls and ceiling of a library, 12 ft. long, 10 ft. 6 in. wide, and 8 ft. high, the base-board being 6 in. wide and the border 1^ ft. wide, with paper ^ yd. wide and 8 yd. long, the paper extending from the border to the base-board '/ BOARD MEASURE. 175 BOARD MEASURE. 237. A Board Foot is a square foot of the surface of a board, 1 inch thick, or less. To find the number of board feet in lumber that is more than one inch thick, we must multiply the number of board feet in the surface by the number of inches in the thickness. A board 10 ft. long, 1 ft. wide, and 1 in. thick or less con- tains 10 board feet; but a beam 10 ft. long, 1 ft. wide, and 8 in. thick contains 8 times 10 board feet = 80 board feet. To find the number of board feet in a tapering board, the average width must be found by taking i- the sum of the widths of the two ends. Thus a board 10 ft. long, and 12 in. wide at one end, and 6 in. wide at the other, contains as many board feet as if it had a uniform width of 9 in. (12 4- 6) -f- 2 = 9. 238. The number of board feet = Length (in feet) x Width (in feet) x Thickness (in inches). Note. — When the thickness is one inch or less, the number of board feet is the product of the length and width in feet. 1. How many board feet in a board 15 ft. long, 15 in. wide, and 1 in. thick ? 2. How many board feet would there be in the board (Ex. 1) if it were | in. thick ? 2 in. thick ? 1 1- in. thick ? 3. How many feet of lumber one inch thick will be required for a tight board fence 6 ft. high around a yard 4 rods square ? 4. How much lumber (Ex. 3) will be required for an open board fence, 4 boards high, boards 8 in. wide, and 5 in. apart ? 5. I need 213 planks 4 ft. 8 in. long, 1 ft. wide, and 2 in. thick, to build a sidewalk. How much will they cost at f 13 a thousand ? 176 MEASUREMENTS. 6. How many feet of lumber will it take to build a line fence 168 ft. long, the fence being 5 boards high, and the boards 6 in. wide ? 7. What will be the cost of 10 planks, each 12 ft. long, 10 in. wide, and 3 in. thick, at $ 16 per M. ? 8. Find the cost of a stick of timber 8 in. square, and 40 ft. long, at $ 18 per M. 9. What is the cost of 8 sticks of timber each 36 ft. long, 10 in. wide, 8 in. thick, at $ 12 per M. ? 10. How many board feet of 2-inch planking will it take to make a walk 4 feet long and 3 feet wide ? 11. rind the cost of 7 planks 12 ft. long, 16 in. wide at one end, and 12 in. at the other, at $ .08 a board foot. 12. At f 18 per M., find the cost of flooring a room 21 ft. by 16 ft., allowing i of the lumber for matching.' Note. — Find area of floor and add |. 13. Find the cost of a board 20 ft. long, 22 in. wide at one end, and tapering to 16 in. at the other, and 1^ in. thick, at $30 per M.? 14. At $12 per M., what will be the cost of 2-inch plank for a 3 ft. 6 in. sidewalk on the street side of a rectangular corner lot 56 ft. by 106 ft. 6 in. ? MISCELLANEO US. 239. 1. My dining-room is 15 ft. long and 12 ft. wide; the walls are 10 ft. high. What will it cost to paper the walls and ceiling with paper 1\ ft. wide, if there are 8 yd. in a roll, and each roll costs $.37|. (J^ allowed for openings.) 2. What will a carpet for the dining-room (Ex. 1) cost me at $ .75 a yd., carpet j yd. wide ? MISCELLANEOUS. 177 3. There are three windows in the dining-room. What will it cost to furnish them with shades at $ 1.10 each and sash-curtains at $ 1.37^ each ? 4. I bought a table at $ 14.50, six chairs at $ 2.75 each, a sideboard for f 30, and other furniture for $28.97. I also spent $ 20 for new table linen. What did it all cost ? 5. What was the entire cost of refurnishing my dining- room ? (Ex. 1, 2, 3, 4.) 6. What will it cost to carpet a room which is 24 ft. long, and 18 ft. wide, with Brussels carpet 1 yd. wide, no waste in matching, at $ 1 per yard ? 7. Find the cost of plastering sides and ceiling of a room 26 ft. long, 13|- ft. wide, 13 ft. high, at 9 cents a square yard, allowing 25 sq. yd. for openings. 8. What will it cost to build a cement walk 40 ft. long and 6 ft. wide, at $ 1.25 per square yard ? 9. A field, containing 8 acres, is 60 rd. long. How wide is it ? 10. How many yards of carpet will cover a floor 18 ft. long, 16 ft. wide? Carpet one yard wide, strips to run lengthwise of room. How many yards if the strips run crosswise of the room ? 11. Find the cost of fencing a rectangular corner lot 68 ft. by 130 ft., the street fence costing 54 cts. a yard, and the line fences 25 cents a yard, but only half of the cost of the latter to be charged to the lot. 12. Find the cost of a carpet f of a yd. wide, at $ 1.50 per lineal yard, for a room 20 ft. long and 18 ft. wide, strips running lengthwise, and allowing a waste of ^ of a yard on each strip for matching. 13. What is the breadth of a rectangular lot whose area is 75 sq. ch. and the length 9 ch. ? 178 MEASUREMENTS. 14. What is the circumference of a circle whose radius is 9 ft. ? 15. How many square yards in the above circle ? (Ex. 15.) 16. How many revolutions does the 5-foot driving-wheel of a locomotive make in going 30 miles ? 17. Find the area of a triangle whose base is 5 ft. and altitude 3 ft. 18. If the circumference of the earth is 25000 miles, what is the diameter ? 19. The circumference of a circle is 18 ft. What is its radius ? 20. How many square yards in a triangle whose base is 48 ft. and whose altitude is 24 ft. ? 21. Find the area of the gable-end of a house whose width is 25 feet and whose ridge is 10 feet 6 inches higher than the base of the gable. 22. If the diameter of the earth is 8000 miles, what is the circumference ? 23. How many board feet in 10 planks 18 ft. long, 15 in. wide, and 2 in. thick, and what will they cost at $40 per M. ? 24. Find the cost of 10 joists, 3 in. by 12 in., 16 ft. long, at $ 25 per M. ? VOLUMES. 240. Anything that has length, breadth, and thickness is called a Solid or Volume. 241. A Rectangular Volume is a solid having six rectan- gular faces. 242. A Cube is a solid having six equal square faces. 243. A Cubic Inch is a cube 1 inch long, 1 inch wide, and 1 inch thick. VOLUMES. 179 m. top 3 in. wide 244. The Volume or Solidity of a body is the number of cubic units that it contains. 1. How many cubic inches in a block 4 in. long, 3 wide, and 2 in. thick ? The block is made up of two layers, each 1 inch thick. In the layer there are 4 times 3 cu. in. In the two layers, therefore, there are 2 X (4 X 3 cu. in.) = 24 cu. inches. The multiplier must be considered as abstract. The three dimensions must have the same unit. The length, breadth, and thickness of a rec- tangular solid are its di- mensions. 3 cu. in. 12 cu. in. 24 cu. in. K. — v-^^y^ ^ ^^ [k x.A-^§< milir-~^"~ ^ lilt N - V \ N~^ l|r 'i! l^J ™.:iiiii 245. Length x breadth x thickness = Solidity. Solidity -j- either dimension = the product of the other two. Solidity -i- the product of two dimensions = the other. 2. Find the number of cubic feet of air in a schoolroom 32 ft. square and 12 ft. high. 3. How high is a room that is 24i ft. long, 20 ft. wide, and contains 4410 cu. ft. ? 4. A cubic foot of ice weighs 56\ pounds. How much will a load of 22 cakes weigh, each cake measuring 2 ft. square and 1 ft. thick ? 5. The capacity of a rectangular box is 480 cu. in. The box is 8 in. wide and 5 in. deep. How long is it ? 6. A schoolroom is 25 ft. long, 18 ft. wide, and 12 ft. high. If 60 pupils are seated in it, how many cubic feet of air are allowed for each child ? 7. A man sold 3 blocks of Vermont marble, each 8 ft. long, and 6 in. x 6 in. at the ends. How much did he receive for the marble at $ 3.50 per cubic foot ? 180 MEASUREMENTS. y 8. A hot-house bed is 3 ft. 9 in. long, and 3 ft. 4 in. wide, inside measure. How deep must it be to contain 25 cu. ft. of earth, and allow 6 in. for the growth of the plants ? 9. How many bricks 8 in. by 4 in. and 2 in. thick will be needed for a wall 60 ft. long, 20 ft. high, and 2 ft. thick, making no allowance for mortar ? 10. How many rectangular blocks 12 in. by 8 in. by 3 in. can be packed into a wagon-box 10 ft. long, 4 ft. wide, and 2 ft. 6 in. deep ? 11. How many cubic yards of earth must be excavated from a cellar 30 ft. long, 21 ft. wide, and 5 ft. 8 in. deep? 12. How many square feet in the surface of a rectangular box 3 ft. 4 in. long, 2 ft. 2 in. wide, and li ft. high ? 13. How many cubes 2 inches on each edge can be sawed from a block of marble 10 ft. 2 in. long, 6 ft. 5 in. wide, and 3 ft. 4 in. thick ? 14. A box is 1.5 in. long, .85 in. wide, and .58 in. deep. What is its capacity in cubic inches ? 15. The altitude of a cylinder is 8 ft. and the circum- ference of the base is 3 ft. What are the cubic contents of the cylinder ? Notes. — Contents of a cylinder = Area of Base x Altitude. Area of curved surface of a cylinder = Circumference of Base X Altitude. This may be seen by cutting a piece of paper so that it will exactly cover the curved surface of a small cylinder. 16. What is the area of the curved surface in the cylinder mentioned in Example 15 ? 17. How much tin will be required to make 2 doz. cylin- drical shaped cans, with a diameter of 4 in. and altitude of 7 in., allowing tin for the curved surface and the two circular ends ? WOOD MEASURE. 181 18. How many cubic inches of water will the 2 doz. cans (Ex. 17) contain ? 19. What will it cost to dig a cellar 36 ft. long, 24 ft. wide, and 6 ft. deep, at 20 cts. a cubic yard ? WOOD MEASURE. 246. A pile of wood 8 feet long, 4 feet wide, and 4 feet high makes a Cord. One of the 8 feet in length of a cord of wood is a Cord Foot. Note. — This may be illustrated by placing side by side 8 books of equal size. One of the books represents a cord foot. How many cords of wood in the following : 1. A pile 18 ft. long, 4 ft. wide, 8 ft. high? 2. A pile 50 ft. long, 8 ft. wide, 6 ft. high? 3. A pile 19 ft. long, 2 ft. wide, 5^ ft. high ? 4. A pile 16 ft. long, 4i ft. wide, 7 ft. high? 5. What is the cost of a pile of wood 10 ft. long, 4 ft. wide, and 8 ft. high, at $ 4-J- a cord ? 6. How high must a pile of 4-foot wood be piled to contain 10 cords, if the pile is 50 ft. long ? 7. How many cords of wood can be piled in a shed 24 ft. long, 18 ft. wide, and 12 ft. high ? 8. How many cords of building-stone in a pile 18 ft. long, 6^ ft. wide, and 3 ft. high ? 9. At $ 3.50 a cord, what will be the cost of a pile of stone 15 ft. long, 4i ft. wide, and 5 ft. high ? 10. How many cubic feet in a cord of 2-foot wood? 3-foot wood ? 18-inch wood ? 182 MEAStTREMBNTS. CAPACITY OF BINS. 247. 1. A bushel fills 2150.42 cubic inches of space. How many bushels of wheat can be contained in a bhi 5 ft. X 5 ft. X 4 ft. ? 5 X 5 X 4 X 1728 - 2150.42. Note. — A bushel fills 1^ cu. ft. of space nearly. 2. A wine gallon fills 231 cubic inches of space. How many gallons of water can be contained in a rectangular tank 10 ft. by 8 ft. by 4 ft. ? Note. — A cubic foot of space contains y/jS gal. = 7 1 gal. nearly. Find the contents in bushels : 3. Of a bin 6 ft. long, 5 ft. wide, and 4 ft. higl>. 4. Of a wagon-box 10 ft. long, 42 in. wide, and 22 in. high. 5. Of a box 3 ft. by 21 ft. by 2i ft. 6. How high must a bin 8 ft. long and 5 ft wide be built to contain 120 bushels ? Find the contents in gallons : 7. Of a tank 8 ft. by 6 ft. by 2^ ft. 8. Of a cistern 6 ft. by 5 ft. by 41 ft. 9. Of a tank 5 J ft. square and 6 ft deep. 10. How many barrels of water will a cistern contain that is 6 ft. by 6 ft. by 7 ft. ? 11. A circular cistern is 5 ft. in diameter and 6 ft. deep. How many barrels of water will it hold ? Note. — Area of base x altitude. 12. How deep must I build a bin that is 6 ft. square, to hold 90 bushels of wheat ? 13. How deep must I build a tank that is 5 ft. square to hold 40 barrels ? LONGITUDE AND TIME. 248. A Meridian is an imaginary line running from the north pole to the south pole. All places on a meridian have the same time. Note. — The meridians of Greenwich and Washington are the meridians that run through Greenwich and Washington. 249. Longitude is distance east or west from some stand- ard meridian, as Greenwich or Washington. When two places are on the same side of the standard meridian, their difference in longitude is found by subtraction. When on opposite sides, their difference in longitude is found by addition. 1. What is the difference in longitude between two cities, one of which is 20° west longitude, the other 30° east longitude ? 20° + 30° = 50°. Ans. 2. What is the dijfference in longitude between two places, one of which is 40° E., the other 70° E. ? 70° _ 40° = 30°. A71S. Note. — No two places can have a difference in longitude exceed- ing 180°. If, in finding difference in longitude by addition, the sum exceeds 180°, subtract the sum from 360° to lind the true difference. The earth turns upon its axis from west to east once in 24 hours, thus -^^ of its entire circumference, or 15° of lon- gitude, passes under the sun in 1 hour. Since the earth turns at the rate of 15° every hour, in 183 184 LONGITUDE AND TIME. 1 minute it turns ^^^ of 15°, or 15', and in 1 second ^ of 15', or 15". Hence, The earth rotates 15° in i hour, 15' in i minute, and 15" in I second. 3. The difference in longitude between two cities is 18° 30'. What is the difference in time ? 15 I 18^ 30' Solution. — Since the earth turns 15° in 1 hr., 15' in 1 min,, 15" in 1 sec, the time 1 hr. 14 min. can be found by dividing the number of degrees, minutes, and seconds by 15. 4. The difference in time between two cities is 54 min. 19 sec. What is their difference in longitude ? Since the earth turns 15° in 1 hr,, 15' in 54 mm. ly sec. i min., and 15" in 1 sec, the distance in 15 degrees, minutes, and seconds may be found 13° 34' 45" ^y multiplying the number of hours, mi and seconds by 15. DateUneNoT^S 6 AJM. ^ 507.32 7 A.M. 166° ^^ 8AJI. 150°,---'r' ° 165° 6 A.M. T'^-.v^^^^ 150° 4 A.M. BA.M. ISBV/"^ \ \ / / N. 135° 3A.M. 10A,M. 120=/ \^ \ \ / / / \ 120° 2 A.M. llAJI 106°/^ ^^\?\\\ \ / / ^^^ J\ 106° 1A.M. '^i~- ' ^^^ Nov.23 90" 12 Midnight Nov.2a 1 ^-^ 3 PJM. 75 "Y"- ^.■'^^/W^^/ \V\^\^^ ^"""~~-y 76°nPJtt. 2 P.M. eo°V^ y^ / / \ \ ^v y 80° 10 P.M. 8 P.M. 45X. / / \ \ ^/45°9PJI., *PJL 30^^-->^,/____^^ \,.,^^^*ii3S^ 6PJL 15° g^ 16'7PJt. 6P Eiimel Nc .M. kleridlan T.22 Diagram showing the Difference in Time at Different Meridians. STANDARD OR RAILROAD TIME. 185 Let the figure on page 184 represent the earth rotating on its axis. When it is 6 p.m., November 22, on the prime meridian, it is noon of the same day at 90° west longitude, and midnight at 90° east longitude. It is therefore a.m. above 90° west longitude and p.m. below. How do we know where it is p.m. and where a.m. at a given place on the earth's surface ? How long does it take a given point on the earth's surfa\}e to move once around its circle ? | Through how many degrees does the earth rotate in one hour ? In one minute ? In one second ? The student will notice that in travelling around the earth from west to east there is an apparent gain of one day, since in this case the traveller is going with the sun ; in going around from east to west there is an apparent loss of one day. From the above considerations we see that there must be somewhere a line at which November 23, and in general each new day, must begin on the earth. This line is taken 180° from the prime meridian, and is called the date line. On crossing this line, ships either gain or lose a day, and must correct their calendar accordingly. Is it possible to have a day last 48 hours ? Which way is the ship sailing if such is the case ? If a person were to travel around the earth from east to west in 120 days, reckoning by local time in various places, in how many days would he actually make the trip ? STANDARD OR RAILROAD TIME. 250. The railroad companies have divided the country into four time belts, extending north and south. All places in each belt take the time of the meridian which passes 186 LONGITUDE AND TIME. through, or near the middle of the belt. The belts are as follows : Eastern, Central, Mountain, and Pacific. The standard meridian for the Eastern belt is the 75th, for the Central belt the 90th, for the Mountain belt the 105th, and for the Pacific belt the 120th. These standard meridians are 15 degrees apart. There- fore, when it is noon in the Eastern belt, it is 11 a.m. in the Central belt, 10 a.m. in the Mountain belt, and 9 a.m. in the Pacific belt. In going westward into another time belt, the traveller sets his watch back one hour. In travelling eastward, he sets his watch ahead one hour. When it is noon on the standard meridian of each belt, it is called noon at all places in the belt. Note, — Time reckoned by this method is not true solar time, but it secures a uniformity of time which is very desirable. The time in general use is Kailroad or Standard time. WRITTEN EXERCISES. 187 Oral. 5. When it is 5 p.m. Mountain time, what is the time in the Pacific belt ? 6. When it is 11 a.m. Pacific time, what is the Central time? 7. In travelling from San Francisco to New York, how many times do I change my watch, and do I set it ahead or back ? 8. When it is 4 a.m. at Augusta, Me., what is the stand- ard time at St. Louis ? 9. When it is 1 p.m. Mountain time at Denver, what time is it at Washington, D.C. ? 10. What is the Pacific time at San Francisco when it is 5 P.M. at Chicago ? Written. 11. The longitude of St. Paul is 93° 4' 55" west, of Philadelphia is 75° 10' west. What is the difference in longitude ? 12. The longitude of New York is 74° 3" west, of Paris is 2° 20' 12" east. What is the difference in longitude ? 13. New York City is 74° 4" west from London. When it is noon at London, what is the true time at New York ? 14. The longitude of Boston is 71° 4' west, and Chicago is 87° 36' west. Chicago is how far due west from Boston, if there are 51.27 miles in one degree at their latitude ? 15. A person travelled until his watch was 3 hours too fast. In what direction and how far did he go ? 16. What is the difference in standard time between Boston and Chicago? 17. If a person goes from New York to San Francisco, will his watch be too fast or to slow, and how much ? 18. The difference in longitude between two places is 17° 54' 55". What is the difference in time ? 188 LONGITUDE AND TIME. 19. The longitude of San Francisco is 122° 26' 15" west, and that of Cincinnati is 84° 26' west. When it is 9 a.m. at San Francisco, what is the time at Cincinnati ? 20. The longitude of Boston is 71° 3' 30" west, and that of Paris is 2° 20' 12" east. When it is 30 min. past 2 p.m. at Paris, what is the time at Boston ? 21. Chicago is 87° 38' west. When it is 27 min. 36 sec. past 11 A.M. at Chicago, it is 10 min. past 12 m. at Washing- ton. What is the longitude of Washington ? 22. St. Louis is 90° 15' 15" west longitude. A gentleman arriving there from Boston, 71° 3' SO" west, finds that his watch, which was set at Boston, is not right. What change must he make ? 23. Mr. Jones started from Philadelphia, and travelled until his watch was 1 hour 30 min. slow. How many- degrees did he travel, and in what direction ? 24. When it is 12 o'clock noon at Chicago, what time is it in a place 60° 30' 30" west of Chicago ? 25. Two men start from the same place, and travel in the same direction, one going 3 degrees and the other 5 degrees per day. They travel until their difference in time is 4 hours. How many days are they travelling ? 26. What is the difference of longitude between two places when their difference of time is 6 hr. 15 min. 12 sec. ? 27. The difference in time between two places is 8 hr. What is the difference in longitude ? 28. The longitude of New York is 74° 3" west. When it is 6 P.M. at New York it is 49 min. 35 sec. past 11 p.m. at Berlin. What is the longitude of Berlin ? 29. The longitude of Philadelphia is 75° 10' west. When it is 10 A.M. at New York it is 50 min. 55 sec. past 6 a.m. at San Francisco. What is the longitude of San Francisco ? EEVIEW. 189 REVIEW OF DENOMINATE NUMBERS. 251. 1. Define a simple number; denominate number; compound number. 2. For what is linear measure used ? 3. For what is square measure used ? 4. For what is cubic measure used ? !- Give tables. 5. For what is liquid measure used ? 6. For what is dry measure used ? 7. For what is Troy weight used? Give the table. Avoirdupois weight ? Give the table. Apothecaries' weight? Give the table. 8. How many grains in a pound Troy ? Avoirdupois ? 9. How many grains in an ounce Troy ? Avoirdupois? 10. What is a long ton, and how is it used ? 11. How many days in a common year? a leap year? What is the solar year ? Explain leap year. When does the civil day begin and end ? 12. What is the use of circular measure ? Define circle, circumference, diameter, radius, arc. Give the table. What is the measure of an angle ? What is a degree ? A quad- rant ? How do we find circumference ? How do we find diameter ? What is a right angle ? 13. What is a surface? a square? a rectangle? a tri- angle ? 14. Give the rule to find the area of a square; of a rectangle ; of a triangle ; of a circle. 15. Define solid, rectangular solid, cube, cylinder. How do we find the volume of a rectangular solid ? Of a cylinder ? 16. What is reduction ? Reduction ascending ? Eeduc- tion descending ? 190 DENOMINATE NUMBERS. 17. Define a denominate fraction. Give the different kinds of reduction of denominate fractions. 18. How do we add • compound numbers ? subtract ? multiply ? divide ? 19. Give the common method of finding the difference between dates. How do we find the exact difference ? 20. What is longitude? How do we find difference in longitude between two places on the same side of a prime meridian ? On opposite sides ? 21. How do we find difference in longitude when differ- ence of time is given? How do we find difference-of time when difference in longitude is given ? 22.- What is standard time? What are the names of the four time belts ? In passing west into a time belt, how does the traveller set his watch ? In travelling east ? 23. How do we find length when area and breadth are given ? 24. How do we find length when volume, thickness, and width are given ? 25. What are the dimensions of a rectangular solid? 26. Define cancellation, even number, odd number, prime number, composite number. 27. When are numbers prime to each other ? 28. How many cubic feet in a cord of wood or stone ? How long, wide, and high is a cord of wood ? What is a cord foot ? 29. For what is board measure used? What is a board foot ? Give the rule for finding board feet. 30. How do we find the capacity of bins ? Of cisterns ? THE METRIC SYSTEM LINEAR MEASURE. 252. The standard unit of Linear Measure in the Metric System is the Meter. It is determined by taking one ten- millionth part of the distance from the earth's equator to either of its poles, measured on a meridian. It is equal to 39.37 inches. QUESTIONS. 253. 1. What denomination in the English linear meas- ure is most nearly like the meter ? 2. Draw a line one meter long. 3. Hold your hands one meter apart. 4. A meter is about how many feet long ? 5. How many meters long is your schoolroom? Wide? High? 6. About how many meters in a rod ? HOW THE TABLE IS MADE. 254. Divide a meter into ten equal parts. One of these parts is a Decimeter. Dec is a Latin stem meaning tenth. About how many inches long is a decimeter? Show with your hands the length of a decimeter. What part of a meter is a decimeter? 191 192 METRIC SYSTEM. 255. Divide a decimeter into ten equal parts. One of these parts is a Centimeter. Cent is a Latin stem meaning hundredth. What part of an inch is a centimeter ? Show its length. How many centimeters in one meter ? What part of a meter is a centimeter ? 256. Divide a centimeter into ten equal parts. One of these parts is a Millimeter. Mill is a Latin stem meaning thousandth. What part of a meter is a millimeter? How many millimeters in a meter? What part of an inch is a millimeter ? 257. Ten meters make one Dekameter. Deka is a Greek stem meaning ten. How many rods in a dekameter? How many feet? How many dekameters long is your schoolroom ? 258. Ten dekameters make one Hektometer. Hekto is a Greek stem meaning hundred. How many meters in one hektometer ? How many feet long is a hektometer ? 259. Ten hektometers make one Kilometer. Kilo is a Greek stem meaning thousand. How many meters in one kilometer ? How many feet ? What part of a mile ? 260. Ten kilometers make one Myriameter. Myria is a Greek stem meaning ten-thousand. How many meters in one myriameter ? How many feet ? How many miles ? 261. These statements may be combined in the following table : 10 Millimeters (mm.)= 1 Centimeter (cm.) = .3937 -f in. 10 Centimeters = 1 Decimeter (dm.) = 3.937 + in. 10 Decimeters = 1 Meter (m.) = 39.37 + in. 10 Meters = 1 Dekameter (Dm.) = 32.808 + ft. 10 Dekameters = 1 Hektometer (Hm.) = 19.927 + rd. 10 Hektometers = 1 Kilometer (Km.) = .621 + mi. 10 Kilometers = 1 Myriameter (Mm.)= 6.213 + mi. REDUCTION. 193 262. 1 Myriameter = 10 Kilometers = 100 Hektorneters = 1000 Dekameters = 10000 Meters = 100000 Decimeters = 1000000 Centimeters = 10000000 Millimeters. BEDDOTION. 1 Millimeter = .1 Centimeter = I .01 Decimeter = g .001 Meter = g .0001 Dekameter = J .00001 Hektometer = .000001 Kilometer = ^1 .0000001 Myriameter. 263. The following series of numbers read from the top downward is reduction ascending; read from the bottom upward is reduction descending. All metric numbers may be reduced in this way. 75689132. mm. = 7568913.2 cm. = 756891.32 dm. = 75689.132 m. = 7568.9132 Dm. = 756.89132 Hm. = 75.689132 Km. = 7.5689132 Mm. = s a s a . s d s All these numbers might be read thus : 7568913 2. QUESTIONS. 264. 1. How can a metric number be reduced to higher denominations ? To lower ? 2. Eeduce 12345678 mm. to cm.; to dm.; to m.; to Dm. ; to Hm, ; to Km. ; to Mm. 3. Eeduce 9.6538714 Mm. to Km.; to Hm. ; to Dm.; to m. ; to dm. ; to cm. ; to mm. 4. Reduce 7 Mm. to lower denominations. 5. Reduce 7 mm. to higher denominations. 194 METRIC SYSTEM. 6. Keduce 6307.1 m. to Km. ; to cm. 7. Reduce 31 meters to inches. 8. Write 2 Mm. as meters; 7 Km.; 6 Hm.; 8 Dm.; 5 m. 3 dm. ; 2 cm. ; 9 mm. Write them all as one number. 9. Reduce 1 Mm. to feet. 10. Write 7 Mm. and 6 mm. in one number, as meters. Reduce it to higher denominations ; to lower. 11. Reduce .075 Km. to cm. 12. Reduce 8 Dm. and 6 m. to Mm. ; to mm. 13. Write 75 Km. and 62 dm. in one number as meters ; as cm. ; as Mm. 14. State the value of each figure in 30769.543 m. 15. A ship sails 100 Mm. in one day. How many miles does it sail ? 16. Give the table of Metric Linear Measure. 17. Name the standard unit. 18. How is it determined ? 19. What is the scale of the Metric system ? 20. Name in order the Latin and Greek stems used in the table. SURFACE MEASURE. 265. The standard unit of surface measure is the Are (pronounced like the English air). The Are is a square whose side is one dekameter. It is therefore a Square Dekameter. QUESTIONS. ' 266. 1. An are is how many meters long ? Wide ? 2. How many square meters does the are contain? 3. An are is how many inches long ? Feet ? 4. The are is about how many rods long ? SURFACE MEASURE. 195 5. About how many square rods does it contain ? 6. About how many ares equal one acre ? 7. How many ares does the floor of your schoolroom contain ? 8. Name all the surfaces you can think of that contain about one are. 267. The table of surface measure, like that of linear measure, is made by prefixing the Latin and Greek stems to the standard unit, thus : 10 Centares (ca.) = 1 Deciare, da. 10 Deciares = 1 Are, a. 10 Ares ^ = 1 Dekare, Da. 10 Dekares = 1 Hektare, Ha. Note. — The denominations of the above table are little used, except the are, the hektare, and the centare, which are employed chiefly in measurements of land. 268. Draw a square whose side is one meter. How many square meters does it contain ? It is how many decimeters on a side ? How many square decimeters does it contain ? How many square decimeters make one square meter ? 269. Draw a square whose side is one decimeter. How many square decimeters does it contain ? How many centi- meters long and wide is it ? How many square centimeters does it contain ? How many square centimeters in one square decimeter ? In the same way find how many square millimeters in one square decimeter. How many sq. Meters = 1 sq. Dekameter ? How many sq. Dekameter s = 1 sq. Hektometer ? How many sq. Hektometers = 1 sq. Kilometer ? 196 METRIC SYSTEM. 270. The answers to the above questions form the follow- ing table of surface measure, which is used for all ordinary surface measurements : 100 sq. Millimeters (sq. mm.) = I sq. Centimeter, sq. cm. 100 sq. Centimeters = 1 sq. Decimeter, sq. dm. 100 sq. Decimeters = 1 sq. Meter, sq. m. 100 sq. Meters = 1 sq. Dekameter, sq. Dm. 100 sq. Dekameters = 1 sq. Hektometer, sq. Hm. 100 sq. Hektometers = 1 sq. Kilometer, ' sq. Km. QUESTIONS. 271. 1. Which denomination of this table is like the are ? 2. Like the centare ? 3. Like the hectare ? 4. How far to the right must the decimal point be moved to reduce sq. m. to sq. dm. ? 5. How many places to the left must the decimal point be moved to reduce sq. m. to sq. Dm. ? 6. To reduce sq. mm. to sq. cm. ? 7. To reduce sq. mm. to sq. dm. ? 8. Eeduce 5555 ca. to Ha. 9. Eeduce 3333 Ha. to ca. 10. A field 134 m. long and 7 Dm. wide contains how many sq, m. of land ? 11. How many ares ? 12. How many Ha. ? 13. How many sq. Dm. ? 14. How many sq. Hm. ? 15. How many sq. cm. ? 16. How many sq. cm. in an oblong 643 cm. long and 2.5,m. wide ? VOLUME MEASURE. 197 17. How many sq. mm. ? 18. How many sq. Km. ? 19. One hectare equals about how many acres ? VOLUME MEASURE. 272. The unit chiefly used in measuring wood and stone is the Stere (pronounced stair), which is a cube whose edge is one meter. What denomination in the English volume measure is most nearly like the stere ? How many cubic meters does the stere contain ? How many decisteres ? How many centisteres ? How many millisteres ? QUESTIONS. 273. A cube whose edge is one meter long contains how many cubic meters ? It is how many dm. long ? Wide ? High? How many cu. dm. does it contain? How many cu. dm. = 1 cu. m. ? A cube whose edge is 1 dm. contains how many cu. dm. ? How many cm. long is it ? Wide ? High ? How many cu. cm. does it contain ? How many cu. cm. = 1 cu. dm. ? A cube whose edge is 1 cm. contains how many cu. cm. ? How many mm. long is it ? Wide ? High ? How many cu. mm. does it contain ? How many cu. mm. = 1 cu. cm. ? 274. From the answers to the above questions make the following : TABLE OF VOLUME MEASURE. 1000 cu. Millimeters (cu. mm.) = 1 cu. Centimeter, cu. cm. 1000 cu. Centimeters = 1 cu Decimeter, cu. dm. 1000 cu. Decimeters = 1 cu. Meter, cu. m. QUESTIONS. 275. 1. How may cubic millimeters be reduced to cubic centimeters ? To cubic decimeters ? To cubic meters ? 2. How many places to the right must the decimal point be moved to reduce cubic meters to cubic millimeters ? 198 METRIC SYSTEM. 3. Reduce 7 cu. m. to cubic millimeters. 4. Reduce 5 cu. mm. to cubic meters. 5. How many steres in one cubic meter ? 6. A pile of wood is 30 dm. long, 3 m. wide, and 18 dm. high. How many cubic meters does it contain ? 7. How many steres ? (Ex. 6.) 8. How many cubic millimeters ? (Ex. 6.) 9. How many cubic centimeters of air in an empty box 2 m. by 12 dm. by 75 cm. ? 10. How many cubic decimeters ? (Ex. 9.) 11. How many steres of stone in a wall 30 m. long, 5 dm. thick, and 250 cm. high ? CAPACITY MEASURE. 276. The metric capacity measure takes the place of both the liquid and the dry measure of the English system. The standard unit of capacity measure is the Liter (pro- nounced leeter), which is a cube whose edge is one decimeter. QUESTIONS. 277. 1. The liter is what part of a meter wide ? High ? Long ? 2. What part of a cubic meter does it contain ? 3. About how many inches wide is it ? High ? Long ? About how many cubic inches does it contain ? 4. Show with your hands how wide, high, and long a liter is. 5. What denomination of English dry measure corre- sponds most nearly to the liter? 6. Make a full-sized picture of a liter. 7. What object the size of a liter do you know ? CAPACITY MEASURE. • 199 TABLE. 278. The table of capacity is formed similarly to the other metric tables, and is as follows : 10 Milliliters (ml.) = 1 Centiliter, cl. 10 Centiliters = 1 Deciliter, dl. 10 Deciliters = 1 Liter, 1. 10 Liters = 1 Dekaliter, Dl. 10 Dekaliters = 1 Hectoliter, HI. 10 Hektoliters = 1 Kiloliter, Kl. 10 Kiloliters = 1 Myrialiter, QUESTIONS. Ml. 279. 1. How many liters in 1 myrialiter? In 1 ml. ? 2. How many milliliters in 1 Ml. ? 3. Reduce 12345678 ml. to higher denominations. 4. Read the above number, giving each figure the name of the denomination it represents. 5. Reduce 154.67 cl. to Kl. 6. Reduce .012346 Ml. to dl. 7. How many liters equal one cubic meter? 8. A bin is 2.5 m. wide, 6.4 m. long, and 17 dm. deep. How many liters of oats will it hold ? How many HI. ? How many Kl. ? 9. A tank is 3 m. long and 3 m. wide. How many dm. deep must it be to hold 50 HI. of water ? 10. A stone containing 1 stere, if dropped in a pond, would displace how many liters of water ? 200 METRIC SYSTEM. MEASURES OP WEIGHT. 280. The Gram is the unit of weight. It is equal to the weight of a cubic centimeter of distilled water at its greatest density. TABLE. 10 Milligrams (mg.) = 1 Centigram, eg. 10 Centigrams = 1 Decigram,, dg. 10 Decigrams = 1 Gram, g- 10 Grams = 1 Dekagram, Dg. 10 Dekagrams = 1 Hektogram, Hg. 10 Hektograms = 1 Kilogram, Kg. 10 Kilograms = 1 Myriagram, Mg. 10 Myriagrams = 1 Quintal, Q. 10 Quintals* = 1 Tonneau, T. or Metric Ton. The weight of 1 gram is 15.432 grains. QUESTIONS. 281. 1. How many grams in 1 metric ton ? 2. How many mg. in 1 metric ton ? 3. Eeduce 1 mg. to T. 4. Reduce 1 T. to mg. 5. Reduce 9876543215 mg. to higher denominations. 6. Read the above number, giving each figure the name of the denomination it represents. 7. Recite the table of weight. 8. Spell the name of each denomination 9. Reduce 7.42 quintals to centigrams. 10. Reduce 543 mg. to Mg. MEASURES OF WEIGHT. 201 11. One gram equals 15.432 grains. How many grains in 1 Kg. ? 12. One pound Avoirdupois contains 7000 gr. How many pounds are equivalent to one Kg. ? 13. Mr. Smith weighs 100 Kg. How many pounds does he weigh ? 14. How many grams does a cubic meter of distilled water weigh ? 15. Would a cubic meter of any other substance weigh the same as a cu. m. of distilled water ? State your reason. 16. How many kilograms of water will a tank 4 m. x 3 m. X 12 dm. hold ? REVIEW QUESTIONS. 282. 1. How many tables in the Metric System ? 2. Name the standard units in the order in which they have been given. Repeat them until you can say them as rapidly as you can talk. 3. Name the prefixes in the same way. 4. Name and describe the unit of capacity measure; of weight ; of length ; of volume ; of surface. 5. Repeat the tables. 6. The stere is the unit of what measure ? The meter ? The are ? The gram ? The liter ? 7. How can metric numbers be reduced to higher 'denominations ? to lower ? 8. How many things are to be committed to memory in the Metric System ? 9. What is 39.37? 15.432? 10? These are the only numbers that need be remembered. GENEEAL REVIEW. 283. 1. Define fraction; numerator; fractional unit; terms; reduction of fractions. 2. Change 217 to 20ths. 3. Give the principle upon which reduction of fractions is based. Illustrate. 4. Add 25^, 14|, 7|. 5. Give the rule for reducing fractions to their least common denominator. 6. A man owned J of a foundry and sold ^ of his share for $ 1200. What was the foundry worth ? 7. Eeduce to simple form (15f - 3J) x (2i + 5f). '• 6 + 8J • 9. Reduce to least common denominator six thirty- fifths, nine twentieths, and five sixteenths, and arrange the results according to value. 10. A man having $ 130 used | of it. How much of it remained ? 11. C and D can do a piece of work in 24 days, D can do it alone in 45 days. How many days will C require to do it? 12. The numerator of a fraction is 6510, the denominator 66495. lleduce the fraction to its lowest terms. 202 "WRITTEN EXERCISES. 203 13. If 7 be added to each term of the fraction f, will its value be increased or diminished, and how much ? 14. Two men are 140 miles apart, and travel towards each other, one at the rate of 3J miles an hour, and the other at the rate of 4|^ miles an hour. In how many hours will they meet ? 15. Define decimal fraction; an account; currency. 16. What will 6827 feet of lumber cost at $10.50 per M. ? 17. 8.7625 + 31.735-17.382569 = ? 18. AVrite in words 365. 8752. 19. Find the cost of 7896 pounds of hay at f 16 a ton. 20. Express in figures two hundred sixty-five and five thousand one hundred ten millionths. 21. Change .875 to a common fraction in its lowest terms. 22. How is a bill receipted? 23. Give a rule for dividing a decimal by 10, 100, 1000, etc. 24. Eeduce 3.25, 12.364, and .56087 to a common denominator. 25. When will a fraction reduce to a perfect decimal ? 26. 7.6875 -- 187.5 x (5|- + 2f ) = ? 27. How is the place for the decimal point in the j^roduct determined ? 28. Give the abbreviations of Creditor and Merchandise. 29. Name the seventh decimal order. 30. James Harris, of Denver, Col., sold for cash to Preston White, on Nov. 4, 1901, 42 lb. of sugar at 10 cents ; 3 lb. Y. H. tea at f .60; 4 gal. molasses at $ .75 ; 48 yd. sheet- ing at $ .14 ; 1 box starch 46 cents, and 8 doz. eggs at $ .24. Make the bill in due form. 204 GENERAL REVIEW. 31. Define a square; a circle. 32. Write the table of cubic measure. 33. For what purposes are the following used: Troy weight ? Dry measure ? 34. The last war with England commenced June 18, 1812, and ended Feb. 17, 1815. How long did it continue ? 35. A jeweller made 3 lb. 2 pwt. 2 gr. of gold into rings weighing 5 pwt. 10 gr. each. How many rings were there ? 36. Reduce 2 mi. 6 ch. 3 rd. to links. 37. Eeduce to integers of lower denominations £|, and .25256 T. 38. Change 4 S 5 3 2 3 8 gr. to a decimal of a pound. 39. Find the result of 4.8 bu. + 2| bu. + .8125 pk. -f- 2f pk. + } bu. 40. A grocer bought 35 casks of molasses, each contain- ing 44 gal. 2 qt. 1 pt. How much did they all contain ? 41. A ship in 8° north latitude sailed due south until it reached 12° south latitude; find the distance it sailed in statute miles. 42. Find the value of -^-^ of a ton. 43. Reduce -^ of a year to integers of lower denomi- nations. 44. Reduce f of a lb. Troy to integers of lower denomi- nations. 45. Express 120 rd. 2 yd. 1 ft. 6. in. as the fraction of a mile. 46. Reduce 45 sq. rd. 2 sq. ft. 9 sq. in. to the fraction of an acre. 47. What part of a day are 6 hr. 13 min. 20 sec. ? WRITTEN EXERCISES. 205 48. What part of 4 gal. 2 qt. 1 pt. are 1 gal. 1 qt. Ipt.? 49. At 25 cents an ounce, what is the value of 18 oz. 10 pwt. 12 gr. of silver ? 50. How much will it cost to fill a bin with corn at $.45 a bushel, if the bin is 10 ft. square on the bottom and 4 ft. deep. 51. A cistern measures inside the walls 8 by 6 by 9 ft., and lacks 1^ ft. of being full. How many gallons does it hold ? 52. How many busliels will a box hold of the same dimensions as in Ex. 51 ? 53. How many cords of wood in a pile of wood that is twice the length, height, and width of an established cord ? 54. Find the total weight of 5 car-loads of coal, weigh- ing respectively 14 T 18 cwt. 63 lb., 17 T. 4 cwt. 85 lb., 13 T. 19 cwt. 26 lb., 15 T. 10 cwt. 43 lb., and 14 T. 7 cwt. 90 1b. 55. How many bricks 8 in. by 4 in. by 2 in. will it take to pave a street ^ mile long and -^^ mile wide, laying the bricks on the longest narrow face ? How many if they are placed on end ? 56. How much wood in three piles, the first of which contains 10 cd. 6 cd. ft. 4 cu. ft., the second 12 cd. 12 cu. ft., the third 17 cd. 1 cd. ft. ? 57. A family consumes daily 6 lb. 14 oz. of bread. If each loaf weighs 1 lb. 6 oz. and costs 7 cents, how much does bread cost the family for the month of August ? 58. Find the sum of f mi., -| fur., ^ rd., and f ft. 59. From 6^ mi. take 4 mi. 140 rd. 4 yd. 60. A man has a bin 6 ft. long, 4 ft. wide, 3 ft. deep, 1 filled with wheat. If he sells 10 sacks, each containing 2 bu. 1 pk. 5 qt., how much is left ? 206 GENERAL REVIEW. 61. From a piece of land 20 rods long, 180 ft. wide, were sold 4 lots, each 50 ft. wide, 150 ft. long. What part remained ? 62. A merchant bought two casks of wine, each contain- ing 41 gal. 3 qt., at $1.80 per gallon. One-seventh of it leaked away. He sold 9 kegs, each containing 5 gal. 1 qt., at 30^ a pint, and the remainder at 40/ a pint. How much did he gain ? 63. A coal-dealer bought 34,160 lb. of coal at $2.50 per long ton. He sold 8 loads, each 1 T. 4 cwt. 60 lb., at $ 3 per ton, and the rest for $3.25 per ton. How much did he gain? 64. Change 3,895,504" to higher denominations. 65. Three quadrants of a circle are equal to how many- seconds ? 66. Through how many degrees does the minute-hand of a clock pass in 2i hours ? Through how many does the hour-hand pass in the same time ? 67. How many minutes elapse between four o'clock Friday afternoon and nine o'clock the following Monday morning ? 68. A boy was exactly 10 years old when the United States declared war against Mexico, May 13, 1846. How old was he at the time of the first bloodshed of the Civil War, April 19, 1861? 69. A train leaves New York at six o'clock Monday evening, and travels an average of J of a mile a minute. When will it reach Buffalo, a distance of 410 miles ? 70. A cistern that holds 50 bushels is 6 ft. square. How deep is it ? 71. A pile of wood is 6 ft. high and 4 ft. wide. How long must it be to contain 3 cords ? WRITTEN EXERCISES. 207 72. A man has a circular garden with a diameter of 36 feet. How many rods of fencing will be required to enclose it ? 73. How many square yards in the above garden? (Ex. 72.) 74. A city lot is 35 ft. front and 125 ft. deep. Find the area. 75. A meadow contains 8| acres. Its width is 35 rods. Find the length of it. 76. How many yards of carpeting j of a yd. wide will be required for a room 18 ft. wide and 20 ft. long, if the strips run lengthwise, and there is a waste of 6 in. in each strip for matching patterns ? 77. The platform in a schoolroom is 30 ft. long and 11 ft. wide. What will be the cost of oil-cloth, at 85 cents a square yard, to cover it ? 78. How many feet, board measure, in 6 boards 16 ft. long, 10 in. wide, 1 in. thick ? 79. Find the cost of 10 Norway sidewalk planks 16 ft. long, 12 in. wide, 2 in. thick, at $18 per M. 80. A class-room is 15 ft. long, 12 ft. wide, 10 ft. high. Find the cost of plastering it at 20 cents a yard. 81. A room is 30 ft. wide, and 40 ft. long, and 16 ft. high. Find the number of square yards of plastering in it after making allowance for wainscoting 3 ft. high, 8 windows, 4 ft. by 8 ft., and 6 doors, 3 ft. 6 in. by 7 ft. 6 in. 82. What will it cost to paper a kitchen 12 ft. by 11 ft. and 9 ft. high, with 10-cent paper, if each roll covers 4 sq. yd.? 83. Find the cost of papering a room 16 ft. long, 12 ft. wide, 9 ft. 6 in. high, with paper 18 in. wide, 8 yd. in a roll, at 50 cents a roll, if 20 sq. yd. be allowed for doors, windows, and base-boards. 208 GENERAL REVIEW. 84. If a shingle is 4 in. wide, and lies 5-| in. to the weather, how many shingles will it take to shingle one side of a roof that is 32 ft. long by 22 ft. wide, allowing an extra course at the eaves ? How many for both sides ? 85. What would be the cost of the shingles for both sides of the roof in Ex. 84 at $3.25 per M. ? 86. The product of two numbers is lyij- ; one of the num- bers is |. What is the other ? 87. What fraction multiplied by -f- will equal -^ ? 88. How many square yards of carpet will be required to carpet a room that is 27 ft. by 33 ft. ? How many yards of carpet will be required if the carpet is 30 in. wide ? 89. If a hotel uses 3 pounds of coffee a week, what would be paid for coffee at 38 cents a pound for January, February, and March, 1896 ? 90. When it is noon at New York, 73° 59' 9" W., what is the time at Chicago, 87° 36' 42" W. ? What is the time at New York when it is noon at Chicago ? 91. When it is noon at Greenwich, what is the longitude of a place whose time is 8.30 a.m. ? 92. A and B start at a given point, and travel in opposite directions. A travels until his longitude is 30° 40' greater than it was, and B travels half as far as A. What is the difference in time between the places they are then in ? 93. What part of a pound Avoirdupois is a pound Troy ? 94. What part of an ounce Troy is an ounce Avoirdupois ? 95. A druggist bought opium at $ 8 a pound Avoirdupois, and sold it at 75^ an ounce Troy. What was his profit on 10 pounds ? 96. How much heavier is a pound of iron than a pound of gold ? WRITTEN EXEBCISES. 209 97. What is the difference in the areas of two fields, one being 5 Hm. long and 8 Dm. wide, the other 8 Hm. long and 14 Dm. wide ? 98. In a cubic dekameter how many cubic millimeters ? 99. A rectangular field is 5.4 Hm. long and 1.5 Hm. wide. How many hektares does it contain ? 100. Three fields have an area respectively of 19 A. 146 sq. rd., 12 A. 73 sq. rd. 15 sq. yd., and 9 A. 127 sq. rd. 26 sq. yd. What is the total area ? 101. Find the volume and the area of the curved surface of a cylinder whose diameter is 8 in., and whose altitude is 11 in. 102. How many square feet of glass in 6 windows of 8 panes each, each pane being 14 by 12 inches ? 103. What will it cost to pave a street 2^ miles long and 2 J rods wide, at $ 12 per square rod ? 104. How many yards of oil cloth 8 ft. wide will cover a floor 32 by 22 ft., if the strips run lengthwise, allowing 1 yard for waste ? 105. How many yards of crash 18 inches wide will cover a floor 18 ft. square, allowing If yards for waste ? 106. How many sq. yards to be plastered in the ceiling and walls of a room 26 by 20 ft. and 12 ft. high, there being 3 windows 6 ft. by 2 ft. 8 in., and 2 doors 8 by 3 ft. ? 107. How many square feet of boards in a tight fence 10 rods long and 6 ft. high ? 108. How many sq. rods in a garden 231 ft. long and 165 ft. wide? 109. What will it cost, at 25^ a sq. foot, to construct a slate blackboard 34 ft. 3 in. long and 4 ft. 6 in. wide ? PERCENTAGE, 284. Oral. How much is J of 20 ? 5 is ^ of what ? 5 is how many hundredths of 20 ? Questions of Relation may be solved by means of hun- dredths; thus, a. How much is ^% of 20 ? 6. 5 is j\\ of what ? c. 5 is how many hundredths of 20 ? Another name for hundredths is per cent; thus, -f^-^ is 25 per cent, -^i-o = ^ P^^ cent, .16 = 16 per cent, .05 = 5 per cent. The sign of per cent is %. 25 per cent is 25%, 6 per cent is 6%, | per cent is ^%. Read questions a, 6, and c, using the name per cent where necessary. Write questions a, b, and c, using the sign % in its proper place. Solve questions a, b, and c, using decimal per cent, 210 WRITTEN EXERCISES. 211 285. Percentage is a process of solving questions of rela- tion by means of hundredths. Written. 1. How much is 3% of 400 ? Solve the above, form question &, and solve it. 2. How much is 10% of 200 ? Solve the above, form question c, and solve it. Solve the following questions, then form questions h and c, and solve them : 3. How much is 4 per cent of $ 200 ? 4. 30% of 500 is how much ? 6. How much is 50% of 90 ? 6. A boy earned $4.00, and spent 10% of it for a book. What was the cost of the book ? (Question a.) 7. A farmer had 100 sheep and sold 20% of them. How many sheep did he sell ? Solve the following, form questions a and c, and solve them : 8. 15 is 10% of what number? 9. 160 is 80% of what number ? 10. 50 is 25% of what number ? 11. A boy lost 20 cents, which was 5 per cent of all his money. How much money did he have ? (Question h.) 12. A farmer sold 150 sheep, which was 50% of his entire flock. How many sheep were in the flock ? Solve the following, form questions a and h, and solve them : 13. $ 20 is what per cent of $ 100 ? 14. What per cent of 60 is 15 ? 15. 40^ is what % of $4.00 ? 212 PERCENTAGE. 16. A boy earns $ 5.00 a week, and saves $ 2.00 of it. What per cent of his money does he save ? (Question c.) 17. A dealer bought a gross of pencils, and sold 36 of them. What per cent of his pencils did he sell ? What per cent remained unsold ? Change the following fractions to others having 100 for a denominator: i; |; /^; |; -^\', J-^; -i; |; -\; ^\. Change the above fractions to decimal hundredths. Eead as hundredths : .05; .186; .331; .24^; .27 J; .2725; ; .1. Write the above in hundredths as common fractions. .5; .1. 286. E-ead the following Questions of Relation : Question a. How much is 5% of 200 ? Ans. 10. Question b. 10 is 5% of what ? Ans. 200. Question c. 10 is what % of 200 ? Ans. 5%. These three kinds of questions form the basis of a great variety of practical computations, which are classed under the general head of Percentage. 287. Every question in percentage involves three ele- ments : the Rate per cent, the Base, and the Percentage. The Rate per cent is the number of hundredths taken. In question a, what is the rate per cent ? The Base is the number of which the hundredths are taken. In question a, what is the base ? The Percentage is the result obtained by taking a cer- tain per cent of a number. In question a, what is the percentage? ,/wwex -y^ Hi 'Vm^-^-^^-^aaw -ux#|,t^ WRITTEN EXERCISES. 213 How much is 8% of $200 Solution. —8% of $200 = 200 x .08 = $16. We now have the three elements, as follows: 8 % is the rate, $ 200 is the base, and $ 16 is the percentage. ■ Since $200 x .08 = $ 16, the percentage ; $ 16 -~ .08 = §200, the base ; And $ 16 -^ -1200 = .08, the rate. 288. Therefore, when any two of these elements are given, the other may be found, thus: Base X Rate = Pernenta^e; Percentage -^ Pasp — Pt^tft 289. Tell which elements are given, and which one is required, in question a ; in question h ; in question c. 290. Find the percentage and form questions h and c, but do not solve them. 18. 6% of 100 is what? 19. How much is 25% of 200? 20. How much is 40% of 250? 21. What is 4% of 50 men? 22. 20% of 80 is what? 23. 15% of f 40 = ? 24. What is 3% of 400 gallons? 25. What is 90% of 200 pounds? 26. 60% of 200 miles = ? 27. 10% of 15 inches = ? 28. What is the base in each of the above questions? 214 PERCENTAGE. 291. Care should be taken to express the decimal rate per cent properly, as hundredths. Every fractional part of 1% must be written at the right of the hundredths place. 1% = .01 12^% = .12^ or .125 9% = .09 J% = .00^ or .005 10% = .10 10tL% = .107 90% = .90 33i% = .331 100% = 1.00 81% = .08 J or .0825 900% = 9.00 J% = .OOi or .0025 125% = 1.25 ^% = .001 or .00125 29 2. Express decimally: 1. 7% 6. 6i% 11. 101% 16. ^% 2. 6% 7. 12i% 12. 110% 17. f% 3. 2% 8. 15|% 13. 250% 18. -S-% 4. 12% 9. 37i% 14. 200% 19. 1% 5. 78% 10. H% 15. 127|% 20. ^^ 293. It is often convenient to change the rate per cent to the common fraction form ; thus : 100 100 2 ;^0 ^ 4 Change to common fractions in lowest terms : 1. 25% ■ 5. 16f% 9. 150% 13. f% 2. 50% 6. 33^% 10. 225% 14. |% 3. 75% 7. 37i% 11. 175% 15. f% 4. 20% 8. 87^% 12. 236% 16. i% What per cent of a number is i of it? i? |? J? f ? 1 ? _9 ? 1 ? 31F • iTF • ^ • WRITTEN EXERCISES. 215 294. Express in both the decimal and the common fraction form : 1. 25% /6. 6|% 9. 108% 13. f% 2. 60% 6. 6i% 10. 150% 14. 1% 3. 18% 7. ^i% 11. 125% 15. \% 4. 1% s. 66|% 12. 137^% 16. tV/ 295. Per cent is commonly used in the decimal form, but many operations may be much shortened by using the common fraction form. Solve, using first the decimal, then the common fraction form, and note the difference : 17. How much is 25% of $ 324 ? 18. Find 12^% of 960 sheep. 19. What is 16|% of 366 men? 20. Find 331% of 12 oranges. 21. 50% of 4 tons is what ? 22. 20% of f 300 = what? Question a, Oral. 23. What is yf (^ of 800 ? 27. 50% of 144 men ? 24. What is y^^ of 900 ? 28. 20% of 15 eggs? 25., 16|% of 48 apples ? 29. .07 of 500 ? 26. 33i% of m sheep? 30. 12J% of $16? Written. 296. Rate and base given, to find percentage. Base X Rate = Percentage. 1. What is 40%^ of $ 120-? 2. How much of 12^% of 1600 lb.? 8. 18j-\% of 365 is what? 216 PERCENTAGE. 4. From a flock of 60 sheep, 10% were sold. How many- were sold? The question is, "How much is 10% of 60?" 5. How much is 100% of 50 bushels? 6. A man having 50 bushels of wheat sold 20 per cent of it. How many bushels did he sell? 7. A man had $ 1500 in the bank and drew out 40% of it. How much remained in the bank ? Note. —100% represents all he had in the bank. 100% -40% = 60 %, the part that remained. The question then becomes, How much is 60% of $1500? 8. A farmer having 320 acres of land sold 15% of it to one man and 25% to another. How many acres did he sell? 9. A wholesale grocer had 480 bbl. of A sugar, and sold 12:^% of it. How much remained unsold? 10. How much is .5% of 80 ? 11. How much is i% of $ 4000 ? Question b, Oral. 1. 5 is ^25^ of what ? 4. 12 is 8% of what ? 2. 5 is 25% of what ? 5. 30 is 12^% of what ? 3. 40 is 10% of what? 6. 15 is 50% of what? Written. 297. Percentage and rate given, to find base. Percentage -^ Kate = Base. 7. $125isl2|-% of what? • 8. 150 bu. is 33^% of what? 9. 240 is 120% of what? 10. $ 1644 is 40% of what ? 11. 75is3J% of what? WRITTEN EXERCISES. • 217 12. 289 is 50% of what? 13. 25% of my property is $5000. What is the value of my property ? The question is, " $ 5000 is 25 % of what ? " 14. I sold a horse for $S1, which was 90% of what it cost me. What did the horse cost me ? Question c, Oral. 1. What part of 45 is 15 ? 2. What per cent of 45 is 15 ? 3. What per cent of 80 is 60 ? 4. What per cent of 90 is 30 ? 5. $40 is what per cent of $60? ' 6. 12 yd. is what per cent of 36 yd. ? 7. 14 bu. is what per cent of 56 bu. ? 8. f is what per cent of |^ ? Written. 298. Base and percentage given, to find rate. Percentage -i- Base = Rate. 9. What per cent of $ 240 is $ 80 ? 10. 150 is what per cent of 900 ? 11. What % of a long ton is a short ton ? 12. What % of 5 days is 6 hours ? y^ 13. 5 cwt. is what % of 3 tons? 14. $ 28.16 is what % of $ 7040 ? 15. What per cent is i of 2i ? ^ of f ? f of 7^ ? 16. My salary is $ 1600 and my expenses $ 1200. What % of my salary are my expenses ? The question is, " $ 1200 is what % of $ 1600 ? " 299. The sum of the base and percentage is called the Amount. 218 ' PERCENTAGE. 300. The difference between the base and percentage is called the Difference. Find the amount in the following : 1. How much is 10% of 20 ? Find the difference. 2. 20 is what per cent of itself ? 3. If 20 is increased by 10% of itself, the amount is 22. What per cent of 20 is 22 ? Solution. — The base ... 20 is 100 % of 20. The percentage . 2 is 10 % of 20. Therefore the amount . 22 is 110% of 20. Ans. 4- 100% +10% = ? 1 + 10%=? 5. If 20 is diminished by 10% of itself, the difference is 18. What per cent of 20 is 18 ? Solution. —The base . . . 20 is 100% of 20. The percentage . 2 is 10 % of 20. The difference . 18 is 90% of 20. Ans. 6. 100% - 10% = ? 1 - 10% = ? 7. What number increased by 10% of itself equals 220 ? Solution. — Since 220 is 10% more than the required number, 220 is 110% of the required number. The amount, 220, is now treated as the percentage, and 110% as the rate: and the question becomes, 220 is 110% of what number? (Question &.) 220-1.10 = 200. Ans. Amount -^- (1 + rate) = Base. 8. What number diminished by 10% of itself equals 180 ? Solution. — Since 180 is 10 % less than the required number, 180 is 90% of the required number. The difference, 180, is now treated as the percentage, and 90 % as the rate; and the question becomes, 180 is 90% of what number? (Question ft. ) 180 - .90 = 200. Ans. Difference -h (1 — rate) = Base. WRITTEN EXERCISES. 219 9. What number increased by 25% of itself equals 290 ? 10. What number diminished by 25% of itself equals 243 ? 11. After selling 20% of his sheep, a farmer had 400 sheep left. How many had he at first? 12. The population of a certain city has increased 12% in two years. If it now numbers 56000, what was it at the beginning of the two years ? 13. A clerk's salary was increased 6J%. If he now receives $ 850, what was his original salary ? 14. By selling goods at $630 I lose 121%. What did I pay for them ? 15. $580 is 10% less than what number? 16. I sold goods at $450, which was 120% of the cost. What was the cost? 17. After withdrawing 45% of my money from the bank, I still have $1300 on deposit. How much had I in the bank at first. 18. A farmer increased his flock of sheep by 12J%, and then had 900. How many had he at first ? 19. A man, after spending a month in the Adirondacks, finds that his weight is 210 pounds, which is an increase of 5%. What was his weight before he went to the Adiron- dacks ? 20. A regiment lost 12|% of its men in an engagement, and had 560 left. How many men were there before the engagement ? 21. A owes C 33J% more than he owes B. If he owes C $ 800, how much does he owe B ? 22. 1227.83 is |-% less than what number? 23. 4 is 20% less than what number? 220 PERCENTAGE. 24. A city lot cost $3600, which is 55% less than the cost of the house. What was the cost of the house ? 25. A farmer raised 1500 bu. of corn, which was 33^% less than the number of bushels of wheat raised. How many bushels of wheat had he ? 26. In the year 1896 a merchant's profits were $ 1836.25, w^ich was 25% more than his profits of 1895. What were his profits in 1895 ? PROFIT AND LOSS, 301. Oral. State the question only. 1. How much is a 10% profit on goods that cost f 200? 2. I bought goods for $400, and sold them at a loss of 5%. How much did I lose ? 3. If I buy goods at f 400, and sell them at $ 600, what per cent profit do I make ? 4. If I buy at $400, and sell at $350, what % do I lose ? 5. By selling a house for $1600 I gain 33^-%. What is the cost? ? 6. John sold his skates for 64 cents, and thereby lost • 5%. What did he pay for them ? 302. All computations in Profit and Loss come under the rules of Percentage. The cost corresponds to the base, and the gain or 4oss is a percentage of the cost. The selling price is the amount when there is a profit, and the difference when there is a loss. PROFIT AND LOSS. 221 303. Written. 7. If I buy eggs for 10 cents a doz., and sell them for 12^ cents, what per cent do I gain ? 8. A grocer bought tea at 18 cents per pound, and sold it at 30 cents per pound. What was the rate of gain ? 9. Find the profit on a bicycle that cost f 75, and was sold at an advance of 30%. 10. Find the selling price of a horse bought at $ 88.65, and sold at 3J% below cost. 11. Find the rate per cent of loss on a cow bought for $80, and sold for $60. 12. Find the rate per cent of profit on a car-load of Cortland wagons sold for $1090, and bought for $1000. 13. Find the cost of a herd of cattle sold at 12|-% above cost at a profit of $ 240. 14. A man bought books for $194, and sold them at a gain of 32%. What was the gain ? 15. I sold a house and lot that cost $11,225 at a loss of 5^%. What was the loss ? 16. Mr. A., by selling his horse at a profit of 14%, made $32.20. What did the horse cost? 17. By selling sljgar at one-half cent per pound profit, a grocer makes Ife per cent. What does he get per pound for his sugar ? 18. An agent gained $.09 by selling twine 25% above cost. What did it cost him ? 19. Find the cost of cotton sold at 16 1% above cost at aprofit of $211.25. 222 PERCENTAGE. 20. By selling flour at a loss of 14|^%, a grocer loses $ 13.45. What was the cost ? 21. A farm that cost $2675 was sold for $3745. What was the gain per cent ? 22. Hats that cost $43.50 a dozen are sold for $4.50 apiece. What is the rate of gain? 23. By selling boots for $206.40 a merchant gained ^. What did they cost him ? 24. By selling corn for $92.61, a man gained 12^%. What did it cost him ? 25. I sell a horse for twenty per cent less than my ask- ing price, and yet make twenty-live per cent profit. I asked $ 200. What did the horse cost me ? 26. My height is 6 feet 1^ inches, my neighbor is 5 feet 10 inches. What per cent am I taller than he is ? 27. A farmer sold 160 acres of land for $2944, which was 8% less than it cost. What did it cost an acre ? 28. By selling a horse for $160, I lose 20%. What would have been the selling price had I gained 20% ? 29. 14f % was gained by selling tea at $ .45 a pound. What did it cost a pound ? 30. Mr. Brown sold a lot for $4300, and by so doing made 11^%. What did he gain ? 31. If I buy oranges at the rate of 3 for 3 cents, and sell them at the rate of 2 for 5 cents, what per cent profit do I make ? 32. A jeweller sold two watches at $24 each. On one he gained 20%, and on the other he lost 20%. What did both watches cost him ? COMMISSION. 223 COMMISSION. 304. Oral. 1. A certain agent receives for his services 2% of the value of the goods which he sells. How much will he receive for selling f 1000 worth of goods ? 2. A purchasing agent receives for his services 3% of the value of the goods purchased. How much will he receive for purchasing f 2000 worth of goods ? . 3. How much must I pay my agent for selling f 3000 worth of potatoes if I pay him 5% ? 4. At 5% how much will a collecting agent receive for collecting $800? 5. How much must I pay my broker for selling $1000 worth of stocks, if I pay him 1% of their value ? 305. An Agent is a person who transacts business for another. 306. Some agents are known as Brokers, or Commission Merchants, according to the kind of business transacted. 307. The compensation of an agent is called Commis- sion, or Brokerage. The commission of a purchasing agent is usually a cer- tain per cent of the value of his purchases. The commission of a sales agent, or of a collector, is usually a certain per cent of the amount collected. 308. The merchandise sent to a commission merchant to be sold is called a Consignment. 309. The sender is the Consignor, and the person to whom the goods are sent is the Consignee. 310. The commission is the percentage, and the amount collected or invested is the base. 224 PERCENTAGE. 311. Written. 6. An auctioneer charges 5% commission for selling $864 worth of goods. What is the amount of his com- mission ? 7. Sold 850 barrels of flour at $5.25 a barrel, and charged 2^% commission. Find my commission. 8. What is an agent's commission for selling 6840 lb. of butter, at 19 cents a pound, commission l|-% ? 9. A dealer sells real estate for a commission of 2%. How much must he sell during the year to secure an income of $75 per month? 10. A broker in New York received J of one per cent commission for negotiating a sale of 500 one thousand dollar bonds. What was his commission? 11. A real estate man made $50 by receiving 2 J per cent instead of his regular commission of 2 per cent. What did his sales amount to? 12. An agent's fee for collecting bills is 3%. If he receives $86.25 as his commission, how much money has he collected ? 13. An agent collected $1864 from a sale of some pictures, and received $ 4.66 as his fee. What was the rate of commission ? 14. An agent having sold 1250 velocipedes at $8 apiece, invested his commission of If % in a new stock company. How many shares at $ 25 each did he take ? 15. What 'per cent does an agent charge who receives $223 for buying $5575 worth of produce ? 16. A man sends his agent $6120 to invest in flour, after deducting his commission of 2%. How much money is spent for the flour, and how much for the agent's commission ? COMMISSION. 225 17. A man is paid 5% for collecting $235.75, How much must he pay over to his employer? 18. A merchant sent his agent $3150 with which to buy flour after deducting his commission of 5%. At $4 per barrel how many barrels did the agent buy ? 19. An agent sold iron for $9872. He received $ 163.70, which included a freight charge of $52.64. What rate of commission did he receive? 20. Received as net proceeds from a sale of cotton $1025.70, after paying my agent 2i% for selling. What did the sale amount to ? 21. An auctioneer sells 15 tables at $1.45 apiece, 22 chairs at $ 1.12^ ajjiece, and some pictures for $ 8.70, on a commission of 5|-%. What were his commission and the net proceeds of the sale ? 22. My agent in Boston sold a number of bicycles at $85 each. After deducting his commission of 3|-%, he returned to me $ 5759.60. How many bicycles did he sell ? 23. An agent who sold 150 lots at $ 233J each, charged $ 262.50 for his services. What rate of commission did he get? 24. A collector pays over to his principal $23358.39^, after deducting a commission of 4-|^%. How much was the entire collection ? 25. If I send my agent $ 367.20, with instructions to buy tea at 30 ct. a pound, and he charges 2% for buying, how many pounds of tea should I receive ? 26. A real estate agent charges me two per cent for sell- ing my property in Boston. He remits me $ 5880. What was his commission ? 226 PERCENTAGE. 27. A commission merchant in Kew York charged $36 foi: insuring my goods, $ 14 for cartage, and $ 50 commis- sion at 2i per cent for selling them. How much money should he remit to me ? 28. Sent my agent $ 2050 to invest in coal at $ 4 per ton, after deducting his commission of 2i per cent. How many tons of coal could he buy ? 29. A cotton broker received $ 2531.71 with which to buy cotton at $ .12 a pound. He charged 2^ % commission. How many pounds of cotton did he buy, and what was his commission ? 30. A real estate agent receives $ 162,193.50 from a com- pany to invest in land. If he charges 5% commission, how many acres of land can he buy at $9 an acre ? What is his commission ? 31. An agent sold 12000 lb. of cotton at 10 cents a pound. He invested the proceeds in lumber at $ 25 per M. If his commission for selling was 4%, and for buying 2%, how many feet of lumber did he purchase ? 32. A grain-dealer received $4820.40 with which to buy wheat at 60^ a bushel after deducting his commission of 3%. How much wheat did he purchase ? 33. How much stock can be bought for $ 10827, allowing li% brokerage ? 34. A speculator sent $ 7308 to his agent in St. Paul, directing him to invest in wheat. How many bushels of wheat could he buy at 90^ a bushel, after deducting his commission of 1^% ? 35. An agent has bought 1170 bbl. of flour at $ 4.50 per barrel. How much money must his principal remit to him to pay the cost of the flour and his commission of 2%? INSURANCE. 227 INSURANCE. 312. Insurance is security against loss. The insurance company agree, for a specified amount, to be paid at stated periods (usually once a year), to pay a definite sum to the insured or his estate in case of loss. Some of the different forms of insurance are Life, Fire, and Accident. 313. The stated sum paid for insurance is called the Premium. It is always a certain per cent of the insurance. 314. The written contract between the insurance company and the insured is called the Policy. 315. Life Insurance. The following are the principal forms of policies issued by life insurance companies : 316. Life Policies, with continuous premiums, or with premiums limited to 5, 10, 15, or 20 years ; also, single pay- ment life. These policies are payable at death only. 317. Endowment Policies, with continuous premiums for 10, 15, 20, 25, 30, 35, or 40 years; also, single payment endowment for either of the periods mentioned, and 10 payment endowment for 15, 20, 25, or 30 years. These policies are payable at the end of the stated period, or at death, if it occurs before the end of the period. There are various other forms of policies, but the above are the most common. 1. For an annual premium of $22.63 a certain company will insure a man 30 years of age for $1000, payable at* death (Life Policy). The premium might be stated as 2.263%. What would a $5000 life policy cost the man each year ? 228 PERCENTAGE. The following table may be used to determine the premium rates per $ 1000 for ages 20-44 inclusive : Whole Life Policies. Endowments. i < 20 IS ■^ o ^1 fe S m hi hi O < 20 24 33 28 83 3811 296 05 100 00 6314 45 29 35 05 28 62 21 17 70 24 82 29 39 38 84 30154 10011 63 27 45 43 35 21 28 79 21 22 18 15. 25 32 29 97 39 60 307 21 100 23 63 40 45 58 35 37 28 97 22 23 18 62 25 84 30 58 40 39 313 08 100 35 63 54 45 74 35 54 29 17 23 24 1911 26 38 3121 4121 319 14 100 49 63 69 45 90 35 72 29 39 24 25 19 63 26 95 3187 42 05 325 41 100 63 63 84 46 07 35 91 29 63 25 26 20 17 27 54 32 55 42 93 33189 100 78 64 01 46 25 36 12 29 88 26 27 20 74 28 15 33 25 43 84 338 58 100 93 6418 46 44 36 35 3014 27 28 2134 28 78 33 i)8 44 78 345 50 10110 64 37 46 65 36 59 30 42 28 29 2197 29 44 34 74 45 75 352 64 10127 64 56 46 87 36 85 30 73 29 30 22 63 3012 35 53 46 76 360 02 10145 64 76 4710 3713 3107 30 31 23 32 30 83 36 34 47 81 367 04 10164 64 98 47 35 37 43 3144 31 32 24 05 3158 37 19 48 89 375 51 10184 65 20 47 62 37 76 3183 32 33 24 82 32 36 38 07 50 01 383 63 102 00 65 44 47 92 3812 32 25 33 34 25 63 3317 38 99 5117 392 02 102 28 65 71 48 24 38 50 32 71 34 35 26 49 34 01 39 94 52 38 400 68 102 51 65 99 48 58 38 92 33 21 35 36 27 39 34 90 40 93 53 64 409 63 102 76 66 29 48 95 39 37 33 76 36 37 28 35 35 83 4197 54 94 418 87 103 03 66 62 49 36 39 87 34 36 37 38 29 36 36 81 43 06 56 29 428 42 103 33 66 99 49 82 40 42 35 01 38 39 30 43 37 84 44 20 57 70 438 29 103 65 67 40 50 32 4102 35 73 39 40 3157 38 92 45 39 5917 448 49 104 01 67 85 50 87 4168 36 52 40 41 32 78 40 07 46 65 60 71 459 05 104 41 68 34 5148 42 41 37 38 41 42 34 07 4129 47 97 62 33 469 97 104 86 68 89 52 15 43 22 38 32 42 43 35 45 42 58 49 37 64 02 48126 105 36 69 51 52 88 4410 39 35 43 44 36 91 43 94 50 84 65 79 492 88 105 92 70 20 53 71 45 07 40 48 44 2. For an annual premium of ^47.10 the same company- will insure the same man for $ 1000, payable in 20 years, or at death, if the insured dies before 20 years (20-year endow- ment policy). What would a $2500 20-year endowment policy cost the man each year ? 3. What is the premium on a life policy for f 2500, age 37, continuous annual premium ? On a 10-payment life policy ? On a 15-year endowment ? On a single-payment life policy ? INSURANCE. 229 4. A man 25 years of age is insured for $ 5000, 20-pay- ment life policy. He dies at the age of 34. How much more is paid to his estate than his insurance cost him ? 5. What will be the cost of a 10-year endowment policy of $3000, age 20, on the annual dividend plan, the first dividend of $ 10.50 per $ 1000 being paid when the second premijim is due, each succeeding dividend increasing 1% of the premium, the last one being paid at the end of the 11th year ? soiiimoN. Age 20. Dividend. Net Cost. 4 1 10 11 31.50 34.50 37.50 40.50 43.50 46.50 49.50 52.50 55.50 58.50 300.00 268.50 265.50 262.50 259.50 256.50 253.50 250.50 247.50 244.50 2608.50 — 58.50 = 2550.00 = cost. If the full premium of $ 300 is paid each of the ten years, and the profits are allowed to accumulate so that at maturity (i.e. at the end of the ten years) the policy is worth $ 3627 1$ 3000 being the face value of the policy, and $ 627 the accumulated profits), what per cent is paid on the money invested ? This form of insurance is sometimes called Semi-Tontine Insurance. 318. Annual Dividend. Since it is impossible to exactly foretell the actual cost of insurance, it is necessary to make the premiums large enough to meet all contingencies, such as increased death rate and reduced rate of interest. Con- sequently, there usually arises a surplus, caused by over- 230 PERCENTAGE. payments. This surplus is distributed annually to each policy holder in proportion to his over-payment, and con- stitutes the Annual Dividend. 319. Semi-Tontine. Under this plan the surplus, instead of being paid annually as dividends, is allowed to remain with the company at interest, and accumulate for a period of 10, 15, or 20 years, as may be selected. Those who sur- vive, and keep their policies in force until the end of the period, receive both the surplus arising from their own policies, and their equitable share of the surplus arising from policies discontinued by lapse or death during the period. 320. Fire Insurance is security against loss by fire. 321. 1. I keep my house insured for $4000. I pay the insurance company 1% annually. How much do I pay annually ? 2. If my house (Ex. 1) burns down at the end of three years, how much shall I receive from the company more than I have paid them ? 3. If I pay $75 per annum for insuring my house at 1%, for how much is it insured? 4. At 2% what will be the cost of insuring $10000 of merchandise at | value ? 5. My household goods are insured at the rate of $.45 per annum on each $100. What premium do I pay if the amount of insurance is $ 1200 ? 6. For a premium of $ .90 on each $ 100 the same com- pany will insure household goods for a period of three years. How much is saved on $1200 insurance on this plan, as compared with the plan in Example 6 ? INSURANCE. 231 322. Accident Insurance. The following is one form of accident insurance : For a quarterly premium of $ 5, a person may be insured for $ 10000 in case of death, or a weekly indemnity not to exceed $50 will be paid, not to exceed 52 weeks, in case of total disability caused by acci- dent while a passenger on a public conveyance propelled by steam, electricity, or cable. One half the above amounts will be paid in case of an accident due to any other cause. 323. 1. After paying seven such premiums a person is thrown from his bicycle, and injured so that he is disabled for 5 weeks. How much more does he receive than he has paid out ? 2. If he had been injured in a railroad accident, and dis- abled for the same time, how much more would he have received ? 3. What will be the premium on a 30-year endowment policy, age 29, for $4000 ? 4. If the total dividends amount to 30% of the total premiums paid, how much more does the man receive at the end of the 30 years than he has paid ? 5. What will it cost to insure a house worth $2500 at |- of its value, for 3 years, at f % ? 6. Injured a country store for $ 5000, and goods for $10,000, at 30^ on $100. $1 is paid -for the policy. What does the insurance cost ? 7. What will it cost to insure a mill for $5000, the rate being one and one-half per cent for 3 years ? 8. How much will I save by insuring my property for $ 5000 at f of one per cent for 3 years, rather than taking an annual policy for i of one per cent ? 232 PERCENTAGE. 9. Tt costs me to insure my house f 22.50 when the rate is J of one per cent. What is the amount of my policy ? 10. A stock of goods is insured for one-half the value, the premium being $ 30, and the rate y^ of one per cent. What is the value of the goods ? 11. The semi-annual premium per one thousand dollars on my $6000 life-insurance policy is $26. What does it cost me a year ? 12. A person who pays $ 12 semi-annually for accident insurance is disabled by an accident for 13 weeks, during which time he receives $ 10 a week. If he had paid three premiums, how much more does he receive than he has paid out ? 13. Paid for insuring a house for ^ of its value, $151. The rate being 75^ on $100, and the policy costing $1, what was the house worth ? 14. To insure a house at -J of 1% cost me $20. What was the house worth ? 15. Paid $18 for insuring goods worth $9000. What was the rate ? 16. A merchant pays $75 a year insurance on his stock of goods at 1J%. What is the value of his stock of goods? 17. A block worth $30000 is insured for | of its value at 2%. How much does the owner lose in case of its total destruction by fire ? 18. For how much must a cargo of wheat worth $23400 be insured, at 2i%, so that the owner, in case of loss, may recover both the value of the cargo and the premium ? Note. — The value of tlie wheat = 97^% of the amount insured. TKADE DISCOUNT. 233 TRADE DISCOUNT. ^<^i.2ms^^ 324. The deduction of a percentage from the price of merchandise is called Commercial Discount. It is used largely by manufacturers and wholesale mer- chants. The greatest discounts are for large purchases and cash payment. 325. The List Price is the price given in the price-list. 326. The Net Price is the list price less the discount. 1. If I can purchase books at 25% off for cash, what must I pay for books listed at ^ 80 ? Solution. — 100 % - 25 % = 75 %. 75 % of $ 80 = $ 60. Ans. 2. At what per cent above cost must a merchant mark his goods so that he may allow a discount of 25% from the marked price, and still make a profit of 10% ? Solution. — Selling price = 110% of cost. This selling price is 75% of the marked price. The question is, "110% is 75% of what?" 1.10 -=- .75 = 1.46| %, therefore the marked price is 46f %.above cost. 3. Find the sum to be paid on a bill of $264 with 10% off for cash. 4. What is the net price of a bill of goods, the list price of which is $56, subject to discount of 25% ? 5. What must be paid on $935, if 15% and 10% off are allowed ? Solution. — Deducting 15% is the same as allowing 85% of the bill. 85 % of 935 = 794.75. 90 % of 794.75 = 715.28. Ans. Note. — When two or more discounts are allowed, the first is deducted, the second computed on the remainder, and deducted from it, etc. 6. Which is the better for the buyer, 40%, or 25% and 15% off? 234 PERCENTAGE. 7. Find a single discount on a bill of $300 equal to 20% and 5% ofp. 8. A discount of $4 was allowed on a bill, which was then paid with a check for $36.- What rate per cent was taken off? 9. Consulting my price-list, I find I can buy goods which are marked $450 at a discount of 20% and 5% olf for cash. How much will the goods cost me, and how much discount do I receive ? 10. Bought furniture amounting to $520 on credit for, 6 months, or 5% "discount for cash. What ready money will pay the bill? 11. What is the cash value of a bill of books amounting to $40, on the face of which a discount of 20% and 5% is made ? 12. The net amount of a bill of goods is $ 359.10. What is the gross amount, the rate of discount being 10% and 5% ? 13. A set of Encyclopaedias, whose catalogue price is $100, can be bought at a discount of 2 tens and 5% off for cash. How much less than the catalogue price will they cost? Note. — The expression 2 tens and 5% means 10%, 10%, and 5%. 14. B offers me some hammocks for $450 with a dis- count of 20%, and 4% off for cash, and A offers me the same goods at a discount of 2 tens and 4% off. Which is the better offer, and how much ? 15. A dealer sold goods at 10% below his asking price, but still made a profit of 20%. What per cent above cost had he marked the goods ? 16. A merchant marked carpeting that cost him 60 cents a yard so that he could allow a discount of 10% and still make a profit of 20%. At what price did he mark it? TAXES. 2B5 17. A book-dealer sold a stock of books for $ 1140, at a discount of 10% from the marked price, and finds that he has made a profit of 14%. What did he pay for the books, and what was their marked value ? 18. Find the net amount of a bill for $386 subject to the following discounts, 20%, 10%, and 5%. TAXES. 327. A tax is a sum of money levied upon property and persons for public use. Note. — A tax upon persons is called Capitation or Poll Tax. It is levied in some localities upon men of full age, without regard to their property. It is usually but a small amount upon each person. The practice is going out of use. 328. Property is of two kinds, Real and Personal. 329. Eeal Property is immovable property, as lands and buildings. 330. Personal Property is property that is movable, as money, securities, household goods, horses, cattle, etc. 331. A tax assessed upon property is a Property Tax. 332. Assessors are officers chosen to make a list of tax- able property, estimate its value, and apportion the tax. 333. A tax is a percentage upon the assessed valuation of property. The tax on $ 1 is the rate. 1. The valuation of property in a certain town is $1,500,000, and the rate is 11%?. What is the tax? 2. The tax to be raised in a certain village is $37,500. The taxable property is $2,500,000. What is the rate? What will be A's tax on $15,000 real estate, and $3000 personal ? 236 PERCENTAGE 3. The property of a town is assessed at $1,250,000. The tax to be raised is $15,975. There are 650 polls, assessed at $1.50 each. What is B's entire tax, if his property is assessed at $2500, and he pays the poll- tax? Rule. — Deduct the amount of poll-tax, if any, from the whole tax. Divide the remainder by the assessed valu- ation. The quotient will be the rate. To find each person's tax, midtiply the assessed valua- tion by the rate, and to the product add the poll-tax^ if any. 4. The officers of a certain town find that all the town expenses for the year 1896 will amount to $46,000. The tax-roll shows real estate valued at $2,000,000, and per- sonal property at $300,000. What is the rate of taxa- tion? 5. A certain town votes to raise a tax of $14,250, be- sides the collector's commission of 5%. What is the rate of taxation if the property valuation is $ 1,000,000 ? What is the collector's commission, and what is A's tax, on property valued at $ 4500 ? 6. If the assessed valuation of a village is $2,384,564, and there are 750 polls at $ 1.50 each, what must be the rate of taxation to meet an expense of $29,807.05? What is B's entire tax, if his property is valued at $3875, and he pays for 1 poll ? 7. What is the valuation of my property, if my tax, 15 mills on a dollar, amounts to $ 30 ? 8. What is my entire tax, if I pay a poll-tax of $1.68, and my property is valued at $24,750, when the rate of taxation is $16.28 on $1000? DUTIES. 237 9. The annual tax-rate for the State of New York for the year 1896 was 2.69 mills on the dollar. The amounts to be raised by tax are as follows : $ 961,116 for general expenses, §4,062,903 for free schools, §2,360,103 for the canals, and §4,368,712 for the State care of the insane. What was the assessed valuation of the property of the entire State? The tax-rate for 1895 was 3.24 mills. What was the entire tax of 1895? DUTIES. 334. Duties are taxes on imported goods, levied by the government, and collected at custom-houses. A port con- taining a custom-house is called a Port of Entry. 335. An Ad Valorem Duty is a certain rate on the value of goods at the place from which they were shipped. 336. A Specific Duty is a fixed sum charged upon an im- ported article, without regard to its value. Allowances are made as follows, in collecting specific duties : for Tare, which is weight of box, cask, etc. ; for Leakage, which is loss of liquids in barrels or casks ; and for Breakage, which is loss of liquids in bottles. 337. The Gross Weight is the weight of articles before any allowances are made. 338. The Net Weight is the weight after the allowances are made. 1. At 30 per cent ad valorem, what is the duty on goods valued at §725? 2. What is the duty on 10 gross of silver spoons, valued at §4.50 a dozen, at 30% ad valorem? 238 PERCENTAGE. 3. A. Mark's Sons imported from Lyons 1560 yd. of silk invoiced at 87^ ^ per yard. What was the duty at 25 ^ a yard, and 30% ad valorem ? 4. If the average rate of duty under the McKinley law was 49.58 per cent, and under the Wilson law it is 37 per cent, what is the difference in revenue on $ 1,000^000 worth of dutiable imports ? 5. A merchant bought goods in London invoiced at £> 450. At the custom-house in New York he paid an ad valorem duty of 18%, and a specific duty of $325. What was the entire cost of the goods in United States money ? 6. Imported from England 5 cases of cloths and cash- meres, net weight 95 lb. ; value as per invoice £ 375 10s. What is the duty, the rate being 50^ per pound, and 35% ad valorem ? QUESTIONS. 339. 1. What is the meaning of the term per cent? How is per cent written? 2. Define Base, Percentage, Rate per cent. Amount, Dif- ference. 3. Tell how to find percentage when base and rate are given. To find base when percentage and rate are given. To find rate when percentage and base are given. 4. Tell how to find base when amount and rate are given. When difference and rate are given. 5. Define Commission, Brokerage, Insurance, Premium, Policy, Taxes, Eeal Estate, Personal Property. 6. What is trade discount? List price? Net price? Give the rule for finding net price. MISCELLANEOUS REVIEW. 239 MISCELLANEOUS REVIEW OF PERCENTAGE. 340. 1. Find 8% of 750. Q>\% of ^12.75. |% of 912. -{^% of 2140. • 2. A man gave his son 42% of his money, his daughter 25% of it, and his wife 16-|% of the remainder. If the son received $ 9350 more than the daughter, what did each receive ? 3. A dealer sold a horse and carriage for $ 637, which was 40% more than cost. If the horse cost -f- as much as the carriage, what did each cost ? 4. What per cent is gained or lost when one-half an article is sold for what the whole cost? When f of an article is sold for what one-half cost ? 5. A merchant pays $ 35 for a suit of clothes. What must he ask for it, so that he may drop 16% from his asking price, and still make 20% on the cost? 6. 14.35 is j7^% of what number? 7. A man spent 20% of his salary for board and 15% of what was left for clothes. If he spent $ 132 more for board than for clothes, how much did he spend for each ? Ni 8. What number diminished by'16|% is 605 ? 9. What number increased 35% is 382.5? 10. Sold a load of wheat weighing 3240 lb. at 68^ a bushel of 60 lb., thereby making a profit of Q>\ per cent. Required the cost of the wheat. 11. On a certain day the sun rose at 5 o'clock and 43 minutes, and set at 6 o'clock and 25 minutes. What per cent of the day was in sunlight ? 12. The salary of a certain teacher of arithmetic is $ 1600. His real estate tax is $ 90 ; his water tax is $ 25 ; gas bill, ^ 15 ; coal bill, ^ 45 ; other expenses, $ 325. What per cent of his salary does he save ? 240 PERCENTAGE. 13. The Oswego Starch Factory employs 700 operatives. The population of Oswego numbers 22000. What per cent of the population are employed in the starch factory ? 14. I is 25% more than what fraction? 15. A dealer lost 10% of his capital, then gained 20% of the remainder, when he had $ 2160. How much had he at first? 16. Goods bought for $400 are marked to sell at an advance of 40%, but are finally sold at a reduction of 25% from the marked price. What is the per cent of gain? What is the gain ? 17. An article is sold for $ 2.80, this being an advance of 2^0/0. Find the cost. 18. A merchant buys sugar at an average price of 4 cents a pound, and sells at a profit of 8%. How many pounds must he sell to clear % 500 ? 19. If by selling an article for 59 cents a dealer gains 10% more than by selling for 55 cents, what is the original cost? 20. 15% of an estate is invested in city bonds, 40% in real estate, 25% in railroad stock, and the remainder, % 5000, is deposited in a bank. What is the estate worth ? 21. Define Percentage, Base, Profit and Loss, Commission. 22. Give the five formulas of percentage. 23. Express as a decimal f per cent. 24. A man having a yearly income of % 1500 spends 80% of it the first year, 75% of it the second year, ^2\clo of it the third. How much does he save in 3 years ? 25. 25% of 200 bushels is 2i% of how many bushels ? 26. A man sold 80 acres of land for % 1472, which was 8% less than it cost. What did it cost an acre ? MISCELLANEOUS KEVIEW. 241 27. What terms in Profit and Loss correspond to base and amount ? 28. Find the cost of fruit sold for $ 207.70, at a gain of 15%? 29. At what price must hats that cost $ 1.12 each be marked in order to abate 5%, and yet make 25% profit ? 30. What is the base in commission ? 31. A commission merchant sells 225 bu. of corn at $ .65 a bushel, and 360 bbl. of apples at $ 2.40 per barrel, com- mission 5%. Find the commission and the net proceeds. 32. The net proceeds are $3800, the rate 10%. Find the amount of sales and the commission. 33. Find the rate, the commission being $ 125, and the sum invested $ 2500. 34. A merchant owning f of a cargo valuld at $ 44000 insures f of his share at 2^%. What premium does he pay? 35. A man having $ 400 paid 62^% of it for a carriage. How many dollars had he left ? 36. An agent charged $ 432.46 for selling goods at $ 49424. What was his rate of commission ? 37. A man sold four horses for $ 100 each. On two he gained 25fcy and on the other two he lost 25%. Did he gain or lose on the transaction, and how much ? 38. If f the number of girls in a certain school exceed the boys 10%, and the girls number 275, what is the num- ber of boys ? 39. A farmer's sheep increased 10% each year for 2 years, when he had 242. How many had he at first? 242 PERCENTAGE. 40. My New York agent buys for me 40 pieces of silk, 32 yards in a piece, at $ 5 a yard. He charges 1^% com- mission. How much money will it require to purchase the silk and pay his commission ? 41. A commission merchant in Boston has sold goods for me to the amount of $6932. He has charged l\-% commission, $ 18.50 cartage, and $ 12.15 for storage. How much is due me ? 42. A boy bought oranges at the rate of 3 for 5 cents, and sold them at the rate of 2 for 5 ^. What was his rate of gain ? 43. Ten per cent of a number is 32 less than eighteen per cent of the same number. What is the number ? 44. I paid $28,871 for insuring my house for |3850 for three years. What was the rate of yearly premium ? 45. A stock of goods valued at $6300 is insured for | its value at |%. What will be the owner's loss if the goods are totally destroyed by fire? 46. A man's income is $ 1720, which is 16|% of the sum he has invested. What sum has he invested ? 47. From a cargo containing wheat, 1620 bu., or 7%, was washed overboard. What number of bushels remained ? 48. A stock -dealer sold 38 head of cattle, which was 4% of his entire herd. How many had he left ? 49. In an orchard containing 820 trees, 20% of them were pear-trees, and the remainder were plum-trees. How many plum-trees were there in the orchard ? 50. From a cask of wine containing 65 gal. all but 15% was sold. How many gallons were sold? 51. In a school containing 875 pupils 32% of them are boys and the remainder girls. How many girls are there ? MISCELLANEOUS REVIEW. 243 52. There is a loss of $ 500 on a house and lot sold for $ 5000. What is the per cent of loss ? 53. An agent reports that he invested the money re- mitted him in wheat, which he sold at an advance of 15% ; then investing the proceeds in a second quantity, he was forced to sell at a loss of 12i%. He now deducts ^100 for expenses and commission, and remits ^5333.75 to his employer as the balance due him. Find the loss to the employer. 54. $90 are paid as premium for insuring a block for three-fourths of its value. If the rate of insurance is f %, what is the value of the property ? 55. In 1896 New York State had a population of 5,998,000, and New York City had 1,515,000. What per cent of the population of the State lived in New York City ? 56. During the war of 1861-1865, the State of New York paid $ 40,000,000 in bounties to her volunteers. Her population at that time was (in round numbers) 4,000,000. What was the average cost to each inhabitant ? 57. If I sell 6 horses for what 8 horses cost, what is my rate of gain ? 58. Sold wheat for $ 73.54i, by which a gain of 15% was made. What did the wheat cost, and what sum was gained ? 59. In a school containing 1160 pupils, 638 are girls and the remainder are boys. What per cent are boys ? 60. A hall is 42 ft. wide, and 294 ft. long. What per cent of the length is the width ? 61. In a certain battle 22|% more than i of the soldiers were killed. If the loss was 110 men, what was the original number? 244 PERCENTAGE. 62. In an orange-grove S^% of the trees were ruined by frost. If 1100 remained uninjured, how many were destroyed ? 63. A merchant sold a lot of goods for ^ 550, thereby gaining 10%. Find the cost of the goods. 64. A man sold a watch for $32, thereby losing 20% on the cost. Find the cost. 65. If a man owns 66| per cent of a factory, and sells 33-J per cent of his share for $ 1800, what is the value of the factory ? 66. A sold 30% of his steamship to B; B sold 60% of his purchase to C ; C sold 75% of his share to D for $ 27,000. What was the value of the vessel ? What was each one's share in dollars after the sales had been made ? 67. After a discount of 30% had been made upon the catalogue price of a book, it was sold for $ 1.75. What was the catalogue price ? 68. Bought a horse for $ 120, and sold him for $ 135. What part of the cost was the gain? What per cent? 69. Bought tea at 60 ct. a pound. What must I ask per lb. so as to abate 10% and still make a profit of 25% ? 70. A merchant's profits for 1895 were 13402.84. If they were 6|% less than in 1894, what were they in 1894 ? 71. In one week John solved 75 problems, correctly. If he failed in 16|% of the number attempted, how many were there in all ? SIMPLE INTEREST. 341. 1. I borrow $ 500 for 1 year, and at the end of the year I repay the money and 6% for the use of it. How much do I pay for the use of $ 500 ? 2. How much must be paid for the use of $50 for 1 10 ' year at 5% ? At 7 3. How much at 5% per annum must I pay for the use of $ 1000 for 1 year ? For 3 years ? 4. I loan James Barnes $500 at 6%. At the end of 2 years he pays me in full. How much does he pay me ? Money that is paid for the use of money is called Inter- est. The money for the use of which interest is paid is called the Principal, and the sum of the principal and interest is called the Amount. Interest at 6% means 6% of the principal for 1 year. 12 months of 30 days each are usually regarded as a year in computing interest. Oral. 6. What is the interest of $ 100 for 3 years at 6% ? Solution. — $ 100 Principal. .06 Rate. $ 6.00 Interest for 1 year. 3 018.00 Interest for 3 years. 245 246 SIMPLE INTEREST. 6. What is the interest of f 80 at 5% for 2^ years r 7. What is the interest of $ 1000 at 5% for 2 yr. 6 mo. ? 8. What is the interest of $100 at 6% for 1 year? For 1-1- ? For 2 yr. 6 mo. ? For 3 yr. 3 mo. ? For 1 yr. 6 mo. ? When the time does not include days, find interest as follows : Principal x Eate x Time = Interest. 9. What is the interest of $297.62 for 5 yr. 3 mo. at Solution.— $297.62 m $17.8572 Note. — Final results should not r 1 include mills. Mills are disregarded 4464S ^^ ^^^^ ^^^^^ ^' ^^^ called another 892860 ^^°^ '^ ^ ^^ "^^^^• $93.75 Ans.. Find the interest of : 10. $384.62 at 6% for 2 yr. 12. $ 250.50 at 8% for 5 yr. 11. 1 463.75 at 7 % for 3 yr. 13. $ 685.20 at 4% for 6 yr. 14. $ 596.15 at 5% for 2 yr. 3 mo. 15. $ 386.42 at 5^% for 6 yr. 5 mo. 16. $ 950.16 at 10% for 41 yr. 17. $ 283.25 at 6% for 2 yr. 8 mo. Find the amount of : 18. $ 284.10 for 3 yr. 2 mo. at 7%. 19. $ 364.24 for 1 yr. 1 mo. at 6%. 20. $ 282.50 for 5 yr. 9 mo. at 5i%. 21. $ 298 for 4 yr. 3 mo. at 6%. 22. $ 389 for 7 yr. 10 mo. at 5%. THE SIX PER CENT METHOD. 247 23. $ 894 for 5^ yr. at 5^%. 24. A man buys a house and lot for $ 2800. He pays -f of the amount in cash, and the remainder after 1 yr. 4 mo. with 5% interest. Find the amount of the second payment. 25. Required the simple interest and amount of ^787.875 for 7 yr. 7 mo. at 7%. 26. Find the interest on a note for $ 12500 for three months at 8%. 27. A man paid his city tax five months after it became due. His tax was $ 560. In accordance with city ordi- nance, 1% is added for each i month the taxes are over- due. He pays to the city collector of taxes, who adds 5% collection fee. How much did he have to pay ? THE SIX PER CENT METHOD. 342. By the 6% method it is convenient to find first the interest of $ 1, then multiply it by the principal. 1. If $.09 is the interest of $1 for a certain time, what is the interest of $ 2 for the same time ? of f 10 ? of | 25 ? 2. The interest of $ 1 at 6% for a certain time is $ .034. What is the interest of $ 36.25 for the same time ? Explanation. — The interest of $36.25 is S6^^-q times the interest of $1. At 6% the interest of 1 1 for 1 year = . . . . | .06 for 1 month = J^ of $ .06 = $ .00|. for 1 day = j\ of f .00^ = $ .000^. 3. What is the interest of $50.24 at 6% for 2 yr. 8 mo. 18 da. ? Solution. — The interest of $ 1 for 2 yr. = 2 x $ .06 = $ .12 for 8 mo. = 8 x $.00^ = .04 for 18 da. = 18 x $ .000^ = .003 The interest of $ 1 for 2 yr. 8 mo. 18 da. = $ .163 The interest of $ 50.24 is 50.24 times $ , 163 = $ 8. 19 248 SIMPLE INTEREST. 4. What is the interest of $ 1 for 2 months? For 6 days ? \jRule. — Find the interest on $ 1 for the given time, and multiply it by the priyicipal, considered as an abstract number. Or, multiply the number of dollars by the number of days, and divide by 6. The quotient will be the interest in mills. rind the interest at 6% of : 5. $ 382 for 6 mo. 24 da. 6. $ 58.63 for 1 yr. 5 mo. 17 da. 7. f 256 for 3yr. 5 mo. 8. f 249.83 for 1 yr. 2 mo. 15 da. 9. $ 51 for 236 da. 10. $ 74 for 2 mo. 19 da. 11. f 1500 f or 1 yr. 3 da. 12. f 287.15 for 2 yr. 11 mo. 22 da. Interest at any rate per cent may be found as follows : At 7%, find interest at 6%, and add -|- of itself. At 5%, find interest at 6%, and subtract \ of itself. At 8%, find interest at 6%, and add f or ^ of itself. At 4%, find interest at 6%, and subtract f or ^ of itself At 41%, find interest at 6%, and subtract \ of itself. Find the interest and amount of the following : 13. 8 2350 for 1 yr. 3 mo. 6 da. at 5%. 14. , $ 125.75 for 2 mo. 18 da. at 7%. 15. $ 950.63 for 3 yr. 17 da. at 41%. 16. ^ 336.48 for 90 da. at 71%. THE SIX PER CENT METHOD. 249 17. $ 738.53 for 2 yr. 2 mo. 24 da. at 8%. 18. $ 5000 for 6 mo. 19 da. at 4%. 19. $ 867.35 for 1 yr. 3 mo. 27 da. at 9%. 20. $ 260.50 for 5 yr. 21 da. at 10%. 21. $ 3050 for 3 yr. 3 mo. 3 da. at 12%. 22. $ 625.57 for 1 yr. 2 mo. 15 da. at 3%. 23. A grocer's bill for $ 84.36 is paid 8 mo. 12 da. after it becomes due, with interest at 5%. How much is paid ? 24. Find the interest at 7% on $ 37200 for 5 days. 25. A note for $ 125 was dated March 1, 1894. What was due Aug. 5, 1895, int. at 6% ? 26. Find the amount of $ 460.50 for 2 yr. 7 mo. 15 da. at 5%. 27. What is the amount of a note for $ 360 due in 3 mo., interest at 5% ? 343. On short-time notes, it is customary to compute interest for the actual number of days, using the 6% method. Find the amount of : 28. I 684.23 from June 5, 1895, to July 23, 1895, at 6%. 29. 1 846 from Jan. 6 to March 9, 1896, at 5%. 30. $ 2064.28 from April 13, 1894, to June 3, 1894, at 8%. 31. $ 1428 from May 12, 1892, to June 9, 1892, at 6%. 32. $ 324 from April 1, 1896, to June 4, 1896, at 7%. 33. $ 3500 from Feb. 9, 1895, to March 12, 1896, at 4i%. 34. $ 862.15 from May 25, 1893, to July 22, 1893, at 6%. 35. What is the amount of a note of $384.16 at 6%, given June 11, 1896, and paid Aug. 12, 1896 ? 250 SIMPLE INTEREST. 36. A note of $395.80 dated April 5, 1896, was paid Aug. 4, 1896. What was the amount ? 37. On Dec. 9, 1894, John Smith borrowed $ 484, agree- ing to pay interest at o%. He paid the debt in full on March 3, 1895. What did he pay ? 38. What is the amount of $ 58.24 at 7% from April 23, 1893, to July 22, 1893 ? 39. A bill of $312 with interest at 5% was paid at the end of 90 days. What was the amount ? 40. What is the interest of $ 30000 at 8% for 7 days ? 344. rind the interest, using the best method. PRINCIPAL. TIME. RATE. 41. $364, 3yr., 8%. 42. $ 692.15, 1 yr. 3 mo., 9%. 43. $342, 62 da., 6%. 44. $ 243.50, 2 yr. 5 mo. 18 da.. 7%. 45. $ .392, 1 yr. 3 mo. 15 da., 4%. 46. $ 150.16, 7 yr. 2 mo. 27 da.. 4^95.. 47. $ 284.10, 1 yr. 8 mo. 18 da.. 6%. 48. $ 1400, 2 yr. 1 mo. 12 da., 7%. 49. $124, 5 yr. 3 mo. 29 da., 6%. 50. $48, 33 da.. . 6%. 51. $124, 112 da.. 5%. 52. $315, 45 da.. ^fo. 53. $214, 93 da., 8%. teXACT INTEREST. 251 345. Find the amount of : 54. $365 from April 1, 1895, to July 5, 1897, at 6%. 55. $250 from July 3, 1891, to April 21, 1893, at 9%. 56. f 582 from Sept. 4, 1896, to July 8, 1897, at 8%. 57. $346.18 from May 10, 1893, to March 10, 1895, at 6%. 58. $287 from Jan. 1, 1895, to July 1, 1897, at 4^%. 69. $ 1684 from July 17, 1896, to Sept. 5, 1898, at 1\%. 60. $2500 from April 16, 1873, to Oct. 11, 1881, at 5%. 61. $ 186 from Feb. 12, 1896, to March 4, 1896, at 6%. 62. $346 from March 11, 1895, to Feb. 11, 1896, at 6%. EXACT INTEREST. 346. When the time includes days, interest computed by the 6% method is not strictly exact, by reason of using only 30 days for a month, which makes the year only 360 days. The day is therefore reckoned as -^\-^ of a year, whereas it is -^\-^ of a year. Rule. — To compute exact interest, find the exact time in days, and consider 1 day's interest as -g-i-g- of 1 year's interest. 1. Find the exact interest of $358 for 74 days at 7%. Solution. — $ 358 x .07 = $25.06, 1 year's interest. 74 days' in- terest is ^-^ of 1 year's interest. -^ of $ 25.06 = $ 5.08. Ans. Find the exact interest of : 2. $324 for 15 d. at 9%. 3. $253 for 98 d. at 4%. 4. $624 for 117 d. at 7%. 5. $ 153.26 for 256 d. at ^%. 6. $ 620 from Aug. 15 to Nov. 12 at 6%. 252 SIMPLE INTEREST. 7. f 540.25 from June 12 to Sept. 14. at 8%. 8. $7560 for 90 days at 5i%. 9. Find the exact interest at 5% on a note dated Jan. 14, 1896, and paid March 31, 1896, for $832. 10. Find the exact interest on $ 800 for 219 days at 41%. 11. A city treasurer deposits $387,913.56 in the banks at 2% per annum. What interest will the city receive in 5 days ? 12. On June 4, 1895, a coal-dealer bought of the D. L. &W. R. R. 235 tons of chestnut coal at $4.10 per ton. At 6% what will be the exact interest on the amount on Jan. 1, 1896 ? PROBLEMS IN INTEREST. 347. To find the Rate, when Principal, Interest, and Time are given. 1. What is the rate when the interest of $ 250 for 4 years is $60? $10^S60 Solution. — The interest on the 6 timP^ 1 ^ -ao/. principal at 1% for 4 years = flO. /^ — ^ /o gi„(je |iQ ig ^i^g interest at 1 %, f 60 must be the interest at as many times 1 % as $ 10 is contained times in $ 60, which are 6 times. Therefore the rate is 6 times 1 % = 6 %. ^ Rule. — Divide the given interest by the interest of the principal for the given time at 1%. 2. A man borrowed $4625 for 5 yr. 8 mo. 18 da., and paid $ 1586. 37|- for the use of it. What was the rate of interest ? 3. If $30.40 is paid for the use of $960 for 7 mo. 18 da., what is the rate per cent ? 4. At what rate per cent must $1450 be loaned for 4 yr. 5 mo. to yield $ 576.37^ ? PROBLEMS IN INTEREST. 253 5. At what rate will $ 1730 amount to ^2048.32 in 4 yr. 7 mo. 6 da. ? 6. 4 yr, 7 mo. 6 da. after its date a note for ^1730 amounted to $ 2048.32. What was the rate of interest ? 7. At what rate % must $ 5600 be invested for 1 yr. 4 mo. to bear $560 interest ? 8. A Kansas farmer has a mortgage on his farm for $ 1250. What rate of interest does he pay, if the interest for 2 yr. 6 mo. equals ^ of the debt ? 9. At what rate must $2800 be invested to yield a semi-annual interest of $ 112 ? 10. At what rate will $600 yield $ 198 in 5 years and 6 months ? 11. At what rate will any sum double itself in 20 yr. ? 348. To find Time, when Principal, Interest, and Rate are given. 1. In what time will $250 gain $ 60 at 6% ? Solution. —The interest of |250 for 1 year at 6% = $ 15. Since $ 15 is the interest for 1 year, $ 60 is the interest for as many years as $ 15 is contained times in f 60 = 4 years. Rule. — Divide the given interest by the iyiterest of the principal for 1 year. 2. In what time will $ 600 yield $ 91.50, interest at 6% ? $ 600 X .06 = $ 36, interest on the principal for 1 year. $91.50 -^ 36 = 2.5416 + years. Reducing the decimal part of the time to months and days, we have 6 mo. 15 da. The answer is 2 yr. 6 mo. 15 da. Note. — A decimal less than .5 of a day is not counted, but .5 or more is counted another day. 3. In what time will $530 gain $92.75 interest at 5% ? 4. In what time will $ 400 yield $ ^5 interest at 51% ? 254 SIMPLE INTEREST. 5. In what time will $500 gain $15 at 6% ? 6. In what time will $4625 yield $1586.38 at 6% ? 7. In what time will $1730 amount to $2048.32 at 4% ? 8. The face of a note was $960, rate of interest 5%, and the interest $ 30.44. How long did it run ? 9. I borrowed $1284 at 41%, and kept it until it amounted to $ 1421.067. How long did I keep it ? • 10. For how long will $ 2700 have to be invested to amount to $2976.25 at 5% ? 11. A man received $9.73 interest on $556 at 7%. What was the time ? 12. In what time will any sum double itself at 6% ? 349. To find Principal, when Interest or Amount, Rate, and Time are given. 1. What principal at 6% will gain $60 interest in 4 years ? $.24)$ 60.00(250. 4g Solution. — Since 1 dollar in 4 years will T^ gain $.24 interest, it will take as many ^9/. dollars to gain .^60 interest as $.24 is con- — — tained times in $60, or $250. ^ Rule. — Divide the given interest hy the interest of $ 1 for the given time and rate. 2. What principal at 6% will amount to $310 in 4 years ? Solution. — Since $1.24 is the amount of $1 for 4 years, $.310 must be the amount of as many times $ 1 as $1.24 is contained times in 310 = $250. 3. What sum invested at 5% will give a yearly income of $500? J PROMISSORY NOTES. 255 4. What principal will yield $25 in 6 mo. at 5% ? 5. What principal in 3 yr. 6 mo. at 5% will yield $ 92.75 interest ? 6. What sum of money will produce $ 1586.37^ in 5 yr. 8 mo. 18 da. at 6% ? 7. What principal will yield $318.32 in 4 yr. 7 mo. 6 da. at 4% ? 8. What principal will pay $ 1556.77-5 interest in 2 yr. 9 mo. at4i%? 9. The amount is $1093.921 time 2 yr. 3 mo. 27 da., rate 5%. What is the principal ? 10. It required $407.65 to pay a loan at 8% for 7 mo. 24 da. What sum was loaned ? PROMISSORY NOTES. 350. A Promissory Note is a written promise to pay a sum of money at a certain time. , 351. At least two parties must be named in the note, the Maker and the Payee. The Maker makes the promise to pay to the Payee the sum named in the note. This sum is called the Face. The owner of a note is called the Holder. Each State has a lawful or legal rate of interest. If no rate is fixed in the note, the legal rate is under- stood. (See Appendix.) 352. Interest higher than the legal rate is Usury. 353. A note is Negotiable when payable to the bearer, or to the order of the payee. It is called negotiable because it can be negotiated ; i.e. bought and sold. 256 SIMPLE INTEREST. 354. The two forms of notes given below are negotiable : NOTE 1. $510^ Chicago, 111., Jan. 8, 1901 Two months after date, I promise to pay to the order of Charles M. Warner Five hun- dred ten and ~ Dollars, for value received, at the First National Bank. U. B. Smith, $216^ Des Moines, la., Juli/ 10, 1901 One ifear after date, for value received, I promise to pay Asa D. Tucker or bearer, Two hundred sixteen and -— Dollars, with interest. John T. Brown. Note. — A note may be payable on a given day ; as, On March 15 after date, I promise to pay, etc. A note may be payable on demand ; as. On demand I promise to pay, etc. 355. A note made payable to the payee only is called a non-negotiable note. Note. — When a note is payable in a State in which three days of grace are allowed, maturity is three days after the expiration of the interval named in the note. 356. 1. Write a negotiable note, bearing interest. Indorse it with payee's name, and find the amount at maturity. To whom must the maker pay the money ? PKOMISSORY KOTES. 257 2. Write a non-negotiable note payable on a specified date, and find the amount due at maturity, 3. Write a negotiable note, and find the amount of it. 4. Write a non-interest-bearing demand note. 5. Write a non-negotiable interest-bearing note. 6. Write a 6% note, dated June 15, 1901, payable in 1 year without interest, with yourself as payee, and your teacher as maker, and find the amount of it to the present time. 7. Write a negotiable note, using the following : Date, Jan. 16, 1894; Time, 6 months; Face, $1684.96; Payee, Andrew Jackson; Maker, Silas Wright; Interest at 6%. Indorse it, showing that the maker has transferred it to another. What is the amount of the note, if paid in full Nov. 11, 1894 ? 357. A note should contain : 1. The face in figures at the left upper corner. 2. The place and date at the right upper corner. 3. The time of payment. 4. The words "Value Received." 5. The face written in words in the body of the note. 6. The place at which it is payable. 7. The words " with interest," if agreed upon. 358. A note is said to mature on the day on which it is due. 359. A note that does not contain the words " with inter- est " bears interest from maturity, if not paid at that time. When does interest begin in Note 1 ? In Note 2 ? 360. In many of the States the maker is allowed three days (called Days of Grace) in which to pay a note, after the time named in the note has expired. In these States, the date of maturity falls on the last day of grace. 258 SIMPLE INTEREST. Days of grace are not allowed in California, Connecticut, District of Columbia, Idaho, Illinois, Maryland, Massachu- setts, Montana, New Jersey, New York, North Dakota, Ohio, Oregon, Pennsylvania, Utah, Vermont, and Wisconsin. Note. — When the holder of a note transfers it to another, he is usually required to indorse it, i.e. to write his name across the back. This is required as an order to the maker to pay the money, when due, to the new holder. An indorser is also resiDonsible for the pay- ment of a note in case the maker fails to pay it when due. PARTIAL PAYMENTS. 361. Payments in part of a note or other debt are Partial Payments. The Supreme Court of the United States has adopted the following rule for finding the amount due on a note after partial payments have been made. UNITED STATES RULE. Find the amount of the principal to the time when the ])ay- ment or sum of the payments equals or exceeds the interest then due. Deduct from this amount the payment or payments. Treat the remainder as a new principal^ and so proceed until the date of settlement. Note. — When a partial payment of a note or other contract is made, the holder writes upon the back of it the sum paid, with the date of payment. Sums so written are called indorsements. The common form of indorsement is as follows • Received on the within, July 16, 1896, $ 1. A note for $1820 was given Jan. 1, 1892, and settled July 13, 1894. The following payments were indorsed upon it: May 25, 1892, $250; Jan. 25,1893, $45; April 7, 1893, $ 375 ; July 13, 1893, $ 750. How much was due on the day of settlement, interest at 6% ? PARTIAL PAYMENTS. 259 First write the note, and properly indorse the payments upon the back of it. YB. MO. I>A. PAYME>TS. 1892 5 1892 1 25 1 $250 $1820 Principal. .024 4 24 $43.68 1820.00 .024 $1863.68 1st Amount. 250.00 1st Payment. Ite 1 189\ 5/ 4 $45. ^lr61^Ua^i>[ew Principal.^ — ■ — ^ JK $64,..5&-tntefest exceBds-Ea^yment. 1893 4 1892 5 7 25 $375 $1613.68 .052 10 12 $83.91 1613.68 .052 $420 $1697.59 Amount. 420.00 Sum of 2d and 3d Payments. 1893 7 1893 4 13 7 $750 $1277.59 New Principal. .016 3 6 20.44 1277.59 .016 $1298.03 Amount. 750.00 4tli Payment. 1894 7 1893 7 13 13 Settled. $548.03 New Principal. .06 1 $28.88 548.03 .06 $576.91 Ans. Note. — The $45 payment, being less than the interest ($64.55),' is not deducted from the amount of the second principal ($1678,23). If this were done, and the remainder treated as a new principal, a portion of it ($19.55), being interest, would draw interest, which is not legal. Therefore, interest must be taken on $1613.68 until the date of the next payment (10 mo. 12 da.). The sum of the two payments, being greater than the interest, is subtracted from the amount. 260 SIMPLE INTEREST. Write in proper form on paper a note for each of the following, indorse the payments, and solve: 2. Date, Jan. 1, 1874, at New Orleans, La. Face, $1000. Interest at 6%. Indorsements: July 7, 1874, $400; Oct. 19, 1874, $300; Dec. 1, 1874, $100. What remains due Jan. 1, 1875? 3. Face, $900. Date, March 1, 1886. Interest at 9%. Indorsements: Aug. 10, 1886, $300; Sept. 1, 1886, $100; Jan. 1, 1887, $50. What was due March 1, 1887 ? 4. Face, $2000. Date, Jan. 20, 1892. Interest at 6%. Indorsements: May 20, 1892, $100; July 20, 1893, $100; Sept. 10, 1893, $700; Oet. 20, 1894, $75. Settled Oct. 20, 1895. What was due ? 6. ^300 Binghamton, N.Y., Oct. 12, 1889 On demand, for value received, /promise to pay S. D. Cleveland^^^^^.^^,^.^^^,^,^,^^ov order, Three hun- dred Dollars, with interest. J. H, Van Alstyne. The following payments were made on this note: June 27, 1891, one hundred fifty dollars; Dec. 9, 1892, one hundred fifty dollars. What was due Oct. 9, 1895? 6. On. a note for $573.25, at 6%, dated June 10, 1888, were the following indorsements : May 20, 1889, $50; July 10,1890, $16.50; April 5, 1891, $14.30; July 14, 1892, $250. How much was due Sept. 20, 1893? 7. A note of $850 was dated June 21, 1892, bearing interest at 6%. On this note were the following indorse- ments : Sept. 15, 1892, $150.90; Nov. 21, 1893, $45; Jan. 15, 1894, $256.88. What remained due June 21, 1894 ? merchants' rule. 261 8. Find what was due June 1, 1896, on a note for $1928, with 41% interest, dated Jan. 1, 1891, and bearing the following indorsements: March 1, 1891, $300; Oct. 16, 1893, $40; Feb. 4, 1894, $800; Dec. 16, 1895, $500. 9. On a note for $832.26 dated Aug. 3, 1889, due in 6 months, the following payments were indorsed: $350, Oct. 5, 1890; and $468.37, May 15, 1892. How much was due Dec. 12, 1893, interest at 7% ? 10. Face, $2950. Date, July 1, 1885. Interest, 7%. Indorsements: Oct. 1, 1885, $750; Jan. 15, 1886, $600; July 1, 1886, $900; Dec. 1, 1886, $300; March 1, 1887, $450. What was due July 1, 1887? MERCHANTS' BULB. 362. When notes and accounts are settled within a year after interest begins, and upon which partial payments have been made, it is customary for business men to make use of the following rule : Find the amount of the entire debt at date of settlement. Find the amount of each payment at date of settlement. Subtract the amount of the payments from the amount of the debt. $648^ Dubuque, la., Ma^ i, 1896 For value received, / promise to pay^ ^ D. McCarthy/ ^ Co.., or bearer. Six hundred forty-eight Dollars on demand, with interest. Charles U. White. 262 COMPOUND INTEREST. Indorsements: June 1, 1896, f JoO; Aug. 1, 1896, $200; Oct. 1, 1896, $300. Interest at 6%. What was due Dec. 1, 1896 ? Solution, — $648 in 7 mo. amounts to $670.68 $150 in 6 mo. amounts to $154.50 $200 in 4 mo. amounts to $204.00 $300 in 2 mo. amounts to $303.00 661.50 $9.18 2. On a note of $1186.48, with interest at 5%, dated April 4, 1890, these payments were indorsed: July 10, 1890, $250; Aug. 4, 1890, $300; Dec. 8, 1890, $150; Jan. 2, 1891, $ 75. How much was due Feb. 4, 1891 ? 3. On Oct. 16, 1896, John D. Wilson gives his note for $483.98, with interest at 6%. He pays the note in full, March 28, 1897, having made a payment of $350 on Jan. 2S, 1897. How much does it require to settle the note ? COMPOUND INTEREST. 363. Compound Interest is interest on unpaid interest, as well as on the principal, at the end of regular interest periods. Note. — Interest is compounded annually, semi-annually, or quar- terly, according to agreement. Compound interest is not authorized by law. It is customary for savings banks to allow interest on interest when it has been on deposit for a full interest period. 1. Find the compound interest of $ 350 for 2 years and 6 months at 6%. Solution. — $350.00 Principal. 21.00 Interest for 1st year. $371.00 Amount taken as new principal. 22.2 6 Interest for 2d year. $393.26 Amount used as new principal. 11.80 Interest for 6 mo. $405.06 Amount for 2 yr. 6 mo. 350.00 1st principal. $55.06 Compound interest for 2 yr. 6 mS. REVIEW OF INTEREST. 263 Note. — When the interest is compounded semi-annually, the rate is one-half the annual rate for each period. When quarterly, one- fourth, etc. When no interest period is mentioned, interest is compounded annually. 2. What is the compound interest of $830 for 3 years at 5 per cent ? 3. What is the amount of $650 for 4 years at 4% in- terest, compounded semi-annually ? 4. What is the compound interest of $365 for 2 yr. 7 mo. 18 da. at 6%, compounded semi-annually? 5. What is the compound interest on $640 for 4 years at 5% ? 6. What is the interest, compounded quarterly, on $538.25 for 2 yr. 6 mo., rate 4% ? 7. What is the interest, compounded annually, on $683.48 for four years at 6% ? 8. What is the compound interest on $437.50, for 3 yr. 6 mo., at 5%, compounded semi-annually ? REVIEW OF INTEREST. 364. 1. What is simple interest ? Compound interest? A promissory note ? A negotiable note ? 2. Define payee, holder, signer or maker. 3. Describe two common methods of computing interest. 4. Prove that, at 6%, 6 cents is the interest on $ 1 for 1 year. 5. Prove that 5 mills is the interest gn $ 1 for 1 month. 6. Prove that ^ mill is the interest on $ 1 for 1 day. 7. Why is interest not accurate when computed by the 6% method? 264 INTEREST. 8. Find the interest on $50000 for 252 days by the 6% method, then by the exact interest method. Which is more favorable to the payee ? 9. When does a note mature ? 10. What elements must be given when we find inter- est? Rate? Time? Principal? 11. How do you find the rate? The time? The prin- cipal ? 12. What are days of grace ? 13. Does the maker of a non-interest-bearing note ever have to pay interest ? Explain. 14. What use is made of compound interest? 15. Find the compound interest, then the simple interest, at 6% on $ 25000 for 5 years, and note the difference. 16. When a note is not paid at maturity, why is it to the holder's advantage to require a new note ? 17. What is the effect of a payee's indorsement ? 18. When a partial payment is made that does not equal the interest due, why is not the payment subtracted from the amount ? 19. Solve a problem in partial payments by both the United States and the merchants' rule. Which is more favorable to the payer ? 20. Find the compound interest on $ 1420.80 for 1 yr. 9 mo. at 6%, computed semi-annually. 21. Find the ameunt of a debt of $5672.00 for 4 years at 4% compound interest. 22. Find the interest on $720 at 6% for 2 yr. 8 mo. 22 days. REVIEW OF INTEREST. 265 Find the interest on : 23. $ 675.20 for 3 yr. 5 mo. at 7%. 24. $ 754.30 for 1 yr. 4 mo. 15 da. at 51%. 25. $ 564.11 for 2 yr. 3 mo. 18 da. at 4%. 26. A county in Missouri owes $ 85,640. In how many days will the interest at 6% amount to $ 897.22 ? 27. Find the interest at 8% on $3960.36 for 9 mo. 20 da. 28. Find the amount of $ 2536.48 for 1 yr. 3 mo. 18 da. at 7%. 29. The interest on $ 600 for 3 yr. 6 mo. was $ 126. What was the rate ? 30. The interest on a note for $460.50 at 5% was $ 60.44. What was the time ? 31. The interest on a certain sum was $ 96.04, the rate 6%. Find the principal. 32. The amount due on a 6% note due in 1 yr. 5 mo. 4 da. was $ 135.708. What was the face of the note ? 33. Find the exact interest on a note for $ 600, dated Aug. 5, 1895, and due July 1, 1896, interest at 6%. 34. Find the amount and simple interest of $623.74, one half of which is to be paid in 2 yr. 3 mo. at 4%, the other half to be paid in 3 yr. 5 mo. at 6%. 35. A note for $ 146.20, dated June 5, 1869, was paid July 11, 1872, with interest at 6 per cent. What was the interest ? 36. A man borrowed, Dec. 25, 1877, $ 137.40 at 6% interest, and kept it until Jan. 15, 1880. What was the interest ? 266 INTEREST. 37. Payments were made on a note of $ 1800 dated Jan. 12, 1891, as follows : March 6, 1891, $ 300 ; April 15, 1891, $ 190 ; July 3, 1891, f 565 ; Oct. 15, 1891, $ 700. What was due Dec. 21, 1891, interest at 6% ? 38. When must $ 1600 be put at interest at 6%, so that it will amount to $1800 on Jan. 1, 1898 ? 39. Find the amount of $ 375 for 2 yr. 8 mo. 16 da. at 40. Find the amount at simple interest of $ 1200 from April 4, 1895, to the present time. 41. A note for $ 728 is dated Nov. 16, 1894. March 8, 1895, there was paid on it $ 25. Find the amount due on Jan. 4, 1896, interest at 6%. 42. Find the amount at simple interest of $ 1184.63 for 1 yr. 4 mo. 17 da. at ^% ? 43. Write your own promissory note for $ 200, with interest, payable in 60 days from to-day. When does it becoijie due ? Find the amount due at maturity. 44. Find the exact interest on $843.20 from April 10, 1895, to March 15, 1896, at 4^%. 45. Upon a note for $ 950, dated Syracuse, N.Y., Jan. 1, 1894, i 150 was paid Aug. 16, 1894 ; $ 25 March 1, 1896; and i 200 April 16, 1896. How much is due to-day ? 46. .$645 was paid as interest on $ 2000 for 3 yr. 7 mo; What was the rate ? 47. $30 was paid as interest on $600 at 6%. What was the time ? 48. A house that cost $ 5000 was rented for $ 500, and $ 100 was paid for annual taxes and repairs. What rate of interest did the investment yield ? O^RUE DISCOUNT. 267 49. A person investing a certain sum of money at 6% for 1 yr. 6 mo. found at the end of that time the invest- ment amounted to $ 545. Find the sum invested. 50. A man bought a horse for $ 150, paying $ 70 in cash, and the balance on time at 6%. He paid at the time of settlement $ 83.60. How much time elapsed before that date ? 51. H. C. Harmon loaned $250 for 1 yr. 3 mo. 27 da., which amounted to $ 269.875 at the time of payment. Find the rate of interest. 52. A person having a certain sum of money invested, and drawing compound interest at 6%, found at the end of 2 yr. 2 mo. that it amounted to $ 567.418. What was the sum invested ? 53. A sum of money was borrowed Jan. 30, 1895, and $ 419.60 paid in full Nov. 24, 1895. The rate of interest being 6%, how much of this was interest ? 54. A man owes $4600 at 7%, and each payment of in- terest amounts to $ 161. How often does he pay interest ? TRUE DISCOUNT. 365. Oral. 1. What will be the amount of $100 at 6% one year from to-day ? 2. What is the value to-day of a debt of $106, due in one year, when money is worth 6% interest? 3. How much money paid to-day will cancel a debt of $112, due two years hence, money being worth 6%? 4. What is the present worth of $ 105, due in one year without interest, when money is worth 5% interest? 5. When money can be loaned at 7%, which is worth the more, $ 100 at the present time, or a note of $ 107 without interest, due in one year ? 268 INTEREST. 6. What sum should be deducted from a debt of $ 108, due without interest in one year, in consideration of its being paid now, when money can be loaned at 8 % ? 366. True Discount is a deduction of interest for the pay- ment of a debt before due. 367. The Present Worth of a debt due at a future time is a sum which will amount to the debt if put at interest till that time. The debt is therefore the amount of the present worth for the given time. 368. The true discount is the difference between the debt and its present worth. It is the interest of the present worth for the given time. 7. What is the present worth and the true discount of a debt of $ 582.40, due in 8 months without interest, when money is worth 6%? Solution. — $ 582.40 4- f 1.04 = $ 560, present worth. $582.40 - $560 = $22.40, true discount. Since $1.04 is the amount of $1 for 8 mo., $582.40 is the amount of as many dollars as $ 1.04 is contained times in $ 582,40 = $ 560. Rule. — To find the present worth, divide the debt by the amount of $ 1 for the given time. To find the true discount, subtract the present worth from the debt. 8. What is the present worth and true discount of $400, due in one year, when money is worth 5% ? 9. A father wills his two sons $3000 each, to be paid in three years from the time of his death. What is the value of the legacies at the probate of the will, if money is worth 6%? 10. What is the present worth of $450, due in two years at 5% ? BANK DISCOUNT. -269 11. What is the present worth of $250.51, payable in 8 months, money being worth 6% ? 12. Which is better, to buy flour for $ 5 cash, or for »f 5.25 on 6 months' time, when money can be borrowed at 13. Find the present worth of $ 750 for 6 months, money being* worth 6%. 14. What is the present worth of $ 600, due in 1 year without interest, money being worth 6% ? 15. Write the note which would be given for the above debt. 16. A man wishing to buy a house and lot has his choice between paying $ 5400 in cash, or $ 4000 in cash and $ 1700 in two years. With money at 6%, which is the most ad- vantageous for him ? 17. Which would be more profitable, and how much, to pay $4000 cash for a house, or $4374.93 in 3 yr. 6 mo., money being worth 7 % ? 18. I can sell my house for $2800 cash, or $3000 and wait 6 months without interest. I choose the latter. Do I gain or lose, and how much, money being worth 6% ? 19. What is the present worth of a debt of $385.31, due in 5 months 15 days, at 6% ? BANK DISCOUNT. 369. When the holder of a negotiable note wishes the money before it becomes due, he may take it to a commercial bank ; and if the banker is satisfied that the parties to the note are responsible, he will pay the holder the amount due after deducting the discount. By this act the bank becomes the holder of the note, and at its maturity the maker must pay to the bank instead of to the payee. 270 INTEREST. 370. The Maturity Value of a note is the amount due at maturity. The Bank Discount is the simple interest on the maturity value, reckoned from the day of discount to the day of maturity. 371. The maturity value less the bank discount is called Proceeds, or Avails. The time from the day of discount to the day of maturity is called the Term of Discount. 372. The maturity value of a note not bearing interest is the face, and the maturity value of an interest-bearing note is the face plus the interest. Note 1. — Only short-time notes are discounted at banks, usually not exceeding 4 months. Note 2. — Banks generally require that the paper which they dis- count he made payable at some bank. Note 3. — Banks usually reckon discount for the exact number of days in the term of discount, although the time in a note may be expressed in months. Banks usually regard the year as 360 days. Note 4. — In States having days of grace, the day of maturity is the last day of grace. Note 5. — If the day of maturity falls on Sunday or a legal holiday, the preceding day is the day of maturity in most States. In some States, however, the note does not fall due until the day following. Note 6. — When a note is discounted at date, the term of discount is the time of the note ( -I- 3 days of grace in States having days of grace). Unless otherwise stated, a note is to be discounted at date. 1. 1555^ Buffalo, N.Y., Jan. 15, 1896 Two months after date, for value received, /promise to pay^, Richard Turner, ^^.^^^^.^.^^^oy order, Three hundred twenty-jive and ^ Dollars at the Third National Henry P, Warner, BANK DISCOUNT. 271 If this note was discounted at 6% at a bank on the day it was made, how much did the bank deduct ? How much were the proceeds ? Solution. — The term of discount is from Jan. 15 to Mar. 15 = Jan. Feb. Mar. 16 da. 4- 29 da. + 15 da. = 60 da. The bank discount is the interest of $325j3^*q for 60 da., at 6 % = $ 3.25. The proceeds = $ 325.24 - $ 3.25 = 1 321.99. 2. Copy the above note, and properly indorse the payee's name. 3. What would be the bank discount and proceeds of the above note if it contained the words " with interest " ? Solution. — Maturity value = face + interest, or $325.24 + -$3, 25 = $328. 49. The bank discount = 6 % of $328.49 for 60 da. = $3.28. The proceeds = $328.49 - $3.28 = $325.21. 4. ^387'^ Boston, Mass., June 27th, 1901 Three months after date, / promise to pay to the order of_____^,^Jawes Gr. Rogers Three hun- dred eighty-seven and ~ Dollars, value received, at the First National Bank, with interest at 5%. G-eorge Price. Discounted July 27 at 5%. Day of maturity, Sept. 27. Solution. — Maturity value = f ace + interest for 90 da., at 5% = $392.34. Discount at 5% from July 27 to Sept. 27, 62 da. = $3.38. Proceeds $392,34 - $3.38 = $388.96, 272 INTEREST. 5, $648.15 Buffalo, N.Y., Jan. 31, 1900 One month after date, I promise to pay to the order of ^.^^James B. Strong ^.^^^^^^^^^.^^Six hun- dred forty -eight and ~ Dollars, at the Shoe and Leather Bank, value received, with interest at 6%. Discounted at date at 6%. The above note is interest-bearing, therefore the discount must be computed on the amount at maturity. 6. $3000 Detroit, Mich., Oct. i, 1901 Ninety days after date, for value received, I promise to \)^y Jerome K. iVz;ro?i,___,__.,_or order. Three thousa7id dollars, at the First National Leroy O. Bondy, 7. $438.29 ' St. Louis, Mo., Feb. 7, 1902 Two months after date, for value received, I promise to pay_______^i^^A 2%omjoso?^,_______or order. Four hundred thirty-eight and ^ Dollars, at the Chemical National Bank. M. H. Winthrop, Discounted March 10 at 6%. BANK DISCOUNT. 273 8. $7.89§^ Cleveland, O., Mar. 4, 1900 Four months after date, I promise to pay to the order of _______ TF! W. W oodf or d,^^.^.^.^^^^,^.^..^^^ Seven hundred eighty -nine and ^ Dollars, value received, at the City Bank. ^^. -r, ,^ •^ Otis It. Young. Discounted May 4 at 6%. 9. $4920 Brooklyn, N.Y., Apr. 5, 1902 Ninety days after date, for value received, / promise to pay to the order oi^.^.^^,.^.^^.^^.^Dewitt Xo7i^______^ Four thousand nine hundred and twenty Dollars, at the Merchants' Bank, with interest. . Elizabeth R. Prentiss, Discounted at date at 6%. 10, $1312 Boston, Mass., May 2, 1901 Sixty days after date, for value received, / promise to pay to the order of Edgar JY. Wilson . One thousand three hundred and twelve Dollars, at the First National Bank. Frank L. Barker, Discounted May 10 at 4%. 274 INTEREST. 11. $2142.84 Albany, N.Y., Dec. 15, 1900 Five months after date, for value received, I promise to pay Charles R. >S'H/i?igr,,_^.,,._^^_^or order, Tivo thousand one hundred forty-two ^-^ Dollars, with interest, at the Park Bank. M. H, Dixon. Discounted Feb. 27, 1901, at 6%. 12. $2000 San Francisco, Cal., June 10, 1900 Three months after date, for value received, /promise to ^cxry^^^^^^^^^^^^^^^ Seymour D. Wileox,^^^^.^^^^^^,^^^,^^^^OT order, Two thousand Dollars, at the Citizens' Bank. P. J. Reed. Discounted July 10 at. 8%. 13. $2500 Syracuse, N.Y., July 6, 1901 Two months after date, I promise to pay to the order oi.^^^^^.^^^^^Rohert M. Beecher, ^Tivo thousand jive hundred Dollars, at the Third National Bank. Value received. John Q. Adams. Discounted at 6% at date. BANK DISCOUNT. 275 14. A note for $ 135 is given for 90 days, and discounted the day it is given at 6%. What are the proceeds ? 15. William Johnson gave John Doe a note payable to the Binghamton Trust Co., time 60 days, amount $ 204.60. Write this note. After 20 days Doe put the note in the bank. What are the proceeds of the note ? In the following problems, write the notes in full, and properly indorse them, using any names for payer and payee. 16. Find the bank discount of $400 for 3 months at 8%. 17. What are the proceeds of $250, with interest at 6%, discounted at bank for 60 days at 6%? 18. What will be the proceeds of a note for $ 175 drawn at 4 mo., with interest at ^j%, if the bank discount is 10% per annum ? 19. On the first day of January, 1896, a farmer gave his note at 90 da. for $525, with interest at 6%. When did the note become due, and what were the proceeds of the note if discounted at a bank at 1% a month on the tenth day of February ? 373. To find the Face of a note, when the Proceeds, Time, and Rate are known. 1. What must be the face of a 60-day note, without grace, which after being discounted at 6% will give $500 as proceeds ? Solution. — The bank discount of $1 at 6 % for 60 da. = $.01. The proceeds of $1 = §1.00 - $.01 = $.99. Since |.99 is the proceeds of $1, $500 must be the proceeds of as many dollars as $.99 is contained times in $500 = 505.05 +. Aiis. 276 INTEREST. Therefore, 1505.05+ must be the face of a 60-day note which will give f 500 as proceeds after being discounted at 6 %. Rule. — Divide the x>roceeds by the jyroceeds of $1. 2. A person must use $ 250 today. For how much must he make a bank note for three months that will give $ 250 proceeds, without grace ? 3. What must be the face of a 60-day note, payable at a Boston bank, upon which I can realize f 350 after it is dis- counted at 6% ? 4. If you buy goods for $ 1200 cash, how large a not« payable in 90 days, at 6% bank discount, must you make that the proceeds shall pay for the goods ? Without grace. 6. Find the face of a 60-day note that will yield $800 when discounted at bank at 7%, with grace. 6. How large a note must I make at a bank for 30 days to pay a debt of $ 475, without grace ? 7. Wishing to borrow $495 at a Chicago bank, for what sum must I make my note at 60 da., with interest at 6%, in order to obtain this amount? Discount at ^% a month. 8. The proceeds of a Buffalo note at 60 da., when dis- counted at a bank at 6% per annum, is $ 742.50. What ig the face ? REVIEW OF DISCOUNT. 374. 1. Define Discount; True Discount; Bank Dis- count ; Proceeds ; Present Worth. 2. How is the present worth found ? The true dis- count ? The bank discount ? The proceeds ? 3. What is the term of discount, and how is it found? The day of discount, and how found ? 4. How is the bank discount of an interest-bearing note found? BANK DISCOUNT. 277 5. How do you find the face of a note when the pro- ceeds, time, and rate are given ? 6. When does a Rochester, N.Y., note mature if given for 1 month from Jan. 31 ? 7. State a point of difference between true discount and bank discount. 8. What kind of notes only can be discounted at banks ? 9. Bought a city lot, and agreed to pay $ 546.94 at the end of 2 yr. 6 mo., without interest. E-eceiving some money unexpectedly after 6 months, I wish to pay cash. How much ought I to pay, money being worth 6% ? 10. What is the present value and true discount of $ 973.52, due in 1 yr. 7 mo. 24 da. hence, without interest, money being worth 8%? 11. A man has an offer of $2000 cash for his house, or $2100 payable in 8 months. If money is worth 8%, which is the better, and how much ? 12. Find the discount and proceeds of a note for $13,500 payable at a bank in 90 days after date without grace, dis- counted at 5%. 13. For what sum must C. F. Norton draw his note on a Bingham ton bank, that when it is discounted at 4% for 60 days he will have $800? 14. A man owes me $ 2540 due in 2 years 3 months, with- out interest. If he pays it at once, what discount should I allow him ? 15. Find the discount and proceeds of a note on a Brook- lyn bank for $350, given May 12, 1896, for 4 months, and discounted at 6%, July 15. 278 INTEREST. 16. %860 St. Louis, Mo., May 5, 1901 Three months after date, for value received, I promise to pay R. B. White, or order, Mght hundred sixty Dollars, with interest, at the First National Bank. H. U. Barrett. Discounted June 11 at 6%. 17. For what sum must I draw a four months' note so that the proceeds will be §800, discounted without grace at 6%? 18. I sell my horse for $216, and take a note due in 6 months without interest. If money is worth 6% per an- num, what is the present value of my note ? 19. For what sum must I give my note for 60 days at a bank in order to receive $650 proceeds, money being worth 8% ? 20. Find the face of a note, discounted for $ 2558.40 at 8%, for a term of 72 days, without grace. STOCKS AND BONDS. 375. Many kinds of business require so much capital that several persons must unite to raise the necessary amount. 376. The Capital to be raised is divided into Shares, usually of $100 each. Shares are then sold until the required amount is raised. Each purchaser of shares is a Stockholder, and receives STOCKS AND BONDS. 279 a Certificate of Stock, which shows the number of shares purchased, and their value. This value is called the Par or Nominal Value. 377. The Market Value of stocks is the price for which they are sold. 378. The value of stocks depends upon the profitable- ness of the business. When the business is very profitable, the shares are worth more than par ; they are then above par, or at a Premium. When the business is unprofitable, the shares are not worth their par value. They are then below par, or at a Discount. 379. 1. The capital of .a company is $100,000. Into how many shares of $ 100 each can this be divided. 2. A stockholder owns 25 shares of stock. How many dollars of stock has he ? 3. If at the end of a year there has been a net profit of $ 10,000, what per cent profit has been made ? $ 10000 is what % of $ 100000 ? The profits are divided among the stockholders, and are called Dividends. Note 1. — Dividends are usually declared semi-annually or quarterly. Note 2. — When a 10 % dividend is declared, each stockholder receives 10 % of the par value of his shares. 4. What will be A's dividend if he owns 35 shares ? When there is a loss, each stockholder is required to pay his share of the loss. This is called an Assessment. 5. What would be A's assessment to meet a 2% loss ? A person who buys or sells stocks for others is called a Stock-broker, and his commission is called Brokerage. 280 INTEREST. Note 1. — Brokerage is usually ^ % or ^ % of the par value. Note 2. — In all stock transactions, dividends, assessments, broker- age, premium, and discount are computed on the par value. Note 3. — Shares are sometimes issued at $200, f 250, $50, $25, or $ 10 each, but unless otherwise stated $ 100 is considered the par value of a share. 6. What is the market value of 10 shares of bank stock, when sold at par ? 7. What is the market value of 50 shares of railroad stock, at 10% premium? Solution. —The market value of 1 share is $ 100 + $10 = $ 110. The market value of 50 shares is 50 times $ 110. 8. What is the market value of 18 shares of mining stock at 15% below par? Solution. —The value of 1 share is 1 100 - $ 15 = $ 85. The value of 18 shares is 18 times 1 85. Stock Quotations are the published prices of stocks. When railroad stock is quoted at 108, it means that it sells for 8% above par in the stock market. When it is quoted at 92, it is selling at 8% below par. 9. If I buy stock at 98 and sell it at 101, what gain do 1 make on 10 shares ? Note. — Stock at 98 means $ 98 for a $ 100 share, and stock at 101 means $ 101 for a $ 100 share. 10. When stock is quoted at 85, what is the value of a share ? What is the value of 1 dollar of stock ? 11. What must I pay for 10 shares of stock at 95, if I pay the broker ^% for doing the business ? Solution. — Cost of 1 share = $ 95 + Brokerage $^ = $95J or $95.25. $ 95.25 X 10 = $ 952.50. Ans. STOCKS AND BONDS. 281 12. If I sell 10 shares of stock at 110, and pay the broker ^%, what do I receive ? Solution. — 1 share brings $110-$^ = $ 109f , or $ 109.75. , 10 shares bring 10 times $ 109.75 = $ 1097.50. Ans. 13. A man invested $4500 in street railway stock at 10% discount. How many shares did he purchase ? 14. If I invest $2100 in bank stock at 105, how many shares do I purchase ? 15. A capitalist bought 80 shares railroad stock at 87^, and 60 shares mining stock at 114. Find the cost. 16. $ 18200 will purchase how many shares of stock selling at 140 ? 17. A stock company declared a dividend of 2|-%. What does A receive, who owns 1500 shares of $ 10 each ? 18. How much is gained on 50 shares of railroad stock purchased at 98 and sold at 102 ? 19. Bought stock at a discount of 2%, and sold it at a discount of 3%. Did I gain or lose, and how much on 20 shares ? BONDS. 380. To meet extraordinary expenses, governments. States, cities, villages, counties, towns, and incorporated companies sometimes borrow money. The securities given by such corporations are called Bonds. Bonds bear a fixed rate of interest, payable annually, semi-annually, or quarterly. They are bought and sold in the same manner as stocks. Bonds are known by the rate of interest they bear : Vir- ginia 6's are bonds of the State of Virginia, bearing 6% ; U. S. 4's of '97 are U. S. bonds bearing 4% interest, and maturing in 1897. 282 INTEREST. 381. A Coupon is an interest certificate attached to a bond. At the expiration of any interest period, the coupon is ci^t off and used in collecting the interest, being worth the amount of interest due on the bond for a specified period. 382. 20. What will be the cost, including brokerage at i%, of 200 shares of C, B., and Q. R.R. bought at 67| ? Solution. —Cost of 1 share = $67| + $ i = $68|. Cost of 200 shares = 200 x $ 68^. 21. How much, including brokerage at |^%, must be paid for $ 5000 of U. S. 4's at llOf ? Solution. —$1 of bonds costs $1.10| + .00^ = $1.11. $5000 worth will cost 5000 times $1.11. 22. What must I pay for f 8275 of stock at 10% dis- count ? 23. What is the cost, including broker's commission of ^%, of 150 shares of railroad stock bought at 89^ ? 24. I buy stocks at 5% discount, and sell at 5% pre- mium. What per cent profit do I make on the investment? 25. March 10, 1896, Western Union Telegraph stock was quoted at 84|^. How many shares could be bought for i 1020, brokerage i per cent ? Solution. — Cost of 1 share, $ 84| + | = $ 85. As many shares can be purchased as $ 85 is contained in $ 1020. 26. How many shares of stock at 10% premium can be purchased for $ 2200 ? 27. I invested f 5100 in N.Y. and N.H. railroad stock at 170. How many shares did I purchase ? 28. If I invest $42400 in 5% bonds at 106, what is my yearly income ? Solution. —$42400 -7- $1.06 = 140000, par value. How much is 5% of $40000? STOCKS AND BONDS. 283 29. If I invest $ 21008 in 5% bonds at 104, what will be my annual income ? 30. What will be my yearly income if I invest $11100 in 5% bonds at 92, brokerage ^% ? 31. A man invests $9500 in Virginia 6's at 94|, broker- ^-ge -4- % . What is his quarterly income ? 32. What will be my annual income if I invest $ 5050 in 4% water bonds at 1% premium ? 33. What is my dividend on 80 shares of electric-light stock, when a 5% dividend has been declared ? 34. What sum must be invested in Chicago 5's at 92 to yield an income of $ 600, brokerage i% ? Solution. — $ 600 -e- .05 = $ 12000, par value. How much is 92^% of $12000? 35. How much must I invest in 4% bonds at 8% pre- mium, to secure an annual income of $ 200 ? 36. How much must be invested in city 3J's at 8% dis- count, to secure an income of $ 350 ? 37. How much telegraph stock must I sell at 11^% dis- count, brokerage -^^j to realize $ 8800 ? 38. I invested through a broker $ 5450 in stock at 108-J, brokerage -J-^. How much did I purchase ? 39. I sell through a broker enough stock at 4J% premium to realize $ 10,475, brokerage i % . How much do I sell ? 40. What rate of interest do I receive on my investment if I buy 7% stock at 112? Solution. — Each share of stock costs $ 112, and yields $ 7 interest. $ 7 is what per cent of $ 112 ? 41. Stock yielding 7% annually is bought at 111-^. What annual rate of income will it yield on the investment ? 284 INTEREST. 42. What rate will 6% bonds pay on the investment if bought at 112 ? 43. What is the rate on Des Moines 4's at a premium of 44. What is the rate of income on 6's at 90, no brokerage ? 45. Which is the better investment, 6's at par, or 5's at a discount of 12^% ? 46. How much must I pay for 5% stock to secure annu- ally 7 % on my investment ? Solution. — 1 share of 5 % stock yields $ 5 interest annually ; this $ 5 is 7 % of the cost of one share. Therefore the question is, $ 5 is 7 % of what? 47. At what price must 5% stock be purchased so that it will yield 4% on the investment? 48. How much must I pay for 5's to make my invest- ment yield 6%? 49. What must I pay for city 6's that my investment may yield 8% annually? 50. How much must I pay for 1 share of 3% stock, that the dividend may be 4% of the purchase price ? 51. How much will be my income if I invest $2300 in 4% bonds at 115 ? Solution. — $2300 -r- .| 1.15 = $ 2000 par value. How much is 4 % of 12000? 52. What sum invested in Tennessee 6's at 85 will yield an annual income of $ 1800 ? 53. How much money must I invest in 6% stock at 80 to secure an annual income of $ 3186 ? 54. I want an income of $ 1500. How much shall I in- vest in 5% stocks at 25% premium to secure that amount? 55. How much must a man invest in a 5% stock at 120 to yield him an annual income of $ 2500 ? MISCELLANEOUS. 285 MISCELLANEOUS. 383. 1. At what premium should an 8% stock sell to yield a 6% income? 2. A man bought stock at 3|-% discount and sold it at 2% premium, paying a brokerage of ^% in both cases. His net profit was f 680. How much money did he invest ? 3. A man invested his money in 6% railroad stocks, and received $ 300 semi-annually. What was the sum in- vested? . 4. Which is the better investment, and how much, a 4% stock bought at 85, or a 6% stock bought at 120 ? 5. What rate on the investment do 7%. stocks pay when bought at a premium of 8 % ? 6. What sum must be invested in U. S. 6% bonds to yield an income of $ 1000 ? 7. What sum must be invested in U. S. 6's at $ 92^ per share to yield a quarterly dividend of $ 300 ? 8. At what price should 8% bonds be bought to make the income from the investment equivalent to that from 6 % bonds at par ? 9. Which is the better investment, 4% bonds at 86, or 6% bonds at 105? 10. How much must I pay for a 4% stock that the in- vestment may yield me 6% ? For a 7% stock that the investment will yield 5%? 11. If 25 shares of stock paying 8% are sold at 150, and the proceeds loaned at 6%, will the income be increased or diminished, and how much ? 12. Bought bonds at 125 and sold them at 110, thereb}' losing $ 600. How many ^ 1000 bonds did I buy? 286 INTEREST. 13. How many dollars of stock can I buy for $ 105,000 if stock is quoted at 120? How many shares ? What per cent do I receive on my investment if the stock bears 6% ? 14. What is the cost of 200 shares of D., L., and W. E.R. at 162|? If it pays a quarterly dividend of 2%, what is the yearly income from this investment? What rate does it pay on the investment ? 15. B invests $ 1680 in a stock selling at 112. What does he receive from a dividend of 4% ? 16. An estate derives an annual income of $3600 from stock that pays 7^%. How many $25 shares does the estate own? AVERAGE OF PAYMENTS. 384. 1. The use of $ 5 for 2 mo. equals the use of $1 for how many months ? 2. The use of $ 10 for 6 mo. will balance the use of $ 5 for how many months ? Solution. — The use of $ 10 for 6 mo. = the use of $ 1 for 60 mo. The use of $ 1 for 60 mo. = the use of $ 5 for ^ of 60 mo. — 12 mo. 3. How long may $ 20 be kept to balance the use of $5 for 20 mo. ? $ 50 for 10 mo.? 4. A credit of $ 10 for 8 mo. equals a credit of $ 20 for how many months ? 5. The interest of $500 for 1 year equals the interest of $ 100 for how long ? Prove this. 6. I pay a debt of $ 20 four months before it is due. How long after it is due should my creditor allow a debt of $ 40 to remain unpaid ? A person owing two debts due at different times may pay both at an intermediate time without loss to himself AVERAGE OF PAYMENTS. 287 or his creditor, by paying one of them before it is due and the other an equivalent time after it is due. 385. The process of finding the time when several debts due at different times can be equitably discharged at one payment is called Average of Payments. 386. The date of such payment is called the Average Time, and the time to elapse before the payment is made is called the Average Term of Credit. Note. — The time to elapse before any debt becomes due is called a Term of Credit. 387. When the terms of credit begin at the same date. 1. On Jan. 8, A bought goods on the following condi- tions : $ 300 due in 2 months. $ 200 due in 4 months. ^ 100 due in 6 months. How long after Jan. 8 may the debt be equitably dis- charged at one payment ? Solution. — A credit of $ 800 for 2 mo. = a credit of $ 1 for 600 mo. A credit of $ 200 for 4 mo. = a credit of $ 1 for 800 mo. A credit of $ 100 for 6 mo. = a credit of $ 1 for 600 mo. A is entitled to a credit of $ 1 for 2000 mo. A credit of $ 1 for 2000 mo, = a credit of $ 600 for ^^ of 2000 mo., or 3| mo. = 3 mo. 10 d., average term of credit. Jan. 8 + 3 mo. 10 d. = April 18, equated time. Ans. Short method. 2 mo. X 300 = 600 mo. 4 mo. X 200 = 800 mo. 6 mo. X 100 = 600 mo. 6^0 )2000 mo. 3^ mo. = 3 mo. 10 da. Jan. 8 + 3 m. 10 da. = April 18. 288 INTEREST. ^ Note. — One-half a day or more is called another day. Less than ^ day, not counted. Call 50^ or more $ 1.00, Less than 50 ^, not counted. Rule. — Multiply each debt by its term of credit. The sum of the products divided by the sum of the debts will be the average term of credit. 2. Gates Thalheimer sold a bill of goods on the following terms: $325 due in 60 days, $ 175 due in 90 days, and $ 185 due in 4 months. What is the average term of credit ? 3. A merchant bought $ 1000 worth of goods, and agreed* to pay for them as follows : f 100 cash ; $ 300 in 3 mo. ; $ 250 in 4 mo. ; and the balance in 5 mo. In what time could he equitably pay the entire amount ? 4. On the first day of April, 1895, a man gave 3 notes, one for $ 250 due in 30 da., one for $ 375 due in 40 da., and one for $ 425 due in 60 da. What is the average term of credit, and when could they have all been paid at once ? 5. D. McCarthy & Co. sold goods amounting to $4000, payable as follows : ^ in 3 months, J in 4 months, and the balance in 5 months. What was the average term of credit ? 6. A merchant sold goods on the following terms : \ pay- able in 2^- months, \ in 3i months, \ in 5J months, and the balance in 6 months. What was the average term of credit ? 7. Equate the following payments : $ 580.75 due in 30 days, $ 650.25 due in 60 days, $ 450.36 due in 90 days, and $ 600 due in 5 months. 8. On the 1st of May a merchant bought goods amount- ing to f 1500, agreeing to pay for them as follows : $ 521.35 on the 10th of June, $398.84 on the 16th of July, $ 199.60 on the 15th of August, and the balance on the 1st of Sep- tember. Upon what date can he pay the whole amount ? AVERAGE OF PAYMENTS. 289 9. Jacob Amos sold a bill of flour amounting to $ 2500, payable as follows : $ 500 due in 4 months, $ 600 due in 5 months, and the balance due in 6 months. What was the equated time ? 10. A purchased a farm for $ 3000, agreeing to pay for it as follows : $ 500 cash, $ 600 in 5 months, $ 1000 in 8 months, and $ 900 in 1 year. He decides to give a note for the whole amount. When was the balance to be paid ? 388. When the terms of credit begin at different dates. 1. A purchased goods of Dey Bros. & Co., as follows : Jan. 8, 1895. $ 200-on 2 months' credit. Feb. 16, 1895. $ 400 on 3 months' credit. April 4, 1895. $ 300 on 4 months' credit. Find the average time. Note. — First find the date when each item is due. $200 due Mar, 8. 200 400 due May 16. 400 X 69 da. = 27600 300 due Aug. 4. 300 X 900 149 da. = 44700 72300 72300 - 900 =: 80^ :da. = 80 da. March 8 + 80 da. = June 27, average time bt is due March 8 , and the last Aug. 4. The average time, therefore, will be between these dates. $ 200 due March 8 has no longer time to run. $400 due May 16 has 69 days after March 8. $ 300 due Aug. 4 has 149 days after March 8. A is therefore entitled to a credit of $ 1 for 72300 da. after March 8, which is equal to a credit of $ 900 for 80 da. after March 8. Rule. — Find the date on which each debt becomes due, and "using the earliest of these as a standard date, reckon the time to each of the others. Multiply each debt by its time, and divide the sum of the products by the sum of the debts. 290 INTEREST. The quotient will he the average term of credit, which add to the standard date to find the average time. 2. Four notes are due as follows : March 4, $ 165; April 15, $325.50; May 9, f 94; June 6, $465. What is the average date of payment ? 3. A retail dealer bought the following bills of goods on 4 months' credit : April 4, $ 480 ; April 26, $ 185.65 ; June 1, $ 480.16; July 6, $ 196. What is the average time for payment ? 4. Bought goods as follows : Jan. 1, $ 250 at 3 mo. ; Feb. 1, I 500 at 4 mo. ; March 11, $ 106 at 60 da. What is the average date of payment ? 5. Mr. B owes $ 1000, due in 5 months; in 2 months he pays $ 600. How long after the expiration of the 5 months should the remainder be paid ? Solution. — $ 600 has been paid 3 months before due, which equals a credit of f 1 for 1800 months. He is entitled to a like credit for $ 400 after it is due. j^^y of 1800 mo. = 4| months. Ans. 6. A lady purchased a piano for $ 500 on 6 months' credit. If she pays $ 200 cash, how long after the expira- tion of the 6 months should the balance be allowed to run ? 7. May 1, 1896, a man buys a store and fixtures for $ 2650, giving his note payable in 6 months without interest. June 15, he pays $ 500 ; Aug. 1, $ 750. When should the balance be paid ? 8. G. L. Hoyt purchased goods of Mann & Hunter to the amount of $3000: $1200 to be paid June 2, 1896; $600 to be paid July 5, 1896 ; $ 200 to be paid Aug. 15, 1896. The balance will become due Aug. 30, 1896. At what date must a note payable in 3 m. be drawn that it may become due at the average date ? AVERAGE OF PAYMENTS. 291 QUESTIONS. 389. 1. Define discount; present wortli; true discount. Tell how to find present worth and true discount. 2. Define bank discount; proceeds; day of maturity; term of discount. Tell how to find bank discount and proceeds. Tell how to find face of note when proceeds, time, and rate are given. 3. What is a stock company ? What are stocks ? Bonds ? Shares ? 4. Define par value ; market value. 5. What is a stock certificate ? 6. Define dividend; assessment. 7. Upon what are premium, brokerage, dividends, and assessments reckoned ? 8. What is the average of payments ? Equated time ? Average term of credit ? KATIO AND PROPOETION. 390. Oral. 1. 5 bears what relation to 10 ? Aris. 5 is ^ of 10. 2. 10 bears what relation to 5 ? Ans. 10 2 times 5. 3. What part of 16 is 4 ? 4. How does $ 7 compare with $ 14 ? 5. John has 20^ and Mary 5^. What is the relation of John's money to Mary's ? Of Mary's money to John's ? 6. What is the relation of 15 to 3 ? Of f 8 to $16? Of 28 men to 7 men ? Of 2 bushels to 2 pecks ? 391. Ratio is the relation between two like numbers. It is found by dividing one by the other ; thus : The ratio of 4 to 8 is 4 -- 8 = f The sign of ratio is ( : ). It is the division sign with the line omitted: The ratio of 6 to 3 is expressed thus, 6:3. It may also be expressed fractionally, thus, J. 392. The Terms of a ratio are the two numbers com- pared. The first term of a ratio is the Antecedent, and the second the Consequent. In the ratio 6 : 12, 6 is the antecedent, and 12 the conse- quent. 292 \ EATio. 293 393. A ratio formed by dividing the consequent by the antecedent is an Inverse ratio. 12 -T- 6 is the inverse ratio of 6 : 12. 394. The two terms of a ratio taken together form a Couplet. 395. Two or more couplets taken together form a Com- pound ratio. A compound ratio may be changed to a sim- ■ pie ratio by taking the product of the antece- 5 : 5 r = 9d : loO dents for a new antecedent, and the product of 4 : 5 J the consequents for a new consequent. Antecedent -=- Consequent = Ratio. Therefore, Antecedent -~ Ratio = Consequent ; and, Ratio x Consequent = Antecedent. Multiplying or dividing both terms of a ratio by the same number does not change the ratio. The ratio 12:6 = 2. The ratio 3 x 12 : 3 x 6 = 2. The ratio 12 -- 3 : 6 -^- 3 = 2. Find the ratio of: 7.-56:7 11. 3bu. :3pk. 8. 20:300 12. 1:4 9. $55:f330 13. 12 : ^ 10. What is the ratio ^-^j to ^^ ? Note. — Fractions with a common denominator have the same ratio as their numerators. Prove this in Ex. 10, by multiplying both terms by 10. 17. TV:if = ? n--^j = '^ if:ff = ? 18. f:|=? 3:5 = 9 4:3 = 9 2:5 = 9 19. Find the inverse ratio of 75 to 25. Of 15 to 225. 20. 16 : (?) = 1 14 : (?) = 2. 21. (?):5 = 4. (?):8 = l. 22. Find the value of the compound ratio, p. ' ^ [• • 14. H :16 15. i- 1 16. H :5f 294 PROPORTION. PROPORTION. 396. Oral. 23. Give three fractions having the same value as |. 24. Give two numbers that have the same ratio as 5 to 10. 25. Give a fraction equal to |. 26. Give a ratio equal to 3 : 4. 27. How does the ratio of 5 men to 10 men compare with the ratio of $5 to ^10? 28. How does the ratio of 8 lb. to 4 lb. compare with the ratio of 40^ to 20^? 29. Name two numbers that have the same relation as 5 to 10. As 4 to 24. As 8 to 16. As ^ to \. 30. What number has the same relation to 5 as 12 to 3 ? 31. Find a number whose ratio to 4 equals 3 : 6. 32. Give three ratios equal to $100 : foO. 33. Give any two ratios that equal each other, and express their equality. 397. An equality of ratios is a Proportion. Thus, 4 : 2 = 12 : 6. The ratio of 4 to 2 equals the ratio of 12 to 6. A proportion is usually expressed 'with the sign (::) between the ratios ; thus, 4 : 2 : : 12 : 6. This is read 4 is to 2 as 12 is to 6. A proportion has four terms, of which two are antece- dents and two are consequents. Each term is a propor- tional. 398. The first and fourth terms are called Extremes, and the second and third terms are called Means. Note. — In the proportion 2 : 6 : : 6 : 18, the two means are the same number, 6. The 6 is called a mean proportional. SIMPLE PROPORTION. 295 Principle. — The product of the extremes equals the product of the means. Rule. — To find an extreme, divide the jn^oduct of the means by the given extreme. To find a mean, divide the product of the extremes by the given mean. Supply the missing term : 34. 1:836::25:( ). 39. 10 yd. : 50 yd. : : f 20 : (| ). 35. 6 : 24 : : ( ) : 40. 40. 15 lb. : 60 lb. : : ($ ) : $ 12. 36. ( ):15 37. 25 :( ) 38. 6:4::i : 60 : 6. 41. i da. : ( da.) : : 12 : 6. :4:8. 42. ( men) : 75 men : : ^ 50 : $ 150. ( ). 43. $f:$3f::( ) : 5. SIMPLE PROPORTION. 399. An equality of two simple ratios is a Simple Pro- portion. It is employed in solving questions having three given terms, two of which have the same relation to each other as the third to the required term. 44. If 12 bushels of oats cost f 4, what will 60 bushels cost ? Solution. — There must be the same relation between the cost of 12 bu. and the cost of 60 bu. as exists between 12 bu. and 60 bu. We place $4 for the third term. The 12 : 60 : : $ 4 : ($ ) answer will be the fourth. We must now ac\^A form a ratio of 12 and 60 that shall equal — — — = $20. the ratio of f4 to the answer. Since the third term is less than the required answer, the first must be less than the second, and we have 12 : 60 for the first ratio. The product of the means divided by the given extreme will give the other extreme, or $20. Ans. By analysis, — Since 12 bu. cost $4, 1 bu. will cost $1^, and 60 bu. will cost $20. Ans. 296 SIMPLE PROPORTION. Rule. — Consider the required answer as the fourth term^ and place the number that is like it for the third terni. Place the two remaining terms as folloivs : If the answer is to be larger than the third term, the second must be larger than the first. If smaller, the second must be smaller than the first. Divide the product of the means by the given extreme. Cancel when possible. 45. If 10 sheep cost $35, what will 23 sheep cost? What will 6 sheep cost? 46. If 5 men can do a piece of work in 9 days, how long will it take 15 men to do the same work ? Solution. — Place 9 days for the third term, because it is like the required answer, thus, : : 9 da. : ( da.) Since 5 men can do it iii 9 days, 15 men can do it in less time. Therefore, since the answer is to be smaller than the third term, place 5 men for the second, and 15 men for the first. Multiplying and dividing we have 3 days. Ans. 47. If 14 horses eat 36 tons of hay in a certain time, how many tons will 13 horses eat in the same time ? 48. If it costs $400 to lay 80 rods of street-car track, how much will it cost to lay 3|- miles at the same rate ? 49. If a pole 8 ft. high casts a shadow 41 ft. long, how high is a tree which casts a shadow 48 ft. long ? 50. If a man walks 280 miles in 8 days, how many days ought it to take him to walk 420 miles ? 51. If it costs $ 13.20 to supply a new arithmetic to each of a class of 24 pupils, what will be the expense of furnish- ing one to each of a class of 19 ? 52. How far can a train run in 3 hours,^ if it can run 160 Km. in 4 hours ? WBITTEN EXERCISES. 297 53. How many men will be required to do in 10 days what 15 men can do in 30 days ? 54. What will 8 tons of coal cost, when 17|- tons cost $ 78.75 ? 55. If a certain sum of money yields $ 360 interest in one year, what would the interest of the same sum be for 15 months ? 56. If $ 800 yield $ 48 interest in a certain time, how large a sum will yield $ 216 in the same time ? 57. If the interest of f 3600 for a certain time is $ 216, what will be the interest of $ 800 for the same time ? 58. If a garrison of _240_ soldiers have a supply of food sufficient for 150 days^ how .leng would the same food last if the garrison were increased to 6p0 men ? 59. In the above example, how long would the food last if, 80 men were sent away ? 60. Find the cost of 1^-f- bushels of wheat, if -f- bu. cost 61. If 120 shoemakers make 40 dozen pair of shoes in a certain time, how many shoemakers would it require to make the same number of shoes in one-half of the time ? 62. If a train runs 140 miles in 4 hr. 30 min., what is the rate per hour ? 63. When 5 tons 1250 lb. of coal cost $ 24.75, what will be the cost of 18 tons 500 lb. ? 64. If a 16-foot board 9 inches wide contains 12 sq. ft., how wide must a board of the same length be to contain 20 sq. ft. ? 65. It takes 26 yards of carpet 1 yard wide to cover a floor. How many yards will it take if the carpet is but 27 inches wide ? 298 COMPOUND PROPORTION. 66. The ratio of Simon's pay to Matthew's is |. Simon earns f 18 per week. What does Matthew earn ? 67. 25 men can do a piece of work in 70 days ; but after 30 days, 15 of them refuse to work. In how many days can the rest complete the work ? COMPOUND PROPORTION. 400. An equality between a compound and a simple ratio is a Compound Proportion ; thus, 8:4 ) . . >- : : 12 : 20 is a compound proportion. Find the fourth term. Solution. — First changing to a simple propor- 3:6) tion, we have, 4.8j''^-( ) 3x4:6x8::3:( ). Then divide the product of the means by the given extreme, using cancellation. Thus, ^^^x^^l2. Ans. 1. If 5 men earn $72 in 8 days, how much can 10 men earn in 6 days ? Solution. — Since the answer is to be in dollars, place $ 72 for the third term, and arrange the 5 men : 10 men ) . . a? 70 . / n *®^°^^ °^ ^^^^ couplet according 8 days : 6 days f • * ^ ' ^ y as the answer should be greater or less than the third term if it depended on that couplet alone. Since 5 men earn ^ 72, 10 men can earn more, so we place 10 men for the second and 5 men for the first ; and since they earn $ 72 in 8 days, they will earn less in 6 days, so we place 6 days for the second term, and 8 days for the first. Dividing the prodilct of the means by the extremes^ we have, 9 2 iZ^.xJix_6^$108. Ans. WRITTEN EXERCISES. 299 By analysis. Since 5 men in 8 days earn $ 72, 1 man in 8 days will earn $ ^^. 1 man in 1 day will earn $ f . 10 men in 1 day will earn $ ^-^-. 10 men in 6 days will earn ($ 108). Rule. — Consider the answer as the fourth term, and place the number that is like it for the third. Arrange the couplets as if the answer depended on each couplet alone, as in simple proportion. Divide the product of the means by the product of the extremes. Cancel when possible. 2. If four horses eat 10 bushels of oats in 5 days, how many bushels will be required to feed 5 horses for 2 days ? 3. If 10 men working 8 hours a day can do a piece of work in 12 days, how many days would it take 6 men, work- ing 10 hours a day, to do the same amount of work ? 4. If a wheelman rides 144 miles in 3 days of 6 hours each, how many miles can he ride in 5 days of 9 hours each ? 5. A section of a street 33 feet long and 20 feet wide can r7^^ *•"" be paved with 15840 stones, each 9 inches long and 8 inches ^ ( ^ wide. How many stones 12 inches long and 10 inches wide ^ #9^ . will it take to pave a street 12 rods long and 16 feet wide ? X- 6. If it costs $84 to carpet a room 24 feet long and 21 *^-^^|» A simple ratio ? V A compound ratio ? 3.y Tell how to find ratio when antecedent and conse- quent are given.^ To find consequent when antecedent and 306 PARTNERSHIP. ratio are given. To find antecedent when consequent and ratio are given. 4:/ Define proportion.*^ How many terms in a simple proportion?*^ Name them. 5.^ Give the principle of proportion. 6. "^ What number is placed for the third term?^ The second ? ^ The first ? How is the fourth term then found ? 7.' What is a compound proportion? What number is placed as the third term ? » How is each couplet theii arranged ? How is the fourth term found ? 8/ What is a partnership ? * A company ? ♦ 9. Define capital stock J*^ dividends. ^ 10. Tell how to find each partner's share of the profit or loss when the capital of each is invested for the same tim5. ' When the capital of each is not invested for the same time. INVOLUTION. 405. 1. 3x4x2 = what ? 2. 3 X 3 X 3 = what? Note, — In Example 2 the factors are equal ; in Example 1 they are unequal. The product of equal factors is a Power. 3. What is the product of 4 taken 3 times as a factor ? 4. What is the product of 6 taken twice as a factor ? 5. What is the product of |- used three times as a factor? 6. What is the product of .6 used twice as a factor ? 406. The process of finding powers is Involution. 407. A power is named according to the number of its equal factors. The product of two equal factors is the Second Power, or Square, of the equal factor. The product of three equal factors is the Third Power, or Cube, of the factor. Note. — The second power is called a square because the area of any square figure is the product of two equal factors, length and breadth ; and the third power is called a cube because the solidity of any cube is the product of three equal factors, length, breadth, and thickness. 307 13. 2.5' 16. (})' 14. 1.1^ 17. (^y 15. .002' 18. (2^)^ 308 INVOLUTION. 408. A small figure at the right and above a number to show how many times it is to be used as a factor is called an Exponent. Thus, 42 zz: 4 X 4 is 4 to the second power, or the square of 4 ; 2^ = 2 X 2 X 2 is 2 to the third power, or the cube of 2 ; 3* = 3x3x3x3 is 3 to the fourth power, or the fourth power of 3. Eead : 8^ 15', 5', f^'^^, |, 84» 16'. 409. Find the powers : 7. 53 10. 6' 8. 2' 11. 1^ 9. 252 12. .01^ 19. Find the square of 36. 20. Find the cube of 15. 21. Find the 4th power of 6. 22. Find the 5th power of 3. 23. State the difference between a power and any other product. 24. Find the difference between the square and the cube of 9. 25. Find the difference between the 4th and 5th powers of 6. 26. Find the difference between the square and the cube of 1 27. What is the difference between the square and the cube of .15 ? 28. Which is greater and how much, the square or the cube of ^? EVOLUTION. 410. 1. What factor is used 3 times to produce 27 ? 2. What are the two equal factors of 64 ? 3. What is one of the three equal factors of 8 ? 4. 36 is the square of what number ? 5. 64 is the cube of what number ? 6. 144 is the second power of what ? 7. 1728 is the cube of what ? 411. One of the equal factors of a power is a Root. One of two equal factors of a number is the Square Root of it. One of the three equal factors of a number is the Cube Root of it. The fourth root of a number is one of its 4 equal factors. The square root of 16 = 4. The cube root of 27 = 3. The fourth root of 16 = 2. 412. The Radical Sign (-y/) placed before a number indi- cates that its root is to be found. The radical sign alone before a number indicates the square root ; thus, V9 = 3 is read, the square root of 9 = 3. 413. A small figure placed in the opening of the radical sign is called the Index of the root, and shows what root is to be taken ; thus, V8 = 2 is read, the cube root of 8 is 2. Read the following : V8l, -^64, -v/Sl, Viil, -^Tm, ^/'9, -s/SM 309 310 EVOLUTION AND INVOLUTION. EVOLUTION AND INVOLUTION. 414. 1. rind the square of 11. The cube of 6. The fourth power of 5. 2. Eind the square root of 49. The cube root of 8. The square root of -f^. 3. 92==? -v/9 = ? 8^ = ? -v^8 = ? 4. Write all the squares from 1 to 100. 5. Write all the cubes from 1 to 1000. 6. Learn the second and third powers of numbers from 1 to 12. SQUARE ROOT. 415. The square of a number is the product of that number taken twice as a factor. Blackboard. 12 = 1. 102 = 100. 1002 ^ i()000^ 92 = 81. 902 ^ gjLOO. 9002 = 810000. From the above illustration it is seen that annexing one cipher to a number annexes two ciphers to the square of that number, as in 1^ = 1 ; 10^ = 100 ; 100^ = 10000. 416. A square contains twice as many figures as its root, or twice as many less one. Squares of even tens. OraL 1. 202=? 3^ 802=? 5^ 702 = ? rj^ 5002 = ? 9^ 6002 = ? 2. 502 = ? 4^ 302=? Q 2002=? 8. 9002=? 417. The square of a number composed of tens and units may be found as follows : 24 = 20 + 4 = 2 tens + 4 units. 242= (20 + 4) X (20 + 4). SQUARE ROOT. 311 20 + 4= 24 20 4-4= 24 (20 X 4) + 42 = 96 202 + (20 X 4) = 480 202 + 2 X (20 X 4) + 42 = 576 From the operation, we find that, The square of the tens . . 202 =: 400 2 times the tens by the units, 2 x (20 x 4) = 160 The square of the units . . 42 = 16 400 + 160 + 16 =^6 418. Prin-ciple. — The square of a number composed of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. Formula. — Tens2 + 2 x tens x units + units^. Separate the following into tens and units, and find their squares : 15, 25, 74. 419. By reversing the process we may find the Square Root. 10. What is the square root of 1225 ? Solution. — Separating into periods of two figures each, begin- ning at units, we have 12'25. Since there are two periods in the power, there must be two figures in the root, tens and units. The greatest square of even tens contained in 1225 is 900, and its square root is 30 (3 tens). 1225 I 30 + 5 = 35. Tens2, 302 ^ 900 2 x tens = 2 X 30 =60 [325 2 X tens + units = 2 x 30 + 5 = 65 [325 Subtracting the square of the tens, 900, the remainder consists of 2 X (tens X units) + units. 325, therefore, is composed of two factors, units being one of them, and 2 x tens + units being the other. But tlie greater part of this factor is 2 X tens (2 x 30 = 60). By trial we divide 325 by 60 to find the other factor (units), which is 5, if correct. Completing the factor, we have 2 x tens + units = 65, which, multiplied by the other factor, 5, gives 325. Therefore the square root is 30 + 5 = 35. 312 EVOLUTION. 420. Square root may be explained by the aid of dia^ grams. The area of every square surface is the product of two equal factors, length and width. Finding the square root of a number, therefore, is equiva- lent to finding the width of a square surface, its area being given. 421. The following formulas illustrate the principles which underlie the operations of square root: 1. Length x Width = Area, 2. Area -^ Length = Width. 3. Area -f- Width = Length. 1. Find the width of a square whose area is 1296 sq. ft. Fig. 1. 2 X 30 ft. = 60 ft. 2 X 30 ft. + 6 ft. = m ft. Solution. — The great- est square of even tens contained in 1296 sq. ft. is 900 sq.ft. (Square A) \~ b | ^ L£J Its width is 30 ft. 1296 , Fig. 2. sq. ft. - 900 sq. ft. = 396 sq. ft., tlie area of 6, c, and d, considered as one rectangle (Fig. 2), whose width we desire to find. The length of this rectangle is (2 X 30 ft. ) 60 ft. + the length of c. But we cannot know the length of c till we find its width. By trial (Formula 2), we divide the area, 396 sq. ft., by 60 ft., its approximate length. The quotient, if cor- rect, is 6 ft., the width desired. To test the correctness: Add the 6 ft. to the trial divisor, and we have Q6 ft. , the entire length of a, 6, and c, which (Formula 1), multiplied by its width, 6 ft., gives its area, 396 sq. ft. There is no remainder, and the work is correct. Therefore, 30 ft., the width of J. + 6 ft., the width of a, &, and c, = 36 ft., the width of the original square. AREA. 1296 WIDTH. d o 900 896 30 ft. Oft. 396 36 ft. Ans. 2 b c A 900 d sq. ft. 30 ft. 30 ft. 6 ft. SQUAEE ROOT. 313 Notes. — All the numbers in the middle column denote area. 1296 sq. ft. = area of the original square ; 900 sq. ft. = the area of A; and 396 sq. ft., the area of 6, c, and d. The numbers in the left-hand column denote length. 60 ft. = the approximate length (or the trial divisor) of b, c, and d ; and 6Q ft. the exact length, or the complete divisor. The numbers in the right-hand column denote width. 30 ft. = the width of A ; and 6 ft. the width of 6, c, and d ; 36 ft. = the width of the original square. In dividing, to find the width of 6, c, and d, since the divisor is too small, care must be taken that the quotient figure be not too large. SHORT METHOD. 422. 2. Find the square root of 1306.0996. 13'06'.09'96 C 36.14 9 66j 40e^ 396 721 ) 1009 721 7224 ) 28896 28896 Rule. — Beginning at the decimal point, separate the number into periods of two figures each, pointing whole numbers to the left and decimals to the Hght. Find the greatest square in the left-hand period, ayid ivrite its root at the right. Subtract the square from the left-hand period, and bring down the next period for a dividend. Divide the dividend, with its right-hand figure omitted, by twice the root already found, and annex the quotient to the root, and to the divisor. Multiply this complete divisor by the last root figure, and bring down the next period for a dividend, as before. Proceed in this manner till all the periods are exhausted. 314 EVOLUTION. Note 1. — "When occurs in the root, annex to the trial divisor, bring down the next period, and divide as before. Note 2. — If there is a remainder after all the periods are ex- hausted, annex decimal periods. Note 3. — If, after multiplying by any root figure, the product is larger than the dividend, the root figure is too large and must be diminished. Also the last figure in the complete divisor must be diminished. Note 4. — For every decimal period in the power, there must be a decimal figure in the root. Note 5. — If the last decimal period does not contain two figures, supply the deficiency by annexing a cipher. 3. Find the square root of 253009. Solution. — As 25'30'09(5 occurs in the root, we 25'30'09(503 Ans. 25 annex to the trial 25 10)"^ ^^^^''^'' ^^' ^^^/^- 1003)~3009 — ^ other period to the divi- ^ oAAn dend, and divide as before. Thus, — ^^25 Note. — To find the square root of a common fraction, extract the root of each term separately. If both terms are not squares, change the fraction to a decimal, and then extract the root. The result will be the approximate root. Change mixed numbers to improper fractions. 4. What is the square root of y^ i ? ^ = Ans. V144 12 Find the square root of : 14. .06432 23. ^ 15. .005625 24. n 16. .913936 25. Ill 17. 25.6036 26. 4^ 18. 24.3049 27. ^ 19. .612089 28. ffff 20. 329.7643217 29. 36.45f 21. 1684.298431 30. 2863|J 22. 389765268 31. 189J| 5. 8836 6. 15876 7. 370881 8. 46656 9. 820836 10. 29.0521 11. 9.2416 12. 3180.96 13. .007921 SQUARE ROOT. 315 41. 13.2^ Find the square root to four decimal places : 32. .15 35. 4.7 38. 3.67 33. .18 36. 72.5 39. .222 42. .009^^ 34. 17 37. 119 40. 963 43. .003f 44. What is the length of one side of a square field that has an area equal to a field 75 rd. long and 45 rd. wide ? 45. How wide is a-field containing 7056 square rods ? Perform the indicated operations. Note. — Carry decimals to the third place. 46. V3.26 X .0063. 47. 03 X V|4-f 1 V9 V9 3 48. 49. Vi X f 50. Vi X Vi- 51. V3.532 - 6.28. 52. V4 + 62 + 2. 53. VF + (|/- 54. V625 + 1296. 55. V625+V1296. RIGHT-ANGLED TRIANGLES. 423. A triangle having one right angle is a Right-Angled Triangle. 424. The side opposite the right angle is the Hypothe- nuse, as AB. BO is the Perpendicular, and AC the Base. In the triangle ABC, the hypothenuse is 5 inches, the perpendicular 3 inches, and the base 4 inches. \ >^ v \/\ y\ b C 316 EVOLUTION. 425. It will be seen that the square of the hypothenuse is 25 sq. in., which is equal to the square of the base, 16 sq. in., plus the square of the perpendicular, 9 sq. in. Principle. — The square of the hypothenuse equals the sum of the squares of the two shorter sides. Therefore, to find the hypothenuse, take the square root of the sum of the squares of the base and perpendicular. VBase^ 4- Perpendicular^ = Hypothenuse. 426. To find the base or the perpendicular, take the square root of the difference between the squares of the hypothe- nuse and the other side. vHypothenuse^ — Base^ = Perpendicular. VHypothenuse'^ — Perpendicular^^ = Base. 1. The base of a right-angled triangle is 32 ft., and the perpendicular 24 ft. What is the hypothenuse ? Solution. — 322 + 242 ^ iqqq. -v/ieOO = 40 ft. Ans. Or, \/322 + 242 ^ 40 ft. 2. The hypothenuse of a right-angled triangle is 40 ft., and the base 32 ft. What is the perpendicular ? 402-322 = 576. \/576 = 24ft. Ans. Or, V402 - 322 = 24 ft. 3. A 40-foot ladder placed 24 feet from a house will just reach to the top of it. How high is the house ? 4. What is the length of a ladder that will reach the top of a house 40 feet high, when the foot is placed 30 feet from the house ? 5. A rope 150 ft. long fastened to the top of a flag-pole reaches the ground 40 feet from the base. How high is the pole ? SIMILAR SURFACES. 317 6. What is the hypothenuse of a right-angled triangle whose perpendicular is 36 feet, and whose base is 27 feet ? 7. A square farm contains 360 acres. What is the di- agonal distance between its opposite corners ? 8. A telegraph pole 32 feet high casts a shadow 28 feet in length. What is the distance from the top of the pole to the end of the shadow ? 9. The base of a right-angled triangle is 16 m., and the perpendicular is 12.8 m. What is the hypothenuse ? 10. A boy rides his wheel due north at the rate of 15 miles an hour, and another boy starting from the same place, rides due east at the rate of 18 miles an hour. How far are they apart at the end of 5 hours ? 11. What is the length of the diagonal of a floor 16 ft. long and 12 ft. wide ? 12. A crayon box is 6 in. long, 4 in. wide, and 4 in. high. What is the diagonal distance across the bottom ? Between the opposite corners ? 13. A street is 32 ft. wide from curb to curb. A tele- graph pole 40 ft. high stands upon one side of the street, How long must a wire be to reach from the top of the pole to the opposite side of the street at the curb ? SIMILAR SURFACES. 427. Surfaces having the same form without regard to size are Similar Surfaces. Note. — Any two squares or any two circles of different size are Similar Figures. Rectangles, triangles, etc., are similar when their corresponding dimensions are proportional. Oral. 1. What is the area of a square whose side is 2 ft. ? 2. What is the area of a square whose side is 3 ft. ? 318 EVOLUTION. 3. What is the ratio of the two sides ? 4. What is the ratio of the two areas ? 5. Are these ratios equal ? (2 ft. : 3 ft.) (4 sq. ft. : 9 sq. ft.) Solution. — From the illustration it will be seen that the areas are to each other as the squares of the sides ; not as 2 to 3, but as 4 to 9. 2 ft. 8 ft. Principles. — Similar surfaces are to each other as the squares of their corresponding dimensions. Corresponding dimensions are to each other as the square roots of their areas. 6. A circle is 4 inches in diameter ; another is 8 inches in diameter. What is the ratio of their areas ? 7. A circle has an area of 16 square feet; another has an area of 64 square feet. What is the ratio of their diameters ? 8. The area of a rectangle 12 ft. long is 84 square feet. What is the area of a similar rectangle 6 feet long ? 9. Two similar fields have areas of 12 acres and 8 acres respectively ; the larger is 32 rods wide. How wide is the smaller ? 10. The altitudes of two similar triangles are 20 ft. and 10 ft. ; the area of the smaller is 80 square feet. What is the area of the larger ? CUBE ROOT. 428. The cube of a number is the product of that number taken three times as a factor. Blackboard. V = 1. 10^ = 1000. 1003 ^ 1000000. 9^ = 729. 908 = 729000. 9003 = 729000000. . CUBE ROOT. 319 429. Annexing one cipher to a number, annexes three ciphers to the cube of the number, as shown in 1^, 10^, 1003, etc. 430. Cubes of even tens. 1. 103 = ? 4. 403=? 7. 3003=? 2. 303 = ? 5, 803 = ? 8. 8003 = ? 3. 503=? 6^ 2003=? 9. 9003=? 431. The cube of a number composed of tens and units may be found as follows : 24 = 20 + 4 = 2 tens + 4 units. 243 = (-20 + 4) X (20 + 4) X (20 + 4). 20 + 4 = 24 20 + 4 = 24 (20x4) + 42zz: 96. 202 + (20 X 4) = 480 202 + 2 X (20 X 4) + 42 =: 576 20 + 4 = 24 (202 X 4) + 2 X (20 X 42) + 4^ = 2304 203 + 2 (202 X 4) + (20 X 42) = 1152 203 + 3 X (202 X 4) + 3 X (20 x 42) + 43 = 13824 From the operation we find that. The cube of the tens 203 = 8000 3 times the square of tens by units 3 (202 x 4) = 4800 3 times the tens by the square of the units . . 3 (20 x 42) = 960 The cube of the units 43 = 64 8000 + 4800 + 960 + 64 = 13824 432. Principle. — The cube of a number composed of tens and units is equal to the cube of the tens plus 3 times the square of the tens by the units, plus 3 times the tens by the square of the units, plus the cube of the units. 320 EVOLUTION. Formula. — Tens' + 3 x tens^ x units + 3 x tens x units' + units^. 10. Separate the following into tens and units, and find their cubes : 35, 54, 63. 433. By reversing the process, we may find the cube root. 11. What is the cube root of 13824 ? Solution. — Separating into periods of three figures each, begin- ning at units, we have 13'824. Since there are two periods in the power, there must be two figures in the root, tens and units. The greatest cube of even tens contained in 13824 is 8000, and its cube root is 20 (2 tens). 13'824 I 20 + 4 Tenss = 20^ = 8000 3 X tens2 = 3 x 202 = 1200 5824 3 X tens x units = 3 x (20 x 4) = 240 units2 = 42 = 16 3 X tens2 + 3 tens x units + units2 = 1456 (3 X tens2 + 3 x tens x units + units2) x units = 5824 Subtracting the cube of the tens, 8000, the remainder, 5824, con- sists of 3 X (tens2 x units) + 3 x (tens x units2) -f units^. 5824, there- fore, is composed of two factors, units being one of them, and 3 x tens^ + 3 X tens x units + units^, being tlie other. But the greater part of this factor is 3 x tens2. By trial we divide 5824 by 3 x tens2 (1200) to find the other factor (units) , which is 4 if correct. Completing the divisor, we have 12002 + 3 x (20 + 4) + 42 = 1456, which, multiplied by the units, 4, gives the product, 5824, proving the correctness of the work. Therefore, the cube root is 20 + 4 = 24. 434. To find the cube root by the aid of blocks. Finding the cube root of a number is equivalent to find- ing the thickness of a cube, its volume being given. The following formulas illustrate the principles that underlie operations in cube root. CUBE ROOT. 321 Note. — For convenience, Z, 6, «, and v will represent length, breadth, thickness, and volume, respectively. (l)lxbxt = v. (2)v^(lxh)^t. (3) v-^(lxt) = b. (4) v---(bxt) = l 12. What is the thickness of a cube whose volume is 13824 cubic feet ? PRODUCT OF LENGTH AND BREADTH. VOLUMES. THICKN 3 X 202 = 1200 13'824 20 ft. 3 X 20 X 4 = 240 8000 4 ft. 42= 16 1456 5824 5824 24 ft. Solution. — The greatest cube of even tens contained in 13824 cu. ft. is 8000 cu. ft. (Cube A.) Its thickness, therefore, is 20 ft. Sub- tracting 8000 (A) from 13824 leaves a remainder of 5824 cu. ft., which are added in solids of equal thickness to three sides of A, as ^ Fig. 2. e '^ l^ — ^ e ^ — 1 / ^— 1 9 seen in Fig. 2. It now remains to find the thickness of the additions (6, r, d), (e, /, g), and h, which have a uniform thickness. As the solids h^ c, d form the greater part of the volume of the additions (5824 cu, ft.), and the length and breadth of each is 20 ft. (the length 322 EVOLUTION. and breadth of A), hj trial, using Formula 2, we find 5824-^(3x202) =4 ft., thickness of the additions, if correct. Knowing the thickness, which is also the breadth of e, /, g, h, we find the product of the length and breadth of e, /, gr = 3 x 20 x 4 = 240 sq. ft. ; and that of /i = 42 = 16 sq. ft. ; both of which added to 1200 sq. ft. = the product of the length and breadth of all the additions. This product, by For- mula 1, multiplied by the thickness, 4 ft, = 5824 cu. ft., proving the correctness. Therefore, The thickness of a cube whose volume is 13824 cu. ft. is 20 + 4 ft. = 24 ft. The numbers in the middle column (Ex. 12) all indicate volume : 13824 = volume of original cube. 8000 = volume of Cube A. 5824 = volume of the additions (6, c, c?), (e, /, g), and h. The numbers in the left-hand column indicate product of length and breadth : 12JJ0 = 1 xb of solids &, c, d. 240 = Z X & of solids e, /, g. 16 = 1 X b ot cube h. The numbers in the right-hand column indicate thickness : 20 ft. = thicknels of A. 4 ft. = thickness of all the additions. 24 ft. = thickness of original cube. 435. Short method. Rule for finding the cube root: Beginning at the decimal point, separate the number into periods of three figures each ; thus: 16'581'.375. Find the greatest cube in the left-hand period, and write its root at the right. Subtract the cube from the left- hand period, and bring down the next period for a dividend; thus: 16'581'.375 12 - 8 8581 CUBE KOOT. 323 To find the trial divisor, square the root already found with a cipher annexed, and multiply by 3; thus: 16'581'.376 L2 8 20 Trial divisor, 1200)8581 _20 400 3 1200 3 To find the trial figure, find how many times the trial divisor is contained in the dividend; thus: ^ 16'581'.375 |_25 8 20 Trial divisor, 1200 ) 8581 20 400 3 1200 n n To find the correction, multiply the former root by 3, an- nex the trial figure, and multiply by the trial figure; thus : 16'581'.375 125.5 8 2 3 1200 8581 325 Complete divisor, 1525 7625 65 5 325 187500 956375 Continue thus, until 3775 all the periods are ex- 191275 956375 hausted. Note 1. — When there is a remainder after all the periods are exhausted, annex decimal periods, and continue the process as far as desired. The result will be the approximate root. Note 2. — When a cipher occurs in the root, we annex two ciphers to the trial divisor, and bring down the next period. Note 3, — The right-hand decimal period must have three places. 324 EVOLUTION. 13. What is the cube root of 8.414975304? Operation. 8.414'975'304 [ 2.034 I ^ Since occurs in the root, an- nex 00 to the trial divisor, mak- ing 120000 ; bring down the next period. 120000 1809 121809 414975 365427 6362700 24376 12387076 49548304 49548304 Note. — To find the cube root of a common fraction, extract the root of each term separately. If both terms are not cubes, reduce to a decimal and then extract the root. The result will be the approxi- mate root. Find the cube root of : ^ ^ 14. 42875 '•'' 19. 17.373979 ^ "X*^ 15. 884736 * ^ ,^ 20. 450827 ^76 + 16. 4492125 ^ ^^"^ 21. 1879.080904 I'^'^H-^^ 17. 77854483 "^ ^^^ 22. 32.890033664 5,7^^i^ 18. 8.615125 zP^^ 23. 10077696 -z. %\J^ "^ 24. What is the cube root of |||f|i? -f^? xlk? 39^J, Extract the cube root to the third dermal place : 25. 14.323 i^*^^^* 27. .06324i.i.^2 29. 3-1*^ 26. 31982.4- -J^.X J'*'*' 28. .0015 -^^^ ^ 30. 7 » »'^ 31. What is the width of a cube whose solidity is 91125 cubic inches ? i M ^ 32. A cubical cistern holds 50 barrels of water. How deep is it? 33. What is the entire surface of a cube whose side is 9 ft.? 3> 34. ■ ^ ^ , 14 X 32 X 96 X 7 X 163 „ ^' ^^^"^192x21x28x55x8 = - 7. How many times will a wheel 12 ft. 4 in. in circum- ference revolve in going 10 miles ? 8. How many days must a laborer work, at $ 1.12^ a day, to pay for 6 cords of wood, at $ 3.37|^ per cord ? 9. A man was born Feb. 29, 1844, and died Mar. 15, 1880. How many birthdays did he have ? What was his age ? 10. What is the product of 12 millionths multiplied by 12 thousandths ? 464. 1. How many pickets 3 in. wide, placed 3 in. apart, will be required for a fence around a rectangular yard 4 rd. 6 ft. long, and 3 rd. 8 ft. wide ? 2. A farmer has a piece of land containing 7|f acres, fenced in the form of a rectangle, its length being twice its width. What are the dimensions of the rectangle ? 3. Oswego County has an area of 970 square miles, and a population of 71780. What is the population to the square mile ? How many acres could be given to each one of the entire population ? 4. Oswego is in longitude 76° 35' W., Albany, 73° 32' W. What is the difference in their longitude ? When it is noon by the sun in Albany, what o'clock is it in Oswego ? 5. What will be the cost of carpeting a room 18 ft. long and 12 ft. wide with Brussels carpet f yd. wide, at 85^ a yd., the strips to run lengthwise of the room, and allowing 4 in. to be turned under? 6. At $ 25 per thousand, what is the value of 16 planks, each 18 ft. long, 6 in. wide, 2^ in. thick ? DENOMINATE NUMBERS. 357 7. rind the cost of 5 pieces of timber, each 48 ft. long, 9 in. by 12 in., at $ 1.50 per hundred board ft. 8. How many board feet of lumber will be required to fence a lot 80 ft. by 40, the boards being 10 ft. by 6 in., and the fence 4 boards high ? 9. How many board feet will it take to cover the top of a tank 14 ft. long, 6 ft. wide, with.planks 2 in. thick ? 10. A man sold two bushels of strawberries as follows : to Mrs. A. he sold -^^ of the berries, to Mrs. B. f , and the remainder to Mrs. C. How many quarts did Mrs. C. buy ? 465. 1. Two telegraph stations are 18 miles, 224 rods apart. If the telegraph poles between the stations are 8 rods apart, how many poles will be needed, and how much will they cost at 50^ apiece ? 2. What is the value of a triangular piece of land, having a base of 60 chains and an altitude of 40 chains, at $ 60 per acre ? 3. How many times can a dish holding 2 qt. i pt. be filled from a jar holding 3 gal. 2 qt. 1 pt. ? How much will be left in the jar ? 4. Find the cost of carpeting a room 24 ft. long and 18 ft. wide, with carpet 27 inches wide, the strips running lengthwise of the room, the cost of the carpet being $ 1.65 a yard, and no loss in matching the figures. 5. After spending f 46|, I had | of my money left. How much had I at first? 6. A man traded 7 wagons at $ 71^ each for 84 bbl. of flour. What was the flour per barrel ? 7. What is the capacity in liters of a cistern 1.5 meters long, 9 decimeters wide, and 86 centimeters deep ? 358 TOPICAL REVIEW. 8. How many bricks 8 in. long, 4 in. wide, and 2 in. thick will it take to pave a section of street 200 ft. long, 36 ft. wide, the bricks being placed on their edges ? How much will the bricks cost at $ 7.35 per M. ? 9. How many cubic inches in a bin which contains 300 bu. of wheat ? 10. The distance around a circular park is 1|- miles. How many acres does it' contain? 466. 1. How many blocks i|- of a foot long can be cut from a board 22 ft. long ? 2. How many poor families can be supplied with ^ of a ton of coal each from 12 tons ? 3. How many pairs of tray-cloths, each containing } of a yard, can be cut from 15 yards of linen ? 4. In how many months, paying $J per week, will a debt of $ 36 be paid ? 5. J is what part of | ? 6. A 37-gallon cask is f full ; 6^ gallons being drawn off, how full will it be ? 7. If from a piece of cloth containing 96 yd. you sell 24f yd., what fractional part of the piece remains ? 8. llf bushels are what fraction of 15| bushels ? 9. fiis what part of I of J? 10. A man had 700 head of cattle. He sold at one time 50 head, at another 75 head. What fraction of the whole did he sell ? 467, 1. How many cubic feet of stone will it take to build the walls of a cellar 36 ft. long, 24 ft. wide, and 8 ft. high, outside measurement, the walls being 18 in. thick ? How much will the stone cost at $ 4.50 per cord ? DENOMINATE NUMBERS. 359 2. Find the diameter of a wheel whose circumference is 50 feet. 3. If 1 bu. 3 pk. 6 qt. of walnuts cost $ 3.10, what is the price per quart ? 4. What will be the cost of 5 gal. 3 qt. 1| pt. of maple syrup at 75 cents per gallon ? 5. Find the cost of 5362 pounds of coal at $4.50 per ton. 6. How long a time has elapsed since the first message was sent by telegraph, May 29, 1844 ? 7. How much profit will there be in buying 4 bu. 1 pk. 6 qt. of cranberries at $ 2 a bushel, and selling them at 10 cents a quart ? 8. How many days will a 6-ounce bottle of medicine last a patient who takes a teaspoonful three times a day, a teaspoon holding 60 drops or minims ? 9. Multiply 9 mi. 25 rd. 3 yd. 2 ft. by f . 10. Divide 110 mi. 149 rd. 3 yd. 2 ft. 6 in. by f 468. 1. Multiply 25 yards 2 ft. 11 in. by 16. 2. From 6 bu. 6 qt. take 3 pk. 1 qt. 1 pt. 3. What is the difference in time between June 16, 1890, and Feb. 4, 1895 ? 4. What will it cost to build the walls of a cellar that is 26 ft. long and 16 ft. wide, 6-|- ft. deep, the wall being 18 in. thick, at $ 1.50 a perch ? 5. A field is 16 ch. 10 links long and 5 ch. wide. How many acres does it contain ? 6. How many board feet in 24 joists, 10 in. by 2 in. by 16 ft., and what are they worth at $ 11 per M. ? 7. What is a pile of four-foot wood worth that is 16 ft. long and 6 ft. high, at $ 4.50 a cord ? 360 TOPICAL REVIEW. 8. How many grains in 5 lb. of butter ? 9. Eeduce 12 cwt. 80 lb. 6 oz. to the decimal of a ton. 10. Find the sum of 184|, 372^, 19|. PERCENTAGE. 469. 1. Express as % the following: .28; .065; 3.07; .004. 2. Express decimally the following: ^% ; 6^% ; 8%; 125%. 3. From a farm of 144 acres 18 acres were sold. What per cent of the farm was sold ? 4. A grocer sold eggs at 12|- cents a dozen and gained 2h(Jo. What was the cost ? 5. A man's farm cost him % 5400. His crop of potatoes yielded him in cash 8 % of the cost of the farm. What was the value of his potatoes ? 6. If a merchant pays $ .80 a yard for a roll of carpet, and because it became damaged sells it for $ .^^ a yard, what per cent does he lose ? 7. Sent my agent in St. Louis $3017.60, with which he is to purchase flour at $4.00 per bbl., after deducting his commission at 2|- per cent. How many barrels should I receive ? 8. If by selling 36840 ft. of lumber at $21.12 per M., you gain 28 per cent, what would be your gain or loss by selling it at $17 per M. ? 9. If a merchant has marked an article for sale at 50 per cent above cost, what per cent will he deduct from the asking price if he sells the article at cost ? 10. $7884.00 is to be raised by taxation in a certain school district. The taxable property of the district is $584,000. Find the rate of tax, and A's tax, whose property is assessed at $3850. PEECENTAGB. 361 470. 1. From ^ of a week take ^ of a day. 112 2. Reduce — ^ to a simple fraction. 12f 3. Define base and rate. 4. How many hundredths of anything is -^ of it ? J of it? iofit? i^ofit? 5. What is 12% of 1682? 6. Express as common fractions in their lowest terms: 25%, 62i%, 121%, 16|%. 7. A speculator bought 2160 barrels of apples, and upon opening them found 15% of them spoiled. How many barrels did he lose ? 8. A farmer sold 50 sheep, which was 25% of his whole flock. How many sheep had he at first ? 9. My income this year is $4028, which is 24% less than it was last year. How much was it last year ? 10. A commission merchant sells goods to the amount of $ 6895. What is his commission at 3% ? 471. 1. I bought two houses at $ 3500 each, and sold one at a gain of 22%, and the other at a loss of 22%. Did I gain or lose on both, and how much ? 2. If I sell for $ 16 what cost $ 20, what per cent do I lose? 3. If I buy a piano for $450, and sell it for $ 600, what per cent do I gain ? _ ' 4. Define insurance ; premium ; taxes. 5. What will be the cost of insuring a quantity of wheat valued at $8,450, at |y<, ? 6. The premium for insuring a schoolhouse, at the rate of 1^%, was $ 75. For what sum was it insured ? 362 TOPICAL REVIEW. 7. The town of B is to be taxed $3700 to build a bridge. The taxable property is valued at $1,850,000. What will be the rate of taxation, and the tax on Mr. A., whose property is valued at $ 5000 ? 8. What is the duty, at 25%, on 4796 pounds of Eussia iron, worth 10 cents a pound ? 9. What number increased by 25% of itself is 506.25 ? 10. Find the net cost of "a bill of goods amounting to $3750 at 10% discount, and 4% off for cash. 472. 1. An agent sold 4250 yd. of calico at 3 J ^ per yard. What was his commission at 2^ % ? 2. A real estate broker, who charges 4% commission, receives $224 for selling a house. What price is paid for the house ? 3. If $ 8240 is sent to an agent to cover the amount of his purchase and his commission of 3%, what is the amount of his purchase ? 4. An hotel is insured for $90,000 at 2i% for 3 years. What is the annual cost of insurance ? 5. A man's weight is 180 pounds, and he is 20% heavier than his brother. What is his brother's weight ? 6. A bill for hardware amounting in gross to $ 2537.75 is subject to discounts of 40%, 10%, and 5%. What is the net amount ? 7. If you remove the decimal point from the number 6.45, what effect does it produce upon the number ? 8. If from the same number you take the period from after the 6 and place it before the 6, what will be the effect ? 9. At $12.75 a ton what will 3265 pounds of hay cost ? 10. A tree measures 8.2 ft. in circumference. What is the diameter ? PERCENTAGE. 363 473. 1. Find i% of 12.00; ^2^% of 2000 bushels of corn ; 200% of 5 dozen eggs ; J of 1 per cent of 100 tons of coal. 2. What fraction increased by 25 per cent of itself equals fi- ? -i 3. What is the effect upon the quotient when both the dividend and the divisor are multiplied by the same number ? 4. Express as fractions in lowest terms, 81%, 2-^%, 181%. 5. Express as per cent, using the sign, .1352, ■}, 2, 3^^. 6. Express as decimals, ^^^, ^, -i%, 20%, 15^%. 7. What per cent of the number of days in February, 1896, is the number of days in January, 1896 ? 8. My house cost $ 6000, which was 400 per cent more than I paid for the lot. Find the cost of both. 9. After spending $ 14 for a suit of clothes, a man had $ 126 left. What per cent of his money did he spend ? 10. An agent purchased 8^ tons of sugar at 3^ cents per pound on 3% commission. Find the cost of the sugar, in- cluding commission. 474. 1. What is the rate of taxation on ^1000 when ^147000 is raised on $35,000,000? 2. A man selling cloth at $4.20 per yard, gained 20%. Had he sold it at $3.60 per yard, would he have gained or lost, and what per cent ? 3. If f of a mill is worth $10,000, what is i of the remainder worth ? 4. Bought a horse for $160 J, and sold it for |- of its cost. How much did I lose ? 364 TOPICAL REVIEW. 5. Define least common multiple; improper fraction; prime factor. 6. Simplify?!. 7. Find the cost of 10 sticks of timber, each 16 feet long, 14 inches wide, and 10 inches thick, at $ 16.50 per M., board measure. 8. How many gallons will a cistern hold that is 12 ft. long, 8 ft. wide, and 6 ft. deep ? 9. If 9| yards of cloth are worth $24,375, what is the value of 16|^ yards at the same rate ? 10. Name the unit of weight in the metric system, and give the table in which that unit occurs. 475. 1. I spend 65 per cent of my salary, but am able to save $ 980. How much do I spend ? 2. How much must I send my agent, that he may buy at 1^ per cent commission, 400 bbl. flour at $ 6.75 per bbl. ? 3. Given the amount and percentage, write the formula for finding each of the other terms. 4. What are like numbers ? Unlike numbers ? 5. Write an abstract number. Give the definition of abstract number. 6. Write in words 2300406.000960. 7. What kind of number is 4.6 bushels ? 8. A father divided his property as follows : to his son John he gave ^, to his daughter Susan ^, to his wife i, and the rest, which was $ 13,000, to endow a school. What was the value of his estate ? PERCENTAGE. 365 9. I own a house that cost me $3000. It cost me to insure it for 3 years $24. The average yearly cost of repairs is $50. The average yearly tax is 2% of the cost. I can get 5% per annum for the $3000 invested. The house will last 60 years. I receive in rent for the house $300 per annum. If these conditions are constant, how much will I gain or lose in 60 years ? 10. A father is 39 years old and his daughter 13. What per cent of the father's age is the daughter's ? 476. 1. Write these per cents as hundredths : 2%, 6-J-%, 20%, 121%. 2. How many per cent of a number is 0.20? 0.75? ,12i? 1.40? 3. What fractions of a number (in lowest terms) are these per cents : 16|% ? 75% ? 331% ? 100% ? and 175% ? 4. Express as hundredths and as common fractions: i% ; f % ; \% ; 1% ; and 3-V%. 5. Trom a stack of hay 7 T. 11 cwt. were sold, which was 75^% of the whole. How much did the stack con- tain before the sale ? 6. A lawyer collected 65% of a debt of $1260, and charged 5% commission on the sum collected. What did the creditor receive ? 7. If a hat that cost $5 be sold for $9, what is the gain per cent ? 8. How many days from Sept. 16, 1892, to Feb. 12, 1894 ? 9. 874 is 33 J % less than what number ? 10. Eequired the cu. feet of a box 6 ft. 6 in. by 4 ft. 9 in. by 3 ft. 3 in. 366 TOPICAL BEVIEW. 477. 1. "Write the following numbers and add: six thou- sand sixteen and sixty-five thousandths, four hundred one thousand forty-one and one-tenth, six hundred one and nine hundredths, ten thousand one hundred seventeen and nine hundred three thousandths, forty-nine hundred forty-nine and nine-tenths. 2. Write in words 83.4937007^^, 1001001.01, 90019^^. 3. Find the number of which 160 is f. 4. Find the exact number of days from July 4, 1893, to to-day. 5. Multiply 7 lb. 8 oz. 15 pwt. by 15. g -. 18 X 963 X 44 X 27 X 2800 ^ ^ 63 X 88 X 105 X 1926 x 45 " 7. Define commission, also brokerage; and state on what sum, or value, both are computed. 8. Express decimally 274 and y^. Find their product as decimals, and as common fractions, expressing both answers decimally. 9. Fruit was sold at 12|-)^ per quart, which was 200 per cent of its cost. What was the cost per bushel, and what was the rate per cent of profit ? 10. An agent sold 840 bu. of grain at 60^ per bushel. His commission was $15.12. Find the rate of commission. 478. 1. A man owes you a debt of $2160, which he de- clines to pay. Your lawyer succeeds in collecting 70 per cent of the debt, and charges 5 per cent commission for his services. What sum do you receive ? 2. A manufacturer sent $ 1287.50 to a commission mer- chant who charges 3 per cent commission, instructing him to purchase wool at $ 0.33^ per pound. How many pounds of wool will be received ? PERCENTAGE. 367 3. A farm was sold for $ 8000, which was 20 per cent less than its real value. If it had sold at $ 12000, what per cent above its real value would it have brought ? 4. A commission merchant sold for a farmer 6000 lb. of pork at S^^ per pound. He charged 1^% commission for selling, and paid $18.81 for freight. How many feet of pine boards at $ 25 per 1000 ft. could he purchase with the proceeds of the pork, after deducting 1 per cent commis- sion for buying ? 5. Keduce to simple fraction in lowest terms : X 6 6. What per cent of 3 is | ? Of f is f ? Of 80 is 50 ? 7. A drover sold 250 sheep for $1150, which was 15% more than they cost. What was the cost per head of the sheep ? 8. If 20% be lost on a ton of rye straw sold for $ 19.20, what is the cost of the straw per ton ? 9. How many per cent of a number is 0.15 ? 0.06^ ? 0.50? 2.25? 10. What common fraction of a number in its lowest terms is 20% ? 50% ? 6i% ? 66|% ? 160% ? 479. 1. A man sold $8400 worth of merchandise, and had 30% of his stock left. What was his entire stock worth ? 2. A nferchant sold goods at 20% and 5% off, and still made 20% on the cost. What was the cost price of a book that was marked $ 1.00 ? 3. Bought 1000 pounds of butter at 18 ^, and sent it to an agent who sold it at 21)^ on a 5% commission. What was my rate of gain ? 368 TOPICAL REVIEW. 4. Mr. Brown has a flock of 940 sheep in three fields. In the first are 20% of the entire flock, in the second 40%, and the remainder in the third. How many sheep are there in each field? 5. A lady has a salary of ^ 825 a year. She spends 20% of it for board, 35% of it for other expenses, and saves the remainder. What sum does she save ? 6. What per cent of a leap year is the time from Wash- ington's Birthday to the Fourth of July ? 7. The Barber Asphalt Company engaged to pave a street 5 miles long at $55,000 a mile. If the actual cost is $ 130 per rod, what is the gain per cent ? 8. A commission merchant charges 1^% for selling, and 2}% for guaranteeing the payment of the money. His commission on a certain transaction amounted to $384.75. Required the amount of the sale. 9. I bought 1100 tons of coal at $3J per ton. I sold 40% of it at a gain of 50%, 40% of the remainder at a gain of 35%, and lost 10% on the rest. What was my actual gain ? 10. An article bought at 18% below the asking price is sold for the asking price. What is the gain per cent ? INTEREST AND DISCOUNT. 480. 1. Find the amount of $975 for 1 year, 4 months, and 12 days, at 6 per cent interest. 2. Find the interest on $128.45 from March 2,1895, to Dec. 14, 1895, at 6 per cent. 3. A pile of wood 256 feet long, 4 feet wide, and 5 feet high is sold for $160. What is the price per cord ? 4. Define per cent ; interest ; proper fraction. 5. State the difference between a prime and a composite number. INTEREST AND DISCOUNT. 869 '6. Find the cost of 6 gal. 3 qt. and 1 pt. of syrup at 46 cents per gallon. 7. 1521 is how many times 13 ? 8. What is the interest on $1200 for 2 yr. 3 mo. 18 da. at6%? The amount? 9. What is the interest on $1240 from March 3 to Aug. 28, at 6% ? 10. Write the United States rule for computing the amount due on a note when partial payments have been made. 481. 1. In what time will $3960 earn $770 at 5%, simple interest ? 2. If $675, at simple interest, gain $172.80 in 3 years, 2 months, 12 days, what is the rate of interest ? 3. When interest, time, and rate are given, how may the principal be found ? 4. Define the present worth and true discount of a debt. Define compound interest, and make and solve an example to illustrate your definition. 5. A merchant sells goods amounting to $6784.00 on a year's credit. If money is worth 8%, what sum should he accept in payment of the bill 6 months before it becomes due? 6. Write a negotiable promissory note signed by James Fox for $ 875.60 due 90 days from June 19, payable to your- self, at a bank. Name (a) the payed ; (6) the drawer ; (c) the date when the note matures (becomes due). What words in the note make it negotiable ? What does negotiable mean ? 7. If you should sell the note (Ex. 6) to Mr. F. P. Weaver, what indorsement must you write upon it? Where should indorsements be written ? 370 TOPICAL REVIEW. 8. If the note is not paid until Sept. 15, 1895, how much interest will then be due on it? 9. A farmer expended $5460 in improvements on his farm, which was 24^ more than f of the cost of the farm. Find the cost of the farm. 10. Principal, interest, and time being given, how is the rate found ? 482. 1. Find the amount of $496.85 for 2 years, 4 months, and 15 days at 4 per cent. 2. How long will it take $750 at 6 per cent to gain $67.50 interest? 3. A dealer bought 65 lawn-mowers at $4.25 each, and sold them at $3.87J each. What per cent did he lose ? 4. If a cellar is 38 ft. long and 28 ft. wide inside the wall, and the wall is 8 ft. high and 18 in. thick, how many cubic yards of masonry does the wall contain ? 5. What per cent of a number equals f of the number ? What part of a number equals 33|^ per cent of it ? 6. Write decimally, 6% ; one hundred six per cent. 7. A town 6 miles long and 4i miles wide is eqaal to how many farms of 80 acres each ? 8. What number must be subtracted from four hundred sixty-seven thousand six hundred thirty-three to make it exactly divisible by 758 ? 9. Find the amount of $535.20 for 2 yr. 4 mo. 18 da. at 5 per cent, simple interest. 10. Give formula or rule for finding the base when rate per cent and difference are given. Form and write such a problem. INTEBEST AND DISCOUNT. 371 483. 1. Find the interest of $263.75 for 1 yr., 3 mo. 16 da. at 5%. 2. Make a 30-day bank note dated Jan. 20, 1896, for $600, payable at some bank. Find the date of maturity, the discount, and proceeds if discounted on the date of the note. (Make the note on a separate piece of paper, and have it properly indorsed.) 3. What is the present worth of a debt of $ 500 due in 1 yr. 6 mo., money being worth 6% ? 4. In what time will $ 600 gain $ 30 interest at 6% ? 5. What will $300 amount to in 4 years compounded annually at 4% ? 6. An agent says he will insure your house for 3 years at 65. What does he mean by " at 65 " ? 7. Define interest ; principal ; usury ; compound interest. 8. Find the amount of $684.50 for 3 yr. 4 mo. at 7%. 9. Compute the interest of $ 1250 for 2 yr. 5 mo. 12 da. by the six per cent method. 10. What is the interest on a note for $515.62, dated March 1, 1885, and payable July 16, 1888? 484. 1. A note for $710.50, with interest after 3 mo. at 8%, was given Jan. 1, 1884, and paid Aug. 13, 1886. What was' the amount due ? 2. What sum of money will gain $173.97 in 4 yr. 4 mo. at 6% ? 3. What is the legal rate of interest in this State ? 4. Find the exact interest of $950 at 5% for 98 days. 5. What principal will amount to $ 1531.50 in 1 yr. 3 mo. 6 da. at 6% ? 6. At what rate will $1500 amount to $1684.50 in 2 years, 18 days ? 372 TOPICAL REVIEW. 7. In what time will $ 840 gain $ 78.12 at 6% ? 8. How long will it take any sum of money to double itself at 4% ? 9. Find the compound interest of $460 for 1 yr. 5 mo. 24 da. at 6% interest, payable semi-annually. 10. If |- of an acre of land costs $ 15, what will 10|- acres cost? 485. 1. Name four different forms of reduction of com- mon fractions. Illustrate one of them to show that the value of the fraction remains unchanged. 2. Define Simple Interest, True Discount, and Bank Discount. How does bank discount differ from interest? How does it differ from true discount ? 3. Define cancellation, and state the principle of arith- metic that authorizes its use. 4. Find the amount of $ 575.871 at 5 per cent, simple interest, from Aug. 5, 1883, to March 17, 1885. 5. What principal will earn $ 71.68 in 2 years, 4 months, at 6 per cent, simple interest. 6. At what rate, simple interest, will $ 175 amount to $ 203.35 in 3 yr. 7 mo. 6 days ? 7. In what time will f 4260 earn $873.30, at 6 per cent ? 8. A 60-day note for $ 610.25, dated June 12, 1889, was discounted in bank, July 1, at 6 per cent. Find the term of discount, discount, and proceeds. 9. Having purchased a horse for $ 125, you wish to borrow that amount at bank for 6 mo. Write your own note, indorsed by your parent as security, for the sum Avhich, discounted to-day, will give $ 125 as proceeds of the note. INTEREST AND DISCOUNT. 373 10. A stock of goods was owned by three parties. A owned |, B |, and C the remainder. The goods were sold at a profit of $ 4260. What was each one's share of the gain ? 486. 1. A horse is offered me for $350 cash, or for $ 382.50 to be paid in 4 mo. What can I save by paying cash, the rate of interest being 6% ? 2. Which is the more profitable, and how much, money being worth 5%, to buy a house for $5940 on 2 years' credit, or for $ 5219.30 on 6 months' credit ? 3. A note dated June 20, 1893, and bearing interest at 6 per cent, was paid Aug. 15, 1895. The face of the note being $ 68.45, what was the amount paid ? 4. Bought 150 front feet of land at f 40 per front foot, paid $ 116 city taxes, $ 32 county taxes, and $ 320 local taxes. At the end of two years I sold for $ 60 per front foot. Reckoning interest at 6% on the purchase price, did I gain or lose by the transaction, and how much ? 5. A man wishes to pay me $ 3252.56. Not having the money, he borrows it from a bank by giving his note for 48 days at 4%. For what sum does he draw the note? No grace. 6. $545.50 Bufealo, N.Y., Apr. 2, 1896 Sixtt/ days after date, I promise to pay^^,,,,,,,^^^^ Henry Hamilton or order, Five hundred forty - jive and ^ Dollars. Value received. Chas. C. Trowbridge, This note was discounted May 4, 1896. Find the proceeds. 374 TOPICAL REVIEW. 7. Required the simple interest and amount of $ 7231.289 for 3 yr. 8 mo. 15 days at 8%. 8. Face of a note $750. Time 60 da. Eate 6%. To find proceeds. 9. Write the following in a note properly, and find the maturity and proceeds : Face, $ 600 ; date, April 3, 1896 ; due in 90 days; discounted at bank, May 20, 1896, at 6%, with grace. lO. $9000 Saratoga Springs, N.Y., Oct. 3, 1895 Nine months after date, / promise to pay to the order of G-ates ^ 0(9._______iV"me thousand Dollars, at the First National Bank. Value received. S. B. Graves. Find the proceeds, if discounted at 6%, Dec. 3, 1895. STOCKS AND AVERAGE OF PAYMENTS. 487. 1. How many shares of stock at 80 can I buy for $ 2550. 2. I sold two houses for $ 2400 each. On one I gained 10%, on the other I lost 10%. How much did both cost me? Did I gain or lose in the whole trade, and how much ? 3. Find the cost of 40 shares of American Express Co. stock at 105|, brokerage i%. 4. A mining company declares a dividend of 8% per annum on its stock. What is the nominal value of a man's shares who gets $ 864 as his semi-annual dividend ? 5. If the stock of a railway company sells at 5% above par, what will 25 shares cost ? 6. If I invest $21,008 in 5% bonds at 104, what will be my annual income ? STOCKS AND AVERAGE OF PAYMENTS. 375 7. Sugar bought at 5 cents a pound was sold for 6^ cents. What per cent was gained ? 8. What sum invested in 4 per cent stock will yield an annual income of $ 320, if the stock is purchased at par ? 9. What would be the investment, if the stock is worth 15 per cent above par. 10. A man invested his money in 6% railroad stocks, and received $ 300 semi-annually. What was the sum invested ? 488. 1. What sum must be invested in stocks bearing 6^ per cent interest, at 105 per cent, to produce an annual income of $ 1000 ? Solve by cancellation. 2. Define brokerage, certificate of stock, par value, pre- mium (as used in stocks and investments). What are bonds ? Name some of the different classes of bonds. 3. What income will be realized from investing $4190.63 in 5% stock, purchased at 7% discount, if I pay ■J-% for brokerage? 4. What is the value of 31 shares of $ 500 each, sold at a premium of 2^^-^% ? 5. Which is more profitable, to buy 8% bonds at 25% premium, or 6% bonds at 10% discount ? 6. A owes B $3000, due as follows: June 15, $1500; Sept. 10, $400; Nov. 1, $500; Dec. 15, $600. B accepts in settlement Oct. 26 a note for 9 mouths, bearing interest at 6% for the amount of the debt, with 6% interest due him at that date. Find the face of the note. 7. On Jan. 1, 1895, a merchant gave three notes : one for $ 500, payable in 30 days ; one for $ 400, payable in 60 days ; and one for $ 600, payable in 90 days. What is the average term of credit, and what the equated time of pay- ment? 376 TOPICAL REVIEW. 8. E. R. Smith owes J. D. Wilson $2500, due Oct. 12, 1896. If Mr. Smith pays $ 500 Aug. 10, and $ 1000 Sept. 25f when should the balance be paid ? 9. A speculator bought N. Y. C. stock at 98^, and sold it at 97f, and lost $187.50. How many shares did he handle ? 10. Had he retained his stock until a quarterly divi- dend was declared, his dividend would have been $312.50. What was the annual rate of dividend ? 489. 1. State why securities fluctuate in value. 2. Name a corporation. 3. What does a stockholder hold to show that he has stock in a company ? 4. On what does the income from his stock depend ? 5. Why does a corporation issue bonds ? 6. Pind the present worth and true discount of $ 300, due in 10 months, at 6%. 7. Find the bank discount and proceeds of a note of $730, due in 3 months, at 6%. 8. What is the face of a note at 2 months and 18 days, which yields $ 2961 when discounted at a New York bank ? 9. A person owning | of a piece of property, sold 20% of his share. What part did he then own ? 10. At what price should 4|-% bonds be bought to make the income from the investment equivalent to that from 3% bonds at par ? PROPORTION AND PARTNERSHIP. 490. 1. What is ratio? 2. Eead the following : 3 : 15. What does it equal ? 3. What is each of the numbers in the above expression called? PEOPORTION AND PARTNERSHIP. 377 4. What is a proportion ? 5. Is the following expression a proportion ? Explain why. 9 : 12 : : 16 : 24. 6. 24 : ( ) =56:7. Find the omitted term. 7. If 8 men can do a piece of work in 10 days, in how many days can 12 men do it ? 8. If 3 men in 12 days of 10 hours each can build a wall 100 feet long, 14 feet high, and 3 feet thick, how long will it take 4 men working 8 hours a day to build a wall 200 feet long, 16 feet high, and 4 feet thick ? 9. If it takes 5 men 4 hr. 24 min. to manufacture 400 boxes, how much time will 8 men require to perform the same work. 10. If -f- of an acre of land cost $ 15, what will 10|- acres cost ? 491. 1. 50 men in 7 da. at 12 hours a day dig a cellar. How many men will be required to dig a similar cellar in 21 J da. of 8 hr. each ? 2. A and B enter into partnership, A with $1800 and B with 1^900. After 8 mo. B adds $300 to his capital. Divide a profit of $840 between them at the end of the year. 3. A bankrupt owes A $350, B $680.50, C $65, D $500, E $980.50; his property nets $1648.64. How much does each creditor receive ? How much does he pay on a dollar ? 4. What is the ratio of 7 to 8 ? Of 2i to 3i ? Of $ 9 to $6? 6. If 20 men can mow a field in 6 days, in how many days will 30 men mow it ? 6. If 6 horses eat 8 bu. 14 qt. of oats in 9 days, at the same rate how long will 66 bu. 30 qt. last 17 horses ? 378 TOPICAL REVIEW. 7. A and B hired a pasture for $ 40 for the season. A put in 9 cows for 4 mo., and B put in 8 cows for 8 mo. Other conditions being the same, what should each pay ? 8. In what time will $ 10,000 yield $ 1200 interest at 8%. Solve by proportion. 9. If the antecedent is | of ^ of ^%, and the ratio is f of |4- of i|, what is the consequent ? 10. Required the ratio of 6 J cu. ft. to 11| cu. ft. 492. 1. A, B, and C entered into partnership. A put in $ 600 for 8 mo., B | 800 for 7 mo., C $ 1500 for 4 mo. They gained $ 820. What was each one's share of the gain ? 2. A, B, and C found a gold-mine, and after developing it sold it for $64000. They agreed to divide the money according to the time each had worked. A had worked 37 days, B 46 days, and C 39 days; for extra services B is to receive $ 1800, and C $ 1200 additional. How much does each receive ? 3. Three men, A, B, and C, enter into partnership. Out of a gain of $ 1200, C takes $ 500 and B $ 400. A's in- vestment is $ 4500. Find B's and C's investment. 4. Divide $ 450 among three people in the ratio of 3, 4, and 8. 5. Three persons bought a block for $ 21000, of which A paid $ 9000, B $ 8000, and C the remainder. They rented it for $ 1400 a year. What was each man's share of the rent ? 6. Forster, Stull, and Furlong made 8000 pairs of bi- cycle pedals in 1895, which they sold for $ 1.60 per pair. The pedals cost them $ 1.15 per pair. If Mr. Forster put in $ 1000 Jan. 1, Mr. Stull $ 1200 April 1, and Mr. Furlong $ 900 May 1, what would be each one's share of the gain after drawing out the original investment ? INVOLUTION AND EVOLUTION. 379 7. Four men purchased a city block for $ 36,000. The first contributed $20,000, the second $7000, the third $ 4000, and the fourth $ 5000. They sold the land at an advance of 50% on the purchase price. How much was each man's share of the gain ? 8. A, B, and C form a partnership in which A is to furnish no capital, but give his whole time to the business, and have J the profits. B furnishes $ 10,000, and C $ 15,000. Their net profit at the end of a year is $ 8000. What is each partner's share ? 9. A, B, and C gain in business together respectively $ 700, $ 1000, and $ 1500. What was the investment of each if their joint capital was $ 16,000 ? 10. Smith, Brown, and Jones gain in trade $9400. Smith furnished $ 10,000 for 5 months. Brown $ 9000 for 6 months, Jones $ 7000 for 1 year. Apportion the gain. INVOLUTION AND EVOLUTION. 493. 1. Define involution; evolution; a square; cube root. 2. Find the square of 6f ; of 2.35. 3. Find the third power of 123. 4. Find the square root of the fraction fffj. 5. What is the distance around a square field which contains 40 acres ? 6. A man has 640 acres of land. How much more will it cost to enclose it with a fence at $ 4 a rod, in a rectangu- lar form 512 rods long and 200 rods wide, than it would if in the form of a square ? 7. What is the length of one side of a cube which con- tains 8120601 cubic inches ? 8. Find the entire surface of a cube whose volume is 42 cu. ft. 1512 cu. in. 380 TOPICAL REVIEW. 9. The edge of a cube is 42 inches. Find the length of the edge of another cube 4 times as large. 10. If 16 cords of wood be piled in the form of a cube, what will be the length of one of its edges ? 494. 1. What are the length and breadth of a rectangu- lar field which contains 60 acres, the length of which is three times its breadth ? 2. A rectangular farm of 300 A. is 7J times as long as it is wide. How many miles of fence will enclose it ? 3. A bird is 15 feet above a monument 80 ft. high. A boy is 145 ft. from the bird. How far is the boy from the base of the monument ? 4. How far is it between the extreme corners of a box 10 ft. square and 6 ft. deep ? 5. Eind how many acres in a lot in the form of a right- angled triangle whose hypothenuse is 50 rd. and whose base is 40 rd. 6. Find the diagonal of a square piece of land equal in area to a rectangular piece whose dimensions are 80 rd. by 20 rd. 7. Wishing to know the height of a church steeple, I find it casts a shadow 165 ft. I also find that a 10-ft. pole, placed perpendicularly, casts a shadow 12^ ft. What is the height of the steeple ? 8. A house is 36 ft. wide, and the ridge of the roof is ^ 12 ft. above the plates. How long are the rafters ? 9. A steamer goes due north at the rate of 12 miles an hour, and another goes due east at the rate of 15 miles an hour. How far apart will they be at the end of 8 hours ? 10. If a pineapple 5 in. in diameter costs 20^, what should be the cost of a pineapple of similar shape 6 in. in diameter ? MISCELLANEOUS. 495. 1. The sum of two numbers is 2120, and their difference 938. What is each number? 2. J. & E. Eoss, New York, bought of A. L. Covert & Co., Philadelphia, the following articles, June 20, 1881 : 15 Nichols's Geography at f 0.65; 12 Meiklejohn's Literature at 1 0.80 ; 25 Bowser's Geometry at $ 0.75 ; 15 Hawthorne & Lemmon's Literature at $ 1.12 ; 10 Thomas's United States History at $ 1.00. They paid $ 25 in cash, and returned books to the amount of $ 10. Make out a bill showing the entire statement. 3. A city contains 22,000 inhabitants. If each inhab- itant should contribute one cent per week for fifty-two weeks towards the erection of a soldiers' monument, how expensive a monument could be built at the end of the year ? 4. The State of New York has 7746 miles of railroad, which cost $ 588,672,762. Find the average cost per mile. 5. The sum of three numbers is 96: the least is 4|-, and the greatest 37|. Find the other number and the product of the three numbers. 6. $9,000,000 has recently been appropriated for improv- ing the Erie Canal. If it is 352 miles long, how many dol- lars may be expended on each mile ? 381 382 MISCELLANEOUS. » 7. Find the least common multiple of 24, 60, 75, 120. 8. What is the smallest sum of money with which I can purchase oxen at $ 30 each, cows at $ 60 each, or horses at $ 80 each ? 9. Find the difference between the greatest common divisor and the least common multiple of 81, 45, 108, and 135. 10. What is the greatest number that will exactly divide 3640, 12750, and 18755 ? ^ 496. 1. If the ties on the N. Y. C. & H. R.R. are If ft. apart from centre to centre, how many are there from New York to Buffalo, a distance of 450 miles ? 2. If the Empire State express has an average rate of 62 miles an hour, how many hours and minutes will it take to run from Syracuse to Albany, a distance of 150 miles ? 3. Multiply 7f by 17|f . 4. E. C. Stearns & Co. sell 24 bicycles at $ 62|- apiece. What do they bring ? 5. How many times does a bicycle wheel 9J ft. in cir- cumference revolve in going 3 miles, there being 5280 ft. in a mile ? 6. Multiply i^ + i by 121 ^ 2 "T" "3" ^7. A and B can build a house in 30 days : B can do the work alone in 45 days. In how many days can A do it alone ? 8. Write a complex fraction, whose numerator shall be a simple fraction, and its denominator compound. 9. A drover bought 375 sheep at $4|- per head. He sold 200 of them at a loss of 20 cents per head, and gained enough on the rest to balance the loss. What did he receive per head for the rest ? PROBLEMS. 383 / 10. A can do a piece of work in 5 days ; B can do the same work in 8 days. In what time can they do it working together ? 497. 1. A boy paid for a book $.70, which was f of his money. The remainder he spent for marbles at 2^ cents apiece. How much money had he at first, and how many marbles did he buy ? 2. At a school examination ^ of the pupils passed, and 250 pupils failed. How many pupils were examined, and how many passed ? n 3. ^ of a number diminished by -| of it is equal to 5. What is the number ? 4. ^4_. of 1743 is j\\ of what number ? 5. A man after giving ^, \, and -|- of his money in charity had $ 10000 left. How much had he at first ? 6. Four persons own a ship. A owns J of it, B -J- of the remainder, C J of what then remained, and T) the remainder, which is worth $3000. What is the value of the ship ? 7. If I of a number be divided by 4, and -|- of \ of the number be taken from the quotient, the remainder will be 6. What is J of the number ? 8. One person can do a piece of work in 6 days, another can work twice as fast. How long will it take them to do the work together ? 9. A boy was asked how many fish he had caught. He said that the difference between ^ and -| the number was six. How many had he ? 10. A, B, and C can do a piece of work in 5 da. A can do it alone in 12 da., C can do it in 15 da. In what time can B do it ? 384 MISCELLANEOUS. 498. 1. What will it cost at $1.75 a yard to carpet a floor 18 ft. long, 14 ft. wide, with carpet | yd. wide ? 2. How many yards of carpeting 27 inches wide will be required for a room 30 ft. long, 24 ft. wide, if the strips run crosswise, and 6 inches be allowed for matching ? 3. What fraction of a great gross is 3 gross, 5 doz., If units ? 4. At $ .27 per square yard, find the cost of plastering a room 30 ft. by 24 ft. by 12 ft. high, allowing for a base- board 1 foot high, two doors 9 ft. by 3 ft., and 5 windows 6 ft. by 3 ft. 5. Reduce 5 cd. ft. 9| cu. ft. to the fraction of a cord. 6. Reduce 33 gal. 3 qt. 1 pt. 1^-^ gi. to the fraction of a hhd. 7. How much tin will be required to make a pail and cover, the pail to be 6 inches in depth and 7 inches in diameter, and the rim of the cover to be 1 inch deep? 8. At $ 16.50 per M., what will be the cost of 12 sticks of timber, each 14 ft. long, 10 in. wide, and 8 in. thick ? A 9. How many board feet in a plank 16 ft. long, 15 in. wide at one end and 10 in. wide at the other end, and 3 in. thick ? 10. The longitude of New York is 74° 0'3" W., and that of San Francisco 122° 23' W. When it is 1 p.m. at New York, what is the time at San Francisco ? 499. 1. The longitude of Syracuse, N.Y., is 76° 9' 16" W., and that of Berlin, Germany, is 13° 23' 44" E. When it is noon in Berlin, what is the time at Syracuse ? 2. The Oswego River is 24 miles long, and descends 120 feet in that distance. What is the average descent per mile ? XTNITERSIXr •S££AUF025^ PROBLE>fS. 385 3. Add I A., ^ sq. rd., J sq. yd., | sq. ft. 4. Find the cost of 4 T. 7 cwt. 40 lb. of hay at $ 12 per ton. 5. From a cask containing 44 gal. 2 qt. 1 pt. of vine- gar, 8 gal. 3 qt. leaked out. What decimal of the original contents remained? 6. Find the number of square inches in the surface of a block 2 ft. long, 18 in. wide, and 10 in. high. 7. The sun rose in the latitude of New YoVk, April 1, 1896, at 5 o'clock and 43 minutes, and set at 6 o'clock and 25 minutes. It rose April 30 at 4 o'clock and 59 minutes, and set at 6 o'clock and 55 minutes. How much longer was the thirtieth day than the first ? 8. How long and wide must a granary be to hold 4000 bushels of grain, if it is 8 ft. high, and the grain to be placed in bins 6 ft. back on each side of an aisle 4 feet wide ? • 9. A cubic ft. of water weighs 62-1- pounds. How many barrels in a cistern of water that weighs 6 T. 5 cwt. ? 10. Find the cost of 1 bu. 1 pk. 1 qt. and 1 pt. of chest- nuts at 5 ^ per quart. 500. 1. At what rate per cent will $ 2500 gain $ 625 in 3 years, 4 months ? 2. A merchant buys goods at $ 1.20 a yard, and, after keeping them 6 mo., sells them at $ 1.35. What is his rate of gain ? 3. A man buys oranges at 1 ^ each, and sells them at 18 cents a dozen. What is his gain per cent ? 4. Find the amount on $ 836.22 from Feb. 19, 1895, to June 3, 1896, at 6%. 5. 3200 votes are cast for two men; one has a majority of 374. How many votes did each receive ? 386 MISCELLANEOUS. 6. A man borrowed $756.12, June 28, 1872. What must he pay to cancel the debt July 11, 1872, at 6% ? 7. A commission merchant in Minneapolis received $ 6150, with directions to purchase flour. His terms were 2^% on the amount purchased. How many barrels of flour at $ 3 a barrel can he ship to the sender of the money ? 8. A merchant sells goods at an advance of 20%, but loses 5% of his sales by bad debts. What % does he gain ? 9. A bought a carriage at 20% discount with 10% and 5% off, and sold it at the list price. What % profit did he make ? 10. An agent sold some Western land, and paid to the former owner $ 7531.30, retaining f 153.70 as commission. What rate did he charge ? 501. 1. A district schoolhouse cost $8010; the valua- tion of the property of the district is $392,375, and the number of polls assessed at $ 1.25 each is 130. What is the rate of tax, and what was A's tax, who paid for 4 polls, the valuation of his property being $ 6000 ? 2. What sum of money placed on interest at 6% will amount to $ 1567.85 in 1 year, 3 months ? 3. Sold wheat at 72 cents per bushel, and thereby lost 10% of the cost. What was the cost per bushel ? 4. What will be the net cost of stationery billed at $ 850, if the discount is 20% and 10% off ? 5. A house worth $ 7200 is insured for -| of its value, at the rate of 60 cents on $ 100. Find the premium. 6. A man sold a house for $ 4200, which. was 20% more than it cost him. What did it cost ? PROBLEMS. 387 « 7. On a bill of goods listed at $645, choice is given between discounts of 20%, 10%, and 5% off, or a direct V discount of 35 % off. Which is better, and how much ? 8. If a merchant gains 16|% by selling cloth at $1.40 per yard, find his gain on a sale amounting to $ 32. 9. I owe B a bill of $ 1980. If I borrow the money from a bank, what must be the face of a note, due in 60 days without interest, which I must give to the bank, that I may receive the amount necessary to pay him, discount at 6% ? 10. A man sells his house for $8000, and receives in payment a note for 90 days. After 30 days he has the note discounted at a bank at 6%. What does he receive for it ? 502. 1. I was offered $ 160 cash for my buggy, or a note of $ 165 payable in 90 days. I took the note, and dis- counted it at a bank at 5%. Did I gain or lose, and how much ? 2. What is the difference between the true and bank discount on $ 1250 for 90 days at 6% ? 3. If John lends James $ 300 for 4 months, how long ought James to lend John $ 800 to equal the favor ? 4. I have a note of $ 1225, due in 48 days. Needing the money immediately, I get it discounted at a bank at 6 % . How kauch shall I receive, and how much will the bank take ?% No grace. 5. Three men hire a pasture for $ 60. A put in 4 cows "w for 11 weeks, B 5 cows for 12 weeks, and C 8 cows for 5 A weeks. What ought each to pay ? 6. If a man 5 ft. 10 in. high casts a shadow 4 ft. 6 in. long, what is the height of a tree which casts a shadow 85 ft. long at the same time ? X 388 MISCELLANEOUS. 7. Give the inverse ratio of -^ to ^- 8. Required the ratio of £ 21 15s. to £ 6 IBs. 9. A, B, and C entered business with a certain capital, Jan. 1, 1894. Jan. 1, 1896, they find the business to be worth $ 7000, which is a gain of 40% on the original capital. A's share of the gain is 50%, B's share 30%, and C's share 20%. What amount did each invest ? 10. What did each gain in Example 9 ? 503. 1. If 4 barrels of flour will last three persons for 1 year, how many barrels will be required to last 10 persons 10 months ? 2. The shadow of a flag-staff at a certain time of day was 64 feet in length. A line stretched from the top of the flag-staff to the extremity of the shadow measured 150 feet. Required the height of the staff. 3. Messrs. Stevens, Jones, & Payne form a partnership, placing into their business $ 350, $ 450, $ 1500 respectively. They make $ 570 the first year. What share of the profits should each receive ? 4. By selling 3% stock at par, and buying 4% stock at 110, a man increases his income $ 105 a year. How many shares of the 3 % stock does he sell ? 5. A, B, and enter into a partnership. A furnished $ 1200 for 8 mo., B furnished $ 1600 for 9 mo., and C fur- nished $ 1000 for a year. They lose $ 560. What is each man's loss ? 6. What is the length of a walk laid diagonally through a park which measures 60 rods on one street and 80 rods on another ? 7. What will be the difference in ratio of income between 6% stock bought at 120 and 4% bought at 95 ? PROBLEMS. 389 8. A fatKer dying left to his family a certain sum of money, of which the wife received $ 8000, his daughter $ 4000, and each of two sons $ 6000. What part of the whole did. each receive ? 9. If sugar costs 5^ cents per pound and coffee 33 cents per pound, what is the ratio of the cost of the sugar to that of the coffee ? 10. At $.50 per rod, how much will it cost to enclose a field of 80 acres, that is twice as long as it is wide ? 504. 1. If the sale of coal at $ .75 per ton above cost yields a profit of 18|%, how much must the seller add to this price to make a profit of 40% ? 2. At what price must a 4% stock be purchased to yield 5% on the investment? 3. If a pile of wood 32 ft. long, 4 ft. wide, and 4 ft. high, costs $ 32.50, what will be the cost of a pile 64*ft. long, 8 ft. wide, and 8 ft. high ? 4. If 10 men, working 10 hr. a day for 30 da., can build a fence 200 rd. long, how many men, working 6 hr. a day for 10 da., can build 92 rd. of the same kind of fence ? 5. The smaller of two numbers is 36, and one-half of the ratio between it and the larger is 2. What is the larger number? 6. What number has the same ratio to 5 that ^ has to J ? 7. Find the mean proportional between 16 and 36. Between -^^ and 1. 8. What income on his investment will a man realize if he purchases 4% stock at 125 ? 9. If A's capital is $ 3000, and B's $ 2000, how much more should B invest at the end of 6 months that he may share equally with A at the end of the year ? 10. What is the rate per cent of a tax for $52.88J on property assessed at $ 3525.50 ? 390 MISCELLANEOUS. 505. 1. Write your own promissory note for $ 200, with interest payable in 90 days from to-day to any person you choose. 2. On what month and day would your note become due, including days of grace (Ex. 1)? Give one reason why the note is void (worthless). Find the amount due on your note at its maturity. 3. What is the time of day when the time past noon equals the time to midnight? When i the time past noon equals the time to midnight? When the time past noon equals J the time to midnight ? 4. A cask can be emptied by a i-inch faucet in 4 hours. In what time can it be emptied by a li-inch faucet ? 5. Explain the difference between factor and root; be- tween product and power. 6. A and B divide $90 in the rafio of | to |. What is each one's share ? 7. If a tank 131 ft. long, 7 J ft. wide, and 3i ft. deep holds 73J barrels of water, how wide must another tank be that is 9 ft. 9 in. long, 4 ft. 10 in. deep, and holds 89^ barrels ? 8. l + (|y- ^ 4, 3£i^^ "^ J-- U\^, "^ ^ .^ - . 519. A Trapezium is a quadrilateral having no two sides parallel. SOLIDS. 393 Note. — By drawing a diagonal between any two opposite sides of a trapezium, we have two triangles, the diagonal serving as the base of each. The altitude of each is the perpendicular distance from its other angle to the diagonal. V 520. To find the area of a trapezium, multiply the diag- onal by half the sura of the altitudes of the two triangles. ^ Trapezium. 6. The diagonal of a trapezium is 18 ft.; the altitudes of its two triangles are 5 ft. and 3 ft. What is the area ? 7. A farm is in the form of a trapezium. The diagonal distance between the northern and southern corners is 108 rods, and the perpendicular distances from the east and west corners to the diagonal are 52 rods and 36 rods respectively. How many acres in the farm? SOLIDS. 621. A solid whose two bases are equal and parallel, and its other faces parallelograms, is called a Prism. Note. — Prisms take their names from the form of their bases, as triangular, quadrangular, pentagonal, hexagonal, etc., according as the bases have three, four, five, or six sides, etc. ^ 522. To find the contents of a prism, multiply the area of the base by the altitude. 8. Find the contents of a tri- angular prism whose altitude is 10 in., and area of base 7 sq. in. 9. What are the contents of a quadrangular prism whose base is 5 in. by 8 in., and whose altitude is 12 in. ? 10. What are the contents of a hexagonal prism, the area of whose base is 10 sq. ft., and whose altitude is 15 ft. ? A Triangular Priam. A Rectangular Prism. 394 MENSURATION. PYRAMIDS AND CONES. y 523. A solid whose base is a triangle, square, pentagon, etc., and whose sides are triangles meeting at a vertex, is called a Pyramid. Note. — A pyramid takes its name from the form of its base. t/ A solid whose base is a circle, and whose convex surface terminates in a point, is called a Cone. ^ . 524. The Altitude of a pyr- Af\^ amid or cone is the perpendic- c*/ I m ^"^^^ distance from its vertex ^ii ' m *^ *^® centre of its base. #^^™ The Slant Height is the D ^l.j4m^v^ shortest distance from the ^ X C vertex to the perimeter of A Pyramid. A Cone. the base. r..h^ 525. To find the contents of a pyramid or cone, multiply the area of the base by \ of the altitude. To find the convex surface, multiply the perimeter of the base by \ the slant height. 11. Find the contents of a quadrangular pyramid whose altitude is 40 in., and whose sides of bases are 8 in. and 6 in. Solution, — 8 x 6 x ^3^ = 160 cu. in. Ans. 12. Find the convex surface of a regular hexagonal pyramid whose slant height is 16 in., and whose side of base is 4 in. 13. Find the convex surface of a cone, when the circum- ference of its base equals 16 ft. and its slant height 18 ft. 14. Find tlie*Convex surface and volume of a cone whose radius is 4 in. and altitude 6 in. 526. The Frustum of a cone or pyramid is the part which is left after the top is cut off in a plane parallel to the base. SOLIDS 395 Frustum of a Pyramid. Frustum of a Cone. 527. To find the contents of the frustum of a pyramid or cone, multiply ^ of the altitude by the sum of the areas of the two bases plus the square root of their product. 15. Find the con- tents of the frustum of a quadrangular pyra- mid whose altitude is 15 ft., and whose ends are 6 ft. and 4 ft. square. 16. A log 16 ft. long is 30 in. in diameter at one end and 24 in. at the other. Find its cubical contents. 528. A Sphere is a solid bounded by a curved surface, all parts of which are equally distant from the centre. 529. To find the surface of a sphere, multiply the circumference by the diameter. ^ ^ 530. To find the contents of a sphere, multiply the surface by ^ of the diameter. 17. Find the surface of a sphere when the diameter is 16 inches. 18. Find the surface of a sphere when the radius is 3 yd. 19. Find the surface of a sphere when the radius is 5 cm. Find the volume when : 20. Diameter = 25 ft. 22. Radius = 12 ft. 21. Radius = 2 ft. 23. Circumference = 12.5664 in. 24. Radius = 3 dm. 25. Compare the volume of a 4-ft. cube and a 4-ft. sphere. 26. Compare the surfaces of a 4-ft. cube and a 4-ft. sphere- A Sphere. X 396 MENSURATION. REVIEW OF MENSURATION. 531. 1. Find the cubic yards in a cone, the circumference of whose base is 20 ft., and whose altitude is 30 ft. 2. Find the area of a semi-circle when its radius equals 14 ft. 3. Find the area of a square inscribed in a circle of 4 ft. in diameter. 4. The circumference of a circle and the perimeter of a square are each 300 ft. Which has the greater area ? 5. A circle is inscribed in a 6-ft. square. Find the area of the circle. 6. Find the value at $50 an acre of a farm in the form ^ of a trapezoid, the parallel sides of which are 120 rd. and 160 rd. respectively, the distance between which is 80 rd. 7. How many miles does the earth travel in a revolu- tion around the sun, the distance between them being 95,000,000 miles ? 8. If a bin is 8 feet square, how deep must it be to hold 100 bushels ? 9. Find the lateral surface of an equilateral triangular pyramid, the perimeter of the base being 12 m. and the slant height 14 m. 10. Find the volume of a square pyramid, the perimeter of whose base is 16 ft. and whose altitude is 9 ft. 11. What is the volume of the largest cone that can be cut from a pyramid whose base is 6 feet square, and whose slant height is 15 feet ? ] 12. A cylindrical tank is 14 ft. deep and 6 feet in diam- eter. Find the cost of cementing the sides at 90 J? a sq. yard. 13. Find the capacity in gallons of a cylindrical cistern whose inside diameter is 6 feet, and whose depth is 7 feet. 14. Find the capacity in Kl. of a cylindrical cistern whose inside diameter is 4 m., and whose altitude is 5 m. APPENDIX. MARINERS' MEASURES. 532. TABLE. 6 feet = 1 fathom 120 fathoms = 1 cable-length 7J cable-lengths = 1 mile 1.15 statute miles = 1 nautical mile 3 nautical miles = 1 marine league OTHER LINEAR MEASURES (APPROXIMATE). 4 inches = 1 hand 3.3 feet = 1 pace 9 inches = 1 span 5 paces = 1 rod SURVEYORS' LINEAR MEASURE. 533. The Surveyors' Chain is made of 100 links, each link being 7.92 inches long. It is called Gunter's Chain, from the name of the inventor. The steel measuring tape is 100 feet long, each foot being divided into tenths and hundredths. 397 398 APPENDIX. 534. SURVEYORS' SQUARE MEASURE. TABLE. 625 sq. links = 1 sq. rod . . . sq. rd. 16 sq. rods = 1 sq. chain . . . sq. ch. 10 sq. chains 1 or I" ~ ^ ^^^® ... A. 160 sq. rods J 640 acres = 1 sq. mile . . . sq. mi. 1 sq. mi. = 640 A. = 6400 sq. ch. == 102,400 sq. rd. = 64,000,000 sq. 1. GOVERNMENT LANDS. 535. The government lands of the United States are divided by parallels and meridians into Townships, 6 miles square. Each township is divided into 36 square miles, or Sections. Each section is subdivided into half-sections and quarter- sections. In surveying the pub- lic lands, lines 6 miles apart are run from east to west and from north to south, dividing the ter- ritory into square town- ships. An east and west line is established as a Base Line, and a north and south line as a Principal Meridian. A line of townships running east and west is called a Tier, and a line of townships running north and south is called a Range. Any township is designated by its number north or south 1 — 1 — 1 — TOWNSHIP 5 NORTH TOWNi 5HIP 4 NOF TH 3J 33 :a 3J 30 ^ 73 33 33 33 ->- ->- ->- ->— ->— ^^H — j>— ->— ->- ->— ■ 7 z z 7 7 — > z 7 ^ Z O o o O (7) Uz o O C) o -u\- -m- -m- -m- -m- ->ili — m— -m- -rn- -m-- 01 ■*>. co h3 ■" I-- ro co ■t^ Ol BA SE i LIT ME m -co- ■i- -%- -%- 2 r-j * > m m > TO WN5 HIP 4 > SOI TH 1 1 TOWNSHIP 5 SOUTH APPENDIX. 399 of the base line, and its number east or west from the principal meridian. Thus, a township that is in the 15th tier north of the base line, and in the 28th range east of the 4th principal meridian, is designated : T.15K I1.28E. 4th P.M. There being 36 sections in a township, each section is designated by a number. The numbering begins at the N.E. corner, increasing toward the west and east, as shown in the accompanying diagram. TOWNSHIP N SECTION N ONE MILE W 6 5 4 3 2 1 7 8 9 10 11 12 18 17 16 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 X Ul -1 330 A. N.E.^ 160 A. Ul I of S.E.M 80 A. ofS.E.« 40 A. of8.E.Ji 40 A, ONE MILE S Each section is divided into 4 quarter-sections, containing 160 acres each, and named, as shown in the diagram. MISCELLANEOUS MEASURES OF WEIGHT. 636. AVOIRDUPOIS WEIGHT. 14 lb. = 1 Stone 56 lb. Butter = 1 Firkin 100 lb. Grain = 1 Cental 100 lb. Dried Fish = 1 Quintal 100 lb. Nails = 1 Keg 196 lb. Flour = 1 Barrel 200 lb. Beef or Pork = 1 Barrel 280 lb. Salt at N. Y. Works = 1 Barrel 400 APPENDIX. Grain, vegetables, seeds, coal, etc., are sold by the bushel. Grain, seeds, and very small fruits are sold by stricken measure. Large fruits, vegetables, corn in the ear, etc., are sold by heaped measure. The measure should be heaped or rounded as high as 6 inches above the top of the measure. The standard unit for the United States is the "Winchester Bushel. It is cylindrical in form, 18^ inches in diameter, and 8 inches deep. It contains 2150.42 cubic inches. It is customary in estimating the number of bushels that will be contained in a given bin or space, to consider the bushel as occupying 1\ cu. ft. of space, nearly. As there are 1728 cu. in. in 1 cu. ft. there must be as many cubic feet in a bushel, as 1728 is contained times in 2150.42, or 1\, nearly. 537. The following table shows the number of pounds in a legal bushel, of different commodities, in various states : Wheat Indian Corn, shelled Oats Barley Buckwheat . . . Eye Clover Seed . . . Timothy Seed . . Blue Grass Seed O 2; -3 H Q i ^ O 60 50 35 48 52 56 60 45 14 >• M 60 56 33i 48 52 56 60 45 14 ■i J 60 56 32 32 32 i < 60 56 30 46 46 56 o 60 56 32 48 42 56 60 ■jr. 60 56 82 48 42 56 60 6 60 52 35 48 52 56 60 45 14 60 56 30 48 50 56 64 ^ 60 58 32 48 48 56 60 45 60 54 48 50 6 O 60 56 32 48 56 60 O 60 56 34 46 42 56 60 ii z 60 56 32 47 48 56 > 60 56 82 46 46 56 < 60 56 36 45 42 56 60 22 60 52 32 50 40 54 56 56 28 45 56 60 56 60 52 32 48 40 54 60 45 14 60 56 32 48 50 56 60 45 14 60 56 32 48 42 56 60 46 Beans, peas, and potatoes usually 60 lb. ; in N. Y., beans 62 1b. Coal, 80 lb., except Ind., 70 or 80, and Ky. 76 lb. Salt: 111., 50 lb. common, or 55 lb. fine, K. J., 56 lb., Ind., Ky., and Iowa 50 lb., Penn, 80 lb. coarse, 70 lb. ground, or 62 lb. fine. APPENDIX. 401 APOTHECARIES' FLUID MEASURE. 538. APPROXIMATES. 1 fluid drachm = 45 drops of water, or a common tea- spoonful 1 fluid ounce = 2 tablespoonfuls 4 fluid ounces = 1 gill, or 1 small teacupful 4- gill = 4 tablespoonfuls, or a wine-glass 1 pint of pure water = 1 pound 4 teaspoonfuls ' = 1 tablespoonful FARMERS' ESTIMATES. 539. To find the number of bushels in a bin or granary, Divide the number of cubic feet in the bin or granary by 1\. To find how large a bin will contain a given number of bushels, Multiply the number of bushels by 1^. The result is the number of cubic feet in the required bin. To find the number of gallons of water in a cistern or tank, Multiply the member of cubic feet in the cistern or tank by 7|. To find how large a cistern will hold a given number of gallons, Divide the number of gallons by 7|-. The result will be the number of cubic feet in the required cistern. To find how many bushels of shelled com in a given number of bushels of corn in the ear, Divide the number of bushels of corn in the ear by 2. 402 , APPENDIX. To find the number of tons of hay in a mow or stack, Divide the number of cubic feet in the mow or stack by 500. If clover, divide by 550. Note. — Hay should be well pressed down. It is estimated that horses and sheep consume daily about 3 pounds of hay for each 100 pounds of weight. Cows and oxen about 2^ pounds. As food for stock, corn and oats are equivalent to about twice their weight in hay. Cotton- seed meal about 3 times its weight in hay. Net weight of fat beeves is about f of live weight, of fat hogs, I ; of fat sheep, J. BUSINESS FORMS. 540. A written acknowledgment that money, or its equiva- lent value, has been received, is called a Receipt. Receipt on Account. $325y\^^ Chicago, 111., Dec. 1, 1900. Keceived from W. S. Smyth, three hundred twenty-five dollars on account. a t\ td A. D. i'erkins. Receipt in Full. Detroit, Mich., July 8, 1901. Eeceived from Charles Anderson, thirty -seven and -fi^ dollars, in full of account. $37tV7 H. King & Co. Receipt for Rent. .f 97_Y^ Cincinnati, O., Jan. 6, 1900. Received from H. K. Pierce, ninety-seven dollars for rent of dwelling No. 504 McBride St., from July 15 to October 15, 1900. rj r^ ' Henry Quilton. APPENDIX. 403 Order for Money. San Francisco, Cal., March 10, 1899. Messrs. David Hunter & Sons, Little Rock, Ark. Please pay to bearer, Mr. Jacob Schmidt, one hundred fifteen dollars, and charge to the account of Nicholas Grumbach. Order for Goods. TopEKA, Kans., May 30, 1899. Messrs. J. H. Andrews & Co., Denver, Col. Please deliver to George M. White goods to the value of one hundred thirty-two dollars, and charge the same to my account. Jacob Riis. A Check is a written order, addressed to a bank by a depositor, requesting the payment of a certain amount of money to a person named, or to that person's order. Bank Check. Bank of Commercb Onno. humAnoA. 5^1/ytogm. nrr\A -Mr X> ollars liy^rr^. IT SoiinAnrx. 404 APPENDIX. COMPUTING TAXES. 541. A tax list may be extended, with a minimum liability to make errors, by arranging a table showing the tax of units, tens, hundreds, thousands, etc., of dollars. The rate used in the following is ^Yiis mills on a dollar. Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. $1 $.00523 $10 $0.0523 $100 $0,523 $1000 $ 5.23 2 .01046 20 0.1046 200 1.046 2000 10.46 3 .01569 30 0.1569 300 1.569 3000 15.69 4 .02092 40 0.2092 400 2.092 4000 20.92 5 .02615 50 0.2615 500 2.615 5000 26.15 6 .03138 60 0.3138 600 3.138 6000 31.38 7 .03661 70 0.3661 700 3.661 7000 36.61 8 .04184 80 0.4184 800 4.184 8000 41.84 9 .04707 90 0.4707 900 4.707 9000 47.07 Make the table by finding the tax on units of dollars to $9. The tax on the tens, hundreds, etc., of dollars is found by removing the decimal point one or more places to the right. What is the tax at the above rate on ^ 7856 ? Solution. Tax on $7000, as per table, =$36.61 Tax on $800, as per table, = 4.184 Tax on $50, as per table, = .2615 Tax on $6, as per table, = .0313 Total tax on $7856, $41.0868 The same result may be found by multiplying $7856 by the rate. APPENDIX. 405 542. The following table shows the legal rate of interest in each of the states and territories in the United States. TABLE. State. Rate. State. Rate. Alabama S 8 Montana 8 Any, Arizona 7 Any. Nebraska 7 10 Arkansas ....... 6 10 Nevada 7 Any. California T Any. New Hampshire .... 6 6 Colorado 8 Any. New Jersey 6 6 Connecticut 6 6 New Mexico 6 12 Delaware 6 6 New York 6 6 District of Columbia . . . 6 10 North Carolina .... 6 6 Florida 8 10 North Dakota 7 12 Georgia T 8 Ohio 6 8 Idaho 7 12 Oklahoma 7 12 Illinois 5 7 Oregon 6 10 Indiana 6 8 Pennsylvania 6 6 Indian Territory .... 6 10 Rhode Island 6 Anv. Iowa 6 8 South Carolina 7 8 Kansas 6 10 South Dakota 7 12 Kentucky 6 6 Tennessee 6 6 Louisiana ....... 5 8 Texas 6 10 Maine 6 A.J. Utah 8 Any. Maryland 6 Vermont 6 6 Massachusetts « Any. Virginia 6 6 Michigan 5 7 Washington 6 12 Minnesota 6 10 West Virginia 6 6 Mississippi 6 10 Wisconsin 6 10 Missouri 6 8 Wyoming 8 12 When no rate is mentioned in a note or other contract, the rate in the left-hand column may be collected by law. Any rate not exceeding that in the right-hand column may be collected when specified in the note or contract. EXCHANGE. 543. Exchange is a method of making payments between distant places without transmitting the money. If A of Boston owes f 1000 to X of Denver, and B of Denver owes $ 1000 to Y of Boston, B of Denver may pay X of Denver for an order on A of Boston to pay the $ 1000 to Y of Boston. A, therefore, may send this order to Y who presents it to A and receives his money. Thus, the two debts are discharged without the sending of money. Such transactions are usually carried on through banks, which charge a small fee for their services. 406 APPENDIX. 544. The written order, directing one person to pay a certain sum to another, is called a Draft or Bill of Exchange. 545. The signer of a draft is called the Drawer, the person to whom it is addressed is the Drawee, and the person to whom it is payable is the Payee. 546. A draft payable on presentation to the drawee is a Sight Draft. A draft payable at a specified time after pres- entation is a Time Draft. TIME DRAFT. ^ foc]^ SanTranci&co, Oct.. u. 1 8 9_q_ At. t orn. . Am iW 9,1/:^ t. r> ny to tke order n f l?n^)^nt.~T}\jrrrr\n/i SniiAonrrrxnm. Ot\p . h xi rrx Ans A,, Txx/rKo. a/nd. ip o Do liars yalue received, and charge tfic sameioihe account of T o hf . C ZmA/iAK vr o -+- o n ■ A A IT, Kn/rx Q.'Y C o. Three days of grace are usually allowed on time drafts. 547. If the drawee accepts a draft, he writes the word "Accepted" across its face, and signs his name, with the date of acceptance. This is an agreement to pay it, and is called the ** Acceptance " of the draft. DOMESTIC EXCHANGE. 548. Exchange between places that are in the same country is called Domestic Exchange. Note. — When a draft sells for its face, it is at par ; when for less than its face, it is at a discount ; when for more than its face, it is at a premium. APPENDIX. 407 1. What will be the cost of a sight draft for $ 500 at J% premium ? Solution. — Since exchange is $ 100 + $ .0025 = $1.0025 ^^ ^ premium, each dollar of the draft will cost $ 1.0025. Therefore $ 1.0025 X 500 = $ 501.25 ^ $ 500 draft will cost 500 times $ 1.0025. 2. What is the cost of a New York draft of $ 1000, at 1% discount? ^ -j^ QQ ^ Q^_.^ 99 Solution. — Since exchange is at a dis- count, each dollar of the draft will cost $ .99 X 1000 = $ 990 ^ 99^ and $ 1000 will cost 1000 times | .99. 3. What is the cost of a sight draft on Denver for $ 5000 at H% premium? Find the cost of the following sight drafts. 4. On New Orleans at i% discount for $ 498. 5. On Cincinnati at lj% premium for $ 3000. 6. On Boston at i% premium for $ 875. 7. On Buffalo at i% discount for $ 750. 8. What is the cost in Chicago of a draft on New York for $ 1000 payable 2 months after sight, at |% premium ? Solution. — At | % premium, each dollar of the draft will cost at sight $1,005. But since the draft is not payable until 2 mo. 3 da. after sight, the banker allows $ 1 + $ .005 = $ 1.005 ^he bank discount at the legal $ 1.005 — $ .0105 = $ .9945 rate of interest in Illinois for $ .9945 X 1000 = $ 994.50 ^^' ^ "'^^ ^ ^^- ^^^ ^'^^^^^ amounts to | .0105 on each dol- lar. Subtracting this discount from $ 1.005, we have the cost of f 1 of the draft, or $.9945. A draft of $1000 will cost $1000 times .9945, or $994.50. 9. What will be the cost in New York of a Denver draft for f 1000 payable 1 month after sight, exchange being at 1% discount. 408 APPENDIX. 10. Find the cost in Savannah, G-a., of a draft on Phila- delphia for $5000, at 60 days' sight, at 1% premium, and interest at 6%. 11. What is the cost in St. Louis of a $500 draft on New York at 30 days' sight, at 1^% premium, and interest at 6% ? 12. How large a sight draft on Chicago can be bought for $3030, exchange being at 1% premium? Solution. — Since $ 1 of the draft $1 + $.01 = $1.01 win cost $1.01, as many dollars can $3030 -J- $1.01 = $3000 be purchased for $3030, as $1.01 is contained times in $3030, or $3000. 13. What was the face of a sight draft on Boston pur- chased for $1015, exchange being at 1J% premium ? 14. How large a sight draft on Nashville can be pur- chased for $2550, when the exchange is at J % discount? 15. How large a draft on Rochester can be purchased in Boston for $5000 at 60 days' sight, the premium being 1^%, and interest 6%? $ 1 + $ .015 = $ 1.015 Solution. —At li% premium, $ 1.015 — $ .0105 = $ 1.0045 $ 1 of the draft will cost $ 1.015 $ 5000 -- $ 1.0045 = $ 4977.60 + ^^ **^^^" But since it is not payable till 63 days after sight, the Boston banker will allow 6 % discount for that time, which is $ .0105 on each dollar. Deducting this from the sight price of $ 1, we have $ 1.0045, the cost of $1 of the draft. Therefore, since $1.0045 will purchase $ 1 of the draft, $ 5000 will purchase as many dollars as $ 1.0045 is contained times in $5000, or $4977.60 + . 16. How large a draft can be purchased for $2500, 60 days after sight at 1% discount, when interest is 6% ? 17. Find the face of a draft on Charleston at 90 days' sight, that can be purchased in New York for $3500, exchange being at 1% premium, and interest 6%. APPENDIX. 409 18. What is the face of a draft on New York at 90 days' sight, which may be purchased for $2000, exchange being \<^o discount, and interest 5% ? FOREIGN EXCHANGE. 549. Exchange between places in different countries is Foreign Exchange. Drafts drawn on foreign countries are expressed in the money of the country in which they are payable. 550. Foreign bills of exchange are usually made in sets of three, of the same date and tenor, and named ^?'s^, second, and tliird of exchange. These are sent by different mails or routes. When either of the three is paid, the other two are void. 551. Exchange in European countries is done chiefly through large commercial cities, as London, Paris, Hamburg, Amsterdam, etc. 552. Bills drawn on England, Ireland, or Scotland are called Sterling Bills, and the current value of a Pound Ster- ling is quoted in United States money. The foreign exchange of the United States is done chiefly with Great Britain, France, and Germany. Note. — The Secretary of the Treasury of this country publishes annually the values of all foreign currency in United States money. 553. The English pound sterling or sovereign is valued at $4.8665 in United States gold. The French franc is valued at $ .193, or 5.18 francs for $ 1. The German mark is valued at $.238 in United States money, or 4 marks for $ .952. The above are the values when exchange is at par. Their current or commercial values may be above or below par. 410 APPENDIX. 1. What is the cost in New York of a sight draft on Liverpool for £520 12s. 6d. when exchange is $4,875 to the pound sterling ? Solution. —£520 12s. 6d. = £520.625. Since 1 pound is wortli $4,875, £520.625 are worth 520.625 times $4,875, or $2538.046+, the cost of the draft. 2. What is the cost of a bill on Paris for 625 francs, exchange being at 5.20 francs to the dollar ? Solution. — Since 5.20 francs cost $ 1, 625 francs will cost as many- dollars as 5.20 francs is contained times in 625 francs, or $120,192+, the cost of the draft. 3. Find the cost of a bill on Glasgow for £384 15s. 9d., exciiange being $ 4.87 for a pound. 4. What is the cost of a draft on Berlin for 1250 marks, exchange being at $ .95 for 4 marks. Solution. — Since 4 marks are worth $.95, 1 mark is worth ^ of $.95, and 1250 marks are worth 1250 times | of $.95, or $296,875, the cost. 5. How much must be paid for a draft on Frankfort for 648 marks, exchange being $ .95^ per 4 marks ? 6. How much must be paid for a bill of exchange on Havre 1274.28 francs, exchange being 5.18 francs to the dollar ? 7. What is the face of a bill of exchange at sight on London, purchased in New York for $ 3500, exchange being $4.86 for a pound sterling ? Solution. — Since £ 1 is purchased for $ 4.86, as many pounds can be purchased for $3500 as $4.86 is contained times in $3500, or £720 3s. 3d, the face. 8. How large a bill on Paris can be purchased for $2500, exchange being 5.15 francs to a dollar? 9. What is the face of a bill on Dublin which costs $4865 in United States gold, exchange at $4,865? 10. How large a bill on Hamburg can be purchased for $ 3810, exchange 95| ? ANSWERS TO EXAMPLES IN HEATH'S COMPLETE PRACTICAL ARITHMETIC. Article 47. 1. 6116. 16. $235,222. 31. $10,290.28. 46. 99,820 2. 8915. 17. $148,285. 32. 1364. votes. 3. 18,441. 18. 28,007. 33. $240.75. 47. $66,440. 4. 21,365. 19. 125,448. 34. $290.42. 48. 7,793,300 5. $771.40. 20. 4492. 35. $134.90. sq. mi. 6. $1287.873. 21. 1,197,972. 36. $1276. 49. 33,897 ft. 7. $821,191. 22. $48,978. 37. $8944. 50. 11,783 lb. 8. $5717.189. 23. $131,165. 39. $31,843. 51. 4996. 9. 68,605. 24. 42,681. 40. $59,584 52. 39,363. 10. 62,054. 25. 58,593. 41. 541 mi. 53. $ 208.43. 11. 60,088. 26. 463,090. 42. 1861. 54. $316,639. 12. 2,953,660. 27. 491,467. 43. 5,892,906. 55. 429,879. 13. 2,760,578. 28. 3122. 44. $221.50. 56. 944,835. 14. 2,651,844. 29. 890,407„ 45. 1693 lb. 57. $1534.028. 15. $197.37. 30. 103,019. Article 56. 2. 276. 13. 6927. 24. 1036. 35. $22.90. 3. 137. 14. 3698. 25. 964. 36. $ .416. 4. 19. 15. 6499. 26. 3979. 37. $.598. 5. 25. 16. 1273. 27. 1019. 38. $11,361. 6. 66. 17. 156. 28. 585. 39. $13,533. 7. 69. 18. 1474. 29. 297. 40. 3269. 8. 618. 19. 2997. 30. 16,677. 42. 1128. 9. 162. 20. 1338. 31. 4315. 43. 32,376. 10. 67. 21. 7082. 32. 10,388. 44. 31,676. li. 219. 22. 1482. 33. 1450. 45. 16,668. 12c 2146. 23. 996. 34. 411 12,223. 46. 4025. 41! 2 ANSWERS. 47. 16,042. 63. $4126.832. 59. 11,369. 65. $149.36. 48. $.129. 54. 7529. 60. $25,967. 66. 2462. 49. $10,449. 55. 25,350. 61. 41,276. 67. 1925. 50. $1,244. 56. 21,996. 62. $813,875. 68. $a725. 51. $8,936. 57. 6039. 63. $2.33. 52. $19.51. 58. 7724. 64. 166 A. Article 56. • 69. 284 yr. ; 42 yr. 75. 483 mi 81. $5250. 70. $6875. 76. 6855 sq. mi. 82. $2131. 71. 489 boys. 77. 13,108 votes. 83. $1275. 72. 13,289,220. 78. 1,763,100. 84. 841 yd. 73. 12,789 ft. 79. 175 mi 85. 3345. 74. $1,226,274,478. 80. $28,500. Article 65. 27. 648. 41. 23,000. 55. $103.88. 69. 31,680 ft. 28. 2065. 42. 53,883. 56. $98,615. 70. $9.88. 29. 2912. 43. 703,676. 57. $164,565, 71. $101.08. 30. 1698. 44. 5,538,972. 58. $89,948. 72. 7040. 31. 2982. 45. 785,928. 59. $ 5.625. 73. $ 18,624. 32. 4232. 46. 1,489,686. 60. $ 18.66. 74. 26,880 lb. 33. 4428. 47. 4,384,740. 61. $11,488. 75. 1344 hr. 34. 2630. 48. 5,838,760. 62. $29.04. 76. 896 cu. ft. 35. 43,130. 49. 2,558,070. 63. $13.75. 77. $151.50. 36. 34,008. 50. 1,111,104. 64. $114.75. 78. $75. 37. 7580. 51. $5.04. 65. $12.96. 79. 320 qt. 38. 11,530. 52. $14.35. 66. 1584 pens I. 80. 864 sq. in. 39. 35,232. 53. $23.64. 67. 3360 bu. 81. 23,688 lb. 40. 34,562. 54. $61.25. 68. 22,464. 82. 1296 units. Article 67. 1. 1250. 9. $395. 18. 8580. 2. $36.40. 10. $469. 19. 20,160. 3. 5080. 11. $360. 20. 19,750. 4. 3090. 12. 240,000. 21. 78, ,750. 5. 7860. 13. $293,000. 22. 97,200. 6. 2800. 14. 439,800,000 23. 131,200. 7. 3600. 15. 2,873,200,000. 24. 356,400. 8. 28,400. 17. 7360. 25. 278,600. ANSWERS. 413 26. 386,100. 55. 268,830. 83. 239,112; 116,268. 27. 1,107,000. 56. 341,478,000. 84. 196,840 ; 308,416. 28. 2,130,000. 57. 342,954. 85. $19,656. 29. 35,880,000. 58. 28,460,432. 86. 28,160 yd. 30. 259,200,000. 60. 68,376. 87. 6290 bu. 31. 8,874,000,000. 61. 151,032. 88. 110,880 ft. 35. 89,148. 62. 196,564. 89. 324,000 1b. 36. 408,498. 63. 180,438. 90. ^1980. 37. 127,050. 64. 199,681. 91. 196,0001b. 38. $1701.56. 65. 160,947. 92. 1512 mi. 39. $7615.11. 66. 331,224. 93. $33,760. 40. 5,678,986. 67. 347,512. 94. 29,344 1b. 41. $123,970. 69. 61,184. 95. $661.50. 42. 4,992,232. 70. 35,424. 96. $21,000. 43. 162,582. 71. 21,588. 97. $2080. 44. 65,492,908. 72. 42,133. 98. 4,498,660. 45. 224,016. 73. 41,956. 99. 8760 hr. 46. 42,739,736. 74. 32,186. 100. 36,266,000 A. 47. $1525.68. 75. 48,885. 101. $1,043,619. 48. 350,649,186. 76. 46,064. 102. $612.11. 49. 122,688. 77. $4914. 103. $1043.75. 50. 93,788,068. 78. $38.10. 104. 10,878 da. 51. 71,800. 79. 7446 mi. 105. 116 mi. 52. 154,629,780. 80. $203.40. 106. 200 mi. 53. 270,060. 81. $301,224. 107. 6,043,500 1b. 54. 418,460,000. 82. $98,022. 108. $180.60; $33.60. Article 82. 6. 511. 17. 14,960. 28. $30.296r\. 38. $35.45. 7. 5541. 18. 6179. 29. $24.40. 39. 41,097; 3. 8. 1172. 19. 11,669. 30. $73.10. 40. $9962; 9. 901. 20. 17,460|. 31. $31.88. $7969§. 10. 412f. 21. 4976|. 32. $30,303. 41. 4315|. 11. 796^. 22. 10,711f. 33. 62|. 42. $300. 12. 349|. 23. 10,940^. 34. 507 bbls. 43. 48 miles an 13. 1293. 24. $120,001. 35. $5.25. hour. 14. 331^. 25. $24,001. 36. $12.08. 44. $5.10. 15. 6882. 26. $30,001. 37. $13.64. 45. 775 bu. 16. 4049. 27. $90,001. 46. Rose, $15.09; James, $60.36. 47. $.96. 48. 965. 4V 1: ANSWERS. Article 83. 7. 384.92. 10. 387. 13. 28.006. 16. $2.89. 8. 296.48. 11. 28. 14. 198.751. 17. $1,398. 9. 169. 12. 39.642. 15. $ 13.805. 18. $29.84. Article 85. 2. 161^. 18. 2576|f. 34. 260,985Hff. 50. 23 da. 3. 807^V- 19. 7113||. 35. 72,632^110. 51. $127. ' 4. 604. 20. 13,085/3. 36. 7445^W^. 52. 54 cu. ft. 6. 322^1. 21. 38,039if. 37. 20,331}|H- 53. r The latter. I $.50. 6. 380^^^. 22. 4406|f 38. 9465i|ff. 7. 483f^. 23. $6.04. 39. 849 carriages, , 54. 45 horses. 8. 413|f. 24. $7.63. 40. 990IH. 55. f 54 mi. per 1 hour. 9. 779if. 25. $ .849. 41. $968. 10. 686||. 26. $2,125. 42. 129 pk. 56. 15 yr. 11. 472H. 27. $6848.76|f. 43. 752 bu. 57. 318. 12. 573ff. 28. $8235.11. 44. 42 bbl. 58. 14 da. 13. 69H|. 29. $7511.23. 45. 32 bu. 59. $156.25. 14. 3960^3^. 30. $2605.102. 46. 36 hr. 60. $6.50. 15. 778H. 31. 28,166^\%%. 47. 206. 61. $2.76. 16. 5413||. 32. 13,185|Mf- 48. $3116. 62. Tlie former. 17. 6729^|. 33. 13,187Mff. 49. $5.50. 63. 27. Article 88. 2. 10. 3. 14. 4. 14. 5. 20. 6. 10. 7. 6. Article 89. 1. 35. 3. 1. 5. 29. 7. 203. 9. 300. 2. 10. 4. H. 6. 4^. 8. 93 • 10. 9|. Article 91. 21. $240. 27. $2.25. 33. 15 yr. 38. 88 times. 22. 1140 mi. 28. 64 yr. 34. 71 da. 39. 28^. 23. 6 da. 29. 24hr., 5^|hr . 35. , 40^. 40. rC,$ 17,000; Id, $6985. 24. 10 hr. 30. $2.10. 36. , 14 hr. 25. $ 3240. 31. $7.65. 37, . $1722. 41. $156. 26. 284 A. 32. $11.58. ANSWERS. 415 Article 99. 13. 3, 3, 7. 20. 2, 2, 2, 2, 3, 3, 13. 27. 3, 3, 6, 7, 11. 14. 2, 2, 3, 7. 21. 2, 2, 2, 2, 5, 5, 7. 28. 2, 2, 2, 2, 3, 3, 7, 13. 15. 2, 5, 5, 5. 22. 2, 3, 5, 7, 11. 29. 2, 3, 5, 5, 7, 11. 16. 2, 3, 5, 7. 23. 11, 13, 17. 30. 3, 3, 5, 5, 7, 11. 17. 2,2,3,53. 24. 2,3,5,7,11. 31. 2,2,2,2,2,2,2,503. 18. 2, 2, 2, 2, 3, 3, 5. 25. 7, 7, 11, 13. 32. 3, 3, 31, 37. 19. 2, 2, 3, 131. 26. 2, 3, 3, 5, 5, 7. Article 102. 2. 4788. 8. 7|. 14. 200 da. 20. 4f|. 3. 2. 9. 12 bu. 15. $62.50. 21. 80. 4. 18. 10. 20 yd. 16. 80. 22. 18. 5. 5^. 11. 21^. 17. 100 bu. 23. 1^ jar. 6. 3f 12. llHbu. 18. 40 1b. 24. 5.tons. 7. /j. 13. 25 sacks. 19. 25 cords. Article 107. 2. 12. 4. 15. 6. 12. 8. 16. 10. 8. 12. 15. 14. 11. 16. 9. 3. 21. 5. 56. 7. 20. 9. 12. 11. 2. 13. 9. 15. 68. Article 108. 18. 270. 21. 24. 24. 108. 27. 140. 30. 9. 19. 60. 22. 56. • 25. 60. 28. 144. 20. 21. 23. 38. 26. 60. 29. 45. 31. 16 bushels ; Wheat, 2 boxes ; Barley, 3 boxes ; Oats, 8 boxes. Article 114. 2. 270. 4. 240. 6. 720. 8. 420. 10. 7560. 12. 630. 3. 36. 5. 756. 7. 72. 9. 420. 11. 1400. 13. 192. 14. 60 quarts ; 15 ; 12 ; 10. 15. 40 minutes ; 8 ; 5 ; 4. Article 115. 2. 5, 5, 7, 29 ; 2, 2, 2, 3, 3, 7, 19 ; 2, .3, 3, 5, 5, 7 ; 2, 3, 7, 11, 13. 3. 36. 4. 2, 3, 5, 11, 7 ; 13, 7, 7, 3, 3, 3 ; 19, 5, 3, 3, 2, 2, 2. 5. 2, 7, 11, 17, 31. 6. 7. 10. 96. 11. 44. 12. 4. 13. 30. 14. 58. 15. 90. 16. 101. 17. 42. 18. 2 A. ; 7 lots ; 9 lots j 11 lots. 416 ANSWERS. 20. 42,336. 22. 5040. 24. 9240. 26. 5040. 21. 1260. 23. 36,086. 25. 97,020. 27. $120. 28. $ 1.50 ; 30 nickels, 16 dimes, 6 quarters, 50 3-cent pieces. 29. 1 hr. 32 miii. 35. 84f. 39. 60|f. 43. 4 da. 30. 3168 feet. 36. 224. 40. If. 44. 140 1b. 33. 18. 37. 6f 41. 90 bu. 45. 30 1b. 34. 10. 38. 1^. 42. ^cord. 46. 8 cents. Article 126. c 31. I|. 33. ^ I 35. ^-h- 37. t¥i- 39. m- 41. Mi 32. M- 34. t\%- 36. x¥o- 38. m- 40. HI- 42. %m Article 127. 23. f 30. if- 37. tIi 44. i. 50. \\. 24. f 31. m 38. \l. 45. M|. 51. f 25. i- 32. h 39. m- 46. \. 52. I. 26. A- 33. H- 40. x^3- 47. f 53. i. 27. if- 34. U- 41. h- 48. iJ. 54. f 28. i 35. ii^ 42. f 49. f 65. M- 29. i- 36. f 43. i. Article 128. 41. 1^. 49. ¥/ 56. -W- 63. W- 70. ^'^5. 43. ^p. 50. ¥i^ 57. ^K 64. 14^35.. 71. -^LSjOA 44. W. 51. ^1^ 58. AfJ^. 65. ^^K 72. ^fl^; A235 45. W-. 52. ^01 59. Yi^. 66. 4||i 73. ^xV-; ¥^- 46. ¥^. 53. ¥f 60. W. 67. Hl^. 74. $i.9^^Q, 47. 13^51_ 54. W 61. Y/. 68. ^Vi^- 75. HP- 48. If^. 55. 131 62. V^. 69. -^/j^. Article 129. 33. 20|. 40. 20f. 47. 60f. 54. 1490||. 34. m- 41. 29|. 48. 74M. 55. 1647 fV 35. iif. 42. 9/,. 49. 92||. 56. 1326|f. 36. 3t\. 43. 19i|. 50. 31|. 57. 1000. 37. m 44. lOff. 51. 159^. 58. 250. 38. 2. 45. 21. 52. 18^. 59. 292|f. 39. 5^. 46. im- 53. 53^. 60. 1393ii 61. 123 bu. ANSWERS. 417 » Article 131. 2. ^, A. 7. M, H- 12. M, M, H. 3. li H- 8. M, fi. 13. M, M, l|. 4. A, A. 9. if, ^5, t',. 14. f^\, ^0^, ^. 5. ft, ft. 10. ,\, ^%, i§. 15. ft, 60, ^|. 6. ii ii 11. H, ii M- 16- IS, IS, ti M. 17. M, H, H, M- 27. M, H, ^TT- 18. ttVt^, ^^%. Ul m- 28. ^^„ ^ ^- 34. I is larger. 26. if, li ^^. Article 132. 21. If 30. Iflf. 39. 28J|. 47. $117f. 22. HI. 31. 2f^\. 40. 28f^. 48. $39f 23. li|. 32. lO^V^- 41. 75||. 49. 45^^ hr. 24. If. 33. lOff. 42. 65f^. 50. llff tons; 25. 2^. 34. 9fi 43. 78|^ mi. $65^^. 26. 2^^. 35. 6f|. 44. 111/^ A. 51. IQl^f lb. 27. 21 36. 8. 45. 112| yd. 52. 45| tons. 28. If. 37. 11||. 46. 41^V. 46. 43 mi. 64. 7f. 81. 21^3,. 47. Geo. $^3^ greater. 65. /^. 82. If. 48. 41 cords. 66. j\%. Article 144. Decimal Fractions. 1. Seven tenths. 3. Seven thousandths. 2. Seven hundredths. 4. Seven hundred-thousandths. 5. Three thousand sixty -five hundred-thousandths. 6. Sixteen thousand nine hundred eighty-four hundred-thousandths. 420 ANSWERS. 7. Ten thousand sixteen hundred-thousandths. 8. Fifty-four ten-millionths. 9. Thirty-five and eighteen thousand six hundred-thousandths. 10. Five ten-thousandths. 11. Five hundred-thousandtlis. 12. Four and ninety-eight thousand six hundred twenty-five hundred- thousandths. 13. Thirty-eight thousand six hundred ninety-four and six hundredths. 14. Nine and ninety-eight million four hundred sixty-three thousand four hundred-millionths. 15. Two hundred thirty-five and eight hundred fifty thousand sixty- two millionths. 16. One hundred and one hundred four millionths. 17. Nine and one million six hundred thirty-two thousand two ten- millionths. 18. Three thousand five hundred forty-three and four million five hundred thirty-six thousand nine hundred eighty-two ten-millionths. 19. Thirty and three million three hundred three thousand three hun- dred three ten-millionths. 20. Three hundred three and three hundred three thousand three hundred three millionths. 21. Nine and nine hundred ninety-nine thousand nine hundred ninety- nine millionths. Article 145. 22. .4, .17, .05, .325, .005, .015, 19.724. 23. .7504, 16.0125, .0006, .5000. 24. .17211, .00004, .00015, 18.00216, .00112. 25. .29, .029, .0029, .00029, 1.1, 1.01, 1.001, 1.0001, 1.00001. 26. 324.000126, 4582.36242, .000017, .00005, 24.0003406. 27. .000010, .00824, .31, .00216, .00007846, 4.00015. 28. .8. 32. .00289. 36. .3. 40. .000001. 29. .16. 33. .028654. 37. .01. 41. 1.000. 30. .615. 34. .0000563. 38. 500.5. 42. .00005. 31. .2123. 35. 15.005. 39. .0027. 43. .0275. Article 148. Reduction of Decimals. 44. .500000, .017000, .125600, .000155, 29.803000. 45. .80062, 305.24000, 70.50000, 3.85263. 46. .1000000, .0001000, 1000.0010000, 1.0100385. 47. .26000, .13682, 9.40000, 25.00000, 8.63521. ANSWERS. 421 Article 149. 48. ^• 51. 1^. 54. ^,. 57 . tV.- "0. 34^QQ*I)Q|^. 49. ^' 52. I 55. UU' 58 - w^h- 61. 2^TTr^\^jj. 50. H- 53. ^. 56. 2J. 59 Article 150. '. 2S^'^. 62. 1084^^^. 64. h 66. |. 68. i 70. f 72. m- 65. ^' 67. i. 69. f. Article 152. 71. |. 73. .8. 77. .1875. 81. .83f 85. .425. 89. 66.66f. 74. .625. 78. .5|. 82. .2916|. 86. .75. 90. 25.125. 75. .75. 79. .6. 83. .875. 87. 12.5. 91. 16.25. 76. .66f 80. .5. 84. M^\. Article 153. 88. 33.33^ ;. 92. 16.25125. 93. 1657.822. 97. 161.1095. 101. 40314.039415. 94. 1914.69356. 98. 105682.1451. 102. 5.55655957. 95. 204.474. 99. 221.212. 103. 58.1933. 96. 1944.425. 100. 278.1223. Article 154. 104. 2.854. 109. 6.14994. 114. .000099. 119. 75.621. 105. 37.644. 110. 847.638. 115. 9.9001. 120. 999.995. 106. 25.05017. 111. .09999. 116. 9.825. 121. 1.19983. 107. 15.2599. 112. 999.99. 117. 43.698519. 122. 7.66. 108. 32.15596. 113. 19.99795. 118. 2.977. 123. .620005. 124. 3.21375. 128. .099999. 125. .3456. 129. Neither. 126. 9.9, 9.99, 5.02, 8.95. 130. Reduce | to thousandths. 127. 999999.900001. Article 155. 1. .608. 6. 90.978. 11. 6.76. 16. .110889. 2. .00075. 7. 36.704. 12. 1440.45. 17. .0000080184. 3. 25.50. 8. 1.01101. 13. 1171.9052. 18. 957.32. 4. 15.2. 9. 17.329. 14. .1. 19. 9.23. 5. .2756. 10. 112.073. 15. 4.0625. 20. 1111. 422 ANSWERS. 32. 17,288. 33. 11,682. Article 157. 34. 1768.4. 35. 3,752,500. 36. 3784. 37. 31,240. 38. 1. Article l58. 39. .000049. 40. .000099999996. 41. 181.3259. Article 159. 1. .45. 6. 12. 11. 2389.636+. 16. 100. 2. .0006. 7. 331.487+ 12. .0066. 17. .01. 3. 1200. 8. 800,000. 13. 29. 18. 1,000,000. 4. 1. 9. .25. 14. 6000. 19. .000001. 5. .004. 10. .763. 15. 1000. 20. .1865. 18. $15.40. 19. $7,535. 20. $145.86 21. 22. 23. Article 162. $318.3345. 24. $236.28. $20,405. $27.30. 25. $193.25. 26. $53.01. 27. 28. 29. $47,364. $857,255. $ 35.30409. Article 166. 10. $125. 11. $5^. Article 168. 22. 166 1 bu. 23. 200 lb. 24. 228 doz. 25. 80 qt. 8. $8.25, $24.66|, $12.00, $12.00, $4.00, $10.00. 9. $98.75. Article 169. 8. .1; .24; .379, .1000; .00085; .020079. 9. 1006.000502. 10. 315001.0011; 38.007; 8270942.005; 1.7. 11. 421.0005; 1027.27; 99.0000099. 12. .1 ; .02 ; .003 ; .0004 ; .00005 ; .000006 ; 12.17 ; 42.32 ; 78.589 ; 200.2001. 13. .At>. h^uh, :^, i^^hn^ ^iku^ 2%, h h 811, 91^5, 4001^1, W4V 14. .625, .4, .16, .9375, .0468+, .375, 20.5957+, 8.004, 4.0625, 708.655. ANSWERS. 423 15. .15, .0775, .188, .6465+, .0000525, .78875, 16. 665.456711. 17. 3628.52791. 18. 61870.29177. 19. 12387.56776. 20. 688.6634. 31. 594. .107875, .125, .08^, .224, .04504, .3775, .38^, .3811+. 21. 910.88127. * 26. 365.48. 22. 406.368661. 27. 199.98. 23. 2.51. 28. I is .4| greater. 24. .533. 29. $2.16. 25. 26. 30. 1,999,999.999998. 0849 acres in all : 293.8349 left. 32. .9999. 42. 62.34f cents. 52. 146.46786+. 33. 2.3836. 43. $9.28^. 53. 3,360,000. 34. 147. 44. $.21|. 54. 1.51015. 35. .001. 45. 5.115365472. 55. 32,320.03. 36. 62.5. 46. 417,000. 56. .70458+. 37. 181.87548. 47. 47.6. 57. 8.41424+. 38. 27.2544. 48. 1719.523+. 58. 16.16366+. 39. 109.090908. 49. 14. 59. 4.5733+. 40. $44.74|. 50. 122,733.333+. 60. .00006. 41. $76.80. 51. 156.414+. 61. 35,002 - 15 = 2333.46f. 62. 206.52. 63. $3600. 64. $120. Article 175. 65. $2.00. 1. $88.90. 2. $8,155. 3. $410,185. 4. $6.39. 5. $143. 6. $102.15. 8. $417,175. Article 176. 1. 70.0542 yd. 11. $15.97. 22. 204.986363. 33. $23.75. 2. 29.875 yd. 12. $6.76. 23. 5.0964+. 34. 30. 3. 27 yd., 13. $317.61. 24. 10. 35. 3000 thou- $.3,375. 14. $66.54|. 25. .9724. sandths. 4. 2.105 yd. 15. $45.70. 26. 38.996. 36. 100. 5. $13,255. 16. $17.68. 27. 487.541+ bu. 37. .0025. 6. $7.99. 17. If 29. 36. 38. $136f|. 7. $.98 gain. 18. If. 30. A and Beach 39. 2|§f^i. 8. $818.80. 19. y«^\. $200, C $386. 40. i^-,. 9. $56.80. 20. \\. 31. 18. 41. $3972.90f 10. $13.11. 21. $64,000. 32. .36. 42. $16,000. 43. 16 children. 45. Increased ^§5. 44. A $10, B $15. 46. diminished -^. 47. 2856. 18. 25. 50. m. 51. 24. 424 ANSWERS. 2. 646 in. 3. 773 ft. 4. 204,978 in, 5. 81,701 sq. yd. 6. 25,679,196 sq. in. 7. 5045 links. 8. 790,596 cu. in. ^ 9. 1036 cu. ft. 10. 127 pt. 11. 33,805 minims. ^12. 507 pt. 13. 528 qt. 14. 146,794 grains. 15. 73,774 oz. 16. 94,851 grains. 17. 101,658 sec. 18. 18,035 far. Article 209. 19. 2,400,540 sec. 20. 1,817,332 sec. 40 quires. 672 pt. 12,816. 87,648 hr. 891 in. 26. $197.12. 27. 1296 sq. in.; 4356 sq. ft. ; 46,656 cu. in. 696 hr. 1155 cu. in. 12,390,400 sq. 108,900 sq. ft. 36 oz. ; 48 oz. $2.40 loss. 21. 22. 23. 24. 25. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. yd. 47. 48. 49. 50. 35,840 oz. $9. 18 centuries. 2 da. 2 lir. 180 degrees. 2160 degrees. $24. 225,932 in. 35,640 ft. 19,138,464. 76,2051 cu. in. 7424 cu. ft. 69,056 oz. 21,972 gr. 1947 gi. 24,000 sheets. 39,180 min. 3. 5. 7. 9. 10. 11. 12. 13. 14. 20. 22. 24. 1. 2. 3. 4. 5. 6. 7. Article 210. 3 mi. 4 fur. 20 rd. 5 yd. 2 ft. 8 in. 4. 6 mi. 240 rd. 3 A. 28 sq. rd. 5 sq. yd. 3 sq. ft. 6. 16 cu. yd. 9 cu. ft. 3 cu. in. 58 cd. 8. 2 T. 3 cwt. 16 lb. 3 lb. 7 oz. 7 dr. 16 gr. Apoth. 60 gal. 3 qt. 3 gi. 5 bales. 3 wk. 6 da. 5 hr. £6 2 far. 23° 30' 23". 17 T. 16 cwt. 82 lb. 21 bu. 2 pk. 4 qt. 28|| cords. $ 13.20. 5220 minims. $2.80. 120,300 times. 50.5 degrees. 30 degrees. (a) $1.17i; (b) $515f. 3 lb. 7 oz. 18 pwt. 4 gr. Troy. 15. 8 pt., 1 flu. oz. 7 flu. dr. 1 minim. 16. 1 mi. 50 ch. 25 links. 17. 560 bu. 1 qt. 18. 5 lb. 7 oz. 16 pwt. 19 gr. 19. 6 lb. 10 oz. 4 dr. 1 sc. 12 gr. 21. 43 Gr. gro. 10 gr. 7 doz. 4 pens. 23. 3 sq. rd. 17 sq. yd. 1 sq. ft. 55 sq. 8. 9. 10. 11. 12. 13. 14. 15. Article 211. . $297.12. ■ 2iffmi. 27^ ft. 40,224^ ft. 3/2¥tr mi. 48 qt. 6pt. $2.60. 16. 17. 18. 19. 20. 21. 22. 4^ mi. 319 pt. 93 boxes. $3.60, gain. $9.60. 67 cents. $1.41, gain. 30 mi. ; 2J mi. ; ANSWERS. 425 24. 25. 26. 27. 3. 4. 5. 6. 7. 8. 9. 10. 11. .488 12. 13. 14. 4. 5. 6. 7. 8. 9. 10. ^1.28. 58| cents. $1.20. 10 cents. $ 12.96. 29. 30. 31. 21 forks. 6pt. 1280 rd. $1.05. 32. 33. 34. 35. 160 rd. 56 ft. or 18| yd. 9^ ft. 228 ft. 213 rd. 1 yd. 2 ft. 6 in. 133 sq. rd. 10 sq. yd. 108 sq. in. 8 oz. 11 pwt. 10^ gr. 8 cwt. 57 lb. 2 oz. 4f dr. 2 qt. 3^ gi. I fur. 31 rd. 1 ft. 10 in. II da. 6 hr. 1 qt. 2 gi. 68 sq. rd. 8 sq. yd. 2 sq. ft. sq. in. 4 cwt. 72 lb. 12 oz. 12.8 dr. 10 degrees 53 min. 24 sec. 4 fur. 5 rd. 1 yd. 3.6 in. Article 213 15. 16. 17. 3 fur, 33 rd. 3 yd. lOf in. 5 mo. 4^ da. 2 qt. I pt. 18. 3 oz. 4|ff dr. 19. 102 sq. rd. 25 sq. yd. 8 sq. ft. 51f sq. in. 20. 2 pk. 5 qt. l|f pt. 21. 43 gal. 1 pt. 2^ gi. 22. 86 sq. rd. 4 sq. yd. 5 sq. ft. 127^5^ sq. in. 23. 6s. 8d. 24. 1 fur. 13 rd. 1 yd. 2 ft. 6 in. f mi. f A. f yr. fib. f T. .43748 mile. .0533+ month. .3125 gal. 11. 12. 13. 14. 15. 16. 17. 18. Article 214. .4267 A. .61625 lb. .361 year. I mile. l%l month. .24 T. .3824 cord. 19. 20. 21. 22. 23. 24. 25. .03026 circle. .61626 mile. Alb. If gal. jh mile- r^ cord. tV? gr. gro. To find what part one denominate number is of another : 2. t^^. 5. .7586+. 8. ^\V 10. 3. flif 6. .6420+. 9. £^. 11. 4. AVW- 7. B|. Til- .4964+. Article 215. 4. 102 T. 1 cwt. 84 lb. 12 oz. 5. 77 deg. 16 min. 38 sec. 6. 806 sq. yd. 3 sq. ft. 137 sq. in. 7. 436 yr. 290 da. 20 hr. 44 min. 16 sec. 8. 390 bu. 1 pk. 2 qt. pt. 9. 19 cords 3 cd. ft. 13 cu. ft 10. 18 T. 2 cwt. 92 lb. 16 oz. 11. 50 hr. 39 min. 34 sec. 12. 31 mi. 197 rd. 6 yd. 6 in. 13. 23 A. 132 sq. rd. 30 sq. yd. 2 titSXO* sq. ft. 56 sq. in. 14. 62 cords 5 cd. ft. 3 cu. ft. 19. 67° 33' 46". 15. 19 T. 11 cwt. 15 lb. 6 oz. 20. 11 mi. 43 rd. 5 yd. 2 ft. 16. 501 mi. 28 rd. 2 yd. 1 ft. 6 in. 21. 167 rd. 2 yd. ft. 2^ in. 17. 64 gal. 1 qt. 22. 12 bu. 3 pk. 4 qt. j\ pt. 18. 46° 2' 20''. 23. 12 cwt. 56 lb. 2 oz. 14| dr. Article 216. 2. 5 A. 104 sq. rd. 2 sq. ft. 12. 1 bu. 2 qt. 3. 4 hr. 36 min. 40 sec. 13. 8 oz. 2 dr. 9 gr. 4. 22 gal. 3 qt. 2 gi. 14. 48° 25' 37". 5. 2 A. 3 R. 12 sq. rd. 24 sq. yd. 15. 8 mi. 163 rd. 4 yd. 1 ft. 6 in. Isq.: ft. 36 sq. in. 16. 197 rd. 1 ft. 4| in. 6. 64 da. 21 hr. 29 min. 48 sec. 17. 10° 8' 26". 7. 14 T. 19 cwt. 49 lb. 14 oz. 18. 9|oz. 8. 236 mi. 13 rd. 5 yd. 2 in. 19. 14 bu. 3 pk. 1 qt. f pt. 9. 4 sq. rd. 8 sq. yd. 8 sq. ft. 36 20. 4 lb. 12 oz. sq. in 21. 1 yd. 1.4 in. 10. 2 cwt. 85 lb. 4 oz. 22. 6 da. 22 hr. 20 min. 11. 26 gal. 3 qt. 1 pt. Article 217. 3. 155 yr. 6 mo. 23 da. 9. $ 142.50. 4. 3 yr. 11 mo. 25 da. 14. 116 da. 5. 67 yr. 9 mo. 22 da. 15. May 5, 1907. 8. 138 da. 16. 595 min. Article 218. 2. 160 gal. 1 qt. 1 pt. 3 gi. 102 10. 754 mi. 120 rd. bu. 2 qt. 4 pt. 11. 65 oz. 15 pwt. 15 gr. 3. 607 mi. 169 rd. 11| ft. 12. 363 bu. 4. 451 A. 138 sq. rd: 29 sq. yd. 13. 400 mi. 244 rd. 14 ft. 5. 55 T. 10 cwt. 14. 21 hr. 20 min. 44 sec. 6. 44 lb. 9 oz. 15. 53 gal. 2 qt. 1 pt. 7. 248 gal. 2 qt. 16. 45 A. 10 sq. rd. 17 sq. yd. 4 8. $228.75. sq. ft. 72 sq. in. 9. 18 lb. 12 oz. Article 219. 8. 24 bu. 1 pk. If pt. 7. 75 A. 49 sq. rd. 25 sq. yd. 4. 60 mi. 240 rd. 16 ft. 8. 1 bu. 2 pk. 7 qt. 6. 3 lb. 11 j% oz. 9. 10 A. 14 sq. rd. 5 sq. yd. 6 sq. ft, a 25 doz. 10. 3 mi. Ill rd. 2 yd. 1 ft. ANSWERS. 427 11. 16 min. 47 sec. 12. 3 qt. 1 pt. 3 gi. 13. 11 mi. 35 rd. 3 yd. 2 in. 14. 27|da. 15. 15 packages. 16. 66| sq. rd. 17. 12 T. 16cwt. 901b. 15 oz. 18. Tifbu. 19. 35^^ packages. 20. 54^ da. 21. 58ff sacks. 22. 171 ill jars. 23. 13 hr. 57 miu. 37f sec. 24. 26 oz. 6 dr. 2 sc. 7 gr. 25. 168 cords, 5J cd. ft. 26. 917 T. 4 cwt. 70 lb. 12 oz. 27. 6 bu. 1 pt. 28. 80 T. 6 cwt. 50 lb. 29. 2 gal. 3 qt. 30. 12 A. 14 sq. rd. 12 sq. yd. 2dr.l29| sq. in. 31. 81 mi. 22 rd. 4 yd. 1 ft. 3| in. 148 gal. 3 qt. 1 gi. 3 lb. 4 oz. 8 pwt. 9f gr. 745 mi. 126 rd. | ft. or 4f in. 17 bu. 2 qt. If pt. 1 yr. 6 mo. 17 hr. 14 min. 25 32. 33. 34. 35. 36. sec. 1. 2 and 7. 2. 3 X 8 ; 6 X 4 ; 2x2x2x3. 3. 3, 3, 5, 7, 7. 5. 7 yd. 6. 2520. 7. 15. 8. 180. 9. 3 yd. 10. $4.50. 11. \. 12. 4 times. 13. $300. 14. $192. 15. 24 ft. 16. 1280||. 31. 113.0976o 32. 176,715. 4 33. 314.16. 34. 706.86. 35. 452.3904. 36. 804.2496. 37. 198,943+. Article 220. 17. 680f A. 18. AV^. 19. 9,699,690. 20. 50 A. 21. 36 bu. 3 pk. 6 qt. 22. 89 lb. 2 oz. 6 pwt. 4gr. 23. m- 24. f 25. 277 bales, 2 bun- dles, 10 quires, 18 sheets. 26. 241 rd. 27. 480 A. 28. 2/^ mi. 29. 6493/x. 30. 5600 rails. 31. 32. 33. 34. 35. 36. 19 gr. 37. 38. ■ 39. 40. 41. 42. 43. 44. 7igr. 14701 ft. 3138.135 Stat. mi. 33 mi. 57.6 sq. rd. 6 lb. 1 oz. 17 pwt. .39^ or .3916+. 3 lb. 10.84 oz. $ 254.826. .7955+. 13 bu. 3 pk. 4 qt. 16 lb. 7 oz. 13 pwt. Article 233. 38. 20,106.24. 12076.3104. 7854. 40 sq. yd. 180 area. 78.54 sq. ft. 39. 40. 41. 42. 43. 44. 157.08 ft. diam. 45. 37.6992 ft. c cumference. 46. 12^ ft. radius. 47. 87^ A. 48. 28^!^ A. 49. 140 rd. 50. 7272 sods. 4ii8 ANSWERS. 61. 181^ ft. deep. 57. 18.4+sq. rd. 62. 36 rd. 52. 1620 tiles. 58. 3.183+ ft. 63. 300 sq. ft 53. 14,400 shingles. 59. 9.4248 in. 64. 84 sq. ft. 54. 49 sq. yd. 60. 50.92+ A. 65. 24 sq. in. 55. 688 sq. yd. 61. Semicircular plat 56. 21 min. has twice the area. Article 234. 3. $22.20. 9. 61 yd. 17. $43.80. 4. 45 sq. yd. 10. $36. . 18. $1.80. 5. 5 strips ; 6 strips ; 11. 45 yd. 19. ^^ yd. ; 6 strips. 12. 28 yd. turned under. 6. 6 strips ; 7 strips ; 13. 64 yd. 20. $91,875. 8 strips. 14. 34 yd. 21. 430 yd. 7. 26| yd. in length. 15. 42 yd. 22. 21 yd. be 8. 38 yd. 16. 45^ yd. 23| yd. carpet. Article 235. 2. $87.42i 7. $27.40. 11. $24.16f. 3. 174|sq. yd. 8. $58.50. 12. $ 167.84f 4. $4.00. 9. $30.32. 13. $7,565. 5. $26.40. 10. $26.60. 14. 87f sq. yd. 6. SS^sq.yd. Article 236. 1. $6.75. 3. 12 rolls. 5. $8.10. 2. 36 strips ; 6 rolls. 4. 15 rolls. Article 238. 6. 11 rolls. 1. 18|bd. ft. 5. $25,844. 10. 240 bd. ft. 2. 18f bd. ft. ; 37^ 6. 420. 11. $7.84. bd. ft. ; 28J bd. ft. 7. $4.80. 12. $6,804. 3. 1584 bd. ft. 8. $3.84. 13. $1,425. 4. 704 bd. ft. 9. $23.04. 14. $13,944. Article 239. » 1. $6.76. 6. $146.64^. 9. 21| rd. 2. $22.50. 6. $48. 10. 36 yd. ; 32 3. $7.42^. 7. $11.53. 11. $43.89. 4. $109.97. 8. $33.33|. 12. $82.40. i yd. border ; ANSWERS. 42S 13. 8Jch. 17. 7isq. ft. 21. 131i sq. ft. 14. 56.5488 ft. 18. 7957 i§i mi. 22. 25,142f mi. 15. 28.2744 sq. yd. 19. 2Hft. 23. 450 ft. ; $ 18.00. 16. 10084.03+ revo- 20. 64 sq. yd. 24. $12. lutions. Article 245. 2. 12,288 cu. ft. 8. 2|ft. 14. .7395 cu. in. 3. Oft. 9. 64,800 bricks. 15. 5A¥^cu.ft. 4. 4950 lb. 10. 600 blocks. 16. 24 sq. ft. 5. 12 in. 11. 137^V.. 17. 2714.3424 sq. in. 6. 90 cu. ft. 12. 30|| sq.ft. 18. 2111.1552 in. 7. $21. 13. 24,147^ cubes. 19. $38.40. Article 246. 1. 4| cords. 4. 3^ cords. 7. 40^ cords. 9. $9.23. 2. 18f cords. 5. $10,625. 8. 2^-i-g cords. 10. 64; 96; 48. 3. 1^^-s cord. 6. 6|ft. Article 247. 2. 2393.76+. 5. 15.06 bu. 8. lOOOff 11. 27.97fbbl. 3. 96.42 bu. 6. 3.73+ ft. 9. 1357f. 12. 3.11+ ft. 4. 51.56+ bu. 7. 897H-. 10. 59|fbbl. 13. 6.73+ ft. Article 250. 11. 1.7° 54' 55". 21. 72° 2' west. 12. 76° 20' 15". 22. Set back 1 hr. 16 min. 47 sec. 13. 3 min. 59 sec . past 7*A.M. 23. East 22^ degrees. ' 14. 847.664 mi. 24. 57 min. 58 sec. past 7 a.m. 15. 45° west. 25. 30 days. 16. Ihr. 26. 93° 48'. 17. 3 hr. fast. 27. 120°. 18. 1 hr. 11 min. 39|i sec. 28. 13° 23' 42": East. 19. 32 min. 1 sec . past 11 A.M. 29. 122° 26' 15" West. 20. 36 min. 25^ sec. past 9 a.m. Article 264. 6. 6.3071 Km. ; 630,710 cm. 12. .0086 Mm.; 86,000 mm. 7. 1220.47 in. 13. 75006.2 m. ; 7,500,620 cm. ; 8. 27,685,329 mm. 7.50062. 9. 32808.33^ ft. 14. 3 Mm. 7 Hm. 6 Dm. 9 m. 10. 70000.006 meters. 5 dm. 4 cm. 3 mm. 11. 7500 cm. 16. 621.3+ mi. 30 ANSWERS. Article 271. 1. Sq. Dm. 3. Sq. Hm. 5. Two. 7. Four. 2. Sq. m. • 4. Two places. 6. Two. 8. .6556 Ha. 9. 33,330,000 ca. 15. 93,800,000 sq. cm. 10. 9380 sq. m. 16. l(50,7i 50 sq , cm. 11. 93.80 A. 17. 16,07i 5,000 sq. mm. 12. .9380 Ha. 18. .0000160750 sq. Km 13. 93.80 sq. Dm. 19. 2^ A. 14. .9380 sq. Hm. Article 275. 3. 7,000,000,000. 6. 16.2 cu. m. 9. 1,800,000 cu. cm. 4. .000000005. 7. 16. 2 steres. 10. 1800 cu. dm. 6. One. 8. 16,200,000,000 mm. Article 279. 11. 37.5 steres. 5. .0015467 Kl. 8. 27,200 J 272 HI. ; 9. 5| dm. 6. 1234.6 dl. 27.2 Kl. 10. 1000 1. 7. 1000 1. Article 281. 9. 74,200,000 eg. 12. 2.2+ lb. 14. 1,000,000 gr. 10. 5,430,000,000 Mg :. 13. 220+ lb. 16. 14,400 Kg. 11. 16,432 gr. Article 283. 2. ^f^. 29. Ten-millionths. 48. Ih 4. mh 30. $18.10. 49. $4.63+. 6. $ 14,400. 34. 2 yr. 7 mo. 29 da. 50. $ 144.64. 7. 92f|. 35. 133^^. 51. 2692||f. 8. /^V 36. 16,6751. 62. 845f. 9. mim;^' 37. ISs.id.; 5 cwt. 53. 8cd. 10. $48f. 5 lb. 1.92 oz. 54. 76 T. 1 cwt. 7 lb. 11. 51 f days. 38. .39375. 55. 1,568,160; 12. ^h- 39. 9 bu. 3 qt. ^ pt. 3,136,320. 13. Increased ^j. 40. 1561 gal. 3 qt. 1 pt. 56. 40 cd. 14. mn hr. 41. 1383.2 miles. 57. $10.85. 16. $71.68. 42. 9 cwt. 33 lb. 5i oz. 68. 7 fur. 13 rd. 14 ft. 17. 23.114931. 43. 21da.21hr.36min, 6 in. 19. $63.17. 44. 2 oz. 13 pwt. 8 gr. 59. 1 mi. 286 rd. 5 yd. 20. 265.006110. 45. fffofamile. 6 in. 21. 1- 46. imm- 60. 14^ bu. 26. .00529+, 47. ^. 61. U' ANSWERS. 431 62. $40.92f. 76. 54f yd. 89. $14.82. 63. $14,925. 77. $31.16|. 90. 5 min. 29f sec 64. 3 c. 2° 6' 4". 78. 80 bd. ft. past 11 A.M. ; 54 min 65. 972,000. 79. $5.76. 30^ sec. past 12 m. 66. 900°; 75°. 80. $16. 91. 52° 30' W. 67. 3900 mill. 81. 282i| sq. yd. 92. 3 hr. 4 min. 68. 24 yr. 11 mo. 6 da. 82. $1.15. 93. m- 69. Tuesday 4. K >A.M. 83. $8. 94. ill- 70. Iff ft. 84. 4704 ; 9408 95. $29f. 71. 16 ft. shing les. 97. 7.2 sq. Hm. 72. 6f rd. 85. $30.57+. 98. 1,000,000. 73. 113.0976. 86. 3A. 99. 8.1. 74. 4375 sq. ft. 87. A- 100. 42 A. 27 sq. rd. 75. 40 rd. 88. 99 yd. ; 118f yd. ; 10| sq. yd. 101. Volume 552.9216 cu. in. ; area 276.4608 sq. in. 102. 56 sq. ft. 105. 73f yd. 108. 140 sq. rd. 103. $216.00. 106. 169|sq. yd. 109. $38.53^. 104. 33 yd. 107. 990 sq. ft. Article 285. 1. 12 ; 12 is 3 % of what ? Atis. 400. 2. 20 ; 20 is what per cent of 200 ? Ans. 10 %. 3. $8. 6. $.40. 9. 200. 12. 300 sheep. 15. 10%. 4. 150. 7. 20 sheep. 10. 200. 13. 20%. 16. 40%. 5. 45. 8. 150. 11. $4. 14. 25%. 17. 25%; 75%. Article 290. 18. 6. 20. 100. 22. 16. 24. 12 gal. 26. 120 mi 19. 50. 21. 2 men. 23. $6. 25. 180 pounds. 27. l^in. Article 292. 1. .07. 6. .0625. 11. 1.01. 16. .005. 2. .06. 7. .125. 12. 1.10. 17. .0075. 3. .02. 8. .1575. 13. 2.50. 18. .004. 4. .12. 9. .375. 14. 2.00. 19. .00625. 5. .78. 10. .04625. 15. 1.275. Article 293. 20. .009. 1. i- 3. i 5. I 7. f. 9. H. 11. If. 13. zh 15. 2. i 4. i. 6. i. 8. |. 10. 2|. 12. 2/5. 14. t^. 16. X^. 482 ANSWERS. Article 294. 1. 2. 3. 4. .25; i .60 ; |. .18; ^. .01 ; T*^. 5. .06|; ^. 9. 1.08; 1^. 6. .0625 ; ^J^. 10. 1.50 ; l^. 7. .07125; ^V 11. 1.25; 1^. 8. .66| ; |. 12. 1.375 ; If. Article 295. 13. 14. 15. 16. .0075 ; ^jf. .004; ^i^. .00125 ; ^^. .007 ; TT^,. 17. 18. .$81. 120 sheep. 19. 61 men. 21. 20. 4 oranges. 22. 2 tons. $60. Article 296. 1. 2. 3. $48. 200 lb. 67.16. 4. 6 sheep. 7. $900. 5. 50 bu. 8. 128 A. 6. 10 bu. 9. 420 bbl. Article 297. 10. 11. .4. $5. 7. 8. $ 1000. 450 bu. 9. 200. 11. 2307.69+. 10. $4110. 12. 578. Article 298. 13. 14. $20,000. $90. 9. 10. 331%. 16|%. 11. 89f%. 13. 8^<%. 12. 5%. 14. r/o- Article 300. 15 16 . 75 );25%; 10o/„. %. 9. 10. 11. 12. 13. 232. 324. 500 sheep. 50,000. $800. 14. $720. 19. 2001b. 15. 644|. 20. 640 men. 16. $375. 21.' $600. 17. $2363^. 22. 1234. 18. 800 sheep. 23. 24. 25. 26. $ 8000. 2250 bu. $ 1468.99. Article 301. 1. $20. 2. $20. 3. 50%. 4. 12i %. 5. $1200. 6. 67+ cents. Article 303. 7. 25%. 14. $62.08. 21. 40%. 27. $20. 8. 66|%. 15. $583.70. 22. 243^4^0/^. 28. $240. 9. $22.50. 16. $230. 23. $172. 29. 39| cents. 10. $85,695. 17. 5| cents. 24. $82.32. 30. $430. 11. 25%. 18. 36 cents. 25. $128. 31. 150%. 12. 9%. 19. $1267.50. 26. 5%. 32. $50. 13. $1920. 20. $94.15. ANSWERS. 433 1. $20. 2. $< Article 304. 3. $150. 4. $40. 5. $5. Article 311. 6. $43.20. 9. $45,000. 12. $2875. 14. 7. $111.56^. 10. $1250. 13. \%. 15. 8. $19.49+. 11. $2000. 16. Investment, $ 5882.35 ; commission, $117.65. 17. $223.96J. 18. 750 bbl. 19. \W. 21. Commission, $2,829; net proceeds, $52,371. 22. 70 bicycles. 25. 1200 1b. 27. 23. 1%. 26. $120. 28. 7 shares. 4%. 20. $1052. $ 1900. 500 tons. 24. $24,459,052+. 29. 20,633^ lb. ; commission, $55.71. 30. 17,163^ A.; commission, $7723.50. 31. 45,176tV ft. 32. 7800 bu. 34. 8000 bu. 35. 33. $10,706,551+. $5370.30. Article 321. 1. 2. $40. $3880. 3. $7500. 4. $150. Article 323. 5. $5.40. 6. $540. 1. 2. 3. 4. 5. $90. $215. $ 122.92. $ 1418.68. $15. 6. $46. 11. $312. 7. $75. 12. $94. 8. $6.25. 13. $30,000. 9. $3000. 14. $6000. 10. $10,000. Article 326. 15. i%. 16. $5000. 17. $10,400. 18. $24,000. 1. 2. 3. 4. 5. $60. 46f%. $237.60. $42. $715.28. 6. 4%. 10. $494. 7. 24%. 11. $30.40. 8. 10%. 12. $420. 9. Cost, $342; 13. $23.05. discount, $ 108. 14. B's $4.32 better than A's. 15. 33i%. 16. 80 cents. 17. Cost, $ 1000 ; marked value, $ 1266.6i 18. $264,024. Article 333. 1. $18,000. 6. 2. Rate, .015 ; A's tax, $ 270. 3. $31.50. 4. .02. 7. 5. Rate, .016 ; collector's com- 9. mission, $750 ; A's tax, $67.50. Rate, .012, nearly ; B's tax, $48. $2000. 8. $404.61. $4,369,083,271.38; $14,155,829.80, nearly. 434 ANSWERS. Article 338.* 1. $217.50. 3. $799.50. 5. $4266.865. 2. $162. 4. $125,800. 6. Article 340. $687,079+. 1. 60; $.79681; 5.^ ill; 6.68f. 3. Carriage, $318.50; horse, 2. Son, $23,100; daughter, $136.50. $13,750; wife, $3025. 4. 100% gain ; 20% loss. 5. $50. 13. 3^0/^. 20. $ 25,000. 6. 3280. 14. |. 23. .00625. 7. Board, $330; 15. $2000. 24. $1237.50. clothes, $198.80. 16. 5%; gain, $20. 25. 2000 bu. 8. 726. 17. $2.24. 26. $20. 9. 283^. 18. 156,250 1b. 28. $180it. 10. $34.56. 19. 40 cents, original 29. $1.47,1^. 11. 50.5+%. cost. 12. 68|%. 31. Commission, $50.51; net proceeds, $959.74. 32. Amt. sales, , $4222|; commission, $422f. 33. 5%. 45. $2131.50. 56. $10. 34. $309.37i 46. $10,320. 57. 33i%. 35. $150. 47. 21,522f bu. 58. Cost, $63.95; 36. 1%. 48. 912 head. gain, $9.59|. 37. $26flost. 49. 656 plum trees. 59. 45% boys. 38. 200 boys. 50. 55.25 gal. sold. 60. 14f%. 39. 200 sheep. 51. 595 girls. 61. 198 soldiers. 40. $6480. 52. 9^^:%. 62. 100 trees. 41. $6814.70. 53. $66.25 63. $500. 42. SOo/o. 54. $16,000. 64. $40. 43. 400. 55. .25, nearly. 65. $8100. 44. \%. 66. Value of vessel, $200,000 ; A's share, $140,000 ; : B's share, $24,000 C's share, $9000; D's share, $ 27,000. 67. $2.50. 69. 88| cents. 71. 90 problems. 68. i; 12r/o. 70. $3645.90. Article 341. 10. 46.154. 15. 136.37. 20. $371,840. 24. $1792. 11. $97,387. 16. 427.572. 21. $373.99. 25. $1206.105. 12. $100.20. 17. 45.32. 22. $541.35f. 26. $250. 13. $164,448. 18. $347,075. 23. $1164.435. 27. $646.80. 14. 67.066. 19. $387.9156. ANSWERS. 436 Article 342. 5. $12,088. 7. 1^52.48. 9. 1 2.006. 11. 090.76. 6. $6.15. --. 8. $18.11. 10. 1. 974 J. 12. $61.80. 13. Int., $148.83^; Aint., $2498.83f 14. Int., $1,907; Amt., $127,657. 15. Int., $ 130.355 ; Amt., $ 1080.985. 16. Int., $6,309; Amt., $342,789. 17. Int., $131.95; Amt., $870.48. 18. Int., $110,555; Amt., $5110.555. 19. Int., $103,431; Amt., $970,781. 20. Int., $131,769; Amt., $392,269. 21. Int., $1192.55; Amt., $4242.55. 22. Int., $22,676; Amt., $648,246. 23. $87,312. 25. $135,708^. 26. $520,940. 27. $364.60. 24. $36^. Article 343. 28. $689,703. 32. $327.97. 35. $388.07. 38. $59.25. 29. $853,402. 33. $3671.938. 36. $403.65. 39. $315.90. 30. $2087.22. 34. $870.34. 37. $489,646. 40. $46.66|. 31. $1434.426. Article 344. 41. $87.36. 45. 2+ cents. 48. $207.43. 51. $1,929. 42. $77.87. 46. $48.93. 49. $39.66. 52. $1,771. 43. $3.5.34. 47. $29.26. 50. $.264. 63. $4,422. 44. $42.04. Article 345. 54. $414,518. 67. $384,259. 60. $3560.76. 55. $290.50. 58. $319,287. 61. $186.65. 56. $621,317. 59. $1953.44. 62. $365.03. Article 346. 2. $1,198. 5. $5,912. 8. $102,526. 11. $106,277. 3. $2,717. 6. $9.07. 9. $8,775. 12. $33,418. 4. $14. 7. $11.13. 10. $21.60. Article 347. 2. Q%. 5. 40/,. 8. 8%. 10. 6%. 3. 6%. 6. i%. 9. 8%. 11. 6%. 4. 9%. 7. 7|%. 436 ANSWERS. Article 348. 3. 3 yr. 6 mo. 7. 4 yr. 7 mo, 6 da. 10. 2 yr. 16+ da. 4. 2 yr. 6 mo. 8. 7 mo. 18+ da. 11. 3 mo. 5. 6 mo. 9. 2 yr. 4 mo. 14 da. 12. 16f yr. 6. 5 yr. 8 mo. 18 da+. Article 349. 2. $250. 5. $530. 7. $1730. 9. $980. 3. $ 10,000. 6. $4625. 8. $12,580. 10. $387.50. 4. $ 1000. Article 361. 2. $245,389. 5. $54,376. 7. $478,116. 9. $160,177. 3. $512.86. 6. $406,944. 8. $616,677. 10. $140,065. 4. $ 1380.615. Article 362. 2. $464,081. 3. $143,547. Article 363. 2. $130,828. 4. $61,523. 6. $56,311. 8. $82.55. 3. $761,578. 5. $137,924. 7. $179,397. Article 364. 8. 6% method. 21. $6635.437. 24. $57.04. 27. $255,223. 15. $955,639. 22. $117.84. 25. $51,898. 28. $2767.30. 20. $155,034. 23. $161,485. 26. 63+ da. 29. 6%. 30. 2 yr. 7 mo. 14+ da. 38. Dec. 1, 1895. 48. 8%. 31. $ 1600f . 39. $436. 49. $500. 32. $ 124.999+. 41. $751.93. 50. 9 mo. 33. $32,646. 42. $1258.21. 51. 6%. 34. $92,001; $715,741. 43. $202. 52. $500. 35. $27,193. 44. $35.34. 53. $19.60. 36. $16,946. 46. 9%. 54. Every 6 months. 37. $ 102.894. 47. 10 mo. Article 368. 8. Pr. worth $380.9523 ; 12. $486.56+ cash. 17. More profitable $ 19.0477. 13. $728.1553. to purchase on time. 9. $2542.37+ each. 14. $566,037. 18. Gain $112.62. 10. $409.0909. 16. Cash. 19. $374,987. 11. $240,876. ANSWERS. 487 Article 372. 6. $648.13. 9. $4918.89. 13. $2475. 16. $8.00. 6. $2962.50. 10. $1304.57. 14. $132,975. 17. $249.98. 7. $436.24. 11. $2168.22. 15. $203.24. 18. $172.55. 8. $781,605. 12. $1972.44. 19. Day of maturity, March 31, 1896. Proceeds, $523.99. Article 373. $253.81. 4. $1218.27. 6. $477.39. $353.53. 5. $809.92. 7. $495.05. 8. $750. Article 374. 9. $488.34. 15. Proceeds, $ 346.56 ; discount, 10. Present value, $ 860 ; true dis- $ 3.44. count, $113.52. 16. Proceeds, $864.90. 11. Cash, $6.33 better. 17. $816.33. 12. $168.75; $13331.25. 18. $209.71. 13. $805.37. 19. $658,783. 14. $302.11. 20. $2600. Article 379. 3. 50 shares. 15. $ 13,840. 17. $330. 19. $20 loss. 14. 20 shares. 16. 130 shares. 18. $200. Article 382. 22. $7447.50. 31. $150. 40. e^/o. 48. 83f 23. $ 13,425. 32. $200. 41. 6|fo/o. 49. 75. 24. loir/o- 33. $400. 42. 5t\%. 50. 75. 25. 12 shares. 34. $11,100. 43. ^m%' 51. $80. 26. 20 shares. 35. $5400. 44. ^%. 52. $25,500. 27. 30 shares. 36. $9200. 45. 6's, n 53. $42,480. 28. $2000. 37. $ 10,000. better. 54. $37,500. 29. $ 1010. 38. $5000. 46. 71^^. 55. $60,000. 30. $600. 39. $ 10,000. 47. 125. * Article 383. 1. 33^ % premium. 5. Q\n- 10. 66f ; 140. 2. $ 13,600. 6. $16666f. 11. Increase $ 25 per 3. $ 10,000. 7. $18,500. annum. 4. 6% stock. /t% 8. 133^. 12. 4 bonds. better. 9. 6 % bonds at 105. 438 ANSWERS. da 13. $ 87,500 ; 875 shares ; 5 % on investment. 14. Cost, $32,500 ; $ 1600, yearly income ; rate on investment, .0492+. 15. $60. 16. 1920 shares. Article 387. 2. 84 da. after sale. 5. 4| mo. 8. July 19. 3. 3 mo. 20 da. 6. 4 mo. 3 da. 9. 5 mo. 11 da. 4. Average term, 46 7. 94 da. 10. In 7 months. El.; 1 time. May 17, 1895. Article 388. 2. May 4. 4. May 11. 6. 4 mo. 3. Sept. 11. 5. 41 mo. 7. Feb. 7, 1897. 8. Average date, July 13 ; note dated April 13. Article 395. 7. 8. 12. h 16. m- 19. i; 15. 8. ^■5' 13. 48. 17. 1 ; 4 ; ^. 20. 32; 7. 9. h 14. /^. 18. H; tf; 21. 20 ; 2. 10. 3. 15. 1- M;t. 22. |. 11. 4. Article 398. 34. 20,900. 37. 50. 40. $3. 42. 25 men. 35. 10. 38. |. 41. I da. 43. 1. 36. 150. 39. $100. Article 399. 45. $80^; $21. 51. $10.45. 57. $48. 62. 311 mi. 46. 3 da. 52. 120 Km. 58. 60 da. 63. $80.30. 47. 33f tons. 53. 45 men. 59. 225 da. 64. 15 in. 48. $ 5600. 54. $36. 60. $2. 65. 34fyd. 49. 85^ ft. 55. $450. *61. 240 shoe- 66. $27. 50. 12 da. 56. $ 3600. makers. Article 400. 67. 100 da. 2. 5bu. 4. 360 mi. 6. •' $ 100. 3. 16 da. 5. 45,6191 stones. 7. ^ t09^V cd. 8. 531 rd. 11. $1.84. 14. 145fm. 17. 3 ft. 9. 31^ da. 12. 56 bu. 15. 8hr. 18. 50 da. 10. 12 men. 13. $3,444. 16. 16yr.8mo. Article 403. 2. Wilson, $ 900 ; Mead, $ 600. 4. A, $ 312^ ; B, $ 500 ; C, 3. Jones, $ 750 ; Smith, $ 1250. $ 387|. ANSWERS. 439 5. A, $6§ ; B, $8 ; C, $9i. 8. A, § ; $6400 ; B, f ; $9600. 6. A, $300 ; B, $360 ; C, $240. 9. 2d = ^ ; 1st = ^ ; /= ^. 7. $1800; $2700. 10. X, $6250 ; F, $3750; Z, $5000. 11. Each man's gain is |^^, or -j^, of his stock. Since j\ represents the gain of the first man, ^f will represent his stock and gain. Tbe question now becomes: $570 (A's stock and gain) is f| of what? 570 -^ if = $480, A's stock. His share of total stock is ||§, or y%, and his share of total gain is y% of $150, or $90. In like manner, find the second man's stock and gain. Ans., Stock of 1st, $480. Gain of 1st, $90. Stock of 2d, $320. Gain of 2d, $60. 13. A, $3000 ; B, $2100. 16. A, $1928//^; B, $1355|§| ; 14. $576; $540; $480. C, $3615|f|. 15. A, $1250; B, $2490. 17. A, $1440 ; B, $1368. 18. Scott, $604,196+; White, $1040.559+; Watson, $755,244+. 19. A, $2000;' B, $4000; C, 21. A, $1373.684+ ; B, $1526.315. $1800. 22. A, $23.62; B, $20.67; C, 20. A, $95; B, $133. $30.71. Article 409. 10. 7776. 11. 1. 12. .00000001. 13. 15.625. 14. 1.21. 15. .000000008. 16. ,V. 17. li 18. 15|. 19. 1296. 20. 3375. 21. 1296. 22. 243. 24. 648. 25. 6460. 26. f 27. .019125. 28. 3^. Article 422. 9. 10. 11. 12. 5. 94. 6. 126. 7. 609. 8. 216. 25. f. 26. 2.0275+. 27. .5773+. 28. |i. 29. 6.0380+. 30. 53.5098+. 31. 13.7573. ,32. .3872+. 906. 5.39. 3.04. 56.4. 33. 34. 35. 36. 37. 38. 39. 40. 13. .089. 14. .253+. 15. .075. 16. .956. .4242+. 4.1231+. 2.1679+. 8.5146+. 10.9087+. 1.9157+. .4711+. 31.0322+. 17. 5.06. 21. 41.0402+. 18. 4.93. 22. 19742.4737+. 19. .782+. 23. f. 20. 18.1594+. 24. |. 41. 3.6429+. 49. ^. 42. .0977+. 43. .0585+. 50. J. 51. .749. 44. 58.094+ rd. 52. 6.480+. 45. 84 rd. 46. .143+. 47. .03651+. 48. i. 53. .960+. 54. 43.829+. 55. 61. 440 ANSWERS. Article 426. 3. 32 ft. 6. 45 ft. 9. 20.489+ m. 12. 7.211+ in. 4. 50 ft. 7. 339.4112 rd. 10. 117.153+ mi. 13. 61.224+ ft. 5. 144.5683 ft. 8. 42.5205 ft. 11. 20 ft. Article 427. 6. 16 to 64. 7. 4 to 8. 8. 21 ft. 9. 26.12+ rd. 10. 320 sq. ft Article 435. 14. 35. 16. 16£ ;. 18. 2.05. 20. 76.67+. 22. 3.203+. 15. 96. 17. 427 19. 2.59. 21. 12.34. 23. 216. 24. .70+;. 76+; 26. 31.739+. 29. 1.440+. 32. 71.3 in. 25+; 3.39^ ; J. 27. .398+. 30. 1.912+. 33. 54 sq. yd. 25. 2.428+. 28. .114+. 31. 45 in. 3^ .579+. \ Article 436. 2. 612 cu. in. 4. 181.9+ lb. 6. 13||)?. 3. 9.9+ in. tliick. 5. 7.15+ ft. Article 438. 51. 800 trees. 53. $151,109. 55. $245,898. 57. $.20 loss. 52. $6275. 54. $922^. 56. $480. 58. 18f%. 59. $.51 A. 67. 52 yd. 76. 11. 60. 28f%. 68. 46^ 32' ; 133° 28'. 77. i- 61. .0911. 69. 812igal. 78. /^V 62. 35x^^%; 70. 2 oz. 12 pwt. 79. 60. 3142800.722 bbl. 71. 1760 steps. 80. $307685f. 63. $27.42+. 72. .5928+ 81. A, 79^ mi. ; B, 64. 31| yd. 73. .27|. 70^ mi. 65. 180 sq. ft. 74. |. 82. 1 yr. 3 mo. 16 da. 66. 18.61+ ft. 75. $1.12. 83. $256,875. 84. 4 mo. 24 da. 88. $20,000. 92. $2336.66|. 96. Cash. 85. $200. 89. $440.83^. 93. $2060. 97. $1152.52. 86. $200,518. 90. $337,534. 94. $450. 98. $2470.83^. 87. $725.07. 91. $83.82. 95. $1.8349. 99. $.215. 100. A, $300 ; B, $297. 104. 60 da., Mar. 4, 1890. 101. $48,346. 105. $570. 102. June 8, '95. 106. 12 bonds. 103. $75,037+. ANSWERS. 441 107. Simple int., $ 1300 ; 4% bonds, $ 1214.44 ; 5% bonds, $ 1200. 108. $355,020.57+. 114. | ; |^. 120. if. 109. 37|%. 115. 8 horses. 121. $316.64+. 110. 6|%. 116. l 122. 6. 111. 6%, at 125. 117. IjW/oft. 123. $12. 112. Lost $62.50. 118. 8f oz. 124. $ 96, $ 60, $ 144. 113. 180 1b. 12 oz. 119. 12 hr. 125. 87 1^ on the dollar; $3360. 126. $ 2105^f ; ^ 1578f f ; $ 12ii3^^j ; $ 1052f f . 127. $240; -$120; $80. 128. Scranton, $ 1555 ; Morris, $930 ; Jackson, $620. 129. $12; $28; $20. 130. 5.656+ ft. 131. Shorter sides, 17.204+ ft. ; longer sides, 18.867+ ft. ; room, 23.494 ft. 132. 21 ft. 137. 6.3+ ft. 142. 240 rd. ; $ 588. 133. 24.495+ ft. 138. 210 sq. ft. 143. 24,200 revolutions. 134. $3.8816+. 139. 34 ft. 144. 2%. 135. 34.7. 140. $11,120.89|. 145. $213,334. 136. 6463.8-» 141. 69.641ft. 146. $823.20. 147. 2567%. 148. 1%. 149. $94.15. 150. $80.75. Article 440. 6. 99 bu. 7. 775^. 8. 376. 9. 240 L. CM. Article 441. 3. $8.95. 5. 575 bbl. 7. 13 yr. 10. 37 mi. 4. $5340. 6. 11, 7, 5, 3. 8. 1560 bu. Article 442. 1. Twenty-three million four hundred fifty-six thousand seven hundred eighty-nine. 2. G. C. D., 25 ; L. C. M., 1500. 3. $43.35. 4. 30,060,290. 6. 2, 2, 3, 11, 11. 8. 72. 5. 60 qt. 7. $19.00. 9. 60001b. Article 443. 1. 456 mi. 4. $1750. 7. f. 2. 204,060,402. 5. 5182. 8. 19 marbles. 3. 987. 6. $1060 gain. 9. 150 eggs. Article 444. 1. 45 J?. 4. ■5^. 6. 21 f. 9. ^ij^. 2. 8yr. 5. $3^. 7. $9.00. 10. i^^^,or5^. 442 ANSWERS. Article 445. 1. $84,000. 3. Increased!. 5. f. 2. 2tV da. 4. 27,853. 6. 7|§ cords. 7. 2, 3, 7, 3, 2, 11, 5. 8. 1. 9. 78. 1. MDCCCXCIV. 3. 9 boxes. 4. 160 sheep. 9. hhhh 1. $495.63. 2. 450 sheep. Article 446. 2. 1, 2, 3, 5, 7, 11, 13, 17. 5. mi 7. $34^^. 6. 4ibbl. 8. 8|. 3. 2||. 4. |. Article 447. 5. 63 cows. 6. 1511. 10. |. 9. $6611.11|. 10. $3.04. 2. Hi 3. h Article 448. 4. B, H, M- 6. 17^^. 8. 24^5^. 5. $f 7. 1. 10. $15,975. 2. 271^ yd. 3. 2. 4. if lb. 5. 9 persons. 6. 20. Article 449. 7. 77^ 8. 3fda. 1. $4608. 4. f. 2. Increased ^83. 5. 7y\. 3. 5V 6. 23ibu. Article 450. 8. 32|, 9. y-i. Article 451. 1. 61f cents. 3. 42^f§. 2. 32^ mi. 4. 20f | da. 7. One thousand nine hundred fifty seventy-three million seven. 9. $4345. 10. $187. Article 452. 3. 43HA. 4. 2 -2. 2. 2. 3. 3. 7. 9. 36 ft. 10. $4900. 10. m^M^ 188 150 160 ttt^ ttS^ Jt^' 5. mi 6. 29, 707, 578. ninety ; four thousand forty ; 1. 157|fyd. 2. 6.225. 7. $27,168+. 8. Sixty-eight and six hundred forty-two ten-thousandths. 9. $1.95. 10. $22.75. 5. 1000. 6. $1.1856. ANSWERS. 443 Article 453. 1. $1200. 3. hf. 2. 999999.999999. 5. 7,095,000 ; 63.015 ; 700.07.' 6. Six hundred forty-two and sixteen ten-thousandths ; one hundred and one hundredth. 7. 600700.19+. 8. 8295.9056. 9. $60,552. 10. $8.06. Article 454. 1. 2. 3. \\\. 4. $.1875. 6. 192 1b. %. 5. 216 sheep. 7. 23^3 da. Article 455. 8. $250. 10. $161.30f. 1. 2. 3. 4. 5. ^1 is 3^6^ greater. 6. .013; 400.05; 250^. 7. ^V.. 60 ft.; 19 lots. 8. $3731. $6^; 12|T. 9. $4.58.' .75; .8; .625; .85; .0555+. 10. $4.26. Article 456. .000515 ; 400.063? 1. 2. 3. 3^. 4. .4375. 7. .05^^^. 1,000,000. 5. X^-^-^^. 8. .8f|. 3.43. 6. 1.0419479325. 9.0. 10. .6559. Article 457. 1. 80.65. 2. Ten thousand twenty and forty-two thousand twenty-four hundred- millionths ; seven hundred two millionths ; eighteen hundred-millionths ; thirty thousand and thirty hundred-thousandths ; ten thousand twenty hundred-millionths. 3. 2da. 2hr. 5. 220.89906. 7. $408.54. 9. .22|. 4. $2.85. 6. 999.999. 8. ^^. 10. 1000 1b. Article 458. 1. $38. 4. 118-/g. 7. 2V 9. $2912. 2. 240.464501. 5. $5376.70. 8. .904. 10. $ 24.64. 3. 2.26;2^2j^ sq. yd. 3. 2 lb. 5 oz. 18 pwt. 3 gr. 6. 54 sq. ft. 9. \% ; h ; iV 444 ANSWERS. Article 460. 1. I or 1|. 2. 15 da. 3. 4 fur. 8 rd. 1 yd. .7225 ft. 4. 4^ sq. yd. 5. 6 A. 118 sq, rd. 24 sq. yd. 2 sq. ft. 22f sq. in. 6. $16.91|; $9. 7. 10 A. 8. 216. 1. .754+. 2. 9 T. 16 cwt. 61 lb. 12| oz. 3- -sT^W^Tiru* Article 461. 4. 216^ yd. 5. 14 oz. 2pwt. 23fgr. 6. 1513.69875. Article 462. 1. $1.22. 4. 3pk. 4qt. 7. $67.46|. 2. $287^1^. 5. 172fbu. 8. $46.40. 3. $16. 6. $261. Article 463. 39|^. 280 rd. 2 yd. 4 in.- 248 boards. 1. 518 pickets. 2. 50 rd. X 25 rd. 3. 74 to the square mile ; 8|f A. 6. m- 7. 4281^V 8. 18 da. Article 464. 3° 3'; 12 min. sec. before 12. .$31.16|. $9. 12 7. $5237.50. 8. $161.77. 10. 16,995 far. 9. 20° East. 10. 19fbbl. 9. 10. 9 birthdays ; 36 yr. 16 da. .000000144. 7. $32.40. 8. 480 boards. 9. 16^. 10. 34|qt. 1. $374. 2. $7200. 3. 6 times ; 1 qt. 4. $105.60. 1. 32 blocks. 2. 42 families. Article 465. $84. $6,451. 1161 liters. Article 466. 20 pairs. 12 mo. 8. 64,800 bricks ; $476.28. 9. 645,126 cu. in. 10. 114.5+ A. 24 1 gal. m- 9. 10. 1. 1368 cu. ft.; $ 48.09f . 2. 15.915+ ft. 3. $.05. Article 467. $4.45^^. $ 12.0645. $ 5.325. 16 da. 9. 5 mi. 143 rd. 2 yd. 7^ in. 10. 128 mi. 281 rd. 1 yd. 2 ft. 2 in. ANSWERS. 445 2. 1 fur. 85 rd. 3 yd. 2 in. 5 bu. 1 pk. 4 qt. 1 pt. Article 468. 3. 4 yr. 7 mo. 18 da. 4. $46.09^t; $68.68^. 5. 8^1^ acres. 6. $7.04. 7. $13.60. 8. 35,000 gr. 9. .64 ton. 10. 576||. Article 469. 1. 28%; 610/,; 307o/„; |o/^. 2. .005; .0625; .08; 1.25. 3. \^%. 4. $.10. 5. $432. 6. 18f<%. 7. 786 bbl. 8. $ 18.42 gain. 9. 38^%. 10. Rate, .0135; A's tax $51. 975. 1. 28fhr. 2. If 4. .50 ; .25 ; .20 ; .10. Article 470. 5. 201.84. 6. i;f; i;i; 7. 324 bbl. 8. 200 sheep. 9. $5300. 10. $206.85. 1. Neither. 2. 20%. 3. 33^%. 5. $67.60. 1. $3,705. 2. $5600. 3. $8000. Article 471. 6. $6000. 7. Rate, .002 ; A'j tax $ 10. Article 472. 4. $637.50. 5. 150 1b. 6. $1301.86575. 8. $119.90. 9. 405. 10. $8240. 9. $20.81^,. 10. 2.61+ ft. Article 473. 1. $ .03 ; 4 bu. ; 120 eggs ; 16 cwt. 2. M- 3. Not changed. 5. 13tV^%; 14r/o;200o/„; 10/^. 6. .0025; .2; .0025; .20; .155. 7. 93H%. 8. $7200. 9. lOo/o. 10. $612.85. Article 474. 1. Rate, .0042. 3. $4000. , 6. 4ff. 2. Gain, 2f%. 4. $32iV 7. $30.80. 8. 1\ gal. = 1 cu. ft.; 4810 gal. 9. $42.1875. 10. Gram. 446 ANSWERS. Article 475. 1. 11820. 2. $2740.50. 6. Two million three hundred thousand four hundred six and nine hundred sixty millionths. 7. Denominate; 8. $60,000. 9. $1920, gain. 10. 33^%. Mixed. Article 476. 1. .02; 06J; .20; .121 3. I; |; l; 1; If. 2. 20%; 75%; 121%; i40o/„. 4. ^h I ■5h I jh ; sh •: t^^o. 5. 10 tons. 7. 80%. 9. 1311. 6. $ 778.05. 8. 614 days. 10. 100i| cu. ft. Article 477. 2. Eighty-three and four million nine hundred thirty-seven thousand seven and ^ ten-raillionths ; one million one thousand one and one hundredth ; ninety thousand nineteen and four hundredths. 3. 240. 8. 27.8; .1875; 5.2125. 5. 115 lb. 11 oz. 5 pwt. 9. $2 per bushel ; 100%, profit. 6. If 10. 3 0; Article 478. 'o- 1. 2. 9. $ 1436.40. 3750 lb. 15%; 61%; 3. 4. 50% 20%. 5. 2iffl|, 18,760 ft. 6. 22|% ; 226 o/o. Article 479. ; 50 10. 7. $4. 1%; 6210/,. 8. $24. ill; .V.;l;if. 1. 2. 3. 4. $ 12,000. $ .63^. io|%. 188 ; 376 ; 376. 5. $371.25. 6. Mj%Y/o- 7. 32H%. Article 480. 8. $10,260. 9. $909f 10. 21ffo/,. 1. 2. 3. $946.75. .$6,037. $4.00. 6. $3,161. 7. 117. Article 481. 8. 165.68; $1365.60. 9. $36.166f. 1. 3 yr. 10 mo .20 da. 2. 8%. 5. i $6523.08. 9. $12,000. Article 482. 1. 2. 3. $544.05. 1 yr. 6 mo. m%- 4. 611 cu. yd. 5. 100%; 1. 6. .06 ; 1.06. 7. 216 farms. 8. 705. 9. $598,978. ANSWERS. 447 1. $17.07. 2. $697.00. Article 483. 10 mo. 5. $350,957. 8. $844,216. 9. $185. 10. $104,413. 1. §845.021. 2. $669.11/^. 4. $12,753. Article 484. 5. $1423.327+. 8. 25 yr. 6. 6%. 9. $42,169+. 7. 1 yr. 6 mo. 18 da. 10. $367^. 4. $622,425. 5. $512. Article 485. 6. 410/0. 7. 3 yr. 5 mo. 8. $606.08. 9. $128,865. 10. A, $1704; B, $1597|; C, $958^. 1. $25. 2. 2d; $308. 3. $77.29. 4. $1812 gain. Article 486. 5. $3270. 6. $542,955. 7. Int., $2145.282; amt., $9376.57. 8. $742.50. 9. Matures July 2, 1896; proceeds, $595.70. 10. $8685. Article 487. 1. 31|. 2. Both cost $4848^1 ; lost, $48ff. 3. $4240.10. 5. $2625. 7. 30%. 9. 9200. 4. $21,600. 6. $1010. 8. $8000. 10. $10,000. 1. 16153.84. 3. $225. 4. $15817.75. 5. 6% bonds. Article 488. 6. $3030.32. 7. Average term, 62 da. ; equated time, Mar. 4. Article 489. 8. Nov. 30. 9. 250 shares. 10. 5 % per annum, 6. Present worth, $285.11 ; discount, $14.29. 7. Bank discount, $10.95; proceeds, $619.05. 8. $3000. 9. \. 10. 150. Article 490 6. 3, missing terra. 8. U} da. 7. 6|da. 9. 2 hr. 45 min. 10. $367^. 448 ANSWERS. Article 491. 1. 24 men. 2. A, $540; B, $300. 3. 64 cents on a dollar; A, $224; B, $435.52; C, $41.60; D, $320: E, $627.52. 4. I; f ; li 7. A, $14.40; B, $25.60. 9. if. 5. 4 da. 8. 1 yr. 6 mo. 10. f |. 6. 21 da. Article 492. 1. A, $240; B, $280; C, $300. 2. A, $18,500; B, $24,800; C, $20,700. 3. B, $6000; C, $7500. 4. $90; $120; $240. 5. A, $600; B, $5331; c, $266|. 6. Foster, $200; Stull, $180; Furlong, $120. 7. $10,000; $3500; $2000; $2500. 8. A, $4000; B, $1600; C, $2400. 9. A, $3500; B, $5000; C, $7500. 10. Smith, $2500; Brown, $2700; Jones, $4200. Article 493. 2. 3. 4. 1. 3. 4. V^ ; 5.5225. 5. 1,860,867. 6. 169.7+ rd. X 56.57- 109.54+ ft. 5. 15.362+ ft. 6. 320 rd. 7. 201 in. $576. 8. 10,584 sq. Article 494. - rd. 2. 4^ ] 3| A. 7. 132 ft. 56.56+ rd. 8. 21.63+ ft. in. mi. 9. 66.5+ in. 10. 12.7- ft. 9. 15.3.69 mi+. 10. 34|| cents. Article 495. 1. 2. 3. 4. 1529; 591. 29.90. $11,440. $75,997. 5. 6. 7. Other number, 53| ; 912.4|. $25,568^23-. 600. Article 496. 8. $240. 9. 1611. 10. 5. 1. 2. 3. 1,425,600 ties. 2 hr. 253^ min. 13li|. 4. 5. $1509. 6. 8^. 1650 times. 7. 90 da. Article 497. 9. $4H. 10. 3rVda. 1. 3. 4. $.84; 6 marbles. 30. 5. 448. 6. 2. 875 examined ; $80,000. 7. 48. $12,000. 8. 2 da. 625 passed. 9. 36 fish. 10. 20 da. ANSWERS. 449 Article 498. 1. $73.50. 5. ^ cords. 8. $18.48. 2. 1141 yd. 6. ^-^ hhd. 9. 50 bd. ft. 3. f 7. 280.9078 sq. in. 10. 46 min. 28^ sec. past 9, a.m. 4. $52.92. Article 499. 1. 2. 3. 7. 8. 1 min. 48 sec. past 6, a.m. 5 ft. 106 sq. rd. 25 sq. yd. 4 sq. ft. 126 sq. in. 1 hr. 14 min. 51 ft. 5^ in. long, 16 ft. wide. 4. 5. 6. 9. 10. $52.44. .803+. 1704 sq. in. 47 If bbl. $2.07|. Article 500. 1. 2. 3. 60%. 4. 5. 6. $900,088. 7. 2000 bbl. 1787; 1413. 8. 14%. $757.76. 9. .46^^o/^. 10. 2%. Article 501. 2. 3. 4. $ 1458.465. 80 ^. $612. 1. Rate, .02 ; A's tax, $ 125. 5. $36. 6. $3500. 7. Direct, $2193 better. 8. 9. 10. $4.57|. $ 2000. $ 7920. Article 502. 1. Gained $293| 2. 28^. 3. 1| months. 4. Discount, $ 9.80 ; Proceeds, $1213.20. 5. A, $18^; B, $25; C, 16f. 7. f|. 6. llOjVft. 8. 3|. 9. A, $2500; B, $1500; C, $1000. A's gain, $1000; B's gain, $600; C's gain, $400. Article 503. 1. 11^ bbl. 2. 135.66+ ft. 3. Stevens, $86f| ; Jones, $lllff ; Payne, $37m. 4. 165 shares. 5. A, $ 149.33 ; B, $224 ; C, $ 186.66f. 6. 100 rd. 9. J. 7. 4 % stock, yf^ % higher. 10. $240. 8. Wife, \ ; daughter, I ; sou, \ each. 450 ANSWERS. Article 504. 1. $1.60. 3. $260. 5. 144. 7. 24; \. 9. $2000. 2. 80. 4. 23 men. 6. 10. 8. 3^%. 10. 1|%. Article 505. 3. 6 P.M. ; 8 P.M. ; 2.24 p.m. 8. 3f 4. fhr. 9. $93.75. 6. A, $40; B, $50. 10. 14.14+ w. 7. 8|f| ft. wide. Article 516. Mensuration. 1. 432 sq. rd. 2. 120 sq. ft. 3. 150 sq. ft. Article 518. 4. 175 sq.ft. 5. 21i bd. ft. Article 520. 6. 72 sq. ft. 7. 29^25 A. Article 522. 8. 70 sq. in. 9. 480 sq. in. 10. 150 sq. ft. Article 525. 11. 160 cu. in. 13. 144 sq. ft. 12. 192 sq. in. 14. 90.478+ sq. in. ; 100.5312 cu. in. Article 527. 15. 380 cu. ft. 16. 63.879+ cu. ft. Article 530. 17. 804.504+ sq. in. 22. 7238.2464 cu. ft. 18. 113.0976 sq. yd. 23. 33.5104 cu. in. 19. 314.16 cu. cm. 24. 113.0976 cu. dm. 20. 8181.25 cu. ft. 25. Cube, 64 cu. ft. ; sphere, 33.5104 cu. ft. 21. 33.5104 cu. ft. 26. Cube, 96 sq. ft. ; sphere, 50.2656 sq. ft. Article 531. 1. 11.78 cu. yd. 2. 307.8768 sq. ft. 3. 8 sq. ft. 4. Area of square, 5625 sq. ft. ; area of circle, 7162+ sq. ft. 5. 28.2744 sq. ft. 9. 84 sq.m. 12. $26.39. 6. $3500. 10. 48 cu. ft. 13. 1484.40 gal. 7. 596,904,000 miles. 11. 137.922 cu. ft. 14. 62.832 Kl. 8. 25.3+ in. ANSWERS. 451 3. $5075. 4. $494.61. 5. $3045. 6. $878.38. 3. $1873.915+. 5. $154.71. Article 548. Appendix. 7. $731.25. 11. .$504.75. 9. $984.50. 13. $1000. 10. $4997.50. 14. $2261.306. Article 553. 6. $246. 8. 12875 francs. 16. $2552.322. 17. $3519.354. 18. $2036.483. 9. £1000. 10. 4000 marks. t(^i ( THIS BOOK IS DUE ON THF. LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. Map i?o «/x "«n Jig ig^ .. nl^ 0^ Q\ ^^^ , 1 V "I A ' f IV '■' '-^ X'O I f 1 1 s) LD 21-100m-7,'39(402 w