IN MEMORIAM FLOR1AN CAJO StAA^ ^v? COMPLETE ARITHMETIC BY SAMUEL HAMILTON, Ph.D. *» SUPERINTENDENT OF SCHOOLS, ALLEGHENY COUNTY, PA. AND AUTHOR OF "THE RECITATION" NEW YORK •:• CINCINNATI •:■ CHICAGO AMERICAN BOOK COMPANY Copyright, 190S, 1900, by SAMUEL HAMILTON. Entered at Stationers' Hall, London. HAM. COMPLETE ARITH. w. p. 3 ORI PREFACE A complete arithmetic should meet all the ordinary de- mands of the elementary school. It may omit that which is non-essential and all matter that properly belongs to text- books for secondary schools ; but it should include a full treatment of all important topics taught in the elementary school, and a limited treatment of those of minor importance. This book is intended for a complete arithmetic, to be used either with or without the author's Elementary Arith- metic. The work is divided into three parts. Part One, after giving a complete treatment of the fundamental opera- tions, covers the work ordinarily found in the -sixth year. Part Tioo, after reviewing the subjects of Bills and Accounts, Denominate Numbers, and Practical Measurements, covers the work of the seventh year. Part Three covers the work of the eighth year. The aim of this book is threefold : (1) To give the pupil skill in the art of computation. (2) To make him a good mathematician. (3) To give him a working knowledge of modern business methods. The first necessarily suggests an abundance of graded work. The second requires both inductive and deductive thought. The method, therefore, is inductive in the development of all mathematical principles, and deductive in their applica- tion. It requires also that all solutions shall be clear and 918185 4 PREFACE concise, and that the statements of all definitions, rules, and principles shall be as brief and comprehensive as possible. The third aim demands a practical treatment of all sub- jects from the standpoint of actual business methods. With these ends in view, attention is invited to the following : 1. The large number of graded problems under each sub- ject and in the reviews. 2. The abundance of exercises for oral drill. 3. The study of problems and processes. 4. The treatment of Fractions, Practical Measurements, and the Comparative Studies in Percentage. 5. The problems arising out of business conditions. 6. The treatment of Promissory Notes, Banking, Com- mercial Discount, Exchange, and Stocks and Bonds accord- ing to the actual methods of modern business. The author gratefully acknowledges his indebtedness to many prominent educators and friends for valuable aid, dis- criminating criticisms, and helpful suggestions. SAMUEL HAMILTON. CONTENTS PART I — SIXTH YEAR Fundamental Processes . . . Arabic System of Notation and Numeration . . Roman System of Notation and Numeration .... United States Money . . . Addition Subtraction Multiplication Division Combining Processes . . . Factors and Divisors .... Tests of Divisibility . . . Factoring Greatest Common Divisor Least Common Multiple . . Cancellation PAGE 7 8 11 12 13 19 23 29 36 37 38 39 40 41 42 Fractions 44 Fractional Units .... 44 Reading and Writing Frac- tions 40 Reduction 47 Addition and Subtraction . 52 Multiplication 58 Fractional Parts of Fractions 05 Division 08 Complex Fractions .... 75 Fractional Relations ... 70 Review 78 Problems for Analysis ... 84 Decimal Fractions 90 Notation and Numeration . 91 Common Fractions and Deci- mals 94 Addition and Subtraction . 90 Multiplication 98 Division 101 Changing Fractions to Deci- mals 100 Review 107 Business Applications of Decimals 108 Simple Accounts Denominate Numbers . . . Standard Units Reduction Addition and Subtraction Multiplication and Division . Review of Denominate Num- bers Practical Measurements . . . Length Surface Painting and Kalsomining . The Right Triangle . . . . Volume Practical Applications Lumber .... Review Problems . Percentage .... Commission . . Commercial Discount 112 115 115 110 119 121 122 123 123 124 128 128 130 132 134 138 141 147 148 Interest 151 Review of Percentage and In- terest Receipts and Checks . . . . General Review 155 157 159 PART II — SEVENTH YEAR Bills and Accounts . . . . 166 Receipts 166 Ordering Goods 1H7 Receipted Bills 108 Accounts 170 Ledger Accounts .... 173 6 CONTENTS PAGE Denominate Numbers . . . 175 Reduction 175 Foreign Money 18 i Addition and Subtraction . 184 Multiplication and Division . 186 Review Problems 188 Practical Measurements . . . 192 Length and Surface . . . 192 Lines and Angles . . . . 194 Triangles 196 Quadrilaterals 198 Rectangles 199 Plastering and Painting . . 201 Roofing and Flooring . . . 2<>2 Papering and Carpeting . . 2i)4 Areas of Triangles, Quadri- laterals, and Circles . . . 2(16 Solids 212 Lumber 217 Concrete, Stone, and Brick- work 220 The Cylinder 221 Bins, Tanks, and Cisterns . 223 PAGE Approximate Measurements . 224 Review Problems .... 225 Analysis 231 The Equation 231 Percentage 237 Review 247 Gain and Loss 249 Review ........ 253 Commission and Brokerage . 255 Insurance 259 Commercial Discount . . . 264 Commercial Bills .... 2i>8 Local and State Taxes . . . 269 Duties or Customs .... 272 Interest 275 Simple Interest 275 Problems in Simple Interest . 283 Annual Interest 286 Exact Interest 287 Compound Interest .... 288 Savings Accounts .... 289 Investments 291 Promissory Notes .... 292 Partial Payments of Notes . 298 PART III — EIGHTH YEAR Banks and Banking .... 303 Bank Discount 307 Exchange 315 Stocks and Bonds 324 Stocks 324 Bonds 331 Test Problems in Percentage . 335 Ratio and Proportion .... 337 Ratio 337 Simple Proportion .... 338 Partitive Proportion and Part- nership 341 Problems for Oral and Written Analysis 345 Longitude and Time .... 349 Government Land Measures . 356 Powers and Roots 358 Extracting Roots .... 360 Square Root Mensuration Regular Polygons .... Solids Similar Surfaces Similar Solids Specific Gravity Review Metric System Agricultural Problems . . . Test Problems General Review Optional Subjects Present Worth and True Dis- count Foreign Exchange .... Compound Proportion . . . Cube Root Reference Tables Index 361 367 367 368 376 378 879 381 382 392 401 406 421 422 425 426 433 437 COMPLETE ARITHMETIC PART I — SIXTH YEAR FUNDAMENTAL PROCESSES NOTATION AND NUMERATION A unit is a single thing ; as, one, one cent. A number is a unit or a collection of units. Numbers are used to tell how many ; they are expressed by figures or letters. The figures we now use are of Hindu origin, but the Arabs were the first people to introduce them into Europe. They are Naught One Two Three Four Five Six Seven Eight Nine 012 34567 89 These ten figures are, therefore, called Arabic numerals. The figure o is called naught, zero, or cipher, and has no value. The Arabic notation is a method of expressing numbers by means of figures. Numeration is a method of reading numbers expressed by means of figures or letters. Arithmetic is the science of numbers and the art of com- puting by them. 7 8 FUNDAMENTAL PROCESSES ARABIC. SYSTEM OF NOTATION AND NUMERATION Any number containing but one figure, simply stands for so many ones; thus, 9 stands for 9 units, or ones. In any number containing two figures, the first place at the right is called ones ; the second place is called tens ; thus, in 25 there are 2 tens, or 20 ones, plus 5 ones, or 25 ones. 25 is read, "twenty-five." 1. Name the ones and tens in the following numbers and then state how many ones each number equals : 15 25 30 75 82 60 72 45 70 99 20 10 39 47 90 In any number containing three figures, the third place from the right is called hundreds; thus, in 325, the 3 stands for 300 ones. 325 is read, "three hundred twenty-five." 2. Name the ones, tens, and hundreds in the following and then state how many ones each number equals: 125 329 879 801 600 650 803 132 400 ' 904 109 705 105 550 900 901 502 999 570 809 In any number containing four figures, the fourth place from the right is called thousands. 4635 is read, " four thou- sand, six hundred, thirty-five." 3. Name the places in each number and then read: 2135 6005 6910 5604 5000 4025 2684 8709 7009 8900 For convenience in reading large numbers, in the Arabic system, the figures are generally separated by commas into groups of three figures each, called periods. The first period, counting from the right, is units; the second, thousands; the third, millions; the fourth, billions; the fifth, trillions ; etc. NOTATION AND NUMERATION 9 The following table shows the arrangement of these periods, and the three orders of figures in each period: TRILLIONS' BILLIONS' MILLIONS' THOUSANDS' UNITS* PERIOD PERIOD PERIOD PERIOD PERIOD O U (A c c c zz « « (/> o C I o 15 c o E c o of (A T3 to *U — w» *D -o — * TO j* C "O c o 2 T3 15 c o 0) E c o 0) (4 to 3 E c ~ C c ™ C C ■f™ c C O c 2 S 3 0, "C 3 12 4604 3261 5427 7173 7504 8675 7259 5050 7891 5009 8795 6985 > 10 8753 7406 5030 4287 6474 3745" 6758 2834 6793 5783 2758 5268 -20 4326 6498 7406 7205 6471 7777 6734 5065 5872 4987 2050 5989 7583 8439 3458 6034 6579 7481 ! 3586 9823 7295 2985 2068 5479 t 10 2734 7984 8376 3046 6579 8705 1 4725 2030 2794 1154 2068 3074. 1 6050 5984 6384 3683 5432 5547 1 q 7438 3749 6589 4594 4280 6875 | 12 5006 5308 7405 5181 5683 ADDITION 17 Making Change. Business men make change by the adding method. Thus, if a purchase is made for 11.57, and $2.00 is given in pay- ment, the clerk will probably say : " One dollar fifty-seven cents, sixty, seventy, seventy-five, two dollars," laying down each time the piece of money that makes the sum named. Acting as clerk when the following purchases and pay- ments are made, give the exact language you might use, if the purchaser were actually present to receive the change: Cost of Purchase 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. $1.34 2.95 2.11 3.17 1.15 4.12 2.24 3.54 1.12 4.02 2.79 2.36 3.09 2.71 3.00 Amount Given Cost of Purchase $2.00 16. $1.95 5.00 17. 7.23 3.00 18. 3.G7 4.00 19. 2.85 1.50 20. 5.02 4.50 21. .91 3.00 22. .79 3.75 23. G.01 1.50 24. 7.11 5.00 25. 3.95 4.00 26. 2.01 5.00 27. 0.79 4.00 28. 7.04 3.00 29. 8.31 3.50 30. 2.98 Amount Given $2.00 10.00 5.00 4.00 6.00 5.00 5.00 10.00 8.00 5.00 10.00 7.00 8.00 10.00 4.00 HAM. COM PL. A KITH. 2 18 FUNDAMENTAL PROCESSES Written Work 1. Chicago is 468 miles, by rail, west of Pittsburg, and New York is 445 miles east of Pittsburg. What is the rail- road distance from Chicago to New York, via Pittsburg? 2. A man purchased a farm for $6500 ; built a barn on it at a cost of $1980; a house for $1825; and spent on im- provements on the land, $971. What was the cost of the farm and improvements ? 3. The area in square miles of the main body of the United States is 3,088,519; the outlying possessions are Alaska, 590,884 square miles; Hawaii, 6449; Porto Rico, 3435 ; Philippines, 115,026 ; other outlying possessions, 761 square miles. Find the total possessions in square miles. 4. The average annual production of corn in the United States for a number of years was $854,000,000; of hay, $467,000,000; the production of cotton, $406,000,000; of wheat, $395,000,000; and of oats, $254,000,000. Find the total average annual value of these five productions. 5. The values of the ten leading exports of the United States for the year 1906 were: cotton, $413,137,936; meat and dairy products, $208,586,501 ; iron and steel and machinery, $172,555,588 ; copper, $90,773,151 ; petroleum, $85,738,866 ; wood, $77,255,225; flour, $58,399,727 ; corn, $52,840,269; wheat, $49,158,650 ; livestock, $45,614,748. What was the value of all these products ? 6. The values of the ten leading imports for the same year were: silk and silk manufactures, $100,052,211; raw and manufactured fibers, $99,635,731 ; hides, $83,884,981 ; sugar, $79,015,471;* chemicals, $78,647,978; coffee, $72,252,465; cotton goods, $68,911,371; wool including woolen goods, $61,040,335; india rubber, $58,664,651; jewelry, $46,047,021. Find the value of all these imports. SUBTRACTION 19 SUBTRACTION 1. Subtract by 2's from 97 to 1 ; by 3's from 75 to 9. 2. Subtract by 4's from 78 to 2 ; by 6's from 98 to 2. 3. Subtract by 7's from 101 to 3 ; by 8's from 106 to 2. Subtraction is the process of finding the difference between two numbers, or of taking one number from another. The minuend is the number from which we subtract. The subtrahend is the number to be subtracted. The difference, or remainder, is the result of subtraction. The difference added to the subtrahend equals the minuend. The sign — , called minus, indicates subtraction. Only like numbers can be subtracted. Give differences at sight : 4. 5. 6. 7. 8. 9. 3-2 12-7 11-9 10-4 11-2 14-3 4-1 9-4 7-4 19-3 16-8 13-9 5-4 9-8 9-5 17-9 15-6 16-7 7-3 11-4 17-8 11-3 14-5 15-7 8-4 9-7 12-6 12-9 12-8 13-8 7-2 5-1 11-5 16-9 14-7 18-9 10. Subtract each of the following numbers from 20 ; then, from 30, 40, etc. 4 6 11 17 13 14 7 8 12 10 15 14 16 11. Subtract each of the following numbers from 100. 40 70 60 25 45 75 44 64 84 37 57 68 12. Take 27 from 65, thus Subtract in like manner: 13. 72 14. 84 15. 91 48 36 45 65-20 = 45; 45- 7 = 38. 16. 63 17. 48 18. 82 24 32 59 20 FUNDAMENTAL PROCESSES A clerk's sales book shows the following sales and amount given by customers. Give differences at sight : Sales 19. $1.55 20. 1.15 21. 3.17 22. 1.78 23. 3.15 24. 1.79 25. 2.34 Amount GIVEN $2.00 1.50 4.00 2.00 3.50 5.00 3.00 Sales 26. $ .49 27. 1.23 28. 1.06 29. 2.14 30. .99 31. 1.29 32. .87 Amount given $ 2.00 5.00 2.00 3.00 2.00 10.00 5.00 Written Work 1. From 632 take 374. Minuend 632 = 500 + 120 + 12 = 5 hundreds, 12 tens, 12 ones Subtrahend 374 = 300 + 70 + 4 — 3 hundreds, 7 tens, 4 ones Difference 258 = 200 + 50 + 8 = 2 hundreds, 5 tens, 8 ones Since 4 ones cannot be. taken from 2 ones, take 1 ten = 10 ones, from 3 tens; 10 ones + 2 ones = 12 ones; 12 ones — 4 ones = 8 ones. Since 7 tens cannot be taken from the 2 tens remaining, take 1 hun- dred = 10 tens, from the 6 hundreds; 10 tens + 2 tens = 12 tens; 12 tens — 7 tens = 5 tens. 5 hundreds — 3 hundreds = 2 hundreds. Test. — 374 + 258 = 632. By adding the difference to the subtrahend, pupils can quickly discern whether the answer is correct. Explain the steps in finding each remainder : 2. 800 3. 9004 4. 9080 5. 7001 6. 9040 594 7907 5987 4908 5879 206 1097 3093 2093 3161 Subtract : 7. 30984 8. 54009 9. 81704 10. 41711 11. 50000 24987 31047 54270 39111 42001 SUBTRACTION 21 The following methods of subtraction are also convenient: I. By adding (the method used in making change). l. From 842 take 385. Think: AVhat number added to 5 will make 12? (7.) Write down 7; carry 1 to 8 tens in minuend. What number added to 9 (8 + 1) will make 14? (5.) Write down 5; carry 1. What number added to 4 (3 + 1) will make 8 ? (4.) Write down 4. 842 385 457 II. By subtracting from 10. 2. From 653 take 378. Borrow 10, subtract 8, and add 3, thus : 10 — 8 in subtrahend = 2 ; 2 + 3 in minuend = 5. 10 — 7 in subtrahend = 3 ; 3 + 4 (5 — 1) in minuend = 7. 5—3 = 2. The steps in the general method of sub- traction are borrow, add, subtract. In this they are borrow, subtract, add. Write, subtract, and test four problems in 3 minutes : 85980 9. 1070.01 15. 590680 21. 6459871 71409 340.97 289796 2987598 753 378 275 4. 57004 20098 10. 11. 12. 13. 14. 1590.10 210.89 16. 17. 18. 19. 20. 998076 433011 22. 23. 24. 25. 26. 5798371 3099384 5. 39702 21308 1953.01 391.54 598801 303397 8342901 5217809 6. 70001 39005 * 401.97 207.58 743019 556601 7654321 3456780 7. 98235 60104 8701.49 511.10 831001 397018 6543903 4239001 8. 80021 51037 * 800. 67 610.34 458995 233450 4932459 2013307 22 FUNDAMENTAL PROCESSES PROBLEMS 1. The minuend is 6389, and the difference is 4360 Find the subtrahend. 2. From the sum of 3645 and 5796, subtract their dif- ference. 3. In 1900 the population of New York State was 7,268,894 ; of Pennsylvania, 6,302,115. How much greater in population was New York than Pennsylvania? 4. In 1906 Iowa raised 373,275,000 bushels of corn ; Mis- souri, 228,522,500 bushels. How much did the corn crop in Iowa exceed the crop in Missouri ? 5. A and B together owe me $7650; B owes me $4675. After each pays me -1*1600 on account, find the amount each one still owes me. 6. A retail merchant bought goods to the amount of 61457. After selling stock from these goods to the amount of $975, he found that the remainder of the goods unsold had cost him 6473. How much had he gained or lost ? 7. Mr. Adams bought a farm for $8670; he expended in improvements on barn and house, 81790; on stock and farming utensils, $2080. How much more did he pay for the farm than for improvements, live stock, and utensils? 8. The surface of the earth contains 196,907,000 square miles, of which 144,500,000 square miles are water. How much of the surface of the earth is land ? 9. A father divided 623.675 among his sons, giving to James 66750 and to Henry 6 5000 less than the part remaining after James was paid ; to Frank he gave the remainder. How much did each receive ? MULTIPLICATION 23 MULTIPLICATION 1. Count to 72 by 2's; by 3*s ; by 4's; by 6's; by 9's. 2. Count backwards from 48 by 2's; by 3's; by 4's; by 6's ; by 8's. 3. Count forwards to 96 by 3's ; by 4's ; by 6's. 4. Count to 25 by 5's ; to 60 by 4's ; to 99 by 9's. 5. Build the multiplication tables by addition, thus : 2 2 2 2 2 2 [2 times 2 = 4 2 2 2 ; then write it in this form • 3 times 2 = 6 4 times 2 = 8 6. Drill on these tables until pupils thoroughly know them : 1 2 3 2's 3's 4's 5's 6's 7's 8's 9's 10' s 11' s 12s 4 6 6 9 12 8 12 16 20 24 10 12 14 21 16 24 32 40 48 56 64 •72 80 88 96 18 27 36 45 54 63 72 81 90 99 108 20 30 40 50 22 24 15 20 25 30 35 40 45 50 18 33 44 55 36 48 60 72 84 96 108 120 132 144 4 8 24 30 36 42 48 54' 60 66 72 28 35 42 49 56 63 70 77 84 5 10 15 6 12 18 60 70 80 90 100 110 66 77 8* 00 110 121 7 14 21 28 32 36 8 10 24 27 9 18 10 20 as 40 11 22 44 55 60 12 24 36 IS 120 132 The fust row of figures at the top stands for the different tables. By multiplying each of the numbers in the first left-hand row by each of the numbers in the top row, the tables can all be made. Thus, in the table of the twos, the products are directly below the number of the table, etc. 24 FUNDAMENTAL PROCESSES 7. How many units are there in 4? 4x15 means that we are to take $5, four times to find the product; this may be found in two ways : $5 + $5 + $5 + #5 = $20, orby multipli- cation, which is a short form of addition ; thus, 4 x §5 = $20. Multiplication is the process of taking one number as many times as there are units in another number. The multiplicand is the number multiplied. The multiplier is the number by which we multiply. The product is the result of multiplication. The sign x indicates multiplication ; it is read, " times," when the multiplier precedes the sign, and, " multiplied by," when the multiplier follows the sign. Read each statement and then give products. 8. 4x$12 10. 12 x 7 yards 12. 11 x 4 pounds 9. 9 x 6 horses 11. 5x8 ft. 13. 7x8 bushels 14. In the above statements, how many times are $12 taken? 8 bushels? 7 yards? 15. Name the multiplicands in the above statements ; the multipliers. A concrete number is a number used with reference to a particular object ; as, 5 days, 10 pounds, 8 inches. An abstract number is a number used without reference to a particular object ; as, 5, 8, 20. Name the abstract and the concrete numbers in the follow- ing statements and then give products : 16. 12 x 7 days 18. 11 x $7 20. 6 x 11 17. 15x10 19. 9 x 12 ft. 21. 12x5^ The multiplier is always regarded as an abstract number. The multiplicand may be either abstract or concrete. 22. In problems 16-21 are the products like the multipli- cand or the multiplier ? The product and the multiplicand are like numbers. MULTIPLICATION 25 Oral and Written Analysis 1. How many eggs are there in 6 dozen? Since there are 12 eggs in one dozen, 1 doz. = 12 eggs; in 6 doz. there are 6 x 12 eggs=72 eggs. 6 doz. = 6x 12 eggs=72 eggs. 2. How many trees are there in an orchard if then; are 11 rows and 10 trees in each row ? 3. James raised 7 bushels of potatoes on an average from each of 10 rows. How many bushels did he raise? What is the cost of : 4. 10 quarts of cherries at 8^ per quart? 5. 9 quarts of milk at 7^ per quart? 6. 8 bushels of apples at $2 per bushel? 7. A twelve-pound cheese at 12^ per pound? 8. 3 pecks of apples at 25^ per peck? 9. How far does a boy ride on his automobile in 4 hours at the rate of 9 miles per hour? 10. How many miles are there in 4 streets, if the streets average 12 miles ? 11. There are 32 quarts in a bushel. Find the number of quarts in 13 bushels. 12. How far does an automobile run in 4 hours, if it averages 14 miles per hour? 13. Find the cost of posting 18 letters at 2^ each. 14. Find the cost of a 7-pound turkey at 13^ per pound. 15. A lady purchased 2 dozen oranges at 40^ per dozen. How much did they cost? 16. It takes John 15 minutes to walk to school. How many minutes will be required to walk to school 60 times? 17. Frank used 12 tablets, at 10^ each, in a school term. How much did they cost? 26 FUNDAMENTAL PROCESSES Written Work 1. Multiply 146 by 3. Multiplicand 146 3x6 ones = 18 ones, or 1 ten and 8 Multiplier 3 ones. Write 8 in ones' place and Product 438 carry the 1 ten. 3x4tens = 12 tens ; „ +. n - . n. -.~ . ~~ 12 ten s+ the 1 ten, carried from ones' Test.-146 + 14b + 146 = 438 place = 13 tenB , or j himdred and 3 tens. Write 3 in tens' place and carry the 1 hundred. 3x1 hundred = 3 hundreds ; 3 hundreds + the 1 hundred = 4 hundreds. The product is 438. Find products : 2. 139 6. 674 10. 307 14. 137 18. 427 22. 507 3 3 5 6 7 6 - 3. 135 7. 278 li. 342 15. 673 19. 784 23. 196 _2 J 6 5 7 8 4. 603 8. 147 12. 281 16. 901 20. 249 24. 379 4 5 4 7 8 7 5. 205 9. 219 13. 309 17. 419 21. 907 25. 583 2 4 _6 _6 2 _8 26. Multiply $1.25 by 3. $1.25 3 Place the decimal under the decimal poir point in the product directly it in the multiplicand. $3.75 27. $2.53 30. $8.09 33. $3.27 36. $2.41 39. $3.19 9 9 8 10 9 28. $6.08 31. $2.25 34. $1.04 37. $3.74 40. $8.92 _9 10 11 12 10 29. $9.09 32. $1.29 35. $3.05 38. $5.05 41. $9.08 9 11 10 12 12 Ul'LTIl'LK ATIOX 27 Multiplication by Larger Numbers Multiplying integers by 20. 100. 1000. etc. ln x 12=12<); 100x2=200; 1000x2 = 2000. Any integer may he multiplied by 10, 100, 1000, etc., by an- nexing to the integer as many naughts as there are naughts in the multiplier. Multiply each number by 10, by 100, by 1000, writing only the products : 16, 409, 290, 301, 205, 250, 175, 791. Written Work l. Multiply 72 by 36. Multiplicand 72 72 In practice, Multiplier 36 qq the in the 1 l * 1 1st partial product 2d partial product 432 = 2160 = 6 x72 30 x 72 .on second partial product is - jXU omitted, and Entire product 2592 = 36 x 72 2592 21 (JO is written as 216 tens. Find products : 2. 150 x 40 4. 805 xl6 6. 304 x 71 3. 107 x 35 5. 500 x70 7. 691 x 74 Multiply and test : 8. 6425 ■ a. 245 Form 100 problems by mul- 9. 1024 b. 344 tiplying each multiplicand by 10. 8720 0. 564 each of the multipliers, thus: n. 9652 d. 746 8 a. 245 x 6425 = ? 12. m\o . by . e. 804 8 5. 344 x 6425 = ? 13. 7894 /■ 961 15 i. 968 x 7695 = ? 14. 8465 9- 869- Write, solve, and test each 15. 7695 16. 8425 h. i. 796 968 problem in 1| minutes. 17. 9476, ■J- 898 28 FUNDAMENTAL PROCESSES 18 How much will 20,000 bricks cost at $7.75 per thou- sand ? Suggestion : 20,000 = 20 thousand = 20 M. 19. A ranchman sold 125 head of cattle at an average of $42.75 per head, and 625 sheep at $3.85 per head. Find the total amount of his sales. 20. The cost of drilling an oil well was 35/ per foot for drilling and 65^ for the tubing. If the well was drilled 1177 feet and tubed 700 feet, find the total cost. 21. The freight rate on corn in car-load lots from Omaha, Neb., to New York City is 20^ per hundred pounds. Find the freight on a car of 42,000 lb. 22. A freight train of 32 cars is laden with corn. The cars contain an average of 700 bushels of 56 lb. each. Find the weight of the corn in pounds. 23. A commission merchant sold 1275 barrels of apples at the rate of $3.25 per barrel, charging $.325 per barrel for selling. How much was realized from the sale of the apples after the charges for selling were deducted? 24. A man owned a farm of 142 acres, worth $72.50 per acre ; 5 city lots worth $ 1875 per lot ; and a business house worth $6350. Find the value of his entire property. 25. Frank lives 87 rods from the schoolhouse. How many rods does he walk in going to school 140 days, if he returns home each day for his dinner ? 26. Two trains leave a station at the same time. One travels west 38 miles per hour ; the other travels east 45 miles per hour. How far apart are they in 9 hours ? 27. A dealer bought a car load of coal, 42,000 lb., at $1.90 per ton of 2000 lb. If the freight was 70^ a ton, and he retailed the coal at $3.25 a ton, find his profit. DIVISION 29 DIVISION 1. How many times is the number 6 contained in 24? Division is the process of finding how many times one number is contained in another, or of separating a number into equal parts. The dividend is the number to be divided. The divisor is the number by which we divide. The quotient is the result of division. The remainder is the part of the dividend remaining when the quotient is not exact. The sign + indicates division, and is read, " divided by." Give quotients : 24 + 6 84 + 7 64 + 8 49 + 7 99 + 11 72 + 9 108 + 9 42 + 6 68 + 9 72+12 81 + 9 72+8 90 + 9 96 + 8 77 + 11 Division is indicated in three ways : 14 + 2; 2)14; and-M-. How many times are : 2. 2 inches contained in (may be taken from) 12 inches? 3. 4 yards contained in (may be taken from) 12 yards? If both the dividend and divisor are concrete, they must be like numbers. Compare 18 + 2 with J of 18 ; 15 + 3 with } of 15. How many : 4. Cents are | of 48 f ? 48^+8=— cents. 5. Cents does one orange cost if 4 oranges cost 20^? In separating a number into equal parts, the divisor is always an abstract number and the quotient is like the dividend. This kind of division is called partition. In the following, point out the problems in partition : 6. $120 + 10 7. \ of 40^ s. 24 ft. +2 ft. 30 FUNDAMENTAL PROCESSES Remainder in Division 34-r-5 = 6, and 4 remaining. $39 -4- 5 = $7, and $4 re- maining. Give quotients and remainders : $26 -=-6; 79 -h 8; $48 -4- $9; 37-*- 4; 84^-*- 8^; 49-4-7. Written Work l. Divide 236 by 3. Divisor 3)236 Dividend 78 Quotient Remainder 2 Test. — 3 x 78 = 234 ; 234 + 2 = 236 23 tens -=-3 = 7 tens, and 2 tens (20 ones) remaining. Write the 7 in tens' place. 20 ones + 6 ones = 26 ones ; 26 ones -4-3 = 8 ones, and 2 ones remaining. Quotient 78; remainder 2. We think: "3 in 23, 7 times, and 2 remaining; 3 in 26, 8 times, and 2 remaining." Find quotients : 2. 344 -*- 3 3. 763 ft. -4- 6 ft. 4. 466 i -4- 9 Divide $6.48 by 3. 3 )$ 6.48 Place the decimal point in the quotient directly $2.16 under the decimal point in the dividend. Divide and test : 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. $203,751 $678.34 $209.07 $390.08 $720.93 $379.38 $297.34 $427.84 $918.07 $847.12 by a. 2 b. 3 c. 4 d. 5 e. 6 /• 7 9- 8 L 9 i. 10 • 11 Form 100 problems by divid- ing each dividend by each of the divisors, thus : 5 a. $203.75 -f- 2= ? 5 6. $203.75-*- 3 = ? 9e. $720,934-6 = ? Write, solve, and test two problems in 1 minute. LONG DIVISION 31 Dividing by Larger Numbers 1. Divide 50, 90. 150, 600, 1 by 10. 2. Name the quotients when 130, 170, 1200, 2000, 100, is each divided by 10. 3. Divide 500, 600, 1500, and 2500 by 100. Removing one naught from the right of. a number divides the number by 10 ; removing two naughts, divides it by 100; remov- ing three naughts divides it by 1000 ; etc. 4. Divide 225 by 20, thus : 2 1 )22 '5 o tens j s contained in 1115 22 tens 11 times, with 5 20 remaining. 5 -=- 20 = 5 %. 5. Divide 2375 by 20 ; by 200. Divide each number by 20 ; by 50 ; by 80 ; by 500. 6. 37,845. 8. 90.200 10. 409,805. 12. 390,075. 7. 50,240. 9. 74,079 11. 790,086. 13. 985,000. LONG DIVISION 1. Divide 4310 by 21. Steps 205 1. Divide 43 by 21. Write the quotient 91 VPvFT) figure 2 over the figure 3 of the dividend. ' 2. Multiply 21 by 2. _ 3. Subtract 42 from 13. HO 4. Bring down the next figure. Is 21 105 contained an integral number of times in 5 remainder H? Write O in the quotient. Test: 21 x 205 = 4305 5> Brin S dowu the next fi S ure and P ro " ," _ '* - ,01 n ceed as before. Write 5 in the quotient. 4305 + o = 4310 Divide and test : 2. 252-21 7. 2214 -s- 21 12. 1326-*- 51 3. 525 -j- 21 8. 4601-4-22 13. 1922-62 4. 724-22 9. 1271-f-31 14. 2193-5-51 5. 642-31 10. 1344-42 is. 7010-4-91 6. 345-4-31 11. 1024-*- 32 16. 6874-81 32 FUNDAMENTAL PROCESSES Divide and test 17. 1364 by i >2 25. 6207 by 76 33. 8538 by 94 18. 1395 by 31 26. 6572 by 68 34. 7646 by 87 19. 1728 by 42 27. 7010 by 91 35. 8544 by 79 20. 2193 by 51 28. 7284 by 92 36. 9584 by m 21. 2583 by 63 29. 6874 by 81 37. 7001 by 84 22. 3034 by 74 30. 6986 by 83 38. 8200 by 77 23. 4345 by 65 31. 7044 by 86 39. 7909 by 96 24. 5072 by 59 32. 8406 by 92 40. 8549 by 78 41. 1009 42 $11.17 715)* 7986. 55 43. 544 395)398555 805)437920 395 715 4025 3555 836 3542 3555 715 1215 3220 3220 Since 35 does not coi itain 715 3220 395, the second figure in the 5005 quotient is 0. 5005 Divide and test : 44. 6464341 ' a. 268 Form 100 problems by di- 45. 7846760 b. 354 viding each dividend by each 46. 5864548 c. 676 of the divisors, thus : 47. 8645341 d. 758 44 a. 1 3464341 -h 268 = ? 48. 9624872 1 e. 865 44 b. I 3464341 -f- 354 = ? 49. 7784100 ■ by ■ /. 984 49 c. ' r784100 -676 = ? 50. 6810404 g. 789 Write ', solve, and test each 51. 7904025 A. 897 problem in 2 minutes. 52. 4867045 i. 509 53. 3234567 J. 890 MULTIPLICATION AND DIVISION 33 Problems of Two or More Operations 1. If 48 barrels of flour cost $324, how much will 275 barrels cost? Cost of 4S bbl. = $ 324.00 Study of Problem Cost of lbbL = 1824.00 -«-48 = * 6.75 i. What is given in Cost of 275 bbl. = 275 x $6.75 = $1856.25 this problem? a. Number of barrels in each purchase, b. Cost of 48 bbl. 2. What is required? a. Cost of 1 bbl. b. Cost of 275 bbl. 3. How do you find what is required? a. Divide cost of first pur- chase by the number of barrels, b. Multiply the cost of 1 bbl. by the number of barrels purchased. Note. — The purpose of these studies is threefold : 1. To train the pupil to see and understand the conditions of a problem. 2. To give that logical, analytic grasp of conditions that forms the basis of all mathematical power. 3. To direct the teacher in his efforts to attain these ends. 2. If 2675 bushels of wheat cost $2728.50, how much are 196 bushels worth? 3. A water tank holds 8640 gallons. If it receives 728 gallons per hour by one pipe and discharges 512 gallons by another, in what time will it be filled ? 4. Two steamers sail towards each other from opposite sides of the Pacific Ocean. If the distance across is 9872 miles, and one sails at the rate of 285 miles a day, and the other 332 miles per day, in how many days will they meet ? 5. The daily pay of a railway conductor is $3.45. If he works 310 days in a year, and spends on an average $65 per month, how much has he left at the end of the year? 6. The receipts of a street railway for 365 days were $119,685.23. Find the average daily profits if the total expenses were $96,478.02. 7. A and B divide an estate of $9875 between them. If A receives $275 more than B, how much does each receive? 34 FUNDAMENTAL PROCESSES 8. A man sold 128 acres of land at #70 an acre, and 96 at $90 an acre. He invested the money in town lots at $550 each. How many did he buy ? Study of Problem 1. What is given in the problem? $8960 value of 1st farm 2 What is required? $8640 value of 2d farm 3. What is the first step in the $17600 value of both solution? the second? the third? the fourth? $17600 -r-$ 550 = 32, no. of lots bought. 9. If it costs 40 cents to ship a 10-gallon can of milk from Hickory to Pittsburg, how much does the railroad realize in 5 days, from a shipment of 135 cans per day? 10. A shipper pays 20 cents per barrel, per month, cold storage charges on apples, and 15 cents per firkin on butter. Find the charges for three months on 45 firkins of butter and 328 barrels of apples. 11. A locomotive in making a certain trip uses 18 tons of coal. If a trip is made in 2 days, how much coal will the engine consume in 190 days? 12. A dealer buys three boxes of oranges for $3.50, $2.75, and $2.50, respectively. If he sells 10 dozen at 50 cents per dozen, 9 dozen at 40 cents per dozen, 12 dozen at 35 cents per dozen, and the remaining 5 dozen at 25 cents per dozen, find his gain. 13. An opera sale of tickets is as follows: 450 @ $1.50; 380 @ $1.00; 520@$.75; 310@$.50; and 240 @ $.25. Find the total sale of the tickets, and the average cost of each ticket. 14. A steamboat consumes 23 tons of coal per day. Find the cost of the coal, at $5.85 per ton, for a trip of 39 days. COMPARISON COMPARISON Comparison, as here used, indicates the relation of two simi- lar numbers, expressed by the quotient of the first number divided by the second. 1. Compare 10 and 5; 12 and 4; 16 and 8; 20 and 5; 24 and 6. 2. 20 is how many times 4? 30 is how many times 6? How does 40 compare with 4? 3. What is the quotient of 48 divided by 8? by 6? by 4? 4. Compare 200 and 50; 400 and 100; 500 and 250. 5. 125 is what part of 250? of 500? of 375? of 625? 6. Compare 48 feet and 2 yards: 75 feet and 5 yards. Xote. — Change yards to feet, or feet to yards ; then compare. 7. 6 feet is what part of 3 yards? of 10 yards? Written Work 1. When 4 pounds of butter cost 80 cents, how much will 12 pounds cost? Xote. — 12 pounds = 3x4 pounds: hence, 12 pounds will cost 3 x 80 cents. 2. Find the cost of 10 barrels of apples, when 2 barrels cost 64.50. 3. At 3 pounds of coffee for $1, how much will 15 pounds cost ? 4. How much will 30 yards of silk cost when 3 yards cost 63.75? 5. When 2 doz. oranges are selling for 60 cents, how much will 8 dozen cost? 6. Find the cost of 20 barrels of cement, when 5 barrels cost 66.25. 7. How much will ••'><> dozen eggs cost, when •*• dozen sell for si? 36 FUNDAMENTAL PROCESSES COMBINING PROCESSES A parenthesis ( ) or a vinculum ' " groups together several numbers and shows that the operations within the groups are to be performed first ; thus, 6 — (3+ 2) = 6 — 5 = 1 ; (5 + 3) x 2= 8x2= 16; 5 + (3x2) = 5 + 6=ll; 32-^4 + 3 = 8 + 3 = 11; 4x2-3=8-3 = 5. When no parenthesis or vinculum is used, the signs x and -=- indicate operations that are to be performed before those indicated by either + or - ; thus, 4 + 8 x 3 = 4 + (8 x 3), or 28 ; 5 + 12 -=- 6 = 5 + (12 -*- 6), or 7. In an expression like 12 -s- 6 x 2, mathematicians are not agreed as to which sign shall be used first. To avoid ambi- guity, the parenthesis should be used in such expressions. Thus, (12 -s- 6) x 2 =4 ; but 12 + (6 x 2) = 1. Find the value of : !. 4 x 12 - 16 + 4. 5. (240 + 98) x (688 - 425). 2.7 + 8x7-26. 6. (56-18) x 11 + 4-6 x 4. 3 . (14 + 8 - 6) x 9. 7. (84 - 7 x 6 + 9 x 4 - 6) + 9. 4 . (87-65 + 96) x 24. 8. (56 + 7) x 12 + 97-7 x 9. 9. 6+10x5 + 8 + 2-4-2 + 8. 10. 7 x 5~+4 + 8^T6 + 2 - 3 x 4. 11. (6-+ 2 x 3) + 4 + (3 x 6) + 2 + 2 x (3 + 5 - 2). 12. 36- 6x4 + 2x6+ (40 + 5) +9 + 3x6. 13. 10 + 20-5 x 3 + 6x2-=- 3 + 5x6. 14. 3x(4+5-2) + 4+5x(4x5 + 2)+5. 15. 3 x (6 + 8) + 7 x (8 + 2) - 3 x (6 + 3) + 15 - 7. 16. 175 - 8 x (19 - 10) - 25 -*- 5 + 6^7 -9 + 3. FACTORS AND DIVISORS 1. What two numbers will give 6 as a product? 8 as a product? 10 as a product? 2. What are 2 and 3 in relation to 6? 4 and 2 in relation to 8? 5 and 2 in relation to 10? An integer or an integral number is a whole number. The factors of a number are the integers whose product is the number ; thus, 5 and 2 are factors of 10. 3. Name two factors that produce 24, 32, 40, 56, 49, 72, 96. A factor of a number is an exact divisor of the number ; that is, it is contained in the number an integral it umber of times. 4. Name the exact divisors of 54, 81, 48, 36, 66, 64, 63. 2 x2 = 4 5. Observe the two equal factors that pro- 3 x 8 _ 9 duce 4; 9; 16. 4 x 4 = 10 2x2x2=8 6. Observe the three equal factors that pro- 3 x 3 x 3 = 27 duce 8 ; 27 ; 64. 4 x 4 x 4 = 64 Instead of repeating a factor, a small figure called an exponent may be written to the right and a little above the number to show how often it is used as a factor ; thus, 33 =3x3x3 = 27: 2 4 =2x2x2x2 = 16. 7. What number will divide 9 and 10? 21 and 25? Numbers are prime to each other when they have no com- mon factor ; thus, 9 and 10 are prime to each other. 37 38 FACTORS AND DIVISORS Even numbers are numbers that contain the factor 2. Odd numbers are numbers that do not contain the factor 2. 8. What are the factors of 7? of 11? Observe that 7 and 11 have no exact divisors except themselves and one. A prime number is one that has no exact divisor except itself and one ; thus, 5, 2, and 3 are prime numbers. 9. Name all the prime numbers to 31. 10. What are the factors of 15? Observe that 15 can be divided by 3 and 5. It is composed of other factors than itself and one. A composite number is one that has other exact divisors than itself and one ; thus, 6 and 10 are composite numbers. 11. Name all the composite numbers to 50. TESTS OF DIVISIBILITY 1. Divide 12, 24, 26, 38, and 50 each by 2. What is the ones' figure in each of the dividends? Divide other num- bers ending in 2, 4, 6, 8, or by 2. A number is divisible by 2, if the ones' 1 figure is 2, 4, 6, 8, or 0. 2. Divide 15, 25, 40, 125, 150 each by 5. What is the ones' figure in each dividend? Divide other numbers end- ing in 5 or by 5. A number is divisible by 5, if its ones' figure is 5 or 0. 3. Divide 36, 69, 48, 72, 162, 369 each by 3. Notice that the sum of the digits (that is, of the figures) in each number is divisible by 3. Divide by 3, other numbers the sum of whose digits is divisible by 3. A number is divisible by 3, if the sum of its digits is divis- ible by 3. FACTORING 39 4. Divide 18, 27, 279, 819, 639 each by 9. Notice that the sum of the digits in each dividend is divisible by 9. Divide by 9, other numbers the sum of whose digits is divis- ible by 9. A number is divisible by 9, if the sum of its digits is divisible by 9. 5. Select the numbers that are divisible by 2 ; by 3 ; by 5 ; by 9. 86 96 123 918 515 3672 94 72 321 819 450 1909 FACTORING 1. Give the two factors that produce 15. 2. If one of them is given, how may the other be found ? To separate a number into two factors, take any exact divisor for one factor and the quotient of the number by this factor for the other. Factoring is the process of separating a number into its factors. A prime factor is a prime number used as a factor ; thus, 3 and 5 are the prime factors of 15. Written Work 1. Find the prime factors of 126. Divide by the least prime factor ; divide the quotient by the next smallest prime factor, etc.. until the last quotienl is a prime number. The divisors and _ o o "_io£ the last quotient are the prime factors; thus, 2, 3, 3, and 4 are the prime fac- 0*i tors of 126. 2 x3 2 x 7 = l:2i; 2 3 3 126 63 21 40 FACTORS AND DIVISORS Find the prime factors of : 2. 125 6. 945 10. 2431 14. 25600 3. 210 7. 2934 li. 7200 15. 64640 4. 225 8. 4620 12. 7700 16. 97125 5. 400 9. 3822 13. 6525 17. 78000 GREATEST COMMON DIVISOR 1. Name a number that will exactly divide both 16 and 24 ; 15 and 25 ; 14 and 27. A common divisor of two or more numbers is a number that exactly divides each of them ; thus, 4 is a common divisor of 16 and 24. 2. Is 4 the greatest number that will exactly divide 16 and 24? What is the greatest number that will exactly divide 16 and 24 ? The greatest common divisor (g. c.d.) of two or more numbers is the greatest number that exactly divides each of them; thus, 9 is the g.c.d. of 27 and 36. 3. Name the g. c. d. of 24 and 36 ; of 32 and 40. Written Work l. Find the greatest common divisor of 56, 98, 154. 56 98 154 As the g.c.d. of two or more numbers is the 28 4^> 77 product of all their common prime factors, divide the numbers by their common prime factors. In the same way divide the quotients until they are prime to each other. The divisors 2 and 7 are all the common prime factors of the numbers. Hence, the g.c.d. of 56, 98, and 154 is 2 x 7, or 14. 2 7 4 7 11 g. c. d=2 x7, or 14. LEAST COMMON MULTIPLE 41 Find the p-.c.d. of: 2. 42, 63, 189 7. 84, 56, 210 3. 54, 216, 360 8. 22, 110, 132 4. 48, 60, 96 9. 42, 84, 175 5. 84, 252, 512 10. 17, 68, 85 6. 21, 48, 78 11. 432, 720, 864 LEAST COMMON MULTIPLE 1. Name a number that will exactly contain 6 and 9 ; 8 and 12; 7 and 9. A common multiple of two or more numbers is a number that is exactly divisible by each of them ; thus, 36 is a com- mon multiple of 6 and 9. 2. Name the least number that is exactly divisible by 6 and 9; hy 8 and 12. The least common multiple (l.c.m.) of two or more num- bers is the least number that is exactly divisible by each of them ; thus, 18 is the 1. c. m. of 6 and 9. 3. Name the 1. c. i». of 6 and 8 ; of 9 and 12 ; of 8 and 12. Written Work l. Find the 1. cm. of 18, 32, and 40. 18 = 2x3x3 The I. c. m. of two or more numbers is QO_ O y •) x O x 9 x 9 the product of all their prime factors, tn \ r. -^ r each factor being used as often as it 40 = x x 2 x 5 occurs in any number. 1. c. m. = i 5 x o 2 X 5, or 1440. o occurs 5 times as a factor in 32. It must, therefore, be used 5 times in the 1. cm. 3 occurs twice as a factor in 18; it must, therefore, be used twice in the 1. c. m. 5 occurs once as a factor in 1<>; it must, therefore, be used once in the 1. c. m. Hence, the 1. c. 111. of is, 32, and ID is L )5 x 3 2 X 5 = 1110. 42 FACTORS AND DIVISORS 2. Find the ]. c. m. of 12, 36, 54, and 63. '** — - Since 12 is a divisor of 36 the I.e. m. 3 )1§ '4l §§ of 36, 54, and 63 is also a multiple of o )b 9 ^1 i2. 12 may therefore be rejected from 2 3 7 the work. 1. c. m. = 2 2 x 3 3 X 7 = 756. Divide any two of the numbers by a common prime factor. Then divide the quotients in like manner until the quotients are prime to each other. The product of the divisors and the last quotients is the 1. c. m. Find the 1. c. m. of : 3. 24, 48, 72 10. 48, 64, 72 4. 36, 70, 105 li. 144, 180, 240 5. 32, 40, 48 12. 85, 51, 255 6. 25, 35, 56 13. 120, 225, 540 7. 30, 60, 105 14. 98, 42, 126 8. 32, 48, 96 15. 180, 216, 120 9. 45, 70, 90 16. 100, 110, 440 CANCELLATION 144 _:- 36 = 4. We may separate the dividend 144 into the factors 9 and 16, and the divisor 36 into the factors 9 and 4. We may, therefore, write 144 h- 36 = 4 as follows : (9x16) -(9x4) = 4. By striking out the common factor 9 in both dividend and divisor, the problem is: (16 -j- 4) = 4. Striking out equal factors from both dividend and divisor does not change the quotient. When the product of a number of factors is to be di- vided by the product of another set of factors, the usual way is to write the dividend above a line and the divisor below, and strike out equal factors : CANCELLATION 43 Thus, ^=2xl = Z = 2i. Cancellation is the process of shortening operations in division by striking out equal factors from both dividend and divisor. Written Work 1. Divide 3 x 6 x 8 x 20 by 11 x 4 x 20. 2 3x6x8x2p_36_ os Write the dividend above and the divi- 11 X 4 X 20 ~ 11 ~ ! T ' sor helow a line. First cancel the 20 from dividend and divisor. Then cancel the factor 4 from 8 in the dividend and from 4 in the divisor, leaving 2 in the dividend and 1 in the divisor. As there are no other factors common to dividend and divisor, you have 3x6x2, or 36, divided by 11, or ff, which equals 3 T 3 T . Note. — When equal factors in the terms are canceled, the factor 1 always remains, but as it does not affect the product, it need not be written. Divide : 2. 27 x 56 x 38 x 50 by 19 x 35 x 40 3. 5 x 51 x 36 x 63 by 17 x 9 x 54 x 10 4. 25 x 72 x 64 x 28 by 40 x 96 x 21 x 4 5. 69 x oG x 45 x 27 by 23 x 45 x 63 x 9 6. 72 x 48 x 84 x 28 by 24 x 48 x 42 x 14 7. 148 x 64 x 57 x 12 by 114 x 32 x 48 8. By selling butter at 24 ^ per pound a lady receives enough money to buy 48 pounds of coffee at 20^ per pound. How many pounds of butter does she sell ? 9. A man worked 16 days of 10 hours each at 20^ per hour, and spent the money he received for corn at 40 ^ per bushel. How many bushels of corn did he get ? FRACTIONS FRACTIONAL UNITS When we say 8, 6 ft., $2, 6 rd., 7 mi., 5 in., what are the units of measure ? Observe that in each case the number and its unit of measure are of the same denomination ; thus, 1 ft. is the unit of measure in 6 ft. A unit, therefore, is any single quantity with which another quantity of the same kind is measured or com- pared ; as, 1 is the unit of 10 ; 1 ft. is the unit of 8 ft. ; 1 yd. of 2 yd. ; 1 mi. of 12 mi. ; 1 acre of 5 acres, etc. A fractional unit is one of the equal parts into which an integral unit has been divided ; as, J^, -|, ^, ^, y 1 ^, etc. A fraction is one or more fractional units ; as, |, f , |, |, •|, etc. The terms of a fraction are the numerator and the denomi- nator. The denominator indicates the size of the fractional unit ; it is written below the line, and shows into how many parts the integral unit has been divided. Thus, in the fraction |, 5 is the denominator, and shows that some unit has been divided into 5 parts. The numerator indicates the number of fractional units; it is written above the line, and shows how many parts are taken. Thus, in the fraction |, 4 is the numerator, and shows that 4 parts have been taken. 44 FRACTIONAL UNITS 45 1. What is the fractional unit in jj ? { ? | ? f ? 2. Read the following fractional units in order of their size, beginning with the largest: -, 1 ,., |, J, y 1 ^, £, |, and .,'j. 3. The use of the numerator and the denominator in T 9 2 yd. may be explained thus, -^ yd. = 9 x ^ 2 - yd. As the integral unit is the basis by which we measure whole numbers, so the fractional unit is the basis by which we measure fractions of the same kind. 4. Name the unit of 4 ft.; 5 mi.; f; §. 5. Which is the larger, | or 1 ? | or 1 ? | or 1? Explain how much larger in each case. 6. What is the difference in value between the fraction | and an integral unit ? A common fraction is a fraction that has both terms ex- pressed; as, f, f, \. A proper fraction is a fraction less in value than 1 ; as, ^, 7 3 4 1 1 3 pfp JT 4' 5' 9 1 T 1 3' euo " An improper fraction is a fraction equal to or greater in value than 1 ; as, |, f, |, f , f , ^g -, etc. A mixed number is a number expressed by a whole number and a fraction ; as, 3^, 12|. Change each of the following to integral units, or to mixed numbers. Thus, | = 1^. 10. ^ 11. V 12. 17- 7. 10. 8. f 9. 6 13. ¥ 16. ¥ 14. 9 17. ¥ 15. ¥ 18. ¥ 46 FRACTIONS READING AND WRITING FRACTIONS Read l. 2. 3. 4. 5. ! 6. 15 40 8 9 7. 4 10 1 1 12 8. 45 ft 1 6 9. 38. 110 H 10. 1 1 5 21? 11. 12. 13. 14. 15. s$12§ To bu - 6| bbl. 1 Qfi 6 1 oz. = 7 4iii °f a ton. Write in figures : One fourth. Three fifths. Six ninths. Three fourths. Seven tenths. 1. 2. 3. 4. 5. 11. 12. 13. 14. 15. 16. 17. 18. 6. Eighteen twentieths. 7. Eight thousandths. 8. Sixty seventieths. 9. Nineteen forty-thirds. 10. Eighty-nine thousandths. Eighty-nine three hundredths. Twelve and three fourths. Six and three fourths. Five and one half. Five ninths of three fifths. One thousand ninety-fourths. Nine hundred three thousandths. Ten and three fourths. 19. Four hundred ninety and six thousand twenty-four ten-thousandths. Write in words : 20. 21. 22. 23. 4 5 L 9 11 1^ i 24. 25. 26. 27. A 35 ¥6 11 85 1 05 ITS 28. 29. 30. 31. 45- 9 - "22 2 - UU 1000 - LVU1 2000 KEDICTIOX OF FRACTIONS REDUCTION OF FRACTIONS Reduction is the process of changing the form of a number without changing its value. 1. Divide both terms of the fraction | by 2. How does | compare in value with l ? 2. Multiply both terms of the fraction | by 2. How- does | compare in value with | '! How may we obtain | or •^2 from | ? Multiplying or dividing both terms of a fraction by the same number does not change its value. Changing a fraction to higher terms. 1. Explain why a fraction is expressed in smaller fractional units when it is changed to higher terms. 2. Explain why changing a fraction to higher terms does not change the value of the fraction. Change : 3. i to 12ths 6. ^to^ths 9. lJto72d& 4. | to 24ths 7. | to 56ths 10. | to 63ds 5. f to 18ths 8. f to 81sts n. | to 96ths 12. f =^ = A 16 ' 13 -9_ — _?_ = J- 17 J-«. in — en — on ■*■'• 5 _ J_ - f_ 6 "~ 36 " Y2 - •> 9 TO - 60 — 90 ■ L/ *8 "4'0 ~6¥ * -io K ? ? 14. A = A =A 18- ■ i IT — I? _ 58" *""' 12 — 96 — 7 2 15. n = st = TTfO ^•" _6_ — 9 — 8T ~ 10¥ "• 11 ~" 132 ""99 Written Work l. Change f to 27ths. 27 -i- 9 = 3 Since multiplying both terms of a frac- 5 5x3 15 ti° n by the same number does not change q = q S = 27 ^ vame ' multiply both terms of the frac- tion by the quotient of 27 -*■ 9, or 3. 48 FRACTIONS Change 2. 3. 4. 5. 6. 7. 8. | to 20ths I to 56ths l| to 96ths T 9 3 to 78ths 11 to 276ths 9. 10. 11. 12. if to 132ds 13. l| to 72ds 14. -^ to 135ths 15. Changing a fraction to lower terms or to lowest terms. 1. Explain why a fraction is expressed in larger fractional units when it is changed to lower terms. Explain also why changing a fraction to lower terms does not change its value. Change : ff to 12ths l| to 275ths if to 372ds f! to 494ths !! to 765ths fi to 415ths |f to 315ths 2. A to 4ths 6. 7. 12 || to Cths |f to 8ths |f to lOths £f to 12ths |§ to 8ths 8. 9. 10. 11. 12. 13. || to Gths ff to 9ths ff to lGths 4 50 ? _ 4 9 12 — 8¥ 14. IS. 16. 17. 18. 19. _?_ 21 ? 9 9 8 9 t ? 9 21 63 _4_8_ 10 8 13^ 144 24 64 35_ 5 6 40 Y2 Written Work A fraction is expressed in its lowest terms when the nu- merator and the denominator are prime to each other. l. Change || to lowest terms. Since dividing both terms of a fraction by the same number does not change its value, we may reject by cancellation all the factors common to both terms, leav- ing the factors 7 and 9. Hence, ££ = J. Or we may, in one step, divide both terms of the frac- tion by their g. c. d., 6. m_ n_ 7 u ~%7l : 9 Or g.c.d 6 42 ■f-6 7 54 *-6 9 REDUCTION OF FRACTIONS 49 2. Change |§£ to lowest terms. 357 + 3 = 119 + 7 = 17 r ff cd-21 35T - 21 17 483 + 3 161 + 7 23 ur 'S- c - a --^ 483 + 21 = 23 Cancel all the factors common to both numerator and denomi- nator. Or, divide both numerator and denominator by their greatest common divisor. Change to lowest terms : 3. 4. 5. 6. 7. 8. 9. 10. 11. Changing a mixed number to an improper fraction, or an improper fraction to a mixed number. 18 24 12. 1 2 1 "13 2 21. 125 325 30. 4£5 53 39. 4 14 999 25 55 13. 54 12 22. 3&5 605 31. 75 S2 5 40. 1 2ii 189 42 49 14. 18 28 23. 480 66 32. 615 94 5 41. 4 3 5 630 72 81 15. 42 4 8 24. 18 2 196 33. 4i2 504 42. in 1 96 21 36 16. 3£ 42 25. 264 333 34. 67 2 936 43. 2 1 6 270 .24 28 17. 58 T4 26. 315 34 5 35. 256 92 4 44. 546 5 88 ^5 18. 128 176 27. 200 450 36. 551 62 1 45. 396 4 :; 2 22 Y2 19. 94 144 28. £28 624 37. 294 476 46. 561 783 84 96 20. 81 96 29. 288 444 38. 322 504 47. 837 945 Change to an improper fraction at sight : 1. If 4. 31 7. u 10 2. n 5. 5| 8. 3-2- °15 3. 01 ~3 6. 2| 9. 8 to 12ths 11AM. 1 "Mil.. AJOIH. — 4 10. 1 2 to 3ds 11. 8 to 6ths 12. 10 to 5ths 50 FRACTIONS Written Work l. Change lllf to an improper fraction. 1 _ 5. In 1 there are § ; in 111 there are 111 -i-ii iii x JL - - JL5.5. times |, or £§ s , which added to f equal £56 I 8 -- JL5 8 " 5 "- Hence ' * 11 * equals^. J,- "T" 6 — 5~ In small numbers the work may be done mentally, only the result being written. Change to improper fractions : 2. 12f 8. ™H 14. 268Jft 20. 391ft 3. 15^ 9. 9* 15. 324f§ 21. 18** 4. 22| 10. 103f| 16. 502* 22. 901^ 5. 80ft 11. 118|i 17. 109^ 23. 100^ 6. 48 T i 12. 1,,J 66 18. 600JJ 24. 390|| 7. 56i| 13. * dit 'l0 5 19. 305f 25. 231H 26. Change -1| 4 to a mixed number. In 1 there are ?, and in if 8 there are as 12 8 _ 128 -7- 7 = 181 many times 1, as 7 is contained times in 7 128, or 18f. Hence, if* is equal to 18$. Every fraction is an indicated division. Change to integers or mixed numbers : 27. 31 15 33. w 39. 34 45. 2 3^8. 15 28. _5_2JL 11 34. J70 35 40. 876 38 46. 14 4 75 29. 955. 7 35. 18 41. 24 47. 5286 48 30. 13JL 16 36. 625 "2 5 42. 862 "84 48. 4646. 50 31. 261 8 37. ,5 7 6 24 43. 534 96' 49. 2200 12 32. w 38. .240 16" 44. 121A 24 50. 4 03 2 "36 REDUCTION OF FRACTIONS 51 Changing to least similar fractions. A common denominator of two or more fractions is a num- ber that contains all the denominators of the fractions an integral number of times ; thus, 24 is a common denominator of ^, ^, and J. The least common denominator (Led) of two or more fractions is the least number that contains all the denomina- tors of the fractions an integral number of times ; thus, 12 is the Led. of }, ^, and J. The 1. c. d is the least common multiple of the denominators. Similar fractions are fractions that express the same unit value. They must therefore have a common denominator. By inspection : 1. Change |, |, and ^ to similar fractions having the least common denominator. It is evident by inspection that 18 is the least common multiple of 2, 3, and 9. It is therefore the least common denominator of the given fractions. Changing the given fraction to ISths. we find that \ = T \ ; t=\t\ and s = jo. Hence, the fractions h §, and | may be changed to the similar fractions ^, ff, and i§. Change to least similar fractions 2. 3. 4. 5. 6. 1 x 0_ 2x9 9 : 18 2x6. 3x6 12 = 18 5 x 2 _ 9 x 2~ 1° 18 t 3' 1 5 4' 12 7. 1 5' 3 7_ 10' 40 12. 2 3" 1 19 6" 36 17. 1 6' 3 JL 7-14 2 3' 1 2_ 5' 15 8. 1 3' A»A 13. 1 3" 1 1 2* 6 18. 1 3' 1 • • 2 h 3' 1? 9. 2 5' 3- 9_ "8"' ¥5 14. 3 7- 1 1 14' - - 19. 2 5' 2 3_ ;;• -JO i 6' 3 17 - • 21 10. 1 9' 1 -5_ 6' 18 15. 3 5 7 If 1 ', 20. 4 7" 5 1 8"' 5 6 1 9' _4_ _§_ 18' 3 6 11. 3 5' 7 JL 10' 2 16. 1 1 i r 53 21. ■ l 3 52 FRACTIONS By factoring the denominators : l. Change |, f , and ^ to similar fractions having the least common denominator. 4=2x2 In finding the I.e. m., use each factor O = -i X o as fteu as it occurs in any one number. 10 = 2 x 5 1. c. m. = 2x2x3x5 = 60 3 x 15 = 45 4 X 15 60 ,The least common multiple of the denominators is 5 x 10 50 ^' w °i cn * s the l eas t common denominator of the £ 1 ft == ~fU\ gi yen fractions. Changing the given fractions to o X a n i 60ths ' we find that I = M; I =»5 and & = M- 9 x 6_54 10 x 6~60 Change to least similar fractions : 2 4 2 3 1 o 2 19 _5_ U 11 _L L J 29 *' 4' 5' 8' 2 °' 5' 20' 12' 30 x *' 10' 12' 20' 3~0 3 2 _5_ _7_ 1 o 1 1 _9 11 is 3 _9_ Q3 9H °* 3' 16' 12' 4 *' 7' 2' 28' 18 xa " 1?' 16' °8' ^2 1 4_7_i _8_ 9. in 1 _9_ _8_ _3_ ic J_ 11 3 5 1 *' 10' 5' 15' 20 1Ul 4' 10' 35' 16 - LO- 2 1' 35' Y0' 120 5 2 4 15 -n _7_ J>_ 11 li 17 5 8 _&_ 31 a * IT' 5' 8' 6 xx * 18' 15' J 9' 30 - L/< 6' 9' 2 1' 63 6 & 12 11 io 1 _5_ _9_ 31 in 8 14 Q 7 ° - 9' 2 5' 18f ""' 18' 2 1' 2 8' 54 xo - 9' 2 5' °T5 7 2 5 _8_ 13 31 11 IS 12 iq 9 16 19 GJ '• 7' 9' 11 ■ LO - °2' 18' A 6' 24 x *' 22' 3^' 66' 132 ADDITION AND SUBTRACTION Size and kind of fractional units. 1. Why is it not possible to add 4 ft. to 5 oz. ? 2. What is the sum of \ and | ? of | and § ? 3. What is the size of the fractional unit of £ and | ? 4. What is the kind of fractional unit in each ? 5. Are the fractional units of | and | alike in size and kind ? ADDITION AND SUBTRACTION 53 6. What is the size of the fractional unit in $| and | ft.? What is the kind of unit in each ? 7. Why can we add or subtract $ £ and $-§, or £ ft. and 2 ff 9 8. What kind of fractions can be added or subtracted ? Before fractions can be added or subtracted they must be ex- pressed in similar fractional units. Fractions having the same kind of units or having related units can be added; thus, f and J, or f yd. and \ ft. (= T \ yd.) can be changed to similar fractions and then added. But fractions whose units are unrelated, as f yd. and \ oz., cannot be added, because they cannot be changed to similar fractions. 9. 5.] ft. and 2 yd. = ft. Why must you change yards to feet before adding ? 10. Add the fractions in the following list that are similar: \ da. |rd. 3 8 2 3 hr. ft. 3 fin -9- tV min - i min. ft. S in-- ird. Add quickly : 11 1 + 1 ■"■■ 2^4 15. £_i_ 1 5^2 19. £ + | 23. 5 4-1 6 ' 3 12 2.4-1 ±di - 3 ' 6 16. 4.1 3^2 20. f+1 24. 1 4- 8 3 • 6 13 3 _i_ 1 17. 2 . 1 3^9 21 -Ui **■■ 5^2 25. Bj.5 4 + ff 14 1 4_ 2 J-*. 4 -t" 3 18. 5 _i_ 3 1+2 22 I 1 4- 2 12 ' 3 26. i+^ Written Work l. Find the sum of f, |, and |. The least common denominator is 36. Chang- ob = 1. c. d. j n g tj ie gj ven fractions to 36ths, we find that | = || ; 3 = |i ; | = 3|, The sum of these fractions 2 3 24 ! 27 8 9 32 — 83 — 011 -T5 36" 8S — Oil 1. What is the first step in the work? 2. "What is the second step ? 3 6 3 6 3. What is the third step? 54 Add: o 5. 7 A 7 3 2 _9_ _9 3 2. g> $, f /• ■$, ^ 10 •"• ^' 12' 32' 4 FRACTIONS 3 4' 2 "5"' 9 10 9 T' 9 14 3 1 4 2 5' 3 7' 1 9 2 3' 3 5' 1 5 4' 6 1 ' 6' 2 3' J7_ 14 12' 15 o 3 9 1 o 2 9 3 13 1 _5_ _3_ 11 3. y, -J4, 2 **• Y' 14' 4 A 8' 12' 16' 2 4 a. 5 11 3 a 2 3 1 14 JL 2 11 25 4 - Y' "2 5' f 9 5' 7' 9 ■'■*• 14' 7' 3 5' 4 9 c 2 1 1 in 2 3 1 5 K 1 1 J_ 12 5 - 31 ?i 2 - 10- 3' 5' 4' 6 4' 5' 10' 2 5 c 5 7 5 -n 12 7 14 16 1 -3 1 -7_ 6 - 8' 12' 6 ■"" 6' 3' 12' 15 ±0 - 4' 8' 5' 12 17. Find the sum of 3^, 2$, and 7ft. -jqq_] (J Since the numbers are mixed numbers, the integers and fractions are added separately, and their sums are united. The sum of \, f, and T 4 o is f^,or l T 3 gV The sum of 3, 2, and 7 is 12. The sum of 12 and 8i 45 100 72 i T s^ i s 13/^. 12 + ftj = 12 + lftft = 13^ Add : 18. 3f, 7f, 9ft 24. 20f, 12Jf, 5|f 19. 18ft, 3y, 9ft 25. 14ft, 32^, 23ff 20. 8f, 10ft 16ft 26. 2f, 7f, lift, 14ft 21. 7f , 12ft, 24ft 27. 90ft, 60ft, 73ft 22. 16|, 30f,45ft 28. 3-i, 4f, 6f, 8ft 23. 50ft, 48ft, 16J| 29 - 84 T2' 36 i 33 f' 39 f 30. A bicycler rode 8| miles the first hour, 7| miles the second hour, 6ft miles the third hour, and 8^ miles the fourth hour. How many miles did he ride in the four hours ? 31. Find the distance around a field 80| rods long and 60| rods wide. 32. Find the sum of the improper fractions f , ff , ft' io- 33. What number is that from which if 32ft is taken, the remainder is 23y 9 ft ? ADDITION AND SUBTRACTION 55 Subtracting fractions. What kind of whole numbers can bo added ? what kind of fractions ? In adding like fractions we find the sum of the numer- ators ; in subtracting like fractions we iind the difference of the numerators. l. From | take T 3 g 72 = 1. c. d. J 63 3 18 12 5 1 _ 1 7 ~7 2 - 2 4 Written Work Since fractions must be made similar before they can be subtracted, f and -Ag are changed to 72ds. The difference between $| and \\ is f|, or \\. 1. 2, 3. Observe the ihret steps in subtraction of fractions: Change the fractions, if necessary, to like fractions. Take the difference of their numerators. Change the difference to its simplest form. Find differences 2. 3. 4. 5. I _ 3 9 4 JL _ 11 10 1^ J 5L 12 18 2 5 27 11 12 6. 7. 8. 9. 5 5 96 1 1 — 24 1 3 15 -H 2:? 2 4 14 15 3 108 _ 1 3 36 10. 11. 12. 13. i_i _ in 12 13 2 3 36 13 2 5 21 _ 12 63 3 5 15 5 6 1 S 12 14. A boy wishes to buy a pair of skates costing 8^, but he has only 6f. What part of a dollar is lacking? 15. From | take f. 16. What fraction added to | will give || ? 17. From -| of an acre of land, subtract | of an acre. 18. What part of a teacher's salary remained after he had spent 1, A^-. and 1 of it ? 7 56 FRACTIONS 19. The minuend is |f, and the remainder f. What is the subtrahend? 20. If a boy spends I of his money one day, \ of it the next day, and has $1£ left, how much money had he at first: Subtracting mixed numbers. Written Work 1. From 1\ take 3$. 7\ = 6 + | + \ = 6|. The integers and fractions are 36 =1. C. d. subtracted separately. The least common denomi- 74 = 6| 45 nator is 36. Changing the fractions to 36ths we find 3 5 20 that f= H, and $=${}. tt ~ ** = H- 6-3 = 3, o^ — |J which added to f£ = 3f|. Hence, the difference 3B between 7J and 3f = 3f£. 7w subtracting mixed numbers subtract the integers and frac- tions separately. Find differences : 2. 7f-3 T \ 8. 6311-24^ 14. 60^-35^ 3. 9f-2f 9. 71^-19* 15 - 71 f!- 54 yf 4. 18H-8| io. 78JJ-18H 16. 39if-18ii 5. 30i-20if 11. 92||- 29 il 17 - 82 H- 45 M 6. 45i-24f 12. 82f|-29f| 18. 29^-llfi 7 . 50^-llf 13. 95|f-47ii 19. 20fi-15fi 20. If I pay a grocery bill of $221 a wa ter bill of $3£, and a gas bill of $5§, how much shall I have left from 2 twenty- dollar bills ? 21. The sum of three numbers is 150. The least number is 151 and it is 63| less than the greatest. Find the other number. ADDITION AND SUBTRACTION 57 22. What fraction added to the sum of £, f, and -^g will make f ? 23. If 5 is added to each term of the fraction f, is the value of the fraction increased or diminished, and how much? 24. Two boys undertake to save $50 apiece. When one of them lacks $8$ of having $50, both together have $ 84|. How much has each ? 25. A traveling man's grips, when starting out, weighed as follows: 12$ pounds and 19| pounds. Find the weight of both, and the difference in their weight. 26. James lives 1 1 miles east of the schoolhouse, and Harry l^g miles west of the schoolhouse. Find the sum of the distances walked by both each day and the distance James walks farther than Harry. 27. Eight women do different parts in making a finished garment. The cost of the different parts is 1}^, 2|^, 41 £ 3|£ 41^, 4f| £ l£ and T 7 ^. Find the cost of making one garment. 28. Four automobiles finish a race in the following time: 8 T 4 5 hours, 8| hours, 7| hours, and 9^0 hours. Find the dif- ference between the time of the winner and each of the others. 29. The average cost per mile of a fleet of freight boats on the Great Lakes was : coal 13f £ crew 12f £ repairs 17f£ supplies 7^. The average cost per mile of a freight train hauling the same freight was: coal 74 §£ crew 37^, repairs 19-^6 supplies 4|^. Find the saving per mile by water. 30. A drayman hauls freight by the ton from the depot to a village. Find the amount hauled in 8 loads weighing respectively : 1| tons, ^ tons, If tons, 2y\ tons, 2f tons, If tons, 21 tons, 1^ tons. 58 FRACTIONS MULTIPLICATION OF FRACTIONS To multiply a fraction by an integer. 1. 1 + 1 + 1 + 1 = how many whole units ? 2. ^ + ^4-^ + 1 = how many fractional units ? 3. How many, then, are 4x1? 4x|? 4. Does it make any difference in the process of multi- plication in problem 3 whether the multiplicand is a whole number or a fraction ? 5. What term of the fraction A is multiplied by 4 ? 6. | may be multiplied by 4, thus : 4 x ^ = — - — , or - = 2| . o o o Give products : 7. 6 x § li. 5 x f 15. 5 x y\ 19. 16 x I 8. 10 X | 12. 12 X f 16. 15 X f 20. 14 X f 9. 9xf 13. 7x| 17. 12 x I 21. 13 xf 10. 12 x I 14. 7 x I 18. 8 x I 22. 11 x T \ 23. In multiplying the above fractions by a whole num- ber, did we increase the size or the number of the fractional units ? 24. How, then, may any fraction be multiplied by an integer without increasing the size of its fractional units ? 25. 2 x | of a square = how many eighths of the square? How many fourths of the same square ? Draw figures to illustrate. 26. How does | of a square compare in size with | of the same square ? Draw figure to illustrate. MULTIPLICATION 59 27. 3 x | of a square = how many ninths of the square? how many thirds of the same square ? Draw figures to illustrate. 28. How does | of a square compare with § of the same square ? 29. How much larger is the fractional unit in fourths than in eighths ? in thirds than in ninths ? 30. How can we increase the size of the fractional unit in | without decreasing the number of fractional units? in | ? 31. How, then, may any fraction be multiplied by an in- teger without increasing the number of its fractional units ? Then < 3x l=db'°4 In what two ways, then, may we multiply a fraction by an integer ? Multiplying the numerator or dividing the denominator of a fraction by a number multiplies the value of the fraction by that number. Give products : 32. 8 x | 40. 15 xf 48. 25 x A 33. 9x^ 41. 11 x A 49. 15 x A 34. 11 x f 42. 9 x* 50. 18 x fl 35. 6x?k 43. 16 x & 51. 12 x Hh 36. 10 x ^ 44. 7vU 1 A u:i 52. 36 x }| 37. 9xf 45. 3 x& 53. 42 x Jf 38. 12 xf 46. 4 x |f 54. 39 x*f 39. 18 x& 47. 5. x -y- 55. 64 x A 60 FRACTIONS Written Work Since multiplying the numerator of a 1. Multiply g^ by 5. fraction multiplies the fraction, 5 times A v JL. — 15 — 1 _3_ _7_ _ 3 5 _ 1 a_ A 32 — 32 — 32 S2 — 32 — *52- When possible, use cancellation. Multiply : 2. - 2 8 T b7 7 11. |Jby72 20. fi by 96 3. f by 10 12. ii by 108 21. fe by 54 4. f£byl5 13. T ^by57 22. j* by 28 5. f by 8 14. II by 144 23. ^ by 105 6. J{by27 15. ft by 135 24. ^ by 96 7. 11 by 35 16. ^ by 21 25. { ° by 121 8. 11 by 28 17. £ by 16 26. if by 48 9. ff by 144 18. T faby72 27. &-by69 10. H by 70 19. T Ve by 24 28. J| by 126. Finding fractional parts of an integer. 8 x I means that f is to be taken as an addend as many times as there are units in the multiplier. Thus, 3_i_.a_i_3i.a_i3i3_i_jj_i_3 4 ' 4 ' l+l + l ' 4T4T4' | of 8 means that 8 is to be divided into 4 equal parts and 3 of these parts are to be taken. What is the first step in rinding the fractional parts of the whole num- ber? the second step? While the process of finding fractional parts of a whole number is classed as multiplication, the multiplicand at no time is taken as an ad- dend, but is partitioned, that is, divided into equal parts, and a certain number of these parts is taken. The sign x is read " of " when the number preceding it is a simple fraction ; as, § x $6 is read "| of $6." MULTIPLICATION 61 | x 12 mo. means, therefore, 3 x (J of 12 mo.) or 3 x 3 mo. = 9 mo. 3 Or, - x 12 mo. = — t^- mo. = 9 mo. 4 £ Find : 1. j of #80 6. ^ofGOhr. 11. ].]xl24 2. | x42 7. {^ of 30 da. 12. ^ x 175 3. | x 63 8. I of 47 13. |$ x 450 4. fg x 48 9. I x 100 14. fj x 720 5. | x 20 10. | x 95 15. |f x 108 Written Work 1. If the Welsh mills turned out in a certain year 576000 tons of tin plate and the American mills || as much, find the output of the American mills for that year. 2. A certain post office in one year handled 53678996 pieces of mail, of which f were letters. Find the number of letters handled at that post office. 3. The records of a blast furnace show 1179360 tons of iron made during the year. If | of this iron is sold as No. 1 iron, find the number of tons of No. 1 sold. Multiplying a whole number by a mixed number. Written Work l. Find the cost of 48| gallons of molasses @ $.88. $.88 qp.88 48f=40 + 8 + f 48| .66 = | of .88 ^ 7.04 =8x.88 704 35.20 =40 x .88 352 #42.90 =48| x.88 #42.90 62 FRACTIONS Find the cost of: 2. 27§ tons of hay at #12 a ton. 3. 31| yards of cloth at $ .24 a yard. 4. 10| ounces of gold at # 18.75 an ounce. 5. 196 T 9 7 ounces of silver at $.46 an ounce. 6. 97| bushels of apples at $.70 a bushel. 7. 68| bushels of berries at #2.65 a bushel. 8. 147^ bushels of potatoes at $.85 a bushel. 9. 84^ pounds of prunes at $.12 a pound. 10. 257 1 feet of curbing at $.38 a foot. 11. A street-car conductor collects on an average $3.60 per hour. How much does he collect in 11| hours? 12. In 4 days a carpenter worked 7| hours, 8 hours, 7^ hours, and 6| hours. How much did the work cost at 45^ per hour ? 13. A ditch costs $.27 a rod. Find the cost of 227J rods. 14. The "Pennsylvania Special" averages 58 miles per hour. How far does it travel in 16| hours? 15. A roller in a steel mill rolls 118| tons, 221^ tons, 193^ tons, and 180| tons in four days. If he is paid $.12 per ton, how much should his pay envelope contain for the four days' work? 16. A car load of corn contains 821| bushels. How much is it worth at $.57 a bushel? 17. Three men in a day cut respectively 2^ cords, If cords, and 2| cords of wood. How much does each receive for his work at $1.25 per cord ? 18. At $26 per ton, how much will the rails for 18 miles of railroad cost, if it takes 158| tons per mile ? MULTIPLICATION 63 19. Two men work respectively 23] days .and 27| days in a month. How much more does one earn than the other, it' each receives $2.25 per day ? Multiplying a mixed number by an integer. In the problem 9 x 3|, which number is the multiplier? the multiplicand ? The multiplier usually precedes the sign X • Give products : 1. 9 x 9i- 8. 14x3$ 15. 11 x7f 2. 6x| 9. 11 x5| 16. 25x4§ 3. 8 x 5J 10. 6x8| 17. 13x31 4. 7 x 8f 11. 18x3f 18. 12 x 15| 5. 9 x 8£ 12. 20x4f 19. 15 x 7f 6. 10 x 10 T 3 o 13. 16x3| 20. 8x9| 7. 15x31 14. 9xll-J 21. 9 x6ft . Written Work 1. Multiply 14§ by 9, 143 9 9x t = ¥ = 3f. 3| = 9x| 9x14 = 12U. 3f + 126 = 129f. 126 =9x14 . 129f = 9 x 14 .3 8 Multiply: 2. 12f by 28 8. 44|1 by 14 14. 47f by 96 3. 11$ by 45 9. 78| by 63 15. 609f by 21 4. 14 A by CO 10. 89-^ by 105 16. 105| by 49 5. 25 11 by 75 11. 715f by 45 17. 290 T % by 45 6. 85f| by 68 12. 101$ by 63 18. 21 3f by 78 7. 64 \l by 56 13. 205f by 75 19. 735| by 21 64 FRACTIOXS Find products: 20. 213 x 609f 25. 596 x 56f 30. 379 x 49 T ^ 21. 612 x 48§ 26. 972 x 32f 31. 49 x 465 1| 22. 842 x 95| 27. 96 x 325f 32. 786 x 49| 23. 728 x 34 T % 28. 856 x 98| 33. 9872 x 36^- 24. 96 x 207 § 29. 54 x 657| 34. 4398 x 94| 35. There are 16^ ft. in a rod. How many feet are there in 12 rods ? 36. Find the cost of 35 tons of railroad iron at $38| a ton. 37. At i$lf each, how much will 40 "General Histories" cost? 38. If I sell 7 apples for 10 cents, how much shall I re- ceive for 7 dozen apples ? 39. When oranges are sold at 3 for 10 cents, how much will 3 crates carrying 180 each cost ? 40. Stephenson's locomotive weighed 4| tons. Find the weight of a modern freight mogul weighing 40 times as much. 41. A mail boy averaged 13-|^ per hour for 117 hours worked during the month. Find his earnings for the month. 42. A residence is lighted by 21 incandescent lights. The cost of each light per day is 1 f ^. Find the total electric light bill for a year of 365 days. 43. The average shipment of ore from a mine per week of 6 days is 347 tons. If $12^ is the average profit realized from each ton of ore, find the net profit for 40 weeks. 44. A mail carrier's deliveries average 23 1 pounds. How many pounds of mail does he deliver in 2 trips each day for 312 days? MULTIPLICATION 65 45. Iii a certain shoe factory a pair of shoes is finished every 3| minutes. Find the number of days of 10 hours each required to make 4600 pairs. 46. An establishment consumed in one year 8450 pounds of twine at 9^ per pound ; 36| dozen bottles of ink at $1.90 per dozen ; 6 gross pens at 90 ^ per gross ; 500 pads of paper at f $ per pad. Find the total cost of the purchase. 47. A real estate agent sold 6 pieces of land containing respectively : 10§ A., 121 A., 18f A., 26j% A., 30§ A., 3| A., at 1250 per acre. Find the amount of the sale. 48. A town has a population of 3600. If the average amount of water used by each person is 6| gallons per day, find the number of gallons of water used in 90 days. 49. If a motorman receives 9\ per hour, how much does he earn in 7 weeks of 6 days each, working 10 hours per day ? Finding fractional parts of a fraction. 1. What is i of 9 hours ? § of 9 feet ? 2. What is i of 9 tenths ? § of 9 tenths? 3. 1 of 9 tenths means that we are to take l of 9 parts of a unit that has been divided into 10 equal parts. 4. | of T 9 7 = how many tenths ? 5. If ^ of t 9 q = ^, how much is § of T 9 a ? Find mentally : 6. loft 11. 7 of £1 9 OI TS 16. 2 3 Of -9- 01 16 7. fof^ 12. 11 of £± 12 U1 33 17. 5 off 8 4 of 14 o. 8 ui 21 13. _9_ of 4 10 U1 ?3 18. 1 1 12 off 9. foflf 14. _4. Of A5 15 Ui 5 3 19. :» 1 D of| 10. * Offl 15. 3 of H 8 Ui 3 5 20. B 8 Of ft HAM. COMPL . A KITH. — 5 66 FRACTIONS 21. What does the expression ^ mean ? 22. Can T 9 7 be separated into 3 equal parts ; thus, T ^-, T 3 ^, ^, without changing the size of the fractional unit ? 23. To what fractional unit must we change f before we can separate it into 4 equal parts ; that is, take \ of | ? 24. Is |§ the same in value as f ? 25. When || is separated into 4 equal parts, how many fractional units are there in each part ? 26. Then 1 of | = how many twentieths ? 27. Observe that - of ? = L*!* or A. 4 5 4x5' 20 28. What 18 f Of |? Ifl0ff=,3_,3 0ff = 3x JL or JL 29. Observe that - of - = !L*J*, or — . 4 5 4x5' 20 A fractional part of a fraction equals a fraction times a fraction. Written Work l. Find | x |f. 7 X 16 7x 16 _ 112 _ 2 To find I x if means to find J of |f. 8 x 21 " 8 x 21 == 168 "~ 3 * of H = A and J of h = 7 x A = W, or*. Or, 2 ? y=? $ x &l 3 3 A fractional part of a fraction is found by multiplying the numerators for the numerator of the product and the denomina- tors for the denominator of the product. Indicate the operation, and cancel when possible. A fractional part of a fraction is sometimes called a com- pound fraction. Thus, \ of \ is a compound fraction. MULTIPLICATION 67 An integer may be expressed in fractional form, thus, — 2 8 = 8. |of8 = - 3 off = | = 6. Find products : r> 3 v IB a 2r ' v 14 lO ?5 y ij 14 %k X ^M o 5 v 38 7 21 v 33 in 21 y 15.5 15 :; s X MS4I 3 * 19 X 3 5 7- 44 X 5 6 ■ LJ " 6 2 X 162 XO ' 43 A 130 d. 15 v 14 o IS v (!5 12 1 3 y 3 4 ic 54 3 x 102 4 - 2% X 2 5 B - 2 6 X 7 2 X ^' 68*39 xo * 5 12 A 2 4 5 e 5 1 v 8 Q 1 1 v 4 6 i -a 7 1 y 3 7 17 111 y 110 5 - 6 4 X IT 9 - 6 9 X 121 13- 74 X 38 L/ ' 220 x 4 1 1 18. Find I of 12|. Note. — Change the mixed number to an improper fraction. Find : 19. § of 7j 23. 3| X 4 5 27. 2| x 7 1 20. f 0fl2| 24. 7| X| 28. 8| x5| 21. I ofl3f 25. 5 T %x{| 29. 7| x3| T r of 10$ 26. 8| x I 30. 4 T ^ x 3§ Find the value of : 31. 4$ x 6f 8 5 6 A Q /-A 3$ 48 30 ^^ QA 4fx6f = |-xy = T , or 30. 32. 6| x2 T 3 T x7| 38. 35^x22^x12 33. 8^ x2| x2| 39. 51fx 19^x121 34. 101 x 11 x 7} 40. 6f x 6| x I x 2& 35. 14j x 17| x 8| 41. 20) x 20f x 20{i x 10 36. 27*fx42fx6J 42. 7§.]xl5 x2|£xl2$ 37. 2i x 3| x 4| x 5| 43. 172.] x 2./ 5 x 3£ 22 68 FRACTIONS 44. The schedule of a train between two cities is 12| hours, and the train's speed is 35^ miles per hour. Find the distance between the places. 45. Find the number of tons of sugar cane on 19 1 acres, if the average number of tons per acre is 11 § tons. 46. An automobile's average rate of speed is 25 ^ miles per hour. How far does it travel in 3| hours? 47. Estimate the value for one season of a Vermont sugar camp at <$ J^- per pound if 6-| pounds are obtained on an average from each of 1275 maple trees. DIVISION OF FRACTIONS Dividing a fraction by an integer. 1. 15 -5- 5 means that 15 is to be separated into 5 equal parts of 3 units each ; and one of the equal parts taken ; thus, 15-^5 = 3. Explain in the same way what 18 -*- 9 means. 2. \^ -5- 5 means that ^| is to be separated into 5 equal parts of j 2 g each, and one of the equal parts taken ; thus, || -i- 5 = T 2 g. Which term of the fraction was divided by the integer? A fraction may be divided by an integer by dividing the numerator of the fraction by the integer. Divide : 3. § by 4 5. 15 j- 5 2 6 * ° •>■ IfbyS 4. if by 6 6. -3-S-^ 9 3T • v 8. |f by 7 9. In | -s- 5, can you separate | into 5 equal parts in which the fractional unit is thirds? DIVISION 09 10. Can you divide § by 5 by dividing the numerator by the integer an integral number of times ? 11. | -i-5 means that | of | is to be taken. Since -\ of 2 — - 2 - then 2- — 5 = - ? - 12. | = yV Can {5 be separated into 5 equal parts of T 2 ^ each ? Then § -h 5 = ^. Observe that multiplying the denominator of | by 5 changes the fractional unit from thirds to fifteenths and takes one of the 5 equal parts into which § has been changed. A fraction may be divided by an integer by multiplying the denominator by the integer. Solve the following problems by the more convenient method and give your reasons : 13. fft + 7 16. J^6 19. T %-*-25 14. 1-8 17. T %% -=- 9 20. iff +- 15 is 25. _s_ 1 fl la _fi_ -s- 3 21 P -*- 7 Written Work 1. Divide || by 8. 04 o " o 1° a the division is performed by divid- er. - — = — ing the numerator by 8. 25 25 ~ . ^ . „ In b the division is performed by multi- b. — — - — =— = — plying the denominator by 8, and changing ZQ X o 200 ^5 tn e fraction to its lowest terms. Q In c the division is indicated and the quo- c. 3 = — tient is found by cancellation. 25 x 8 25 A fraction may be divided by an integer either by divid- ing the numerator or by multiplying the denominator of the fraction by the integer. 70 FRACTIONS Find quotients: 2. | + 5 11. 3. | -5- 9 12. 4. ft + 7 13. 5. f^l8 14. 6. 11 -=- 5 15. 7. T 8 2°o-25 16. 8. 1| -4- 7 17. 9. II -5- 8 18. 10. T 9 Y -=- 5 19. 29. Divide 3| by 5. 21^« 65 ■ ° 31 • y _8_ ^_ 24 2 1 ' 12 13 16 19 • "' 16 ^40 36 ^_ 19 37 * Xii 12 ii 18 3 5 ^.42 4 3 * *^ 31 = 25 °8 8 ' 25 . 8 5 25 8xp 5 8 20. 21 28 -18 21. 12 83 -30 22. 96 113 -64 23. 51 61 -38 24. 132 16 3 -48 25. 111 12 7 -52 26. 11 1 112 -37 27. 256 "2 7 5 -80 28. 180 19T -HOO Note. — Small mixed numbers are frequently reduced to improper fractions and then divided by the same principle as proper fractions. 30. 31. 32. 33. 34. 35. 2f-6 41-1-7 4 T « T h, 13 922 ^2 5 51 3 48 23 17J + 8 36. 37. 38. 39. 40. 41. 28f - 25 284 h- 18 32- 8 - 21| -4- 25 181 + 26 19J T V45 21 48. Divide 1286f by 9. 42. 43. 44. 45. 46. 47. 10 T 2 3 + 24 29f + "12 22ft 30| 191 58 23 26 211 _=_ 56 9)1286f 142f| Note. — The division may be performed by changing both numbers to sevenths, but it is a much shorter process to perform the work as indicated, changing the remaining mixed number, 8f, to an improper frac- tion, V, and dividing it by 9. Thus, \ of - 6 ? 2 - = If- DIVISION 49. 25681^ -s-7 54. 8002 fa -12 50. 9863^ -=- 6 55. 1428*1 -5-12 51. 6532^h-8 56. 10935| -4 52. 6879^ -5- 5 57. 9720| -5-11 53. 36370|^-11 58. 7090^ -5-9 71 Dividing fractions by first making them similar. 1. What is the unit of measure in eaeh of the following : 8 ; 6 yd. ; 5 ft. ; 4 in. ; 8 rd. ; 5 oz. ; 2 lb. ; 10 mi.? 2. What kind of units may be added or subtracted ? 3. Since both the dividend and the divisor of concrete numbers must represent like units, to divide 24 yd. by 3 ft. we must either change yards to feet, thus, 72 ft. -5- 3 ft. ; or feet to yards, thus, 24 yd. -=- 1 yd. 4. In | -5- 1 are the fractional units alike in size and kind? Then how often is | contained in |? 5. Since | is contained 3 times in |, how can we divide one fraction by another when the denominators are alike? A fraction may be divided by another fraction by chang- ing both fractions to a common denominator and dividing the numerators. 6. Divide f by f . 3 _ JL . 5 _ 1 ¥ — 12' 6 — 12 JL _=_ 1 _ Q _:_ 1 () — JL 12 * 12 — V ' ±XJ — i7 Find quotients : 8. X -=-1 12. o 9 . n • 6 T 5 2- !_ 2 • 3 9 _ 3 5 1 3 15 3 JL 11 16- tf + J 17 5. _;_ 1 - 1,3 - 16 • 3 ■ L/# 1? ' 9 10. U + f !*•♦ + ♦ 18 - H + l 72 FRACTIONS By inverting the terms of the divisor and multiplying. 1. How many half inches are there in 1 in.? in 2 in.? 3 in. ? 4 in. Then how many times is ^ contained in 1? in 2? in 3? in 4? 2. If 1 -j-'J = 4, what does 2 + \ equal ? 3 - \ ? 12 + \ ? 3. What does 1 -f- l equal ? 3 -=- l ? 6 -=- i ? To divide any number by a fractional unit multiply the num- ber by the denominator ; thus, 12 h- \ = 12 x ^ = ^ or 48. 4. 5. 6. 13. 14. 15. 16. 12 — 1 15 16 ? 1 _ 9 3 — • 7. 8. 9. 10 -^ - 1 = ? io. 20-^i = ? 11. 25h-£ = ? 9 - iW 9-h T V = ? 12. 18-^1 = 1 -s- 1 = how many ? 1 -=- | = how many ? 1 -s- -Jj = how many ? 1 -=- | = how many ? 17. 1 -=- j 5 q = how many ? 18. 1 -f Jj = how many ? 1 -=- § = how many ? 1 -=- | = how many ? 19. 20. Observe that the number of times each of the above frac- tions is contained in 1 equals the number of times the numerator is contained in the denominator. The number of times a fraction is contained in 1 is called the reciprocal of the fraction. Thus, f is contained in 1, | times. Hence, | is the reciprocal of §. 1 divided by any fraction equals the fraction inverted. Find quotients : 21. 8^| 23. 4 + | 25. 10-5-1 27. 25 22. 6H-| 24. 15-^-f 26. 12 -f 28. 14 29. | -=- | = how many ? 2 _t 3 • 3 — 2 v 4 _ 8 Since 1 " 4 — 3 X 3 — 9 ° U . 3 _ 4 2 ^ 3 _ • ? — 3> 3 • ¥ ~ | of |, or |. DIVISION 7:; Any number is divided by a fraction by inverting the terms of the divisor arid multiplying. 30. Work if -r- | by both methods. What principle does the second method introduce that the first does not? Note. — The process should be shortened by cancellation when possible. Written Work Change the mixed numbers to improper fractions before dividing. Divide : 1. 18 by f 20. 13 _ 18 . 26 ■ 2 7 39. 119. 7 T3 8 • US' 2. 12 by | 21. 5 2 2 3 5 40. 85 .51 87" ' 1J5 3. 25 by| 22. 2 7 _ 32 l. 9 ' 16 41. 3 9 .13 6 4 *24 4. 32 by if 23. 2 5 _ 3 6 . 15 16 42. 993 . 95 ^"4 * -8 5. 40 by f § 24. 28 39 _ 42 65 43. 45| -4- 5« 6. 48 by 1| 25. 65 _ 7 2 5 8 44. 6* + 8 | 7. 172- 8 9 26. 13 3 _ 1 4 4 95 192 45. 128 ■*- 17/2 8. 235- 4 5 27. 99 _ 12 5 209 2 r. 46. 160 by f of | 9. 770- - 11 15 28. 81 _ 1 12 45 "72 47. | of 640 by 6£ 10. 882- _ 3 Y 29. _4_9_ _ 1 II s 91 132 48. 198 by 12f 11. 1035- _ 5 6 30. 243 2 56 189 32 49. 8| + 4J 12. 984- _ 3 31. 1 96 _ 22 5 _ 56 3T5 50. 10| -*- 6f 13. 3 _ I 2 3 32. 185 _ 22 4: 111 896 51. 101^31 14. 2 3 _ 3 1 33. 175 _ 282 125 "375 52. (f Of f) -4- (J Of |) 15. 1 8 _ 3 4 34. 23 _ 231 2 33 53. (f of £) -*-(f of f) 16. 9 _ 10 _ 3 5 35. 143 _ lYO l. 6 5__ ' 136 54. (4 ( > f «) + (A 17. 13 _ 1? 6 Y 36. 114 -A « " f H) 18. 7 _ 12 2 3 37. 8 3 S4 _ 5 5 6 55. (H^W+H 19. 15 _ 16 _ 1 1 24 38. 55 _ 7 9 " 22 56. («ofH)+(jofft: 74 FRACTIONS Miscellaneous 1. At $1|- per day how long will it take a laborer to earn $671? 2. Divide | of | of J by f of * of Jf 3. A man who owned T 7 y of an estate sold | of his share. What part of the estate did he then own ? 4. A man who owned f of a store sold | of his share for $1406.25. What was the value of the store? What part had he left ? 5. One man has $35| ; another has $62|. If each gives to the other § of what he has, how much more will the one then have than the other ? 6. If | of 10 bushels of oats cost |3f, how much will 20| bushels cost ? 7. What number must be multiplied by £ of 3| to give a product of 32 -^ ? 8. A storm moves eastward at the rate of 18^ miles per hour. In how many hours after the storm is first observed in Chicago should it be due in Pittsburg, 468 miles east ? 9. Kerosene oil weighs 6| pounds to the gallon. Find the number of gallons in a tank car of oil weighing 39200 pounds. 10. A steam threshing machine averages If bushels of wheat per minute. How many minutes at the same rate will it take to thresh 358| bushels? 11. The average cost of the education per pupil in a certain city is $51. If the total cost is $16480, find the number of pupils attending school. 12. A contractor employs 6 men for 27| days and pays them $363. Find the average daily wages. COMPLEX FRACTIONS 75 COMPLEX FRACTIONS A complex fraction is a fraction which has a fraction or a mixed number in either or both of its terms. Thus, -2 , — , £, !i2 are complex fractions. ' o' 3' 4' 41 r 8 5 ^? Such examples are simplified by the principles of division of fractions. Thus, £ = - ■*■ 3 = -. They rarely occur except in ad- o — O vanced courses of study. Written Work l. Find the quotient of 2J divided by § . 3 6 2 Simplify : 1 4 of 21 ,„ |x6| o 9 7 5 5 12. A_ a 9 7. 5 *" -5 12. ?i if 3 _8. °3 • 8 11 , , 44 3. 1| 8. | X | 13. |0f-£ 4 W 9 . _LJ 14. A Of! 5 S + 2i 10 Six I' 8-gbtt) 6 (-fe-Dx(t + |) u (i°f8j)-j 16 (21x31)-,- If 76 FRACTIONS FRACTIONAL RELATIONS Finding what part one number is of another. 1. What part of 12 is 4? What part of 18 is 5? Ex- press the answers also in the form of division. Thus, T 4 2=4-12; t \=5h-18. 2. What part of | is f ? of | is f ? How, then, can we find what part one number is of another? Divide the smaller number by the larger number. Written Work 1. What part of 208 is 96? cm • 908 96 -5- 16 _ 6 We divide the smaller number 96 " 208 — 16 " 13 ky 208, and reduce the resulting frac- tion 2 9 jfg to its lowest terms, T 6 j. 2. What part of | is -^ ? 3 7 _ . 3 §_3 We divide r 3 g by J by inverting the 16 8 ~ 16 7" 14 divisor of | and multiplying. The result 2 shows that r \ is fa of |. What part of 3. 90 is 16? 6. | is |? 9. 4 is 31? 4. 120 is 50? 7. lis^? 10. 13|islf? 5. 200 is 18? 8. \l is if? li. 12| is If? 12. 250 pupils belong to a school, and only 128 are present. What part of the whole number is present? 13. One year the population of a city was 220,000, and the next year 250,000. What fraction of the second year's population was the first year's? 14. The product of two numbers is 29,160, and one of the numbers is 27. What part of the other number is 27? FRACTIONAL RELATIONS 77 Finding a number when a fractional part of it is given. l. If | of a number is 30, what is the number? If three fourths of the number is 30, one fourth of the number is one third of 30, or 10, and four fourths of the number, or the whole number, is 4 x 10, or 10. Hence, 30 is | of 40. Written Work 1. 360 is T 6 y of what number ? i if aon en Since 36 ° is rV 0T a number, T V of the number = \ of 350, or 60 , . . . . . J! , , , ,„ _,, „ v A- of it is 1 of 360, or 60, and H j I ot the number = 17 x 00, or 1020 T I .. . , , " , .,„. . ' 17 17 of it is V oi »360, or 1020. 2. If | is | of a number, what is the number ? Since £ of the number is J, \ of the number = \ of |, or 5 7 ¥ , £ of it is \ of |, or ^V- aQ d ? of | of the number = Ix 2 7 ? = |f, or li the number is 1 x 2 7 ? , which equals §£, or 1\. Find the number of which : 3. 84 is | 5. 3| is I 7 -Z is -L /. 9 it> 16 4. 196is T 4 T 6. 12|isi§ 1 2 jo 6 8. T g 1» y 9- H i8 H 10. 5fis^ 11. If T 9 3 of a man's salary is $900, what is his salary ? 12. A man lost in speculation $2700, which was -j 3 g of his entire fortune. What was his fortune? 13. If I.]- of the number of books in a library is 9922, how many books are there in the library ? 14. There are 30,205 women in a certain town, which is y 7 ^ of the number of men. How many men are there ? 15. An author spent $2100 for a piece of property, which was y 7 ^ of what he was paid for a novel. How much did lie receive for the novel? 78 FRACTIONS REVIEW OF FRACTIONS 1. | of 30 is f of what number ? 2. If 6 is added to both terms of the fraction f, will the value be increased or diminished, and how much ? 3. Change iff to its lowest terms. 4. The sum of two numbers is 43^. One of the num- bers is 18|. What is the other ? 5. If 51 tons coal cost $28.27, find the cost of 12§ tons. 6. If 2£ acres of land cost $110, how much will 12| acres cost at the same rate ? 7. The dividend is 165, and the quotient is 6|. What is the divisor ? 8. Five tubs of butter contain, respectively, 27^ lb., 30| lb., 24 Jg lb., 32| lb., and 34| lb. How many pounds are there in the five tubs ? 9. Multiply the sum of § and f by their difference. 10. If 10 men can build a wall in 35 days, how long will it take 25 men to do the work ? 11. There are 27 1\ square feet in 1 square rod. How many square rods are there in 43560 square feet ? 12. If a man travels 2|- miles in f of an hour, at the same rate how far could he travel in 1\ hours ? 13. A man owning -^ of a mine sold his interest for $48700. Find the value of the mine at that rate. 14. Find the difference between 1\ x 3| and 2\ -4- 3|. 15. A merchant owned f of a store, and sold | of his share for $5760. Find the value of the whole store at that rate. 16. How much will 18000 stamped envelopes cost at $21^ per thousand ? REVIEW OE FRACTIONS 70 17. When oysters yield 1{ gallons to the bushel, how many bushels will be required to till a 10-gallon tub ? 18. One buyer offered | of the cost of a property, another | of the cost. The difference in their offers was $186. Find the cost of the property. 19. Find the cost of 18 yards of cloth if 3| yards cost $9. 20. A clothier paid $180 for 12 suits of clothing, and sold them at $19|- a suit. How much did he gain ? 21. E, who owns § of a factory, sells | of his share for $3560. What is the value of the factory ? 22. What is the distance from Pittsburg to Philadelphia if | of the distance is 265^ miles ? 23. The product of two fractions is | ; one is ^, what is the other ? 24. If C's wages are $3| a day, and his daily expenses $1|, how many days must he labor to save $28 ? 25. A traveler walked 25| miles the first day, 15| miles the second day, 19| miles the third day, 20| miles the fourth day, and 22| miles the fifth day. How far did he travel in the 5 days, and what was the average rate per day ? 26. A field is 40| rods long, which is f of its width. What is its width and what is the distance around the field ? 27. Find the cost of 10A cords of wood at $3 a cord, and 8| cords at $4 a cord. 28. If 12| bushels of apples cost $5, how much will 15| bushels cost ? 29. A farmer raised 22f> bushels of potatoes. He sold | of them to one merchant, and \ of the remainder to another. Find the number of bushels he had left. 30. Reduce \% to a fraction whose denominator is 320. lb 80 FRACTIONS 31. Find the value of a mill if f of f of it is worth 13750. 32. If % the trees in an orchard are peach trees, | apple trees, % cherry trees, and the remaining 21, plum trees, how many trees are there in the orchard ? 33. What number is it whose f exceeds its § by 40 ? 34. Divide into 5- equal parts the product of the sum and difference of 1^ and 1|. 35. A real estate dealer sold some lots for §12360, gain- ing 1 of the cost. If he had sold them for $10500, would he & 5 have gained or lost, and how much ? 36. A piece of land was sold at $ 90 an acre, which was a gain of | of the cost. How much did the land cost per acre ? 37. My board and room cost $32 per month. What does each cost if § of the cost of the room equals § of the cost of the board ? 38. A father left $30000 to his two children, giving the daughter | as much as the son. What was the share of each? 39. The owner of % of a mill sold | of his share for $4800. How much at this rate would a man who owns ^ of the mill get for 2 3 o of his share? 40. One fourth of a certain number minus 3| equals -fa. What is the number ? 41. A carpenter, working 9| hours a day, built a shed in 16 days. How many hours a day, at the same rate, must he work to build it in 18 days? 42. A merchant sold 10 dozen hats of a certain kind at $2.75 each, and a number of dozens of another kind at $2 each, receiving $474 for all. How many dozen did he sell at $ 2 each ? REVIEW OF FRACTIONS 81 43. A man engaging in trade lost | of the money invested, then gained $495, after which he had $2551. How much was his capital at first ? 44. Simplify ^ + (£+1). 5 Ui 9 45. A hardware merchant bought a bill of hardware at auction for \% of its value, and retailed it for f of its value. If his gain was $48.75, how much did he pay for it? 46. A boy bought lemons at the rate of 4 for 5 cents, and sold them at the rate of 3 for 5 cents. If he made $6 in 2 weeks of 6 days each, what were his daily average sales ? 47. A path leading to the top of a hill has a rise of 9 inches in 90 feet. What is the elevation of the hill in feet, if the path is f of a mile long ? 48. A teacher taught 8* months, and after spending f of his salary for board had left $204. How much did he earn per month? 49. Divide § of 9 by A of 8|. 50. Two men, A and B, each bought farms, A's farm cost- ing 2| times as much as B's. Find the cost of each, if both cost $58,000. 51. Divide § of | of | of 3| by f of ^ of 8^. 52. A ship is worth $120000, and the owner of f of it sells \ of his share. Find the value of the part he has remaining. 53. Find the value of (18| - |) - (20^ - 1$). 54. I purchased 160 acres of land at $60 an acre, and sold | of it at $70 an acre. Find the number of acres I had left, and my gain on the number of acres sold. 55. A real estate agent bought land for $7200, and sold it so as to gain ^ of the cost. If the gain was $6 per acre, how many acres did he buy ? HAM. COMPL. ARITH. — O 82 FRACTIONS 56. A man paid § of his indebtedness the first year, § of the remainder the second year, f of what then remained the third year, when he found that he still owed $ 1296. Find the amount he owed at first. 57. If a man can do a piece of work in 6| days by work- ing 9 hours a day, how long will it take him to do it, working 8 hours a day ? 58. Mr. Williams has ^ of his money in government bonds, | in the bank, and $ 520 in cash. How much is he worth ? 59. A real estate agent bought 2 houses ; his income from the one was $480, which was | of the income from the other. How much was his income from both houses ? 60. Four brothers agreed to pay the mortgage on their father's farm. The first payment made was ^ of the mort- gage, the second ^ of the remainder, and the third | as much as the first and second. If the difference between the first and third payments was $300, how much was the mortgage? 61. A merchant bought 250 barrels flour at $4.80 per barrel. He sold 10 barrels which were damaged for | of the cost. On the sale of 150 barrels he gained 15 cents per barrel. If his total gain was $28.50, at what price did he sell the remaining number of barrels ? 62. A merchant buys 36 pairs men's shoes, at $2^ per pair ; 24 pairs women's shoes, at $1| per pair; and 30 pairs slippers, at $| per pair. What is the amount of his bill ? 63. If the men's shoes in example 62 are sold for $3 per pair, the women's shoes for $2| per pair, and the slip- pers for $1| per pair, how much does the merchant make ? 64. A cubic foot of water weighs 62| pounds. A barrel contains about 4* cubic feet. Find the weight of a barrel of water. REVIEW OF FRACTIONS 83 65. A student, in a half year at college, spent his money as follows: tuition, |50; books, $9| ; 18 weeks' boarding, at $2| per week; incidental expenses. $41|. What were his expenses for the half year? 66. T 9 Y of a barrel of oil, containing 51 gallons, was sold at 13 cents per gallon; the remainder of the barrel at 14 cents a gallon. If the oil cost 11| cents per gallon, what was the gain ? 67. A merchant bought a stock of goods for 16000. He sold | of it at a gain of l of the cost. \ of it at a gain of \ of the cost, and the remainder at a loss of \ of the cost. How much did he gain or lose ? 68. A hall is 21f feet long and 10] feet wide. At 8£ cents per foot, how much will it cost to put a molding around this hall? 69. Find the perimeters of each of five rooms, the dimen- sions being as follows : 221 ft. x 16 ft., 14f ft. x 15* ft., 12§ ft. x 17 ft., 181 ft. x lof ft., m ft. x 24 ft. 70. From a certain number 18| + '2~^ was subtracted, leaving a remainder of 9 T 5 g. What was the number? 71. From New York to Chicago by the Baltimore and Ohio railroad the distance is 1052 miles. From Chicago to El Paso by the Santa Fe railroad it is 1630 miles, and from El Paso to Mexico City by the Mexican Central railroad it is 1224^ miles. What is the distance from New York to Mexico City, and how long would such a journey take, traveling 31 1 miles per hour ? 72. A freight train runs from Kansas City to St. Louis, 288 miles, traveling 16 miles per hour. How long does it take it to make the trip? How long would it take a pas- senger train, running \ as fast, to make the same trip? PROBLEMS FOR ANALYSTS Pupils should be taught : (1) To express themselves accurately and rapidly ; (2) to do this work without the aid of pencil and paper ; (3) to give clear analytic statements in the solution of a problem. 1. The coal for the school buildings in a certain town cost $94.50. How many tons were purchased at $3 per ton ? 2. A shoe dealer buys shoes at $21 per dozen pairs and retails them at $2.50 a pair. How much does he gain on each pair ? 3. A laborer earns $18 in 12 days. At that rate how much can he earn in 80 days ? 4. What is f of f of 24 ? 5. A lady buys 10 yards of ribbon and uses 6| yards. What part of the ribbon has she left ? 6. If eggs are bought at the rate of 4 dozen for $1.08 and sold at 30 cents per dozen, what will be the gain on 3 dozen ? 7. A man buys a farm of 102 acres and divides it into lots of 6 to the acre. How many lots are there if £ of the farm is laid out in streets ? 8. How many sheep, at $5 per head, may be purchased with the money from the sale of 10 head of cattle at $42 per head? 9. A contractor buys 8000 bricks at $12.50 per thousand. Find the amount of his bill. 84 PROBLEMS FOR ANALYSIS 85 10. If I pay 6 cents for the use of §1 for one year, how much should I pay for the use of §150 for - years/ 11. If I buy goods at $1.37 per yard and sell them at 11.50 per yard, how much do I gain on 12 yards? 12. A boy earns 75 cents per day and pays f of his wages for board. At that rate how much can he save in 26 days ? 13. My father pays §252 per year rent. How much is that per month ? 14. A lady purchased 2 handkerchiefs at 35 cents each ; 6 yards of ribbon at 12 cents per yard ; 6 yards of cloth at §1.12 per yard. How much change should she receive from a ten-dollar bill ? 15. A man buys a farm for §2000 and pays | of the cost, giving his note for the balance. For how much does he give his note ? 16. A farmer buys fertilizer at §28 per ton and retails it at §1.90 a hundred pounds. How much does he gain on 35 tons ? 17. At 16 cents per pound, how many pounds of steak does a woman get if the amount of the purchase is 83 cents ? 18. A huckster bought 6|- pounds of butter at 16 cents per pound, and 6| dozen eggs at 18 cents- per dozen. How much did he pay for both? 19. A huckster buys chickens at 8^ cents per pound and sells them at 12 1 cents per pound. How many pounds must he purchase and sell in order to gain §25? Solution. — On each pound he gains 4£ cents. To gain $1 he must sell 24 pounds. To gain $25 he must sell 25 times 24 pounds, or 600 pounds. 20. A street car conductor earns 23 ^ cents per hour. How much does he earn in 10 hours ? 86 PROBLEMS FOR ANALYSIS 21. If the conductor averages 10 hours per day, how much will he earn in 80 days ? 22. If a box of 40 dozen oranges is purchased for $5 and retailed at 40 cents a dozen, what is the entire gain ? 23. What fraction of a gallon is a pint ? a gill ? a quart ? 24. A man sold | of his farm for $1800. At the same rate, how much would he receive for the whole farm ? Solution. — § of the amount received for the farm =$1800. £ of the amount received for the farm = \ of $1800, or $900. f , or the amount received for the farm = 3 times $ 900, or $2700. 25. In a certain school there are 40 pupils in the grammar grade, which are f of the number in the intermediate grade. How many pupils are there in both grades ? 26. A man sold a piano for $360, which was | of what it cost him. How much did it cost him ? 27. There are 112 cubic feet in | of a cord of wood. How many cubic feet are there in 1 cord ? 28. A farmer sold | of his farm for $1521. At that rate, what was the value of § of the farm ? 29. There are 198 cubic inches in $ f a gallon. How many cubic inches are there in 2 gallons ? 30. If | of a ton of hay costs $16, how much will 2| tons cost? 31. A lady paid 84 cents for | of a yard of silk. At the same rate, how much would she pay for |- yard ? 32. If ^ of an acre produces 160 bushels of potatoes, how many bushels will an acre produce ? 33. A man saves $220, which is f of his yearly salary. What is his yearly salary? PROBLEMS Foil ANALYSIS ST 34. If it takes 4 men 9 days to do a piece of work, how long will it take 6 men, at the same rate, to do the work ? 35. If | of a farm is valued at $ 1240, what is the value of ^ of the farm ? 36. A lady spends § of her income for hoard, and 1 of the remainder for clothes and travel. If she saves $160 per year, what is her income ? 37. A farmer has 600 bushels of wheat, which is \ of | of what his neighbor has. How many bushels has his neighbor ? 38. A house was sold for 11800, which was § of its cost. What was the loss ? 39. If 20 men can dig a ditch in 25 days, how long will it take 5 men ? 40. A farmer sold 36 head of cattle, which was 6 more than ^ of all he owned. How many had he remaining ? 41. If a man works | of a day and receives $1.50, how much should he receive for f of a day ? 42. I purchased an overcoat for $45, and found I had % of my money left. How much had I at first ? 43. James and John have together $40, and John has seven times as much money as James. How much does each have ? Solution. — Once James's money = James's money. 7 times James's money = John's money. 8 times James's money = amount of both, or Once James's money = $5. 7 times James's money = 8-35, or John's money. 44. There are two numbers whose sum is 40; one number is A of the other. What are the numbers? 88 PROBLEMS FUR ANALYSIS 45. A man invests \ of his money in a mill, | in a farm, and the remainder, which is $900, he deposits in a bank. How much is he worth ? 46. A town has 3000 people. The number of pupils in the school is \ of the remaining population. How many pupils are there in the school ? 47. In a farm of 500 acres the woodland is § of the cleared land. How many acres are there of each ? 48. A miller has 360 bushels of wheat, § of the number of bushels of wheat equals | of the number of bushels of corn. How much is the corn worth at 50 cents per bushel? 49. If | of a clerk's salary for the year is $800, how much is I of his salary ? 50. | of 21 is -^ of what number ? 51. § of 20 is T 4 3 of what number ? 52. | of I of a number is 16. What is the number ? 53. I of C's farm equals ^ of D's, and C has 100 acres. How many acres has D ? 54. There are two numbers: § of the first is f of the second. The first number is 18. What is the second ? 55. A and B agree to do a piece of work for $80. If A works 7 days and B 9 days, how much should each receive? Suggestion. — Together they work 16 da. Hence, A should receive r \ of $ 80, and B T 9 S of $ 80. 56. Sarah earns $10 a week, and ^ of what Sarah earns is § of what Edna earns. How much does Edna earn ? 57. | of T 3 ^ of 160 is T 8 T of what number ? 58. If | of | of a farm costs $2500, how much does the farm cost ? PROBLEMS FOR ANALYSIS 80 59. I of my money is invested in coal land, \ of the remainder in building lots, and the remainder, which amounts to $800, is in the bank. How much have I invested in coal lands ? 60. A man sold a horse for 8120, winch was \ less than the horse cost him. Find the cost. 61. If 24 is | of a number, 21 is what part of that number ? 62. I spend \ of my monthly salary for board and room, \ of the remainder for clothing, and save the remainder, which is $20. What is my salary? 63. If f of a piece of work can be done in 9 days, how long will it take to complete the work after f has been done ? 64. By selling land at $120 an acre, I gain J of the cost. Find the cost. 65. If | of the value of a farm is $000, what is the value of | of the farm ? 66. If 4| tons of coal cost $27, how much will 3 tons cost ? 67. What is the difference between \ of \ and § of 1 ? 68. If I can do a piece of work in 7 days, what part can I do in 1 day? in 3 days? in § of a day? in 1| days? 69. A fruit grower sold 200 bushels of apples, which was A of his crop. How much did he realize from the sale of his crop at $1.20 per bushel ? 70. An heir gets f of an estate and invests f of his share, and still has $1600. What is the value of the estate ? 71. A tank holds 120 gal. and is f full. § of the quantity is drawn off. How many gallons will it take to fill the tank ? 72. One man bids ^ of tlie cost of an article ' another man bids § of the cost of the article. The difference between their bids is 70 cents. Find the cost of the article. DECIMAL FRACTIONS 1. What is the largest common fractional unit ? 2. Show that an integral unit may be divided into any number .of fractional units. 3. Name the different fractional units from \ to 2V in order of their size. 4. What fractional unit divides the integral unit into 10 equal parts ? into 100 equal parts ? into 1000 equal parts ? The divisions of an integral unit into lOths, lOOths, lOOOths, etc., are called decimal divisions. There are three ways by which decimal divisions may be expressed : (1) by words, as nine tenths ; (2) by common fractions, as j 9 q, y| 7 ; (3) by decimals, as .9, .75. A decimal fraction is any number of lOths, lOOths, lOOOths, etc. of an integral unit. When expressed after a decimal point and without a written denominator, it is usually called a decimal. A decimal point is a period placed after ones' place and before tenths' place. 5. What is the largest decimal unit? the second largest? the third largest? In any decimal system 10 x 1 unit in any place =1 unit of the next higher place. 6. Show that United States money is a decimal system. 90 NOTATION AND NUMERATION 91 NOTATION AND NUMERATION OF DECIMALS 1. Since the first decimal division of an integral unit is tenths, what is the first place to the right of the decimal point? 2. What is the second place called ? the third place ? 3. Five tenths is written .5 ; five hundredths is written .05 ; five thousandths is written .005, etc. Write 7 tenths, 6 hundredths, 8 thousandths. 4. Express decimally ^, ^, TT ^ iooo' Too' Tooo- Every decimal contains as many decimal places as there are naughts in the denominator of the equivalent fraction. Table of places and names of integral and fractional units : CO a © s H a o 5 3 CO .3 CO *j 13 -a a GO 3 d GO CO S CO +3 CO 3 CO J3 O a .« CO J3 .4 c« CO CO tn o - -a i so T3 3 a T3 H •C — * 13 +J £3 CO o c* CO eS CO co =s o CO 2 •r* "2 — 1 CO C5 -5 K CO S '3 XI ■o CO © "3 •3 .2 a 8 co H X a O co Q CO H 3 S J3 r £ 8 § CO H 5 7 8 4 5 5 • 7 4 8 9 8 5. In .555 what figure stands for tenths? for hun- dredths ? for thousandths ? It is read 555 thousandths. 6. Is the decimal point named in reading a decimal ? Observe that the decimal is read as an integer and that the last figure is given the required denomination. Read : 7. .25 li. .101 15. .60745 8. .05 12. .0045 16. .678705 9. .005 13. .4045 17. .0065 10. .375 14. .0002 18. .60005 92 DECIMAL FRACTIONS 19. 50.0745 is read 50 and 745 ten-thousandths. How are both the integer and the decimal read ? How is the deci- mal point read ? What name is given to the last decimal place ? 20. When is the decimal point read ? When is it not read ? 21. What determines the value of any figure in a decimal ? 22. In writing .5, .45, .075, .0075 as common fractions, what figure in each decimal tells us the size of the denomi- nator ? A mixed decimal is a whole number and a decimal ; as, 4.625. 23. What number in a mixed decimal is always read first? 24. How do the number of places in any decimal compare with the number of naughts in the denominator when the decimal is expressed as a common fraction ? Read : 25. 45.075 28. 72.003745 26. 50.3007 29. 1001.1001 27. 290.25387 30. 794.3085 34. 5 thousandths calls for how many decimal places? What part of the decimal (5 thousandths) stands for the numerator of the fraction? what part of this decimal stands for the denominator? 35. Name the numerator and the denominator in the fol- lowing decimals: .05, .0006, .000025, .045. In writing a decimal write the numerator, and point off from the right as many decimal places as there are naughts in the denominator. 31. .00875 32. .008090 33. 2.004890 DOTATION AND NUMERATION 93 Written Work Write : 1. 34 hundredths. 2. 675 ten-thousandths. 3. 16 and 75 millionths. 4. 400 and 45 thousandths. 5. 6006 and 66 ten-thousandths. 6. 89 and 5 thousandths. 7. Seven hundred forty-six ten-thousandths. 8. Nine hundred and 84 millionths. 9. 5 million 9 and 4 hundred 9 ten-millionths. 10. 5095 millionths. 11. 8 and 17 ten-thousandths. 12. 125 millionths. 13. 896 and 301 hundred-thousandths. 14. One thousand and one thousandth. 15. 18051 and 957 thousandths. 16. 97 and 3 ten-thousandths. 17. 9864 millionths. 18. '2135 and 32 millionths. 19. One and one millionth. 20. One million and one tenth. 21. 90 thousand and 71 thousandths. 22. 1830 and 11684 hundred-thousandths. 23. 429 thousand and 46 ten-thousandths. 24. 7035 and 97 hundredths. 25. 67375 and 35 hundred-thousandths. 26. 5815 hundred-thousandths. 27. 375 and 69 thousandths. 94 DECIMAL FRACTIONS 28. 419863 and 23456 millionths. 29. 81 and 921 hundred-thousandths. 30. 2986 and 298643 ten-millionths. 31. 3020 and 302 hundred-thousandths. 32. 70 and 7 hundredths. 33. 8 thousand and 8 thousandths. 34. 645 million and 9 millionths. COMPARISON OF COMMON FRACTIONS AND DECIMALS 2. Then observe that 5 tenths = 50 hundredths = 500 thousandths. .5 = .50 = .500. 3. 50 hundredths may be written .50, or .500. Does adding naughts to the right of a decimal change the value of the decimal? 4. What is the difference in value between $.5 and 1.50? In writing decimal parts of a dollar, we always write two places for cents even if the last place is a naught. 5. Compare in value \ and T 5 ^, T 5 ^ and -^$q. 6. Does canceling the same number of naughts from both numerator and denominator change the value of a fraction ? 7. Since .5 = .50 = .500, does canceling naughts from the right of a decimal change the value of the decimal ? 8. Observe that canceling naughts from the right of a decimal really means canceling naughts from the numerator and the denominator. Thu S ,.50 = ir.5 P = l|. COMMON FRACTIONS AND DECIMALS 95 9. .400 is read 400 thousandths. How else may it be read ? 10. Is the unit in .4 the largest decimal unit in which .400 can be expressed ? Read first as given, then as if the naughts at the right of the decimal were canceled : 11. .040 13. .7500 15. 10.0057 12. 26.0050 14. 8.0090 16. 20.0900 Changing decimals to common fractions. Write as common fractions and change to lowest terms : l. .25 3. .045 5. .50 7. .0025 2. .7 4. .025 6. .75 8. .0775 9. Give the steps in changing a decimal to its fractional equivalent. A complex decimal is a decimal and a fraction united; thus, .16 1 is read 16 1 hundredths. * 3== 100x3 300 6' 10. Explain why multiplying both terms of the fraction i-^ bv 3 does not change the value of the fraction. 100 J ° Change to common fractions: 11. .371 15. .831 19. •62£ 23. lOl 12. .41 1 16. • 06£ 20. •33£ 24. .14f 13. .66| 17. .04 \ 21. .31^ 25. •58i 14. .241 18. .081 22. •03£ 26. .88J 96 DECIMAL FRACTIONS ADDITION AND SUBTRACTION OF DECIMALS 1. How must integers be written before they can be added '? subtracted ? 2. What change must be made in \ and | before they can be added or subtracted ? In adding or subtracting decimals, tenths must be placed under tenths, hundredths under hundredths, thousandths under thousandths, etc. Written Work 3. Add .8085 and .005. 4. Subtract .005 from .8085. .8085 .8085 .005 .005 .8135 .8035 Add as indicated and test : 5. 6. 7. 8. 9. .45 + 72.5 + 8.557 + 87. = — 10. 8.07 + 5.07 + 0.039 + 6.5 = — 11. 62.093 + 89.09 + 16.909 + 11.9 = — 12. 40.937 + 20.98 + 41.005 + 0(15 = — 13. 4- + + = — Subtract examples 14 to 17 and add the remainders : 14. 15. 16. 17. 40.275 9.0098 219.75 28.7 39.009 6.7849 8.95 12.5 18. + + + = — Add: 19. 3.7 20. .001 21.- .65 5.06 12.3 .001 8.023 15.0248 10.1 9.04 18.0149 25.004 ADDITION AND SUBTRACTION 9' Add: 22. 1.1 23. 120.2601 24. 36.15 4.01 230.81002 9.00999 1.0101 .05673 128.37 5.055 26. 3.7 27. 16,08753 25. 166.6 .27 185.057 7.0425 .0616 127.0348 28.318 .010912 216.253 142.0101 1.940054 456.03456 Add : 28. 12.015, 26.01102, 126.0592, 134.00876. 29. 100.001, 9.99, 149.0492, 7.077. 30. 2.2, 28.18, 140.027, 284.0295. 31. 318.003, 33.33, 495.0485, 12.0012. 32. Find the weight of four silver bars weighing as fol- lows : 15.75 pounds, .125 pounds, 14.3125 pounds, and 16.875 pounds. 33. Find the number of acres in four fields containing, respectively, 4.125 acres, .3125 acres, 8.8 acres, and 9.85 acres. 34. Find the sum of one hundred twenty-five and seven hundredths, eighty-nine and two hundred thirty-five thou- sandths, one hundred twenty-seven ten-thousandths, and sixteen and four tenths. 35. A farm cost * 4225.50; stock, $745.25; buildings, ■$1825.75; and implements, $358.45. What was the total cost? 36. How many square feet are there in four floors measuring, respectively, 245| square feet, 278| square feet, 174.375 square feet, and 168.3125 square feet? HAM. COMPL. A KITH- 7 98 DECIMAL FRACTIONS Find differences : 37. .75 .1825 38. .3216 .275 39. 4.205 1.7856 40. 15. 5.007 41. 38. 18.276 42. $45.67 12.09 43. 1251 87.432 44. 249| 178.625 45. 230.4897 116.5988 46. 100.001 99.9 47. 1001.101 900.909 48. 105.55 79.067 49. 1.1 .999 50. 5.05 .6565 51. 8.25 .0085 52. 1000.00 999.99 53. (9.5- 2.25) + (15.28-12.056) + (22.089- 19.063). 54. (11.001- 1.99) + (17.0107-14.014) + (29.3-23.2867). 55. The difference between two numbers is 1001.101, and the greater is 1101.011. What is the smaller number? 56. A is 35.875 years old, B is 48.25 years old, and C's age is 25.5 years less than the age of A and B combined. How old is C ? 57. To the sum of .808 and 80.8 add their difference. MULTIPLICATION OF DECIMALS 1. Multiply .05 by .5. 2. What is the numerator in .05 ? in .5? 3. What is the denominator in .05 ? in .5 ? What shows the denominator in a decimal ? 4. Multiply the numerators in .5 x .05 ; thus, 5x5 = 25. 5. Multiply the denominators in .5x.05; thus, 10x100 = 1000. 6. Write the result as a common fraction ; thus, T f -| ^ . 7. What term of the fraction is expressed by the decimal point ? MULTIPLICATION 99 Observe that there are as many decimal places in each deci- mal as there are naughts in the denominator of the equiva- lent common fraction. Find the product of the following decimals by multiplying the numerators and the denominators, separately, and ex- pressing the result as a decimal ; thus, • 04x - 7 =i^ x ii=iIro=- 028 9. .2x.06 12. .35x.05 15. .02 x .42 10. .04 x .5 13. .07 x .05 16. .06 x .004 11. .25 x .05 14. .03 x .02 17. .05 x .009 The product of two decimals contains as many decimal places as the sum of the decimal places in both factors. Written Work l. Multiply .25 by 5. f a \ 5 x 5 hundredths = 25 hundredths or 2 tenths and 5 hundredths. Write the 5 in hundredths' place, •"" and carry the 2. 5x2 tenths = 10 tenths; 10 2. tenths + 2 tenths = 12 tenths, or 1 aud 2 tenths. 1.25 Hence, 5 x .25 = 1.25. 2. Multiply .26 by .12. (h) cyo 1. What is the sum of the decimal places in the two factors ? 2. The product, then, must contain how many .12 52 places? 26 .0312 When the product has not enough decimal places, supply the deficiency by prefixing naughts. 100 DECIMAL FRACTIONS Find products : 3. .25 x .22 20. .1232 x .961 37. .4986 x .086 4. .17x.28 21. .2592x8 38. .006x20 5. .027 x .03 22. 65.65 x .65 39. 38.2 x .75 6. 27 x .12 23. 75.002 x 16.04 40. .0045 x .05. 7. .35x42 24. 275 x .007 41. 9.876 x .786 8. 8.7x9.22 25. .0018x720 42. 362.9 x .0076 9. .085x50 26. 1500 x .004 43. 119.8x2.74 10. .027 x 18 27. 124 x .064 44. 20.08 x .006 li. 1.005 x .011 28. 326 x .096 45. .375 x 2.027 12. 26.8 x 34 29. 627 x .78 46. 98.64 x 4.096 13. 28.25 x 12 30. 246 x .3 47. .069 x 8.92 14. 324.6 x 81 31. 29.4 x .08 48. 6.34 x 2.34 15. 39.10 x 18.4 32. 9.86 x 3.8 49. 12.34 x .004 16. .0214 x .016 33. 39.75 x .27 50. 6.08 x .0001 17. 12.134 x. 0025 34. 8.708x6.8 51. 1.002x1.004 18. 15.684 x 8 35. 368.9 x 8.5 52. .05 x .005 19. .1232 x 345 36. 2009 x .006 53. 6.876 x 4.37 54. How much must be paid for 85 acres of land at $45.75 per acre ? 55. Three brothers divided an estate worth 19600. The first received .125 of it, the second .375 of it, and the third the remainder. How much did each receive ? 56. A contractor furnished 2,626,000 bricks at $7.75 a thousand, and a laborer for 65 days at $2.75 a day. What was the amount of his bill ? 57. If there are 39.37 inches in a meter, how many inches are there in 12 meters ? how many yards ? DIVISION 30] Multiplying by moving the decimal point. 1. Multiply 6.385 by 10; by 100 ; by 100TT. 10 x 6.385 = 63.85 1- How do you murtfpjj a ribjrnWi 6j 10? inn a oor />oc c -. How may you multiply it decimal b<' 100 xb.38o = 638.5 U)? by lu() , by U)()(( , 1000 x 6.38o = 638o 3 How is the Ya]ue of a nuillber a ff ec ted by moving the decimal point one place to the right? two places? three places ? Moving the decimal point one place to the right multiplies the number by 10 ; two places by 100 ; three places by 1000. 2. Multiply 6.1234 by 10 ; by 100 ; by 1000. 3. Multiply .0342 by 10 ; by 100 ; by 1000. 4. Multiply 1.3412 by 10; by 100 ; by 1000. DIVISION OF DECIMALS Dividing a decimal or a mixed decimal by an integer. 1. Find J of 48 hundredths; of 64 hundredths. 2. Find \ of .25; of .35; .45; .75. 3. Find I of 6 and 36 hundredths; 12 and 24 hundredths. 4. Find 1 of 12.36; 24.42; 48.06; 54.06. Observe that in each problem a decimal or a mixed decimal when divided by an integer is simply separated or partitioned. Give quotients at sight : 5. J of .16 9. i of 6.6 13. 6.42-6 17. .006-5-3 6. iof.25 io. J of 8.08 14. 12.04-4 18. .024-5-6 7. iof.08 li. -J of 10.10 15. 15.05-S-5 19. .008-5-4 8. J of .04 12. i of 12.06 16. 24.18-5-3 20. .105-5-5 102 DECIMAL FRACTIONS Written Work l. Divide 12t).685 by 15. 2. Divide 174.44 by 28. _ 1 379 How many times is 15 con- 6.23 15)20.685 tainedin20? in 5.6 ? in 1.18? 28)174.44 15. in .135? ' 168 ' ~~F~f> In practice we divide as in the second example, placing the " •" decimal point directly above 56_ 1.18 the point in the dividend, be- 84 1.05 fore beginning to divide, and $4 J35 dividing as in integers. .135 When the divisor is an integer, division simply separates or partitions the dividend into equal parts. Thus, ^ of 20 and 085 thousandths (20.085) = 1 and 379 thousandths (1.379). A decimal or a mixed decimal is divided by an integer by 'placing a decimal point above or below the decimal point in the dividend, before beginning to divide, and dividing as in the division of integers. 3. 96.16-4-8 16. 12.312-4-27 4. 849.6-^-6 17. 2.25-4-15 5. 72.84-^-12 18. 809.6-4-16 6. 22.5-15 19. 256.25-4-25 7. 80.96-4-16 20. 96.064-4-32 8. 2.5625 -f- 25 21. 7010.5-4-35 9. .96064-4-32 22. 61.472-4-68 10. 701.05-4-35 23. 27.8142-4-307 11. 2.268-4-27 24. 425.92-4-605 12. 2.867-4-47 25. 901.57-4-97 13. 36.54-4-42 26. .2322-4-86 14. .666-4-74 27. 34.356-4-409 15. 6.675-4-89 28. 45.76-4-650 DIVISION 103 Making the divisor an integer. 1 fi 25 _i_ 1 25 = 5 Study of Problems 2. 62.5 -s- 12.5 = 5 1- What is the first quotient? the secoml ? 3. 625.-5- 125. = 5 the third? 2. What was done to the first problem to make the second? to the second to make the third? 3. How is a decimal affected by moving the decimal point one place to the right ? two places ? 4. How did moving the decimal point to the right the same number of places in both dividend and divisor of each problem affect the quotient? Multiplying both dividend and divisor by the same number does not change the quotient. Written Work Since multiplying both dividend and divisor by the same number does not change the value of the quotient, make the divisor an integer before beginning to divide. 1. 6.48 -4- .4 = 64.8 ■*■ 4. 1- Make the divisor an integer by ±\a\ a moving the decimal point one place ^ ' ' to the right in both dividend and divisor. 2. Show that this does not affect the quotient. 3. Solve, placing the decimal point directly below the point in the dividend, before beginning to divide. 2. 57.6 -r- .024 = 57600 -5- 24. 1. Make the divisor an in- 2400. teger by moving the decimal 24^57600 point three places to the right ' ,' in both dividend and divisor. 4o — — 2. Solve, placing the deci- °" mal point directly above the 96 point in the dividend, before 000 beginning to divide. Divide as in integers, placing the decimal point directly above or below the decimal point in the dividend, before begin- ning to divide. 104 DECIMAL FRACTIONS The use of the caret in division of decimals. Many teachers prefer to mark off by a caret as many deci- mal places from the right of the decimal point in the dividend as there are decimal places in the divisor, and divide as in integers, placing the decimal point directly below or above the caret in the dividend. Thus, . 8)5.68 = .8 )5.6 A 8 7.1 It is evident in the above problem that if both the divi- dend and divisor were changed so as to make the divisor a whole number, the decimal point in the dividend would be in the place occupied by the caret, and that the decimal point would be placed in the quotient immediately after the num- bers to the left of the caret had been used in the process of division. The use of the caret determines the position of the decimal point in the quotient, and at the same time retains the iden- tity of the problem. Thus, l. Divide 96.8 by .004. 2. Divide 1.2864 by .032. .004 )96. 800 A .032)1.286,4(40.2 24 200. 1 28 64 64 Mark off by a caret the same number of decimal places from the right of the decimal point hi the dividend as there are decimal places in the divisor. Divide as in integers, placing the decimal point in the quotient immediately after all the numbers to the left of the caret have been used in the process of division. DIVISION Find quotients : 3. 4.05 -.27 19. 1000 - .001 4. .252-=- .14 20. .2375 -j- .095 5. 8.398 -h 3.8 21. 177.8028-72.87 6. 2.173 -s- 1.06 22. 145.908 -=- 1.26 7. 144 -.12 23. .0656 -=-.004 8. .144-12 24. .1701-63 9. 31. 36 -.056 25. 85. 75 -=- .0049 10. .41912 -.338 26. .025641 -.7.77 11. 3. 125 -.25 27. .0022-200 12. .3105-15 28. 222 -.002 13. .5 -.625 .025 -=-.00025 14. 6.705-^.009 30. .0003-1.5 15. 139.195 -*- 14.35 31. $1. 05-1.005 16. 46.5 -.1875 32. .685-5-500 17. .00522 -.29 33. .01058-5-46 18. .001705.-5-. 31 34. 125.625-5-1.005 105 The division is frequently not exact. In such cases the sign + may be placed after the decimal to show that the division is not com- plete; thus, 1-7 = .142 + Find the sum of the quotients : 35. 36. 37. 1- .1 = 3- - .03 = .18-=- 72 = .1- 10 = 30- - .3 = .04-=- 50 = .25-5- 50 = 6- =-.006 = 2 -.025 = 2.5 -5- .5 = 16- - .04 = 20 -.002 = 25-4-2.5 = 60- - 300 = 200-12.5 = .15-=-. 15 = .6-: - 30 = 64 -.016 = 1.5 h- 15 = .66- - 1.1 = 64- 160 = .15-5-2.5 = .9- -.009 = .4h- 400 = 106 DECIMAL FRACTIONS 38. Divide $.10 by 1100. 39. If a stone cutter earns $3.75 a day, how many days will it take him to earn $311.25 ? 40. If 4275 acres of land cost $1731.375, what is the price per acre ? 41. At $. 22 a dozen, how many dozen eggs can be bought for $19.47? 42. If 16 stoves are sold for $292, what is the average price per stove ? 43. Divide $.18 by $20. 44. If the wheel of a bicycle is 9.25 feet around, how many times does it turn in going a mile ? 45. The product of two numbers is .9375. One of them is .75. What is the other ? 46. There are 30 \ square yards in one square rod. How many square rods are there in a plot containing 559.625 square yards ? 47. A merchant, in closing out his stock of goods, sold .37^ of the stock the first month, .35 the second month, and the remainder, $5500 worth, the third month. What was the value of his stock of goods ? Changing a common fraction to a decimal. Written Work Since A = 4 -=- 5, to change a fraction to a decimal, consider it a problem in division of decimals. Thus, -| = 4-3-5 = 5)4.0 0.8 Change to decimals and test : ■■■•8 *' 12 5< ff 7 - T6 9 ' 16 2-3 4-5- fill «19 ir»17 *' J *' 16 ** 25 8 32 10 - 2~5 REVIEW OF DECIMALS 1Q7 When the division does not terminate, the quotient may he shown as a complex decimal. Thus, ^ = 7)3.000 or as an 0.4281 7)3 000 u.-*z82- incomplete decimal, J ' 0.428 11. t*t 12. 9 31 13. 45 7 3 14. 29 53 13. .45^ + 0.42^ 5 4- 8.7£ + .95^ = 14. 86.55 +9.05^4- 9.87| +.00-| =■ 15. .875 + 0.75 + 7.7 + . 41 g =- 16. + + + 1 6(J Change to complex or incomplete decimals of not more than four places : 15. lj 19. |f 23. 16. A »■ !i 24. & 17. V- 21. £ 25. 2 7 18. ff 22. £- 26 . is REVIEW OF DECIMALS 1. Addf, .045, .12|, 18f, .675. 2. Subtract f from .5 of 3|. 3. Multiply (36.7 - 4|) by 6.7. 4. Subtract 6| from 11.065. 5. Take .0031 f rom 6. 6. Divide .047f by 2.3|. Add as indicated and test : 7. 8. 9. 10. 11. 8.375 4- .025 + 6.24f +.87] =— 12. .05f +6.041 +98.005| +.05| =- 108 DECIMAL FRACTIONS BUSINESS APPLICATIONS OF DECIMALS In all business transactions three things must be considered : (1) The quantity of the commodity bought or sold. (2) The price per unit at which it is bought or sold. (3) The total amount paid or received for the commodity. Quantity is measured by standard units established by custom or law ; thus, the pound is a unit of weight ; the foot or the yard, a unit of length ; and the gallon or the barrel, a unit of liquid measurement. The price per unit is the amount of money paid or re- ceived for a standard unit of the commodity ; thus, when butter is sold at 25 ^ per pound, the standard unit is the pound and the price is 25 cents. 1. What standard is used in measuring grain ? butter ? eggs ? milk ? cloth '? hay ? oil ? 2. What unit is used in measuring values in money ? How many cents are there in $1 ? in $% ? •$ \ ? $>§ ? 50 ^ is what part of $1 ? 20 f is what part of ffl ? 25 i ? 10 ^ ? 5 1 ? Parts of $1 that Should be Known U =T*o<>f$l 25^ =iof$l. 2^ -^ of$l 33^ = £of$l 2^ = T V of$l 37.^ = fof$l 4^ =£, of$l 40? =fof$l 5^ =^ of $1 50^ =,Vof$l 6^ = T V of$l 62^ = | of $1 Sh? = T \ of$l 66|^ = |of$l 10^ =1 V of$l 75^ =fof$l 12^= | of$l 80^ =iof$l 16^=i of$l 83^ = £<>f$l 20^ = i of $1 87^ = |of$l. BUSINESS APPLICATIONS OF DECIMALS L09 Finding the total cost when the quantity purchased is given and the price of a unit is an even part of $1. Written Work Business computations may be shortened by knowing the relation that the price of a unit bears to $1 or to $100. 1. How much will 44 bushels of potatoes cost at $.25 per bushel ? Decimal Method Short Method $ .25 = price ±)$U 44, no. of bushels "$TT" 1 qq At $ 1 each, 44 bu. would cost $44. $iT00 = total cost Atneach,the y costiof«44,or«ll. Find the cost of the quantity at $1. Divide this by the quantity that can be purchased for $ 1. Find cost of : 2. 60 bu. apples at 33^ per bushel. 3. 25 lb. butter at 25^ per pound. 4. 960 yd. calico at 6^ per yard. 5. 50 lb. lard at 121^ per pound. 6. 80 lb. rice at 12 1 -fl per pound. 7. 120 yd. ribbon at 371** per yard. 8. 500 books at 40^ each. 9. 1200 doz. eggs at 25^ per dozen. 10. 600 bu. oats at 33^ per bushel. 11. 1600 bu. coal at 6|^ per bushel. 12. 86 qt. cherries at 6| ^ per quart. 13. 2500 bu. corn at 40^ per bushel. 14. 160 lb. beef at 10 4 per pound. 110 DECIMAL FRACTIONS 15. A merchant buys 240 lb. coffee at 12| t per pound, 300 lamp chimneys at 8^ each, 6000 qt. milk at 41^ per quart, 560 bu. potatoes at 50^ per bushel. Find total cost. 16. A farmer sells 1260 heads of cabbage at b$ per head, 250 bu. potatoes at 50^ per bushel, 2240 lb. beans at 6| $ per pound, 600 qt. cherries at 8^ per quart, 1200 qt. strawberries at 12|^ per quart. Find total receipts. 17. A milk dealer bought 300 bu. corn at 50 ^ per bushel, 6000 lb. bran at \$ per pound, 6000 lb. hay at f>l per hundred pounds. He sold 4000 gal. milk at 6\ tf per quart. How much did he make ? Finding the quantity purchased when the total cost is given and the price of a unit is an even part of $1. Written Work 1. How many yards of calico, at Q\$ per yard, can be purchased for $100? Decimal Method Short Method 100 x 16 yd. = 1600 yd. 1600 . since 6\f = $ f L, $ 1 pays for 16 yd.; $.0625)!$100.0000 A and $100 pays for 100 x 16 yd., or 1600 yd. Multiply the quantity purchased for $1 by a number equal to the number of dollars invested. BUSINESS APPLICATIONS OF DECIMALS 111 Find the quantity of each article if a grocer invested 2. $1G0 in sugar at Q\ f per pound. 3. $120 in sugar at 4^ per pound. 4. $6.00 in rice at 10 ^ per pound. 5. $100 in cloth at 50^ per yard. 6. $13.00 in gingham at 5^ per yard. 7. $3.00 in cheese cloth at 2^ per yard. 8. $ 800 in milk at G\ i per quart. 9. $ 100 in meat at 12| t per pound. 10. $4.00 in collars at 12| ^ each. 11. $1.00 in bananas at 20 ^ per dozen (find number of bananas). 12. A hotel keeper purchased $100 worth of sugar at G\ ^ per pound, $250 worth of potatoes at 33£ ^ per bushel, $60 worth of soap at 2\ ^a cake. Find number of pounds, bushels, and cakes purehased. 13. A farmer sold to a merchant 10 bu. apples at 40 ^ per bushel, 20 qt. beans at 10 $ per quart, 16 bu. potatoes at 75 ^ per bushel. He invested | of the proceeds in cloth at 25 9 per yard and the balance in coffee at 12* ^ per pound. How much of each did he purchase? 14. A grocer bought coffee at 12| ^ a pound, and sold it for $39, thereby gaining $G.50. How many pounds did he buy ? SIMPLE ACCOUNTS FOR BOYS AND GIRLS An account is a statement of the receipts and disburse- ments of any person. There are two sides to an account : the first, or debit side, on which are entered all receipts; the second, or credit side, on which are entered all disbursements, or amounts paid out. Dr. indicates the debit side of an account ; Cr. indicates the credit side. The balance is the difference between the debit and credit sides. September 1, 1906 Cash on hand Dr. Cr. Sept. 1 $12 10 Sept. 1 Note-book, $.15; pencil, if .05 $ 20 Sept. 4 Arithmetic 50 Sept. 5 Geography 1 00 Sept. 7 Copy-book, $.10; ink and pens, $.08 18 Sept. 15 History 1 00 Sept. 17 Worked one day 1 00 Sept. 25 Car fare 50 Sept. 29 Tools 1 60 Sept. 30 Balance, Cash on hand A $13 8 12 10 $13 10 Continue the balance of each month through the following months to September, 1907. Note to Parents. — Children should be encouraged to keep their own personal accounts. 112 SIMPLE ACCOUNTS FOR BOYS AND GIRLS 113 1. October. Oct. 3, Bought 1 pair of shoes, $2.50. 1 hat, $1.50. Oct. 8, Repairs to bicycle, $.75. Oct. 15, Earned $1.50. Oct. 17, Worked for Mr. Black and received $.15. Oct. 25, Saturday outing, $.60. 2. November. Nov. 5, Bought a sled, $.95. Nov. 9, Bought a cap, $.75. Nov. 15, Shoveled snow off Mrs. Graham's walk, $.30. Nov. 17, Sawed kindling wood for Mr. Goff, $.50. Nov. 26, Bought a knife, $.25. Nov. 30, Ran errands, $.35. 3. December. Dec. 3, Bought 1 pair of skates, $.75. Dec. 10, Received from Mr. Black for work in store, $1.00. Dec. 17, Expense for school supplies, $.17. Dec. 21, Received from Mrs. Williams for carrying in load of coal, $.30. Dec. 22, Bought Christmas presents, $3.75. Dec. 25, Christ- mas gift from Uncle James, $1.00. Dec. 29, Expense for having skates sharpened, $.10. 4. January, 1907. Jan. 5, Received from Mrs. Jones for fixing doorbell, $.15. Jan. 8, Bought 1 pair mittens, $.50. Jan. 15, Delivered bills around town for Mr. Black, $.50. Jan. 25, Bought necktie, $.25. Jan. 30, Bought " History of French Revolution," $.75. 5. February. Feb. 6, Worked on Saturday for Mr. Black, $.75. Feb. 11, Shoveled snow from sidewalk for Mr. Hart, $.25. Feb. 16, Ran errands, $.40. Feb. 20, Helped unload car of feed, $1.00. Feb. 26, Copied 2 leases for Mr. Irwin, $.75. Feb. 28, Bought pair of gloves, $1.25. 6. March. March 1, Cleaned yard for Mrs. Williams, $.50. March 6, Bought 2 pairs of socks, $.30. March 11, Bought new umbrella for mother, $1.75. March 15, Repaired fence for Mr. Jones, $.25. March 27, Car fare, $.30. March 30, Sold my old bicycle fur $5.00. HAM. COMPL. ARITH. -8 114 SIMPLE ACCOUNTS FOR BOYS AND GIRLS 7. April. Apr. 1, Burned paper and refuse for Mr. Hart, $.25. Apr. 8, Made garden for Mrs. Black, $.50. Apr. 10, Whitewashed cellar for Mrs. Goff, $.35. Apr. 15, Wheeled load of coal for Mr. Brown, $.35. Apr. 25, Bought 4 collars and 2 pairs of cuffs, $.90. Apr. 30, Bought neck- tie, $.25. 8. May. May 3, Bought straw hat, $1.00. May 7, Mowed lawn for Mrs. Jones, $.25. May 13, Repaired Mr. Brown's sidewalk, $.40; May 30, Bought baseball, $.50. May 31, Received a reward of $5.00 for finding a pocket- book containing $50, which I returned to owner. 9. June. June 1, Made $.20 selling papers. June 6, t Worked a day for Mr. Black, $.75. June 10, Delivered package, $.25. June 17, Bought ball bat, $.50. June 20, Wheeled a trunk for Mr. Hart, $.25. June 29, Bought 1 pair of baseball shoes, $1.00. 10. July. July 4, Fireworks, $.50. July 6, Received from Mr. Black salary for week, $5.00. July 12, Bought 2 shirts, $1.50. July 13, Received week's salary, $5.00. July 15, Bought outing suit, $6.50. J^ily 20, Received my salary, $5.00. July 25, Expense for small articles, $.95. July 27, Received my week's salary, $5.00. July 30, Re- ceived for overtime, for month, $7.50. 11. August. Aug. 3, Salary, $5.00. Aug. 8, Bought 1 pair of tan shoes, $2.50. Aug. 10, Received salary, $5.00. Aug. 15, Bought fishing tackle, etc., $3.75. Aug. 17, Re- ceived week's salary, $5.00. Aug. 31, Expenses for 2 weeks' vacation, $15.75. Sept. 1, Balance, Cash on hand, . Make out a statement at close of year, showing total receipts and disbursements, and proving final balance. DENOMINATE NUMBERS 1. Write from memory the following tables : Liquid Measures, Dry Measures, Avoirdupois Weight, Time Measures, and Measures of Length or Distance. 2. 1 yr. = mo. = da. = hr. = niin. = sec. 3. 1 mi. = rd. = yd. = ft. = in. 4. 1 T. = cwt. = lb. = oz. 5. 1 bu. = pk. = qt. 6. 1 gal. = qt. = pt. The standard or principal units of measure are as follows : Liquid — gallon. Length or distance — yard. Dry — bushel. Avoirdupois — pound (i6oz.). Time — day. All other measures are determined from the above unit measures. Thus, the ton is 2000 times 1 pound (16 oz.). The hour is ^ of the day, the period of one revolution of the earth on its axis. A denominate number is a concrete number whose unit is a measure established oy custom or law; as, 10 feet, in which 1 foot is the unit of measure, or 5 pounds, in which 1 pound is the unit of measure. A simple denominate number is a number of one denomi- nation ; as, 12 rods, 2 ounces, 5 days, etc. A compound denominate number is composed of two or more concrete numbers that express one quantity; as, 6 yards, 2 feet, 4 inches. Here yards, feet, and inches are used to express but one quantity. ll . 116 DENOMINATE NUMBERS REDUCTION OF DENOMINATE NUMBERS Change : 1. 5^ yd. to feet. 2. 90 in. to feet. 3. 3 yd. 2 ft. to feet. 4. .5 rd. to inches. 5. 25 ft. to yards. 6. 5.5 hr. to minutes. 7. .5 mi. to rods. 8. 3.5 gal. to pints. 9. ^ day to minutes. 10. .25 bu. to quarts. 11. | pk. to quarts. 12. 2 lb. 8 oz. to ounces. 13. | cwt. to pounds. 14. .5 yd. 1 ft. to inches. 15. .75 mi. to rods. 16. .25 bu. to pints. 17. 3.5 pk. to quarts. 18. 2 yd. 1.5 ft. to inches. 19. 3.5 min. to seconds. 20. 48 qt. to pecks. 21. 64 pt. to bushels. 22. 64 oz. to pounds. Written Work l. Change 3 gal. 3 qt. 1 pt. to pints. gal. 3 4 qt. pt. 3 1 12 + 3 15, number of quarts. _2 30 + 1 31, number of pints. Observe that 4 qt. is really the multi- plicand and 3 the multiplier in finding the first product; and that 2 pt. is really the multiplicand and 15 the multiplier in finding the second product. In considering the num- bers abstractly, however, either factor may be regarded as the multiplicand and the arrangement as indicated saves time. REDUCTION 11' 2. Change .875 gallon to pints, etc. .875 3.500, number of qt. 1.00, number of pt. Siuce there are 4 qt. in a gallon, in .875 of a gallon there are .875 of 4 qt., or :5.."> qt. Since there are 2 pt. in 1 qt., in .5 of a quart there is 1 pt. The answer is 3 qt. 1 pt. Change 3. 15 lb. 8 oz. to ounces. 4. 96 ft. 5 in. to inches. 5. 5.5 bu. to quarts. 6. 3.5 pk. to pints. 7. 18 cwt. 25 lb. to pounds. 8. 23 hr. 16 min. to minutes. 9. 8.3 mi. to yards. .75 yd. to inches. 4| T. to pounds. 7| min. to seconds. 10. 11. 12. 13. 6.5 L. T. to pounds. 14. 63.5 gal. to pints. 15. £ bu. to quarts. 16. 10| bu. to pecks. 17. Change 266 quarts to bushels, etc. 8 )266 There are | as many pecks as quarts, 4 )33, 110. of pk. + 2 qt. that is, 33 pk. + 2 qt. There are \ as 8, no. of bu. -f- 1 pk. many bushels as pecks, that is, 8 bu. + ., , „ 1 pk. Hence, 266 qt. = 8 bu. 1 pk. 2 qt. 8 bu. 1 pk. 2 qt. Change to higher denominations: 18. 312 inches. 19. 6625 yards. 20. 5281 feet. 21. 2043 seconds. 22. 1033 ounces Av 28. 43920 in. 29. 6875 sec. 30. 56.5 pk. 31. 684.5 rd. 23. 347 cwt. 24. 6095 pounds. 25. 16857 rods. 26. 11097 qt. (Dry). 27. 952 pt. (Liquid). 33. How many gallons of milk will a family consume in 75 days, if they use 2 qt. 1 pt. daily ? 34. How much is received for H bushels of chestnuts at 8 cents a quart ? 32. 964| min. 118 DENOMINATE NUMBERS 35. How much will 15 turkeys, averaging 14^ lb. each, cost at 18 cents a pound ? 36. If 100 tons of coal are bought by the long ton, at 12.24 a ton, and sold by the short ton at the same price, how much is gained ? 37. At 20 cents an hour, how much will a man earn in 26 days, working each day from 8 A.M. to 5 p.m., allowing 1 hour for lunch ? 38. If a flour mill grinds wheat at the rate of 1 pint in 5 seconds, in how many hours and minutes will it grind 21,600 bushels ? 39. A train goes 104 miles in 3 hours and 15 minutes. What is the rate per hour ? 40. At 2 cents a foot find the length in miles and rods of a telephone wire that costs $4672.80. 41. If a man's step averages 2 ft. 6 in., how far will he travel in taking 6600 steps ? Relations of denominate measures. 1. | pk. is what decimal part of a bushel? f pk. = 6 qt. 6 qt. = — bu. = .1875 bu. 4 32 2. 3 ft. 2 in. is what fractional part of a rod ? 3 ft. 2 in. = 38 in. 1 rod = 198 in. 3 ft. 2 in. = T 3 9 8 ? , or |f rd. Find the fractional part : 3. 2| hr. is of 1 day. 6. 1| pt. is of 3 qt. 4. 71 ft. is of 1 rod. 7. 2-| in. is of 10 ft. 5. 3| qt. is of 1 gallon. 8. 1^ qt. is of 2 gal. ADDITION' AND SUBTRACTION 119 Find the fractional part : 9. 15 hundredweight is of 1 ton. 10. 3.5 quarts is of 1 bushel. 11. 280 rods is of 1 mile. 12. 37 pounds 8 ounces is of 1 hundredweight. 13. 440 yards is of 1 mile. Find the decimal part : 14. 16 hours 48 minutes is of 1 day. 15. 1 foot 8 inches is of 1 rod 2 in. 16. 180 pounds is of 1 ton. 17. 16 minutes 48 seconds is of 1 hour. 18. A machinist works 10 hr. per day in summer and 8| hr. per day in winter. If his wages in summer are $3.35 per day, at the same rate find his wages per day in winter. ADDITION AND SUBTRACTION 1. Find the sum of 2 gal. 3 qt. 1 pt., 4 gal. 1 qt. 1 pt., 7 gal. 1 pt., 5 gal. 3 qt. gal. qt. pt. ^ 3 The sum of the pints = 3 pt. = 1 qt. and 1 pt. 4 11 The sum of the quarts + 1 qt. carried = 8 qt. = 7 12 gal. qt. 5 3 The sum of the gallons + 2 gal. carried = 20 gal. 20 o r Add: 2. 14 bu. 2 pk., 5 bu. 6 qt., 7 qt. 1 pt., 9 bu. 6 qt. 3. 5 T. 11 cwt., 4 T. 15 cwt. 60 lb., 11 T. 80 lb., 19 T. 3 cwt. 64 lb. 4. 9 yr. 120 da. 8 hr., 12 yr. 104 da. 17 hr., 14 da. 5. 3 wk. 6 da. 15 hr., 4 wk. 3 da. 9 hr., 7 wk. 5 da. 14 hr. 7 3 3 3 15 1 P k. 6 qt,; 2 4 16 Subtract : mi. rd. yd. ft. 7. 80 120 12 57 215 14 120 DENOMINATE NUMBERS 6. From 7 bu. 3 pk. 3 qt. take 3 bu. 1 pk. 5 qt. bu. pk. qt. 1 pk., or 8 qt., + 3 qt. = 11 qt.; 11 qt. - 5 qt. 6 qt.; 2 pk. - 1 pk. = 1 pk.; 7 bu. - 3 bu. = 4 bu. gal. qt. pt. 8. 23 1 9 3 9. From 18 hr. take 9 hr. 16 min. 45 sec. Finding the difference in time between two dates is the most practical application of subtraction of denominate numbers. 10. Find the difference in time between November 15, 1903, and August 12, 1905. Aug. 12, 1005, is represented as the 12th day yr.^ mo. da. of the 8th mouth of 1905, and Nov. 15, 1903, 1905 8 12 as the 15th day of the 11th month of 1903. 1903 11 15 1 mo., or 30 da., + 12 da. = 42 da.; 42 da. - 1 8 27 15 (la - = 27 c,a -' 1 vr -> or 12 mo -' + 7 mo - = 19 mo.; 19 mo. — 11 mo. = 8 mo.; 1904 yr. — 1903 yr. = 1 yr. Subtract : yr. mo. da. yr. mo. da. li. 1908 7 12 12. 1905 9 1 1901 9 15 1890 8 15 13. How many years, months, and days old is each pupil in the room? 14. General Robert E. Lee was born January 19, 1807, and General Ulysses S. Grant April 27, 1822. How old was each at the close of the Civil War, April 9, 1865? How much older was General Lee than General Grant? 15. How old is a man to-day who was born July 3, 1882 ? MULTIPLICATION AND DIVISION 121 MULTIPLICATION AND DIVISION 1. Multiply 3 \vk. 5 da. 9 hr. by 7. wk. da. hr. o r g 7 x fl hr. = 63 hr. = 2 da- and 15 hr.; 7x5 da. = 35 da.; 35 da. + 2 da. = 37 da. = 5 wk. and 2 1 da.; 7 x 3 wk. = 21 wk.; 21 wk. + 5 wk. = 26 wk. 26 2 lo Hence, the answer is 26 wk. 2 da. 15 hr. Multiply : 2. 3 gal. 2 qt. 1 pt. by 3. 3. 12 bu. 3 pk. 3 qt. by 6. 4. 15 T. 5 cwt. 12 oz. by 10. 5. 27 wk. 3 da. 14 hr. by 9. 6. 23 mi. 124 rd. 11 ft. 4 in. by 12. 7. Divide 54 T. 15 cwt. 72 lb. by 12. 54 T. -r- 12 = 4 T. and 6 T. remain- ing; 6 T. = 120 cwt. ; 120 cwt. + io\c\J Ye to 15 cwt. = 135 cwt.; 135 cwt. h- 12 = 11 cwt. and 3 cwt. remaining ; 6 cwt. = 300 lb.; 300 lb. + 72 lb. = 372 lb. ; 372 lb. -*- 12 = 31 lb. T. cwt. lb. 11 31 Divide : 8. 18 wk. 5 da. 21 hr. by 5. 9. 188 gal. 1 pt. by 7. 10. 88 bu. 3 pk. 4 qt, by 9. 11. 61 yr. 11 mo. 18 da. by 11. 12. 86 T. 3 cwt. 44 lb. by 6. 13. Find the cost of 19 gross of pencils at 10^ a dozen. 14. A man digs 4 rods, 2 yards of ditch in a day. How many rods, etc., can he dig in 6 days ? 15. How many packages, weighing 5 ounces each, can be made from 5 pounds of candy ? 122 DENOMINATE NUMBERS REVIEW 1. If a watch gains 18 seconds in a day, how much too fast will it be in three weeks ? 2. How many barrels, each holding 2 bushels and 3 pecks, will be required to pack 88 bushels of apples ? 3. How many bushels of potatoes are necessary to plant 8| acres, allowing 6 bu. 1 pk. to the acre? 4. A merchant sells linseed oil at 12 ^ a pint that cost him 56^ a gallon. Find his profits on 45 gallons 3 quarts. 5. 5 car loads of coal weigh : 57,698 lb., 49,875 lb., 63,545 lb., 49,897 lb., and 54,273 lb. Find the number of tons, hundredweight, and pounds in all. 6. 4 men buy a plot of land that has 222 feet 8 inches street frontage. Allowing for an alley 20 feet in width in the center, what is the width of each man's lot if they divide the plot equally ? 7. A force pump in a coal mine lifts 76| gallons of water to the surface per minute. Find the number of gallons pumped out in one day. 8. If 3 pounds 4 ounces of coal are consumed in generat- ing power to lift 5 gallons of water in problem 7, find the number of tons of coal consumed each day. 9. A Kentucky farmer clipped 241^ pounds of mohair from 70 Angora goats. Find the average clip from each goat and its value at $.37*- per pound. 10. An automobile runs 2| miles in 5 minutes. At that rate, find the distance in miles, rods, and feet it runs in 1 hour 35 minutes. 11. A pencil factory makes 6| gross of pencils per hour. Find the number of dozen made in 26 days of 9^ hours each. PRACTICAL MEASUREMENTS MEASURES OF LENGTH 1. Measure the length of your desk ; the length of the room ; the length of the blackboard ; the height of the window from the floor. 2. In what are these short lengths measured? To the Teacher. — Secure a tape measure 50 feet long. 3. Measure the distance around the schoolroom in feet and fractions of a foot. How many yards is it around the room ? 4. Measure the distance around the school grounds in rods, feet, and inches. 5. Take 16| ft. of the tape measure and measure 10 rods along the public road or street. 6. 320 x 16| ft, = how many feet? 1760 x 3 ft, = how many feet? 7. How many feet equal a mile ? how many yards ? 8. James walks 1| miles to school each day. How many rods does he walk in going to and from school ? 9. How many rods equal 5280 ft, ? | of a mile? 3560 ft.? 10. Mary walks f of a mile to school each day. How many miles does she walk in going to and from school in 180 days? 11. Henry walks .8 of the number of miles Mary walks. Find the distance Henry walks in a term if he attends 160 days. 123 124 PRACTICAL MEASURED ENTS MEASURES OF SURFACE Observe that the straight lines AB and CD cannot meet, however far they may be extended. Such lines are called parallel lines. A C- -B ■D Lines that meet, making a square corner, form a right angle. A figure that has four straight sides and four right angles is called a rectangle. A rectangle having its four sides equal is called a square. 1. Name six different rectangles in the schoolroom. Are the opposite sides of a rectangle parallel lines ? 2. How many dimensions has every rectangular surface ? How does a surface differ from a line ? 3 in. .c 1 V f \ 3. Draw, on a scale of £, a rectangle 3 inches long and 2 inches wide. Divide it by lines into square inches. How many square inches are there in the first row ? in the second ? in the rectangle ? What is the unit of measure in this surface ? Observe that 2x3x1 sq. in. = 6 sq. in. The area of a rectangle is found by multiplying its unit of measure by the product of its two dimensions^ when expressed in like units. 4. Draw a rectangle 2 ft. by 6 ft. Divide it into sq. ft. 5. Draw a square a foot on a side. Mark off the sides into 12 equal parts and connect them by straight lines. How many square inches equal a square foot? MEASURES OF SURFACE 125 6. Draw oil the blackboard a line 4 feet long. From each end draw lines in the same direction 3 feet in length, making square corners with the 4-foot line. Connect by a straight line the ends of the 3-foot lines. 7. Are the sides of the figure straight ? Are the corners equal in size? Find the area of the figure. 8. What is a right angle ? a rectangle ? (p. 124). 9. Show by a diagram the number of square feet in a square yard. 10. Draw a diagram on a scale of 1 inch to 3 feet to rep- resent a rectangle 24 ft. long and 18 ft. wide. Find its area. Draw diagrams on scales suitable to the size of your tablet or slate and rind the surface of each of the following : 11. A rectangle 20 ft. by 24 ft, 12. A flower bed 16 ft. by 8 ft. 13. A floor 16 ft. long and 14 ft. wide. 14. A wall 15 yd. long and 5 yd. high. 15. By actual measurement find the number of square feet in the floor, the door, the blackboard, and the walls of the schoolroom. 16. In what denominations did we find the lengths and widths of the problems just given? Land is measured in acres, square rods, square feet, etc. 17. Measure a square rod on your playground. How long is it ? how wide ? 18. Measure the length and width of your school grounds in rods and feet. 19. Since 16^ feet equal 1 rod, how many yards equal 1 rod ? How many square yards equal 1 square rod ? 126 PRACTICAL MEASUREMENTS 20. Since 16 1 feet equal 1 rod, how many square feet equal 1 square rod? 21. A field is 70 rods long and 40 rods wide. How many- square rods are there in it? how many acres? 22. Memorize this table : 144 square inches (sq in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30 \ square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) 1 A. = 160 sq. rd. = 4840 sq. yd. = 43,560 sq. ft. Change : 23. 2700 sq. yd. to sq. ft. 26. If A. to sq. rd. 24. 50 sq. ft. to sq. in. 27. 800 sq. yd. to sq. rd. 25. 1600 sq. rd. to A. 28. 5f A. to sq. ft. 29. A farm is 90 rods long and 60 rods wide. Find the number of acres in it. Find its cost at 160 per acre. 30. A lot 100 ft. square has a house 36 ft. by 42 ft. located on it. The remaining space is lawn. Find the number of square feet of lawn. Draw diagram. 31. A concrete sidewalk in front of the lot is 4 ft. wide. Find its cost at 19 ^ per square foot. 32. Find the cost of a flagstone walk, 135 ft. long and 6 ft. wide, at 21 i per square foot. 33. City lots are sometimes sold by the square foot. Find the cost of a lot in Pittsburg 21 ft. by 70 ft. at $27.50 per square foot. MEASURES OF SURFACE 127 34. A farm 160 rods long and 12<> rods wide is sold in two pieces, § of it at 160 per acre, and the remainder at $ 50 per acre. Find the amount of the entire sale. 35. An Iowa farmer owns a farm a mile square. How many acres has he ? Find its value at $85 per acre. 36. A western wheat held 100 rods long and 80 rods wide yields 880 bushels of wheat. Find the average yield per acre. 37. City lots are usually sold by the front foot. Find the cost, at $20 per foot front, of a lot 25 ft. front by 120 ft. deep. Find the cost per square foot. 38. A four-room school building has a slate blackboard 24 ft. by 4 ft. in each room. Find the total cost of the blackboard at 23^ per square foot. 39. The area of a field in the form of a rectangle is 8 acres. If one side is 32 rods, what is the other? These diagrams represent pieces of land. The dimensions are given in rods, and the corners are all square. 40. Divide the first piece into 3 rectangles and find (1) how many square rods there are in each ; (2) the perimeter of each ; (3) the area of the entire piece in acres. 41. Divide the second piece into rectangular lots, and find (1) the perimeter of each ; (2) the area of each ; (3) the area of the entire piece. 20 6 14 4 6 5 4 7 25 7 5 '° 13 5 a 9 7 12 5 3 6 7 128 PRACTfCAL MEASUREMENTS PAINTING AND PLASTERING Painting, plastering, and kalsomining are generally meas- ured by the square yard. In some localities an allowance is made for doors and windows, but there is no uniform rule in practice. 1. How much will it cost to paint a ceiling 18 ft. long and 15 ft. wide at 10/ per square yard? 2. How much will it cost to kalsomine a hall 30 ft. long, 9 ft. wide, and 15 ft. high, at 5/ per square yard ? (Observe that the perimeter of the hall is 78 ft.) 3. How many square yards of plastering are there in a room 21 ft. long, 18 ft. wide, and 12 ft. high, making no allowance for openings? 4. How much will it cost, at 15/ a square yard, to plaster a room 24 ft, x 191 ft. x 15 ft.? 5. A public hall is 120 ft. x G6 ft. x 22| ft. How much will it cost to paint the walls and ceiling at 10/ per square yard? THE RIGHT TRIANGLE 1. Draw on the blackboard a rectangle 12 inches long and 8 inches wide. Connect the opposite corners by a straight line. This line is called the diagonal of the rectangle. 2. Into how many parts have we divided the rectangle? Shade one of the parts with chalk. How many angles are there in each part ? how many right angles ? A triangle is a surface bounded by three straight lines. THE RIGHT TRIANGLE 129 A right triangle is a triangle having one right angle. The base of a triangle is the side on which it is assumed to stand. The altitude of a triangle is the line that meets the base line at a right angle. To the Teacher. — As an aid in drawing have each pupil, if possible, gei a right triangle as here shown. 3. Point out the base and altitude in the triangles at the right. o 4. Fold a rectangular piece of paper, as ABCD, on its diagonal. Observe : (1) That the rectangle ABCD and the triansfle ABB have the same base and altitude. (2) That the area of the triangle is just I the area of the rectangle. Hence, the area is | of 4 x 2 x 1 sq. in. = 4 sq. in. m c * ■■"v. n InllTTThfc. ■o IhTttk c »~ T-> ip ''M ilUlHiit^ ^ A Base 4 in. B The area of a right triangle eqvals the unit of measure mul- tiplied by \ the product of the base and altitude. Draw on a scale suitable to your paper and find the ana of the following right triangles in square inches : 5. Base 10 in., altitude 8 in. 7. Base 25 in., altitude 18 in. 6. Base 12 in., altitude G in. 8. Base St! in., altitude 21 in. 9. Find the area of a field in the form o( a right triangle whose base is 80 rods and altitude 40 rods. HAM. COMPL. AKITU. 9 130 PRACTICAL MEASUREMENTS MEASURES OF VOLUME 1. How many dimensions has the cube ? Name them. 2. What dimensions has a line ? What dimensions has a surface ? a solid ? 3. Name the different units of measure in which the length of a line may be expressed. m4. Name the ^-= — : -» different units of square measure in which surface may be expressed. 5. What cubic unit have we in the first cube ? 6. If a cube is 1 foot on an edge, what is the cubic unit ? 7. Draw a square 1 foot on a side. Show that it contains 144 square inches. MEASURES OF VOLUME 131 3 in. 8. Observe that 144 cubes 1 inch on an edge can be placed on a surface 1 foot square. How many layers of such cubes will it take to make a cube 1 foot on an edge ? 9. How many cubic inches equal 1 cubic foot ? 10. How many surfaces has a cube ? 11. Show that all the surfaces of an inch cube are the same in area ; of a 2-inch cube ; of a 9-inch cube. 12. Examine carefully the fig- ure. Observe : (1) That the surface of the face upon which it rests con- tains 9 square inches. (2) That the first layer of units of volume contains 9 cubic inches. (3) That the whole solid, if 6 inches high, contains 6x9 cubic inches, or 54 cubic inches. 13. How many 1-inch cubes are there in the first layer ? how many in the solid ? 14. What is the shape of the surfaces of the solid? Is each surface a rectangle? A rectangular solid is a solid whose surfaces are all rectangles. 15. Observe that the number of inch cubes in the solid is equal to the product of its three dimensions. 16. What is the unit of measure in the solid ? Observe that 3 x 3 x 6 x 1 cu. in. = 54 cu. in. The contents, or volume, of a rectangular solid equals the unit of measure multiplied by the product of its three diynensions. 132 PRACTICAL MEASUREMENTS To the Teacher. — Secure 144 1-inch cubes. 17. Build a cube 2 inches on an edge. 18. Build a cube 4 inches on an edge. 19. Compare the 4-inch cube with the 2-inch cube. 20. Give the different units of measure of surface ; of length ; of contents. 21. Find the contents of a box 3 ft. long, 2 ft. wide, and 11 ft. high. 22. Observe the cube. What is its length? width? height? 3ft. 23. How many 1-foot cubes does it contain ? A cube 3 ft. on an edge is called a cubic yard. 3f 24. Memorize this table : 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) A cart load of earth is considered 1 cubic yard. PRACTICAL APPLICATIONS Excavations are estimated by the cubic yard. l. Find the cost, at 30^ per cubic yard, of excavating a cellar 36 ft. in length, 24 ft. in width, and 4 ft. in depth. 36 x 24 x 4 x 1 cu.fi. = 3456 cu. ft., the contents of the cellar. 3456 cu. ft. ^ 27 = 128, number of cu. yd. 128 x $.30 = S 38.40, cost of excavation. PRACTICAL APPLICATIONS 133 This diagram shows the outline of a cellar 5 ft. deep. Its dimensions are given in feet. 2. Find its square feet. area in 4 4 14 14 30 12 12 30 3. Find the length of its walls. 4. What is the cost of excavating it at 32 ^ per cubic yard? 5. How much will it cost to cement the floor at $ .90 per square yard ? 6. If a boy inhales 24 cubic inches of air at a breath, how many times must he breathe to inhale 1 cubic foot ? 7. 29 pupils and their teacher occupy a schoolroom 30 ft. in length, 24 ft. in width, and 12 ft. in height. What is the average number of cubic feet of air for each person ? 8. A city lot of 37| ft. by 120 ft. is to have a layer of earth 1 ft. thick over its surface. Find the number of loads needed and its cost at 25 ^ per load. 9. A dining room is 13 ft. by 18 ft. and has a rug on it 9 ft. by 15 ft. Find the surface not covered by the rug. 10. If the rainfall on a certain day was 21 inches, find the number of cubic inches that fell on a lot 25 feet wide and 100 feet long. Find the number of gallons. 11. Find the cost of digging a ditch, 60 rods long, 3£ feet wide, and 6 feet deep, at 60 ^ per cubic yard. 134 PRACTICAL MEASUREMENTS 12. Memorize : 1 gallon = 231 cu. in. 1 bushel = 2150.42 cu. in. 1 bushel = 1] cu. ft. (nearly) 13. Compare a 3-inch cube with a 4-inch cube. If a 2-inch cube weighs 6 ounces, how much will a 4-inch cube of the same material weigh ? 14. A bin is 8 ft. long, 6 ft. wide, and 4 ft. deep. Esti- mate quickly about the number of bushels of wheat or oats it will hold. 15. A farmer has a tank 12 ft. long, 8 ft. wide, and 6 ft. deep. How many gallons of water will it hold ? 16. How much larger is a farm 80 rods square than a farm 60 rods square ? Draw diagrams on a suitable scale to represent this. 17. The base of a rectangular tank is 48 sq. ft. and the volume is 192 cu. ft. Find the height. 18. What is the area of each surface of a cube 8 ft. on an edge ? the entire surface ? MEASUREMENT OF LUMBER A board 1 foot \i K y ^? square and 1 inch c ,- v ^ thick or less is a board I in. foot. .{i 4 ^- A board foot is the vnit in measuring N lumber. l. Draw on the blackboard a figure to represent a board 4 feet long, 1 fout wide, and 1 inch thick. MEASUREMENTS 01 LUMBEB ! 4ft. 2. Show that this board contains 4 hoard feet. 3. How many board feet are there in a sill 4 ft. long, 1 ft. wide, and 4 in. thick? Observe : (1 ) That the sill is rft; equal to 4 boards 1 ft. long, 1 ft. wide, and 1 in. thick. ( -1 > That each board contains 4 hoard fe< Hence, the -ill contains 1x4x1 board foot = 16 board feet. 27//? number of board feet in a piece of lumber equals the number of hoard feet in one surf 'ace multiplied by the number of inches in thickness. Find the number of board feel in the following: 4. A plank 12 ft. long, 12 in. wide, and 2 in. thick. 5. A board 12 ft. long, 9 in. wide, and 1 in. thick. 6. A plank 15 ft. long, 12 in. wide, and 3 in. thick. 7. A plank 16 ft. long, 18 in. wide, and 2 in. thick. 8. A sill 20 ft. long, LO in. wide, and 6 in. thick. 9. A .-.ill 30 ft. long and 12 in. squar< Buying and selling lumber. Lumber is usually .sold at so much per 1000 (M.) board feet. Find the cost of : 1. 5000 ft. poplar at 8 40 per M. 2. 500 ft. hemlock at 3 24 per M. 3. 10,850 ft Georgia pine at 8 24 per M. 4. 8000 ft. white pine at $50 per M. Small bills of lumber are usually estimated al 90 much per board foot. 136 PRACTICAL MEASUREMENTS 5. Show that lumber at $ 40 per M. = $.04 per board foot; at $27 per M. = $.027 per board foot. 6. Make out a receipted bill to Henry James for the following : 365 ft. hemlock at $ 25 per M., 780 ft. white pine at $40 per M., 980 ft. yellow pine at $29 per M. The dimensions 10 ft. by 6 in. by 10 in. are commonly written 10' x 6" x 10". Estimate the cost of the following at $28 per M. : 7. 4 sills 4" x 8" x 24' 11. 60 joists 3" x 8" x 20' 8. 6 girders 6" x 10" x 16' 12. 90 studding 2" x 6" x 16' 9. 2 posts 6" x 9" x 10' 13. 90 planks 2" x 10" x 14' 10. 8 beams 3" x 8" x 20' 14. 60 rafters 2" x 4" x 24' 15. Observe the dimen- sions of the school build- ing. What is the height of the sides of the building ? 16. Find the number of board feet of siding needed for the sides and the two ends of the same height as the sides, making no allow- ance for openings. 17. The triangular parts at the top of the house, in front and in back, are called gables. Each gable can be divided by a line through the center of its base into two right triangles. How many board feet of siding are neces- sary for the two gables ? 18. Find the cost of painting the siding at 10 cents per square yard. S- 28ft.-- ■-# MEASURING WOOD 137 MEASURING WOOD A pile of wood, of 4-foot sticks, 8 ft. in length and 4 ft. in height, is called a cord of wood. 4x4x8x1 cu. ft. = 128 cu. ft. = 1 cord of wood. 1. How man) r cords are there in a pile of 4-foot wood, 160 feet long and 4 feet high ? 2. Two men cut several piles of 4-foot wood that measure in all 640 feet in length and 4 feet in height. How many cords do they cut and how much do they receive for the Avork at $5.50 per cord? Wood is frequently cut for house purposes into short lengths from 16 inches to 2 feet. The price of such a cord varies according to the length of the sticks. The number of cords of short wood in a pile is found by dividing the number of square feet in one side by 32. 3. At a school building there is a pile of 16-inch wood 80 ft. long and 4 ft. high. Find its cost at $1.50 per cord. 4. Two men cut 4 cords of 2-foot wood each day for 16 days. Find the cost of the cutting at 70 cents per cord. 5. One side of a pile of 2-foot wood contains 400 square feet. Find the number of cords it contains. 138 PRACTICAL MEASUREMENTS REVIEW OF PRACTICAL MEASUREMENTS 1. How many tiles 12 in. square will be required to lay a floor 36 ft. by 15 ft. ? 2. What is the length of a board walk that is 4 ft. 8 in. wide and contains 1350 sq. ft. ? 3. How many cubic yards of earth must be removed in digging a cellar 36 ft. long, 26 ft. wide, and 8 ft. deep ? 4. Find the cost of covering the floor of a hall 45 ft. long and 30 ft. wide with matting, a yard wide, at 70 cents a yard. 5. How many times will the wheel of an engine 9 ft. in circumference turn in going 3000 miles ? 6. Find the cost of 30 boards 16 ft. long, 12 in. wide, and 1 in. thick, at 5 ^ a board foot. 7. At $.80 a. bushel what is the value of a bin of wheat 16 ft. long, 8 ft. wide, and 4 ft. deep ? 8. What is the number of gallons in a tank 12 ft. long, 10 ft. wide, and 8 ft. deep ? 9. How much will it cost to cement the floor of a cellar 50 ft. long and 28 ft. wide at $1.08 a square yard? 10. At 7^ a square yard, how much will it cost to paint the four sides of a building 50 ft. long, 20 ft. wide, and 15 ft. high ? 11. My farm is in the form of a rectangle, and contains 40 acres. What is its width, if its length is 128 rods ? 12. What will be the cost of plastering the ceiling of a room 22 ft. by 18 ft. at 11 ^ a square yard ? 13. A rectangular field contains 5 acres. If its length is 80 rods, what is its width ? REVIEW OF PRACTICAL MEASUREMENTS 139 14. How many cakes of soap 4 in. by 3 in. by 2 in. can be packed in a box whose inside dimensions are 2 ft., 3 ft., and 4 ft. ? 15. Find the cost of digging a cellar 42 ft. long, 30 ft. wide, and 6 ft. 3 in. deep, at 40 cents a cubic yard. 16. How much flooring 1 inch thick will be required to lay the first floor of a house 22 ft. by 36 ft., no allowance being made for waste, and how much will it cost at $30 per M. ? 17. The length of a field is 80 rods, and its width is 30 rods. How many acres are there in the field ? 18. What is the number of bushels in a bin 20 ft. long, 16 ft. wide, and 8 ft. deep ? 19. A tank 9 ft. square and 8 ft. deep contains how many gallons ? 20. A building lot 100 foot front contains 15,000 sq. ft. What is its depth ? 21. A baseball ground 160 yd. by 170 yd. has a tight board fence around it 8 ft. high. How much will the paint- ing of the outside of the fence cost at 5| cents a square yard ? 22. The area of a right triangle is 560 sq. ft., and its alti- tude is 28 ft. What is the base of the triangle ? 23. How much will it cost to excavate a street 800 ft. long and 50 ft. wide, to a depth of 18 in., at 36 cents a load ? 24. A plot of ground in the form of a square is 100 ft. on each side. A straight walk 8 ft. wide divides it into 2 equal parts — a lawn for flowers and a garden for vege- tables. In the lawn there is a flower bed 5 ft. by 8 ft. Draw the plot. 140 PRACTICAL MEASUREMENTS 25. Find the perimeter of the plot ; of the lawn ; of the garden ; of the flower bed ; of the walk. Find the area in square yards : 26. Of the plot. 28. Of the flower bed. 27. Of the lawn. 29. Of the walk. 30. How much will it cost to fence the plot at $3f per rod ? 31. How much will it cost to pave the walk at 11.55 per square yard ? 32. How much will it cost to spade the flower bed at 5 cents per square yard ? 33. How much will it cost to sod the lawn, excluding the flower bed, at $0.25 per square yard ? 34. A board 16 ft. long contains 9 sq. ft. Find its width. 35. A room is 20 ft. long, 16 ft. wide, and 10 ft. high. How much will it cost to plaster the walls and ceiling at 20 cents a square yard ? 36. How many gallons of water are there in a tank 12 ft. long, 8 ft. wide, and 6 ft. deep, if it is half full ? 37. Find the cost of 40 boards, each 14 ft. long, 18 in. wide, and 1 in. thick, at 8 20 per M. 38. A city 5 miles long and 3 miles wide is equal in area to how many farms of 160 acres each ? 39. How many sods 16 in. square will be required to turf a lawn 106 ft. 8 in. long and 50 ft. wide ? 40. What will be the cost of painting the outside of a house 48 ft. long, 30 ft. wide, and 20 ft. high, at 18 cents a square yard ? PERCENTAGE Per cent means by the hundred or hundredths. The sign for it is % . We may express the per cent of a number either as a common fraction or a decimal. Thus, 6% = r fcj = -06 ; 6% of 500 means ^ of 500, which equals 30; or, expressed decimally, .06 of 500 = 30. 2% of a number mean's T § ff , or .02, of the number. 25'/o of a number means r 2 ^,or .25, oi the number. 1. What term in common fractions corresponds to the number before the sign % ? to the sign % ? 2. What expresses the numerator and what indicates the denominator of the fractions represented by the following : 1%? 20%? 40%? 90%? 6%? 30%? 75%? 100 % ? 3. Find 6 % of 100. jfo of 100 = 6 ; or .06 x 100 = = 6. 4. 5% of 100 5. .05 of 100 6. T ^ of 100 7. 6 % of 150 8. .06 of 150 9. .10 of 100 10. 10 % of 100 16. 8% of 75 11. 25 % of 100 17. .08 of 75 12. .25 of 400 18. i!o of 75 13. 3 % of 60 19. 331 of 300 14. .03 of 60 20. S3 1 ioo " f 300 15. t§o of 60 21. 33$% of 300 111 142 PERCENTAGE Changing per cents to equivalents. Since 5% = .05 = T ^ = 2V these expressions may be called equivalents. l. Give the fractional and decimal equivalents for 10 % ; 6%; 4%; 20%; 25%. Read the following equivalents : 2.^,20%, .20,1 s |7±, 87Wf .3TJ, I 3. 1Q0 , 12|%, .12J, J 6> ^ 80% , .80, I 4. ^, 40%, .40, f 7 . 87i 87 i % , , 87i , 1 8. Change the fractions -|, §, |, | to their equivalent decimals and per cents ; also ^, |, |, |. i= 5)1.00 .20, or 20% I = 2 x .20 = .40, or 40% | = 3 x .20 = .60, or 60% | = 4 x .20 = .80, or 80% *.= ! = 5 _ j — 7 _ j — 8)1.00 .12.1, or 3 x .12V = . 5 x .12* = . 7 x .12| = . 37£, or 37*% 62|, or 62J% 87*, or 87J% Change to their equivalent decimals and per cents : 9 l v. 2 13. f 17. 3 10 21. | 25. f 10. £ 14. i 18. a 22. | 26. f 11. § 15. f 19. ■A 23. i 27. | 12. 1 16 - tV 20. i 24. $ • 28 ' 16 Give the products rapidly: 29. 2x.33£ 32. 5x .121 35. 4x.l2i 38. 4x.04l 30. 5x.l6| 33. 7x .12| 36. 6x.l2£ 39. 3x.l6| 31. 3x.l2i 34, 3x .81 37. 2x.l5 40. 4x.l6| PERCENTAGE 143 Memorize the following table: * = 50% i = 20% f = 83 : \% rV = 8i% i = 33) ; % f = 40% 1=13*0 A = 41|% f = 66f% f = 60% t = 37|% xV = 6i% i=25% | = 80% 1 = 62-> % ^f = 5% 1 = 75% * = 16f% | = 87.V% 2^ = 4% Name rapidly the fractional equivalents of the following per cents : 41. 50% 46. 20% 51. 37|% 56. 90% 42. 331% 47. 40% 52. 621% 57. 12.1% 43. 66|% 48. 60% 53. 87 1 % 58. 16f% 44. 25% 49. 16% 54. 10% 59. 45. 75% 50. 83|% 55. 30% 60. 70% Write the equivalents of the following in decimals, thus 1% = .01; 32% = . 32; |% = .00|; etc. 61. 1% 67. 1% 73. 50% 79. 13% 62. 32% 68. 3% 74. 1% 80. 131% 63. 1% 69. 11% 75. 6% 81. 1% 64. 2% 70. \% 76. \°Io 82. 100% 65. 16£% 71. 4% 77. 7% 83. 12S% 66. « 72. 43£% 78. \% 84. 127% 144 PERCENTAGE Finding a given per cent of a number. Recite the following thus : Look at " 66f%," think "§": l. 66f % of 18. 12. 37|% of $7200. 2. 331% of 90. 13 121% of $6400. 3. 50% of $500. 14. 75% of $4800. 4. 25% of $2000. 15. 66|% of $999. 5. 75% of 16 inches. 16. 80% of 60 sheep. 6. 20% of 100 yards. 17. 60% of 75 horses. 7. 40% of 60 feet. 18. 40% of 90 miles. 8. 60% of 40 miles. 19. 871% of $160. 9. 80% of 75 gallons. 20. 621% of $240. 10. 16f% of $6000. 21. 37|% of $880. 11. 831% f) f $1200. 22. 121% f 24. Written Work 1. A man had 100 cows many did he sell? 25% =.25 and sold 25 % of them. How 100 cows • 25 As 25% = 500 result is 25, 200 .25 we multiply 100 by .25. The the number sold. 25.00 number sold Find results decimally: 2. 50% of 750 4. 40% of 8.75 6. 32% of 1000 3. 25 % of 85.5 5. 28 % of 840 7. 75 % of 980 8. John earns $21.60 per month, and spends 75% for clothes. How much do his clothes cost him ? PERCENTAGE 145 9. There are 780 pupils in school, and 40% are males. How many are males ? ■ 10. If a man buys a horse for $150 and sells it at a profit of 20 %, how much does he gain ? In each of the preceding problems we have two terms, a per cent and a whole or a mixed number. The per cent in each problem is the multiplier, and is called the rate. The whole or the mixed number is the multiplicand, and is called the base. The product is called the percentage. The base is that number of which some per cent is to be taken ; as, 5 % of $ 200 (base}. The rate is the number of hundredths taken ; as, 5% (rate) of 80 horses ; that is, ^^ of 80 horses. The percentage of a number is the result obtained by tak- ing any per cent of it ; as 10 % of 200 acres is ^q of 200 acres, or 20 acres (percentage). 11. What is 75% of 85.12? Multiplier Multiplicand Product Rate Base Percentage 75% of $5.12 = ( ) Decimal Method Study of Problem 75 % = .75. What is the base ? $5.12. What is the $5.12 = base rate? 75 %- -r To what do the base and rate correspond — - — — in simple multiplication? Multiplicand ^" uv and multiplier. 3o84 To what does percentage correspond in £3.8400 = percentage simple multiplication? Product. How is the product found in simple Fractional Method multiplication? Multiplier y. multiplicand. 75 % = f How is the percentage found? Rate X | of $5.12 = ¥3.84 base ' The percentage of a number equals the product of the base by the rate. HAM. COMPL. A KITH. — 10 146 PERCENTAGE Find : 12. 6 % of 1200. 17. 8 % of 400. 13. 33| % of 6 months. 18. 7 % of 400 horses. 14. 60 % of 30 days. 19. 3^ % of 99. 15. 1 % of 100 acres. 20. 6 % of 150 lb. 16. 5 % of 100 acres. 21. 1| % of $75. 22. 80 is the base, 25 % is the rate, find the percentage. 23. A house costs $2500, and the damage by fire is 8%. Find the amount of the damage. 24. John owes his tailor $80, and pays 37 1 % of the debt. How much does he still owe ? 25. Mary spells 90 % of 80 words correctly. How many does she miss ? 26. A boy buys apples at $1 per bushel, and sells them at a profit of 20%. How much profit is that per bushel ? 27. G| % of 3680 equals what number ? 28. A man bu} T s a farm for $2500, and sells it for 25 % more than it cost him. For how much does he sell the farm ? 29. A man earns $180 per month, and puts 33^ % of it in the savings bank. What is his deposit each month ? 30. If 37| % of a man's farm is in timber, and the total area is 240 acres, how much timber land has he ? 31. A teacher who earned $1200 a year spent 66| % of her salary. How much did she save ? 32. Mr. Scott's horse is valued at $250 and Mr. Hill's at 60% of this. What is the value of Mr. Hill's horse ? 33. The population of a town of 9672 inhabitants increased 12-i-% [ n a vear# What was the increase in population? COMMISSION 147 COMMISSION An agent is a person who transacts business for another. Commission is the sum charged by an agent or commission merchant for his services. The net proceeds is the sum left after the commission and other expenses have been paid. l. A real estate agent sold a house for 15000, retaining 5% of this sum for his services. How much did he receive? How much did the owner receive ? $5000 = selling price. .05 — rate charged by the agent. $250.00 = amount charged by the agent. $5000 - $250 = $4750, amount received by the owner. A commission merchant made the following sales. Find his commission for each day at 5 %. 2. Monday, $1800 5. Thursday, 11400.80 3. Tuesday, $1594 6. Friday, $1528 4. Wednesday, $1954 7. Saturday, $2370.60 8. Find his total commission for the week. 9. A real estate agent sells a house and lot for $6750, charging 2% commission. Find his commission and the net proceeds. 10. A traveling salesman sold $50,000 worth of goods in a year at a commission of 8%. If his expenses for the year were $2200, how much had he left? 11. An agent rents 12 houses at $40 per month. If he receives 5% for collecting the rents, how much is remitted to the owners each month ? 148 COMMERCIAL DISCOUNT COMMERCIAL DISCOUNT Wholesale merchants and manufacturers usually publish printed price lists of their goods. The prices in these lists are higher than the wholesale prices and are subject to deductions called trade discounts or commercial discounts. Note. — A discount is any deduction from a fixed price. Sometimes several discounts are allowed. The first is a discount from the list price; the second, a discount from the remainder, etc. The net price is the price less all trade discounts. Find the selling price of goods marked : 1. $15, less 20%. 7. $40, less 60%. 2. $20, less 40%. 8. $48, less 25%. 3. $6, less 50%. 9. $6.80, less 25%. 4. $25, less 20%. 10. $4.50, less 331%. 5. $7.50, less 20%. 11. $9.60, less 16§ %. 6. $12.50, less 40%. 12. $4.80, less 371%. Written Work Find the selling price of goods marked : 1. $168.75, less 25%. 2. $1374, less 16|%. 3. $1872, less 331%. 4. $278.40, less 371%. 5. $3030 less 40%. Find the cost of : Discount, 20% 11. 60 readers @ $ .40 150 geographies @ $ 1 78 grammars @ $ .60 6. $225.65, less 20%. 7. $875.50, less 30%. 8. $278.90, less 10%. 9. $2378.50, less 4%. 10. $6775.20, less 5%. Discount, 4 % 12. 160 lb. rice @ $.06 300 lb. sugar @ $.041 200 lb. coffee @ $.16 COMMERCIAL DISCOUNT 149 13. Find the net price of a bill of goods for $ 75.40, trade discounts 20%, 10%. List price, $ 75.40 1 ess °0 °! 15 08 Observe that the second discount is ,,. . , — Ql . ., , reckoned on the first remainder. As there tirst remainder, 00. o'l . ,. ,, are only two discounts, the second reinam- Less 10 %, 6.03 der is the net price. Net price, $51.29 Find the net price of articles listed at : 14. $100, less 20%, 10%. 17. $ 10.75, less 40%, 5%. 15. 375.50, less 25 %, 5 %. 18. $ 6.80, less 25 %, 10 %. 16. 290.80, less 40%, 10%. 19. 112.75, less 331%, 10%. Find the net price of the following bills of goods : 20. 36 dozen boys' caps @ $ 6, discounts 25 %, 20 %. 21. 50 buggies @ $ 120, discounts 20%, 15%. 22. 75 sets harness @ $ 40, discounts 30%, 10 %. 23. 25 grain drills @ $ 95, discounts 40 %, 5%. 24. 12 rubber hose, each 50 feet long at 15^ per foot, discounts 30%, 15%. 25. Mr. Austin buys a wagon listed at $95, less 20% . 15%. Find the amount paid for the wagon. 26. A merchant buys 12 stoves listed at $45, less 40%, 10 %. Find the net amount of the bill. Compare this with the net amount of the bill with only one discount of 50 %. 27. A hotel keeper buys 675 yards of carpet at $1.25, less 20 %, 5 %. Find the cost of the carpet. 28. Compare the net price of an article listed at $500, discounts of 20 %, 10 %, with the net price of a similar article listed at $500, discounts of 10 %. 20 %. 150 COMMERCIAL DISCOUNT Solve according to conditions : 29. The Packard Hardware Co. bought for cash from Jas. M. Armstrong Co., Chicago, 111., 4 doz. Acme lawn mowers @ $30 a dozen, 50 lb. lawn seed @ 15^ a pound, 2^ doz. brushes @ 40^ a dozen. Trade discounts: 20%, 10%. Terms: 30 days net; 2% cash in 10 days. Cost of bill of goods = ,1128.50. ■1128.50 less trade discount of 20% = 1102.80; $102.80 less trade dis- count of 10% = f92.52, net price of bill if paid in 30 days. If the bill is paid within 10 days from date of purchase, the buyer gets a further dis- count of 2%. This is called a cash discount. $92.52, less 2% for cash within 10 days, = $90.68. 30. Jamison and Redmond, South Bend, Ind., bought for cash from the Acme Buggy Co., Cincinnati, O., 72 buggies @ $105, 50 sets harness @ $45, 15 sleighs @ $60, 40 robes @ $20. Trade discounts: 30%, 15%. Terms: 30 days net ; 3 % cash in 10 days. 31. James Cubbison, Greenville, O., buys for cash from Arbuthnot, Stevenson & Co., Pittsburg, Pa., 5 doz. hand- kerchiefs @ $3.60; 5 bolts muslin, 40 yd. each, @ 8^; 5 bolts prints, 42 yd., @ 7 ^. Trade discount : 33^%. Terms: 30 days net ; 2 % cash in 10 days. 32. S. H. Gardner Co., piano dealers, Detroit, Mich., order from the Harmonic Piano Co., Chicago, 111., 2 Har- monic pianos #266 @ $600, less 40%, 10% trade discount. Terms : 90 days net ; 10 % off 10 days. Find the cash price. Find the net price if paid in 30 days. Notk. — The sign #, when placed before a number, is read " number." 33. M. L. Smith, tailor, Brockton, Mass., orders from Bender & Co., New York, importers, 3 pieces suiting, 22 yd. each, @ $3.15. Terms: 30 days net; 2% off 10 days. Find the net amount of bill if paid within 10 days. INTEREST 1. Mr. Johnston pays the liveryman $6 for the use of a horse and buggy for two days. What does he get in ex- change for the $6? 2. Mr. Daniels pays $6 for the right to pasture his cow in a field for two months. What does he get in exchange for the $6? 3. Mr. Watson pays $6 for the use of #100 for one year. What does he get in exchange for the $6 ? 4. In the first two examples money is paid for the use of something that is not money. For what does Mr. Watson pay the money in the last example ? Interest is money paid for the use of money. Interest corresponds to the percentage in percentage. 5. How much does Mr. Watson pay for the use of the money? What is the $6 called ? 6. On what is the interest reckoned ? The $100 is called the principal. The principal is the sum on which the interest is paid. The principal corresponds to the base in percentage. The rate of interest is a certain number of hundredths of the principal paid for the use of the principal for one year. Time is always a factor in interest. Interest, then, is the product of three factors: principal, rate, and time. The amount is the sum of the principal and the interest, 151 152 INTEREST Interest for Years and Months 1. What part of a year are 6 months ? 4 months ? 3 months ? 2 months ? 1 month ? 2. If the interest for a year is $100, what should it be for 6 months ? for 4 months ? for 3 months ? for 2 months ? for 1 month? Written Work 1. What is the interest on -1200 for 2| years at 6 % ? $200 principal . . 11 1 he interest for 1 year is .06 of the principal, or .$12. The in- il2.00 interest for one year terest for 2\ years is 2\ x $12, or 21 $30. $30.00 interest for 2| years Multiply the 'principal by the rate and the product by the number of years. The year is usually considered as 360 days, that is, 12 months of 30 days each. Find the interest on : 2. $300 at 5% for 1 year. 4. $150 at 6.} % for 3 years. 3. $800 at 8 % for 2 years. 5. $700 at 4| % for 4 years. Find the interest of : 6. $250 fori | years at 4%. 11. $500 for 21 years at 4| %. 7. $75 for 2 years at 8 % . 12. $960 for 9 mo. at 6 %. 8. $100 for 3f years at 7%. 13. $900 for 2f years at 7 %. 9. $ 80 for 41 years at 5 % . 14. $654 for f year at 6 % . 10. $40 for 21 years at 6-1 % . 15. $ 220 for J year at 8 % . INTEREST FOR YEARS, MONTHS, AND DAYS 153 Find the interest at 6 % on : 16. $100 for 6 months. 19. 1624 for 120 da. 17. $500 for 4 months. 20. $170 for 8 mo. 18. $150 for 2 yr. 2 mo. 21. $355 for 130 da. Interest for Years, Months, and Days 1. What part of a month (30 days) are 15 days? 12 days ? 20 days ? 3 days ? what part is 1 day ? 2. If the interest for 1 year is $360, what is the interest for 1 month ? If the interest for 1 month is $ 30, what is the interest for 1 day ? for 15 days ? for 12 days ? Written Work l. Find the amount of $ 200 at 6 % interest for 2 yr. 7 mo. 12 da. Principal = $200 Rate = .06 Int. for 1 yr. =$12.00 Int. for 2 yr. = 2 x $12.00, or $24.00 Int. for 7 mo. = T 7 5 of % 12.00, or 7.00 Int. for 12 da. = j$, or f, of $1.00. or .40 Int. for 2 yr. 7 mo. 12 da. = $31.40 Principal = $200.00 Amount for 2 yr. 7 mo. 12 da. = $231.40 Study of Problem a. What is the first step in the work ? the second step? b. How do we find the interest for 1 month? for 7 months? for 12 days? c. What new term is introduced in interest? For what length of time is rate of interest always considered? 151 INTEREST Find the interest and amount of : 2. 1300 for 3 yr. 6 mo. at 6%. 3. 1 250 for 2 yr. 4 mo. at 7 %. 4. $ 160 for 1 yr. 3 mo. at 5 %. 5. $ 50 for 1 yr. 8 mo. at 5 % . 6. $ 800 for 3 yr. 2 mo. at 6 %. 7. $50.80 for 9 mo. at 10%. 8. $16 for 8 mo. at 6%. 9. $75 for 8 mo. at 6 %. 10. 1420 for 10 mo. at 10 %. 11. $40.50 for 1 yr. 1 mo. at 6 %. 12. $ 300.40 for 5 mo. at 7 %. 13. $ 100 for 7 mo. at 7 ft, . 14. $500 for 11 mo. at 6 %. 15. $1000 for 1 mo. at 6%. 16. $60.(30 for 8 mo. at 8%. Find the interest and amount of : 17. $250 at 8 % for 3 yr. 5 mo. 20 da. 18. $75.80 at 5 % for 4 yr. 1 mo. 16 da. 19. $ 1500 at 6 % for 2 yr. 9 mo. 15 da. 20. $125.50 at 4 % f or 4 yr. 11 mo. 12 da. 21. $ 1140 at 5| % for 4 yr. 8 mo. 24 da. 22. $ 912.60 at 5 % for 2 yr. 10 mo. 11 da. 23. $ 3209 at 6 % for 3 yr. 7 mo. 21 da. 24. $634.50 at 8 % for 11 mo. 12 da. 25. Henry Boydson borrows $ 275 Sept. 1, 1906, at 6 % interest, and settles the note Jan. 1, 1908. Find the amount of the note at settlement. REVIEW OF PERCENTAGE AND INTEREST 1. A boy has $30 in a savings bank and deposits a sum equal to 10% of it. What is the total amount he has in bank? 2. Mr. James's salary is $1200 per year and he saves 33^% of it. How much does he spend? 3. A boy spends $8 for an overcoat and 37.] % of that sum for shoes. How much does he spend for shoes? 4. In a school of 45 pupils, 33^ % of the pupils are boys. What is the number of girls ? 5. Find '2;')% of .05; of .5; of 5.5; of .25. 6. John earns $50 during his vacation, and Margaret 25% as much as John. How much does Margaret earn? 7. A farmer sold a horse that cost him $ 80 at a loss of 20%. Find the selling price. 8. What is the interest on $150 for 2] years at »!% ? 9. Find the interest on $100 for 00 days at 5%. 10. My father borrows $75 from his neighbor and promises to pay it in 4 months at 0%. Find the amount my father must pay at the end of four months. 11. A huckster buys eggs at $.20 per dozen. For how much per dozen must he sell them to gain 20 % ? 12. If I borrow $50 from Mr. James for 6 months at 0%, how much interest must I pay him? 155 156 REVIEW OF PERCENTAGE AND INTEREST 13. A grocer sold flour last week at $1.20 per sack and this week at 10 % advance on last week's selling price. Find the price of flour per sack this week. 14. A huckster buys 150 dozen eggs at $.20 per dozen and sells them to a merchant at a gain of 25 %. The merchant sells them at a gain of 20 %. How much does the merchant receive for the eggs ? 15. If I buy cloth at $1.50 a yard, for how much must I sell it to gain 33£ % '■ 16. A grocer buys goods to the amount of $1200, 10 % off for cash. He sells them for $1500 cash. How much does he gain ? 17. A mother took two boys and a girl to a store to buy clothes. The first boy's suit cost $10 less 10% for cash. The second boy's suit cost 66| % of the cash price of the first boy's suit. The girl's coat cost 331% f the money paid for both boys' suits. How much did the mother pay for the children's clothes ? 18. Find 25 % of 200, and divide the result by .00J. 19. A man buys a house and lot for $5000. It costs every year $25 for repairs and $50 for taxes and insurance. He rents the house for 8% of its cost. How much has he left after paying expenses ? 20. A coal dealer bought 300 tons of coal for $600. The freight, storage, and delivery cost 331 cj f the cost of the coal. What was the retail price per ton if he sold it at a gain of 12^% ? 21. A real estate agent purchased a house for $1250. For how much per month must he rent the house to make 6 % after paying each year $18 for taxes and insurance and $15 for repairs ? RECEIPTS AND CHECKS John Watson pays James Adams $ 35.50 for work for one month, and asks Mr. Adams for a receipt. Write the receipt to show that the money was paid by Mr. Watson and received by Mr. Adams. $ Rochester, jY. Y., fwyva /, 1907. Eeceifceli from Dollars for 1. What must every receipt show? 2. Write the receipt your grocer would give you in pay- ment of $18.50 on account by your father or mother. 3. Your school district pays the National Book Company, New York, $ 25.75 for school books on Sept. 15, 1907. Make out the receipt of the National Book Company. 4. Henry Smith received I 3.65 from James Brown for 3 months' water rent. Make out a receipt for the amount. 5. Ralph Taylor pays II. W. Henderson |5 for a month's tuition. Write the receipt Ralph Taylor should receive. 157 158 RECEIPTS AND CHECKS 6. Write a receipt for $ 75 which Nelson Page paid Edgar Poe for balance due on a buggy. 7. Make out and receipt the bill for the following articles bought by James Thomas from Jos. Home & Co. : 3 shirts @$1.75 2 neckties @ $.75 6 collars @ .20 4 pairs cuffs @ .20 8. Presuming that you are a collector for the Gazette- Times, Pittsburg, Pa., make out a receipt to a subscriber who has paid you $ 2.60 in full of account. A check is an order on a bank where a person keeps a deposit, ordering the bank to pay money. Stub Check c/t*. 875 J No. 875 I Seattle, Wash., fan. fO, 1907 oo &\>t gufcon Rational Bank of Seattle. PAY TO THE i Order of- fame*, lO-avci fan. /O, '07 / &vxiu-4,&v-&K—~~~~~lJoUars ' 100 3ov jCa-6-cyv i Z0-. $. ?)1oov&. 1. Name the different things stated in this check. 2. Observe that this check is payable to the order of James^ Ward. He orders it paid by writing his name across the back of it. This is called indorsing the check. 3. Write the check your father would give your teacher in payment of $3.50 for September tuition. 4. Emil Smith borrows from Joseph McLean $240 to attend school and pays the same in 2 yr. 4 mo. 18 da. at 0%. Write the amount of the check that would pay the note. GENERAL REVIEW 1. The remainder is 92,568 and the minuend is 202,660. Find the subtrahend. 2. The dividend is 364,450 and the quotient is 9850. What is the divisor ? 3. Add 3.5, .035, 45.006, and 2.06. 4. Write decimally twenty-five and sixty-one thousandths; one hundred twenty-five and five tenths ; and three hundred and two ten-thousandths. 5. What number multiplied by one hundred seventy- nine is equal to 848,818 ? 6. From 2.0011 take 1.9892. 7. Explain the difference between \°J and \. 8. Add 1 2f, 1 ||, and 4f . 9. Find 1%, -i-, 1%, fa £>%, 50%, and ^ of 100. 10. The multiplicand is 1325 and the multiplier is .0416. What is the product ? 11. If 38 dozen eggs cost $11.40, what is the cost per dozen ? 12. A building is 46 ft. 3 in. wide, and twice as long as wide. Find the distance around the building. 13. From 86 miles and 3 inches, take 46 miles and 8 inches. 159 160 GENERAL REVIEW 14. A man and his son together earn $72 per month. If the man's earnings in 6 months amount to $300, how much are the son's earnings in the same length of time ? 15. A man bought 48 head of cattle, at $36 per head, and sold them at a gain of 25%. What was the total amount received for the cattle ? 16. Find the interest on $370.50 for 4 yr. 8 mo. at 6%. 17. Divide 48| by 21f 18. Divide .65 by 6.5. 19. Reduce 187^ rods to the fraction of a mile. 20. How much will it cost to ship a car load of wheat containing 42,000 lb. from Fargo, N.D., to Chicago, 111., if the freight rate is $.06 per bushel ? (60 lb. = 1 bu.) 21. A train leaves Chicago at 8:15 a.m., and arrives at Pittsburg at 8:20 p.m. The distance is 468 miles. Find the number of miles per hour the train travels. 22. The steel rails on the Bessemer railroad weigh 100 pounds to the yard. Find the number of tons necessary to lay 5 rods of single track. 23. How much does an architect receive, at 4* %, for the plans of a house that cost $8350? 24. A man's salary is $150 per month. He spends 40 % of it for clothing and other expenses. How much does he save in a year ? 25. A man purchases 80 acres of land for $6400, and sells them at 25% gain. How much does he receive per acre? 26. Frank Stewart borrows $250 Sept. 15, 1906, at 6% interest. Find the amount of the note if paid March 15, 1908. 27. Find the area in acres of a street 7 miles long and 66 feet wide. GENERAL REVIEW 161 28. A town lot is 43 ft. 3 in. wide and 120 ft. deep. How much is it worth at 75 $ per square foot? 29. A western farmer harvests 8960 bu. wheat from a field 320 id. long and 160 rd. wide. If he sells the wheat at 60^ per bushel, how much does he realize from each acre? 30. In 1 hour 20 minutes and 40 seconds, a train travels 60 miles. At that rate how long would the train be in traveling 1200 miles? 31. The average wages of a steel mill employing 3000 men are $2.50 per day. If a 10% reduction in wages is made, how much per day will the company's pay roll be reduced? 32. In a certain class the salary of the teacher for a year is $500. The books and supplies cost $90.65 ; fuel, $40 , repairs and other expenses, $75.30. There are 35 pupils in the class. Find the average cost per pupil for the year. 33. Reduce to improper fractions: 6.25; 3.375; 4.66| ; and 2.05. 34. If it costs $72 to carpet a room 18 ft. long and 18 ft. wide, how much will it cost to carpet a room 36 ft. long and 36 ft. wide, with the same quality of carpet? 35. Mt. Rainier is 14,363 ft. high. Reduce the height to miles and the fractions of a mile. 36. How many cubic inches are there in a bin 9 ft. 7 in. long by 8 ft. 3 in. wide and 4 ft. 9 in. deep? how many cubic feet? 37. A grocer bought 225 bu. apples at $.50 per bushel. He sold 150 bu. at $.75 per bushel. The remainder, which were damaged, he sold at $ .40 per bushel. Did he gain or lose and what per cent? HAM. COMPL. ARITI1. 11 102 GENERAL REVIEW 38. What is 331% of 24? of 84.80? of 862.50? 39. A piece of land 30 rods wide and 480 rods long was sold at 862.50 per acre. Find the amount of the sale. 40. Time, 3 months; rate of interest, 5 % ; money borrowed, 8100. Find amount to be paid. 41. If f of a bushel of potatoes cost 8.40, how much will 1\ bu. cost? 42. A piece of land 40 rods long in the form of a rec- tangle contains 5 acres. Find its width in rods. 43. A farmer sold 12| acres of land at 855| per acre. How much did he receive for the land? 44. Houser Brothers sold the following bill of goods to William Pool : 12 lb. sugar 10 cans tomatoes @ 8.061 @8.15 6 lb. rice 11 lb. prunes 2 pair boots 1 overcoat 1 pair shoes @ I.07J @ 8.071 @ 83.50 @ 813.50 @ 84.00 Pool at the same time sold Houser Brotl 85 bu. potatoes 50 bu. corn 16 lb. butter @8.65 @ 8.421 @$.24 10 lb. butter @ 8.28 Houser Brothers gave Mr. Pool the balance in cash. Make out the account. 45. A painter worked 17^ days. After spending |- of his wages for board he had 815 left. Find his daily wages. GENERAL REVIEW 163 46. I owe -Frank Morrison, the grocer, $32.50 and pay him $23.75. Write the receipt that Mr. Morrison should give me. Note. — When a debt is not paid in full, the receipt should read " On account." 47. A cellar 24 ft. by 32 ft. is to be excavated to an average depth of 5| ft. Find the number of cubic yards to be removed. 48. Express 22| yards as rods, feet, and inches. 49. The width of a rectangle is 20 rods and the area is 560 square rods. Find the length. 50. What is the difference between a square and a rec- tangle ? 51. Give the rule for rinding percentage. On what is gain or loss always reckoned ? 52. A man's farm and personal property cost $5600. The first year he cleared 12| % of the money invested. The sec- ond year, on account of floods, he lost 5 % of the cost of the property. How much was his gain in the two years? 53. Of a bill of $155 sent to a collector, 80% was col- lected and the collector retained $12.40. What per cent did he charge for collecting? 54. A boy receives $1.20 per day and a man $2.50 per day. How long will it take the boy to earn as much as the man can earn in 30 days? 55. The perimeter of a rectangle is 72 rods. The width is 12 rods. Find the length. 56. Estimating that 300 cu. ft. of air is required for each pupil, how many pupils, including the teacher, should occupy a room 40 ft, long, 30 ft, wide, and 12 ft. high? 164 GENERAL REVIEW 57. Divide one thousand and one thousandth by one and one thousandth. 58. What is the interest on $375 for 270 days at 6 % ? 59. Reduce f of a mile to lower denominations. 60. A boy deposited half of his money in the savings bank; ^ of the remainder he spent for clothes; and he had $3 remaining. How much had he at first ? 61. Reduce .025 cwt. to lower denominations. 62. A man bought two city lots costing him -13500 and $ 4100 respectively. He sold them at a gain of 25 %. What was the gain in dollars ? 63. How many gallons of water will a tank contain that is 11 ft. long, 3^ ft. wide, and 4 ft. deep ? 64. A barn floor is 20 ft. wide and 45 ft. long. How much will it cost to cover it with plank 2 inches thick at 1 20 per thousand board feet ? 65. Divide j by f of f . 66. John and James have together 165 acres of land, but James has twice as many acres as John. How many acres has each ? Suggestion. — 165 acres = twice John's + once John's or 3 times John's. 67. What fractional part of a day are 10 hours, 50 min- utes, 40 seconds? 68. Divide nine ten-thousandths by one hundred twenty- five thousandths. 69. What is the interest on $ 180 for 4 yr. 8 mo. at 5f % ? 70. I made $1.95 by selling 15 dozen eggs at $.31 per dozen. What was the cost of the eggs per dozen ? GENERAL REVIEW 165 71. Find the net proceeds from the sale of 145 books, at |2 each, on which a commission of 33^% is paid. 72. A father divided his farm of 202 A. 16 sq. rd. equally among his four sons. How many acres did each receive ? 73. .21 of a mile is equal to how many feet? 74. An automobile that cost $2675 was sold at a loss of 28 %. For how much was it sold ? 75. What is the cost of 18 planks 20 ft. long, 12 in. wide, and 2 in. thick, at 120 per M? , 76. \ of 7 is what part of 9 ? 77. f of a farm is worth $7500. What is 20% of the farm worth ? 78. What is the cost of a car load of bituminous coal weigh- ing 84,000 pounds at $2.65 per ton ? 79. A farm in the form of a rectangle containing 120 acres is 60 rods wide. How long is it ? 80. Express decimally the quotient of £ ■*- .35. 81. If | of a ton of hay is worth $12, how much are 33,000 pounds of hay worth ? 82. What is the value of a pile of 4-foot wood 48 ft. long and 6 ft. high, at $4.50 per cord ? 83. A dairyman owns a cow that averages 3 gal. 2 qt. 1 pt. of milk daily. If he sells the milk at $.06 per quart, how much will he realize from the cow during the month of May ? 84. I can buy an automobile at one store for $3000, with discounts of 25 %, 10 %; or at another store for $3000 with only one discount of 35%. Which is the cheaper? PART II — SEVENTH YEAR BILLS AND ACCOUNTS RECEIPTS John Bentz rents a house in Boston, Mass., from James Smith for one year for $240, rent payable the first day of each month in advance. 1. Every receipt should state (1) the place and date of payment; (2) who pays the money ; (3) who receives the money; (4) for what the money is paid ; (5) the amount both in figures and in writing. 2. Every receipt in full should state in full to date. Write the receipt given Mr. Bentz for September's rent. 1. Providing John Bentz fails to pay the rent for August when due, but pays on September 1 the rent for both August and September, write the proper receipt. 2. Write the receipt for the tuition for the term of your school that any non-resident pupils would have to pay. Write the receipt in full to date for each of the follow- ing bills which I owe : 166 ORDERINCx GOODS 167 3. John Thompson for milk, $6.75. 4. Frank Jones for coal, $16.85. 5. Smith & Co. for books, $3.75. ORDERING GOODS These forms of orders should be studied carefully, as they come into almost daily use in business life. A FURNEE AND KENNERDELL, KlTTANN.NG, Pa., fat. W, / vEA,Lean- fSaok, (HvwLJbaAvy, /OO IM-a^kinato-n ^qioaA&, cftzw- 1fr>ik- Jbe,av cfvi&: ffl&a&t Qs/i ifi at &n&& 6-if S > &n / n^ / iflv-ayyvuv ^vt^Ujkt : 300 Rois, §'vlr,i&v&. "Uoii 1 1 1 1 ;ily, ~j :?c If ofon. Jch W-at^yyv. // / 6 / 50 30 00 20 fO 00 Observe : 1. The place and date of sale. 2. The names of the buyer and the seller. 3. The name, quantity, and price of each arti- cle. 4. The entire amount of each separate item. 5. total amount of the bill. 6. The receipt of the bill. The RECEIPTED RILLS it;o A bill is a written statement in detail of goods sold or of services rendered. A bill is receipted when the words "Received payment" are written at the bottom of the bill, either by the seller or by some person authorized by him. Note. — When the person authorized signs the name of the seller, he should always write on the next line below the word " by " or " per " and his own name or initials. When a person purchases anything on time, the purchaser is called a debtor. When the seller extends the time of payment to any one, the seller is said to give credit, and therefore is called a creditor. Some abbreviations used in business: Acct. %, account mdse., Amt., bal., Co., Cr., Dr., do. CO, amount balance company creditor debtor the same No. (#), paymt., pd., per, pc, rec'd, merchandise number payment paid by piece received The symbol # means pounds when placed after a number; but number if placed before a number. Thus 6 j means 6 pounds but *6 means Number 6. Make receipted bills for the following transactions, per- forming all necessary operations : l. Carter Bros., Elkins, W. Va., purchase from Bindley Hardware Co., Pittsburg, Pa., the following: 3 dozen locks @ $4.80, 67 kegs of nails @ $4.10, 6 dozen lanterns @$6.25, 1300 feet steel tracks @ 16^, and 7 lawn mowers @ $4.25. 170 BILLS AND ACCOUNTS 2. John Dunn & Son, Akron, Ohio, bought from Thomas Townsend & Company, Cleveland, Ohio, 36 barrels of flour @ $4.80, 4 boxes of prunes @ $1.65, 500 pounds of coffee @ llf^, 7 boxes of yeast @ 75^, 50 pounds of Huyler's cocoa @ 32^. 3. James Brown, Lincoln, Neb., purchases from May & Co., St. Louis, Mo., 22 bunches bananas @ $1.75, 32 boxes oranges @ $3.15, 17 boxes lemons @ $'2.80, 29 crates cran- berries @ $2.25, 6 boxes grape fruit @ $2.90, and 35 bbl. apples @ $2.75. 4. James Sweitzer, Peoria, 111., bought from Swift & Co., Chicago, 1587 pounds of dressed beef @ 7-^, 267 pounds of mutton @ 9^, 933 pounds of pork @ 5|^, and 180 pounds of lard @ 12£ 5. Lyle Bros. & Co., Dubuque, la., bought of the De- laney-Brown Lumber Co., Grand Rapids, Mich., 28215 feet oak boards at $32 per M., 147820 feet hemlock at $27 per M., 92629 feet No. 1 white pine at $60 per M., 63605 feet poplar boards at $35 per M. Note. — $32 per M. equals $.032 per board foot. 6. Mrs. James Thorpe bought of B. Altman & Co., New York, 1 pair gloves at $2.75, 5 yd. ribbon at 39^ a yard, I dozen handkerchiefs at 25^ each. ACCOUNTS In the study of bills we simply found how much the debtor owed to the seller, or what one party owed to another for services rendered. In an account we have a business transaction covering a period of time in which there is both a debtor's bill and a debtor's payments. ACCOUNTS 171 Form of Account Pittsburg, Pa., TTicwf /, / c /07. Plv. cfawvu&f- BarucL, U^/lt.alVna, Z&. Ucl: Jo JOSEPH HORNE CO., Dr. 3v. Ofa. / &o CLmawnt ieM,deA,&cL $ffO 29 ti fO " 20 yd. ^Ifc @ $f. 50 SO 00 1 1 25 " 2 Ladies' &uvU @ ¥-5.00 qo 00 //- 28 7 80 238 oq // /¥■ ISu & 127 pt _ 15 gal# 3 117 pt. = 15 gal., 3 qt., 1 pt. q t. i p t. Note. — The dividend and divisor are regarded as abstract numbers. Do not read 127 pt. •*■ 2 = 63 qt. Such a statement would be absurd. ( hange: 2. 225 qt. to gallons, etc. 7. 5675 sec. to minutes, etc. 3. 2550 qt. to barrels, etc. 8. 75000 in. to miles, etc. 4. 1463 pk. to bushels, etc. 9. 36481.62 ft. to leagues, etc. 5. 15000 min. todays, etc. 10. 175680 oz. to long tons, etc. 6. 3184 in. to rods, etc. 11. 9049 in. to rods, etc. 12. 3 qt. 1 pt. to the decimal of a gallon. 2 )1.0 , no. pt. • 5 Since 2 pt. = 1 qt., 1 pt. = .5 qt.; 3 qt. 3. + .5 qt. = 3.5 qt. Since 4 qt. = 1 gal., 4 )^5, n o. qt. 3 - 5 ^ = - 875 S al - .875, no. gal. Change : 13. 45 yd. .6 ft. to the decimal of a mile. 14. 6 cwt. 8 oz. to the decimal of a ton. 15. | ft. to the common fraction of a yard. 1 ft. = 4 yd. 1 ft. = § of i yd., or f yd. 16. | in. to the fraction of a foot ; of a yard. 17. 12 oz. to the fraction of an avoirdupois pound. 180 DENOMINATE NUMBERS 18. 240 rd. to the fraction of a mile. 19. The winner in an automobile race won in 21 hr. 3 min. 3.6 sec. What decimal of a day did it take ? 20. How much did a merchant receive for 3 barrels (42 gallons each) 9 gallons 3 quarts of oil, retailed at 15 cents per gallon ? 21. A grocer bought 150 bushels of potatoes at 60^ a bushel. He lost ^ of them by freezing, and retailed the remainder at 10^ a half peck. How much did he gain ? 22. How many boxes, each holding a quart, can be filled from 3 bu. 1 pk. 7 qt. of blackberries ? 23. At $2.88 a bushel, how many quarts of chestnuts can be bought for $13.50? 24. What is the profit on 9 quires of paper, bought at $2.40 a ream and sold at a cent a sheet ? 25. A rural mail carrier's route is 21 miles 176 rd. 4 yd. in length. Find the number of feet he travels in one delivery of mail. 26. A fruit grower sold in one season 23 bu. crates of cherries at 10 cents per basket; and 45 bu. crates and 17 baskets of strawberries at 13 cents per basket. Find the amount of the sales. A crate contains 32 baskets. 27. A milk dealer put his milk in pint bottles. Find the number of bottles delivered in one evening if he sold 23 gal., 3 qt., and 1 pt. 28. How many 4-ounce packages of soda can be put up from IT., 3 cwt., and 75 lb. of soda ? 29. A huckster bought 3 barrels of apples, each containing 2| bushels, for $8.25 and retailed them at 15^ a half peck. Find his profits. FOREIGN MONEY 181 30. Find the length of ;i double-track railroad laid with 1640 rails, each 30 feet in length. 31. An ocean steamer in making a certain trip consumed l'.>20 tons of coal. If the time was 6 days, 5 hours, and 8 minutes, find the average number of pounds consumed per minute. 32. The report of a cannon was heard 1 minute 5 sec- onds after it was discharged. If sound travels 1120 feet per second, how many miles, rods, etc., was the hearer from the cannon ? 33. A pupil pays 845 tuition in a term of 9 months of 20 days each, and is absent from school 16 days. Counting 6 school hours to a day, find the amount of tuition lost to him by his absence. FOREIGN MONEY English Money The standard unit of English money is the pound, $4.8665. 4 farthings (far.) = 1 penny (y the part of the circumference cut by the lines extended. 4. How do you explain that ^ of circum- ference A contains as many degrees as ^ of circumference B? 5. Observe the figure. Show that the curved lines are simply parts of circumferences of circles B that could be formed about the point B. How do you show that each curved line measures an angle of 30° ? gles on the figure. Show the an- 6. Show that an angle is the difference in direction of two lines that meet at a point and that the angle remains the same, however far the lines may be extended. Angular Measure Angles are measured by an instrument called a protractor. When the center of the protractor is placed at the vertex of the angle to be measured, the size of the angle may be seen on the scale between the lines that A B form the angle. Thus, BOO is an angle of 30°, and AOC is an angle of 150°. Every circumference contains 360 degrees (360°), each degree, 60 minutes (60'), and each minute, 60 seconds (60"). 196 PRACTICAL MEASUREMENTS Table of Angular Measure 60 seconds (' 60 minutes 360 degrees ') = 1 minute (') = 1 degree (°) = 1 circumference (C) The length of a degree at the equator is 691 miles. Draw an angle of 90°; 45°; 60°; 120°; 30°. Kinds of Angles Which one of these angles is a right angle ? Why ? Which is less than a right angle ? Which is greater than a right angle? A right angle is an angle of 90°. An acute angle is an angle less than 90°. An obtuse angle is an angle greater than 90°. TRIANGLES A triangle is a surface bounded by three straight lines. (Tri means three.^) A vertex of a triangle is a point where two sides meet. The base of a triangle is the side on which it seems to rest. The altitude of a triangle is the perpendicular distance from the vertex opposite the base to the base, or the base extended. T III AN(1 LESS 197 Triangles are named in two ways : I. From their angles : (1) Right-angled triangles. (One right angle.) (2) Acute-angled triangles. (All angles less than a right angle.) (3) Obtuse-angled triangles. (One angle greater than a right angle.) Right-angled Acute- angled Obtuse- angled II. From their sides : (1) Equilateral. (Having three sides equal.) (2) Isosceles. (Having two sides equal.) (3) Scalene. (Having no two sides equal.) Equilateral Isosceles Scalene Measuring degrees and angles. 1. How many right angles are there in the square ? 2. How many right angles are there in the rectangle ? 3. Cut from paper a square. Fold it on the line connecting the opposite corners, and cut it into two triangles. Square Rectangle 4. How many degrees are there in each angle of each triangle thus formed? 198 PRACTICAL MEASUREMENTS 5. Every right triangle contains how many degrees? By Geometry it is shown that the sum of the angles in any triangle is equal to 180°. This can also be shown by meas- uring the angles with a protractor. The sum of all the angles of any triangle is equal to two right angles, or 180°. The following numbers in each case represent the size of two angles of a triangle. Find the size of the third angle : 6. 90° and 45° 10. 60° and 40° 7. 90° and 60° 11. 100° 45' and 37° 8. 120° and 30° 12. 75° 10' and 95° 30' 9. 1201° and 601° 13. 100° and 45° 40' KEOIAJSULli QUADRILATERALS A quadrilateral is a surface having four straight sides. (Quadrilateral means having four sides. ) l. Examine the quad- rilaterals. What are the essential features of A ? A square is a quadri- lateral having four equal sides and four right angles. 2. What are the essential features of B ? In what way does figure B differ from figure A ? A rectangle is a quadrilateral having four straight sides and four right angles. 3. Show that the opposite sides of a rectangle must be equal and parallel. Is a square a rectangle ? A parallelogram is a quadrilateral whose opposite sides are parallel. Square AREAS OK RECTANGLES 199 D liaoimoiu 4. Examine these quadrilat- erals. Why are they parallelo- grams ? How do the sides of surface C compare in length ? **°**™ Show that its angles are not right angles. A rhombus is a quadrilateral whose sides are equal, and whose angles are not right angles. 5. Why is surface D a parallelogram ? Show that its angles are not light angles. Show that its sides are not equal. A rhomboid is a quadrilateral whose opposite sides are equal and whose angles are not right angles. 6. Why is surface E a quadrilateral ? Why is it not a parallelo- gram? How many of TBAI'EZOID Tkatezium its sides are parallel ? A trapezoid is a quadrilateral having but two sides parallel. 7. Why is surface F not a trapezoid ? What is its name ? A trapezium is a quadrilateral having no two sides parallel. 8. Describe each of the six quadrilaterals named above with reference to its sides and angles. How many of these quadrilaterals are parallelograms ? Give reasons. AREAS OF RECTANGLES Finding the area of a rectangle. Find the area of a rectangle 4 yd. long and 3 yd. wide. How long is this rectangle ? how wide ? What is the unit of measure ? How many such units are in the first row? in the second ? in the entire surface ? 4yd. *fsq. yd 200 PRACTICAL MEASUREMENTS If the length and width of a rectangle are expressed in inches, the unit of measure is 1 sq. in.; if expressed in feet, the unit of measure is 1 sq. ft. ; if expressed in rods, the unit of measure is 1 sq. rd. If the length and width are expressed in related units, as feet and inches, or yards, feet, etc., the dimensions must be changed to like units before finding the area, that is the number of square units it contains. Written Work 1. Find the area of a flower bed 20 feet 8 inches in length by 10 feet 6 inches in width. Length = 20§ ft. ; width = 10| ft. Area = 20§ x 10£ x 1 sq. ft., or - 6 3 2 x ^ x 1 sq. ft. = 217 sq. ft. Tlie area of a rectangle is found by multiplying its unit of measure by the product of its two dimensions when expressed in like units. Find the areas of rectangles having the following dimen- sions : 2. 20.5 ft. by 12 ft. 6. 115 ft. by 54 in. 3. 21 ft. by 6.9 ft. 7. 45 yd. by 7 ft. 4. 72 yd. by 401 yd. 8. 108 in. by 3 ft. 5. 6 yd. 1 ft. by 3 yd. 2| ft. 9. 54 ft. by 108 in. 10. How many square yards are there in a lawn 45 feet long and 36 feet wide ? 11. A square ball-park 600 feet on a side is inclosed with a tight board fence 9 feet in height. Find the outside sur- face of the fence in square yards. 12. Compare in area a surface 8 inches square and a sur- face 2 inches square ; a surface 20 rods square and a surface 40 rods square. 13. Bricks are generally 8 in. x4 in. x 2 in. in size. Esti- mate the number necessary to lay a sidewalk 100 ft. long and 5 ft. wide, if the bricks are laid on the flat side. Find the cost of the bricks needed at $13.75 per thousand. PLASTERING AND TAINTING 201 14. A surveyor finds a field in the form of a rectangle to be 680 ft. long and 330 ft. wide. Find its area without changing feet to rods. 15. A field in the form of a rectangle contains 1200 sq. rd. and the length is 40 rods. Find the width. 16. How many lots, each 30 ft. by 120 ft., can be made from a plot of ground 120 ft. in depth and containing 10800 sq. ft. ? (Make a diagram.) PLASTERING AND PAINTING In plastering, painting, and kalsomining, the unit of meas- ure is the square yard. In some localities an allowance is made for openings and baseboards, but there is no uniform rule in practice. Any allowance should always be specified in the contract. There are either 50 or 100 laths in a bundle. A bundle of 100 is generally estimated to cover 5 square yards of surface. Written Work 1. How many square yards of plaster are necessary to cover the ceiling of your classroom ? 2. Find the cost of painting both sides of a tight board fence, 150 ft. long and 8 ft. high, at 15 ^ per square yard. 3. Allowing nothing for openings, how much will it cost to kalsomine the walls and ceiling of a room 20 ft. long, 16 ft. wide, and 12 ft. high, at 6^ per square yard ? 4. A store room is 75 ft. long, 20 ft. wide, and 15 ft. from floor to ceiling. It has a door in the rear 7 ft. by 3£ ft., and a window 8 ft. by 3 ft. How many bundles of laths, each containing 100, are required for the sides, rear, and ceiling, making full allowance for openings ? 202 PRACTICAL MEASUREMENTS 5. How much will it cost to plaster the walls and ceiling of a store room, 40 ft. by 18 ft. and 12 ft. high, at 6^ per square yard for lathing, and 18^ per square yard for plas- tering, deducting ^ the area of 2 doors, each 9 ft. by 4 ft., and of 4 windows, each 6^ ft. by 3| ft. ? 6. A building 90 ft. by 24 ft. contains 3 stories, each 13 ft. high. The first story is plastered on the sides and rear. The second and third stories each have 3 windows in the front, each 8 ft. by 3^- ft., and each 2 windows in the rear, each 8 ft. by 3 ft. If the ceilings are sheet iron, find the cost of the plastering, at 33^ per square yard, deducting for all openings. 7. In modern business buildings metal laths are used. Estimate the cost of metal laths, for the building in example 6, at 25^ per square yard. ROOFING AND FLOORING In roofing, tiling, and flooring, the unit of measure is the square of 100 square feet. Written Work 1. Each of the two slopes of a roof is 60 ft. long and 20 ft. wide. Find the cost of covering them with tar paper at •$5.60 per square. 2. The floor of a hallway 30 ft. by 12 ft. is inlaid with 2-inch square tile. Find the number necessary. In roofing with slate, each course of slate is part- ly overlapped. Each slate as here shown is 10 in. by 16 in. and has 4 in. ex- posed to the weather. nil ■hi It •!:? Hi Ujll*!' t lllp! in . ! 1 fe :ii ryi | If 1 1 r in pywi 11 ROOFING AND FLOORING 203 3. How many square inches of each slate are exposed? 4. If a 10-inch by 16-inch slate is exposed 4 inches to the weather, find the number of slates necessary to lay a square (10 ft. by 10 ft.)- 5. If slate 10 in. by 16 in. is laid 6 in. to the weather, find the number necessary to lay a square. Find the weight of a square of slate at 4 ; ]- lb. per square foot. 6. Each slope of a roof is 40 ft. by 20 ft. Find the number of slates, 10 in. by 16 in., exposed 4 in. to the weather, required for this roof, allowing nothing for breakage. Find the cost of the slates at $5.50 per square. There are 250 shingles in a bunch. Shingles average 16 inches in length and 4 inches in width. The exposed surface of a shingle laid 4| inches to the weather is, therefore, 18 square inches. Without waste 8 shingles will lay one square foot, and 800 shingles will lay 100 square feet, or 1 square. Allowing for waste, 4 bunches, or 1000 shingles are estimated to lay a square. 7. Allowing nothing for waste, how many bunches of shingles are required to cover a barn roof 35 ft. in width on each side and 70 ft. in length. Find the cost at $4.00 per thousand shingles. 8. Adding \ for waste, estimate the cost at $3.50 per thousand of 157 bunches of shingles required to cover the roof in example 7. Flooring is frequently estimated by the square. 9. How much will it cost, at $5.00 per square, to lay the floor of a hall 30 ft. by 60 ft., adding l for waste? 10. Estimate the number of squares of flooring required for two floors of a store room 25 ft. by 60 ft. 204 PRACTICAL MEASUREMENTS PAPERING AND CARPETING The unit of measure in wall paper is the single roll, which is 8 yards in length and usually 18 inches in width. A double roll is 16 yards in length. In approximating the number of rolls, paper hangers generally deduct from the perimeter of the room the width of the doors and windows. The remaining number of feet divided by H ft. (18 in. = 1| ft.) gives the number of strips required for the surface of the wall. Dividing the total number of strips by the number that can be cut from a double roll gives the number of double rolls required. Fractional parts of a roll are not sold. The ends of the rolls are generally sufficient to paper the surfaces above and below the doors and windows. Border is sold by the linear yard. Carpet, matting, and border are sold by the linear yard. Oil cloth and linoleum are sold by the linear yard or by the square yard. Ingrain carpets are usually 1 yard wide, other carpets are generally 27 inches wide. Liberal allowance must be made for loss in matching. Written Work 1. Estimate the number of double rolls of paper required for a ceiling 18 ft. by 22 ft., strips running lengthwise. 22 ft. = length of one strip. 16 yd. = 48 ft. ; 48 ft. -4-22 ft. = 2, the number of whole strips in a double roll. 18 ft. -4- li ft. = 12, the number of strips required. 12 -=- 2 = 6, the number of double rolls required. 2. A dining room 15 ft. by 22 ft. is 11 ft. from baseboard to ceiling. It has four openings 3^ ft. by 7 ft. Estimate the paper required for it, strips on ceiling running lengthwise. 3. The dining room in problem 2 lias a plate rail extend- ing around it between the openings. Find the cost of this rail at 80^ per foot. PAPERING AND CARPETING 205 4. How much carpet 27 in. wide, laid the long way of the room, is required for a room 18 ft. long and 15 ft. wide, allowing 12 in. on each strip except the fust for matching ? 6 yd. = the length of one strip. 27 in. = 2\ ft. ; and 15 ft. ■*- 2\ ft. = % therefore 7 = the number of strips. 7x6 yd. = 42 yd. 6 x 12 in. = 72 in., or 2 yd., the waste on 6 strip.'-. 42 yd. + 2 yd. = 44 yd. of carpet required. 5. Explain why it takes fewer yards of carpet to cover a room 18 ft. by 27 ft. with ingrain carpet (1 yard wide) than with Brussels carpet (27 inches wide). Laying the carpet the long way of the room, how many yards of each would it take, if 10 in. were allowed on each strip, except the first, for matching? 6. The widths of certain floors are 15 ft., 13^ ft., 15| ft., 18 ft., 16 ft, Estimate the number of strips of ingrain carpet necessary to cover each room. 7. Estimate the number of strips of Brussels carpet necessary to cover each room described in example 6. 8. Find the cost of covering a kitchen 13| ft, by 12 ft. with linoleum at $1.60 per yard double width, if £ of a yard is allowed for matching and the linoleum is laid the long way of the room. 9. Estimate the difference in cost between covering a room 18 ft. by 20^ ft, with Axminster carpet 27 inches wide, at $1.45 per yard, laid lengthwise, allowing 12 inches on each strip except the first for matching, and covering the room with ingrain carpet at 85^ per yard, laid in the same way, allowing 12 inches on each strip, except the first, for matching. 206 PRACTICAL MEASUREMENTS ib. >, ^. = 35yd. Li. C=6.2832rd. 14.22 = 10 ft. L2. C = 94.248 in. 15. D =10 yd. 19. A circle 20 ft. in diameter is inscribed in a square. What is the area of one of the corners within the square, but outside the limits of the circle ? 20. A circular fountain 20 ft. in diameter is surrounded by a cement walk 4 ft. wide. How much will the walk cost at $1.50 per square yard? Xote. — Find the difference between the areas of the two circles, the first bounded by the circumference of the fountain, and the second, by the circumference of the walk. 1. 2> = 10 rd. 2. R = 10 rd. 3. 2> = 18 ft. 7. R = 40 rd. 8. 2> = 25 yd. 9. R = 40 ft. 16. R = 125 ft, 17. 2) =120 yd. 18. 22=19;yd. 212 PRACTICAL MEASUREMENTS SOLIDS 1. How many faces has this solid? What is their shape? How do they compare in size ? A cube is a solid bounded by six equal square faces. 2. Every 1-inch cube rests on how many square inches of surface ? 3. Show that 144 one-inch cubes may be placed on 1 square foot of surface. 4. How many cubes would make 12 such layers? 5. Show that 1728 cubic inches equal 1 cubic foot. 6. How many 1-inch cubes can be put into a cubical box whose edge is 3 inches? 7. How many 1-foot cubes can be placed on 9 square feet of surface? (Make diagram.) 8. How many cubes are there in three such layers? 9. Show that 27 cubic feet equal 1 cubic yard. Learn this table of solid or cubic measure : 1728 cubic inches ( cu. in ) = 1 cubic foot (cu. ft.) 27 cubic feet (cu. ft.) = 1 cubic yard (cu yd.) 128 cubic feet (cu. ft.) = 1 cord of wood or tanbark 100 cubic feet (cu. ft.) = 1 cord of stone 1 cubic yard of earth equals 1 load. The unit of cubic measure is a cube whose edge is one of the linear units ; thus, a cube each edge of which is one inch in length is a cubic inch. SOLIDS 213 A cubic foot is a cube whose edge is one foot. A cubic yard is a cube whose edge is one yard. A cord of wood or tanbark is a pile of 4-foot wood or tan- bark 8 feet long and 4 feet high. A cord of short wood is a pile of short lengths 8 feet long and 4 feet high. The number of cords of short wood in a pile is found by dividing the number of square feet in one side by 32. Surface of Rectangular Solids How is the surface of each face found ? How many faces has this solid ? Show that the sum of the faces in this solid is the surface of the solid. A rectangular solid is a solid bounded by six rectangular surfaces. Written Work Find the entire surface of : Rectangular Solids 1. 12 ft. by 8 ft. by G ft. 2. 20 ft. by 10 in. by 10 in. 3. 1G ft. by 2 ft. by 1^ ft. 4. 10 ft. by 8 ft. by 7 ft. 5. G ft. by 5 ft. by 5 ft, 6. 13 ft. by 8 ft. by 3 ft. 7. 20 ft. by 9 ft. by 7 ft. Cubes 8. 4 inches on an edge. 9. 12 inches on an edge. 10. 2 feet on an edge. 11. 121 inches on an edge. 12. 14 inches on an edge. 13. 11| feet oil an edge. 14. \\\ inches on an edge. 214 PRACTICAL MEASUREMENTS Volume of Rectangular Solids 4 in. Scale : \ inch= 1 in. Each cube in the solid represents one cubic inch. How man)" cubic inches are there in the first layer ? How many such layers does this solid contain? How many cubic inches does the solid contain ? Observe that the product of the three dimen- sions expresses the number of cubic units. The volume of a solid is the num- ber of cubic units it contains. If the dimensions are expressed in inches, the unit of measure is 1 cubic inch : if expressed in feet, the unit of measure is 1 cubic foot ; if expressed in yards, the unit of measure is 1 cubic yard. If the dimen- sions are expressed in related units, as feet and inches, or yards and feet, they must first be changed to like units. Written Work 1. Find the volume of a rectangular solid 8 ft. 6 in. square and 12 ft. 4 in. in length. Thickness = 8.5 ft. : width = 8.5 ft.; length = 12J ft. Contents or volume = S.5 x 8.5 x 12£ x 1 cu. ft., or 891.08$ en. ft. The volume of a rectangular solid is found by multiply ing the unit of measure by the product of its three dimensions when expressed in like units. Find the contents or volume of the following solids: 2. 10 ft. by 6 ft. 3 in. by 4 ft. 5. 1 yd. by 2 ft. by 18 in. 3. 12 ft. by 9 ft. 6 in. by 6 ft. 6. 68 in. by 1 ft. by 10 in. 4. 10 ft. square and S ft. high. 7. 5 yd. by 1 1 yd. by 2 ft. SOLIDS 215 8. How many loads of earth must be removed in excavat- ing for a cellar 30 ft. by 24 ft. and 8 ft. in depth ? 9. Estimate the number of cakes of soap 3 inches by 2 inches by 2 inches that can be packed in a box 3 feet by 2 feet by 2 feet. 10. A schoolroom is 40 ft. by 28 ft. by 16 ft. How many cubic feet of air space are there for each of 39 pupils and their teacher ? 11. How many cords of 4-foot wood are there in a pile 40 ft. long and 4 ft. high ? 12. Estimate the number of cords of 18-inch wood in 3 piles each 60 ft. long and 4 ft. high. 13. How many cubical boxes 3 ft. 6 in. on an edge can be placed in a storage room 14 ft. in length, width, and height ? In a certain township, the piles of 20-inch wood in the yards of four schools were as follows : 14. Sykes, 15. Graham, 16. Wilson, 17. Clark, Estimate the number of cords at each school and the value of the wood at $1.85 per cord. 18. Find the number of loads of earth removed in exca- vating for a cellar 16 feet wide, 30 feet long, and 6 feet in depth. of Piles Length of Piles 60 ft. 40 ft. Height of Piles 4 ft. 6 ft. <2 u 50 ft. 60 ft. 5 ft. 4 ft. \\ 72 ft. 45 ft. 5 ft. 6 ft. \l 36 ft. 60 ft. 5 ft. 4 ft. 216 PRACTICAL MEASUREMENTS 19. A pile of tanbark is 8 ft. wide, 9 ft. high, and 100 ft. long. Find the number of cords. 20. A cubical block of granite 2 ft. on an edge is what part of a cubical block of granite 6 ft. on an edge ? When possible, use cancellation in the following problems : 21. Cape Cod cranberries are shipped in a crate whose inside dimensions are 20 in. x 10|^ in. x 6| in. How many cubic inches are 'there in a crate ? 22. Sweet potatoes are sometimes sold in a box 19^ in. x llf in. x 10 in. How much does this differ from a bushel (2150.4 cu. in.)? 23. Colorado apples are sometimes shipped in a box 18 in. X 11 J in. x 11 in. How many cubic inches more or less than a bushel does such a box contain ? 24. Colorado apples are sometimes shipped in a box 16 in. x 11 1 in. x 8| in. How many cubic inches does such a box contain ? 25. California celery is shipped in crates 24^ in. x 22 in. X 20£ in. How many cubic inches are there in such a crate ? 26. California dates are sold in boxes 17| in. x 10 in. x 9^ in. How many cubic inches are there in such a box ? 27. Figs are packed solid in a box 12 in. x 9 in. x If in. How many cubic inches are there in such a box ? 28. The standard California orange box is now 24 in. x 11^ in. X 12 in. Tangerines are shipped in boxes 24 in. x 12 in. x 6| in. Which is the larger, and how many cubic inches larger is it ? 29. How many cubic inches are there in a box 5| in. x 8| in. x 2| in. ? Find the number of cubic inches in a box 81 in. x 12i in. x 44 in. LUMBER 217 One Board Foot LUMBER Measurement of lumber. Lumber is any kind of sawed timber as boards, planks, sills, etc. The unit of lumber measure is the board foot; it is a board 1 foot long, 1 foot wide, and 1 inch thick. Draw it. Note. — Boards less than 1 inch in thickness are measured as if they were 1 inch thick ; boards over 1 inch in thickness are measured by their actual thickness in inches and fractions of an inch. 1. How many board feet are there in a board 1 foot wide, 1 inch thick, and 3 feet long? 5 feet long? 9 feet long ? 2. How many board feet are there in a board 6 inches wide, 1 inch thick, and 3 feet long? 10 feet long? In a board 6 inches wide, \ inch thick, and 12 feet long? 3. How many board feet are there in a sill 5 feet long, 1 foot wide, and 4 inches thick ? 5 ft. Written Work 1. Find the number of board feet in a sill 18 ft. long, 10 in. wide, and 8 in. thick. 10 in.= | ft. One surface = 18 x | x 1 hoard foot, or 15 board feet. The sill contains 8 x 15 board feet, or 120 board feet. The number of board feet in a piece of lumber is found by multiplying the number of board feet in one surface by the number of inches in thickness. 218 PRACTICAL MEASUREMENTS Find the number of board feet in the following: 2. 1 board, 10 ft. long, 11 ft. wide, and 1 in. thick. 3. 1 board, 16 ft. long, 11 ft. wide, and | in. thick. 4. 2 boards, each 16 ft. long, 1 ft. wide, and | in. thick. 5. 6 boards, 15 ft. x 2 ft. x 1 in. 6. 4 boards, 16 ft. x 1| ft. x -|- in. How many feet of lumber are there in : 7. 1 plank, 12 ft. long, 1 ft. wide, and 3 in. thick ? 8. 1 sill, 15 ft* long, 1^ ft. wide, and 8 in. thick ? 9. 4 planks, 12 ft. long, 1| ft. wide, and 2 in. thick ? 10. 2 pieces, 18 ft. by 1 ft. by 1 ft.? Find the number of feet of lumber in : 11. 10 planks, each 8 ft. long, 1^ ft. wide, and 3 in. thick. 12. 12 sills, each 20* ft. long and 10 in. square. 13. 20 joists, each 12 ft. long, 12 in. wide, and 3 in. thick. 14. 3 beams, each 40 ft. long and 10 in. by 12 in. 15. 30 scantlings, each 16 ft. long and 2 in. by 3 in. 16. How much will the flooring for two rooms, each 18 ft. x 20 ft., cost at $30 per M.? Buying and selling lumber. Lumber is bought and sold by the thousand board feet. In practice the cost is computed at so much per board foot ,• thus, $20 per thousand feet (M.) is $.02 per board foot. Show that $35 per M. = $.035 per board foot. $60 per M. = $.06 per board foot. LUMBER 219 Written Work 1. Estimate the cost of 378 feet of oak at $26 per M.; of 6389 ft. white pine at $48 per M.; of 972 ft. cherry at $72 per M.; of 693 ft. white ash at $47 per M. Find the cost at $35 per M. of: 2. 50 hoards, 16 ft. long, 12 in. wide, and 1 in. thick. 3. 60 hoards, 12 ft. long, 15 in. wide, and 1| in. thick. 4. 100 boards, 15 ft. long, 6 in. wide, and f in. thick. 5. 75 boards, 18 ft. long, 10 in. wide, and 1 in. thick. 6. 45 boards, 16 ft. long, 5 in. wide, and 1 in. thick. Short forms are used by carpenters, architects, and mechanics ; thus, one mark (') represents feet, and two marks (") represent inches. Find the number of board feet: 7. 120 studding, 2" x 4" x 12'. 8. 400 planks, 2" x 1' x 16'. 9. 300 boards, 1" x 10" x 14'. 10. 600 boards, 1" x 6" x 16'. li. 100 boards, f " x 12" x 16'. 12. 15 sills, 6" x 10" x 20'. 13. 250 joists, 2" x 8" x 24'. 14. 70 sills, 10" x 12" x 30'. 15. 125 sleepers, 3" x 10" x 28'. 16. 200 boards, J" x4'"x 16'. 17. 500 joists, 21" x 8" x 20'. 18. 325 planks, 3" x 14" x 16'. 19. 300 sills, 5" x 8" x 24'. 20. 50 posts, 10" x 12" x 14'. 21. 400 studding, 2" x 3" x 18'. 22. 500 beards, 11" x 10" X 16'. 220 PRACTICAL MEASUREMENTS 23. Estimate the cost of the planks in examples 8 and 18, at $.027 per board foot. 24. Estimate the cost of the sills in examples 12, 14, and 19, at $.032 per board foot. 25. Estimate the cost of the studding in examples 7 and 21, at $.026 per board foot. CONCRETE, STONE, AND BRICKWORK Concrete work is estimated by the cubic yard. Stone work is estimated by the perch, of 24.75 cu. ft., or by the cord of 100 cu. ft. Stones are often sold by the pound. 3200 pounds are estimated to lay 1 perch. In estimating either contract work or cost of labor, in concrete and stone work, the distance around the wall is considered the length. In cases where there are inside corners, however, as at a and b in the figure on p. 226, add, for each inside corner, twice the thickness of the wall. This measures all corners twice. In estimating material, deduct for openings and measure the corners but once. Range work and lintels are measured by the linear foot. Brickwork is estimated by the thousand. Bricks vary in size, but they are usually 8" by 4" by 2". In estimating the number of bricks in a wall, measure the corners once, deduct all openings, and multiply the number of square feet re- maining in the surface by 7 when the wall is 1 brick thick; by 14 when the wall is 2 bricks thick ; and by 21 when the wall is 3 bricks thick. Written Work 1. Find the number of cords of stone in a breakwater 200 ft. long, 14 ft. wide, and 16 ft. high. 2. A building 150 ft. by 130 ft. has a concrete foundation 4 ft. in width and 10 ft. in depth below the structural iron. Estimate the number of cubic yards of material used. THE CYLINDER 221 3. If the cement, the gravel, and the sand are in the ratio of 1, 5, and 2, find the number of loads of gravel and of sand used in the construction of the foundation in example 2. 4. Estimate the contract cost of the concrete work at $7.75 per cubic yard. 5. Estimate the number of ^^.^^..^..^.^ . ^....^^ cubic yards of concrete in this M?-_ . . . . — . : . ; . , . . . retaining wall. ^;Y. ',' '•'.". . '-• / '.^v ■;•;• ■.] 6. The walls of a brick house ^gfc '.•'•'••; .':"•:! ^: ■■■'.. 36 ft. long, 24 ft. wide, and 18 ft. 200' high are 13 in. or 3 bricks thick. Estimate the number of bricks required for the walls, allow- ing for 11 windows averaging 3± ft. by 6 ft., and 2 doors averaging 3|- ft. by 7 ft. 7. A house whose walls are 9 in. or 2 bricks thick is 40 ft. long, 30 ft. wide, and 24 ft. high. Estimate the number of bricks required for the walls, allowing for 12 windows 3 ft. by 7 ft., and 3 doors 3| ft. by 8 ft. 8. The stone work for the foundation of a house 28 ft. by 38 ft. is 1* ft. in thickness and 6 ft. in height to the range work. Estimate the cost of the stone work at $6.30 a perch, and the range work along the two sides and the rear at 60 cents per linear foot. THE CYLINDER Examine this solid. ^SlS=s How many ends or bases has it? What is the shape of each? Are the bases equal and parallel? Describe the shape of the body. A cylinder is a solid whose two bases are equal and parallel circles and whose diameter is | uniform. 000 — — — PRACTICAL MEASUREMENTS The convex surface of a cylinder is the lateral or curved surface. The altitude is the perpendicular distance between its two bases. Examine this cylinder. Observe: 1. That if a piece of paper is fitted to cover its convex sur- face and then unrolled, its form will be that of a rectangle. 2. That the circumference of the base is the length of the rectangle, and the altitude of the cylinder is the width of the rectangle. The convex surface of a cylinder is found by multiplying the unit of measure by the product of the circumference and the altitude. The entire surface of a cylinder is found by adding the area of the bases to the convex surface. • Written Work Find the convex surface of a cylinder: 1. D. 10 in., height 24 in. 4. D. 20 in., height 4 ft. 2. D. 15 in., height 30 in. 5. D. 8 in., height 4 ft. 3. D. 2 ft., height 10 ft. 6. D. 6 ft., height 15 ft. Find the entire surface of : 7. A water tank 12 ft. in diameter and 12 ft. in height. A steam boiler 15 ft., long and 3 ft. in diameter. Find the volume of a cylinder 3 ft. in diameter and 5 ft. high. Observe: 1. That the area of the base is 3 2 x .7854 x 1 sq.ft., or 7.0686 sq. ft. 2. That the first row of cubic units contains 7.0686 cu. ft. 3. That the cylinder contains 5 times 7.0686 cu. ft., or 35.343 cu. ft. 8. 9. BINS, TANKS, AND CISTERNS 223 The volume of a cylinder is found by multiplying the unit of measure by the urea of the base and this product by the height of the cylinder. Find the volume of a gas tank, silo, cistern, etc. : 10. D. 15 ft., height 18 ft. 14. R. 2 ft., depth 8 ft. 11. D. 25 ft., height 30 ft. 15. R. 8 ft,, height 30 ft. 12. D. 16 ft., height 20 ft. 16. D. 1 ft., length 16 ft. 13. D. 20 ft,, depth 15 ft. 17. D. 5 ft., length 12 ft. BINS, TANKS, AND CISTERNS Wheat and other grains are generally sold by weight, but the capacity of bins is often estimated in bushels. The capacity of tanks and cisterns is estimated in gallons or barrels. Note. — The standard bushel in the United States contains 2150.42 cu- bic inches, stricken measure, and 2747.71 cubic inches heaped measure. 231 cu. in. = 1 gal. 31 \ gal. = 1 bbl. when estimating contents. Written Work Find contents in bushels of : 1. A bin 20 ft. by 10 ft. by 5 ft. 2. A box 12 ft. x 9 ft. x 6 ft. 3. A metal trough for watering cattle is 12 ft. long, 3 ft. wide, and 20 in. deep. Estimate the number of gallons it holds. 4. A cistern tank for a windmill pump is 8 ft. in di- ameter and 10 ft. in depth. Estimate the number of barrels of water it holds. 5. The rainfall on a certain day was 1\ inches. Find the number of barrels of water that fell on Mr. Anderson's flower plot which is 20 ft. long and 10 ft. wide. 224 PRACTICAL MEASUREMENTS APPROXIMATE MEASUREMENTS Approximate equivalents of the following measures are 1 bu. shelled grains = 1] cu. ft. 1 bu. apples, coal, roots, corn in ear, etc. = If cu. ft. 1 bbl. in estimating contents = 4i cu. ft. 1 cu. ft. of water = 62] lb. 1 gal. of water = 81 lb. 1 cu. ft. of any liquid = 7.i gal. 1 ton of hay well packed = 450 cu. ft. 1 ton of clover hay = 550 cu. ft. 1 ton of bituminous coal = 42 cu. ft. 1 ton of hard coal = 35 cu. ft. Written Work 1. A water meter registered 900 gallons of water con- sumed in a month. Estimate the weight of the water used, and its volume in cubic feet. 2. The inside measurement of a wagon box is 12 ft. 4 in. by 3 ft. 6 in. by 16 in. Estimate the number of tons, etc., of anthracite coal it would contain; the number of tons, etc., of soft coal. 3. The rainfall on a roof 20 ft. by 30 ft. during April and May was 9.5 in. Find the weight of the water that fell on the roof during that time. 4. A swimming pool is 80 ft. long, 60 ft. wide, and 5 ft. deep. Estimate the number of barrels of water in the pool. 5. Estimate the number of bushels in an oat bin 14 ft. long and 10 ft. wide if the bin is filled with oats to a depth of 6 feet. REVIEW PROBLEMS 225 6. I Tow many tons of hard coal are there in a bin 16 ft. x 12 ft., when the pile is 1 ft. high ? 7. There are 5 ft. of water in a cistern 4 ft. in diameter. How many gallons of water are there in the cistern? 8. Find the weight of the water in a railroad tank 12 ft. in diameter and 16 ft. in depth, if the tank has 12 ft. of water in it. 9. How many bushels of wheat can be shipped in a car whose inside measurements are 36 ft. by 8 ft. 6 in. by 8 ft. ? 10. Estimate the number of tons of clover hay in a mow 60 ft. by 18 ft. by 16 ft. Estimate the number of tons of timothy hay in the same mow. REVIEW PROBLEMS 1. A field containing 20 acres is 61 rods long. How wide is it? 2. The area of the floor of a schoolroom contains 1120 sq. ft. The air in the room occupies 16800 cu. ft. What is the height of the ceiling ? 3. How many tiles 6 in. square are required for a hall 40 ft. by 20 ft. 6 in.? 4. The side of a square is 20 inches. Find its area; its perimeter. 5. The edge of a cube is 18 inches. Find its surface; its contents. 6. How many cubical boxes whose edges are 6 in. can be put into a box 8 ft. 6 in. by 4 ft. 6 in. by 3 ft.? 7. How many cakes of soap 2 in. x 2 in. x 4 in. may be packed in a box 2 ft. long, 1 ft. wide, and 1 ft. high? HAM. t OMl'L. A HI III. — 15 226 PRACTICAL MEASUREMENTS 14 Elii'iiiii I* Miniii nun ; O 8. The edges of two cubes are respectively 10 inches and 12 inches. How much more surface has one than the other? 9. Find the cost of a farm, 480 rods long and 320 rods wide, at 860 per acre. 10. How much will it cost to put a wire fence around this farm at 50 per rod ? 11. A ranchman bought one square mile of land at $10 per acre. He put a fence around it and then divided it by fences into four equal square farms for his sons. Find the entire cost if the fence cost $.40 per rod. 12. Estimate the contract price of building this cellar wall 18 in. thick and 6^ ft. in height at $5.95 a perch. Distance around the wall = 164 ft. Add to this distance twice the thickness of the wall for each of the inside corners, a and \^) 6; that is, twice 2 x 18 in., or 6 ft. Then, the length of the wall with 8 corners counted twice = 164 ft. + 6 ft. = 170 ft. The volume of the wall = 170 x 6 J x U x 1 cu. ft. = 1657.5 cu. ft. The number of perch of stone = 1657£ -h 24| = 66f| perch. The cost of the wall = 66ff x $5.95 = $398.47. Query. — Why are the 8 corners measured twice ? 13. Estimate the number of bricks necessary for a dwell- ing erected on the foundation as given in example 12, if the walls are 3 bricks (13 in.) thick and 20 feet in height, mak- ing an allowance of 150 square feet for openings. Query. — Why are the 8 corners measured once ? 14. Estimate the cost of the face brick for the above dwelling at $16.50 per thousand and $9.00 per thousand for laying. Note. — Consider the face brick as a wall 4 in. thick and measure the 8 corners once. 11 IIIIIIITTTT I 1 MM. I , 40' REVIEW PROBLEMS 227 15. A level lot 60 ft. by 120 ft. has erected on it a dwell- ing otj ft. by 42 ft. If the excavating averages 5 ft. and the removed earth is placed on the lot, to what height will it raise the grade of the lot ? 16. A certain town has a cylindrical water tank 20 ft. in diameter and 45 ft. in height. The gauge shows 30 ft. of water in the tank. Estimate the weight of the water. 17. How many yards of carpet, 27 in. wide, are required for a room 24 ft. by 20 ft. 3 in. ? Strips are to run length- wise. 18. How much will it cost to carpet a room 24 feet square with carpet 27 inches wide, at $1.25 per yard, allowing 10 inches on each strip except the first for matching ? 19. How much will it cost to plaster a room 20 ft. by 16 ft., and 12 ft. to the ceiling, at 20^ per square yard, allowing for one door 3 J ft. by 7 ft., and 2 windows, each 4 ft. by 6 ft. ? 700 ft. 20. This plot of ground is 700 ft. long | Qak'St_ and (500 ft. wide. Find the cost of grading streets 40 ft. in width run through the cen- ter each way, as here shown, at $1.90 per linear yard. Find the amount from the sale of lots 30' x 140' facing on Oak, Center, and Clark Streets, at $ 20 per front foot. ..C/arkSt,. 21. Each of the three sides of a triangle is 50 ft. What is the size of each of its angles ? Draw the figure. 22. The sides of a triangle are 80 fr., 80 ft., and 30 ft. respectively. If the angle opposite the short side is 24°, what is the size of each of the other angles ? Draw the figure. Cet itt >rSt 228 PRACTICAL MEASUREMENTS Oo O 23. The angle opposite one of the equal sides of an isos- celes triangle is 75°. Find the size of the other two angles. 24. One angle in a right triangle is 13.75°. Find the other two angles. 25. The angle opposite the base in an isosceles triangle is 18^°. What is the size of the other two angles ? 26. A railroad com- pany owns a strip of land in the form of a parallel- ogram, 66 ft. wide and 92.78 + rd. long through this farm. Find the area of A and the area of the part owned by the railroad. How can the area of B be found ? 27. A wheel is 3 ft. in diameter. How many revolutions will it make in moving forward 942.48 ft.? 28. The speed of a vessel for 5 hours was 23.17 knots per hour. Find her average speed per hour in statute miles. 29. Lead is 11.35 times as heavy as water. Find the value of a cubic foot of lead at 5 ^ per pound. 30. How many rolls of paper are required for a room 20 ft. X 18 ft., and 13 ft. 3 in. from the top of the baseboard to the ceiling, allowing for 2 windows 3^ ft. in width and one door 3^ ft. wide (papering the ceiling lengthwise) ? 31. The base of a triangle is 30 ft. and its altitude 23 ft. Find its area. 32. A barn is 80 ft. by 50 ft." It is 40 ft. to the base of the gable and 58 ft. to the top of the gable. How much will it cost to paint it at 8 ^ per square yard ? . <. lOOrd.- -65rd Z w <£>\ \ \ \ ,-. "—A \ B cp\ \ \ A -v\ — ^ \ o-\ \l8rd. REVIEW PROBLEMS 229 33. If the slope of the roof of the barn in problem 32 is 32 ft. long, and it projects 1| ft. at each end, how much will it cost to roof it at $ 8 per square? 34. A schoolroom is 40 ft. long and 30 ft. wide. Estimating 450 cubic feet of air to each person, what should be the height of the room to accommodate 39 pupils and their teacher ? 35. Find the cost of a stone wall 30 ft. long, 2| ft. thick, and 6 ft. high, at $5.30 a perch. 36. How much will it cost to cement the floor of a cellar, 40 ft, by 20 ft., at 90^ a square yard? 37. A street 50 ft. from curb to curb is opened for a dis- tance of 300 yards. How much will it cost to excavate it to a depth of 1 foot at 40 ^ per cubic yard ? 38. How much will the curb of this street cost at 26^ per linear foot ? 39. The sidewalk on this street is 12 ft. wide, including a curb of 8 inches. How much will the brick for the walk cost, at $ 9 per thousand, if the exposed surface of a brick is 4 in. x 8 in. ? 40. A farmer built a circular silo 12 ft. in diameter and 24 feet high. Find its contents in cubic feet. 41. How many blocks of ice, 2 ft. x 1 ft. x 1 ft., can be packed in a ear ft, x 8 ft. x 40 ft. ? Ice is .92 as heavy as water. Find the weight of the ice if a cubic foot of water weighs 62| lb. 42. A cistern is 4 ft, in diameter and 6 ft. deep. How many barrels of water will it contain ? 43. Estimate the weight of the water in a tank 8 ft. long, 6 ft. wide, and 2 ft. deep. 44. A vault is 5 ft. square and G ft. deep. How much will it cost to cement the sides and bottom at $ .50 per sq. ft. ? 230 PRACTICAL MEASUREMENTS 45. A circular amusement park is 80 rods in diameter. Find the cost of the boards for a tight board fence 8 ft. high, inclosing the park, at $ 20 per M. 46. A corner lot in Seattle is 25 ft. by 100 ft. At $ 25 per M., what will be the cost for 2-inch plank for a 10-foot sidewalk in front and on the side, including the corner ? Note. — Illustrate by diagram. 47. Mr. Ames owns a 50-ft. lot fronting on a street 60 ft. wide from curb to curb. The law compels him to pay ^ of the cost of paving the street in front of his lot. How much will it cost at $ 2.90 per square yard ? 48. A tank open at the top is 50 ft. long, 4 ft. wide, and 3 ft. deep. How much will the lead for lining it cost, at 8 ^ per pound, estimating 4 pounds to a square foot ? 49. Clay weighs 1.2 as much as the same volume of water. Estimate the weight of a load of clay. j j' 50. Find the cost of painting the sides and ends of this hay barn at 15^ per square yard, and the cost of stain- ing the roof at 12^ per square yard. 51. In excavating for a cellar 60 ft. long, 30 ft. wide, and 8 ft. deep, the material was evenly distributed over a lot 90 ft. by 40 ft. To what depth was the lot covered ? 52. A two-story school building has 8 rooms 30 ft. x 32 ft. and a hallway 28 ft. x 15 ft., on each floor. How much will the flooring for the building cost at $44 per M. ? 53. In digging a sewer 31 ft. in width and 8 ft. deep, 1244| cubic yards of earth were excavated. Find the length of the sewer in feet. Test: 8 + 4 = -4 :12 -4 5. 8- 8 -2 = 6 = 12- -4 ANALYSIS THE EQUATION l. 8=8 2. 8 + 4=12 3. 8 = 12-4 In example (1) we have an equal number on each side of the equality sign. In example (2) we have 8 + 4 = 12 ; but in example (3), in order to preserve the equality, when we take 4 from the left of the equal- ity sign in example (2), we must sub- tract 4 from the number on the right of the equality sign. Thus,8= 12 — 4. 4. 8 = 6 + 2 Observe that a number may be moved from one side of an equation to another by changing its sign. Written Work Change the following so that the first number in each problem will stand alone at the left of the equality sign : 6. 20 - 10 = 10 10. 75 - 20 = 55 7. 40-15= 25 n. 85- 5-10=70 8. 80 + 15 = 95 12. 90 - 10 + 5 = 85 9. 100 + 75 = 175 13. 100 + 10 - 20 = 90 14. First add, then subtract, 5 from each member of the equation 10 = 10. (a.) 10 = 10 (b.) 10 = 10 + 5 = +5 -5 = —5 15= 15 5= 5 The same number may h? added to or subtracted from both sides of an equation without dest roying the equality. 231 232 ANALYSIS Factors and their Product 1. 5 times a certain number is 35. What is the number ? Factors Product 5 x the number = 35 The number = 35 -=- 5 = 7 When the product of two factors is divided by one of the factors, the quotient is the other factor. When one of the factors is unknown, it may be found by dividing the product by the known factor. State the factors and solve : 2. 5 times John's money = $ 40. How many dollars has he? 3. 2 times A's sheep are 60. How many sheep has he ? 4. 6 times B's age is 360 years. How old is he ? 5. l of a number is 150. What is the number? 6. 1.25 times a number is 30. What is the number? 7. .75 of a number is 75. Find the number. 8. ^ of a number is 75. Find the number. I of the number = 75 The number = 3 x 75 = 225 The work may be shortened by calling the unknown factor x. For example, 9. Mr. Brown's profits equal 4 times Mr. Long's profits, and together their profits are 8 125. Find the profits of each. Let x = Mr. Long's profits. 4 x = Mr. Brown's profits 5 x = $125, or the profits of both. x = 825, Mr. Long's profits. 4 x = $ 100, Mr. Brown's profits. 10. Mr. Byers and Mr. Boydson together have 240 acres of land, and Mr. Byers has 40 acres more than Mr. Boydson. How many acres has each? THE EQUATION 233 Let x = the number of acres in Mr. Boydson's farm. x + i() = the number of acres in Mr. Byers's farm. 2 x+ 40 = the number of acres in both farms, or 240 acres. 2 x = 240-40 2x = 200 x = 100, the number of acres in Mr. Boydson's farm. x + 40 = 140, the number of acres in Mr. Byers's farm. Solve first by written analysis, then orally : 11. Four times my money and $6 more is $50. How much money have I ? 12. $80 is $5 more than twice the cost of a bicycle. Find the cost. 13. Harry's age plus | his age plus 6 years equals 30 years. How old is he ? 14. 21 times the number of books in Henry's library, less 5, equals 70. How many books has he ? 15. James spent \ of his money for a top, | of it for a ball, and had 10 cents remaining. How much money had he at first ? 16. After paying | and ] of my debts, I still owed 125. How much did I owe at first ? 17. A merchant lost \ of his capital, then gained | as much as he had left, and then had $10800. How much was his capital at first ? 18. Robert's money, diminished by | and 4 of itself, equals $1.25. How much money has he ? 19. After a fruit dealer had sold § of his apples, and | of the remainder, he had 12 bushels left. How many bushels had he at first ? 20. If 1 of Wilbur's money is increased by \ of f of his money, the sum will be $54. How much money has he? 234 ANALYSIS 21. A banker gave a f interest in a bank to one son, a \ interest to another son, and the remaining interest, valued at $10000, to his wife. What was the value of the bank ? 22. A lot was sold for $360, which was f of what it cost. Find the cost. 23. Mr. Amos sold his farm for $3300, which was § more than it cost him. Find the cost. 24. A typewriter spends | of his income and saves $400. How much is his income ? 25. A suit of clothes was sold for $18, which was \ less than it cost. Find the cost. 26. A merchant sold apples at $1.80 a barrel, which was •| more than they cost him. How much did they cost per barrel ? 27. There are 1200 pupils in a certain school. The num- ber of boys is f of the number of girls. How many girls are there in the school ? 28. The united ages of Alice and Mary are 28 years ; Alice is f as old as Mary. How old is Alice ? 29. A lady paid $30 for a watch, which was \ more than it cost. Find the cost. 30. A house that cost $1200 was sold for \ more than the cost. How much was gained ? 31. A has 45 cents, which is | more than B has. How much has B ? 32. How much will a two-thirds interest in a store cost, when a four-fifths interest sells for $6000? 33. There are 40 pupils in a school, and \ of them are boys. How many girls are there in the school ? 34. If a man owns § of a mill, and sells § of his interest for $3000, what is the value of the mill ? THE EQUATION 235 35. A lady paid $35 for a cloak. £ <>f tin 1 cost of the cloak was \ of what she paid for other clothing. How much did all cost? 36. A house and lot cost $8000. The lot cost f as much as the house. How much did the lot cost ? 37. A traveler went 30 miles in two days ; the first day he went 1^ times as far as the second. How many miles did he travel the first day ? 38. A sold a watch to B for \ more than it cost him ; B sold it to C for $20, thereby losing 1 of what it cost him. How much did A pay for it ? 39. The difference between two numbers is 36, and the greater is three times the less. What are the numbers ? 40. If to | of Mr. Barnhart's salary you add $40, the sum will be | of his salary. How much is his salary ? 41. A merchant sold a dry goods store, receiving | of the price in cash. He invested | of the sum received in a jew- elry store bought at $900. For how much was the dry goods store sold ? 42. What is the value of f of a ship if | of it is worth $48000? 43. A man invested | of his money in a lot. Had he paid $100 more he would have invested | of his money. Find the cost of the lot. 44. Two merchants had a profit of $9600. After paying ^ of it for rent, they divided the rest so that one received |- as much as the other. How much did each receive ? 45. If Wayne can do a piece of work in 6 days, what part of it can he do in 1 day ? If Ray can do the same work in 4 days, what part of it can lie do in 1 day? 236 ANALYSIS 46. What part can Ray and Wayne both do in 1 day ? 47. If Ray and Wayne can do T 5 ^ of it in 1 day, in how many days can they do the whole work, working together ? 48. If 4 men can do a piece of work in 3 days, how long will it take 1 man to do it ? 49. If one man can do a piece of work in 12 days, how long will it take 2 men to do it ? 50. If 8 men can do a piece of work in 2^ days, how long will it take 5 men to do it ? 51. A jeweler sold a watch for $60, and gained | of the cost. What was the cost of the watch ? 52. A horse, sleigh, and harness cost $220 ; the sleigh cost twice as much as the harness, and the horse cost 4 times as much as the sleigh. Find the cost of each. 53. Ira can do a piece of work in 12 days; Baxter can do it in 16 days. If Baxter's wages are $1.50 a day, how much per day should Ira receive ? 54. A man has three houses which together are worth $5700. The second house is worth twice as much as the first, and the third is worth -^ as much as the other two. How much is the third house worth ? 55. If A can do a piece of work in l-i days, B in 3 days, and C in 4 days, in what time can they do it working together ? Suggestion. — Since A does the whole work in § days, he does f of it in a day ; B does i in a day, and C \ in a day. What part of the work do they do together in a day? How long, then, will it take them to do the whole work together? 56. A man bought three automobiles. The first cost $1500, the second cost 1| times as much as the third, and the third cost twice as much as the first. Find the cost of the second and the third. PERCENTAGE The term per cent means hundredths or by the hundred. The sign for it is %. Thus, five hundredths may be written 1 ^ 5 , .05, 5 per cent, or 5%. These are called equivalents. Percentage is the process of computing by hundredths. It is simply an application of decimal fractions. Write both as a decimal and as a common fraction in its low- est terms each of the following per cents; thus, 10% = .10 = JUL — 1 too iU- l. 5% 5. 371% 9. 120% 13. 75% 2. 6f% 6. 14|% 10. 250% 14. 300% 3. 81% 7. 331% 11. 43% 15. ■ 1% 4. 121% 8. 125% 12. 65 % 16. % % Write the following decimals as fractions and per cents: 17. .05 21. .081 25. 1.50 29. .25 is. .20 22. 2.50 26. Ill 30. .16f 19. .331 23. 1.25 27. .031 31. .371 20. .50 24. 1.20 28. .06f 32. .45 Write the following as decimals and as per cents : 33. 1 37. | 41 1 **■■ 9 45 -1- 12 34. \ 38. 1 42 - lV 46. | 35. \ 39. \ 43 - iV 47. § 36. f 40. 1 44. } 48. | 237 238 PERCENTAGE Memorize the following equivalents : 49. What is 50% of 100 ? 40 ? 10? 2? i? I? 50. What is 16f % of 90? 150? 120? 9? 12? 3? 51. What is 331% of 3000? 2000? 50? 75? 100? 52. What is 12|% f 200? 32 ? 96 ? 4 ? 6 ? 1 ? 53. Whatis37±% of 72? 56? 800 ? 2000 ? 40 ? 8 ? 54. What is 20 % of 400 ? 600 ? l? i? 16? 20? l% = iio 14f% = | 2% = 5 1 o 16|% = i H% = £> 20% =i 4%= A 25% = J 5% = A 33|%=i 6i% = tV 37f% = | 6!% = iV 50% =i 8J% = A 62J % = | n % = it 66}*-} io%= T V 75% =| in*=* 8^% = f 12i% = l 871% =| 55. Find 12|% of 16; of 48 800 ; of 220 ; of 404. of 72; of 96; of 168; of 56. Find 5% of 25; of 50; of 75; of 100; of 125. In each example in 56 we have two terms, a per cent and a number. In each case we are to find 5 % of the number. The number of which we take the y|^ (viz. 25, 50, etc.) is called the base. The number of hundredths (5) to be taken is called the rate, and the number of hundredths actually taken, that is, the answer, is called the percentage. The base is the number on which the percentage is computed. The rate or rate per cent is the number of hundredths. We generally express rate as a decimal. The percentage is the product obtained by taking a certain per cent of the base. The sum or amount is the base plus the percentage. The difference is the base minus the percentage. PERCENTAGE 239 Finding a given per cent of any number. l. What is 20% of 300? Think of 20% of 300 as \ of 300, or 60. Find : 2. 2 % of 100 17. 81 % of 480 3. 5 % of 400 is. 9£ % of 660 4. 10% of 500 19. 111% of 729 5. 20% of 800 20. 121% of 648 6. 50% of 1200 21. 16|%of366 7. 40% of 1000 22. 331% of 333 8. 25 % of 360 23. 621 % f 864 9. 30% of 90 24. 66|% of 724 10. 3% of 420 25. 75% of 968 li. 2 % of 500 26. 871 oj of 568 12. 6% of 150 27. 6% of $60 13. 7% of 800 28. 8% of $560 14. 5% of 440 29. 10% of 350 acres 15. % of 550 30. 30 % of 960 sheep 16. 31% of 900 31. 62i % of 856 Written Work l. What is 7 % of 245 ? 66f % of 300 ? (a) 245 base 7<>/ ° of a number e q uals - 07 of ih ^ J n „ Therefore, 7% of 245 is .07 times 245, —M rate or 17.15. 17.15 percentage -« n a 662% f a number equals § of it. ( 5 > ? of £00 = 200 1 of 300 = 200. ^4. given per cent of any number is found by multiplying the base by the rate. PERCENTAGE Find: 2. 4% of 328 9. 80% of 6.75 3. 9% of 1126 10. 331% of 75 4. 11% of 263 11. 60 % of f 5. 15% of 380 12. 14f% of 105 6. 24% of 165.5 13. 75% of f 7. 38% of $77.50 14. 87^ oj f 168 8. 72 % of 328 15. 331 cj of 336 16. In a school of 400 pupils, 45 % are girls. How many girls are there? how many boys? 17. A clerk who received $50 a month had his wages increased 15%. How much were his wages increased? 18. If iron ore yields 63 % of pure metal to the ton, how much iron is there in 40 tons of ore? 19. A merchant, failing in business, paid 85% of his debts. How much should a creditor receive whose claim is $2850? 20. A bill of goods cost $137.50. How much was gained by selling the goods at a profit of 12% ? 21. Compare 48% of $45 and 45% of $48. 22. I owe a debt of $246.50. If I pay 40% of it at one time, and 50% of the remainder at another time, how much do I still owe ? 23. A n automobile cost $ 3500 and the repairs for 2 years were 10% of the cost. If the automobile was sold at 40% reduction from the cost, find the entire loss. 24. The operating expenses of a factory are 45 % of the sales. If the sales for a year amount to $650450, how much are the operating expenses? PERCENTAGE 241 25. 400 men were employed in a factory at daily wages averaging $1.95. If 100 of these men received 33| % of the entire daily wages, rind their average daily wages. Find the average daily wages of the other 300 men. 26. Three newsboys, John, James, and Henry earned to- gether $550 in a year. John earned 40%, James 60% of the remainder, and Henry what remained. Find how much each earned. 27. 40% of a Western farm containing 600 acres is in wheat, 30% of the remainder in corn, 66|% of the remainder in oats and grass. How many acres are there in each crop, and how much remains not cultivated ? Finding what per cent one number is of another number. 1. What part of $10 is $5? What % of $10 is $5? Think $5 is \ of $10, or 50% of $10. What % of: 2. 20 is 10 ? 8. 3| in. is 1^ in.? 3. 35 ft. is 7 ft.? 9. 25 gal. is 64^ gal.? 4. 100 is 16| ? 10. 60 rods is 20 rods? 5. 500 lb. is 100 lb.? 11. 1 mile is 80 rods? 6. 871 is 121? 12. 1 lb. (av.) is 1 oz. (av.) ? 7. ij y d - is i y cL? 13. 1 dollar is 1 dime ? Written Work l. What per cent of 75 is 15 ? The unknown number is the rate. 15 .j_ 75 = .20 = 20% Since the percentage equals the base multi- plied by the rate, the rate must equal the percentage divided by the base. 15 divided by 75 is .20, or 20%. Or, 15 is ff, or \, or 20%, of 75. Test : 20% of 75 = 15. The rate equals the percentage divided by the base. BAM. COMPL. IR1TH. — 16 242 PERCENTAGE What per cent of : 2. 25 is 10 ? 32 is 12? 65 is 39? 196 is $72? 3. 4. 5. 6. 7. 8. 9. 18 621 a. is 6| A.? -2- is -&? 9 i& 9 * 4isf? 125 yd. is 75 yd.? 10. 4 bu. is 1 pk.? 11. 40 is 6§? 12. 75 is 3.125? 13. | is .125? 14. I|is2i? 15. $18 is $45? 16. 10 qt, is 36 qt.? 17. $3 is 18 cents? Out of 350 words, I spelled 315 correctly. What per cent did I make in spelling? 19. From a farm of 160 acres, 24 acres were sold. What per cent was sold? 20. If a man saves $262.50 out of his salary of $1250, what per cent does he save ? 21. A farmer raised 150 bu. of potatoes from 6 bu. of seed. What per cent of the crop was the seed ? 22. A merchant owes $8750 and his assets are $3675. What per cent of his debts can he pay ? 23. A pupil misspelled 35 words out of 80. What per cent did he spell correctly? 24. .875 is what per cent of .3125? 25. I paid $5.25 for the use of $75. What per cent did I pay? 26. A son, on receiving $5000 from his father, bought a farm for $2750, a store for $1875, and deposited the rest in a bank. What per cent of his inheritance did he deposit ? 27. A house rents for $240 a year. The taxes and in- surance are $30. If the property is valued at $3500, what per cent does the owner realize on the value of the property ? PERCENTAGE 243 Finding the number when a per cent of it is given. 1. If £ of a number is 10, what is the number? If 20% of a number is 10, what is the number ? 2. If 33| % of a man's loss is $300, how much does he lose? 3. If 87 ^ % of a man's gain is $70, how much does he gain ? What is the number of which : 4. 10 is 66| %? li. 40 is 16| % ? 18. $48 is 10 % ? 5. $100 is 331 % ? 12. lOt) is 12.] rr/ ? 19 . #12 is 16f % ? 6. 500 is 50 % ? 13. 300 is 37J % ■> 2 o. 30 is 331 % ? 7. 60 is 75%? 14. 500 is 62^ %? 21. 60 is 66f%? 8. $24 is 40 % ? 15. 700 is 874 % ? 22. 120 is 20 % ? 9. 300 is 60 % ? 16. 500 is 25 % ? 23. $10 is 831 % ? 10. 500 is 50 % ? 17. 600 is 75 % ? 24. $50 is 831

or 300 °- zou x - r - duuu Test . A of 3000 _ 250 The base equals the percentage divided by the rate. PERCENTAGE ?in d the number if : 3. 8% of it is $2.40 8. 4. 12% of it is $3.60 9. 5. 12|% of it is $91 10. 6. 37|% of it is $27 11. 7. 331% of it is $42.50 12. 244 6% of it is $72 32% of it is $3.60 45% of it is $3.60 62-i-% of it is $35.50 12. 87|% of it is $28.28 13. A has $3612, which is 87^% of what B has. How much has B? 87|%of B's = $3612, A's. 14. If $14 is 25% of A's salary, find his salary? 15. After a battle 70 % of a regiment, or 644 men, were left. How many men were there in the regiment at first? 16. I drew from the bank $1500, or 83± % of my deposit. How much was my deposit? 17. If a man rents a house for $752 per year, which is 16% of its value, what is the value of the house? 18. A teacher's expenses are $30 a month, and this amount is 37| % of his salary. How much does he save? 19. The number of pupils in attendance at school in a cer- tain town is 576, which is 96 % of the enrollment. What is the enrollment? Finding a number when the number plus the rate of increase is given. Written Work 1. What number increased by 17 % of itself equals 585? 100% of itself = the number 100% of itself + 17%of itself, or 117% of the number = 585, amount 1% of the number = T i y of 585, or 5 100% of the number = 100 X 5, or 500 Divide the sum by one plus the rate. PERCENTAGE 245 What number increased by : 2. 8% of itself is 324? 6. 50% of itself is 69? 3. 30% of itself is 260? 7. 250% of itself is 105? 4. 37} % of itself is 550? 8. 16| % of itself is 1050? 5. | % of itself is 2011? 9 . 70% of itself is 510? 10. 1 gained 35% by selling an article for $4.05. How much did it cost? 11. A laborer had his wages twice increased 10%. If he now receives $2.42 a day, what were his wages before they were increased? 12. A property sold for $4025, which was an increase of 15% of the cost. How much did the property cost? 13. A receives $1600 salary, which is 60% more than B receives. What salary does B receive? 14. W. H. Richmond bought a jewelry store for a certain sum and increased the stock 27 % of the purchase price. He found that the whole investment amounted to $5969. What was the purchase price of the store ? 15. The land surface of the District of Columbia is 60 square miles, which is 500 % more than the water surface. What is the water surface? Finding a number when the number minus the rate of de- crease is given. Writt&a Work 1. What number diminished by 16% of itself equals 168? 100% of the number = the number 100% of the number — 10% of the number, or 84 % of the number = 168 (difference) 1 % of the number = & of 168, or 2 100% of the number, or the number = 100 x 2, or 200 Divide the difference by one minus the rate. 246 PERCENTAGE What number diminished by : 2. 45% of itself equals 55? 6. 18f % of itself equals 325? 3. 18% of itself equals 246? 7. 23% of itself equals 308? 4. 621% ( ,f itself equals 27? 8. 95% of itself equals 25? 5. 50% of itself equals 22.5? 9. 10% of itself equals 4| ? 10. John has $35, which is 12| % less than his brother has. How much has his brother? 11. Mrs. Lee spent $24 for a coat, which was 33| % less than the cost of a suit. Find the cost of both. 12. After losing 8|% of his money, a man had $352 left. How much had he at first? 13. What number decreased by 35 % of itself equals $1300 ? $520? $6500? 14. A school enrolls 249 boys, which is 17 % less than the number of girls it enrolls. How many pupils are there in the school? 15. A lady when shopping spent $15 of her money for a hat, which was 25 % less than the amount she spent for a coat. How much did she spend for the coat ? 16. The fraction T 9 g is 40 % less than what fraction ? 17. If a certain number is increased 40 % of itself, and this sum is diminished by 50 % of itself, the result is 700. Find the number. 18. I sold two lots for $1200 each ; on one T gained 25% and on the other I lost 25 % . Did I gain or lose and how much? 19. The population of a town in 1906 was 14000, which was 121% less than the population in 1907. What was the population in 1907? REVIEW OF PERCENTAGE 247 REVIEW OF PERCENTAGE 1. What is 50% of 200? 2. .05 is what per cent of .25? 3. .25 is 25% of what number? 4. I sold goods for $8.75. My actual loss was $1.25. What was the cost and the per cent of loss ? 5. A man owns a farm valued at $9600. His annual taxes are -$86.40. How much must he make each year to clear 8 % on the cost of his property ? 6. Ten is what per cent of 20? 15 is 150% of what number ? 7. What per cent of 10 days are 30 days? What per cent of 30 days are 10 days ? 8. | of 72 is how many per cent of 120 ? 9. I paid $7200 for a house, $150 for repairs, and $350 for delinquent taxes. I then sold it for $9000. What per cent did 1 make on my money ? 10. In a business college 20 % of the students study book- keeping, 60 % of them study typewriting, and the remaining 68 students study other courses. How many students are there in the school ? 11. The income from an investment which pays 5.}% is $220. What sum is invested ? 12. Mr. Wilson buys a house and lot for $6400. The average expenses per year for taxes are $80; for insurance $12; and for repairs $24. What must be the annual rent that he may have an income of 6 % net on the original cost of the property? 13. What fraction increased by 35% of itself equals J$? 248 PERCENTAGE 14. If my property sells for $7433.25 and I owe $8745, what per cent of my debts can I pay ? 15. Ten thousand boxes of fruit were sold for $9450, which was 33^% less than the cost. What was the cost? 16. If 12| % of 16| % of a number is 2J, what is the whole number ? 17. A section of land was sold for $4000, which was 25% more than it cost. How much did the land cost per acre ? 18. Express as a per cent : | ; fa ; |. 19. A horse and buggy cost $300. If the cost of the horse was 200% of the cost of the buggy, what was the cost of each ? 20. After paying 70 % of his debts, a man found that $3600 would put him out of debt. Find his original indebtedness. 21. Thirty-five per cent of 640 pounds is 5.6% of how many tons ? 22. A bankrupt sold his property for $4100, which was 18 % less than its real value. If the property had sold for $5250, what per cent above its real value would it have brought ? 23. Two adjacent properties sold for $13200. 75 % of the sale of one equaled 90% of the sale of the other. Find the selling price of each. 24. A certain excavation cost $120. What would be the cost of an excavation 20% wider? 25% deeper? 50% longer? 25. What per cent of 121.92 is 15.24 ? 26. What is the difference between \ % of $7000 and 25 % of $7000? 27. I bought 100 bu. apples at 50^ a bushel, but lost 20 bu. by freezing. At what price per bushel did I sell the remain- der, if my entire loss was 4 % of the cost of the apples ? GAIN AND LOSS 249 28. The net profits of a store in two years were 83483. The profits the second year were 15% more than the first year. How much were the profits the first year ? 29. An executor in settling an estate found 7| % uncol- lectable, 12^ % invested in city lots, 40% in cash, 15% loaned, and the remainder, 810,000, invested in the home. The estate was equally divided among four sons. How much did each receive ? 30. A merchant increased his capital the first year 331%, and the second year 25 % of the capital at the end of the first year. He lost 36% of his original capital the third year, and had $11760 left. Was his original capital increased or decreased, and how much ? 31. An estate was worth $8400. Had it been sold for that amount, the creditors would have received 87| % of their claims, but \ of the estate was sold at 18-| % below its value, and the remainder at 12| % below its value. What per cent of the debts did the estate pay? GAIN AND LOSS 1. A dealer bought goods for 11000, and sold them at a gain of 10%. What was the selling price? What was the gain ? 2. Sugar that cost 5P per pound was sold at a gain of 20 % . What was the gain per pound ? 3. A huckster bought fruit for 820, but found he had to sell it at a loss of 20 %. How much did he lose ? 4. A grocer sold butter that cost 20^ per pound, for 25 $. What part of the cost did he gain ? 5. Books that cost 81.50 wholesale were sold at a gain of 10%. Find the selling price. 250 PERCENTAGE Gain and loss are terms used to designate the profits or the losses in business transactions. The cost is the amount paid for an article ; the selling price is the amount received for it. The gross cost of goods is the original cost increased by what is paid for freight, storage, etc. The net proceeds is the amount received for goods after all charges incident to the sale have been deducted. The per cent of gain or loss is always reckoned on the cost or on the sum invested. Written Work 1. A merchant bought goods for $4500 and sold them at a gain of 12 %. How much did he gain ? Comparative Study $4500, cost The amount bought or sold corresponds to what term . 1 2 rate * n f ercentct ff e ? % 5-1-0 00 m in ^ e S 3 * 11 or l° ss corresponds to what term in Per- ' ° l centage ? Find the gain or loss: 2. $75, gain 20%. 6. $356, gain 21 %. 3. $96, gain 381%. 7 . $132.50, gain 28%. 4. $115, loss 15%. 8. $485.60, loss 5%. 5. $227, loss 19%. 9. $880.80, gain 12 1 %. 10. How much is gained by selling a property that cost $3250 at a profit of 8% ? 11. A grocer bought 120 dozen eggs at 18 cents a dozen, and sold them at a profit of 11^%. What was his gain? 12. A real estate dealer bought three lots for $1500, $1800, and $2000 respectively. He sold the first at an advance of 8%, the second at an advance of 10%, and the third at an advance of 12%. What was his gain ? GAIN AND LOSS 251 Finding the gain or loss per cent. Written Work .1. I bought a piece of property for $2500 and sold it for $2750. Find the gain or loss. $■2730 - $2500= $ 250, gain $250 h- $2500 = .10, or 10%, gain Find the gain or loss per cent when the : 2. Cost is $100 and the selling price 1 105. 3. Cost is 1175 and the selling price $210. 4. Cost is $240 and the selling price $280.40. 5. Cost is $476.80 and the selling price $309.92. 6. Cost is $775.50 and the selling price $1008.15. 7. If flour is bought for $4.50 a barrel and sold for $6 a barrel, what per cent is gained ? 8. A farm was bought for $3500 and sold for $4200. What was the gain per cent ? 9. If hats are bought for $27 a dozen and retailed at $2.75 each, what per cent gain is realized ? 10. A grain dealer bought 500 bushels of wheat at 84 cents a bushel, 000 bushels at 80 cents a bushel, and sold it all at 82 cents a bushel. What per cent did he gain or lose ? Finding the cost. Written Work l. What was the cost of a house if the owner, by selling it at an advance of 25%, gained $900? $900 = gain $ 900 -- .25 = $3600, cost 252 PERCENTAGE Find the cost when: 2. 5% loss is $25. 6. J% gain is $16.80. 3. 12± % gain is $37 J. 7. 44 % loss is $1100. 4. 150% gain is $750. 8. 35% gain is $12.25. 5. I % loss is $12±. 9. 16| % loss is $ 37.50 10. A dealer sold a buggy at 25% gain, and received $90 for it. How much did the buggy cost? Selling price = £ of the cost $90 = | of the cost Cost = | of $90, or $72 Find the cost when selling price at: 11. 10 % gain is $ 220. 14. 37-| % loss is $600. 12. 18 % loss is $492. 15. 16| % gain is $1190. 13. 121 % g a i n i s $990. i 6 . 114% gain is $1284. 17. A merchant, after losing 25% of his goods by fire, had $8700 remaining. What was the value of his goods at first? 18. An attorney turned over to his client $1125 after re- taining 10% for his services. What amount of money did the attorney collect? 19. If 8% is lost by selling an article for $1.15, how much did it cost? 20. If dress goods sold at $1.50 a yard yielded a profit of 20 %, how much did the goods cost per yard? 21. Mr. Rice sold his farm at a gain of 5 % and received $ 4200 for it. What would the gain per cent have been had he sold it for $4400? $4200 - 1.05 = $4000, cost $4400 - $4000 = $400, gain - $4000 = .10, or 10%, gain REVIEW PROBLEMS 253 22. I sold a horse for -1240 and thereby lost 20%. What selling price would have given me a gain of 20%? 23. If a merchant loses 10% by selling goods at 45 cents a yard, for what should they have been sold to gain 20 %? 24. A piece of cloth was sold for $32, which was at a loss of 20%. What would have been the loss percent had it been sold for $39? REVIEW PROBLEMS 1. Find the selling price of goods on which there is a loss of 2}, % which amounts to $106.25. 2. How many per cent above cost must an article be marked in order to make 25% after a discount of 20% has been given? 3. I sold | of an acre of land for what £ of it cost. What per cent did I gain or lose? 4. At what price must an article that cost $60 be marked, so that after deducting 25 % from the marked price, a profit of 10% may be realized? 5. A sold a property to B and gained 20 % ; B sold it to C and lost 16| % ; C sold it to D for $2800 and gained 12%. How much did A receive for the property? 6. Find the value of a property that increased annually 4 % on the previous year's value and after three increases was worth $11,248.64. 7. A note was bought for 5% less than its face and sold for 2% more than its face. If $40.25 was gained, what was the value of the note? 8. I bought a lot for $1600, which was 20% less than its real value, and sold it for 20% more than its real value. How much did I gain? 254 PERCENTAGE 9. If 80% of a car load of wheat is sold for what the car load costs, what per cent is gained ? 10. A company sells sewing machines to a wholesaler at 33^% profit, the wholesaler sells to the retailer at 12i% profit, and the retailer sells to the trade at a profit of 20 % . Find the cost of a machine to the company when the retailer's profit on the machine is $9. 11. A merchant marked his goods so as to gain 20 % ■ By giving credit, 5 % of his sales were imcollectable. His gain was $1050. What was the value of his goods? 12. A bankrupt can pay only 80 cents on the dollar. What will be the gain or loss per cent to the retailer who sold him a buggy at 30 % profit ? 13. A merchant marked a lot of goods costing $ 12.000, at 25 % above cost, but sold them at 10 % less than the marked price. What per cent did he gain ? 14. A merchant sold silk at 45 cents a yard above cost, and gained 20%. What was the selling price per yard? 15. Mr. McKay bought 150 shares of W. Va. Lumber Co. stock at $ 140 a share, and 200 shares of telephone stock at $120 a share. He sold the lumber stock at a loss of 10%. For how much did he sell the telephone stock per share, if he gained 10% on the transaction? 16. A and B invested an equal amount of money in busi- ness ; A gained $2000 on his investment, and B lost $1500; B's money was then 65% of A's. How much money did each invest? 17. A bought a horse and sold it to B at a gain of 10% ; B sold it to C and gained 10%. If C paid 131.50 more for the horse than A, how much did the horse cost A ? COMMISSION AND BROKERAGE 255 COMMISSION AND BROKERAGE A person who buys or sells goods or transacts business for another is called an agent, collector, commission merchant, or commission broker, according to the nature of the business transacted. A commission merchant actually receives the goods which he buys and sells for another. A commission broker simply makes the contract between the buyer and the seller for whatever is to be bought or sold, the goods being delivered directly from the seller to the buyer. The commission or brokerage is a certain per cent of the amount of money involved in the transaction. A commission merchant gets a certain per cent on the amount of his sales; a collector gets a certain per cent on the amount collected ; a broker gets a certain per cent of the cost or the selling price. The net proceeds is the amount left after commission and all other charges have been paid. The one who sends the merchandise to be sold is the prin- cipal, the shipper, or the consignor. Selling or collecting through an agent. Written Work l. A Pittsburg commission house sold 275 barrels of apples at $4 per barrel, on a commission of 10%. The freight from Rochester, New York, was $67.50 and the drayage was $11.25. Find the commission and the net proceeds of the sale. Amount of sale 275 bbl. at $ 4 $1100.00 Commission I0%of $1100 $110. Freight 07.50 Drayage 11 ■-•"' 188.75 188.75 Net Proceeds $ 911-25 256 PERCENTAGE Comparative Study The amount bought, sold, or collected in Commission corresponds to what term in Percentage? The rate in Commission corresponds to what term in Percentage ? The commission or brokerage in Commission corresponds to what term in Percentage ? The net proceeds in Commission correspond to what term in Percent- age? 2. A real estate agent sold four lots for $ 250, $ 325, $ 395, and $405 respectively. How much was his commission at 5%? 3. A commission merchant sold 320 barrels of apples at $3.25 a barrel and 16 barrels of sweet potatoes at $4.80 a barrel. Find his commission at 7%. 4. A lawyer collected an account of $385 for a client, charging 5%. How much should he remit? 5. A cotton broker sold 200 bales of cotton of 225 pounds each, at 12 J ^ per pound, charging a commission of 2| %. Find the net proceeds. 6. A lawyer collects 80 % of an account of $ 1125. Find his rate of commission if he charges $ 9. 7. A real estate agent sold 475 acres of coal at $ 85 an acre. His commission was $1211.25. Find the per cent charged. 8. My agent bought 180 barrels of flour at $4.80 per bar- rel. He paid $50 freight and $6 storage. I sent him $ 937.28. What was his rate of commission? 9. An attorney succeeded in collecting 90 % of the amount of a consignment of cotton sold for $6700. He remitted $5874, retaining the balance to pay attorney's fees and $5.25 freight charges. What per cent did he charge for collecting? COMMISSION AND BROKERAGE 257 10. My Chicago broker sells for me 82560 bushels of wheat at 83£ ^ per bushel, charging \ # per bushel brokerage. Find the amount remitted to me. Find the selling price if a commission of: li. 2% =$3.05 14. 41% = $56.25 17. 3f % =$36.25 12. i%= 3.12 15. ?>{% = 75.60 18. 1% = 7.59 13. \%= 1.54 16. 1|%= 61.50 19. 4 mills = 79.00 20. An agent for the Diamond Pneumatic Tool Company received 20% on the sales of 13 pneumatic drills averaging $737.50 each. If his expenses for traveling and freight on machines were $697.50, find his net profits. 21. James Amidon & Co. offered one of their salesmen $2400 per year, $1800 for traveling expenses, and 2% on all sales over $40000; or 6% on all sales if he paid his own expenses. He chose the former, and sold $72000 worth of goods in the year. Did he gain or lose, and how much, by accepting the first offer? 22. For selling a house, a real estate agent received $ 96. 25, which included $7.65 for advertising, and $27.90 for re- pairs. Find the rate of commission, if the house was sold for $3035. Buying or investing through an agent. Written Work 1. An architect charged 2} 2 % for plans and specifications and 2^ % for superintending construction. His commission amounted to $ 810. How much did the building cost ? 2. I telegraphed my agent at Chicago to buy me 10000 bushels of wheat at 79 cents or less. lie bought at 77| cents and charged me | cent per bushel brokerage. Find the amount of the check I should send him. HAM. COM PL. Alt I'l'll. — 17 258 PERCENTAGE 3. An agent's investments in 1905 were $ 80500 and in 1906 $ 87650. His commissions for 1906 were $ 143 more than for 1905. Find his commission for each year. 4. My broker in New Orleans buys 50000 pounds -of cot- ton at 11|^ per pound. His commission is | % and freight, storage, and cartage amount to $ 95.80. How much should I remit ? 5. A house and lot bought for $8500 was sold afterwards for 80 % of the purchase price. If the agent received 2% on each transaction, find his commission. 6. An agent buys a property for his principal for $36790 at 2% commission. The owner puts $674.20 in repairs and afterwards sells the property through the same agent for $43000, at 2% commission. Find the agent's commission, and the principal's rate per cent of gain. 7. An attorney invested for his client $8750 in a mort- gage. The owner of the property paid the attorney 2 % commission for getting the money and $35.75 for examining title and for docket fees. How much did it cost the mort- gager to secure the money ? REVIEW Find the values of the missing terms : 1. Gross Sales EXPENSES Commission Rate of Commission Net Proceeds $675 $11.25 ( ) 8% ( ) 2. $550 $4.00 $22.00 ( ) ( ) 3. ( ) $119.00 ( ) 20% $681 4. $560 00 $56.00 ( ) ( ) 5. ( ) 00 $48.00 ( ) $552 6. $1000 00 ( ) 4% ( ) 7. ( ) ( ) $134.00 2% $6566 8. $2500 $46.00 $50.00 ( ) ( ) INSURANCE 259 9. A fruit grower ships to his commission merchant 600 barrels of apples, which are sold at $ 3.50 per barrel. The agent deducts 143.90 freight charges, $27.75 cartage, 12 ^ per barrel for cold storage, and 5 % commission. Find the amount remitted. 10. My agent sends me a bill for $37800, which includes the cost of some land bought at $100 an acre and his com- mission of 5%. Find the number of acres purchased. 11. My purchasing agent in Chicago sends me a bill for $6776.60, covering cost of mining machinery, commission of 2%, and $65 for extra expenses. Find his commission and the cost of the machinery. 12. My agent retains %\°fo commission, pays freight charges of $16.80, storage and drayage of $9.75, and remits to me $1594.65. Find the amount of gross sales. 13. An investment broker in Denver receives a draft for $ 23935. This includes a commission of 2 % for investing, and an allowance of $ 175 for traveling expenses. Find the amount invested. INSURANCE Insurance is security against loss or damage. A merchant owns a store, uninsured, valued at $ 5000. If it is burned, who will bear the loss ? An insurance company agrees to insure the store for $4000 at 1 % annually. In case the store is totally destroyed by fire, how much is the company expected to pay to the merchant? How much must the merchant pay annually to the company to guarantee this loss? The policy is a written contract between the person insured and the insurance company. The premium is the sum paid for the insurance. The rate is a specified number of cents or dollars per $100 of insurance, or a certain per cent of the sum insured. The term is usually a year or a period of years. Short rates arc rates charged when the term is less than one year. ofjO PERCENTAGE Property Insurance The principal kinds of property insurance are fire insur- ance and marine insurance. Other forms are burglar insur- ance, insurance against bad debts, etc. l. A frame dwelling with a tin roof is insured for $2800 for one year at 1 %. Find the premium. Written Work Comparative Study The amount insured corresponds to what $2800, amount insured term in Percentage ? ^-. , The rate corresponds to what term in ' ' Perppntfuip ? Percentage t $28.00, premium -pj ie premium corresponds to what term in Percentage f 2. A brick house is insured for ¥4000 at 60^ on the $100. Find the rate of premium and the annual premium. 3. If the three-year rate is twice the rate for one year, find the cost of insuring a brick dwelling for $6500 for 3 years when the annual rate is 45^ per $100. 4. A store building is insured for $8500 and the annual premium is $212.50. Find the rate per cent of premium and the annual cost per $100 of insurance. 5. A school board pays annually $45 for $6000 of fire pro- tection on a school building. Find the rate of premium. 6. The premium on a dwelling insured for $5500 is $38.50 for three years. Find the average rate for a year. 7. If the premium on a plate-glass policy is $9.50 and the rate is |%, rind the face of the policy. 8. Mr. Lawrence wrote a check for $31.50 to pay the in- surance on his dwelling for 3 years. If the house cost $2400 and was insured for I of its value, rind the rate for the term. INSURANCE Jtil 9. A farmer insured his house for $2700 at 1]%. his bam for $1200 at \%, and his furniture for $900 at 1%. What premium did he pay ? 10. A drug store is insured for | of its value at 2%. What is the value of the store if the premium is #192 ? 11. The premium on 8000 bushels of wheat, valued at 90^ per bushel and insured at \ of its value, is $57.60. Find the rate of insurance. 12. A jewelry store is insured for $20000 and its con- tents for $27000. The premium is $705. What is the rate of insurance ? 13. A clothier insured his stock of goods, valued at $12000, for 1 year at 1|%. At the end of 6 months he surrendered his policy. If the " short rate " for G months was 90^ per $100, how much premium was returned? 14. A farmer insured his buildings for $3500 at \\% for a term of 3 years. After he had paid the premium for 4 terms the buildings were totally destroyed by fire. What was the farmer's loss ? the company's loss ? 15. A vessel worth $27000 is insured for § of its value at 3|%. In case of shipwreck, what is the company's loss ? What is the owner's loss ? 16. How much insurance, at |%, can be placed on a build- ing for $42 ? 17. A business block valued at $300000 was insured in 4 different companies, the rate of each being 1%. The first company took $50000 ; the second, $60000 ; the third, $90000 ; and the fourth the remainder. After the premiums had been paid four times, the block was damaged by fire to the amount of $120000. What was the loss of each com- pany ? 262 PERCENTAGE Personal Insurance The principal kinds of personal insurance are life insurance and accident insurance. Kinds of life insurance policies : 1. A life policy is one that guarantees a fixed sum of money on the death of the insured. The premiums on a life policy run for life. 2. A life policy with a twenty-year settlement is one that guarantees, after twenty annual payments have been made, either a cash surrender value, or a paid-up policy, payable at death. 3. An endowment policy is one in which the face of the policy and the profits on the premiums are guaranteed to the insured if living at the end of a specified time, or to his estate if his death occurs within the time. 4. A term policy is one in which the face of the policy is paid, provid- ing the insured dies within the time the policy runs. Otherwise nothing is paid. Most insurance companies pay dividends on the premiums already paid, thus lessening the amount of the annual premium. The premium is always so much on $ 1000 of insurance; thus, a premium of f 23.40 means $23.40 on $1000. The age of the insured is always reckoned according to the age at his nearest birthday. This table shows the annual premiums for each $ 1000 insurance in a leading insurance company. Age Ordinary Life 20-Payment Life •20-Year Endowment •20-Year Term 20 $18.95 $27.64 $49.35 $ 12.48 25 21.14 30.05 45.98 13.34 30 23.96 32.98 50.74 14.61 35 27.63 36.62 51.88 16.70 40 32.48 41.18 53.69 20.15 45 39.02 47.09 56.70 25.85 50 47.79 54.98 61.75 35.00 55 60.33 65.81 70.02 60 77.48 81.09 INSURANCE 263 Dividends vary according to the number of premiums that have been paid. It is fair to estimate that the dividends on an ordinary life policy after 20 annual premiums have been paid, will amount to about 25% of the sum of the annual premiums. Written Work Rates as given in the table on page 262. 1. What is the premium on an ordinary life policy of $5000 at the age of 30? 2. A man at the age of 25 takes out a $ 2000 ordinary life policy. If he dies after paying 16 premiums, what per cent of the face of the policy has been paid in premiums ? 3. If, in example 2, the insured had taken a 20-payment life policy, what per cent of the face of the policy would have been paid in premiums ? 4. A young man at the age of 20 took out a 20-payment life policy for 12000. The dividends at the end of 20 years amounted to $142 per thousand. What was the net cost of this insurance at the expiration of this policy, if the interest on the premiums is not considered ? 5. What is the premium on a 20-year endowment policy for $5000 at the age of 40 ? 6. The first annual premium on a 20-year endowment policy for $8000 amounts to $453.60. What is the age of the insured ? 7. If a man 25 years old takes out a 20-term policy for $ 3000, how much will he have paid for his insurance at the close of the term ? 8. A man at the age of 25 took out a 20-payment life policy for $1000. His dividends for the 20 years amounted to $150.45. What was the amount of his premiums, less the dividend? 264 PERCENTAGE COMMERCIAL DISCOUNT 1. I owe a bill of $50 clue in 60 days. As the creditor needs the money he offers to take $40 if I pay at once. What per cent is the reduction ? On what is the reduction reckoned ? 2. A catalogue lists goods at 11.00, -$.50, $.25, subject to a discount of 20 %. Find the net price of each article. 3. In the above examples what numbers correspond to the base in Percentage ? The fixed or list price of an article or the amount of an obligation is always considered the base. Commercial discount is a reduction from the fixed or list price of an article, or from the amount of a bill or obligation. 1. Trade discounts are reductions from the fixed or list price of an article at the time of sale. 2. Time discounts are reductions from a bill or other obligation for payment within a certain time. 3. Cash discounts are reductions made for the immediate payment of a bill of goods sold on time. The net price is the price after trade discounts have been deducted. Written Work 1. Neckties listed at $6.00 per dozen are sold at 50 % dis- count. Find the net price. What is the kind of discount ? 2. An agent buys $100 worth of goods on 60 days' time, or 5% off, if paid immediately. What kind of discounts is he offered ? 3. A merchant offers $100 worth of goods for $90. What is the kind of discount ? What is the per cent of discount ? 4. I pay cash for a bill of goods sold on 60 days' time, and thereby get 10% discount, or $12.50. Find the amount of the purchase. Why is this a cash discount ? COMMERCIAL DISCOUNT 265 5. A suit marked $40 is offered for $28. Find the per cent of discount. What is the kind of discount ? 6. A merchant buys boys' suits at $60 per dozen list price, less 20 %. What is the kind of discount ? 7. A dealer sells $75 worth of goods at 5 % discount, if paid within 60 days, or 10 %, if paid in cash. Explain the different kinds of discount, and the amount saved by paying cash. 8. Why does a cash discount always cut out a time dis- count ? What discount is made without reference to time ? Business houses print on their billheads their terms of credit. For example: 1. "Terms: 60 days net; 3% off 10 da," 2. "Terms: 00 days net; 60 days 2%; 10days5%." 3. "Terms: 30 days net ; cash 5%, etc." 9. Johnston Bros., Toledo, O., purchase of Amidon & Co., Chicago, $300 worth of merchandise. Terms: 60 da. net; 5% off 10 da. Find the discount if paid within 10 days. 10. I buy $600 worth of goods. Terms: 30 da. net; 5 % off for cash. Find the amount I save by paying cash. 11. Explain the meaning of a bill head which reads as follows: "Terms: 60da.net; 10% cash." 12. $200 worth of goods are bought Aug. 10, 1905. Terms: 60 da. net ; 5 % off 30 da. ; 10 % off 10 da. Find the amount saved by payment Sept, 5. 13. Jamison & Son, Baltimore, Md., bought $1500 worth of merchandise from Brown & Co., Philadelphia. Terms: 90 days net; 2% off 60 da.; 5% off 30 da.; 10% off cash.. What was the amount of the bill if cash was paid ? 14. Which is the best discount on a bill of goods for $200 and how much: 10% for cash, 5% for 60 days, or 2% for 90 days? Explain why. 266 PERCENTAGE Successive Trade Discounts Wholesale merchants, publishers, and manufacturers usually have fixed price lists for their goods from which the retailer gets a certain trade discount. If a reduction from these prices is to be made, an extra discount is taken from the former discount price. 1. A music dealer buys an organ on 60 days' time, list price $100 at 40 %, 20 % off. Find the net price. .40 X $100= $40, 1st discount Observe that, when $100 - $40 = $60, 1st discount price more than one discount -.„. ,»..« rti t is given, each succes- .20x$60 =$12, 2d discount . * ' . , , ' " ^ w v ' sive discount is reckoned $60 -$12 =$48, net COSt on the last discount price. 2. Providing the music dealer gets a further discount of 10 % for cash, find the price of the organ. 3. On what discount price was the last discount reckoned? N OTE . — The trade discount is first deducted ; then the cash discount is taken from the remainder. Find the net cost of articles listed at : 4. $90, discount 30%, 10%. 5. $120, discount 25 %, 20%. 6. $240, discount 10%, 331%. 7. $ 35, discount 20 % , 5 % . 8. $12.50, discount 20%, 20%. 9. $348, discount 20%, 10%, 5%. 10. $100, discount 10 %, 5 %, 2 %. 11. $425, discount 37^%, 16%. 12. $400, discount 20%, 10%, 5%. 13. What is the net price of a bill of goods for $5700 after discounts of 25 %, 20 %, and 5 % are allowed ? COMMERCIAL DISCOUNT 26 >i 14. The amount of discounts at 20%, 5% off, is $42. What is the list price of the bill ? Suggestiox. — The total discount expressed in per cent is 20%+ (5% of 80%),or 24 %. $42 is 24 % of what number ? 15. What is the cost of a bill of farming implements listed at $480, discounts 50%, 5%, and freight $3.60 ? 16. Goods marked $50 were bought at 10 % trade discount, 2% off for cash. If sold at $55, what was the rate of gain ? 17. A dealer bought 50 gross of buttons for 25%, 10 % ? 5% off and sold them for $35.91, making a profit of 12%. What was the list price of the buttons per gross ? 18. A grocer is offered a discount of 10 % from one firm on a bill of goods of $1000, and two successive discounts of 5 % by another firm on the same bill. Which is the better offer and how much ? 19. What is the difference between a discount of 15 %, 5 %, on a bill of $2000 and a discount of 5 %, 15%, on the same bill ? Show that the same result is obtained from any num- ber of discounts on a bill in whatever order they are taken. 20. Three firms bid on the glass for a building as follows : (1) $2000, 00%, 20%, off. (2) $2100, 70%, 10%. off. (3) $2400, 80%, 10%, off. Which offer is the best and how much ? 21. A jobber buys merchandise listed at $1500, at 20%, 15 % off, and sells at 15 %, 1 %, 5 % off the list price. Find his profits. 22. I purchased $2000 worth of goods trade discount 20%, cash 5%. I pay cash plus $5.95 freight. What is the entire cost of the goods ? 23. Find a single discount equal to successive discounts of 40%, 20%, 10%. 268 PERCENTAGE COMMERCIAL BILLS 1. On June 10, 1906, James Boydson, Sandusky, O., bought of J. M. Gordon & Co., Cleveland, O., 35 dozen bronze locks at $5.50 per dozen, less 20 %, 10%. Terms: 30 days net; 5% off 10 days. Form of Bill Cleveland. O., fane /O, /-&C 35 J&oa. fduyyvit, Lo-@£& @$5. 60 X&o-a* 5/ to &aoA, t&QA> 5% f. 711. gcyutcni V &0-. 53 50 aa i Comparative Study $600 = principal , m , 1. The principal corresponds to what term in Percentage ? $36.00 =illt. for 1 yr. o. The rate corresponds to what 2f =time in years term in Percentage f $ 27 =int. for 9 mo. <*. The interest corresponds to what 72 =int. for 2 yr. term in Percenta ff e? 1, nn : — : — ;; tt- 1 t. 4. What term is found in interest $99 =int. tor 2 yr. 9 mo. ,, , . , e , . n , 9 J that is not tound in Percentage ? The interest equals the product of the principal, the rate, and the time expressed in years. Find the interest on : 2. $250 for 21 yr. at 6%. 7. $320 for 5 yr. at 5J%. 3. $708 for 4 yr. at 7%. 8. $600 for 3^ yr. at 6%. 4. $650 for If yr. at 4%. 9. $200 for 11 mo. at 6%. 5. $800 for 10 mo. at 6 %. 10. $570 for If yr. at 4%. 6. $260 for If yr. at 4* %. 11. $290 for 3 yr. at 6|%. Method by Aliquot Parts for Years, Months, and Days Written Work 1. Find the amount of $ 960 for 3 yr. 7 mo. 18 da. at 5 % Principal = I960 Rate = .05 Int. for 1 yr. = $ 48.00 Int. for 3 yr. = 3 x $48, or $144.00 Int. for 6 mo. = * of $ 48, or 24.00 Int. for 1 mo. = j\ of $48, or 4.00 Int. for 15 da. = \ of $4.00, or 2.00 Int. for 3 da. = T \ of $4.00, or .40 Int. for 3 yr. 7 mo. 18 da. = $174.40 Principal = 960.00 Amount = $1134.40. SIMPLE INTEREST 277 Find the amount of : 2. $1400 for 3 yr. 3 mo. 12 da. at 6%. 3. $975 for 5 yr. 8 mo. 24 da. at 4* %. 4. $360 for 4 yr. 5 mo. 10 da. at 5.] %. 5. $480 for 2 yr. 9 mo. 15 da. at 7%. 6. $2700 for 6 yr. 6 mo. 20 da. at o%. 7. $1040 for 4 yr. 9 mo. 18 da. at 8 % . 8. $176.45 for 3 yr. 7 mo. 25 da. at C %. 9. $1840 for 5 yr. 4 mo. 27 da. at 4|%. 10. 81875 for 7 yr. 2 mo. 6 da. at 8%. 11. 8200 for 2 yr. 1 mo. 15 da. at 6%. 12. $1200 for 1 yr. 8 mo. 10 da. at 7%. 13. $97.30 for 3 yr. 3 mo. 3 da. at 8%. 14. 83500 for 2 yr. 7 mo. 24 da. at ('.%. Find the interest on : 15. 81575 at 5% from Jan. 3, 1907 to Sept. 5, 1909. 16. 81790.80 at 4* % from Sept. 8, 1906 to Dec. 10, 1910. 17. $2005 at 3.] % from June 12, 1906 to Oct. 8, 1909. 18. $4000 at 4| % for 5 yr. 8 mo. 21 da. 19. $4670 at 5| % from Oct. 10, 1906 to June 5, 1907. 20. $2890 at 6 1 % from April 3, 1905 to Oct. 1, 1908. 21. $290.75 at 6% from May 8, 1905 to Sept. 19, 1908. 22. 89S0.60 at 5' % from Aug. 7, 1906 to June 7, 1912. 23. $759.40 at 1% from May 9, 1907 to June 10, 1909. 24. $800.50 at 4 T \, % for 1 yr. 7 mo. 20 da. 25. 8200.80 at b\% from May 3, 1906 to Sept. 5, 1911. 278 INTEREST The Sixty Day Six Per Cent Method Since the interest on $100 at 6% for 1 yr. is $6, the in- terest for 60 da. Q of a year) = jt of $6, or $1. How much is the interest at 6 % for 60 days on : 1. 1200 3. $300 5. $150 7. $125 2. $ 50 4. $ 6 6. $250 8. $120 Does the interest in each problem equal j^ of the princi- pal? Written Work 1. Find the interest at 6% on $650 for 90 days. Interest for 60 days (2 mo.) = $6.50 Interest for 30 days (1 mo.) = $3.25 Interest for 90 days (3 mo.) = $9.75 The interest of any principal for 60 da. (2 mo.) at 6 % is found by moving the decimal point in the principal two places to the left. Find the interest at 6 % on : 2. $275 for 2 mo. 5. $198 for 3 mo. 3. $ 95 for 3 mo. 6. $475 for 3 mo. 4. $172 for 3 mo. 7. $280 for 3 mo. 8. Find the interest at 6% on $345 for 8 mo. The interest for 2 mo. = $3.45, or T ^ 5 of the principal The interest for 8 mo. = 4 x $3.45 = $13.80 Find the interest at 6% on: 9. $125 for 120 da. 13. $805 for 75 da. 10. $190 for 4 mo. 14. $425 for 2 1 mo. 11. $325.50 for 8 mo. 15. $500 for 30 da. 12. $ 62.50 for 240 da. 16. $280 for 45 da. SIMPLE INTEREST 279 17. $ 600 for 6 mo. 18. $350 for 9 mo. (8 mo. + 1 mo). 19. $450 for 19 mo. (18 mo. + 1 mo.). Find the interest at 6 :J.2>>l Find the interest at 6 $> on : 41. $ 500 for 6 mo. 15 da. 49. $175.50 for 105 da. 42. $250 for 8 mo. 20 da. 50. $ 150 for 9 mo. 12 da. 43. $360 for 5 mo. 10 da. 51. $387.50 for 6 mo. 25 da. 44. $475 for 90 da. 52. $125.50 for 10 mo. 21 da. 45. $900 for 6 mo. 25 da. 53. $345.50 for 6 mo. 15 da. 46. $125 for 4 mo. 19 da. 54. $ 755 for 1 yr. 9 mo. 6 da. 47. $ 325 for 6 mo. 23 da. 55. $ 544 for 5 yr. 3 mo. 5 da. 48. $25.50 for 3 mo. 29 da. 56. 8 80. SO for 2 yr. 15 da. 280 INTEREST 57. $5175 for 4 yr. 10 mo. (60 mo. — 2 mo.). 58. $640 for 3 yr. 2 mo. (40 mo. — 2 mo.). 59. $ 1240.60 for 2 yr. 9 mo. 15 da. From the interest on any principal at 6 %, the interest at other rates may be found by adding or subtracting aliquot parts of the interest at 6 % ', thus, 4 % = 6 % - i of 6 % 7-1 % = 6 % + I of 6 % 41 % = 6 % - J of 6 % 8 % = 6 % + £ of 6 % 5 % = 6 % - 1 of 6 % 9 % = 6 % + 1 of 6 % 7 cj = 6 % + \ of 6 % 10 % = i of 6 % x 10 60. Find the interest at 5 % on $ 360 from May 1, 1905 to March 16, 1907. Time, 1 yr. 10 mo. and 15 da. = 22| mo. Principal = $380.00 Interest of $ 360 for 20 mo. at 6 % = $ 36.00 Interest of $360 for 2 mo. at 6 % = 3.60 Interest of $ 360 for j mo. at 6 % = .90 Interest of $360 for 22$ mo. at 6 % = $40.50 Less interest of $ 360 for 22} mo. at 1 % = 6.75 Interest of $360 for 22 h mo. at 5% = $33.75 Find the interest on : 61. $216 from July 25, 1891 to Sept. 10, 1893, at 6 %. 62. $348 from Jan. 16, 1893 to Feb. 4, 1896, at 7 %. 63. $ 650.40 from April 19, 1903 to March 4, 1906, at 4 %. 64. $1200 from Nov. 2, 1900 to Oct. 19, 1903, at 4| % 65. $1800 from Aug. 25, 1902 to June 1, 1906, at 4| %. 66. $476.25 from Aug. 19, 1902 to June 1, 1906, at 8 %. 67. $1600 from Sept. 25, 1902 to May 18, 1906, at 4| %. 68. $164.88 from July 3, 1903 to Dec. 31, 1905, at 5 %. SIMPLE INTEREST 281 The One Dollar Six Per Cent Method The interest on $1 for 30 da. (1 mo.) = $.005. The interest on $ 1 for 1 da. (^ of $.005) = $ .0001. • Change the time to months and days. Since the interest on $1 for 1 mo. is | of a cent and for 1 da. ^ of a mill, the interest on one dollar ivill be ^ as many cents as there are months and ^ as many mills as there are days. Midtiply the result by a number equal to the number of dollars in the principal. Written Work 1. What is the interest on $240.60 for 2 jr. 3 mo. 13 da. at 6%? 2 yr. 3 mo. = 27 mo. Interest on $1 at 6 % for 27 mo. = $.135 Interest on $ 1 at 6 % for 13 da. = $.002$ Interest on $1 at 6% for 2 yr. 3 mo. 13 da. = $.137$ Interest on $240.60 = 240.60 x $.137 f, or $33.00 Find the interest at 6% on : 2. $ 7450 for 93 da. 14. $ 8790 for 5 mo. 3. $ 8400 for 65 da. 15. $ 8250 for 6 mo. 4. $ 9800 for 40 da. 16. $ 150 for 3 mo. 6 da. 5. $ 8440 for 72 da. 17. $ 180 for 5 mo. 9 da. 6. 8 5500 for 5 da. 18. #195 for 6 mo. 10 da. 7. $ 0750 for 8 da. 19. $ 250 for 8 mo. 12 da. 8. 1 4765 for 25 da. 20. $ 340 for 2 yr. 4 mo. 9. 8 6245 for 110 da. 21. $275 for 3 yr. 8 mo. 10. $8425 for 52 da. 22. $450 for 1 yr. 5 mo. 11. $ 5150 for 3 mo. 23. $ 675 for 3 yr. 7 mo. 12. * 8465 for 4 mo. 24. $64.60 for 2 yr. 9 rao. 13. 8 9640 for 7 1110. 25. $78.40 for 4 yr. 3 mo. 282 INTEREST Find the interest : 26. At 5% on $ 237.50 from Jan. 3, 1906 to Sept. 11, 1908. 27. At 7% on $309.75 from May 5, 1905 to Jan. 12, 1907. 28. At 7| % on $ 7500 from June 12, 1907 to Nov. 1, 1909. 29. At 61% on $ 6225 from Oct. 11, 1907 to Mch. 1, 1910. 30. At A\% on $750 from Feb. 12, 1907 to Aug. 9, 1911. 31. At 8% on $2900 from July 1, 1906 to May 10, 1910. 32. At 4% on $3675 from June 4, 1907 to Apr. 1, 1910. 33. At 3% on $ 290.80 from Nov. 12, 1907 to Apr. 10, 1912. 34. At 5% on $875 from June 11, 1907 to Mch. 11, 1912. 35. At 5|% on $8000 from Apr. 1, 1907 to May 9, 1912. Solve the following problems by both the six per cent methods and compare results as to time and accuracy. Find the interest from the following conditions: Prin. Time Rate Prin. Time Rate 36. $350 105 da. 1% 39. $129 23 mo. 9% 37. $685 4 mo. 8% 40. $750 13 mo. n% 38. $850 87 da, 6% 41. $492 97 da. H% 42. Fin d the amount of $1275.80 for 1 yr . 7 mo. 24 da. at 1%. 43. Find the interest of $4780 from April 1, 1907 to Sept. 18, 1909, at 6|%. 44. On the 16th of September, 1907, I borrowed $3600 at 8 %. How much will settle the loan April 1, 1909 ? 45. July 28, 1907, a broker borrows $3200 at 5% in- terest and on the same day loans it at 7| % interest. If full settlement is made April 1, 1909, how much will the broker make by reloaning the money ? PROBLEMS IN SIMPLE [NTEREST 283 PROBLEMS IN SIMPLE INTEREST Finding the principal. Written Work 1. What principal invested at 4 % per annum will yield an annual income of $200 ? Since the interest on $ 1 for 1 year at 4% is $.04, as many dollars must be invested to yield 8200 per year as $.04 is contained times in $200. $200 h- $.04 = 5000. Hence, $5000 must be invested. TJie principal equals the given interest divided by the interest on $lfor the given time at the given rate. 2. What principal at 4£% will gain $213.75 interest in 4 yr. 9 mo.? 3. What principal at 5% will gain $120.70 interest in 3 yr. 6 mo. 18 da.? 4. What principal at 8% will gain an interest of $163.20 from Sept. 30, 1903 to June 12, 1905 ? 5. A man gave his note April 1, 1901, at 6%. When he settled the note Aug. 13, 1903, he paid $195.25 interest. What was the principal, or face of the note ? Finding the rate. Written Work 1. At what rate must $500 be invested to yield $75 interest in 2 yr. 6 mo. ? Since the interest on $500 for 2\ yr. at 1% is $12.50, it will require a rate of as many per cent to yield $75, as $12.50 is contained times in $75, or 6%. The rate equals the given interest divided by the interest for the given time at 1 % • 2. The interest on $1125 for 3 yr. 4 mo. 24 da. is $229.50. What is the rate per cent ? 284 INTEREST 3. The interest on $1800 for 4 yr. 8 mo. 16 da. is 1424. What is the rate ? 4. At what rate will $2460 give $682.65 interest in 5 yr. 6 mo. 18 da.? 5. A note for $880 was given July 5, 1900, and settled May 2, 1904, for $1081.96. What rate per cent was charged? Finding the time. Written Work 1. In what time will $450 at 0% yield $90 interest? Since the interest on -1450 for 1 yr. at 6% is $27, the required time is as many years as $ 27 is contained times in $90, or 3$ yr. The time in years equals the given interest divided by the interest at the given rate for 1 year. 2. In what time will $275 gain $55 interest at 6 % ? 3. In what time will any principal double itself (that is, gain 100 % of itself) at 5 % ? at 6 % ? at 8 % ? 4. In what time will any principal treble itself (that is, gain 200% of itself) at 5 %? at 6 % ? In what time will it quadruple itself at 8 % ? at 10 % ? 5. A note of $500 at 5% interest was paid May 1, 1905, the interest amounting to $63.75. When was the note given? REVIEW PROBLEMS 1. Find the interest on $80 for 6 yr. 6 mo. 18 da. at 6 %. 2. In what time will $180 at 5 % yield $22.50 interest? 3. In what time will $600 amount to $715.50, at 5|% interest ? REVIEW PROBLEMS 285 4. At what rate will $1650 gain #326.70, in 4 yr. 4 mo. 24 da.? 5. What sum of money loaned at 0% will give a semi- annual income of $13.02 ? 6. Interest $31.80; time 3 yr. 6 mo. 12 da.; rate 5%. Find the principal. 7. The interest of ^ of a principal for 3 yr. 6 mo. at 6% is $19.25. Find the principal. 8. If $186 pays a debt of $150 which has been due for 4 years, what is the rate of interest ? 9. I borrowed $450 at 5% and kept it until it amounted to $525. When did I settle the note ? 10. Dec. 1, 1901, 1 loaned $300 at 5* %. Find the amount due March 16, 1905. 11. Find interest at 6% on $350 from June 1, 1898 to Aug. 13, 1902. 12. If $300 was borrowed April 1, 1902, at 5%, when should the principal and interest be paid that their sum may be $357? 13. The amount of a certain principal at 6 % for a given time is $780, and at 10% for the same time it is $900. Find the principal. 14. If $148 is loaned April 1, 1902, at 5%, when will it amount to $179.45 ? 15. What principal for 3 mo. at 8 r / c will yield the same interest as $5100 for 5 yr. 8 mo. at 6 % ? 16. A farmer bought 75 acres of land at $ 50 an acre, paying ^ cash, and giving his note for the balance, due in 3 yr. 6 mo., with interest at 6%. What was the amount of the note at maturity ? 286 INTEREST ANNUAL INTEREST, OR SIMPLE INTEREST ON UNPAID INTEREST In some states when a note reads with interest " payable annually," simple interest may be collected upon the princi- pal and upon each year's interest from the time it was due until paid. In most states annual interest is not collectable by law. Interest payable annually is simple interest; but interest collected on the principal and on the overdue payments of simple interest is annual interest. Written Work 1. James Brown borrows $1200 at 6% interest, "payable annually." In case no interest is paid for 3 years, 6 months, and 15 days, how much money is necessary to pay the debt ? Simple interest on $ 1200, at 6%, for 3 yr. 6 mo. 15 da. The interest for each year is $72. The 1st annual int., $72, remains unpaid for 2 yr. 6 mo. 15 da. The 2d annual int., $72, remains unpaid for 1 yr. 6 mo. 15 da. The 3d annual int., $72, remains unpaid for 6 mo. 15 da. Interest on $72 at 6% for . . . 4 yr. 7 mo. 15 da. The annual interest on $1200 for 3 yr. 6 mo. 15 da. The principal ... ..... The amount of $ 1200 at annual interest for 3 yr. 6 mo. 15 da. 255 19 274 1200 1474 98 98 00 98 Annual interest is the simple interest on the principal for the given time plus the simple interest on each year's interest for the time it remains unpaid. 2. Find the total interest due on a note of $ 675 for 2 years, 8 months, and 20 days at 6%, with interest payable annu- ally, if no interest has been paid. 3. Find the amount of $6400 for 4 years, 5 months, and 15 days, with interest payable annually at 6 (f . EXACT INTEREST lis; 4. An attorney collects a note of $3750 with annual in- terest on it at 6% for 4 yr. 9 mo. 18 da. Kind the amount collected and his commission on it at 10%. EXACT INTEREST Exact interest is simple interest on the principal reckoned on the basis of 365 days to a common year and 366 days to a leap year. It is used in computing interest on all obligations by the United States government ■; on all foreign securities; and to some extent by city controllers and bankers. Since common interest is computed on the basis of 12 months of 30 days each, or 360 'lays ; and exact interest is reckoned on the basis of 365 days to a common year, or 366 to a leap year, 1 day's exact iuterest is jl-s of a year's common interest. It is evident that the common and exact interest for 1 year are the same. Thus, fff of one year's common interest equals one year's exact interest. They differ only for parts of a year. Written Work 1. Find the exact interest on $2400 for 95 days at 6%. Exact interest for 1 year = 6% of $2400, or $144 Exact interest for 1 day = -353 of $144, or $.394.") Exact interest for 95 days = 95 x $.3945, or $37.48 Exact interest is found by dividing the common interest at the given rate for one year by 365 and multiplying the quotient by the exact number of days. Find the exact interest of: 2. 6800 for 78 days at 6 % . 6. $500 for 90 days at 9 %. 3. $2000 for 92 clays at 7%. 7. $1020 for 74 days at 10%. 4. 62400 for 115 days at 8%. 8. $6500 for 280 days at 6%. 5. 61775 for 100 days at 8|%. 9. $10000 for 61 days at 7%. 10. Find the exact interest on $1020 from Oct. 19, 1905 to April 1, 1907, at 6%. 288 INTEREST Notk. — Why do we find exact interest for a fraction of a year only V The exact number of days from Oct. 19, 1905 to April 1, 1907, is found as follows : Oct., 12 da. ; Nov., 30 da. ; Dec, 31 da. ; Jan., 31 da. ; Feb. 28 da. ; March, 31 da. ; April, 1 da. Total, 164 days. 11. Find the exact interest on $1795.80 from July 7, 1901 to Sept. 1, 1907 at 7%. 12. The United States government paid exact interest at 4 (f on a warrant of $ 650000, 83 days past due. Compute the amount paid. COMPOUND INTEREST Mr. Reed Colburn loans Robert Patterson $200 for 2 years at 6 % . Suppose Mr. Patterson says to Mr. Colburn, at the end of the first year : " I cannot pay you the $ 12 interest due, but will pay you interest at 6 % on the $12 for a year." How much interest should Mr. Patterson pay Mr. Colburn at the end of the 2 years? How does the 12-1.72 inter- est differ from simple interest? Compound interest is interest on both the principal and the unpaid interest added to the principal when due. Interest may be added to the principal annually, semiannually, or quarterly, according to agreement. Written Work 1. Find the compound months at 0%. Principal Interest for 1st yr. atG % Principal for 2d year Interest for 2d yr. at 6 % Principal for 3d yr. Interest for 6 mo. at 6 % Amount for 2 yr. 6 mo. at 6 % Original principal Compound interest for 2 yi iterest on $200 for 2 years 6 $200.00 . 12.00 . , 212.U0 . , 12.72 . . 224.72 . . 6.74 6% • 231.46 . ► o 200.00 . G mo. at ( i% • ■ 31.46 SAVINGS ACCOUNTS 289 Note. — 1. Unless otherwise stated in the agreement, interest is com- pounded annually. 2. When interest is compounded semiannually, consider the rate as A the annual rate, or if quarterly, £, etc. 2. Find the compound interest on $1000 for 2 years at 5%, with interest compounded semiannually. 3. Find the compound interest at 6% on $800 for 1 yr. 5 mo., interest payable quarterly. 4. Find the compound interest on $600 for 9 mo. at 6%, interest payable quarterly. SAVINGS ACCOUNTS Compound interest is no longer allowed on notes. Its only practical application for elementary schools is found in computing interest on savings accounts. Many banks to-day have a savings department. The amounts thus deposited are not subject to check, but draw from 2% to 4% interest which is usually compounded semiannually. The interest periods are generally January 1 and July 1 of each year, although sometimes the interest is compounded quarterly. Thirty days are reckoned to a month. Interest on savings accounts is sometimes calculated from the 1st and 15th of each month succeeding the serend dejiosits. Thus, $ 10 deposited on the 1st of any month would draw interest from date; but $10 deposited on the 2c? of any month would draw interest from the loth: or money deposited on the 16th w r ould draw interest from the 1st of the next month. There is no fixed rule, however, as each bank determines for itself when the interest date begins. No interest is allowed on a fractional part of a dollar, and parts of a cent are omitted on all interest credits. Most banks require notice from a depositor before a savings account may be withdrawn. Amounts withdrawn before the end of an interest period draw no interest for that period. HAM. COMPL. A KITH. — 19 290 . INTEREST Written Work 1. On July 1, 1905, Raymond Wilkinson makes a sav- ings deposit of $400 at 4% interest, payable semiannually. If the interest at each period is added to the deposit, what is the total amount in bank January 1, 1907 ? Deposit July 1, 1905 $400.00 Interest on $400 at 4 % July 1, 1905 to Jan. 1, 1906 . .. 8.00 Amount in bank Jan. 1, 1906 408.00 Interest at 4 % on $408 from Jan. 1, 1906 to July 1, 1906 . 8.16 Amount in bank July 1, 1906 ...... 416.16 Int. at 4 % on $416 (why?) from July 1, 1906 to Jan. 1, 1907 8.32 Amount in bank Jan. 1, 1907 424.48 2. Find the difference between the simple interest on a note of 8200 dated July 1, 1906, due in two years at 4| %, and the interest on $ 200 deposited in a savings bank at 4 %, compounded semiannually, for the same period. 3. A savings account of $150 deposited April 1, 1906, at 3% interest, payable January 1 and July 1, is with- drawn April 12, 1908. Find the amount withdrawn. 4. A savings bank pays 4% interest, calculated from the 1st and the loth of each month succeeding the several de- posits. The deposits are Sept. 1, $20; Oct. 10, $15; Nov. 15, $20; Dec. 10, $25. Find the amount in bank the following January 1, if the interest periods are January 1 and July 1. 5. The Lincoln School had on deposit in the Holmes Sav- ings Bank Jan. 1, 1907, $495.80. The deposits were as fol- lows: Feb. 1, $76.90; March 1, $105.05; April 1, $114.29; May 1, $129.70; June 1, $98.75. Find the amount in the bank Jan. 1, 1908, at 4 % interest, compounded the first of January and July. INVESTMENTS 291 Find the amount in bank from the following deposits Deposit Date Rate Int. Payable Amount in Bank 6. $200 Jan. 1, 1905 3% Jan. 1 and July 1 July 1, 1906 7. $150 Mar. 16, 1906 4% Jau. 1 Jan. 1, 1908 8. $875 May 29, 1906 21 % Jan. 1 Jan. 1, 1908 9. $1200 Aug. 10, 1906 2% Jan. 1, Apr. 1, July 1, Oct. 1 Jan. 1, 1908 INVESTMENTS Compound interest tables are frequently used by insurance companies, building and loan associations, and trust com- panies, to calculate the income, from investments where the interest is added each interest period to the amount invested. The following table shows the amount of $1 at compound interest at the given rates for 10 years. Compound Interest Table Yr. n% 2% n% 3% n% 4% 1 1.0150 000 1.0200 0000 [.0250 in ii 10 1.0300 0000 1.0350 0000 1.0400 0000 8 1.0302 250 1.0404 0000 1.0506 2500 1.0609 0000 1.0712 2500 1.0816 0000 3 1.0456 784 1.0612 0800 1.0768 9062 1.0927 2700 1.1087 17S7 1.1248 6400 4 1 .0618 636 1.0824 3216 1.1088 1289 1.1255 0881 1.1475 2300 1.1698 5856 5 1.0772 840 1.1040 8080 1.1314 0821 1.1592 7407 t.1876 8631 1.2166 5290 6 1. 0984 488 1.1261 6242 1.1596 9342 1.1940 5230 1.2292 5533 1.2653 1902 7 1.1098 (50 1.1486 8567 1.1886 8575 L.2298 7887 1.2722 7926 1.S159 8178 8 1.1264 926 1.1716 5938 1.2184 0290 1.2667 7008 1.3168 0904 1.3685 6905 9 1.1488! 1.1951)9257 1.2488 6297 1.3H47 7318 1.3628 9735 1.4288 1181 10 1.1605 408 1.2189 9442 1.2800 8454 1.8489 1688 1.4105 9876 1.4802 (423 The compound interest on any amount for 4 years at 8% payable semiannually is evidently the same as upon the same amount for 8 years at 4 % payable annually. The amount of any given principal for any given number of years is found by multiplying the principal by the amount of $1 at the given rate for the time as given in the table. 202 INTEREST Written Work 1. Find the amount of $1200 invested for 7 years at 3^ %, interest compounded annually. 2. Find the compound interest at 4 % on $ 10000 invested for 9 years. 3. The amount of $12000, invested for 10 years at 31 %, interest compounded annually, is divided equally among 3 sons. Find each one's share. 4. Find the amount of $1200 for 2 years and 6 months at 4 %, compound interest payable semiannually. PROMISSORY NOTES Mr. James H. Ames, a grocer, Erie St., Buffalo, N.Y., has an account of $52.00 against Robert Patterson for groceries, and Mr. Ames asks Mr. Patterson to give him a note at 6 % interest for the amount of the bill. The note reads as follows : $52.00 Buffalo, JV.T., Mv-. Bf, 1905. c/ux, 'nvcyyttJt^ after date -_.c/--- promise to pay to the order of jtawvea- /if. CL , wu&^ &C££y - 1 w- c i rrrrrrrrrr?ccrr^'rrrrrr^rcc'rrrr^^ / ars. Value received, with interest at 6%. f\ot>evt ^utt&VQycyyv. 1 A promissory note is a written promise to pay to a certain person named in the note, or his order, a specified sum of money at a specified time. PROMISSORY NOTES 293 The Essentials of a Promissory Note : 1. It should state the place where and the time when given. 2. It should promise to pay to a certain person or to his order. 3. It should promise to pay a certain sum of money, expressed both in figures and in writing. 4. It should state when the money is to be paid. 5. It should state by whom the money is to be paid. 6. It should state for value received. (Xot absolutely necessary, but usually written in a note.) 7. It should state with interest and tlie rate, if it is an interest-bearing note. The promissor is called the maker of the note. The person who is to receive the money is called the payee of the note. 1. Who is the maker of the note on page 292? 2. Who is the payee of the note on page 292? 3. Find the amount to be paid when due. 4. The face of the note is the sum written in the note. What is the face of the note on page 292? 5. This note reads "pay to the order of James H. Ames," and mean- that Mr. Ames has the right to sell this note to any one by simply writing his name across the back of the note and delivering it to the purchaser. What words in the above note' give Mr. Ames the right to sell it? When the owner of a promissory note writes his name across the back of it, he is said to indorse the note. If Mr. Ames indorses the note and then sells it to Mr. B.. and Mr. B. indorses it and sells it to Mr. C, to whom does the note belong? A promissory note, therefore, like any other property may be bought and sold ; hence it is called negotiable paper. When a note is made payable to a definite person, it cannot be trans- ferred, and is therefore nut negotiable. 294 INTEREST jlavi'&& CLnd&x^o-ru. Indorsement in Full Promissory notes may be indorsed as follows: Indorsement in Blank (1) In blank : In this form the indorser simply writes his name across the back of the note, thus making the note payable to the holder. (2) In full : In this form the in- dorser designates that the note is to be paid to the order of a definite person. (3) In limited form : In this form the indorser writes ''without re- course " above his name. This means that the holder cannot com- pel him to pay if the maker fails to do so. lAJ-iZkout \,e&cm,'v&&. &aAf to- tk& o'uLeA, o Limited Indorsement / Every indorser in blank, or in full, makes himself liable for the amount of the note if the maker and the previous indorsers /a;7 to pay. Banks are required to notify the indorsers in a manner prescribed by law in case the note is not paid when due. This is called protesting the note. If the note is not protested, the indorsers are released from the liability of payment. Forms of promissory notes. I. As to time : 1. If the words "on demand" are substituted for the words "six months " in the note of Mr. Ames, page 2. c )2, it will then be a " demand note " ; that is, the maker may be called upon to pay it at any time after date. 2. The note of Mr. Ames is a "time note" because it is not to be paid until a certain time named in the note. The time of payment in a note must be definite. A promise to pay " when able " is too indefinite, and not binding. PROMISSORY NOTES 295 II. As to payees : 1. When a note is payable to the order of some particular person, he alone can collect it, or sell the note by indorsement. 2. When a note is payable to some particular person, or bearer, the holder can collect it, or sell it by indorsement. III. As to the number of makers : 1. An individual note is a promise made by one person. 2. " A joint and several note " is a promise made by more than one person. It contains the words " we, or either of us," and is signed by the makers. Maturity of Promissory Notes. A note is said to mature on the last day of the time named in the note. Some states allow 3 days, called " days of grace," from the time a note matures before the payee can proceed to collect the note. In this case three days are added to the time on which the interest is computed. Days of grace are now abolished in most states. The note on page 2i)'J matures May 21, 1906. If a note falls due on Saturday, Sunday, or a legal holiday, it is usu- ally payable on the next succeeding business day. Some states require such notes to be paid on the preceding business day. Interest on Promissory Notes. If either a time or a demand note contains the words "with interest," the note bears interest from date at the legal rate in that state. If the words "with interest" ai - e omitted from a time note, it bears interest from the date of maturity until paid. If the words " with interest " are omitted from a demand note, it bears interest from the time payment is demanded until paid. Written Work 1. Is the promissory note given by Mr. Patterson (p. 292) a demand or a time note? an individual or a joint and sev- eral note ? payable to order or bearer ? 2. Write a promissory note, in which you are the maker, for $125 due in 6 months, payable to the order of Ellsworth Slater, with the legal rate of interest in your state. 296 INTERp;ST 3. Mr. Slater sells this note to Herman Gross, and in- dorses it in full. Write the indorsement on the note. 4. Write a joint and several note for 8250, dated Sept. 24, 1905, due on demand, with interest at 6%, payable to the order of James Harbison. Your teacher and yourself may sign this note as makers. 5. May the two names to the above note be written by the same person ? Why not ? 6. In case James Harbison sells this note to James Brown, but says to Mr. Brown, " I shall not be responsible for the collection of this note," write the indorsement and explain why you use that form. 7. Find the amount paid to the holder of Mr. Harbison's note (Ex. 4), if settled Jan. 4, 1907. Note. — Time from Sept. 24, 1905 to Jan. 4, 1907, 1 yr., 4 mo., 1 1 da. 8. Name the different kinds of negotiable notes : (1) as to time ; (2) as to payee ; (3) as to number of makers. 9. Find the amount to be paid on the following note, if settled March 1, 1907 : The note is legally due June 1, 1906; therefore it bears interest at the legal rate from that date. $300.00 Chicago, 171., iHoA^k /, 1906. &ki&& wuyn&kb after date ■.__ .J __ promise to pay to the order of Qavi&a, 6vy&& Value received. PROMISSORY NOTES 297 10. Find the interest to be paid on the following note, if settled Aug. 10, 1906, with interest at 6 % ' $/75.60 Boston, Mass., fam,. 20, IcLav& c^z^^c^^r^r^^^r2^^r^CzDollars. 100 With interest ; at 6%. /ferny Bxow-n. PARTIAL PAYMENTS OF PROMISSORY NOTES 299 The following payments are indorsed on this note : What amount is due March 2, 1908? Nov. 26, 1907, $50.00. Jan. 2, 1908, $25.00. Solution : Principal Interest on $200 for 1 yr. The amount of the note Nov. 20, 1907 . Payment Nov. 20, 1907 ... Balance = new principal due Nov. 26, 1907 yr. mo. da 1908 1 2 1907 11 26 $200.00 12.00 212.00 50.00 162.00 Interest on $ 1 62 for 1 mo. 6 da. . Amount due Jan. 2, 1908 Payment Jan. 2, 1908 .... Balance = new principal due Jan. 2, 1908 $ .97 162.97 25.00 137.97 1908 1908 3 2 1 2 Interest on $ 137.97 for 2 mo. The amount due March 2, 1908 (I 1.38 $ 139.35 2. What was the amount due on the note on Nov. 26, 1907? 3. How much interest was due Nov. 26, 1907 ? What payment was made ? How much greater was the payment than the interest ? 4. How much was the new principal due Nov. 26, 1907, after the payment of $ 50.00 ? 300 INTEREST 5. How much interest was due Jan. 2, 1908? How much greater was the payment than the interest? Observe : 1. That the interest was computed on the principal to the time of the first payment ; then on the balance, as a new principal, to the time of the second payment; then on the balance, as a new principal, until March 2, 1908. 2. That the interest at each payment was first paid and the balance of the payment was credited on the principal. 3. As the interest must^rs^ be paid, in case the payment does not equal the interest, the interest must be computed until such time as the sum of the payments equals or exceeds the interest. This is the United States rule of partial payments, and is the legal one in most states. Find the amount of the principal to the time of the first pay- ment, and from the amount subtract the first payment. Con skier the remainder as a new principal and proceed as before until the time of final settlement. If any payment does not- equal or exceed the interest, then find the interest to the time when two or more payments equal or exceed the interest. The Supreme Court of the United States has decreed: (1) That the payment on a note must first be applied to cancel the interest then due, before the piincipal may be diminished. (2) That interest must not be charged upon interest. Written Work 1. A note for $1800, bearing 6% interest, was given April 1, 1903, and settled Oct. 22, 1906. On the back of the note were these indorsements: May 10, 1904, $225; June 16, 1905, $50; Sept. 28, 1905, $340; March 10, 1906, $475. Find the balance due on the note at date of settle- ment. PARTIAL PAYMENTS OF PROMISSORY NOTES 301 Principal Interest from April 1, 1903 to May 10, 190-1 Amount due .May 10, 1 : * < » i First payment made May 10, 1901 . l>a lance = new principal due May 10, 1904 Interest from May 10, 1901 to June 16, 1905 51800.00 119 .70 1919.70 225.00 1694.70 $111.85 The interest exceeds the payment and a new principal is not formed. Interest from June 16, 1905 to Sept. 28, 1905, . . 28.81 Interest from May 10, 1904 to Sept. 28, 1905 .... 140-66 Amount due Sept. 28, 1905 18:35.36 Sum of second payment June 16, 1905, and third payment Sept. 28, 1905. $50.00 + 8340.00 • . = 3 90.00 Balance = new principal due Sept. 28, 1905 Interest from Sept. 28, 1905 to March 10, 1906 Amount due March 10, 1906 .... Fourth payment made Marc h 10, 1906 . Balance = new principal due March 10, 1906 . Interest from March 10, 1906 to Oct. 22, 1906 Balance due Oct. 22, 1906 . 1445.36 39.024 1484.384 475.00 1009.384 37.347 $1046.731 2. A mortgage for $960, bearing 6% interest, was given June 20, 1900, and settled Dec. 26, 1904. On the mortgage wore these indorsements: Nov. 2, 1901, $140 ; Jan. 14, 1903, 1200; June 1, 1904, $30; June 20, 1904, $150. Find the balance due on date of settlement. 3. On a claim of $850, dated May 2, 1901, interest 5%, the following payments were made: Aug. 8, 1901, $200; Dec. 14, 1901, $2i order of fo/m ft. 3 Ivo-yyv^o-n, f^OO.*-^- For R&nt to- . This check is made payable to the order of John R. Thompson, which means that in order to receive the money from a bank, or transfer the check to another person, he must write across the back of the check the na'mtT^John R. Thompson." This is called a blank indorsement, because it (jojeajiot state to who m the cheek is made, payable. If John R. Thompson should write across the hack of the check the following : Pay to the order of Marshall Field & Co., Chicago, 111. John R. Thompson, this would be known as a full indorsement, for no one but Marshall Field & Co. could collect or indorse the check. Checks, like promissory notes, may be written in different ways, as follows : 1. Pay to bearer, 2. Pay to cash, collectable by bearer. 3. Pay to James Ogden, or bearer, . 4. Pay to self (collectable by maker only). 5. Pay to the order of self (collectable by indorsement of the maker). 6. Pay to the order of James Ogden (collectable by indorsement of James Odgen only). Note. — The last form of check is the one in general use. 3. How may the check on p. 304 be indorsed in blank ? 4. How may it be indorsed in full ? 5. Suppose Mr. Thompson wishes to send this check to Sage, Allen & Co., Hartford, Conn., in payment of an HAM. COMF1.. ARITH.— 20 306 BANKS AND BANKING account: first, write the check as indorsed by Mr. Thomp- son in blank ; second, write the check as indorsed by Mr. Thompson in full. 6. Give reasons why it will be better for Mr. Thompson to indorse the check in full. 7. When a check is indorsed and sent by mail, what form of indorsement should always be used ? Why ? 8. Give the essentials of a check. Balancing Accounts ; Depositing ; Checking on Accounts : 1. Your deposits in a bank for the month of September are as follows: Sept. 1, currency, $50; silver, $10; check, $15. Sept. 6, currency, $20; silver, $10 ; check, $100 ; gold, $20. Sept. 10, currency, $45; silver, $4.75. Sept. 16, currency, $20; silver, $3.40; check, $80. Sept. 25, gold, $40; check, $40; silver, $10. Sept. 29, check, $80; currency, $80; silver, $35. Make out deposit slips and find amount of deposits for September. 2. Your check book shows the following : Balance in bank Sept. 1, $847.10. No. 1, Sept. 4, Keller Bros., for coal, $15.50. No. 2, Sept. 4, Geo. K. Stevenson & Co., for groceries for August, $49.50. No. 3, Sept. 4, Dr. S. N. Pool, for services to date, $90. No. 4, Sept. 5, cash, $55. No. 5, Sept. 7, Jos. Home Co., for merchandise, $65.30. No. 6, Sept, 11, Midland Lumber Co., for lumber, $93.75. No. 7, Sept. 15, Johnson & Co., for repairs on automobile, $29.35. No. 8, Sept. 19, cash, $25. BORROWING FROM HANKS 307 No. 9, Sept. 24, J. H. McFarland, for interest due on note, 124. Write the checks for the bills paid for September, and find balance in bank. 3. Arriving at Chicago, I find in my mail a check from the Keystone Lumber Co., Pittsburg, Pa., for $415.40, in payment of my salary and expenses for September. I wish to deposit the same to my account in the Colonial Trust Co., Pittsburg, Pa. How should I indorse the check before send- ing it through the mail ? BORROWING FROM BANKS AND COMPUTING BANK DISCOUNT Banks usually lend money on promissory notes drawn in one of three forms : 1. The note is made payable to the indorser, who signs his name across the back of it. 2. A joint and several note is made payable to the order of the bank and signed by both parties as makers. 3. The note is made payable to the order of the bank. The security in the form of stocks, bonds, mortgages, etc., is deposited as collateral. $200.^. Pittsburg, Pa., ifefit. 8, 1905 3~kx,&& vio-nthfr after date J. promise to pay to the order of. ft. A W-at&aw at the Lincoln Rational Bank of Pittsburg Zfw-o- f'ficncLi&cL avid — -^^o^^r^r^^^c^^ryr^^^yr^Dollars too without defalcation, for value received. J. & &kandle,v. 308 BANKS AND BANKING This note matures three months after Sept. 8, or Dec. 8. If the time in the note were " ninety days " instead of " three months," the note would mature ninety days after Sept. 8, or Dec. 7. If Mr. Chandler wishes to borrow money at the bank, he may make out a note as on p. 307 and get Mr. Watson to indorse it. If both men are responsible from a financial point of view, the bank will buy the note and give Mr. Chandler the difference between the value of the note at its maturity and the interest on that value at the legal rate for the exact number of days the bank is without the use of its money. The value of Mr. Chandler's note is the amount the Lincoln National Bank will receive from Mr. Chandler at its maturity. If Mr. Chandler fails to pay. Mr. Watson will be held responsible. The buying of notes by a bank is called discounting notes, and the interest deducted is called bank discount. The proceeds of a note discounted by a banker or a broker is the value of the note at its maturity less the discount. The term of discount is the exact number of days that the borrower has the use of the money. There are two methods, however, of reckoning this term: the first method counts the day of maturity, but not the day of discount; the second counts both : thus, by the first method Mr. Chandler had the use of the money 22 days in Sept., 31 days in Oct., 30 days in Nov., and 8 days in Dec, or 91 days in all; by the second method he had the use of the money 23 days in Sept., 31 days in Oct., 30 days in Nov., and 8 days in Dec, or 92 days in all. Note. — Pupils should solve the problems according to the practice in their vicinity. Answers are given for both methods. When days of grace are allowed they are included in the term of dis- count, but in this book days of grace are not reckoned. Computing bank discount on note on p. 307 : Date of maturity, December 8. Term of discount. 91 days (not including day of discount). Bank discount on 8200 for 91 days at 6% = $3.03. l. How much does the bank pay to Mr. Chandler? How much does Mr. Chandler pay to the bank at maturity?- BORROWING FKo.U BANKS SOP, •^. If Mr. Chandler had borrowed $200 from Mr. Watson for 3 months at 6%, how much would he have paid Mr. Watson when the note became due? 3. How much would Mr. Chandler have received from Mr. Watson at the time he borrowed the money? 4. Find the difference between the discount paid to the bank and the interest he would have paid to Mr. Watson. Comparative Study Banks differ from individuals in lending money, as follows: 1. Banks require the interest on a note to be paid in advance ; indi- viduals demand interest when the note is due, or annually, if for a longer period than a year. 2. Banks compute interest for the exact number of days; individuals compute interest by months and years. 3. Banks lend money for short periods, usually not exceeding four months; individuals lend for longer periods, not exceeding five years in most states. 4. Banks require the maker to give additional security ; individuals may or may not demand security. 5. Interest is computed on the face value of a note; bank discount is computed on the value of a note at its maturity. Bank discount is the simple interest paid in advance on the value of a note at its maturity for the exact number of days the banker is without his money. Given the dates and time of notes, to find the date of maturity. Find the date of maturity of the following : Date Time Date Time 1. June 1 2. July 3 3. Aug. 5 4. Sept. 10 2 months 50 days 100 days 1 month 5. Jan. 2 6. March 3 7. April 1 8. Ma\ 5 3 months 7~> days 70 days 4 months 810 BANKS AND BANKING Find the date of maturity and the term of discount. Date of Time of Date of Date of Time of Date of Note Note Discount Note Note Discount 9. March 1 60 da. April 1 14. Jan. 2, '08 90 da. March 1 10. April 10 3 mo. June 15 15. March 23 4 mo. June 2 11. July 10 4 mo. Sept. 30 16. Oct. 8 60 da. Nov. 1 12. Mav 24 30 da. June 1 17. June 5 90 da. July 10 13. August 5 70 da. Sept. 1 18. Sept. 24 30 da. Oct. 1 Written Work 1. Write a promissory note for $ 300 payable to John Jackson, dated Aug. 3, 1907, due in four months, with inter- est at 6%, and signed by Glenn Campbell. 2. Mr. Jackson indorses the note in example 1, and Mr. Campbell borrows the money from the Park National Bank. Indorse the note in full and find the bank discount and pro- ceeds. The following notes are each discounted on the day of issue. Find the date of maturity and the bank discount. Date of Note Time Face Rate of Discount 3. Aug. 10, 1906 90 da. 9 150 6% 4. June 12, 1906 2 mo. 515 6J% 5. July 2, 1906 3 mo. 1000 u% 6. Jan. 8, 1906 4 mo. 625 n% 7. Mar. 5, 1906 50 da. 570 6% 8. May 8, 1906 70 da. 423.25 5*% 9. Sept. 1, 1906 1 mo. 1200 8J% 10. Dec. 1, 1906 100 da. 387.75 6% 11. Mar. 5, 1906 72 da. 1125 7% DISCOUNTING NOTES 311 Discounting Interest and Non-interest bearing Notes. Business men frequently take notes from their customers due at a future date, and in case they need the money before the notes become due, they sell them to a bank. The bank deducts from the value of each note at maturity the interest (bank discount) for the term of discount. These notes may or may not bear interest. • Mr. James Edwards has two notes that read as follows : $nth& after date J. promise to pay to the order of <^^ryr*^r^^j!am-&a, £dnAXMj£&c^^c^r^c^^>- c/tinetu and i ^^^^r^^o^^o^^c^^^r^^c^^rC^Z>oZZa;»5. ' 100 Value received. ff&nvy (ZmaXaav. $/50^- Columbus, Ohio, ?11awA fO, 1907 100 &ouv i yYuy)ith& after date J. promise to pay to the order of^^yr^c^y^yc^^a-'viv&a, €dwavcU,-c^?^?^c^^I)ollars. Value received, with interest at 6%. 312 BANKS AND BANKING 1. What is the value of the first note at maturity? 2. What is the value of the second note at maturity? 3. Mr. Edwards gets both notes discounted April 20, 1907, at 6 %. Why is the discount on the first note computed on $90? on the second note, on $153? Banks always discount notes on the amount they are to receive at maturity. Discounting the first note on page 311 : Maturity of note, June 2. Value of the note at maturity . . . = $90.00, or face Bank Di scount for 43 da. at 6 % . . = .65 Proceeds April 20 = 9 89.35 Discounting the second note on page 311: Maturity of note, July 10. Value of the note at maturity = % 153.00 ox face + interest for 4 months. Bank Discount for 81 da, at 6% = 2.07 Proceeds April 20 ....=$ 150.93 Written Work 1. A 60-day note for $2500, without interest, dated Jan. 12, 1907, was discounted Feb. 12, 1907. Find the proceeds of this note. 2. Find the proceeds of a 90-day note for $1560, with interest at 6 %, dated March 8, and discounted April 12. 3. Find the proceeds of the note in example 2 without interest. 4. A 90-day note for $ 4500, with interest at 6 %, is dis- counted 30 days after date. Find the proceeds. Suggestion. — Note the difference between a note for 90 days and a note discounted for 90 days. DISCOUNTING NOTES 313 5. Find the proceeds of the note in example 4 without interest. 6. A 120-day note for $3500, without interest, dated June 5, is discounted Aug. 10. Find the proceeds. 7. Find the proceeds in the note of example 6, if the note bears interest at 6 %. 8. The proceeds of a 90-day note, without interest, dis- counted 30 days after date, is $ 990. Find the face. 9. A note for $1200, bearing interest for 3 mo. at 6 %, was dated Jan. 15 and discounted Feb. 20. Find the proceeds. 10. Mr. Boyd gives his note Jan. 10, 1905, to William Savers for $200, payable in 9 months, with interest at 6%. What is Mr. Boyd's note worth on the day of issue? on the day of maturity ? 11. Should Mr. Boyd get the note discounted July 5, on how much money would the bank reckon the discount ? 12. Find the amount Mr. Boyd would receive July 5. What is the money received by Mr. Boyd called with ref- erence to the note ? 13. Write the note given by Mr. Sanders and transfer it by indorsement in full to one of your local banks. 14. Face, $223.50 ; time, 90 days ; rate of interest, 6% ; term of discount, 50 days; rate of discount, 8%. Find proceeds. 15. A business man's bank account is overdrawn ^381.50, and he presents to the bank, May 1, two notes to be dis- counted, at 6%, and the proceeds to be placed to his credit. Face Date Time Rate of Interest $290 Mar. 10 5 mo. Without interest $355 Apr. 20 90 da. 6% Find his balance. 314 BANKS AND BANKING 16. The discount on a note for $400 for 60 days, exact time, is $6.00. Find the rate of discount. 17. A broker buys a $300 note, thirty days before matu- rity, for $297. Find the rate of discount. 18. Mr. James sold a horse for $155, and took the pur- chaser's note, dated Jan. 20, 1905, due in one year, with interest at 4| %. Mr. James sold the note to the Farmers' Bank, Oct. 10, at 7 % discount. How much did he realize ? 19. A note for $1800, at 8%, dated August 1, due in 3 months, was discounted October 6, at 6 °J • Find the proceeds. 20. What should the Merchants' National Bank pay for a note of $1200, bearing &% interest, dated April 12, due in 4 months, if purchased June 1, at 6 % discount ? 21. What are the proceeds of a note for $2500, dated February 10, 1907, and due in 4 months, without interest, if discounted March 24, at 6 % ? 22. A merchant's bank book shows a balance of $1375.50, and he presents at the bank four notes, which are discounted June 1, at 6 %, and the proceeds placed to his credit : Face Date $600 March 4 $1375 April 2 $1050 March 19 $2000 May 29 Find his balance in bank then. 23. For how much must I give my note, discounted at a bank for 60 days, at 6%, to realize $990 ? Note. — The proceeds of $1, discounted for 60 days, at 6% = $.99. 24. For what sum must I draw my note so that when dis- counted at 6% for 90 days I may realize $2758? Time Rate of Interest 90 days 6% 4 months No interest 90 days 5% 3 mo. No interest EXCHANGE Paying Bills at a Distance What is meant by a debtor? by a creditor? How may a debtor pay a bill in a distant city without the actual transfer of cash ? How may a creditor collect a bill in a distant city without the actual transfer of cash ? Exchange is a method of paying or collecting bills at a distance without the actual transfer of money. There are several different ways in which bills may be paid at a distance without the transmission of money: (1) By a postal money order. (2) By an express money order. (3) By a telegraphic money order. (4) By a personal check. (5) By a bank draft (banker's check). Paying by postal or express money order. If you wish to order from Siegel, Cooper & Co.. Chicago, $15 worth of merchandise, unless you have credit there, you will probably send them (1) either a postal money order, or (2) an express money order. The first will direct the postmaster at Chicago, the second some express agent at Chicago, to pay to the order of Siegel. Cooper & Co. i 15. The cost of either of the above orders is the same ; the only differ- ence being that a postal money order is payable to the order of the party or firm upon identification at the place named in the order, while an express money order is payable to the party or tirin upon identifica- tion at any office of the same company where orders are sold. 3 1 5 31G EXCHANGE Money orders may be purchased for any amount up to $100, payable to any person or firm in the United States, or foreign countries where such orders are sold. The rates charged 'in the United States are as follows : $2.50 and under 3? 5? 8? 10? 12? 15? 18 ? 20? 25 (» 30? Over $2.50 and not exceeding $5.00 . Over $5.00 and not exceeding $10.00 . Over $ 10.00 and not exceeding $20.00 . Over $20.00 and not exceeding $30.00 . Over $30.00 and not exceeding $40.00 . Over $10.00 and not exceeding $ 50.00 . Over $50.00 and not exceeding $ 60.00 . Over $60.00 and not exceeding $75.00 . Over $75.00 and not exceeding $ 100.00 . The rates to foreign countries are from 10 ? to $ 1 for the same amounts as domestic orders. This fee of from 3 ? to 30? for domestic orders, and from 10? to $ 1 for foreign orders, to cover the cost of paying the bills at a distance, is called the exchange for issuing the orders. Paying by telegraphic money order. Such orders are drawn by agents of the telegraph company, and direct the agent at some designated office to pay to the person named in the telegraphic message, upon identification, the sum specified. Th3 present rates for sending money by telegraphic order are as follows : For $25 or less, double the cost of a ten-word message, plus 25 ?. Above $ 25, double the cost of a ten- word message, plus 1 % of the amount of the order. Paying by checks. Business men find it necessary to pay bills in their vicinity or at a distance, almost daily, and if their financial standing is good, their checks are generally accepted in payment. In fact, most bills to-day are paid by checks. Sometimes the seller does not know the financial standing of the pur- chaser, and therefore requires the check accompanying the order to be certified ; that is, the cashier of the bank on which the check is drawn stamps the word " certified," with the date and his signature, across the face of the check. The check is thereafter the check of the bank, and is good as long as the bank is solvent. EXCHANGE 317 A certified check is a notice to the payee of the check that the amount named on the face has been taken from the maker's deposit and placed with the bank's funds for the payment of the check when presented. Certified checks are frequently demanded in payment of notes and collections at banks, and in payments where the payee does not wish fco take a personal check. Like other checks, they are mailed daily in payment of bills in all parts of the country. 1. What is a check? What are the essentials of a check? 2. In buying a lot from James Carothers for $800, you are asked to give your certified check for the amount. Write the check on your local bank, yourself being the maker. 3. Moore & Co., Youngstown, Ohio, purchase 'IB 825 worth of furniture at 20% and 10% off from James Boydson & Co., Detroit, Mich., 3% off for cash in 10 days. They send a certified check within ten days on the Diamond Trust Co., of which James Patterson is secretary and treasurer. Write the certified check in payment of the bill. Note. — The secretary and treasurer of a trust company corresponds to the cashier of a bank. 4. William Anderson, 7531 Hermitage Ave., Chicago, 111., receives a check on the First National Bank of Wilkins- burg, Pa., from Freeman Lewis for $730.80 in settlement of an estate. The Commercial National Bank of Chicago charges Mr. Anderson $1.50 for collecting the check. This fee is called the exchange for collecting. A person who cashes a check at a bank in which he is not a depositor is frequently charged an exchange of 10^ and upward, according to the amount of the check. Paying by bank draft. A draft is a check drawn by one bank on another. As New York City and Chicago banks collect exchange on outside checks, nearly all banks keep deposits there, as 318 EXCHANGE well as in most of the other large commercial centers, to accommodate their depositors and others, who have occasion to remit payment for bills in any part of the country. Bauks usually charge exchange on drafts to cover the cost of keeping funds on deposit at these commercial centers. This fee varies from -^% to \% of the face. When the draft is less than $100, a fixed charge is frequently made, varying from 10 f to 50^. The custom of banks is not to charge depositors for drafts. In issuing or collecting a draft, the exchange is either a fee or a certain per cent of the face of the draft. New York and Chicago drafts are usually cashed at any point in the United States without exchange. Drafts on other large cities are cashed without exchange in the territory contiguous to those cities. Brown & Foster, Cleveland, O., buy $2500 worth of merchandise from John Wanamaker & Co., New York ; and $650 worth of machines from the Wheeler Wilson Co., Bridgeport, Conn. The business method of paying these bills is either by a check or by a bank draft. The draft is made payable to the order of the purchaser, who indorses it in full to the payee. For example : Cleveland National Bank Cleveland, Ohio- ^we 2, 1097 , ;y a /o^-O Pay to tlie order of Bvowi, V dfaat&v $2600^. 3w-&nty-(vv-E, /runoU&d V — -c^^c^^c^^r^vyc^DoUaj^s. To (Efje iJHercanttle National Bank, New York, JV*. Y. d. ?Vl. /MwveA,.. Cashier. EXCHANGE :;i'.t This draft simply means that Brown & Foster purchased at the bank where they kept their deposit a draft (banker's check) for the above amount. The Cleveland National Bank had money on deposit at the Mercantile National Bank, and simply checked on its deposit. Had Brown & Foster not been depositors in the Cleveland National Bank, they would probably have been charged T \% exchange. The draft would then have cost them $2502.50. The party who signs a draft is called the drawer of a draft. The party to whose order the draft is drawn is called the payee of the draft. The party who is to pay the money is called the drawee of the draft. Thus, in the draft on p. 318, the cashier of the Cleveland National Bank is the drawer; Brown & Foster the payee; and The Mercantile National Bank the drawee. Written Work 1. Find the cost of a New York draft for $550.25 at ^ % exchange. 2. Mr. Amidon buys $2500 worth of farm implements at 30% and 10% off, and pays by a Chicago draft at \% ex- change. Find the face of the draft and the exchange. 3. Find the cost of sending $80 from Pittsburg to Chi- cago by telegraphic money order, the rate being 25 f for 10 words. 4. I sent $75.80 to Chicago by express money order. How much could I have saved by purchasing a bank draft at 15 cents exchange ? 5. A draft costs $1080, including the exchange at ^%. Find the face. Suggestion. — $ 1080 is 100^% of the face. 320 EXCHANGE 6. Write a draft for $2600, making' one of your local banks the drawer, the First National Bank of Buffalo, N.Y., the payee, and the Niagara Falls Power Co. the drawee. Indorse the draft in full to James Osborne & Co., Syra- cuse, N.Y. 7. Mr. Madison had a note for 81000 discounted for 60 days at 6%, and with the proceeds bought a Chicago draft at yo% exchange, which he mailed to Mandel Bros., Chicago, to apply on account. Find the face of the draft and the cost of exchange. 8. My settlement of an account in New Orleans gives me $26785.50. After investing #13750 of this amount in a land deal on which I pay my agent 2% commission, I pur- chase a New York draft with the balance at ^% exchange. Find the exchange, the commission, and the face of the draft. 9. James Anderson & Son, Helena, Montana, order $790 worth of goods from a Boston firm, and send in payment a New York draft at \ % exchange. Find the cost of the draft. 10. A dealer in San Francisco buys $2000 worth of goods at 30% and 10% off and sells them at an advance of 25% on the cash price. After paying for these goods with a Chicago draft at \ % exchange, find his profit. Collecting Bills at a Distance Bills are collected at a distance in two ways : (1) By a sight commercial draft of a creditor on a debtor. (2) By a time commercial draft of a creditor on a debtor. Bills collected by a sight commercial draft. Tf Letche & Co., grain dealers, Pittsburg, Pa., order from Harris Bros., Chicago, 111., 1 car load of No. 1 oats, Harris Bros, will ship EXCHANGE 321 the car load of oats to Pittsburg to tlic order of themselves and draw a sight draft on Letche & Co., payable to the order of some Chicago bank, and deposit it with the bill of lading for collection. The Chicago bank will then mail the draft, together with the bill of lading, to some bank in Pittsburg. The Pittsburg bank will notify Letche & Co. If the car load of oats is accepted by Letche & Co., they will pay the draft and receive the bill of lading which will entitle them to the oats. This form of draft is commonly known as a commercial sight draft and reads as follows : $^0^. Chicago, III., fusne, 27, 1906 ~Cy^yC^C^yC^C^^Clt av^/ito^yr^^^^^^^y^yc^c^-JPay to the order of /at oAatio-na.1 Bank, <&,Aie-aao, Sow, /funded <$£fty V ^y^^^^os^c^DoUars- J'al//r received, and charge to account of To JUUAe, V &». ) , • n L Zi-aAsVva, Jovoa-., No. /#■ <t&6-Ul how much is remitted to Mr. Johnson ? 6. T. F. Bowman & Co., Chicago, sell to Speer Bros., Seattle, Wash., $4000 worth of merchandise. Terms: 60 days net; 6% off 30 days. Write the banker's check given by the Seattle National Bank and indorsed in full by T. F. Bowman & Co. Find the cost of the banker's check at I % exchange, if paid within 30 days. 7. James Brown, Lansing, Mich., sells $2500 worth of celery on March 1, 1908. to Grimm Bros., Boston, Mass., and draws a draft for 90 days after sight. The draft is accepted March 18, and discounted the same day at 6%. If the cost of collection is \ % exchange, find the proceeds from the sale of the celery. STOCKS AND BONDS STOCKS When an individual or a few persons do not wish to fur- nish all the capital or money required for a business, or to assume all the responsibility, they may secure a charter from the state government to form a corporation or stock com- pany, and choose a board of directors to transact the busi- ness in the name of the firm designated in their charter. A corporation is a company authorized by a charter to transact business as an individual. The capital stock of a company is the amount of stock for which shares are issued. Thus, 1000 shares at $10 each make a capitalization of $10000. The par value of the shares in different corporations varies from % 1 to % 100. The persons who form the corporation determine the number and par value of the shares. Observe that the certificate on p. 325 gives the number and value of the shares. The par value of a share of stock is the amount written on the face of the stock certificate. What is the par value of the stock certificate on p. 325 ? The market value of a stock is the price at which it is selling. A stock is selling at a discount when purchased for less than its pai value, and at a premium when selling for more than its par value. 324 STOCKS 325 STOCK CERTIFICATE Incorporated under the Laws of the State of Pennsylvania . 2 < 20 tf/iat&fr Enorpcnocnt Iron OTompang of pttsimrg This certifies that j/oont&o, W-oo-cL is the owner of. Sw-tnty full paid shares of the Capital Stock of One Hundred Dollars each of the Independent Iron Company. Transferable only on the books of the Company by the holder in person or by an attorney upon the surrender of this certificate. j. &. 711&I/&I, President. B. c/1 #W£A, Secretary. Pittsburg June 1, 1906. A stockholder is one who holds stock in a corporation. An assessment is a sum levied on the par value of each share of stock to defray expenses and losses when the earn- ings are not sufficient. A dividend is a part of the net profits divided among the stockholders, in proportion to the par value of their stock. These dividends are paid yearly, half-yearly, or quarterly, as the board of directors may determine. A stock broker is a person who buys and sells stocks for another. The charge, called brokerage, is usually \ % to \ % on a par value of a hundred dollars. Most brokers belong to some stock exchange. 326 STOCKS AND BONDS Par Value and Brokerage. 1. Mr. James buys 10 shares of railroad stock, par value $100 per share, at $89 per share, brokerage ^%. 1. What is the par value of each share? 2. What is the market value of each share? 3. Show that each share costs Mr. James $ 89.12J. In the study of this subject the following should always be observed : 1. When the par value of a stock is not stated it is always regarded as |100. 2. When a stock is quoted at 90, 110, 78, etc., it always means so many % of the par value. A stock quoted above 100 is said to be above par, or at a premium, and one quoted below 100, below par, or at a discount. 3. Brokerage is always reckoned on the par value and the broker collects brokerage from both the buyer and the seller. \°/ brokerage means $.12^ on a par value of $100, or $.06£ on a par value of $ 50. To find the cost we add the brokerage to the selling price or pur- chase price and then multiply that amount by the number of shares bought or sold. Brokerage is not to be computed,_unless stated in the problem. Written Work 1. Find the cost of 48 shares of railroad stock bought at 95, brokerage \ % • $95 + $ \ = |9oi = cost of 1 share. 48 x $95£ = 14566, cost of 48 shares. Find the cost of : 2. 60 shares of stock at 101, brokerage | %. 3. 88 shares of stock at 102, brokerage \°Jo- 4. 104 shares of bank stock at 116J, brokerage \ %. 5. 120 shares railroad stock at 94|, brokerage \ % • STOCKS 327 6. My broker sold for me 128 shares of mining stock at 156, brokerage |%. What sum should I receive? Find the net amount received from the sale of the follow- ing, including brokerage at \o/ : 7. 125 shares at 09|. 9. 500 shares at 132|. 8. 145 shares at 142|. 10. 1000 shares at 37|. 11. I bought 125 shares of Penn. R. R. stock at 123| and sold it at 129|; brokerage |%. Find the net gain. $ 129f - %\ = $129.25, amount realized from each share $ 123J + $ I = $123.625, cost of each share $5,625, gain on each share 125 x $5,625 = $703.13, net gain Why do we subtract the brokerage when selling stock? Why do we add the brokerage when buying stock ? 12. How many shares of railroad stock at 108 1, brokerage l %, can be purchased for $26100 ? 13. How many shares of stock must be sold at 99|> brok- erage | %, to pay a debt of $793 ? 14. I receive $1375 net profits on stock bought at 58 and sold at 72, brokerage ^ % in each case. Find the number of shares. 15. If I realized $1595 net from the sale of stock, broker- age | %, rind the number of shares sold at $40 per share. 16. How many shares of stock selling at $45 per share, brokerage \ %, can be purchased for $2000 ? (Parts of a share are not sold.) 17. A note for $5000, with interest at 6 %, was paid 1 year 4 months and 18 days after date and the amount invested in stock at 87|. Find the number of shares purchased. 328 STOCKS AND BONDS Premium and Discount. 1. What is the cost of 1 share of stock, par value $100, selling at 10 % discount ? at 40 % premium ? at 30 % discount ? 2. What is the cost of 60 shares of bank stock, par value $50, at 5% premium, brokerage \ % ? Par value of 1 share = $50. Premium of 1 share = .05 x $50 = $2.50. Brokerage on 1 share = .00^ x $50 = $ A2\. Cost of 1 share = $50. + $2.50 + $.12|, or $52.62£. Cost of 60 shares = 60 x $52.62| = $3157.50. Note. — All premiums are reckoned on, and added to, the par value; and all discounts are reckoned on, and subtracted from, the par value. Find the cost of the following stock, brokerage ^ % : 3. 16 shares, par value $50, at 3% premium. 4. 42 shares, par value $50, at 10% discount. 5. 100 shares, par value $25, at 14 % discount. 6. 220 shares, par value $100, at \ % premium. 7. 150 shares, par value $50, at f % discount. 8. My broker purchased for me 256 shares of milling stock, par value 100, at 4-|-% premium, brokerage \°/ . How much did the stock cost me ? 9. How much will 120 shares of railroad stock cost at 114f , brokerage | % ? 10. How much is realized from the sale of 480 shares of gas stock, par value $50, at 2% discount, brokerage \°Jo ? 11. How many shares of stock can be bought for $2475, at 3 % premium, brokerage \ % ? Par value of 1 share = $ 100 Premium on 1 share = $ 3. Brokerage on 1 share = $ .125 Entire cost of 1 share = $ 103.125 $ 24750 - $ 103.125 = 240, number of shares STOCKS 320 Find the number of shares bought for : 12. $5827.50, par value $50, at 3% discount, brokerage Wo- 13. $11970, at 106|, brokerage |%. 14. $ 2165, par value $ 50, at 8% premium, brokerage \%. 15. $ 19677, at 116|, brokerage \%. 16. $10025, par value $50, brokerage \ %. Dividends and Investments. 1. What is the income from $ 1000 loaned for 1 year at 6%? 2. What is the income from $ 1000 invested in a manu- facturing plant that pays 8% in dividends each year? 3. Why is a $100 share of stock that pays $ 12 in divi- dends each year worth more than $ 100 ? 4. Why is a $100 share of stock that pays only $2 in dividends each year worth less than $ 100 ? 5. 1|% dividend payable quarterly is equivalent to what per cent payable annually ? 6. $ 60 per year is the dividend on $ 1200 worth of stock par value. Find the rate of dividend. 7. I receive $120 from a dividend of 6%. What is the par value of my stock ? 8. Mr. Johnston owns 100 shares of stock in a company whose capital is $200000. If a dividend of 8% is declared, find the amount of the check that will pay the whole divi- dend ; the amount of Mr. Johnston's share. Observe: 1. Dividends are always declared on the par valne of a stock. 2. Incomes are always reckoned on how much a stock costs. 330 STOCKS AND BONDS 9. A dividend of 6 % is declared on a stock, par value $100, purchased at $ 120. What % is received on the investment ? 6% of $ 100 = $ 6, income on one share <| 120 = cost of one share $ 6 h-$120 = .05,or5% 10. A share of stock, par value $ 100, is sold at $ 250. What is the per cent of income if an annual dividend of 10% is declared? Find the rate of income when : 11. $150 is paid for 9% stock, par value 1 100. 12. $133^ is paid for 6% stock, par value $100. 13. $75 is paid for 8% stock, par value $50. 14. $ 80 is paid for 5% stock, par value $100. 15. $50 is paid for 4 % stock, par value $100. 16. Would a stock yielding 6% have to be purchased for more or less than par value to yield 8% on the investment ? Explain why. 17. Explain why a stock, par value $100, dividend 12%, yields only 6% on the mone}'" invested when purchased at $200. 18. A man buys 120 shares of stock at 137|, and receives a 6% dividend. He sells it at 141f. Find his net profits on the investment after paying brokerage at -| % each for buying and selling. 19. Which yields the better income and how much, 6% stock at $ 120 or 4% stock at $ 85? 20. The Amidon Asbestos Co. is capitalized at $80000. The gross receipts for a year are $ 170000. The expenses, material, and repairs amount to $ 14(3000. If $ 14000 is put in the surplus fund, what dividend can be declared from the balance ? BONDS 331 21. A gas company declares a dividend of 8% which amounts to $64000. What is its capitalization ? 22. A bank is capitalized at 1100000, and pays S% dividend. How much is the dividend on 3(3 shares? 23. How much must be invested in 0% stock, at 108, brokerage -|%, to yield an annual income of I 366? Since $1 of stock 8 366 -=- $.06 = 6100; $6100, par value yields $ . 06 income, the 1.08^ x$ 6100 = $ 6595.63, sum invested par value to yield § 366 must be as many dollars as $.06 is contained times in $366, or $6100. Adding brokerage, the sum invested must be 1.08$ times $6100, or $6595.63. 24. If a stock paying 3%% semiannual dividend is quoted at 120, how much must be invested in it to produce an annual income of 81400, brokerage \°fo ? What sum must be invested, at \ % brokerage, in : 25. 3i% stock, at 104, to yield an annual income of $245 ? 26. 5% stock, at 109, to yield an annual income of $1675 ? 27. 4^% stock, at 116 J, to yield an annual income of $364.50^? BONDS When corporations need large sums of money to carry on their business, instead of issuing more stock, they frequently issue a series of bonds payable at some future date with in- terest. Bonds are written obligations, under seal, by which cor- porations or governments bind themselves to pay specified sums, at a fixed rate of interest, at or before the time speci- fied in the bonds. The bonds of a business corporation are secured by a mortgage on its property. This mortgage authorizes the 332 STOCKS AND BONDS sale of the property in case the conditions of the bonds are not fulfilled. The bonds of governments are without mortgage. Bonds and stocks are at a premium when they sell for more than the face, or par value ; at a discount when they sell for less than the face, or par value. Coupon ACME GL2ASS COMPANY will pay to Bearer at the Colonial Erust GTo. of $ tttsuttrg, $a. on the first day of June, A. D., 1907, in United States Gold Coin, being six months' Interest on Bond No. 501 fawve& &hui&, Treasurer. A coupon bond is a bond with interest coupons attached. These coupons are detached when the interest is due, and the amount may be collected personally or through a bank. Coupon bonds are payable to the bearer. A registered bond is a bond registered on the books of the corporation issuing it. The interest when due is sent by check to the owner. Registered bonds are payable to the owner or to his assignee. The name of a bond often indicates its rate of interest and the time when the bond is due. Thus, " U. S. 4's, 1907," are United States 4 % bonds, due in 1907 ; " Western Union 7's, 1920," are Western Union bonds, due in 1920, and BONDS 333 bearing ~ bonds is $240 per year. How much have I invested ? 12. I bought 24 shares of mining 'stock at 89 and, after keeping it 3 years, sold it at 39. As no dividends were paid, find my loss, money being worth 6% simple interest. 13. A $1000 5% bond, bought at par, after paying 6 annual dividends, is sold for $.60 on the dollar. Find the loss, money being worth 5% simple interest. 14. A certain stock bought in 1904 at 40^ per share was sold in 1906 at $2.40 per share. Find the gain per cent on the investment. 15. A $1000 5% bond due in 10 years was purchased for 81100. Find the average rate of interest on the investment, if the bond is held to maturity. The total income on the bond for 10 years at 5% = $500. Loss by redemption of bond §1100 - 81000 = $100. Total income in 10 yr. = $500 - $100 = $400. Average annual income = ^ of $400 = $40. Average annual rate = $40 -*- $1100 = 3^%. 16. A §2000 4 % bond due in 5 yr. was bought for $1900. Find average rate of interest, if bond is held to maturity. Note. The purchaser gains $100 when the bond is redeemed. TEST PROBLEMS IN PERCENTAGE 1. Mr. Byers's farm is valued at $18000. He pays 4-^j mills taxes on an 80% valuation of it. Find his tax. 2. An agent buys 30 tons of fertilizer, at $1.50 per hun- dred, 20%, 10% off. Terms: 30 days net, or 2% for cash. If he pays cash, find the cost. 3. A western farmer sells 10000 bu. of wheat, \t per bushel brokerage, at 89^. After deducting freight and drayage of $>67A, find the net amount of the sale. 4. An Iowa farmer buys cattle for $1500, and sells them for $2350. If the grazing and feeding are 20% of the cost, find the per cent of profit on the sale. 5. A merchant has a note against Mr. Johnston for $500, bearing 6 % interest, dated June 1, 1906, due in one year. If he discounts the note at his bank March 1, 1907, at 7 %, find the proceeds. 6. A farmer has the following annual insurance on his prop- erty : house valued at $ 2000, insured at | % ; barn valued at ?2500, insured at T 9 o%; grain valued at $1000, insured at |%; live stock $1200, insured at |%. If twice the annual premium covers the cost of the insurance for 3 years, find the yearly cost if his property was insured for 3 years. 7. Mr. Ames, who keeps a general store, sold goods in one year amounting to $24000. If he has an average of 20% profit on the cost of the goods, find his profits for the year. 335 336 TEST PROBLEMS IN PERCENTAGE 8. A manufacturer, owing to a depression in business, offers goods at 12|% discount, but finally sells at a further discount of 8%. Find the entire per cent of discount. 9. What per cent is gained by buying stocks at 15 °} discount and selling at 5 % premium, brokerage \ °J ? 10. The tax rate in a certain city is 17 mills on the dollar on a valuation of $66390. Find the tax on a property val- ued at $12500. 11. A salesman who received a salary of $2400 and $1500 expenses, sold $75000 worth of goods. In addition he re- ceived 2% on all sales over $60000. What per cent of the selling price of the goods did it cost the firm to sell them ? 12. A man buys through his broker 10 shares of railroad stock (par value $100) at $125 per share. After receiving 5 semiannual dividends of 3% each, he sells the stock at $131^. Find the rate per cent of gain on the investment. 13. By selling a piano at 40 % above cost, a profit of $150 is realized. For how much must the piano be sold to realize a profit of 56 (f ? 14. A collector is given a bill of $1500 to collect at 5%. He succeeds in collecting 90 cents on the dollar. Find how much is due his client and how much is the collector's com- mission. 15. A house and lot cost $6000. The insurance averages $14, taxes $50, and repairs $56 annually. For how much must the house rent per year to realize 6% net on the investment ? 16. A house rents for $40 per month, and it costs the owner on an average $125 per year for insurance, taxes, and repairs. If the property yields him 5% net on the investment, find the cost of the house. RATIO AND PROPORTION RATIO 1. The quotient of 30 -f- 10 is 3. Compare 30 with 3 in such a way as to show how many times 3 is contained in 30. What, then, is the relation of 30 to 3? Ratio is the relation of two similar numbers as expressed by the quotient of the first divided by the second. 2. What is the ratio of 10 to 8 ? of 12 to 4 ? of 3 to 6? of 2 yd. to 8 yd.? of $12 to 83. Since the division of two similar numbers gives an abstract quotient, all ratios are abstract. The sign of ratio is a colon : placed between the numbers. Thus, the ratio of 12 to 3 is written 12 : 3. It is read, l » the ratio of 12 to 3." It may be written also 12 ~- 3, or J^-. The terms of a ratio are the numbers compared. The first is the antecedent ; the second the consequent. 19 # r _ antecedent _ 19 _i_r -- dividend !- numerator consequent divisor denominator Since the antecedent of a ratio may be regarded as the numerator, and the consequent as the denominator of a frac- tion, both terms of a ratio may be multiplied or divided by the same number without changing the value of the ratio. Find the ratio of : 3. 10 to 5 7. 18 to 9 11. 40 to 10 15. 50 to 25 4. 5 to 15 8. 27 to 9 12. 8 to 24 16. 24 to 8 5. 8 to 2 9. 35 to 5 13. 2 to] 17. 4 to 1 6. 2 t0 i" HAM. COHP1 10. J to 1 .. AKITH. — 22 14. :;37 fto| 18. ftoj 338 RATIO AND PROPORTION Written Work Find the value of the following ratios : 1. 125 : 25 5. -2- to -1- 3 lo 12 9. $225 to $2.25 2. 6.25 : 25 6. (3.4 to 16 10. 3 yd. to 3 ft. 3. 1 toj 7. 37| to 200 11. 75% tol2|% 4. i to H 8. &2} to 500 12. 1 mi. to 1 rd. SIMPLE PROPORTION Proportion is an equality of ratios; thus, 12: 6 = 8: 4 or -g 2 - = | is a proportion. Proportion is generally indicated by the equality sign or by a double colon : : between the ratios. Thus, 12:6 as 8 : 4, is written, 12 : 6 = 8 : 4, or 12 : 6 : : 8 : 4. A proportion may be read in two ways ; thus, 12:6 = 8:4 is read, " The ratio of 12 to 6 is equal to the ratio of 8 to 4," or, "12 is to 6 as 8 is to 4." The extremes are the first and the fourth terms of a pro- portion ; the means are the second and the third terms. In 15 : 5 = 12 : 4, the extremes are 15 and 4 ; the means, 5 and 12. Find the product of the means ; then the product of the extremes : 1. 8 : 4 = 10 : 5 3. 24 : 4 = 36 : 6 5. § = f 2. 15:3 = 30:6 4. | = T % 6. | = ^ Observe how the product of the extremes in each proportion compares with the product of the means. In every proportion the product of the means is equal to the product of the extremes. SIMPLE PROPORTION :;:',!) Written Work Find the value of x, the unknown term : 1. 36: 0=24: x Then, 30 times x, or 36 x = 144, and once x, or x = 4. 2. 15 : 25 = x : 40 Then, 25 x = 15 x 40, or 600, and x = 24. 8. 9. 10. 11. 12. 3. 60 : 15 = 75 : x 4. 75: a; =90: 18 5. 40: z=72:18 6. x.: 30 = 8 : 48 7. a: 45 = 7 : 63 3 5 5 8 2 _ q 3 — ' z= 25 < .5 : 1.5=2.5 6.25 : 2.5= x 60 150= 36 X 8 x 1 a; 13. If 8 sheep cost $48, how much will 20 sheep cost ? 8 sheep cost $48 20 sheep cost $ x Since ratio is the relation of two similar numbers, 8 sheep and 20 sheep form one ratio, and $48 and $x, the other ratio. Write as the second ratio $48 : $x. Since 20 sheep cost more than 8 sheep, $x represents a larger sum than $48. Therefore, as the larger number is the consequent of the second ratio, the larger number must be made the consequent of the first ratio. The proportion, therefore, is, 8 sheep : 20 sheep = $48 : $z 8x = $960 x = $120 14. It is estimated that 25 men can build a bridge in 18 days. How long at the same rate will it take 15 men to build it ? 15. How much will 30 bushels of potatoes cost, if 70 bushels cost $42 ? 16. It is estimated that 90 men are necessary to grade a certain street in 45 days. If only 81 men are hired to do the work, how long will it take them? 340 RATIO AND PROPORTION 17. The ratio is }. The first term is | of (6x4). What is the second terra ? 18. A bankrupt's debts are $32000, and his assets $10000. Counting nothing for court costs, how much will be paid on a claim of $6150? 19. If $2.25 is paid to clean 35|- square yards of paper, how much at the same rate will it cost to clean 65| square yards ? 20. A bakery sells 5^ loaves weighing 6 oz. when flour is $4. What size loaves should they sell at 5^ when flour is $6 per barrel? 21. A map is drawn on a scale of 100 miles to | of an inch. What distance is represented on the map by T 5 g of an inch ? 22. If the interest on $500 for 6 months is $15, how much is the interest on the same sum for 1 year 4 months ? 23. If 6.5 tons of coal cost $55.90, how much will 9.25 tons cost at the same rate ? 24. A estimates that he can do a piece of work in 20 days, working 8 hours per day. How long will it take him to do \ of it, working 10 hours per day ? 25. Sound travels 1120 feet per second. How long will it take the sound of a cannon to travel 8 miles ? 26. In a stamp canceling machine 1000 letters were can- celed in one minute and 20 seconds. If the machine was in operation for 5 minutes and 10 seconds, how many letters at the same rate were canceled ? 27. A monument casts a shadow 150 feet long. At the same time a post 3 feet in height casts a shadow 2 feet and 6 inches long. Find the height of the monument. 28. A machine for making pressed brick turns out 7500 brick in 6 days. How large an order for pressed brick can be filled in 25 days ? * PARTITIVE PROPORTION AND PARTNERSHIP ^41 29. An automobile passed 5 mile-posts in 8 minutes 10 seconds. How many miles per hour was it moving? 30. It is estimated that 24 men working 18 days can re- pair a certain street. The contract calls for the work to be completed in 8 days. How many extra men must be employed? 31. It is estimated that GO men can dig a sewer on Main Street in 24 days. The contract time is 40 days. How many men may be discharged and yet have the work com- pleted within the contract time ? 32. A's property is assessed at $2750, on which $37.90 taxes are paid each year. How much tax should B pay on his property assessed at $4375 ? 33. A train of 30 cars of ore contains 1200 tons. How many cars must be added so that the train may carry 2700 tons ? 34. A city's assessed valuation is $5675000. There must be raised in taxes on this valuation $ 85125. How much is Mr. Templetons tax on a property assessed at $15750? PARTITIVE PROPORTION AND PARTNERSHIP Partitive proportion is the process of separating a number into parts proportional to two or more numbers. Written Work l. Separate 180 into parts proportional to 1, 2, and 3. Since the parts are in the ratio of 1, 2, and :5, then 1 + 2 + 3, or G parts = 180. The 1st number = 180 - 6, or 30 The 2d number = 2 x 30, or GO The 3d number = 3 x 30. or 90 Test : 30 + 60 + 90 = 180 ; 30 : 60 : 90= 1 :2: 3. 342 RATIO AND PROPORTION 2. A man and two boys earn $162 and agree to divide it as follows : 3 parts to the man, 2 parts to the first boy, and 1 part to the second boy. How much should each receive? 3. The receipts of a street railway in one month are $15600, and the expenses are to the profits as 1 to 2. Find the expenses and the net savings. 4. Four men own a gold mine valued at $805000. The parts owned by each are in the ratio of 6, |, f, and T 9 g. Find each man's share of the mine. 5. The cost of shipping a train load of 2000 tons of iron ore from Duluth, Minn., to Bessemer, Pa., is $2800. If the lake freight is to the railroad freight as 50 to 90, find each one's share of the freight charges. 6. Two railroads, valued at $6900000 and $23000000, share charges of $299000 for freight carried over both roads in proportion to the valuation of each road. Find the earnings apportioned to each road. Partnership is the associating of two or more persons who agree to combine their money, labor, goods, skill, or " good will " in some enterprise, and to share the profits or losses of the business in proportion to the interest each part- ner owns. The partnership is frequently called a firm, or a house, and derives its name from the persons that compose it; as," Brown & Hamilton." The capital of a partnership is the sum of the investments of the partners. This capital may be money or anything that has a money value, as skill, good will, experience, labor, etc. Gains and losses in a common partnership are usually appor- tioned in proportion to the amount of capital each partner invests and the length of time such capital is invested ; but in case any partner cannot pay his proportionate share of the loss, the re- maining partners are liable for the whole loss. PARTITIVE PROPORTION AND PARTNERSHIP 343 Written Work l. A and B engage in business ; A furnishes $800 and B, 81200; they gain $500. What is each man's share? $800 + $1200 = $2000, entire capital SS,H) $2000 = I, A's share of the capital ® UQ0 = §, B's share of the capital $2000 | of $500 = 8200. A's .share of the gain | of $500 = $300, B's share of the gain Or, $2000 : $800 = $500 : $200 $2000 : $1200 = $500 : $300 The ratio of the whole capital to each partner's investment is equal to the ratio of whole gain or loss to each partner's share of the gain or loss. 2. A, B, and C engaged in manufacturing iron. A in- vested 842000, B $96000, and C his skill, valued at $72000. Their profits the first year were $12600. How much was each man's gain ? 3. M, N, and R formed a partnership; M furnished | of the capital, N §, and R the remainder. They gained $7560. What was each man's share of the gain ? 4. E, F, and G engaged in merchandizing with a capital stock of $28000. E furnished 87000, F $6000, and G the remainder. They gained 14f% on the investment. What was each man's share of the gain? 5. The assets of a firm that failed in business were $3750 ; their liabilities 8 22000. How much will two credit- ors, to whom they owe $7800 and $5400 respectively, receive? 6. A storeroom belonging to Smith, Jones, & Brown was entirely destroyed by fire. They received $9675 insur- ance. What was each man's share, if Smith owned ^, Jones ^, and Brown the remainder of the stock? 344 RATIO AND PROPORTION 7. A and B formed a partnership January 1, and each invested 12500; May 1 A added $500, and B withdrew $500. At the end of a year their gain was $1800. How much should each one receive? A's capital, $2500 for 4 mo. = $10000 for 1 mo. A's capital, $3000 for 8 mo. = $24000 for 1 mo. A's total capital = $34000 for 1 mo. B's capital, $2500 for 4 mo. = $10000 for 1 mo. B's capital, $2000 for 8 mo. = $16000 for 1 mo. B's total capital = $26000 for 1 mo. Total capital of both = $60000 for 1 mo. A , • 34000 As Gram = , or fe 60000 ?0> of $1800 = $1020. -o, ■ 26000 B s sain = , oi & 60000 1 3 3T>> of $1800 = $780. $1020 -f $780 = $1800, total gain. Test: 8. M and N formed a partnership for 2 years. M put in $6400; N put in $3600 and at the end of 6 months added $1400. Their settlement at the end of 2 years showed $7956 profits. How should it be divided? 9. R and S began business as partners April 1, 1904, each investing $5000. On July 1, 1904, R added $3000 and S, $2000. They dissolved partnership January 1, 1905, sharing a profit of $3150. Find each one's share. 10. A, B, and C formed a partnership for 3 years. A put in $10000, B $8000, and C $6000. A withdrew $2000 at the end of 18 months. They dissolved partnership at the end of 2 years with a loss of $4750. As nothing could be collected from C, what proportionate share of the loss should A and B pay? PROBLEMS FOR ORAL AND WRITTEN ANALYSIS 1. Two properties are valued at $1000; J of the value of the first equals f of the value of the second. Find the value of each. 2. At a certain election 1080 votes were cast for A and B ; | of the votes cast for A equaled f of those cast for B. How many votes were cast for each candidate ? Solution. — f of A's vote = | of B's vote. \ of A's vote = i of % of B's vote, or 1 of B's vote. | or A's vote = 4 x 4, of B's vote, or 4. of B's vote. | of B's vote = B's vote, f of B's vote = A's vote. | of B's vote = vote of both, or 1080 votes. B's vote = 600. A's vote = 480. 3. A real estate dealer paid 87200 for two city lots; f of the cost of the first lot equaled ^ of the cost of the second. How much did each cost? 4. A mill and machinery cost $27000; f of the cost of the mill equaled f of the cost of the machinery. How much did the machinery cost? 5. What per cent of a day are 12 hours? 6 hours? 36 hours? 6. | of Frank's money equals £ of Henry's, and Frank has 83 more than Henry. How much has each? 345 346 PROBLEMS FOR ORAL AND WRITTEN ANALYSIS 7. Walter and Philip bought sleds; f of the cost of Walter's sled equaled | of the cost of Philip's; both sleds cost $2.70. How much did each cost? 8. A pair of shoes that cost a dealer $2.50 were sold for $3.50. What was his gain per cent? 9. An estate was so divided between two sons that the share of the elder was to that of the younger as | to ^. If the elder son received $1000 more than the younger, what was the value of the estate ? 10. Brown and Long were partners in business ; Brown furnished | as much capital as Long, and their profits for* the first year were $2250, which was divided in the ratio of the capital invested. What was the share of each ? 11. In a partnership A invested | as much as B, and C in- vested | as much as B ; they shared a loss of $ 2000. How much should C pay ? 12. A piano sold for $360, which was at a loss of 20 %. What was the cost? 13. Moore, Silvens, and Rogers were partners in business and made a profit of $ 4500. Moore owned y 4 ^ of the stock, Silvens •§, and Rogers ^. What was each partner's share of the total profit ? 14. A clerk's expenses are $30 a month, which is 66| % of his salary. How much is his salary ? 15. A stone cutter received $4 a day for his labor and paid $6 a week for his board. At the end of 16 weeks he had saved $212. How many days did he work ? 16. If in an investment | of A's capital equaled | of B's, and A received $900 for managing the business, how should profits of $5100, including cost of management, be divided? PROBLEMS FOR ORAL AND WRITTEN ANALYSIS 347 17. A wagon was sold for I 12, which was 12.] % less than the price paid. What was the cost of the wagon? 18. A carpenter's wages were $3.50 a day, and he paid 50 cents a day for his board. If in 40 days he saved $>99, how many days was he idle ? 19. A real estate dealer bought some lots at #150 each, and twice as many at 8175 each. He sold them at $200 each, thereby gaining $800. How many did he buy in all? 20. A manufacturer pays boys $1, women $1.25, and men $1.75 a day, and his weekly pay roll is $348. He employs three times as many boys as men, and twice as many women as men. How many persons does he employ ? 21. A farmer paid $7200 for two farms of equal size, pay- ing $50 an acre for one and $40 an acre for another. How many acres were there in both farms ? 22. A dealer bought 20 dozen glasses at 50^ per dozen. At what price per dozen must he sell them to make a profit of 20 % on the transaction ? 23. It is estimated that 80 men can make an excavation for a public building in 30 days. After working 12 days, \ of the men were discharged. In how many days could the remainder finish the work ? 24. A father desires that the amount of $5000 for 6 years at °J shall be divided between his son and daughter in the ratio of 8 to 9. Find the share of each. 25. An architect gets a commission of 5% for drawing plans and superintending the construction of a building cost- ing 825000. How much is his commission ? 26. The interest on | of Robert's money and | of Samuel's money for 4| years, at 6%, is $81 and $121.50 respectively. How much money has each? 348 PROBLEMS FOR ORAL AND WRITTEN ANALYSIS - 27. The sum of E's and F's money being on interest for five years, at 5%, amounts to $3000. How much money has each, if E's is f of F's? 28. The amount of a certain principal for 4 years at a certain per cent is $620 ; and for 7 years, $710. Find the principal and the rate per cent. 29. 30 is 6 % of what number? 25 is | % of what number? 30. It is estimated that 15 men can build an embankment of earth in 20 days. If 5 additional men are employed, in how many days can it be built? 31. Harley spent ^ of his money and $5 more for a suit of clothes, and had $11 remaining. How much money had he at first? 32. After paying 25 % of his debts, a merchant found that $240 would pay the remainder. How much did he owe at first? 33. How shall I mark goods that cost $750, so that I can deduct 10 °J from the marked price, and yet make 20 % on the cost? 34. A speculator bought wheat at 80^ per bushel and sold it at 90^ per bushel. How many bushels did he buy if his gain was $2000? 35. An agent remitted $95 as the proceeds of an account he collected. How much did he retain if his rate of commis- sion was 5 % ? 36. A bankrupt, who owed $12000, paid 60^ on the dol- lar. Find A's claim, and his loss if he received $600. 37. 8 men take equal shares in an oil lease, agreeing to give the owner of the land \ royalty on all oil produced. How much greater interest in the oil has the owner than any of the other men ? LONGITUDE AND TIME Meridians are im- aginary lines passing north and south from one pole of the earth to the other. The equator is an imaginary line pass- ing around the earth midway between the poles. These imaginary lines aid in locating places on the earth and in determin- ing differences in time. Observe that the equator is a circumference of a circle ; therefore distances along it are measured in degrees. The prime meridian is a meridian from which time and place, east and west, are reckoned. The meridian passing through the Royal Observatory at Greenwich, England, is the prime meridian in common use. Longitude is the distance east or west of this prime merid- ian measured in degrees. Places east of this prime meridian have east longitude; places west of this prime meridian have ivest longitude. From the time the sun's rays are vertical over any meridian until they are vertical again it is 24 hours. Therefore, any 349 350 LONGITUDE AND TIME point passes through 360° in one rotation of the earth on its axis. Since 360° of longitude pass under the sun's vertical rays during 24 hours, how many degrees pass during 12 hours ? 1 hour ? Since ■£% of 360° or 15° pass under the sun's rays in 1 hour, then 1 hour of time corresponds to 15° of longitude. Since 15° of longitude correspond to 1 hour of time, ^ of 15° or |°, or 15' of longitude, correspond to 1 minute of time, and 15" of longitude to 1 second of time. Table of relation between longitude and time : 360° of longitude correspond to 24 hours of time 15° of longitude correspond to 1 hour of time 15' of longitude correspond to 1 minute of time 15" of longitude correspond to 1 second of time 1° of longitude corresponds to 4 minutes of time 1' of longitude corresponds to 4 seconds of time When the sun's rays are vertical on the 90th meridian, all places on that meridian have noon. The rotation of the earth from west to east makes the sun appear to move from east to west. The Mercator's map on p. 351 shows that when it is noon at Greenwich, it is before noon or earlier at all places west, because the sun's rays are not yet vertical on any meridian west of the prime meridian. It is after noon or later at all places east, because the sun's rays have already been vertical on all meridians east of the prime meridian. Examine the map. What time is it on the meridian of Greenwich ? 45° east of Greenwich ? 45° west of Greenwich ? In traveling from London to New York would a watch be set forward or backward'? about how much? About how much change in time must be made in traveling from Cal- LONGITUDE AND TIME 351 cutta westward to Sun Francisco? from Honolulu east- ward to Cape Town ? 135 105 75 45 15 15 45 75 105 135 East 160 150 I20i°"9 90 w«> 60 30 30 60 Long 90 East 120 Map showing Noon, February 1, at Greenwich What is the difference in degrees between a place 30° east longitude and a place 45° east longitude? What is the difference in time and which has the earlier time ? Table of Longitude of Some Important Places London 0° 5' 48" W. Cape Town 18° 28' 45" E. New York 74° 0' 3" W. Honolulu 157° 50' 30" W. Pittsburg 80° 2' 0" \V. Tokyo 139° 11' 30" E. Washington 77° 3' 00" W. Manila 120° 58' 0" E. Chicago 87° 30' 42" W. Canton 113° 10' 30" E. San Francisco 122° 2.">' 42" W. Berlin 13° 23' 44" E. Boston 71° 3' 50" \V. Rome 12° 27' 11" E. 1 >>• iiver 104° 58' ()" W. Paiis 2° 20' 15" E. Longitudes are given to the nearest seconds. 352 LONGITUDE AND TIME Written Work 1. When it is noon, solar time, at Paris, what is the solar time at New York ? Since the earth rotates 15° in 1 hr., 15' in 1 tnin., and 15" in 1 sec, the differ- ence in time is as many hours, minutes, and seconds as there are degrees, minutes, and seconds in T x 5 of the difference in longitude. The difference in time is 5 hours, 5 minutes, 21| seconds. Since New York is west of Paris, the time in New York is earlier ; that is, when it is noon at Paris, 't is 6 o'clock, 54 min., and 38| sec. a.m. at New York. 2° 20' 15" E. 74° 0' 03" W. 15)76° 20' 18" 5° 5' 91 \U * dl 5 5 hr. 5 min. 21isec. 5rir-.5niin. 2iy s .| 12 Noon NewYork ' n lon 3 J6°20'\& Paris What is the difference in longitude between the two places? the dif- ference in time? In going west from Paris to New York would a trav- eler set his watch forward or backward? how much? Note. — Study this diagram and make a similar one for each problem. Find difference in degrees, difference in time, and which place has earlier time : Places 2. 60° W. and 45° W. 3. 120° W. and 75° W. 4. 15° W. and 45° E. 5. 30° E. and 60° E. 6. 75° W. and 30° E. Places 7. 120° W. and 30° E. 8. 90° W. and 30° E. 9. 135° E. and 30° E. 10. 45° W. and 60° E. 11. 45° E. and 15° E. LONGITUDE AND TIME 353 When the sun's rays are vertical on the meridian of Wash- ington, find the solar time in the following places : 12. Denver 15. Paris 18. Berlin 13. Chicago 16. Rome 19. New York 14. San Francisco 17. Honolulu 20. Pittsburg 21. When it is midnight (solar time) on the last day of the year in Boston, how much of the year (solar time) remains to the people of Honolulu ? 22. The first shock of the earthquake at Kingston, Jamaica (long. 76° 47' W.), Jan. 14, 1907, occurred at 3:25 p.m. What was the solar time at New York ? at Cape Town ? 23. A ship sets sail from Liverpool for New York, Jan. 10, 1907. When in longitude 34° 6' 10" W. its chronometer reads 2:30 p.m. Jan. 15. Find the difference in the read- ings between the ship's time and the meridian time of New York. 24. Berlin meridian time is 6 hr. 44 min. and l\i sec. later than Chicago meridian time. Find the longitude of Berlin. 101 26 difference in longitude 87 36 42 W. (Chicago) 101 26 13 23 44 E. (Berlin) 15 times the difference in time expressed in hours, minutes, and seconds corresponds to the difference in longitude expressed in degrees, minutes, and seconds. Therefore, 15 x 6 hr. 44 min. \\\ sec. corresponds to 101° 0' 26" of longitude. This difference in longitude would not tell us whether Berlin is east or west of Chicago, but as Berlin has faster time than Chicago, it must be east of it. Chicago is 87° 36' 42" west of the prime meridian. There- fore, Berlin must be 13° 23' 44" east of the prime meridian. HAM. COMPL. AIMTII. — 23 hr. min. sec. 6 44 m 15 354 LONGITUDE AND TIME 25. The "Treaty of Portsmouth" between Japan and Russia was signed at Portsmouth, N.H., Sept. 5, 1905, at 47 minutes past 3 p.m., 75th meridian time. What was the cor- responding solar time at St. Petersburg, Russia, 30° 17' 51" E. and at Tokyo, Japan, 139° 44' 30" E. ? Tokyo is east of the 75th meridian 214° 44' 30", therefore its time is 14 hr. 18 min. 58 sec. faster than Portsmouth, which has 75th meridian time. Counting this time forward from 3:47 p.m. Sept. 5, gives 5 minutes and 58 seconds past 6 o'clock a.m. Sept. 6, Tokyo solar time. 26. The President of the United States takes the oath of office at 12 noon, 75th meridian time. Find the solar time and date in each of the following places for the inaugura- tion of March 4, 1909 : Honolulu ; Berlin ; San Francisco ; London. International Date Line The nations have agreed upon the 180th meridian, with slight changes as shown on page 355, as the place where the new day always begins. The calendar is set forward one day on ships crossing this line sailing westward : the calendar is set back one day on ships crossing this line sailing eastivard. 1. A ship sets sail from San Francisco for Manila, Oct. 9, 1906, at 9 A.M., 120th meridian time; it arrives at Manila, Oct. 27, at 9 a.m., meridian time. How long is the voyage ? 2. The same ship sets sail from Manila, Nov. 3, 1906, at 3 p.m., meridian time, and arrives at San Francisco, Nov. 23, at 3 p.m., 120th meridian time. How long is the voyage ? Standard Time The railroads of the United States in 1883 agreed upon a system of standard time and divided our country into four time belts, as shown on the map. In the eastern time belt- all trains keep the time of the 75th meridian, known as STANDARD TIME 355 eastern standard time. In the central belt all trains keep the time of the 90th meridian, known as central standard time. In the mountain time belt all trains keep the time of the 105th meridian, known as mountain standard time. In the Pacific time belt all trains keep the time of the 120th meridian, known as Pacific standard time. Each railway has selected the most convenient towns on its route, as is shown on the map, to change from the standard time of one belt to the standard time of another belt. The time in any belt is 1 hour faster than the time in the belt west of it, or 1 hour slower than the time in the belt east of it. Correct time is telegraphed each day to all parts of the United States from the Naval Observatory at Washington. /~*V. pr»ndca ; • .-• \ — Vandaa • / : u Fort Minneapolis '. i < ' 'peJI"r." 1 .roag Pine _.'.-- . ! V North .'■ K^ l f f ^*°\ oke f-.McCook V « ' * 'PbillipsTaurt* ■ .*v IV-nv ^'sco :' O \\° L. ° : A. rJV ' .".---"-.„ :■.■•■;■' ■St.LpvJS \ > ' "— £. ^vpagc City* Hoieington £ i Dallaa '• STANDARD TIME / BELTS ISO y - / '~"\ j, J^O^J ^few Orleans Galveston 1. What is the difference in time between closing of the election polls at 7 p.m. in New York and 7 P.M. Seattle, on the day for choosing presidential electors ? 2. A telegraph message was sent from Philadelphia at 11 a.m. Oct. 12, l!»0*i, to San Francisco and delivered at 7:45 A.M. San Francisco time. Why could this be true? GOVERNMENT LAND MEASURES Surveyor's square measure is used by surveyors in measur- ing and computing land areas. 16 square rods = 1 square chain 10 square chains = 1 acre 640 acres = 1 square mile 36 square miles = 1 township The Gunter's chain for measuring land is gradually going out of use. In its place surveyors use a steel tape 100 ft. long, divided into feet and decimal parts of a foot. They find the number of square feet in the plot to be measured, and change the result to acres by dividing by 43560, the number of square feet in an acre. The public lands of the United States are surveyed by selecting a north and south line, called a principal meridian, and intersect- ing this by an east and west line, called a base line. Range lines are lines running north and south on each side of the principal meridian, at dis- tances of 6 miles. They divide the land into strips 6 miles wide, called ranges. East and west lines parallel to the base line, and at dis- tances of 6 miles, divide the ranges into townships. A range is, therefore, a row of townships running north and south. The townships in each range are numbered north and south 356 A Group of Townships <;<)\T.KXMK\T I. AND MEASURES 357 from tlir base line, and the ranges are numbered cast and west from the principal meridian. A township is designated by its number and direction from the base line, the number and position of its range, and the name or number of the principal meridian. Tims, Township A is 4 North, Range 5 West of Principal Meridian. i S * 3 2 1 7 a 9 10 II 12 IB 17 76 IS It 13 19 20 21 22 23 24 .30 29 28 27 25 25 31 32 33 34- 35 J6 W/2 section (320 A) n.e.v* of NE. '/* S.E./* of NE.'/* Eti of ■.: . of S£3i of S.E.'/* A Township A Section A township is 6 miles square and is divided into 30 sec- tions each one mile square. Each section contains 640 acres. 1. W. \ Sec. 31, T. 22 N., 4 E. 3d P. M. is read west | section 31, township 22 North, Range 4 east of third princi- pal meridian. 2. Read: S. 1 of S.E. J, Sec. 31. 3. Read : NAY. \ of S.E. J, Sec. 31. 4. Read : N.E. \ of N.E. \, Sec. 31. Locate the tract of land in sections and give number of acres: 5. SAY. \, Sec. 5, T. 4 S., R. 3 W. 6. S.l of N.E. l Sec. 4, T. 15 N., R. 5 E. 7. S.E. \ of N.W. {, Sec.8, T. 125 S., R. 4 W. 8. Find the number of rods of fence required for the tract in problem 7. 9. How many rods of fence are required for the tract mentioned in problem ? POWERS AND ROOTS 1. 3x3 = 9. 4x4 = 1G. 5x5x5 = 125. 2. Name the two equal factors that produce 9. 16. 3. Name the three equal factors that produce 125. 4. What two equal factors produce 25? 36? 49? 81? 5. What three equal factors produce 8? 27? 64? 6. How many times is 5 used as a factor in 5 ? 25 ? 125? 7. How many times is 3 used as a factor in 3 ? 9 ? 27 ? A power of a number is the product obtained by taking the number one or more times as a factor. Thus, 9 is a power of 3, and 8 is a power of 2. The first power of a number is the number itself. The second power of a number is called the square of the number. Thus, 16 is the square of 4. The third power of a number is called the cube of the number. Thus, 64 is the cube of 4. 8. What is the square of 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 9. What is the cube of 2? 3 ? 5 ? 6 ? 7? 10. | x | = |. What, then, is the square of f ? 11. I x | x | = §£. What, then, is the cube of f ? 12. .3 x .3 = .09. .5x.5=.25. What, then, is the square of .3 ? of .5? The square of a fraction is found by squaring both terms, the cube of a fraction by cubing both terms. 358 roots 359 13. 3x3 = 9. The square of 3 is indicated thus, 3 2 ; 4 x 4 x 4 = 64. The cube of 4 is indicated thus, 4 3 . 14. How much is 52? 6 2 ? 7 2 ? 5 3 ? 6 3 ? What is the value of (f) 2 ? (§) 3 ? .4 2 ? .4 3 ? (2\f! An exponent is a small figure placed at the right of, and a lit- tle above the number, to indicate the number of times it is to be taken as a factor ; thus, 4 3 = 4x4x4 = 64. Exponent, 3. Written Work 1. Square 6, 7, 8, 9, 10, 12, 15. 2. Cube 3, 4, 5, 6, 7, 8, 0, 10, 12. 3. Square 30, 50, 60, 80, 120. 4. Cube 20, 30, 40, 50, 100. 5. Find the value of 6 2 , 8 2 , 9 2 , 5 3 , 6 3 , 7 3 . 6. Find the value of (f) 2 , (f) 3 , (f) 2 , (|) 3 , (f) 2 , (ll) 2 . 7. Square .3, .04, .05, .6, .06. Cube .4, .04, .6. Find the value of : 8. 15 2 ll. 22 2 14. .75 2 9. 16 2 12. 25 2 15. (If) 2 io. 18 2 13. 6.5 2 16. (16i) a Find the number of square units in a surface whose side is: 17. 15 in. 20. 8 ft. 6 in. 23. 10 yd. 18. 25 ft. 21. 5 in. 24. 5 yd. 2 ft. 19. 16 yd. 22. 8.5 in. 25. mi. Find the number of cubic units in a volume whose edge is: 26. 8 in. 28. 3 ft. 3 in. 30. 1 yd. 10 in. 27. 2 ft. 29. 42 in. 31. 12 ft. 4 in. 360 POWERS AND ROOTS EXTRACTING ROOTS A root of a number is one of the equal factors that pro- duce that number. Thus, 3 is a root of 9. The square root of a number is one of its two equal factors. Thus, 4 is the square root of 16. The cube root of a number is one of its three equal factors. Thus, 4 is the cube root of 64. 1. What is the square root of 25 ? 36 ? 49 ? 81 ? 100 ? 2. What is the cube root of 8 ? 64 ? 125 ? 216 ? The root of a number is generally indicated by writing the number under the radical or root sign -y/~ ' , and placing a figure called the index in the angle of the sign ; thus, a/27 denotes the cube root of 27. The square root is indicated by V without the index. The root of a fraction equals the root of the numerator di- vided by the same root of the denominator. Find the required root : Thus, V64 = V8 x 8 = 8. /16 3. 4. 5. 6. 7. \V27 a/25 aV«34 8. 9. 10. 11. 12. v 100 V49 VI 25 v 125 a/81 13. 14. 15. 16. 17. V216 a/1000 V343 V.064 18. 19. 20. 21. 22. V512 V144 V169 a/2500 V8100 Some perfect powers and their roots. Memorize: VI=1 a/36 = 6 a/4 = 2 a/49 = 7 a^9= 3 a/64 = 8 Vl6 = 4 a/81 = 9 a/25 = 5 a/100 = 10 aVI25 = 5 aVI = 1 aV8 = 2 a/27 = 3 aV64 = 4 a/216 = 6 a/343 = 7 aV512 = 8 a/729 = 9 a/1000 = 10 SQUARE ROOT 361 Finding the root of a perfect square or cube by factoring. Written Work 1. Find the square root of 1225. When factored 1225 =5x5x7x7. Arranged into two like groups 1225 = (5 x 7) x (5 x 7). V1225 = 5 x 7, or 35. Find the root of each number, as indicated, by factoring : 7. VT84 12. 13. V225 V576 V441 4. 5. 6. V1600 V196 8. 9. 10. 11. V1296 V17G4 ^13824 V4096 V2304 a/42875 14. 15. 16. V3136 a/5832 a/8000 17. 18. 19. 20. 21. V5184 a/15625 a/32768 ID! Hi a/19683 SQUARE ROOT Comparing roots and periods. The squares of the smallest and the largest integers com- posed of one, two, and three figures are as follows : !2 = i 10 2 = 1 00 100 2 = 10000 92 = 8 i 992 = 9801 999 2 = 998001 1. Separate each of these squares into periods of two figures each, beginning at the right ; thus, 99' 80' 01. 2. How does the number of periods in each square compare with the number of figures in the corresponding roots ? The number of periods of two figures each, beginning at units, into which a number can be divided equals the number of figures in the root. Note. — The left-hand period may contain only one figure. 3. How many figures are there in the square root of 4225? of 12544? of 133225? of 810000? 362 POWERS AND ROOTS Written Work 1. Square 25. 25 = 20 + 5, hence it may be squared in two ways, thus: 25 = 20 + 5 The square of 25 has three partial 25 = 20 + 5 products : 125^ 20 x 5 + 52 f(l) 20 2 = 400 1 500= 2Q2+20x5 25 2 = (2) 2(5x20) = 200 \ =625 625 = 20 2 + 2 (20 x 5) + 5 2 [ (3) 2. Find the square root of 625. 5 2 = 25 J 6'25 20 2 = 4 00 Trial divisor, 2 x 20 = 40 _5 Complete divisor, = 45 225 225 20 _5 25 In practice: 6'25 J 25 4 45 225 225 mtm m i Since 625 has two periods, its square root is composed of two figures, tens and ones. Since the square of tens is hundreds, 6 hundreds must be the square of at least 2 tens. Two tens or 20 squared is 400, as shown in figure A ; and 625 — 400 leaves a remainder of 225. The root 20, therefore, must be so increased as to exhaust this remainder and keep the figure a square. The necessary additions to enlarge A, and keep it a square, are the two equal rectangles B and C, and the small square D. B, C, and D contain 225 square units ; and since rj the area of D is small, if 225 is divided by 40, the combined length of B and C, the quotient will indicate the approximate width of these additions. The quotient is 5; the entire length of B, C, and D is 20 + 20 + 5 = 45 units ; and the area of the additions is 5 times 45 x 1 sq. unit, or 225 sq. units. Since these three additions exhaust the remaining 225 sq. units, and keep the fig- ure a square, the side of the required square is 25 units, and the square root of 625 is 25. B :M ±t 20 $z'z SQUA1JK HOOT 363 3. Square 35(30 + 5); then find the square root of 12:1"). Partial Products : (1) 3o 2 =900 (2) 2(5x80) =300 (3) 5 2 = 25 12'25 30 2 9 00 Trial divisor, 2 x 30 = 60 _5 Complete divisor, 05_ ► 1225 30 _5 S3 Study of Problem 1. How many periods in 1225 ? 2. How many figures, then, will be in its root ? 3. What is the largest partial product in 1225 ? 4. What is the square root of 9<>o? 5. What two partial products are contained in the 325 remaining? 6. 325 is composed principally of which partial product? 2(5 x 30). 325 is composed principally of 2 times the first figure of the mot by the second. Hence, if 325 is divided by (2 x 30), the quotient will give approximately the second figure of the root. 7. What is the quotient ? What, then, is probably the second figure of the root? 8. What must be added to the trial divisor (2 x 30) to form the com- plete divisor? Since 5 x 65 = 325, 5 is the second root figure, and 30 + 5, or 35, is the square root of 1225. 9. How does (5 x 60) + (5x5) compare with 5 x 65? 4. Find the square root of 21.16. Separate the number into periods, left and right from the decimal point. . 40 2 = '21.16' 16 00 4.0 + .6 = 4.6 In practice : '21.16' | 4.6 Trial divisor, 2 x 40 = 80 5 16 5 16 16 _6_ Complete divisor, 86 86 516 5 16 Beginning f a right tri- angle is the side opposite the right angle. 1. How many square units are there in the square described upon the hypotenuse '! in the square described upon the perpendicular? in the square described upon the base ? 2. How do the number of square units described upon the hypotenuse compare with the gum of the square units described upon the other two sides ? That this is universally true is shown by the following diagram : iri 5 3 \ 4 Take any right triangle, as 1; lay it off on a piece of cardboard and draw the square on its hypotenuse. Cut this square into the four equal triangles 1, 2, 3, and 4, and the small square 5. as here shown. By changing the position of the triangles 1 and 2 as indicated, we change the first diagram into the second. But the first is the square on the hypotenuse, and the second is the sum of the squares on the other two sides. Since they are equal, the truth of the proposition is evident. The square on the hypotenuse of a right triangle equals the sum oj thesquar* * described on the other two sides. 366 POWERS AND ROOTS Written Work 1. Find the hypotenuse of this right triangle. Hypotenuse 2 = W 2 + 60 2 , or 4756 Hypotenuse = V4756 = 68.9 + ft. 60 ft. ~ Draw figures to a convenient scale and find the unknown side : Base Perpendicular Hypotkxuse 2. 18 in. 27 in. (x) 3. (a;) 4 ft. 5 ft. 4. 24 ft. (a;) 40 ft. 5. 15 yd. 20 yd. (x) 6. (x) 40 in. 60 in. 7. Find the length of the longest straight line that can be drawn on a table 8 ft. by 4 ft. 8. A has a field 40 rd. long and 30 rd. wide. B has a square field whose side equals the diagonal of A's field. What is the difference in the area of the two fields ? 9. Find the longest straight line in a room 16 ft. in length by 12 ft. in width by 10 ft. in height. 10. Two automobiles, A and B, start from the same point. A goes east 10 mi., then north 10 mi. ; B goes west 20 mi., then south 20 mi. Draw figure and find distance apart. 11. Find the length of the diagonal of a square field con- taining 225 sq. rd. 12. Find the side of the largest square that may be in- scribed in a circle 2 ft. in diameter. 13. A fireman's ladder just reaches a window 36 ft. above the ground. How long is the ladder if its foot is 27 ft. from the building? MENSURATION Rectangular surfaces, rectangular solids, and the cylinder have been treated under Practical Measurements. REGULAR POLYGONS A plane is a surface such that a straight line joining any two of its points lies wholly in the surface. A polygon is a plane figure bounded by straight lines. A regular polygon is a plane figure having equal sides and equal angles. Triangle Square Pentagon Hexagon Polygons are named from their sides. A regular polygon of three sides is called an equilateral triangle; one of four sides, a square; one of Jive sides, a pentagon ; one of six sides, a hexagon, etc. Finding the area of a regular polygon. Draw a circle. Divide the circumference into 6 equal parts by points marked at distances apart equal to the length of the radius. Join these points, thus making a hexagon. Connect the opposite points by dotted lines, thus dividing the hexagon into six equilateral triangles. Show by folding together the opposite sides of any equilateral, and the equal sides of any isosceles triangle, that each may be divided into two equal right triangles. The area of any regular polygon equals the sum of the areas of the triangles composing it. 3G8 MENSURATION • 1. Find the surface of the bottom of a hexagonal silo that is 12 feet on a side, the distance from the middle point of the side to the center of the bottom being 10.3 ft. 2. How far from the corner is the center of a square field that is 40 rods on a side? (Draw the figure.) 3. Find the area of an equilateral triangular design that is 15 inches on a side. (Divide into two right triangles.) SOLIDS A solid is anything that has length, breadth, and thickness. The faces of a solid are the surfaces that bound it. The lateral or convex surface of a solid is the area of its sides, or faces. The volume of a solid is the number of cubic units it con- tains. A prism is a solid whose ends are equal and parallel poly- gons, and whose sides are parallelograms. Prisms are named from their bases, as triangular, square, rectangular, pentagonal, hexagonal, etc. Triangular Prism Square Prism Pentangular Prism Rectangular Prism A cylinder is a solid with circular ends and uniform diameter. The ends are the bases, and the curved surface is the convex surface. The altitude of a prism or of a cylinder is the perpendicu- lar distance between the bases. SI'IMWCKS OF SOLIDS 3fi0 Cylinder A pyramid is a solid whose base is a regular polygon, and whose faces are triangles that meet at a point called the vertex. Pyramids are named from their bases, as, triangular, square, pen- tagonal, etc. Pyramid The slant height of a pyramid is the altitude of the triangles that bound it. A cone is a solid whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex. The altitude of a pyramid, or of a cone, is the perpendicular distance from the vertex to the base. The slant height of a cone is the distance between the vertex and any point in the circumference of the base. A globe or sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center. sphere Cone SURFACES OF SOLIDS Surfaces of Prisms and Cylinders Observe : 1 . That if a piece of paper is fitted to cover the convex surface of a prism or a cylinder, and then unrolled, its form will be that of a rectangle, as ABCD. 2. That the perimeter of the solid forms one side of the rectangle, and the altitude of the solid the other side. HAH. COMPL. AKITII. — 24 370 MENSURATION q The convex surface of a prism or of a cylinder is found by multi- plying the unit of measure by the product of the perimeter and the S altitude. Find the convex surface of a regular prism of : 1. 5 sides ; 1 side 10 ft. ; height 5 ft. 2. 3 sides ; 1 side 20 in. ; height 42 in. 3. A steam boiler, diameter 3 ft. ; length 10 ft. Entire surface = ? 4. A water pail, diameter 11 in. ; height 15 in. Entire surface = ? Surfaces of Pyramids and Cones Observe : 1. That the convex surface of a pyramid is composed of triangles. Pyramid Cone 2. That the convex surface of a cone may also be considered as made up of small triangles. 3. That the bases of the triangles in both pyramid and cone form the perimeter of the base of the figure, and the altitude of the triangles the slant height. Hence, TJie convex surface of a pyramid or of a cone is found by multiplying the unit of measure by one half the product of the perimeter and the slant height. CYLINDER AND SPHERE 371 Find the convex surface of a pyramid or a cone if : 1. Diameter of base of cone = 9 ft. ; slant height = 12 ft. 2. One side of a square pyramid = 16 ft. ; slant height = 24 ft. 3. One side of a square pyramid = 5 ft.; altitude = 16 ft. 4. Altitude of square pyramid = 24 ft. ; one side = 14 ft. 5. A church spire is in the form of a hexagonal pyramid, each side being 10 feet, and the slant height 65 feet. Find the cost of painting it at 25^ per square yard. 6. A spire on the corner of a church is in the form of a cone. Its base is 12 feet in diameter and its slant height 24 feet. Find the cost of covering it with tin at $13 per square (100 sq. ft. = 1 square). Comparative Surfaces of Cylinder and Sphere Examine the solids. What is the height of the cylinder? What is the diameter of the cylinder? What is the diameter of the sphere? How does the diameter of each compare with the height of the cylinder? Observe that the dimensions are equal. Geometry shows that the surface of a sphere is equal to the convex surface of a cylinder w T hose height and diameter are each equal to the diameter of the sphere. 372 MENSURATION To show this, wind a hard wax cord around a cylinder 1 in. in height and 1 in. in diameter until its convex surface is covered. Unwind the cord from the cylinder on to a sphere 1 in. in diameter as shown in the illustration. When one half the surface of the sphere is covered with the cord, one half of the convex surface of the cylinder is uncovered. Hence, The surface of any sphere equals the convex surface of a cylinder of equal dimensions. It may also be shown by geometry that The surface of a sphere equals the square of the diameter multiplied by 3.1416, or ttcP (representing the diameter by d and 3.1416 by 7r). Find the surface of : 1. A globe, D. 12 in. 2. A ball, R. l-i- in. 3. A sphere, Z>. 13 in. 4. A ball, I). 4 in. 5. How much will it cost to paint a dome in the form of a hemisphere, 20 ft. in diameter, at 25 cents per square yard ? VOLUME OF SOLIDS Prisms and Cylinders - N iif Scale \ in. = 1 in. Observe : 1. That the solids are all 4 in. high. 2. That the first row in the rectangular prism contains 4 cu. in. 3. That if the first row in each solid contains 4 cu. in." the volume of each solid is 4 times 4 cu. in., or 16 cu. in. PYRAMIDS AND CONES 373 The volume of a prism or of a cylinder is found by multiplying the unit of measure by the product of the numbers corresponding to the area of the base and the altitude. Find the volume of : 1. A prism 4 inches square ; altitude 8 inches. 2. A square prism, side 12 in.; altitude 24 in. 3. A hexagonal silo is 25 ft. high, 12 ft. on a side, and 10.3 ft. from the middle point of a side (measuring at the base) to the center of the base. Estimating 50 cu. ft. to a ton of ensilage, how many tons will the silo contain ? 4. In the rotunda of a building there are 6 cylindrical mar- ble columns, 18 in. in diameter and 18 ft. in height. Esti- mate the number of cubic feet in all. PYRAMIDS AND CONES 1. Fill a hollow pyramid with sand. Empty it into a prism having the same base and altitude. How often must the pyramid be filled and emptied to fill the prism ? The vol- ume of a pyramid, then, is what part of the volume of the prism ? 2. Measure in like manner with a cone the volume of a cylinder having the same dimensions. The volume of the cone is what part of the volume of the cylinder ? 374 MENSURATION Observe : 1. That the dimensions of the pyramid and of the prism are the same, and that those of the cone and of the cylinder are the same. 2. That the volume of the pyramid is \ that of the prism, and the volume of the cone is | that of the cylinder. By geometry, it is shown that the volume of a pyramid is ^ of that of a prism having an equal base and an equal alti- tude. Hence, The volume of a pyramid equals one third of the volume of a prism of like dimensions. The volume of a cone equals one third of the volume of a cylinder of like dimensions. But we have already learned that the volume of a prism or of a cylinder is found by multiplying its unit of measure by the product of the area of its base by its altitude. Hence, The volume of a pyramid or of a cone is found by multiply- ing its unit of measure by one third the product of the altitude and the area of the base. 1. Find the volume of a cone whose altitude is 12 in. and the diameter of the base 8 in. 2. How often can a conical cup 8 in. high and 6 in. in di- ameter be filled from a cylindrical vessel 2 ft. high and 6 in. in diameter? 3. Find the volume of a pyramid whose base is 12 in. square and whose altitude is 30 in. 4. A square pyramid whose side is 18 in. is 32 in. high. Find its volume. 5. Find the volume of a pyramid whose altitude is 12 ft. and whose base is a square 8 ft. on a side. 6. Find the contents of a rectangular pyramid 15 ft. high, the sides of whose base are 10 ft. and 12 ft. respectively. 7. A pile of grain in the form of a cone is 15 ft. in diameter and G ft. high. How many bushels of grain does it contain ? SPHERES 375 8. A concrete mixer, 6 ft. from base to apex, being coni- cal in form, and measuring 3 feet across the base, is filled six times an hour. How many cubic feet of concrete mate- rial may be manufactured with it in a week of six working days of 8 hours each ? 9. A wooden hopper supplying coal to a furnace is in the form of an inverted pyramid. If it is 8 ft. deep and 6 ft. square at the top, how many tons of hard coal will it contain ? 10. A square pyramid, the perimeter of whose base meas- ures 64 inches, contains 2048 cubic inches. Find its altitude. 11. The contents of a cone are 471.24 cu. ft. ; the altitude is 18 ft. Find the diameter. SPHERES Examine the figure: Observe : 1. That the solids formed by the dissected part of the sphere are pyramids. 2. That the radius of the sphere is the altitude of the pyramids. 3. That the combined bases of the pyramids form the convex surface of the sphere. The volume of a sphere is found by multiplying its unit of measure by one third the product of the radius and its convex surface. It is also shown by geometry that The volume of a sphere equals four thirds of the cube of the radius multiplied by 3.1416, or {representing the radius by r and 3.1416 by tt)^^. Find the volume of : 1. A globe 12 inches in diameter. 2. A bowling ball with a radius of 4 inches. 3. A cannon ball with a diameter of 8.2 inches. 376 MENSURATION Comparative Volumes of Cone, Sphere, and Cylinder -/-"--- w. /'i — _J I /': Compare the diameters of the bases and the altitudes of the cone and the cylinder with each other, and with the diameter of the sphere. Ob- serve that the dimensions are all equal. By geometry it is shown that the volumes of these three solids are in the ratio of 1, 2, and 3. The volume of the cone is ^, and of the sphere § of that of the cylinder. SIMILAR SURFACES Similar figures are plane surfaces that have exactly the same shape, but differ in size. Point out similar figures: V C\J f-inch sphere with a 50-inch sphere. 5. What are corresponding lines in spheres? in triangles ? 6. The dimensions of a rectangular solid are 30 ft. by 20 ft. by 12 ft. What are the dimensions of a similar solid 20 ft. in length ? 7. A village has two similar cylindrical water tanks, one 15 ft. in diameter and 24 ft. high; the other 10 ft. in di- ameter. Find the height of the second tank. 8. Make three problems with similar rectangular solids. 9. The volume of a sphere is 1600 cu. in. What is the volume of a sphere with one half the diameter ? 10. A rectangular bin 8 ft. long contains 92 bu. of wheat. How many bushels will a similar bin 10 ft. long contain ? SPECIFIC GRAVITY 1. A piece of timber 12 inches square and 6 feet long floating in water shows 2 inches of the log above water. How many cubic feet of the log are under water ? How many cubic feet of water are displaced by the log ? 2. A cubic foot of water weighs 62| lb. How many cubic feet of water will a log displace that weighs 250 lb. ? If the log in problem 1 is § as heavy as the same volume of water, how much of the log is under water ? Every ohject floating in water displaces its own weight of water. Every object that sinks in water displaces its own volume of water. 380 MENSURATION 3. A piece of wood when floating is | under water. The ratio of the weight of the wood to the weight of an equal volume of water is therefore 1 ^ 2 or ^. What is the ratio of the weight of objects to the weight of the same volume of water if the displacement is \ ? § ? | ? f ? .5 ? Copper is 8.9 times as heavy as an equal volume of water. The ratio of the weight of copper to the weight of an equal volume of water is, therefore, 8.9. Specific gravity is the ratio of the weight of any substance to the weight of an equal volume of water. A cubic foot of water weighs 62^ pounds or 1000 ounces. 4. The specific gravity of ice is .92. Find the weight of a block of ice 2' by 18" by 12". 5. The specific gravity of pure cows' milk is 1.03. Find the weight of 50 gallons of milk. 6. The specific gravity of cork is .24. Find the weight of 10 cu. ft. of cork. 7. The specific gravity of lead is 11.3. Find the weight of 20 bars 16" long, 4" wide, and 2" thick. Find the weight of 1 cu. ft. of each object whose specific gravity is given : 8. Silver 10.50 16. Copper 8.90 9. Milk 1.03 17. Lead 11.30 10. Ice .92 18. Sandstone 2.90 11. Nickel 8.90 19, Iron 7.80 12. Granite 2.70 20. Tin 7.29 13. Mercury 13.59 21. Steel 7.83 14. Gold 19.30 22. Marble 2.70 15. Cork .24 23. Ivory 1.83 REVIEW OF MENSURATION :M REVIEW OF MENSURATION 1. From an artesian well 2 inches in diameter the water flows out at the rate of 1| ft. per second. Find the number of barrels that flow out per hour. 2. One cubic inch of gold could be pounded into how many square inches of gold leaf ^^ of an inch in thickness ? 3. A spherical ball must be how large in order that a cube, each surface of which contains 36 square inches, could be cut from it ? 4. A copper ingot containing 1 cubic foot is to be drawn into a copper wire ^ of an inch in diameter. Find the length of the wire when drawn. 5. A plowman found by measurement that he had plowed a strip 4 rd. in width around a rectangular field 40 rd. long and 20 rd. wide. Find the number of acres he had plowed. Draw figure to illustrate. 6. A western farmer has a pile of corn in the ear, 400 ft. long. The pile at the end is in the form of an isosceles tri- angle, 12 ft. wide at the bottom, and the altitude of the pile is 6 ft. Find the number of bushels, allowing If cu. ft. to a bushel. 7. A solid ball 6 inches in diameter is in a cylinder 10 inches in diameter and 10 inches high. How many cubic inches of water will the cyclinder contain ? 8. What is the convex surface of a piece of stove pipe 2 feet long and 8 inches in diameter ? 9. The inside diameter of a hollow globe is If ft. How many gallons of water will it contain ? 10. The flow of water from the same source through two different pipes depends upon the area of the cross section of the openings. Cm pare the flow through a i-inch pipe with the flow through a |-inch pipe. METRIC SYSTEM OF WEIGHTS AND MEASURES By the United States system of money we may write 5 dollars, 9 dimes, and 7 cents thus, $5.97, because there is a uniform ratio between the dollar and the dime, the dollar and the cent, the dollar and the mill ; the dollar being 10 times the dime, 100 times the cent, and 1000 times the mill. As a mill is y^j of a dollar, a cent T Jo of a dollar, and a dime ^ of a dollar, United States money is based on a decimal system. By the English long measure, 12 in. = 1 ft., 3 ft. = 1 yd., and 5^ yd. = 1 rd. ; thus, we see there is no uniform ratio between the rod and the yard, the rod and the foot, and the rod and the inch, the rod being 5| times the yard, 16| times the foot, and 198 times the inch. By the English measure of weights, 16 oz. = 1 lb., 100 lb. = 1 cwt., 20 cwt. = 1 ton. The ton equals 20 times the hundredweight, 2000 times the pound, and 32000 times the ounce. The metric system was devised by the French govern- ment in an effort to establish a system of weights and meas- ures that would be on a uniform decimal scale, so that a unit of one denomination might be changed to a unit of another denomination by simply moving the decimal point. The meter is the fundamental unit of the metric system. Its length (about 39.37 in.) was meant to be .0000001 of the distance from the equator to the pole. Though an error has since been discovered in the measurement of the distance from the equator to the pole, the standard unit has not been changed. The original standard is a bar of platinum 39.37 inches in length, deposited in the archives in Paris. 382 METRIC SYSTEM 383 From the meter every other unit of measure or weight is derived. Thus, the unit of weight is the gram, which equals the weight of 1 cubic centimeter of pure water. Draw a cube .01 of a meter on an edge and state the length of the edge in inches. The unit of capacity is the liter (leter), which contains 1 cubic decimeter. Draw a cube .1 of a meter on an edge and state the length of the edge in inches. The metric system is now in use in most of the civilized countries except Great Britain and the United States, and in the latter it is in use in some of the departments of the government. It is the official system adopted by Congress for our island possessions. It is univer- sally used by scientists. The United States by a vote of Congress per- mitted its use in I860. Observe : The meter measures length. The square meter measures surface. The cubic meter measures solids or volume. The gram measures weight. The liter measures capacity. Latin prefixes. To express .1 of a meter, .1 of a gram, and .1 of a liter, we prefix deci to each of the words, meter, gram, and liter. Thus, decimeter means -^ of a meter ; decigram, -^ of a gram ; and deciliter, -^ of a liter. To express .01 of a meter, gram, and liter, we prefix centi to each of the words, meter, gram, and liter. To express .001 of a meter, gram, and liter, we prefix milli to each of the words, meter, gram, and liter. Note. — From these Latin prefixes we get our words dime, cent, and mill. 384 METRIC SYSTEM Greek prefixes. To express 10 times a meter, 10 times a gram, and 10 times a liter, we prefix deca to each of the words, meter, gram, and liter. Thus, decameter means 10 times a meter ; deca- gram, 10 times a gram ; and decaliter, 10 times a liter. To express 100 times a meter, gram, and liter, we prefix hecto to each of the words, meter, gram, and liter. To express 1000 times a meter, gram, and liter, we prefix kilo to each of the words, meter, gram, and liter. METRIC MEASURES OF LENGTH Comparison of Fundamental Units of Measures of Length i i I i i i I i i i \ 2 24 36 1 Yard. hum / 2 3 4 5 6 7 8 9 10 I Meter Table of Long Measures 1 millimeter (mm.) = .001 of a meter 1 centimeter (cm.) = .01 of a meter 1 decimeter (dm.) = .1 of a meter (nearly 4 in ) 1 meter = 39.37 in. 1 decameter (Dm.) = 10 meters 1 hectometer (Hm.) = 100 meters 1 kilometer (Km.) = 1000 meters (nearly .6 mi .) In metric long measure 10 times one unit of any denomi- nation equals one unit of the next higher denomination. The denominations most frequently used are given in black-faced type. MEASURES OF LENGTH 385 Approximately : 1 yard = \\ meter 1 mile = 1.6 kilometer The kilometer is used for measuring long distances; the meter, for short distances and for measuring cloth, etc. ; and the millimeter is used in the sciences and to show veiy small measurements, as the thickness of wire, etc. Written Work 1. What decimal parts of a meter are expressed by the Latin prefixes ? What multiples are expressed by the Greek prefixes ? 2. Draw a line one meter long. Show the number of decimeters in a meter ; the number of centimeters ; the number of millimeters. 3. Explain why 5m. = 50 dm. = 500 cm. = 5000 mm. 4. In the metric S3'stem the fundamental operations are decimal or multiple operations. . Thus, 8 m. 5 dm. G cm. 25 mm. are added in this manner. Added in meters: Added in millimeters: 8 m. =8 m. 8 m. = 8000 mm. 5 dm. = .5 m. 5 dm. = 500 mm. 6 cm. = .06 m. 6 cm. = 60 mm. 25 mm. = .025 m. 25 mm. = 25 mm. 8.585 m. 8585 mm. 5. Add 1 m., 3 dm., 6 cm., 3mm. Add 6.5 m., .25 mm.,. 65 dm. 6. The distance between two towns is 5 Km. and 45.V m. After a bicyclist has traveled 3 Km. 57.5 m., how much of the distance remains to be traveled ? 7. The distance from Paris to Calais is 295.32 Km. Express this distance approximately in miles. HAM. COMPL. ARITH. — 25 386 METRIC SYSTEM 8. How many meters of ribbon are necessary to make 150 badges, each 25 cm. in length? 9. The distance from Erie, Pa. to Buffalo, N.Y. is 112.651 Km. Express the distance approximately in miles. 10. Reduce the decimal in the last problem to meters and lower denominations. 11. The distance from New York to San Francisco is 4000 miles. Approximate this distance in kilometers and meters. METRIC MEASURES OF SURFACE Comparison of Fundamental Units of Square Measure • IS a. Yd ISq.M Table of Square Measures 1 sq. millimeter (sq. mm.) 1 sq. centimeter (sq. cm.) 1 sq. decimeter (sq. dm.) 1 sq. meter (sq. m.) 1 sq. decameter (sq. Dm.) 1 sq. hectometer (sq. Hm.) 1 sq. kilometer (sq. Km.) .000001 sq. meter .0001 sq. meter .01 sq. meter 1.196 sq. yd. 100 sq. meters 10000 sq meters 1000000 sq. meters (nearly .4 sq. mi.) In metric measure of surface 100 times one unit of any denomination equals one unit of the next higher denomination. MEASURES OF VOLUME 387 Land Measure. The standard unit used for measuring is the are (ar) con- taining 100 square meters = 119.(3 square yards. Table : 1 centare = 1 sq. meter. 1 are = 100 sq. meters (nearly 120 square yards) 1 hectare = 10000 sq. meters (nearly 2* acres). The square meter is used in measuring ordinary surfaces, such as are found in houses, lots, farms, etc. ; the square kilometer for measuring areas of countries and their divi- sions into states, counties, etc. METRIC MEASURES OF VOLUME Comparison of Fundamental Units of Cubic Measure 1 Ctbic Yard 1 Cubic Meter Table of Solid or Cubic Measures 1 cu. millimeter (cu. mm.) = .000000001 cu. meter 1 cu. centimeter (cu. cm.) = .000001 cu. meter 1 cu. decimeter (cu. dm). = .001 cu. meter 1 cu. meter =1 308 cu. yd. In metric measure of volume 1000 times <<>"' unit of any de- , equals "/<>■ unit <>f th<< next higher denomination. 388 METRIC SYSTEM In measuring wood 1 cu. meter is called a stere. The cubic meter is the practical unit of measure of vol- ume for all purposes. Written Work in Square and Cubic Measures 1. Find the number of square meters in the floor of your schoolroom. 2. At 27 cents per cubic meter, find the cost of excavating a cellar 10 m. by 18 m. by 1| m. 3. How much will it cost to paint one side of a tight board fence 25 meters long and 3 meters wide at $10 per square decameter ? 4. How many square meters of linoleum will be required to cover the floor of a hall 8 meters long and 3 meters wide? 5. How many steres are there in a pile of wood 2 meters high, 2 meters wide, and 6 meters long? METRIC MEASURES OF CAPACITY Comparison of Fundamental Units of Capacity Quart MEASURES (>F WEIGHT :;*!> Table of Measures of Capacity 1 millimeter (ml.) = .001 of a liter 1 centiliter (cl.) = .01 of a liter 1 deciliter (dl.) = .1 of a liter 1 liter = 1.0567 liquid quarts = .908 dry quart. 1 decaliter (Dl.) = 10 liters 1 hectoliter (HI.) = 100 liters (nearly 2.84 bu.) In metric measure of capacity 10 times one unit of any de- nomination equals one unit of the next higher denomination. The liter is used for liquid and dry measures. METRIC MEASURES OF WEIGHT Comparison of Units of Weight I Ounce I Gram lib. I Kilo Table of Measures of Weight 1 milligram (mg.) = .001 of a gram 1 centigram (eg.) = .01 of a gram 1 decigram (dg.) = .1 of a gram 1 gram = .03527 of an oz Avoir. 1 decagram (Dg.) = 10 grams 1 hectogram (Hg.) = 100 grams 1 kilogram (Kg.) = 1000 grams (nearly 2.2 lb.) 1 myriagram (Mg.) = 1O000 grams 1 quintal (Q.) = 100000 grams 1 tonneau (T.) = 1000000 grams ( nearly 2205 lb.) 390 METRIC SYSTEM In the metric measure of iv eight 10 times one unit of any de- nomination equals one unit of the next higher denomination. The gram is the weight of 1 cubic centimeter of water, the kilogram, of 1 cubic decimeter, and the metric ton, of 1 cubic meter; the gram is used by druggists and chemists ; the kilo- gram (usually called the kilo) for weighing small articles; and the metric ton for large, heavy articles. Table of Metrical Equivalents 1 cu. mm. of water weighs 1 mg. and measures .001 ml. 1 cu. cm. of water weighs 1 g. and measures 1 ml. 1 cu. dm. of water weighs 1 Kg. and measures 1 1. 1 cu. m. of water weighs 1 T. and measures 1 Kl. Written Work Things sold in the United States and England by the quart are sold in countries using the metric system by the liter. 1. Estimate the number of liters in a tank 2£ m. long, | m. in width, and ^ m. in depth. 2. A cylindrical tank 6 m. in diameter and 10 m. in height is § full. Estimate the number of liters it contains. Estimate the weight in metric tons. 3. A Paris milkman retailed on an average 110 liters of milk daily at 25 centimes a liter. Find the amount of his sales in our own money for 30 days. 4. The rainfall in a certain place in one week was 1 dm. Find the number of liters that fell on 3 hectares of land. 5. A horse eats 4 liters of oats 3 times a day. How many hectoliters does it eat in 60 days ? 6. From an olive orchard, 4.5 Kl. of olive oil was put up in bottles holding .5 1. How many bottles were used? METRICAL EQUIVALENTS 391 7. Find the amount in United States money from the sale of 2000 HI. of wheat at 10 francs per hectoliter. 8. A German ice dealer retails blocks of ice .8 m. in length, .3 m. in width, and .2 m. in thickness. The weight of ice is .92 that of the same volume of water. Find its cost at 3.5 pfennigs per kilo. Find the amount in our money. 9. Change a cubic meter of water to liters. 10. Find the weight of a barrel of flour in kilos. 11. A stone 8 ft. by 3 ft. by 2 ft. contains how many cubic meters ? 12. If stone is 2.9 times as heavy as the same volume of water, find the weight of the stone in kilo*. 13. The Washington Monument is 555 feet high. Express its height in meters. 14. Find the cost of laying a cement walk .025 Km. in length and 1.5. m. in width, at $1.70 per sq. m. 15. Mr. James bought a tract of land in the Philippine Islands, 3 Km. in length and 2.5 Km. in width, at $15.75 per hectare. Find the cost of inclosing this land with wire fence at 10^ per meter. 16. A hallway is 12 m. in length and 5 m. in width. Esti- mate the number of tiles 1 cm. square necessary to cover it. 17. A railroad in building a retaining wall used 52000 cu. m. of stone. Find its weight in metric tons if stone is 2.7 times as heavy as the same volume of water. 18. A city sewer is 1.3 Km. in length, 1.2 m. in width, and averages 3 m. in depth. Estimate the number of cubic meters of earth removed. 19. A certain kind of cloth costs 90^ per meter + 25 % ad valorem duty. For how much must it be marked in United States mi y to gain 25% on tlie yard? AGRICULTURAL PROBLEMS FEEDING STOCK Table of Digestible Nutrients This table shows the number of pounds of digestible nutrients found in 100 pounds of each kind of feed tab- ulated. A fair price for these nutrients is as follows: Protein, $4 per cwt. Carbohydrates, 40 ^ per cwt. Fat, $1 per cwt. Find the total value of the digestible nutri- ents in 1 ton of the following: 1. Timothy hay. 2. Red clover hay. 3. Dry corn fodder. 4. Corn silage. 5. Wheat straw. 6. Alfalfa hay. 7. Corn, grain. 8. Oats, grain. 9. Wheat bran. 10. Oil meal. 11. Oat straw. 12. Cowpeahay. 13. Corn fodder, green. 14. Red clover, green. Protein is a muscle former, while carbohydrates and fats are fat formers. The ratio of the muscle formers to the fat formers is called the nutri- tive ratio. A nutritive ratio of 1 :3 is a narrow ratio, while one of 1 : 15 is a wide ratio. A desirable nutritive ratio for a dairy cow is about 1 :5.7; for a work horse, about 1:7; for swine, about 1 : 5.9. 392 Feeds PRO- TEIN Carbo- hydrates Fat Corn fodder, green . 1.1 12.1 .4 Red clover, green . 3.1 14.8 .7 Alfalfa, green . . 3.9 11.2 .4 Corn silage . . . 1.2 14.6 .9 Corn fodder, dry 2.3 32.3 1.1 Timothy hay . . . 2.9 43.7 1.4 Red clover hay . . 7.4 38.1 1.8 Alsike clover hay . 8.1 11.7 1.4 Alfalfa hay . . . 10.6 37.3 1.4 Cowpea hay . . . 10.8 38.4 1.5 Wheat straw . . . .4 36.3 .4 Oat straw .... 1.2 38.6 .8 Corn, grain . . . 7.1 66.1 4.8 Oats, grain . . . 9.2 48.3 4.2 Wheat bran . . . 12.0 41.2 2.9 Oil meal .... 30.6 38.7 2.9 Cotton-seed meal . 37.0 16.5 12.6 Whole milk . . . 3.4 4.8 3.7 Skim milk . . . 3.0 5.0 .3 FEEDING STOCK 393 Fat, as a food constituent, is regarded as 2} times more valuable than carbohydrates. Hence, the customary ride for rinding the nutritive ratio is to add to the carbohydrates 2\ times the fat, and divide by the protein. 15. What is the nutritive ratio of green corn fodder? Protein, 1.1; carbohydrates, 12.1; fat, .4. » Tims, .4 lb. fat = 2\ x .4 lb. carbohydrates = .9 lb. carbohydrates; 12.1 lb. carbohydrates + .9 lb. carbohydrates = 13 Lb. carbohydrates; 13 lb. -^ 1.1 lb. = 11.8 +. Hence, the nutritive ratio is 1 to 11.8 +. 16. Find the nutritive ratio of a feed composed of 100 lb. timothy hay and 50 lb. wheat bran. 17. A farmer feeds his sheep 100 lb. of alfalfa hay to 10 lb. of bran. Find the nutritive ratio. 18. An Ohio farmer feeds his dairy cows 100 lb. red clover hay to 20 lb. of wheat bran. Find the nutritive ratio. 19. The best authorities suggest that a dairy cow weighing 1000 lb. and giving daily 22 lb. of milk should be fed 2.5 lb. protein, 13 lb. carbohydrates, and 0.5 lb. fat. Find the nutritive ratio of this feed. A balanced ration is a mixture of different kinds of feed according to a nutritive ratio that will produce the desired end of the feeding, whether that end be growth, fat, milk, or butter. 20. A farmer mixed a balanced ration of 3 parts by weight red clover hay and 1 part oats grain. Find the nutritive ratio. 21. A feeder prepared a balanced ration for his hogs witli the following parts by weight: 1| parts corn grain, 1 part oats grain. Find the nutritive ratio. 22. A balanced ration is as follows : 3 parts by weight timothy hay and 1 part corn grain. Find the nutritive ratio. 394 AGRICULTURAL PROBLEMS 23. A farmer prepared a balanced ration consisting of 150 lb. of alfalfa hay and 100 lb. corn grain. Find the nutritive ratio. 24. Find the nutritive ratio of a balanced ration consist- ing of 200 lb. corn silage and 100 lb. red clover hay. 25. What is the nutritive ratio of a balanced ration that uses 200 lb. of dry corn fodder with 50 lb. of wheat bran ? 26. A balanced ration for a dairy cow should be about 1 to 5.7. Is the nutritive ratio of the following balanced ration too wide or too narrow for dairy feeding? 200 lb. alsike clover hay ; 100 lb. corn grain. 27. A balanced ration for work horses should have a nutri- tive ratio of about 1 to 7. Is the nutritive ratio of the fol- lowing ration too wide or too narrow for work horses ? 200 lb. timothy hay ; 100 lb. oats grain. 28. A dairy man prepared a ration for his cows consisting of 100 lb. corn silage, 50 lb. timothy hay, and 100 lb. oats grain. Find the nutritiv-e ratio. Is this ration too wide or too narrow? 29. A livery man feeds his horses on a ration consisting of 50 lb. timothy hay, 25 lb. corn grain, and 25 lb. oats grain. Find the nutritive ratio. Is this ratio too wide or too narrow? 30. A stockman preparing cattle for market feeds 100 lb. red clover hay to 30 lb. corn grain, 10 lb. oats grain, and 1 lb. oil meal. Find the nutritive ratio. 31. A Kansas farmer feeds his young cattle 100 lb. alfalfa hay to 10 lb. corn grain and 2 lb. cotton seed meal. Find the nutritive ratio. FERTILIZERS 39:. FERTILIZERS Poinds of Fertilizing Constit- uents in One Ton This table shows the num- ber of pounds of plant tood, such as nitrogen, phosphoric acid, and potash in one ton of the article named. All other constituents necessary to plant food can be obtained from the soil, if the water supply is suffi- cient and the heat conditions proper. The leading artificial ferti- lizers are rock phosphate, ground bones, ammonia sul- phate, nitrate of soda, animal refuse called "tankage," fish scrap, and potash salts, espe- cially the sulphate and the muriate of potash. Lime is used on land to neu- tralize the acidity of the soil. Nitrogen in fertilizing material is sometimes, though not often, given as ammonia. 14 lb. of nitrogen correspond to 17 lb. of ammonia. Phosphoric acid contains 43.6% phosphorus! this constituent of ferti- lizer is sometimes, though rarely, given as phosphorus. A ton of timo- thy hay contains 16 lb. of phosphoric acid, which is equivalent to 7 lb. of phosphorus. Potash contains 83% potassium. This constituent is sometimes, though rarely, given as potassium. A ton of timothy hay contains 17 lb. of potash, which is equivalent to 39 lb. of potassium. The value of fertilizing constituents depends largely on the forms in which they occur. An average value per pound is 15 cents for nitrogen. 3 cents for phosphoric acid, and 5 cents for potash. The composition of farm produce varies greatly, so that the same kind of crop may be almosl twice as rich in certain constituents in some cases as it is in other ca-es. The table given here is intended to show the am rage composition of each. Materials Nitro- gen I'llos- PHORIO A. 11. P.)T- A 11 Timothy hay Clover hay . Alfalfa hay Com, grain Corn, stover Bran . . . Oat straw . Milk . . . Butter . . Farm animals Farm manure 19 39 46 34 12 51 13 12 1.6 53 10 16 13 13 13 9 62 7 4 1 37 5 47 44 35 9 39 35 38 3 1 3 10 390 AGRICULTURAL PROBLEMS Through the growth of the leguminous plants, such as clover, alfalfa, beans, peas, etc., it is estimated that the farmer may obtain nitrogen from the air at a cost of about 2| f per pound. 1. How much nitrogen is required to produce each of the following : 6 tons clover hay ? 3000 lb. corn grain ? 5500 lb. timothy hay ? 4 tons alfalfa ? 2. How much potash did a farmer sell in 1500 lb. of but- ter ? in 760 lb. milk ? in 5 steers averaging 1450 lb. each ? 3. A farmer cut on an average 3 tons per acre of clover hay from a 5 acre field. How many tons of manure are required for the field to supply the phosphoric acid in the hay? 4. A farmer sold 50 tons of timothy hay in one season. How many tons of farm manure will balance the loss in potash ? 5. When nitrogen is worth 16^ per pound, phosphoric acid 5^, and potash 0^, find the fertilizing value of 5 tons of farm manure. 6. A commercial fertilizer when analyzed was found to contain by weight 3.4% nitrogen, 6.6% phosphoric acid, and 9% potash. Find its value per ton, when nitrogen is worth 15^ per pound, phosphoric acid 3^ per pound and potash 5^ per pound. 7. The analysis of a certain brand of bone meal showed the following: 1.6% nitrogen, and 27.9% phosphoric acid. What is its value per ton at 15 ^ per pound for nitrogen and Z\$ per pound for phosphoric acid. 8. A farmer sold 500 bu. of corn grain (shelled corn 56 lb. to the bushel) from a certain field. How many tons of farm manure will supply the field with the phosphoric acid removed by the corn grain ? sn;.\Y!\<; plants 39' SPRAYING PLANTS The Bordeaux mixture used for killing fun- us growths — such as Mack rot and scab of apples, black rot and mildew of grapes, brown rot of plums and cherries, etc. — was first discovered in Bordeaux. France. To make a solution for spraying: first, dissolve 4 pounds of copper sulphate, (blue vitrol) in 25 gallons of water; second, dissolve 4 pounds of freshly slaked stone lime in 25 gallons of water until it forms a milk of lime; third, pour the two solutions together and they are ready for use. The solution should be applied to the fruit while small. In this way a coak ing is formed over the fruit through which the fungus cannot grow. Note. — In all directions for spraying, 50 gallons are considered a barrel. 1. For a vineyard of 25 acres of grapes suffering from black rot of the fruit, how many pounds of sulphate of copper and lime are necessary, if 100 gallons of Bordeaux mixture will spray § of an acre ? 2. If sulphate of copper can be boughtat 6/ per pound, and stone lime at \$ per pound, what is the cost of mate- rials for 500 gallons of Bordeaux mixture ? 3. If a man with a team of horses at $3.50 per day, using a geared sprayer, can spray 3 acres of vineyard in a day, and Bordeaux mixture costs 27/ per barrel, using 10 barrels per day, what is the cost of spraying 30 acres of grapes ? 4. 600 bushels of apples from 40 trees without spraying for fungus growth were sold as follows : 100 bu. of perfect fruit at 81 per bushel, and 500 bu. of scabby and wormy fruit at 30/ per bushel. If spraying at 15/ per tree re- versed the quantities of perfect and imperfect fruit, what would be gained by spraying ? 5. The estimated value of a crop of grapes is $400 per acre. Thorough spraying costs* $6 per acre. If without spraying, fungus diseases destroy 30 % of the crop, what is the net value of the spraying of 10 acres ? HAM. SCH. ARITH. — 21 398 AGRICULTURAL PROBLEMS 6. The average yield from 4 acres of sprayed grapes was 4770 lb. per acre, and the yield from unsprayed grapes was 3108 lb. The grapes were sold for 2±^ per pound. The cost of spraying was $8.60 per acre. What was the net gain on the 4 acres as a result of spraying? A Paris green solution is used for killing chewing insects that destroy the plants by eating the leaves; and for destroying the codling moth to prevent wormy fruit. A solution for this purpose is made by mixing i ounces of Paris green with 50 gallons of water, a sufficient amount to spray 20 apple trees. When fungus diseases are to be controlled also, the Paris green is added to Bordeaux mixture. 7. What is the cost of materials to spray an apple orchard of 250 trees, for the apple worm, with Paris green at 32^ per pound, using a barrel of the solution to 20 trees ? 8. Mr. Wagner sprayed 8| acres of potatoes 4 times with Bordeaux mixture containing Paris green to control blight, rot, and insects. His expense account was as follows: 183 lb. copper sulphate at 8^; 204 lb. lime at #1.15 per hun- dred; 10 lb. Paris green at 35 ^; 48 hr. labor for a man at 20^; 40 hr. labor for team at 15 ^; wear of sprayer, $1.50. What was the cost of spraying per application per acre ? 9. The 8| acres mentioned above yielded 1567.5 bu. pota- toes. A portion of the same field that was left unsprayed yielded at the rate of 156 bu. per acre. Mr. Wagner sold his crop for 55^ per bushel. What was his net profit per acre resulting from spraying ? 10. Mr. Gould sprayed an orchard of 240 trees five times during the season, using altogether 3013 gal. Bordeaux mixture containing Paris green. The cost of materials was: Copper sulphate, 256 lb. fit 8^; lime, 347 lb. at \t\ Paris green, \b\ lb. at 35^; and the cost of applying was labor, 3 men, \\ days at $1.50; 1 team,4| days at $1.50; wear on SPRAYING PLANTS 399 spray machinery, $5. Determine the cosl of materials and the cost of applying per gallon and also per tree, including wear on machinery. 11. Thirty-four apple trees sprayed to control scab and codling moth yielded 90 bu. merchantable fruit and 32 bu. culls and windfalls. In the same orchard 21 unsprayed trees yielded 11 bu. merchantable fruit and 40 bu. culls and wind- falls. The merchantable apples were sold for 67^ per bushel and the culls and windfalls for 22^ per bushel. The cost of spraying was 25^ per tree. What was the net gain from spraying per tree? Kerosene Emulsion is used to kill insects that suck the juices of plants. A solution for this purpose is made as follows : first, dissolve 2 ounces of soap in one quart of water by heating until the soap is dissolved ; second, add two quarts of kerosene oil and stir for five minutes; third, add to this mixture 17 quarts of water to make 5 gallons of a 10% mixture. After thorough stirring, it is ready for use. 12. What would be the cost, at 12^ per gallon, for enough kerosene to make 500 gallons of a 10 % kerosene emulsion that might be used on plants in leaf ? To make 128 gallons of an 8 % emulsion ? 13. A farmer has b\ A. of cabbage. He estimates that the yield will be 6500 heads per acre, and that the selling price will be 1\t per head. Cabbage lice threaten a loss of 10%. If by applying kerosene emulsion at a cost for mate- rials and labor of $7.25 per acre, the loss can be reduced to 2 %, what will be the gain from applying the treatment to the b\ A.? 14. A nurseryman used 10% kerosene emulsion to kill plant lice on his roses and young fruit stock. His expendi- ture for kerosene was $4.50. If kerosene was 15^ per gal- lon, how many gallons of the 10% mixture did he use? How many pounds of suap were used in making it ? 400 AGRICULTURAL PROBLEMS The San Jose" scale is a very small insect not larger than the head of a pin. It injures the tree by sucking the juice of the bark. A lime- sulphur solution for killing the insect is made as follows: hist, dissolve 20 pounds of freshly slaked lime and 15 pounds of sulphur in 12 gallons of water, boiling the mixture 1 hour; second, add to this enough water to make 50 gallons of spraying mixture. This forms a hard covering over the tree, thereby preventing the scale insect from doing further harm. 15. If sulphur is worth 3^ per pound and lime is worth \t per pound, and the labor of making lime-sulphur solution is worth 15^ per barrel, what will be the cost of making enough solution to spray an orchard of 252 peach trees, 25 gallons being sufficient for 9 trees? 16. In the above mentioned orchard, the cost of applying is 85 % of the cost of solution. How much does it cost per tree to spray the orchard? If one application per year for ten years prolongs the life of each tree four years and the average annual yield of fruit is worth $1.25, what is the net gain per tree due to prolonged life? 17. A lime-sulphur solution costs 75^ per barrel to make, and a soluble oil preparation for killing scale insects costs $1.25 per barrel. A barrel of the former will cover 18 trees and a barrel of the latter 15 trees. What would be the sav- ing from using a lime-sulphur solution on a ten acre orchard of pear trees set 140 to the acre, the cost of applying being equal and the solutions being equally effective insecticides? 18. The market value of a crop of peaches is $225 per acre. The total cost of production, including interest on in- vestment, cost of marketing, etc., is $175 per acre. What would be the per cent of loss on the market value from fun- gus diseases and insect injuries, if the market value is so reduced as to equal the cost of production ? TEST PROBLEMS 1. J. P. Black & Co.'s bank account is overdrawn $182.50. They place to their credit a 90-day note for 11500 with in- terest at 6%, discounted 30 days after date. They then deposit $84.30 and check on their account to the extent of $341.65. Find their balance in bank, if the custom of this bank in discounting is to count both the day of maturity and the day of discount. 2. A cow is tied in the corner of a square field by a rope 2 rods lonsr. Find the extent of the surface over which it can graze. 3. A cold air register is 18 in. by 30 in. What are the dimensions of a similar register to let in double the volume of cold air? 4. A clothier sold a suit marked $24, at 10% off for cash, making a profit of 20 %. How much did the suit cost? 5. There are 50 persons in a schoolroom 36 ft. long, 30 ft. wide, and 15 ft. high. How many cubic feet of air space are there for each person? The law fixes the minimum of fresh air at 30 cubic feet per minute per person. How frequently must the room be filled with air to meet this requirement ? 6. A municipality paved a street 48 ft. from curb to curb at $1.60 per square yard, under an ordinance that assessed property holders abutting on the street | of the cost- Mr. James owns two lots each 25 ft. wide fronting on the street. He puts in a curb at 75^ per linear foot and a side- walk 12 ft. wide at $1.80 per square yard. How much is his bill for the improvements? lot 402 TEST PROBLEMS 7. A conical glass is 4 inches in diameter and 6 inches high. How often can it be filled from a cylindrical vessel 4 inches in diameter and 12 inches high ? 8. An article was sold at 2") oj advance on the cost. The proceeds were invested in a second purchase which was after- wards sold for $240. The latter sale was at a loss of 20 c / . Find the selling price of the first article. 9. It cost 1280 to fence a field 80 rods long and 60 rods wide. How much less will it cost to fence a square field of equal area with the same kind of fence? 10. How much will it cost to bronze a globe 12 inches in diameter with gold leaf at 5^ per square inch ? 11. The diameter and altitude of a cone, the diameter of the base and the altitude of a cylinder, and the diameter of a globe are each 3 ft. Find the volume of each. 12. A town assessed at $2,450,000 must raise a tax of $7766.25. The poll tax is $825. Find the tax rate, if the collector is allowed 5 % for collecting. 13. When the Florida collided with the Republic off the coast of Massachusetts, the wireless message summoned the Baltic to the wreck from a point 62 miles north and 84 miles east. What was the distance in a direct line ? 14. A merchant sold 20 lb. more than l of his butter. He then reduced the price b$ per pound, thereby reducing his profit $ 2. How many pounds had he at first ? 15. Mr. Adams sold 50 % of his stock at 20 % gain, and 80% of the remainder at 25% gain. What was his total gain, if the stock unsold cost $1000? TEST PROBLEMS 403 16. A cow and a horse cost 8190, and $ the cost of the cow plus $12 equal -^ the cost of the horse. Find the cost of each. 17. A piece of steel in the form of ;i cylinder is 4 ft. long and 2 inches in diameter. How long is it when rolled into a bar 1 inch square ? 18. A rectangular field that contains 40 acres is four times as long as it is wide. Find its dimensions. 19. A man bought a tenement containing 8 apartments for §35,000. His taxes, repairs, and insurance cost him 81157 annually. He rented 4 suites at 840 per month, 2 suites at §35 per month, and 2 suites at §28 per month. What per cent did he realize on his investment ? 20. If a piano is marked 80 % above cost, what per cent discount can be allowed from the marked price to realize 20 % profit ? 21. The edge of a cube is 10 inches. Find its diagonal. 22. Find the shortest distance on the surface of the cube mentioned in Ex. 21 between two diagonally opposite corners. 23. A rectangular box is 8 ft. long, 6 ft. wide, and 4 ft. high. Find the dimensions of a similar box whose length is 12 ft. 24. What is the ratio of the volume of the two boxes mentioned in Ex. 23 ? 25. Find the duty in U.S. money on an invoice of leather from Paris, if the leather costs 12,350 francs and the duty is 35% ad valorem. 26. Rose Adhorn and Co. imported 10 cases of woolen goods from England of 385 pounds each, invoiced at £ 408 per case. Find the total duty at 40^ per pound and 60 %. Which is the better, both being safe investments? 79. Solve the following equations : 6% of $100 = 5% of ($ ). 37*% of 64 = ( %) of 120. 80. SI 650 yields 8530.75 interest in 5 years, 4 months, 10 days. Find the rate. 81. What sum of money, at 6% simple interest, will produce in 2 years, 6 months, the same interest that $900 will produce in 3 years, 4 months, at 5 % ? 82. The slant height of a church spire is 48 feet, and its base is a hexagon 6 feet on each side. Find the cost of painting it at 40^ a square yard. 83. On a bill of goods amounting to $900, I am offered a discount of 25%, or two successive discounts of 15% and 10 % . Which would be more advantageous for me to accept, and how much more ? 84. My gas meter, Jan. 1, registered 11800 cu. ft.; Feb. 1, 35800 cu. ft. I paid the bill before Feb. 10, receiving a dis- count of 2 ^ on the even thousand. At 27 / a thousand cubic feet, how much did I pay for the gas used in January? 85. Simplify: a. JL^^g^ J. gf^ x ^ 86. I imported from Canada 7500 yards of flannel valued at 80^ a yard, and weighing 1480 pounds. Specific duty 22^ per pound, and ad valorem duty 30 % . Find the amount of duty paid. 87. The taxes on a property last year were $42, which were | less than this year. Find the per cent of increase in taxes. 414 GENERAL REVIEW 88. A street £ mile long and 30 feet wide is paved and curbed. The paving costs $ 3 a square yard, and the curb- ing 30 ^ a linear foot. Find the entire cost. 89. Find the sum of the quotients : .01-*-. 001 .1 -10 .001 + . 1 10+. 01 .01 + 10 .01 +.001 90. The specific gravity of milk is 1.032. Find the weight of 18 gallons of milk. 91. A hotel and farm sold for $6000 each. The hotel was sold at a gain of 20%, and the farm at a loss of 20%. Find the gain or loss on both sales. 92. The cost of insuring a dwelling at f % is $33.75 a year, and the cost of insuring the furniture at 1 % is $12.75. Find the amount of each policy. 93. Which produces the greater per cent of income and how much, 4 % bonds at 75 or 5 % bonds at 90? 94. Find the sum of the square roots of : .5625 226.5025 110J .0016 100.2001 148|i 95. I bought stock at 89 J, brokerage | %, and after receiv- • ing a dividend of 4%, sold at 104|, brokerage \% clearing $150. Find the amount of stock purchased. 96. A town expends for improvements $6894. The assessed valuation is $480000. Find the rate levied to cover the expense, including the collector's commission estimated at $306. 97. A swimming pool is 30 meters long, 14 meters wide, and averages 1.5 meters in depth. Find the number of kiloliters of water it contains and its weight in kilograms. GENERAL REVIEW U5 98. A Mexican ranchman purchased a tract of land in the form of a rectangle 10 kilometers in length and 4 kilometers in width. Find the number of acres in it. 99. The oxygen in the air is to the nitrogen as 21 to 79. Find the number of cubic feet of each gas in a schoolroom whose inside dimensions are 30 ft. x 24 ft. x 12£ ft. 100. What sum will cancel a note for $122.50 bearing in- terest at 6%, dated April 10, 1903, and maturing September 4, 1905 ? 101. A garrison of 800 men have provisions for 90 days. A reenforcement arrives at the end of 40 days, and the pro- visions last only 40 days longer. Find the number of the reenforcement. 102. Simplify: ^-A^xfofii-^J- 103. What decimal bears the same ratio to .05 that f does toll? 104. A savings bank pays 4 % interest compounded semi- annually, the interest periods being April 1, and October 1. I deposited $100, April 1, 1903, and an equal amount semi- annually to and including October 1, 1905. What amount had I on deposit April 1, 1906? 105. An insolvent debtor pays 40 cents on the dollar. How much will a creditor receive whose claim is $960, after paying his attorney 10 % for collecting it ? 106. A rectangular field whose width is f of its length contains 7| acres. Find the distance between the opposite corners. 107. I loaned f of a certain sum at 6%, and the remain- der at 5%. The entire income was $322.50. Find the sum loaned. 416 GENERAL REVIEW 108. A gentleman wishes to invest in 4| % bonds, selling at 102, so as to provide for a permanent income of -$1620. How much should he invest ? 109. From one tenth take one thousandth ; multiply the remainder by 10000 ; divide the product by one million, and write the answer in words. no. A drugget 9 ft. by 12 ft. covers 50% of the floor of a room 13| ft. wide. Find the length of the room. ill. A mechanic had his wages twice increased 10%. Find his wages before the first increase, if he now receives $4.84 per day. 112. A house which had been insured for 13000 for 9 years at If % for a term of three years was destroyed by fire. How much did the money received exceed the premiums paid ? •113. A natural gas company declares a semiannual divi- dend of 4 % on a capital stock of $150,000. Find the yearly dividend of a stockholder who owns 36 shares, par value $50 a share. 114. A merchant in Denver, Col., buys a New York draft for $600, at \% exchange, and mails it in payment of a bill in Memphis, Tenn. Find the amount paid for the draft. 115. How long must $120 be at interest at 6% to earn $34.64? 116. If a clerk's wages are $48 a month, when he works 8 hours a day, how much should he receive for 9 months' work, of 10 hours a day ? 117. Each side of a roof is 30 ft. long and 18 ft. wide. How many shingles 16 in. long and 4 in. wide, laid \ to the weather, will be required to cover the roof? GENERAL REVIEW 417 118. City bonds bearing 4% interest are sold at 12% premium. Find the rate per cent the buyer gets on his investment. 119. The capital stock of a company is $50,000. There is a deficit of $ 4000 in the earnings. I own 80 shares. Find the amount 1 must pay if an assessment is levied. 120. A lawyer in collecting a note of $ 3000, compromised by taking 80 % and charged 5 % for his fee. Find his com- mission. 121. A man's income is *- of his capital. His taxes are 2| (f of his income. Find the amount of his capital if he pays $ 24 taxes. 122. The specific gravity of iron is 7.80. Find the weight of an iron bar 12 ft. long and 2 in. square. 123. Find the result of C°5 * 1.25+ .1875) x 96 _ 16>5# (.4 -5 +.17) 124. The inside measure of a cubical box is 4 in. on each side. A sphere 4 in. in diameter is placed in the box. Find the per cent of space unfilled. 125. At $1£ a rod, it cost $240 to fence a square field. Find the cost of fencing a rectangular field of equal area wmose sides are to each other as 1 is to 4. 126. A piano listed at $ 650 was sold at a discount of 40 % and 20 %. If the freight was $3.25 and dray age $5, what was the net cost of the piano? 127. A merchant sold a bill of goods amounting to $ 3600 and took a 90-day note for it. Fifteen days later he sold the note at a bank at 6 % discount. How much did he receive for the note ? 128. A boat in crossing a river 400 feet wide, drifted with the current 300 feet. How far did the boat go? HAH. COM PL. Altllll. — 27 418 GENERAL REVIEW 129. Mr. Brown owed Mr. Smith $2000 which he was unable to pay ; but he gave him two 90-day notes covering the amount. One was for $1000 without interest; the other with interest at 6%. Mr. Smith had both notes discounted at a bank at 7 % on the day they were given. How much cash did he receive ? 130. A merchant owing a bill of $1250 in New York is asked to send a draft in settlement of the account. The merchant has only $868 in bank and holds a note of $900 due in 30 days without interest. This he has discounted at 6 % for 20 days and the proceeds is placed to his credit. He buys a draft at | % exchange, giving his check for the amount. How much does he then have in bank? 131. Mr. Franks has a promissory note of $800 dated July 1, 1906, due in one year with interest at 6%, against Boyd Emerson, on which are the following endorsements : Jan. 1, 1907, $50.00 Jan. 1, 1908, $150.00 Dec. 1, 1907, 25.00 Apr. 1, 1908, 200.00 Write the note with the indorsements and find the balance due Aug. 1, 1908. 132. A San Francisco banker discounts a draft for $3000 payable at Portland, Oregon, 90 days after sight. Exchange -Jg %, discount 8 %. Find the proceeds. 133. The United States government pays exact interest at 5 % from April 1 to Oct. 10 on a claim of $63500. Find the interest the government pays. 134. Find the cost of carpeting a hall 30 ft. by 50 ft with carpet 27 inches wide, laid lengthwise, at $1.10 per yd., sur- rounded by a carpet border 18 inches wide, at $ 1.10 per yd., allowing 6 inches for matching on each strip except the first. GENERAL REVIEW 419 135. A cable message was sent at 6:15 a.m. from New York to London. It was delivered 28 minutes 20 seconds after being sent. At what time was it delivered? 136. Mr. Coll leased some property for three years at $2400 per year. His commission for leasing was 1 % of the first year's rent and his commission for collecting 5 % of each year's rent. Find the agent's entire commission for the three years, if all the rent was collected. 137. A commission merchant was offered $1800 per year salary and 2 % on all sales above $40000, or 5 % on all salts. He chose the latter and sold $85000 worth of goods. Did he gain or lose, and how much, by so doing ? 138. 2000 yards of silk when imported cost 215 ^ per meter. If sold at $2.25 per yard, find the gain. 139. A railroad tank along the line of the Paris and Lyons railway is 3.5 meters in diameter and 6 meters in height. Find the number of kiloliters it contains. 140. The specific gravity of iron is 7.80. Find the weight in kilograms of a bar of iron 1 meter long, 1 decimeter wide, and 5 centimeters in thickness. 141. A bookkeeper's income is $2700 per year. His expenses average $50 per month. If he deposits the bal- ance every six months in a savings bank, how much will he have in the bank, at 4 % interest, compounded semiannually, after 3 deposits ? 142. A ship sets sail at Seattle, Nov. 15, at 1 p.m., 120th meridian time, and arrives at Canton in 21 da. 5 hr. 18 niin. Find the solar time of arrival in Canton. 143. Sound travels 1120 ft. per second. The thunder from a flash of lightning was heard 8 seconds after the flash was seen. How far distant was the cloud ? 420 GENERAL REVIEW 144. At what price must a bank stock paying 6 % annual dividends be purchased so as to net the purchaser 5 % income on his investment ? 145. An agent's commission at 2\ % is $57.85. What must be the amount of a check mailed to cover the purchase ? 146. A railway company declared a If. % quarterly divi- dend. How much did the purchaser pay for the stock, if it yielded him 10 % on the amount invested ? 147. Ames Bros., brokers, bought for me 10 shares of Delaware & Hudson, selling at 129| % premium, par $ 100. Write the check in payment to Ames Bros, for the stock. 148. A clerk made the following deposits in a savings bank, at 4 % interest, payable January 1 and July 1 : January 1, 1906, $500 ; July 1, 1906, $350 ; July 1, 1907, $200. On January 2, 1907 he drew out $100. What was his balance July 1, 1908? 149. In one section of the Bessemer Railroad there was laid in one year 6 miles of double track. The rails weighed 100 pounds to the yard and the market price was $32.50 per long ton. The ties cost delivered 69 cents each and were laid on an average of one tie to every two feet. Find the cost of the rails and ties. * 150. A commission broker was to receive 5 % on the first $ 50000 from a sale of coal land and 2 % for the remaining amount of the sale. His entire commission amounted to $ 4000. Find the total amount of the sale. 151. Mr. Adams bought a property for $ 20000. He ex- pended $4000 in improvements. The repairs each year averaged $ 250, the insurance and taxes 2^ % on | of the original cost of the property. For how much a year must lie rent the property to realize 6 % net on his investment ? OPTIONAL SUBJECTS PRESENT WORTH AND TRUE DISCOUNT A owes B $ 106, due in one year without interest. If A pays the debt to-day, $100 will cancel it, since $100 at 6 % will amount to $106 when the debt is due. $106 is the debt; $100 is the present worth; and $106 minus $100, or $6, is the true discount. The present worth of a debt, due at a future time without interest, is a sum of money which, at a given interest, will amount to the debt when it becomes due. The true discount is the difference between the debt and its present worth. True discount is seldom used in business. Written Work l. Find the present worth and true discount of a debt of $287.50, due in 2 yr. 6 mo. without interest, money being worth 6 %. $.15 = the interest of $1 for 2$ yr. at 6 %. $1.00 + $.15 = $1.15 = the amount of $1 for 2£ yr. at 6%. $287.50- $1.15 = 250, and 250 x $1.00 = $250, the present worth of $287.50. Since the present worth of $1.15 is $1, the present worth of $287.50, which is 250 times $1.15, is 250 times $1 = $250. $287.50 - $250.00 = $37.50, the true discount. Comparative Study- Bank Discount is the interest paid in advance upon the value of a note, or a debt, at maturity; true discount is the interest on the present worth of the note, or debt, for the given time. In bank discount notes may, or may not, bear interest ; in true discount debts are without interest. The present worth corresponds to the principal; the true discount, to the interest: the sum due at a future time, to the amount. 121 422 OPTIONAL SUBJECTS 2. What principal will, in 3 yr. and 6 mo., at 6$>, amount to $344.85 ? (The principal is the present worth ; the interest is the true discount.) 3. Find the present worth of 1517.50, due in 2 yr. and 6 mo., without interest, money being worth G %. 4. A merchant buys goods amounting to $355.25 and agrees to pay for them in 3 mo. What cash sum will pay the bill, money being worth 6 % ? 5. A farmer is offered $5000 for his farm, or $5600, pay- able one half in cash and the balance in 1 yr. without interest. How much is the second offer better than the first, money being worth 6 % ? 6. Find the true discount of $1350, due in 1 yr., 4 mo., without interest, money being worth 6 %. 7. What is the difference between the true discount of $575 due in 2-| yr. without interest, money being worth 0%, and the simple interest of $575 for 2^ yr. at 6 % ? 8. A purchaser is offered a horse for $195 cash, or $206 due in 6 mo. without interest. Which is the better offer and how much ? FOREIGN EXCHANGE Foreign exchange is a method of paying or collecting bills in foreign countries without the actual transfer of money. A bill may be paid in a foreign country by a postal money order, by an international express money order, by a telegraphic money order, or by a foreign draft, called a bill of exchange. Bills of exchange are usually issued in duplicate and numbered first and second of exchange. By sending them by different mails, the payee is almost certain to receive one. Each contains a condition that it shall be void after the other is paid. Bills in foreign countries are collected by commercial drafts in the same manner as in domestic exchange. FOREIGN EXCHANGE 423 A foreign draft, or bill of exchange, is similar to a domestic l>ank draft and is payable in the money of the country on which it is drawn. 'I'll us. a bill of exchange on Paris is payable in francs. Premium and discount. In domestic exchange, there is practically no premium or discount on money, except during financial panics. In foreign exchange, the premium or discount varies ac- cording to the demand for, and the supply of, money. English exchange is quoted as so many dollars to the pound. Thus, a quotation of 4.91 means that a foreign draft for £ 1 will cosl s L91. French exchange is quoted either at so many francs to the dollar, or at so many cents to the franc. Thus, a quotation of 5.8 means that a draft for $1 will purchase 5.8 francs ; or a quotation of 20 J T means that, a draft for 1 franc will cost 2QJ s f. German exchange is quoted at so many cents to the -i marks or to the mark. Thus, a quotation of 98 means that 98 ^ will purchase a draft for 4 marks ; or a quotation of 24.2 means that a draft for 1 mark will cost 24.2 cents. The par of exchange between two countries is the standard value of the monetary unit of one expressed in that of the other. The English par of exchange is $4.8665. A quotation of 4.90 is above par and one of 4.84 is below par. The French par of exchange is about 5.18^, or 19.3. The German par of exchange is about 95.2, or 23.8. The following is a newspaper quotation of commercial foreign exchange : 60 Days Demand Sterling 4.81 .... 4.87 Germany, reichsmarks .93f . . . .94| France, francs 5.21 .... 5.18$ This means that a XI draft payable on demand will cost 8 4.87 ; or $4.81 payable in 60 days. Written Work 1. Find the cost of a demand draft, at the quotation given, for £ 2">. Cos! of £ 1 =8 1.87; cosl of a £ 25 draft = 25 x 1 I. -7 = 9 121.7.".. 424 OPTIONAL SUBJECTS 2. Find the cost of a 60-day draft for 4480 marks. Cost of 4 marks = 93|^; cost of 4480 marks = ^^ x 934/ =$1050. 4 3. Find the cost of a 60-day draft for 3200 francs. Cost of 5.21 francs = $1; cost of 3200 francs = ?=92 x $1 = $614.21. 5.21 Find the cost of a 60-day draft for : 4. £ 25 6. 6500 fr. 8. 635.5 M. 5. 275 M. 7. £ 255.5 9. 398.2 fr. 10. Find the cost of a sight draft on London for £ 400, at the foreign quotation given. 11. Find the face of a demand bill of exchange on Paris for $2500, at the foreign quotation given. 12. What is the cost of a demand draft on Hamburg for 1125 marks, exchange 94| ? 13. What is the cost of a draft on Lyons for 14,000 francs at 5.19? 14. A merchant buys a London draft 60 days after sight for £ 95. If exchange is 4.82, rind the cost of the draft, 15. How large a bill of exchange on Berlin can be pur- chased for $1590, exchange being 98 ? 16. A mechanic has $ 1500 with which to purchase a London draft, If exchange is 4.845, how much is the face of the draft in English money ? 17. An American tourist bought a letter of credit on London for £ 300 at $4.88 and 1 % commission. How much, in United States money, did this letter of credit cost him ? If lie drew £50 in Paris, how many francs did he get in exchange, the pound being valued at 25.2 francs? COMPOUND PROPORTION 425 COMPOUND PROPORTION A simple ratio is a ratio of two numbers ; thus, 9:3 is a simple ratio. A compound ratio is the produet of two or more simple .. ,n qn ra on f- ] 6\ [9:31 . ratios ; thus, (9:3) x (b : 2), or ( - x - j, or j ,, t9 l is a com- pound ratio. A compound proportion is a proportion in which one or f 9*3] both ratios are compound; thus, jV^[ = 18:2 is a com- pound proportion. Written Work 1. If 3 men earn $24 in 4 days, how much can 6 men earn in 3 days? Since the answer is to be dollars, the second ratio in this compound proportion is $24 : $ x. The first ratio in this proportion is compound. If 3 men earn $ 24 in a certain time, G men can earn more in the same time; x, then, repre- sents a larger sum than $24, and the Jim t simple ratio in the compound ratio is 3:6. If a given number of men earn $24 in 4 days, in 3 days they will earn less; x, then, represents a smaller sum than $24, and the second simple ratio in the compound ratio is 4 : 3. Combining these two ratios, the compound ratio is I . \ q 1 and the com- , ,. . f 3 : 6 1 ^ oi m 6 x 3 x $24 «.,„ pound proportion is \ . . ._, j- = $24 : x. 1 hen, x = T~,~~Z = 2. If 6 men earn $75 in 5 days, how much can 12 men earn in 3 days ? 3. If 8 men, in 10 days of 9 hours each, earn $280, how much can 9 men earn in 5 days of 8 hours each ? 4. If 24 men dig a trench 72 rods long, 3 feet wide, and 5 feet deep in 12 days, how long a trench, '2\ feet wide and 3 feet deep, can 18 men dig in »i (lavs? 426 OPTIONAL SUBJECTS CUBE ROOT Comparing Roots and Periods The cubes of the smallest and the largest integers com- posed of one, two, and three figures are as follows : l 3 = 1 103 = loco 100 3 = 1,000,000 9 3 = 729 99 3 = 970,299 999 3 = 997,002,999 1. Separate each of these perfect cubes into periods of three figures each, beginning at the right; thus, 997 '002' 999. 2. How does the number of periods in each cube compare with the number of figures in the corresponding roots ? The number of periods of three figures each, beginning at units, into which a number can be separated, equals the number of figures in the cube root of the number. Note. — The left-hand period may contain one, two, or three figures. 3. How many figures are there in the cube root of 46,656? 1,030,301? 12,326,391? - 4. Cube 25. 25 = 20 + 5 ; hence, it may be cubed in two ays, thus : 25 20 + 5 25 20 + 5 125 = (20 x 5 ) + 5 2 50 tens = 20 2 + (20x5) 625 20 2 +2(20x5)+5 2 25 20 + 5 3125 (20 2 X 5) + 2(20 x 5 2 ) + 5 3 1250 tens = 20 3 + 2(20 2 x5)+ (20x5 2 ) 15625 = 20 3 + 3(20 2 x5) + 3(20 x5 2 ) + 5 8 Or, representing the tens by t and the units by u, we have the formula: ( r" + u y = f + 3 f'u + 3 tu 1 + u\ CI BE HOOT 427 25 3 = = 15,625 Observe thai 15,625, tin- cube of 25, is composed of four partial products : , 1 ) f 3 =2<>3=S (2) Zfiu =3(202 x 5)= 6000 (3) 3fn a =3(20 x5 2 ) = 1500 (4) tf s = 5 8 = 125 5. Find tbe cube root of 15,625, or find the edge of a cube whose volume is 15,625 cubic units. 15'625 20 3 = 8000 Trial divisor, 8x20 2 =l:i<"i 3 x 20 x 5 = 300 5 2 = 25 Complete divisor, 1525 7 625 7 625 20 5 2^> Separate the num- ber into periods of three figures each. Since 15,625 con- tains two periods, its cube root is com- posed of two figures, tens and ones. As the cube of tens is thousands, the largest cube found in 15 thousands is 2 tens, or 20 ones. 20 3 = 8000 (1st partial product, or t 3 ) as shown in figure A; 15,625 — 8000 leaves a remainder of 7625 (3 t-u + 3 tu 2 + u*). The root 20. there- fore, must be so increased as to exhaust this remainder, and keep the figure a per- fect cube. The neces- sary additions to enlarge A and keep it a cube are : First, the three equal square solids B. C, and D (2d partial product, or %Pu)\ second, the three equal rec- tangular solids (p. £28) E. F, and G (8d partial product, or 3 tu*)) and third, the small cube (p. 428) H (4th partial product, or n 3 ). The sum of these three additions is 7625 cubic units ; and, since the square solids B, C, and D (2d partial product, or 3 t*u) contain the greatest part of the additions, their volume is nearly 7r.-_>:> cubic units. If we rr ^^^a-:;v:^25Sg^ -r- — — - — — ; _ +j 1_ __^ — e- — — i , , , , . . , /> a . f f. - -- - shall find their H I ■ ■_ is 9 i • b '.-- I - the surface of --— j j i ".'-:.••-■ . iitions. 1. C.L. E F. G. aud H v - ic each Sraatet] I : - i » ihnne " " cubic Trnxte) vould be feimil* i than H< . -■ ; -. - T -r - - ' unite. : •-■ -- " ii- - ..L"- - - _- . .-. i " - - r* •- • ■ . •- :n . ' --.- I T =^ :ji anc ei e L._ . -. ;:-_.,- ' ami:- [ - - mite wide I - i . • - mac rants, and the surface of tbe ' • ' - ' - __ - -i«ft*iMw - i be UK, ^ ^^ • . ' - ■ - fcb±tf T • - ..iditio: ..•-.••- 25, - ' . ' .Are Tir - ■ . - -■ ^^m ' .' ■ V: -.-- - - - .. ~ * = uii"*rt c.'j'.- - : . - : naust - - " ■Bttfc. li> - - ". - ■ ^^^^^^^^^ - ' - - - ' ■ "'■—' ' - - - " ' I " i I - umbere eaiino; - -->••. be iato two or f/tr*^ - Kfan; l ■ vd. art - .' In -eacli case, one o: . - root of -.j* iiuiD' regarded aB ■e number of unit* iu tfce «o the g CUBE ROOT 129 6. Cube 35 (30 + 5) ; then find the cube root of 42875. (1) f»=30 8 = 27000 Partial I (2) 3 fiu = 3 ( 31 > 2 x 5 ) = 1 3500 products: 1(3) 3f» 2 = 3(30 x 5 2 ) = 2250 (4) tf 3 = 5 3 = 125 42875 42'875 30 Trial divi- 30 3 = = 27000 5 sor, 3 X 30 2 = 2700 15875 35 3 x 30 x 5 = 450 5 2 = 25 Complete divisor, 3175 15875 Study of Problems 1 . How many periods are there in 42,875 ? 2. How many figures are in its cube root? 3. The left period is 42 (42,000). The largest cube found in it is 27 (27,000), the cube of the tens. 4. What, then, is the tens' figure of the root ? ( \/27 =3.) 5. Subtract; annex the next period, and the new dividend is 15.s7.">. The cube, 30, or 3 tens, is to be so enlarged as to exhaust this remainder and yet preserve it a cube. 6. This dividend is composed mainly of what partial product? (The second, 3 fiu) = 3 (30 2 x 5) . 7. What factors in the second partial product are already known? (3 x 30 2 .) 8. If 15,875 is divided by (3 x 30 2 ) as a trial divisor, the quotient (5) will be (approximately) the other factor in the second partial product. 9. What, then, is probably the units' figure of the root? (5.) 10. Observe that the trial divisor, 2700, is equal to three times the square of the root found, considered as tens. 11. The first addition to the trial divisor is 3 x 30 x 5, or 450 (three times the root found, considered as tens). The second addition is 5 2 , or 25. 12. What, then, is the complete divisor? 3 p + 3 tu + «a = 3 (30)2 + (3 x 30 x 5) + 5 2 = 3175. 13. Multiplying 3175 by 5 gives 15,875 ; or, u(3 P + 3 tu + u 2 ) = 3:*u + 3 tu 2 + u*. This exactly exhausts the remainder of 15,875. Hence, the unite' figure of the root is 5, and the cube root of 42.875 is 30 I •>, or 35. 430 OPTIONAL SUBJECTS 7. Find the cube root of 34012.224. (Separate the num- ber into periods, to the left and right from the decimal point.) 34'012.'224' | 32.4 3 3 = 27 Trial divisor, 3 x 30 2 = 2700 3 x 30 x 2 = 180 2 2 = 4 Complete divisor, 2884 012 5768 Trial divisor, 3 x 320 2 = 307200 3 x 320 x 4 = 3840 4 2 = 16 Complete divisor, 311056 1244224 1244224 When the cube root consists of more than two figures, three times the square of the root already found (considered as tens), is used as a trial divisor in finding the next figure of the root. 8. Find the cube root of 5 to thousandth IS. 5. | 1.709+ 1 3 = 1 Trial divisor, 3 x 10 2 = 300 4000 3 x 10 x 7 = 210 7 2 = 49 Complete divisor, 559 3913 Trial divisor, 3 x 170 2 = 86700 87000 Trial divisor, 3 x 1700 2 = 8670000 87000000 3 x 1700 x 9 = 45900 9 2 = 81 Complete divisor, 8715981 78443829 When a ciDher occurs in the root, annex tiro more ciphers to the trial divisor, bring down the next period, and proceed as before. CUBE ROOT 4:11 Separate the number into 'periods of three figures each, to the left and right from the decimal point. Find the largest cube in the left-hand period, and write its root as the first root figure sought. Subtract this cube from fin- left-hand period and annex the next period to the remainder. For a trial divisor, take three times the square of the root already found, considered as tens, or three times the square of the root with two naughts annexed. Divide the dividend by it, and the quotient, or the quotient diminished, will be the second part of the root. To the trial divisor, add three times the product of the first part of the root, considered as tens, by the second part, and also the square of the second part. Their sum will be the complete divisor. Multiply the complete divisor by the second part of the root, and subtract the product from the dividend. When other periods remain, take three times the square of the root already found, considered as tens, for a trial divisor, and proceed as before. Note. — 1." When a number is not a perfect cube, annex periods of naughts, and continue the work as far as desired. 2. Decimals are separated into periods of three figures each, begin- ning at the decimal point and passing to the right. :!. To find the cube root of a common fraction, take the cube root of the numerator and the denominator separately, or reduce the fraction to a decimal and then extract its cube root. Find the cube root of : 1. 2744 7. 373,248 2. 4096 8. 941,192 3. 13,824 9. 1,860,867 4. 19,683 10. fife 5. 32,768 11. T VW, e. 91,125 12. T y^ 13. .729 14. 15.625 15. 39.304 16. 2i^ti.981 17. .003375 18. 3| 132 OPTIONAL SUBJECTS Written Work 1. The volume of a cubical box is 5832 cu. in. What is its edge ? Note. — Since the volume of a cube is a number of cubic units equal to the product of its three equal dimensions, the cube root of the volume of a cube gives the length of its edge. 2. If 9261 cubic inches are built into one cube, what is the area of its base ? 3. Find the depth of a cubical cistern that will contain 2197 cu. ft. 4. What is the edge of a cubical box that will contain 50 bushels of 2150.42 cu. in. each? 5. A box is 16 ft. long, 8 ft. wide, 4 ft. high. What is the edge of a cubical box of the same volume ? 6. Find the edge of a cube whose volume is equal to the volume of three cubes whose edges are respectively 3, 4, and 5 inches. Geometry shows that : The corresponding lines of similar solids are proportional to the cube roots of their volumes. 7. If a ball 10 in. in diameter weighs 125 lb., what is the diameter of a similar ball that weighs 216 lb. ? 8. Of two similar solids, one contains 8 times the volume of the other. The diameter of the smaller is 8| feet; what is the diameter of the other ? Suggestion : . 438 INDEX Decimal fractions, division of, 101-106. multiplication of, 98-101. notation and numeration of, 91-94. reduction of, 95. subtraction of, 96-98. Decimal point, 90. Degree, of angles and arcs. 194, 196, 436. of longitude, 350, 351. Demand note, 294. Denominate numbers, 115— 122, 175-191. Denominator, 44. Deposit slip, 303, 304. Diagonal, 123, 207. Diameter, 210. Difference, in percentage, 238. in subtraction, 19. in time, 120. Digits, 38. Dime, 12, 382, 436. Discount, bank, 307-314, 421. commercial, 264—269. on stocks, 324, 326, 328. true, 421. Discounting notes, 303, 308. Dividend, in division, 29. in insurance, 263. in stocks, 325, 329. Divisibility, tests of, 3S, 39. Division, denned, 29. of Common fractions, 6S-74. of decimals, 101-106. of denominate numbers, 121, 180, 187. of integers, 29-32. sign of, 29. Divisor, 29. Dollar, 12, 3S2, 436. Domestic exchange, 315-323. Dozen, 25, 435. Drachma, 182. Draft, 317-323, 422. Dram, 435. Drawee, 319. Drawer, 319. Dry measures, common, 134, 433. metric, 388, 389. Dutiei or customs, 272-274. Eagle, 436. Eastern time, 355. Endowment policy, 262. English money, 181-183, 436. Equality, sign of, 13. Equations, analvsis bv, 231- 236. Equator. 349. Equilateral triangle, 197-36". Even number, 38. Evolution, 360-36G, 426-432. Exact divisor, 37. Exact interest, 287. Excavations, 132. Exchange, 315, 323. 422-424. Exponent, 37, 359. Express money order, 315, 422. Extracting roots, 860-866, 426- 432. Extremes, 338. Face, of note, 293. of solid, 368. Factor, defined, 37. Factoring, 39, 40. roots extracted by, 361. Factors and divisors, 37—43. Farthing, 181. Fathom, 433. Feeding stock, 392-394. Fertilizers, 395, 396. Figures, notation by, 7. Fire insurance, 260. Firm, 342. Flooring, 202, 203. Fluid dram and ounce, 435. Foot, 433. Feinting of account, 174. Foreign exchange, 422. Foreign money, 181-1S3. Fractional parts, of fractions, 65, 66. of integers, 60. Fractional relations, 76, 77. Fractional units, 44. 45. Fractions, common, 44-83, 106. 107. decimal, 90-111. Franc, 181, 182,436. Free list, 273. French money, 181-183. Full indorsement, 294. Fundamental processes, 7-35. Furlong, 433. Gain and loss, 249-253. Gallon, 115, 223, 224, 433. German money, 182. Gill, 433. Globe, 369, 375, 376. Government bond, 332. Government land, 356, 357. Government revenue, 272- 274. Grace, days of, 295. Grain, 435. Gram, 383, 389. Great gross, 435. Greatest common divisor, 40, 41. Greek money, 182. Gross, 435. Gross cost. 250. Gunter's chain, 356. Hand, 433. Hectare, 387. Hectoliter, 389. Hexagon, 367. Hurher terms, 47,48. Hour, 436. House, 342. Hundreds, S. Hundredths, 91. Hundredweight, 435. Hypotenuse, 365. Improper fraction, 45. Inch, 433. Incomes, from stocks and bonds, 329. Incomplete decimal, H)7. Index of root, 360. Individual note. 295. Indorsements, 15>. 293, 294. Indorser of note, 293. 294. Insurance, 259-263. Integer, 37. Interest, annual, 286, 287. compound, 288-292. exact, 287, 288. simple, 151-154, 275-284. table of, 291. Interest term, 289. Internal revenue, 272. International date line, 354. Inverting terms, of fractions, 72, 73. Investments, 291, 329. Involution, 35S, 359, 362. Isosceles triangle, 197. Italian money, 1S1-1S3. Joint and several note, 295. Kilogram or kilo, 389. Kilometer, 384. Knot, 433. Land measures, 356, 357, 387, 434. Lateral surface, 368. Laths, 201. Leap year, 436. Least common denominator, 51. Least common multiple, 41, 42. Ledger, 173. Ledger accounts, 173, 174. Length, common measures of, 123, 192, 193, 433. metric measures of, 384—386. Letters, notation by, 892— 405. Life insurance, 262. Life policy, 262. Like numbers, 14. Linear measures, 433. Lines, 194. Link, 434. Liquid measures, common, 433. metric. 388. 3S9. Lira, 181, 132, 436. List price. 264. Liter, 383, 389. Load, 212. Long division, 31, 32. Long ton, 435. Longitude and time, 349-355. INDEX 139 Loss, 250. Lowest terms, 48, 49. Lumber, 134, 135, 217-220. .Milker, of check, 305. of imt.'. 298. Making change, 17. Murine Insurance, 260. Mark. 182, 186. Market value of stock, 324. Maturity of note, 295, 808. Means, in proportion, 388. Measurements, practical, 193— 230. Measures, tables of, 198, 196, 212, 224, 884-590, 433- 486. Mensuration, 307-381. Merchants' rule for partial payments, 302. Meridian, 349. Meter. 382. Metric system, 382-391. Metric ton, 389. Mile, 3S5, 434. Mill, 12, 382, 436. Millimeter, 384. Millions, 8, 9. Minim. 435. Minuend, 19. Minus, 19. Minute. 436. Mixed decimal, 92, 101, 102. Mixed number, 45, 49, 50, 56, 57, 61-65. Money order, 315, 316, 422. Months, of year, 436. Mortgage, 331, 332. Mountain time, 355. Multiple, 41. Multiplicand, 24. Multiplication, defined, 24. of common fractions, 58- 68. of decimals, 9S-101. of denominate numbers, 186, 187. of integers, 23-28. sign of, 24. Multiplier, 24. National bank, 303. Nautical mile, 433. Negotiable note, 293. Net price, 148, '-'04. Net proceeds, 147, 250, 255. Non-negotiable note, 293. Notation and numeration, of common fractions, 46. of decimals, '.'1-94. of integers, 7-1 o Notes, promissory, 292-302. Number. 7. symbol, Jf, 169. Numeration, 7-12. 46, '.'I B4, 392. Numerator. II. Obtuse angle, 196. odd number, 88. One dollar six per cent inter- est method, 281, 2S2. < Operation, signs of, 36. Orders for goods, 167. Orders of units, 8. Ounce, 185. Pacific time, 355. Painting. 201. Papering and carpeting, 204, 205. Par value of stock, 324. Parallel lines. 124, 194. Parallelogram. 198, 207. Parenthesis, 36. Partial payments, 298-302. Partition. 29. Partitive proportion, H41-344. Partnership, 342-344. Payee, 293, 319. Peck, 433. Penny, 181. Pennyweight, 435. Pentagon, 367. Per cent, 237. Percentage, 141-146, 237-274, 335, 336. Perch, 220. Perfect square, 360, 361. Perimeter, 193, 369. Periods of figures, 8. Perpendicular, 194. Personal insurance, 262, 263. Personal property, 269. Peseta, 181, 182,436. Peso, 436. Pfennig, 182. Pi (tt), 210, 372, 375. Pint, 433. Plane, 367. Plastering and painting, 128, 201, 202. Plus, 13. Policy, 259. Poll tax, 269. Polygons, regular, 367, 868. Postal money order, 315, 422. Pound (sovereign 1 ). 181. 182, 436. in weight, 115. 435. symbol, $, 169. Powers and roots, 358-366. Practical measurements. 128- 140, 192-280. Premium, on exchange, 259, 262. on policy , 262. on -toeks. 824, 320, 328. Present worth, 421. Price list, 266. Prime factor, 39. Prime meridian, 349. Prime number, 37, 88. Principal, in Interest, 151, 276. consignor), 255. Prism, :o;s-:f7:i. Private bank, 808. Problems in Interest, 288, 284. Proceeds, net. 147, 250, 255. of note. 808. Product, •.'!. Profit and loss, 249-268. Promissory notes, 292-302. Proper fraction. 45, Property insurance, 260, 261. Proportion, 888 844, 126. Protesting a note. 294. Protractor, 195. Public land, 350, 357. Pyramid, 369-371,373-375. Quadrilaterals, L98, 199. Quart, 1 88 Quintal, 889. Quire, 435. Quotations, of stocks and bonds, 333. Quotient, 29. Radical sign, 360. Radius, 210. Range, 356. Kate, of insurance, 259. of interest, 151, 275. of taxation. 269. per cent, 145, 288. Ratio, 337, 338, 425. Real estate or real property, 269. Ream, 435. Receipts, 157, 166, 168, 169. Reciprocal, 72. Rectangles, 124, 198-201. Rectangular solids, 131, 213, 214. Reduction, of common frac- tions, 47-52, 106, 107. of decimals to common frac- tions, 95. of denominate numbers, 116-119, 175-184. Registered bond, 832. Regular polygons, 367, 368. Remainder, in division. 29. in subtraction, 19. Revenue, government, 272- 274. Reviews, 7S-s3, 107, 122, 138, 155, 156, 159-105. 1- 191, 225-280, 247-249, 253, 254, 284,285,885,886,881,401- 420. Rhomboid, P.m. Rhombus, 199. Righl angle, 124,129,194, 196. Right-angled triangle, 128, 206, 865, 306. Rod, i Roll of paper, 204. Roman notation. 11. Roofing and flooring, 202. 208. R 9, 858 360, 426-482. Ruble, 1 tfl 440 INDEX Savings accounts, 289-291. Savings bank, 303. Scalene triangle, 197. Score, 435. Scruple, 435. Second, 436. Section, 192, 357. Selling price, 250. Share of stock, 324. Shilling, 181, 436. Shingles, 203. Shipper, 255. Short rates, 259. Sight draft, 320, 321. Similar figures, 376-378, 432. Similar fractions, 51. Similar solids, 37S, 379, 432. Similar surfaces, 376-37S. Simple denominate number, 115, 175. Simple interest, 275-285. Simple proportion, 338-341. Simplification of complex frac- tions, 75. Six per cent interest method, 278-282. Slxtv-day six per cent interest "method, 278-280. Slant height, 369. Solar year, 436. Solids, 212-216, 36.9-376, 37S, 379. similar, 378, 379, 432. Sovereign, 181, 182, 436. Spanish money, 181-183. Specific duty, 273. Specific gravity, 379, 380. Sphere, 369, 375, 376. Spraying plants, 397-400. Square, of numbers, 35S. of roofing and flooring, 202. (rectangle), 124, 19S, 367. Square common units, 12S, 193, 206. 434. Square metric units, 386. Square root, 360-366. Standard time, 354, 355. State bank, 303. Statements, on bills, 171. Stationers' measures, 435. Stere, 388. Sterling money, 181, 182, 486. Stock broker, 325. Stock company, 324. Stockholder, 325. Stocks and bonds, 324-334. Stonework, 220, 221. Stub of check, 304, 305. Subtraction, defined, 19. of common fractions, 55-57. of decimals, 96-98. of denominate numbers, 120, 184, 185. of integers, 19-22. sign of, 19. Subtrahend, 19. Sum, in addition, 14. in percentage, 238. Surfaces, common measures of, 124, 193, 194, 434. metric measures of, 386, 387. of solids, 213, 369-372. similar, 376-378. Surveyors' linear measures, 434. Surveyors' square measures, 356, 434. Swiss money, 181-183. Tables, interest, 291. of common measures, 126, 193, 196, 212, 223, 224, 356, 433-436. of life insurance, 262. of longitude, 350, 351. of metric measures, 384-390. of money, 181, 182, 436. of specific gravitv, 380. Tanks. 223. Tare, 273. Tariff, 272. Taxes, 269-272. Telegraphic money order, 316, 422. Tens, 8. Tenths, 91. Term of discount, 308. Term policy, 262. Terms, of fraction, 44. of ratio, 337. Thousands, 8. Thousandths, 91. Time, and longitude, 349-355. standard, 354, 355. Time belts, 355. Time discount, 264. Time draft, 320-322. Time measures, 436. Time note, 294. Ton, common, 224, 435. metric, 3S9. Township, 356. Trade discount, 14S, 264. Trapezium, 199, 209. Trapezoid, 199, 208. Triangle, 128, 196, 367. Trillions, 8, 9. Troy weight, 435. True discount, 421. Trust company, 303. Unit, 7, 44. Unit of measure, 175, 199. United States money, 12, 382, 436. United States rule for partial payments, 298-302. Unlike numbers, 14. Vertex of triangle, 196. of pyramid, 369. Vinculum, 36. Volume, 130, 214-216, 368, 372- 376, 378, 379. common measures, 212, 434. metric measures, 3S7, 388. Water, weight of, 224, 390. Week, 436. Weight, common measures, 435. metric measures, 389-391. Winchester bushel, 433. Wood measures, common, 137, 217-220, 434. metric, 3S6. Yard, 115, 384, 385, 433. Tear, 436. Yen, 436. ANSWEES Page 11. — 1. 43; 449; 1,000,502; L492. 2. 90; 1908; 1576; 1801. 3. XLI; LXIII; LXXXIV ; XCIX ; CVII ; CCXVIII ; DLXXII ; DCCXXXV; CMXCVJ ; MCMVII ; MDLXIV ; MDCXVI ; M; CCLX. Page 15. — 2. §3603.18. 3.2,595,237. 4.3,303,261. 5.3,996,337. Page 16. — 7. 5182. 8. 5261. 9. 6684. 10. 5269. 11. 2799. 12. 83,302. 13. 82,276. 14. 76,385. 15. 84,503. 16. 62,872. 17. 71,809. Page 18. — 1. 913 rai. 2. $11,276. 3: 3,805,074 sq. mi. 4. $ 2,376,- 000,000. 5. $1,254,060,661. 6. $748,152,215. Page 20. — 7. 5997. 8. 22,962. 9. 27.4:!4. 10. 2600. 11. 7999. Page 21. —3. 14.571. 4.36,906. 5.18,393. 6.30,996. 7.38,131. 8. 28,981. 9. $329.04. 10. $379.21. 11. $561.47. 12. $194.39. 13. $ 190.39. 14. $190.33. 15.300,884. 16.565,065. 17.295,404. 18. 186,418. 19. 433,983. 20. 225,545. 21. 3,472,273. 22. 2,698,987 23. 3,12-3,092. 24. 4,197,541. 25. 2,304,902. 26. 2.919.062. Page 22.— 1. 2029. 2. 7290. 3. 966,779. 4. 144,752,500 bu. 5. A, 81375; B, $3075. 6. $9 gained. 7.8 4800. 8. 52,407,000 sq. mi. 9. James, 86750 ; Henry, $11,925 ; Frank. $5000. Page 27. —2. 6000. 3. 3745. 4. 12.880. 5. 35,000. 6. 21,584. 7. 51,134. 8 a. 1,574,125; 8 6.2,210,200; 8-'. 3,623,700; 8./. 4,793,050; 8 e . "lr,. .700 ; 8 f. 6,174,425; % n. 5.583,325; 8 ft. 5,114,300; 8 i. 6,219,400; 8./. 5,769,650. 9a. 250,880; 9 b. 352,256; 9 <\ 577,536 ; 9d. 763,904 : 9^. 823,296 ; 9/. 984, 064 ; 9 g. 889,856 ; 9 ft. 815,104 ; 9 «. 991,232 : 9/. 919,552. 10.'. 2.130,4 6 - W- 7 - W- 8. iff*. 9. «jp> 10. tffft. ii. i|i.. 12. iff a. 13. Jfffi. 14. ifff*. 15. HV-*- 16- ^tM-- 17- fit*. 18. HV*- I 9 - ^f 3 - 20. a^M. 21. iff*. 22. ^A 23. ijfi. 24. i^ 4 A 25. ^W. 27. 2 T V 28. 47if 29. 136f 30. 0ff 31. 32f. 32. 15 2 ". ; . 33. 22ff 34. 22. 35. 18f 36. 25. 37. 24. 38. 15. 39. 21ff 40. 23 T V 41. 24. 42. 10ff 43. 5 T 9 ? . 44. 61f 45. 157J§. 46. 45ff. 47. IlOf 48. 92&. 49. 183f 50. 112. p 31yo ci 9 4 3 5 Q 10 3 2 A 1 5 1 2 T K _4_ 9_ K X-dge JJ.. - 13) xJ) is- •*• igi iJj if- <•»• 24' 54' 24" 6 4 8 5_ t _8_ 12-7 ft 8 1_0 1 Q J_B 15 9 Irt 2 3 5 • T5' T5' 3?' •■ 40' 40' ¥5* °- 24' 24' 24' "• 40' 45' ?0" 1W - 1?' 1 ?' 1 J" U12 14 1 lO 24 _6 19 1» 2 3 1 Id. 12 22 19 IB 27 25 21 • T >5' TO' 50' i#s# '36' 3?' 36' x "* 5' 6' 6* trx ' 58' 58"' 58" ±0 ' 45' l3' 43* 1R16 6 1 I? 85 18 2? 10 1615 34 1Q2 4 4J.9 1D - 58"' 58' 58". Xl - 45" 42' 45" i0 - 48' 48' 48"" ±C '' 50 ' 60' 67J" On 3 2 35 1 Ol 4 4 8 4 5 ^ U - 35' 35' 35" ,4± - 60' 60' 60" P,„o KO O 10 16 15 20 q 32 15 2.8 12 A 42 36 32 27 rd 5 e •*"■ *■ 40' 40' 40' 45 - "• 48' 48» 48"' 4?' *" 60' 50' 50' 67" B 80 96 105 100 fi 250 216 425 7 191 385 54.22. 3. §1.75. 4. $.90. 5. §2.80- 8. $ 10.65. 9. $10.60. 10. $28.65. 13. 19. 23. 27. 30. 34. 14,500 lb. 3 mi. 244 id. 17 T. 7 cwt. 119 gal. 14 bu. . $3.84. pk. 1157 in. 5. 10. 27 in. 14. 508 pt. 15. 28 qt. J yd. 20. 1 mi.l ft, 21. 24. 3 T. 95 lb. 25. 52 mi. 28. 221 rd. 4 yd. 1 ft. 6 in. 31. 2 mi. 44.5 rd. 32. 176 qt. 6. 56 pt. 7. 1825 lb. 11. 9600 lb. 12. 465 sec. 16. 43 pk. 18. 1 rd. 4 yd. 34 min. 3 sec. 22. 64 lb. 9 oz. 217 rd. 26. 346 bu. 3 pk. 1 qt. 29. 1 hr. 54 min. 35 sec. min. 33. 16 hr. 4f 46 1 gal. 36. §26.88. 41. 3 mi. 40 rd. 7. 48 8. 3 IS' Page 118. —35. §39.83. 39. 32 mi. 40. 44 mi. 80 rd 3. i. 4. T \- 5. f 6. Page 119. — 9. f. 10. &. 11. f 15. .1. 16. .09. 17. .28. 18. §2.95. 2. 29 bu. 3 qt, 1 pt. 3. 40 T. 11 cwt. 4 lb 5. 16 wk. 1 da. 14 hr. 37. §41.60. 38. 1920 hr. 12. 13. \. 14. .7. 4. 21 yr. 239 da. 1 hr. Page 120. —7. 22 mi. 194 rd 9. 8 hr. 43 min. 15 sec. 11 14. Lee, 58 yr. 2 mo. 20 da.; 3 mo. 8 da. 4 yd. 2 ft. 6 in. (i yr. 9 mo. 27 da. Grant, 42 yr. 11 mo. 8. 13 gal. 1 qt. 1 pt. 12. 15 yr. 16 da. 12 da.; dif. 15 yr. ANSWERS ' I 19 Page 121. 2 10 gal. 3 qt. 1 pt. 3. 77 bu: 2 qt. 4. 152 T. 10 cwt. 7 lb. s oz. 5. 247 wk. 4 da. 6 hr. 6. 280 mi. 216 rd. 1 yd. 1 ft. 8. 3 wk. 5 da. it hr. 9. 26 gal. 3 qt. 1 pt. 10. 9 bu. 3 pk. 4 qt. 11. 5 yr. 7 mo. 18 da. 12. 14 T. 7 cwt. 24 lb. 13. $ 22. so. 14. 26 rd. 1 yd. 15. 10 packages. Page 122. — 1. 6 min. 18 sec. 2. 32 bbl. 3. 55$jj bu. 4. 818.30. 5. 137 T. 12 cwt. 88 lb. 6. 50 ft. 8 in. 7. 110,340 gal. 8. 35.8605 T. 9. 3.451b.; $1.29+. 10. 54 mi. 200 rd. 11. 20,007 doz. Page 126. —20. 272.25 sq. ft. 21. 2800 sq. rd. ; 17* A. 23. 24,300 sq.ft. 24. 7200 sq. in. 25. 10 A. 26. 260 sq. rd. ' 27. 26 r V r sq. rd. 28. 250,470 sq. ft. 29. 33| A.; $2025. 30. 8488 sq. ft. 31. $76. 32. $170.10. 33. 8 40,425. Page 127. — 34. $6450. 35. 640 A. ; $54,400. 36. 17? bu. 37. 8 500; $.16|. 38. $88.32. 39. 40 rd. 40. (3) 2]^ A. ' 41. (3) lf& A - Page 128. — 1. §3. 2.8 8. 3. 140 sq. yd. 4. $29.55. 5.8 181. Page 129. —5. 40 sq. in. 6. 36 sq. in. 7. 225 sq. in. 8. 432 sq. in. 9. 10 A. Page 133. — 2. 028 sq. ft, 3. 120 ft. 4. $37.21. 5. 8 62.80. 6.72. 7. 288 cu. ft. 8. 166$ loads ; $ 41f . 9. 99 sq. f t. 10.810,000 cu. in.; 3506ft gal. 11. 8 462. Page 134. — 13. f| ; 48 oz. 14. 153$ bu. 15. 4308*$ gal. 16.2800 sq. rd. larger. 17. 4 ft. 18. 04 sq. ft. ; 384 sq. ft. Page 135. — 4. 24 bd. ft. 5. 9 bd. ft. 6. 45 bd. ft, 7. 48 bd. ft. 8. 100 bd. ft. 9. 360 bd. ft. 1. 8 200. 2. 812. 3. 8260.40. 4. 8400. Page 136. —6. 868.75. 7. 87.17. 8. $13.44. 9. 82.52. 10. 88.06. 11. $67.20. 12.8 40.32. 13. $58.80. 14.8 26.88. 15.16 ft. 16. 2112 bd. ft. 17. 224 bd. ft. 18. $25.96. Page 137. — 1. 20 cords. 2.8 440. 3.815. 4.8 44.80. 5. 12$ cords. Page 138. —1. 540 tiles. 2. 289$ f t. 3. 277$ cu. yd. 4. $105. 5. 1,760,000 times. 6.8 24. 7. $327.68. 8. 718114 sal. 9.8108. 10. $16.33. 11. 50 rd. 12. 8 4.84. 13. 10 rd. Page 139— 14. 1728 cakes. 15. 8116.07. 16. 792 bd. ft. : $23.76. 17. 15 A. 18. 2048 bu. 19. 4847^2 gal. 20.150 ft. 81. $96.80. 22. 40 ft. 23. 8 800. Page 140. —25. 400 ft. ; 202 ft. : 292 ft. ; 26 ft. : 216 ft. 26. 1111$ sq. yd. 87. 511$ sq. yd. 88. 4$ sq. yd. 29. 88| sq. yd. 80. $90£f 31. $137.78. 32.8.22. 33.8120.07. 34. ^ ft. 35. $23.11. 36. 2154f* gal. 37. $ 10.80. 38. 60 farms. 39. 3000 sods. 40. $62.40. Page 144. — 2. 375. 3. 21.375. 4. 3.5. 5. 235.2. 6. 320. 7. 735. 8. $ 16.20. Page 145 —9. 312. 10. $30. HAM. COM PL. \UIT1I. — 29 450 * ANSWERS Page 146— 12. ft 12. 13. 2 mo. 14. 18 da. 15. 1 A. 16. 5 A. 17. 32. 18. 28 horses. 19. 3 T \. 20. 9 1b. 21. ft If 22. 20. 23. ft 200. 24. §50. 25. 8 words. 26. ft .20. 27.280. 28. §3125. 29. ft 60. 30. 90 A. 31. §400. 32. $150. 33. 1209 people. Page 147. — 2. §90. 3. §79.70. 4. §97.70. 5. §70.04. 6. §76.40. 7. ft 118.53. 8. §532.37. 9. § 135, commission ; §6615, net proceeds. 10. §1800. 11. §456. Page 148. — 1. §12. 2. § 12. 3. § 3. 4. § 20. 5. §6. 6. $7.50. 7. § Hi. 8. §36. 9. §5.10. 10. §3. 11. §8. 12. §3. 1. §126.56. 2. §1145. 3. § 1248. 4. §174. 5. §1818. 6. §180.52. 7. §612.85. 8. §251.01. 9. §2283.36. 10. §6436.44. 11. $176.64. 12. §52.90. Page 149. —14. §288. 15. §267.54. 16. §157.03. 17. $6.13. 18. §4.59. 19. 8 7.65. 20. ft 129.60. 21. §4080. 22. §1890. 23. §1353.75. 24. §53.55. 25. §64.60. 26. §291.60; §21.00 dif. 27. $641.25. 28. Both, §360. Page 150.— 30. $6643. 31. §31.82. 32. §583.20 ; §648. 33. §203.74. Page 152. — 2. §15. 3. §128. 4. §29.25. 5. §126. 6. §15. 7. §12. 8. §25.07. 9. §18. 10. §6.50. 11. §52.50. 12. §43.20. 13. §173.25. 14 §29.43. 15. §15.40. Page 153. — 16. §3. 17. § 10. 18. §19.50. 19. $12.48. 20. $6.80. 21. §7. (19. Page 154.— 2. §63; §363. 3. §40.83; §290.83. 4. §34; §194. 5. $4.17; 854.17. 6. §152; §952. 7. §3.81; §54.61. 8. §.64; §16.64. 9. §3; §78. 10. §35; §455. 11. $2.63; §43.13. 12. §8.76; §309.16. 13. §4.08; §104.08. 14. $27.50 ; §527.50. 15. 85; §1005. 16. §3.23; §63.83. 17. $69.44; $319.44. 18. $15.64; $91.44. 19. $251.25; §1751.25. 20. §24.85; §150.35. 21. §296.78; $1436.78. 22. §130.68; §1043.28. 23. §701.17; §3910.17. 24. §48.22; $682.72. 25. §297. Page 155. — 1. §33. 2. §800. 3. $3. 4. 30 girls. 5. .0125; .125; 1.375; .0625. 6. §12.50. 7. §64. 8. §22.50. 9. § .83. 10. §76.50. 11. 8.24. 12. §1.50. Page 156. — 13. §1.32. 14. $45. 15. $2. 16. $420. 17. $20. 18. 20,UU0. 19. §325. 20. §3. 21. §9. Page 158. — 7. §8.75. 4. §274.32. Page 159. —1. 110,092. 2. 37. 3. 50.601. 4. 25.061; 125.5; 300.0002. 5. 4742. 6. .0119. 8. 8jL 10. 55.12. 11. $.30. 12. 2771 It. 13. 39 mi. 319 rd. 5 yd. 1 ft. 1 in. Page 160. — 14. §132. 15. $2160. 16. §103.74. 17. 2f. 18. .1. 19. T ",V 20. §42. 21. 38104 mi. 22. 2| T. 23. §375.75. 24 §1080. 25. §"100. 26. §272.50. 27. 56 A. Page 161. —28. §3892.50. 29. §16.80. 30. 1 da. 2 hr. 53 min. 20 sec. 31. §750. 32. §20.17. 33. -^-, -% 7 -, ^, §£. 34. §288. 35. 2||ff mi. 36. 048,045 cu. in. ; 375?} cu. ft, 37. 26$% gain. ANSWERS J.".1 Page 162. — 38. 8; $1.60; $20.83$. 39. $5625. 40. $ 101.25. 41. $4.50. 42.2i)rd. 43. $ 707 J. 44. $55.08. 45. $2. Page 163.- 47. 156f cu. yd. 48. 4 rd. 1 ft. 7 Page 280. — 57. $1500.75. 58. $121.60. 59. $207.80. 61. 827.54. 62. $74.30. 63. $74.80. 64. $160.05. 65. $306.10. 66. $144.15. 67. $202.0(1. 68. $20.50. Page 281 . — 2. $115.48. 3. $91. 4. $65.33. 5. $101.28. 6. $4.58. 7. $9. 8. $19.85. 9. $114.49. 10. $73.02. 11. $77.25. 12. $169.30. 13. $337.40. 14. $219.75. 15. $247.50. 16. $2.40. 17. $4.77. 18. $6.18. 19. $10.50. 20. $47.60. 21. $60.50. 22. $38.25. 23. $145.13. 24. $10.66. 25. $19.99. Page 282. — 26. $31.93. 27. $36.56. 28. $1312.19. 29. $966.60. 30. s 151.59. 31. $896.13. 32. $415.28. 33. $38.48. 34. $207.81. 35. $2246.44. 36. $7.16. 37. $18.27. 38. $12.33. 39. $22.25. 40. $77.19. 41. $9.61. 42. $1423.15. 43. $705.53. 44. $4044. 45. $134. Page 283. — 2. $1000. 3. $680. 4. $1200. 5. $1375. 2. 0°/ . Page 284. -3. 5%. 4. 5%. 5. 6%. 2. 3 yr. 4 mo. 3. 20 yr. ; 16f yr. ; 12} yr. 4. 40 yr. 5. Oct. 13, 1902. 1. $31.44. 2. 2 yr. 6 mo. 3. 3 yr. 6 mo. Page 285. — 4. 41%. 5. $434. 6. $180. 7. $275. 8.6%. 9 In 3 vr. 4 urn. 10. $354.31. 11. $ 88.20. 12. Jan. 19, 1906. 13. $600. 14. July 1, 1906. 15. $86,700. 16. $3025. Page 286. — 2. $116.19. 3. $8292.48. Page 287.-4. $4954.20; $495.42. 2. $10.26. 3. $35.29. 4. $60.49. 5. $41.34. 6. $11.10. 7. $20.68. '8. $299.18. 9. $116.99. 10. $88.70. Page 288. — 11. $396.40. 12. $-655,912.33. Page 289. —2. $103.81. 3. $10.45. 4. $27.41. Page 290. — 2. $1.52. 3. $157.99. 4. $80.52. 5. $1056.07. Page 291. — 6. $206.04. 7. $160.60. 8. $909.93. 9. $1233.35. Page 292. — 1. $1526.74. 2. $14,233.12. 3. $5642.40. 4. $1324.90. Page 298. — 1. $103. 2. $255.44. 3. $630.42. 4. $367.44. 5. $129.13. 6. $1219.50. 7. $316.60. Page 301. — 2. $669.20. 3. $295.65. 4. $110.17. Page 302. — 5. $355.64. 6. $9.57. 1. $ 439.70. 2. $2380.37. Page 306. —1. $663.15. 2. $399.70, balance. Page 309. — 2 *203. 3. $200. 4. 30;6<*. 1. Aug. 1. 2. Aug. 22. 3. Nov. 13. 4. Oct. 10. 5. April 2. .May 17. 7. June^lO. 8. Sept. 5. tj 458 ANSWERS Page 310. —9. April 30. 10. July 10. 11. Nov. 10. 12. June 23. 13. Oct. 14. 14. April 2. 15. July 23. 16. Dec. 7. 17. Sept. 3. 18. Oct. 24. 3. Nov. 8; $2.25; $2.28. 4. Aug. 12 ; $ 5.67 ; $5.76. 5. Oct. 2 ;$ 11.50; $11.63. 6. May 8; $15.63; $15.76. 7. April 24 ;$ 4.75; $4.85. 8. July 17 ; $4.53; $4.59. 9. Oct, 1 ;$ 8.50 ;$ 8.78. 10. March 11; $6.46; $6.53. 11. May 16 ;$ 15.75 ;$ 15.97. Page 312. —1. $2487.92 ; $2487.50. 2. $ 1568.89 ;$ 1568.63. 3. $ 1545.70 ;$ 1545.44. 4. $ 4521.82 ; 14521.06. Page 313. — 5. $ 4455 ; $ 4454.25. 6. $ 3468.50 ; $ 3467.92. 7. $ 3537.87 ; $3537.27. 8. $1000. 9. $ 1207.04 ;$ 1206.83. 10. $200; $200. 11. $209. 12. $ 205.62 ;$ 205.59. 14. $224.33. 15. $ 259.21 ;$ 259.10. Page 314. — 16. 9%. 17. 12%. 18. $ 158.76 ;$ 158.73. 19. $1828.04; $1827.74. 20. $1217.22; $1217.01. 21. $ 2467.50 ;$ 2467.08. 22.$6375.81; $6374.98. 23. $1000. 24. $2800. Page 319. — 1. $ 550.80. 2. $ 1575, face ; $ 2.63, exch. 3. $ 1.30. 4. 15?. 5. $ 1078.92. Page 320. —7. Face, $989.01 ; exch., $.99. 8. Exch., $ 12.75; com., $275; face, $12,747.75. 9. $791.98. 10. $311.85. Page 322. — 1. 60 da.; 61 da. 2. $1183 ; $1182.77. Page 323. — 4. $ 11,016.75 ; $11,014.59. 5. $591.80 ; $591.70. 6. $3809.50. 7. $2456.25; $2455.83. Page 326.-2. $6067.50. 3. $8987. 4. $12,142. 5. $11,355. Page 327. —6. $19,936. 7. $8656.25. 8. $20,680.63. 9. $66,187.50. 10. $37,250. 12. 240 shares. 13. 8 shares. 14. 100 shares. 15. 40 shares. 16. 44 shares ; surplus $ 14.50. 17. 61+ shares ; surplus $54,625. Page 328.-3. $825. 4. $1892.63. 5. $2153.13. 6. $22,137.50. 7. $7453.13. 8. $26,784. 9. $13,785. 10. $23,490. Page 329. —12. 120 shares. 13. 112 shares. 14. 40 shares. 15. 168 shares. 16. 200 shares. 1. $60. 2. $80. 5.7%. 6.5%. 7. $2000. 8. $16,000; $800. Page 330.-10. 4%. 11.6%. 12. 4J%. 18.6$%. 14.6}%. 15.8%. 16. Less. 18. $1185. 19. Former, T \% better. 20. 12J %. Page 331. — 21. $800,000. 22. $288. 24. $24,025. 25. $7288.75. 26. $36,556.88. 27. $9446.63. Page 333. — 1. $15,581.25. 2.50. 3. $30,000. 4. $2000. 5. $35,437.50. 6. 12%. Page 334. - 7. 6\ %. 8. 6 %. 9. $ 1487.50. 10. 6£ % ; 5| % ; 5%; 4J%; 4%. 11. $4000. 12. $1584.48. 13. $400. 14. 500%. 16. 5 r y/ . Page 335. —1. $00.48. 2 $635.04. 3. $8870. 4. 30|%. 5. §520.52; $520.42. 6. $36.40. 7. $4000. p458 ANSWERS 459 Page 336.-8. 19$%. 9.23+%. 10. $212.50. 11. 6|%. 12. 16^%. 13. $585. 14. $1282.50; commission, $67.50. 15. $480. 16. $7100. Page 338. — 1. 5. 2. .25. 3. 3. 4. &. 5. 8. 6. .4. 7. fa. 8. \. 9. 100. 10. 3. 11. 0. 12. 320. Page 339.-3. 18f. 4. 15. 5. 10. 6. 5. 7. 5. 8. 10. 9. I. 10. .5. 11. 2.5. 12. 90. 14. 30 da. 15. $ 18. 16. 50 da. Page 340. — 17. 8. 18. $1021.88. 19. $4.17. 20. 4 oz. 21. 41 j mi. 22. $40. 23. $70.55. 24. 4 da. 25. 37 Z sec. 26. 3875 letters. 27. 180 ft. 28. 31,250 bricks. Page 341. —29. 36|| mi. 30. 30 men. 31. 24 men. 32. $60.30. 33. 38 cars. 34. $236.25. Page 342. — 2. $81, man; $ 54, first boy ; $27, second boy. 3. Expenses, $5200; net savings, $10,400. 4. $672,000; $28,000; $42,000; $03,000. 5. Lake, $1000; railroad, $ 1800. 6. $60,000; $230,000. Page 343.-2. A's, $2520 ; B's, $5700 ; C's, $4320. 3. M's, $2520; N's, $3024 ; K's. $2010. 4. E's, $1000 ; F\s, $857i ; G"s, $2142f 5. $1329 r 6 r ; $920 T \. 6. Smith's, $3225; Jones's, $4300; Brown's, $2150. Page 344.-8. $3348 to N ; $40">8 to M. 9. R's, $1053.75 ; S's, $1496.25. 10. A's, if ; B's, |§. Page 345. —1. $600; $400. 3. $3200; $4000. 4. $15,000; $12,000. 5. 50%; 25%; 150%. 6. Frank, $ 18 ; Henry, $15. Page 346. —7. Walter's, $1.20; Philip's, $1.50. 8.40%. 9. $5000. 10. Brown's, $1050; Long's, $1200. 11. $375. 12. $450. 13. Moore's, $1200; Silven's, $1800; Rogers's, $1500. 14. $45 per month. 15. 77 da. 16. To A, $2700 ; to B, $2400. Page 347. — 17. $48. 18. 6 da. 19. 9 lots. 20. 48 persons. 21. 100 A. 22. 600 per dozen. 23. 24 da.- 24. $3200 ; $3000. 25. $1250. 26. $500; $000. Page 348.-27. F, $1440; E, $900. 28. $500; 6%. 29. 500, 5000. 30. 15 da. 31. $32. 32. $320. 33. $1000. 34. 20,000 bu. 35. $5. 36. Claim, $1000 ; loss, $400. 37. & greater. Page 352. —2. 15° ; 1 hr. ; 60° W. has earlier time. 3. 45° ; 3 hr. ; 120° W. 4. 60° ; 4 hr. ; 15° W. 5. 30° ; 2 hr. ; 30° E. 6. 105° ; 7 hr.; 75° W. 7. 150°; 10 hr. ; 120° W. 8. 120°; 8 hr. ; 90° W. 9. 105°; 7 hr. ; 30° E. 10. 105° ; 7 hr. ; 45° \V. 11. 30° ; 2 hr. ; 15° E. Page 353. — 12. 10 hr. 8 min.20 sec. a.m. 13. 11 hr. 17 min. 45£ sec. a.m. 14. 8 hr. 58 min. 29£ sec. a.m. 15. 5 hr. 17 min. 33 sec. p.m. 16. 5 hr. 58 min. \§ sec. p.m. 17. 6 hr. 36 min. 49f sec. a.m. 18. hr. 1 min. 46^ sec. p.m. 19. 12 min. 11| sec. p.m. 20. 11 hr. 48 min. 4 sec. a.m. 21. 5 hr. 47 min. 7^ sec. 22. N.Y., 3 hr. 36 min. 71 sec. p.m. ; Cape Town. 9 hr. 40 min. 3 sec. p.m. 23. 2 hr. 39 min. 35/. sec. 460 . ANSWERS Page 354. — 26 Honolulu, 6 hr. 28 min. 37| sec. a.m. ; Berlin, 5 hr. 53 min. 34j| sec. p.m. ; San Francisco, 8 hr. 60 min. 17£ sec. a.m. ; Lon- don, 4 hr. 59 min. 3G| sec. p.m. 1. 17 da. 2. 21 da. Page 355. —1.3 hr. Page 357.-5. 160 A. 6. 80 A. 7. 40 A. 8. 320 rd. 9. 480 rd. Page 359.-8. 225. 9. 256. 10. 324. 11. 484. 12. 625. 13. 42.25. 14. .5625. 15. 2J$. 16. 272£. 17. 225 sq. in. 18. 625 sq. ft. 19. 256 sq. yd. 20. 72£ sq. ft. 21. 25 sq. in. 22. 72.25 sq. in. 23. 100 sq. yd. 24. 32^ sq. yd. 25. 36 sq. in. 26. 512 cu. in. 27. 8 cu. ft. 28. 34§£ cu. ft. 29. 42| cu. ft. 30. 56# 6 cu. ft. 31. 1876 J y cu. ft. Page 361. —2. 15. 3. 24. 4. 21. 5. 14. 6. 40. 7. 28. 8. 36. 9. 16. 10. 48. 11. 35. 12. 42. 13. 24. 14. 56. 15. 18. 16. 20. 17. 72. 18. 25. 19. 32. 20. 64. 21. 27. Page 364.-5. 22. 6. 24. 7. 26. 8. 31. 9. 33. 10. 43. 11. 51. 12. 63. 13. |. 14. if. 15. |f. 16. .5. 17. .15. 18. 2.1. 19. 2.5. 20. .25. 21. J7.74+. 22. 22.91+. 23. 34. 24. 66. 25. 14.5. 26. 13.35+. 27. 122. 28. 163. 29. 369. 30. 4.56. 31. 9.84. 32. 13|. 33. 23.25. 34. 56.5. 35. 75.12+. 36. .92. 37. .114. 38. .02+. 39. .66+. 40. 335. Page 366.-2. 32.45+ in. 3. 3 ft. 4. 32 ft. 5. 25 yd. 6. 44.72+ in. 7. 8.94+ ft. 8. 1300 sq. rd. 9. 22.36+ ft. 10. 42.42 mi. 11. 21.21+ rd. 12. 1.41+ ft. 13. 45 ft. Page 368. — 1. 370.8 sq. ft. 2. 28.28 rd. 3. 97.425 sq. in. Page 370. — 1. 250 sq. ft. 2. 17^ sq. ft. 3. 108.3852 sq. ft. 4. 613.3974 sq. in. Page 371. — 1. 169.6464 sq. ft. 2. 768 sq. ft, 3. 161.9 sq. ft. 4. 700 sq.ft. 5. 854.17. 6. *58.81. Page 372. —1. 452.3904 sq. in. 2. 28.2744 sq. in. 3. 530.9304 sq. in. 4. 50.2656 sq. in. 5. •$17.45. Page 373. — 1. 128 cu. in. 2. 2 cu. ft, 3. 185.4 T. 4. 190.8522 cu. ft. Page 374. —1. 201.0624 cu. in. 2. 9 times. 3. 1440 cu. in. 4. 3456 cu. in. 5. 256 cu. ft. 6. 600 cu. ft. 7. 284+ bu. Page 375. — 8. 4071.5136 cu. ft. 9. 2ff T. 10. 24 in. 11. 10 ft. 1. 904.7808 cu. in. 2. 268.0832 cu. in. 3. 288.696+ cu. in. Page 377. — 1. 75 ft. 2. 80 rd. 3. 12 ft. 4. 2^. 5. 8 ft. Page 378.-6. $8.67. 7. 21ft. 1. b Page 379. - 2. T \. 3. ^f ,, 4. fr 6. 20 ft, ; 13$ ft, ; 8 ft, 7. 16 ft. 9. 200 cu. in. 10. 179^ bu. ANSWERS 461 Page 380. —4. 172.51b. 5. C'.o.js lb. 6. 1501b. 7 1046.20+ lb. 8. 650.25 lb. 9. 64.375 lb. 10. -",7..") lb. 11. 556.25 lb. 12. 168.75 lb. 13.849.3751b. 14. 1206.251b. 15. 151b. 16. 556.25 1b. 17. 700.25 11.. 18. 181.251b. 19.487.51b. 20.455.6251b. 21.489.3751b. 22. 10K.75 lb. 23. 114.375 lb. Page 381. —1. 27.08 bbl. 2. 1000 sq. in. 3. 10.39+ in. in diam. 4. 3.4724 ini. 5. 23 A. 6. 0257 i bu. 7. 672.3024 cu. in. 8. 4.18+ sq.ft. 9. 18.1:5- gal. 10. 1 to 2|. Page 385— 5. 1.363 in. ; 13.00025 m. 6. 1 Km. 088 m. 7. 177.102 mi. Page 386.-8. 37.5 in. 9. 67.5906 mi. 10. 6 Mm. ; 5 Dm.; 1 m. 11. (1400 Km. Page 388.-2. s 72.00. 3. 87.50. 4. 24 sq.m. 5. 24 steres. Page 390. — 1. 937.51. 2. 188,4961.; 188.496 M. T. 3. $159.23. 4. 3,000,000 1. 5. 7.2 111. 6. 9000 bottles. Page 391. — 7. 838(50. 8. 154.56 pf. ; $.3678. 9. 1000 1. 10. 89 T l T Kg. 11. 1.3591+ cu. m. 12. 3941.4 Kg. 13. 169.164 m. 14. 863.75. 15. 811, 812.50, cost of land; 81100, cost of fence. 16. 000,000 tiles. 17. 140.400 M. T. 18. 4080 cu. m. 19. 8 1.20. Page 392. — 1. 86.10. 2. 80.33. 3. 84.04. 4. 82.31. 5. 83.30. 6. 811.74. 7. 811.93. 8. 812.06. 9. 8 13.48. 10.828.10. 11. s4.21. 12. 812.01. 13. 81.93. 14. 8 3.80. Page 393. — 16. 1 to 7.0+. 17. 1 to 3.8+. 18. 1 to 5.27+. 19. 1 to 5.05*. 20. 1 to 5.8+. 21. 1 to 8.7+. 22. 1 to 13.7+. Page 394. — 23. 1 to 5.08+. 24. 1 to 7.69+. 25. 1 to 8.8+. 26. Too wide: lto7.1+. 27 Too wide ; 1 to 10.09+. 28. Too wide; 1 to 8.2+. 29. Too wide ; 1 to 10.3+. 30. 1 to 6.04+. 31. 1 to 4.06+. Page 396. — 1. 234 lb. ; 01.2 lb. ; 52.25 lb. ; 184 lb. 2. .75 lb. ; 1.14 lb. ; 10.875 1b. 3. 39. tons. 4. 285 tons. 5.812.25. 6.8 23.16. 7.824.33. 8. 36.4 tons. Page 397. — 1. 300 1b. of each. 2.82.60. 3. $62. 4.8274. 5. § 1 140. Page 398—6. 8131.80. 7. $1. 8. $1.14. 9. 814.14. 10. Materials: .000+ per gal. ; 8 .115 per tree. Labor: 8 .01+ per gal. ; 8.13 per tree. Page 399—11. 8.00. 12. 86.25; 81.28. 13. 830.19. 14. 300 gal. ; 7* lb. Page 400. — 15. $9.80. 16. $.072- ; 84.28. 17. $58.33. 18. 22$%. Page 401. — 1. $1007.17. 2. 3.1416 sq. rd. 3. 25.42+ in.; 42.42+ in. 4. $18. 5. 324 cu. ft. ; 10.8 min. 6. $200.72. Page 402. — 7. times. 8. $300. 9. 8 2.88. 10. f 22.62. 11. 7.0086 cu. ft, ; 21.2058 cu. ft. ; 14.1372 cu. ft. 12. :! mills. 13 104.4 mi. 14. 00 lb. 15. $2000. 462 ANSWERS Page 403. — 16. Cow, $50; horse, §140. 17. 150.7968 in. 18. 4ii rd. ; 160 rd. 19. BJ %. 20. 33$%. 21. 17.32 in. 22. 22.36 in. 23. Width, i» ft.; height, 6 it. 24. 1 to 3f. 25. 8 834.40. 26. 813,453. Page 404. —27. 8 1080. 28. 82000. 29. 8181.43; 8181.40. 30. $ 175. 31. 20 A. 32. A's, 320 A. ; B's, 480 A. ; C's, 600 A. 33. 4.31+%. 34. 8630.40; 8630.29; 8639.86; 8639.75. 35. 81119.57; 81118.87. Page 405.— 36. 852.17. 37. 85$ ft. 38. 848.72. 39. 8 570. 40. 73.24+ ft. 41. 25. 42. 887.14. 43. 5 02+%. Page 406. —1. 60%. 2. 28?; $2.80. 3. 830,000. 4. 100%. 5. 6|hr. 6. 6i%. 7. $1.65. 8. 31ft %• Page 407. — 10. M's, 8900; N's, 8800. 11. 240 A. 12. 20%. 13. 10 ft. 15. 300 crates. 16. 8120. 17. 40.15%. 18. 836. 19. 1081%. 20. 6i%. Page 408. — 21. 64 cu. in. 22. 4%. 23. A's, 8450 ; B's, 8600 ; C's, $1050. 24. $1400; 8 1500. 25. 84500. 26. 8 25,000. 27. a. 37$; b. f. 28.31%. 29. 5456 yd. ; 8342.4 yd. 30.8393.67. 31.8 2510. 32.32,768. 33. a. 4* ; b. 10^. Page 409. —34. * 3600. 35. 288 cakes. 36.11,011,011.000011. 37. 3f. 38. 1121.112. 39. 8250,000. 40. 40 A. 41. .4, .375, .28, .5625, .75, .15625. 42. 8 419.85. 43. 24, 15, f, 2. 44. 84800. 45. 84.32. 46. 12$ mills. Page 410. —47. 60 da., 8986.67 ; 61 da., 8986.45. 48. 20^ sections ; 8 2880. 49. $429.84. 50. 8 505. 51. 86.75. 52. July 1, 1906. 53. 8 450. 54. 8 705. 55. A, T 2 3 ; B, £.&. 56. 8353. 57. 8168.48. Page 411. —58. 60 shares. 59. 20 vr. ; 16f yr. ; 121 yr. 60. 8396. 61. 8 313.04. 62. 8 .921 per yard. 63. 8 336. 64. 14011b. 65. 819.20. 66. 8 2949.60. 67. 21 mi. Page 412.— 68. 16.568 rd. 69. 75%. 70. 83$%. 71. 6283.2 sq. ft. 72. $39.90. 73. 13 J*. 74. 2 yr. 4 mo. 75. 1000 cu. ft. 76. $39.25. 77. 8887.50. Page 413. — 78. Second, $15. 79. 8120; 20%. 80. 6%. 81. 81000. 82. s 38.40. 83. First, $13.50. 84. $6. 85. A, 1^ 5 ; B, f. 86.82125.60. 87. 143%. Page 414. — 88. 827,984. 89. 1020.021. 90. 413.875 1b. 91. 8500 loss. 92. 8 4500 ; 81275. 93. Latter by § %. 94.48.55. 95.8 800. 96. 15 mills. 97. 630 Kl. ; 630,000 Kg. Page 415. — 98. 10,000 A. 99. Nitrogen, 1890; oxygen. 7110. 100. $140.14 101. 200 men. 102 4V&. 103. .0125. 104. 8 643.40. 105. $345.60. 106. 50 rd. 107. 8 6000. ANSWERS 463 Page 416. — 108. 836,720. 109. Ninety-nine hundred-thousandths. 110. Hi ft. 111. $4. 112. $2842.50. 113. $144. 114. $601.50. 115. 4 yr. 9 mo. 22 da. 116. $540. 117. 9720 shingles. Page 417. — 118. 34%. 119. §640. 120. $120. 121. $6000. 122. 162.5 1b. 123. 79.5. 124. 47!'|'/ . 125. §300. 126. §320.-.".. 127. $ 3554.40 ;$ 3655. 128. 500 ft. Page 418. — 129. $ 1979.74 ;$ 1979.35. 130. $513.44. 131. $404.22. 132. $2937; $2936.33. 133. $1(370.14. 134. §2(57.48. Page 419. — 135. 11 hr. 33 min. 57 sec. a.m. 136. $384. 137. $1550, gain. 138. $558.33. 139. 57.7269 Kl. 140. 39 Kg. 141. $3213.42. 142. Dec. 7, 9 hr. 51 min. sec. a.m. 143. 1 mi. 22:! rd. (i in. Page 420. — 144. 120. 145. 8 2:171.85. 146. 70. 147. $2298.75. 148. §1032.82. 149. $83,144.91. 150. §125,000. 151. §2090. Page 422. — 2. $285. 3. $450. 4. $350. 5. $441.51. 6. $100. 7. $ 11.25. 8. The former is § 5 Letter. Page 424.-4. §120.25. 5. $64.45+. 6. $ 1247. GO. 7. $1228.96. 8. §148.95. 9. $76.43+. 10. $1948. 11. 12,921 fr. 87.5 c. 12. $ 2GG.48. 13. $2697.50. 14. §457.90. 15. 6489 M. 80 pf. 16. £309 lis. lid. ' . 17. §1478. G4; 1260 francs. Page 425. — 2. $90. 3. §140. 4. 54 rd. Page 431. — 1. 14. 2. 16. 3. 24. 4. 27. 5. 32. 6.45. 7. 72. 8. 98. 9. 123. 10. 1. 11. I'i. 12. r \. 13. .9. 14. 2.5. 15. 3.4. 16. 6.1, 17. .15. 18. 'lh Page 432. — 1. 18 in. 2. 441 sq. in. 3. 13 ft. 4. 47.55+ in. 5. 8 ft. 6. 6 in. 7. 12 in. 8. 17 ft. 9. 5 to 7. 10. 18 ft. high. 11. The base of the bin is 172.8+ in. square ; the height of the bin is 86.4+ in. = =• ASSESSED FOR FAii- pENA UTY X%0 E OK S ON -H -J| N ? S U ON ThVeO.HTH ™1_l -ncrease to jo ceN the gEvENTH daY oA Y AND TO OVERDUE. LD 21-100m-8,'34 F YL 22560 918 !r^> THE UNIVERSITY OF CALIFORNIA LIBRARY /.'"