LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIRT OK V Accession 8t>96$ JUNIOR ARITHMETIC A TEXT BOOK FOR FOURTH AND FIFTH YEAR PUPILS BY CHARLES E. WHITE, PRINCIPAL FRANKLIN SCHOOL, SYRACUSE, N.Y. BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS COPYRIGHT, 1895, BY D. C. HEATH & Co. PBESSWOBK BY S. J. PABKHILL & Co. TYPOGBAPHY BY C. J. PETEBS & SON, BOSTON. PREFACE. In arranging the work of this book, it has been the author's aim to adhere to a proper order of topics, and to treat each topic with proper fulness, not only with reference to its business application, but also with regard to its importance as a basis for more advanced study. The oral, mental, written, and abstract work are arranged in the relative order and proportion that naturally lead pupils to a clear understanding of processes, as well as to habits of rapid and accurate thinking. Much time may be saved by early mastering the four fundamentals, particularly column addition and long divis- ion, frequent exercises in which occur throughout this book. Pupils often attempt to solve problems before they have thought out the processes. This leads to erroneous re- sults, loss of time, and discouragement. The practice of first indicating processes by means of signs encourages the pupil to do his thinking first ; moreover, it often permits a very material shortening of the work by can- cellation. A few pages of merely indicated work are in- troduced, to emphasize the advantages of such operations. The latter can be carried into the written work of the book at the pleasure of the teacher. 3 iv PREFACE. A few short methods, such as multiplying and dividing by the aliquot parts of $1.00, and by the parts of 100 and 1,000, etc., are introduced. Pupils in the fifth year can master them readily, and at a great saving of time. Questions of relations (more fully explained in the body of the book) will make plain a group of difficulties in division of fractions and, later on, prove of great value in percentage and its applications. C. E. W. SYRACUSE, NEW YORK, CONTENTS. CHAPTER I. PAGE. NOTATION AND NUMERATION 1 NOTATION AND NUMERATION OF FEDERAL MONEY 4 ADDITION 5 SUBTRACTION 8 INDICATED WORK, ADDITION 11 INDICATED WORK, SUBTRACTION 12 MULTIPLICATION 12 DIVISION 15 REVIEW WORK 20 INDICATED WORK, MULTIPLICATION 21 FRACTIONS 22 FEDERAL MONEY 25 DRY MEASURE 26 LIQUID MEASURE . 27 AVOIRDUPOIS WEIGHT 28 LINEAR MEASURE 29 ADDITION AND SUBTRACTION OF DENOMINATE NUMBERS .... 30 GENERAL REVIEW PROBLEMS 32 MISCELLANEOUS 44 to 52 CHAPTER II. FACTORS AND MULTIPLES 53 COMMON FRACTIONS 66 LOWEST TERMS 58 MIXED NUMBERS TO IMPROPER FRACTIONS 59 IMPROPER FRACTIONS TO MIXED NUMBERS 60 FRACTIONS TO COMMON DENOMINATOR 62 ADDITION OF FRACTIONS 63 SUBTRACTION OF FRACTIONS 65 LINEAR MEASURE 67 V VI CONTENTS. PAGE. INDICATED WORK, DIVISION 68 INDICATED WORK, CANCELLATION 83 DENOMINATE NUMBERS, ADDITION AND SUBTRACTION .... 87 COMPOUND NUMBERS 105 GENERAL REVIEW PROBLEMS 106 CHAPTER III. BILLS AND ACCOUNTS 117 ADDITION AND SUBTRACTION OF FRACTIONS 120 MULTIPLICATION OF FRACTIONS 120 DIVISION OF FRACTIONS 124 PARTS OF A DOLLAR 128 THE THREE QUESTIONS OF RELATION 131 REVIEW OF FRACTIONS 133 SQUARE MEASURE 138 SHORT MULTIPLICATION 144 PARTS OF 100 AND 1000 145 REVIEW 147 INDICATED WORK AND PARTS OF A DOLLAR 150 QUESTIONS OF RELATION 156 MISCELLANEOUS PROBLEMS 156 to 174 CHAPTER IV. PLASTERING AND PAINTING 175 CARPETING ROOMS 177 PAPERING WALLS 180 SHORT MULTIPLICATION 185 CUBIC MEASURE 194. CAPACITY OF CISTERNS AND BINS 197 MISCELLANEOUS TABLES 199 DECIMAL FRACTIONS 213 REDUCTION OF DECIMALS 218 ADDITION OF DECIMALS 220 SUBTRACTION OF DECIMALS 221 MULTIPLICATION OF DECIMALS 221 DIVISION OF DECIMALS 223 MISCELLANEOUS REVIEW 228 JUNIOR ARITHMETIC. CHAPTER I. NOTATION AND NUMERATION. A Unit is one, or one thing. As 1, 1 orange, f 1. A Number is one or more units. As 1 apple, 10 men, 25. The Unit of a Number is one of its units. The unit of 6 is 1 ; of 6 pecks is 1 peck. Numbers having the same unit are Like Numbers. As 8, 12, 20, unit, 1 ; 5 men, 8 men, 12 men, unit, 1 man. A number whose unit is 1 is called an Abstract Number. As 24, unit, 1. A number whose unit names one particular thing is called a Concrete Number. As 24 men, unit, 1 man. An Integer is a whole number. The system of figures or letters used in expressing num- bers is called Notation. Arabic Notation is used in expressing numbers by figures. (Illustrate.) Roman Notation is used in expressing numbers by letters. (Illustrate.) Numeration is naming the places of figures, and reading numbers. (Illustrate.) NOTE. Teach units of first, second, and third orders, etc., simple and local values of a figure. 1 2 ARITHMETIC. Three figures, standing in a line from right to left, form a Group or Period. Rule. Begin at the right and separate the number into groups, using the comma. Begin at the left and read the number in each group, giving to it the name of that group. No name is given to the number in the last group. TABLE. 3d Group. 2d Group. 1st Group. MILLIONS, THOUSANDS, ONES. 1 I i III III III 724, 618, 327. Seven hundred twenty-four million, six hundred eighteen thousand, three hundred twenty-seven. Copy and read : 1. 2,083 9. 713,006 17. 72,392,809 2. 97,216 10. 16,384 18. 7,007 3. 412 11. 219 19. 8,100,203 4. 88,640 12. 9,287 20. 21,836 5. 9,208 13. 291,011 . 21. 789,030 6. 17,021 14. 17,316 22. 19,200,036 7. 4,903 15. 428,063 23. 410,004 8. 5,007 16. 52,913 24. 7,200,163 Write a number containing three groups. NOTATION. 3 Write in figures : 1. Two hundred thousand, sixteen. 2. Eleven thousand, two. 3. Four million, six hundred eight thousand, three hun- dred seventy-five. 4. Twenty-five thousand, three hundred eighty-seven. 5. Nineteen thousand, seventeen. 6. Twenty-seven million, six hundred fifty-two. 7. Eighty million, six hundred nine thousand, four hun- dred twenty-eight. 8. Four hundred thirty-six thousand, forty-one. 9. Six hundred twenty million, seventeen thousand, four hundred seventy-seven. 10. One hundred fifty-seven million, six hundred eight thousand, four hundred seventy-seven. Separate into groups and read : 13087; 716042; 2730010; 1 26003184 ; 47250627 ; 1002970 ; 17042; 14390023; 11935079; 4000030; 29307070; 14280643. . NOTE. Additional numbers may be dictated. Read these productions of New York State : Corn, 17,512,700 bu. Buckwheat, 4,514,000 bu. Wheat, 9,288,000 bu. Potatoes, 29,688,000 bu. Eye, _ 2,724,000 bu. Tobacco, 6,488,000 Ib. Oats, 2,391,000 bu. Butter, 111,922,423 Ib. Barley, 7,418,000 bu. Cheese, 8,362,950 Ib. Hay, 5,426,757 tons. Hops, 21,628,931 Ib. Milk, 231,965,533 gal. Write in words the largest number of two groups. 4 ARITHMETIC. Read: 1. XLV. 6. MCIV. 11. DCXCIV. 2. CDIL 7. DCCVI. 12. MCCV. 3. ML 8. MXIX. 13. CCXI. 4. XIX. 9. DCX. 14. MMV. 5. LXXXVI. 10. CDLI. 15. DLXVI. Write in Roman Notation : 1. 284 5. 319 9. 2,870 2. 98 6. 1,515 10. 837 3. 1,013 7. 745 11. 1,400 4. 56 8. 47 12. 245 Notation of Federal Money. Federal Money is the currency of the United States. 10 mills = 1 cent. Mills are not coined. The smallest coin used is the cent. Cents are written ct. or /. 100 cents = 1 dollar, written $1. When dollars and cents are written together, a period, called the decimal point, is placed between them. Thus, four dollars and sixty-seven cents is written, $4.67. The first two places at the right of the point are occu- pied by cents, the third place by mills. Nine cents four mills is written, $.094. Four dollars sixty-five cents three mills is written, $4.653. Copy and read : 1. $ 8.40 2. $ 16.30 3. $ 92.003 17.03 9. .105 2.842 .087 256.428 200.504 4007.91 94.02 Add the above. The points must stand in a vertical line. Write in figures and add : 1. Seventy dollars twenty-six cents. FEDERAL MONEY. 5 2. Forty-four dollars six cents seven mills. 3. Eighty cents five mills. 4. Three thousand seven hundred six dollars six cents seven mills. 5. Two hundred four dollars fifty cents three mills. ADDITION. Addition is the process of uniting two or more like numbers into one sum. Add: 1. 2673 2. 837 3. 628 4. 8063 846 2964 4307 259 1025 418 526 8264 92 3825 8279 1287 837 842 428 428 642 29 4273 3064 5. $26.43 6. $715.30 7. $165.00 8. $ 20.863 18.75 21.86 2.93 9.246 4.10 10.163 128.06 7.208 563.13 516.00 28.00 8.096 8.429 129.40 113.82 5208.00 6.804 .926 39.625 128.753 11.31 37.15 476.203 192.097 9. Add seventeen thousand, nine hundred six; four thousand, two hundred eighty-nine ; eight hundred twelve thousand, seven hundred eight ; six hundred two ; forty- two thousand, nine hundred two ; twelve thousand. 10. Find the sum of eleven thousand, six hundred seven- teen ; sixty-eight ; four thousand, twenty-five ; two thou- sand, three hundred nine ; eighty-five thousand. 6 ARITHMETIC. 11. A merchant's sales were $2963.84 in January, $1463.27 in February, $3846.25 in March, and $2016.92 in April. What did his sales amount to in the four months ? 12. Find the sum of all the numbers between and in- cluding 167 and 174. 13. A man had $170 in his pocket, which was $70.75 less than he had in his safe. How much money had he in the safe ? 14. Add nine dollars six cents ; fifteen dollars seventy- two cents ; sixty dollars eighty-seven cents ; fifty-nine cents ; four dollars five cents ; two hundred dollars thirteen cents. Add: 15. 4286 3804 9273 6518 8274 2936 8142 9370 6425 3184 6293 16. 3184 2929 3641 8207 9243 6041 2938 8465 3721 4936 8749 17. Add without copying : 18. 178469 19. 729476 738527 835694 592946 209731 846953 569420 362431 182134 234234 234123 2306 1242 8936 6084 8346 2920 1289 9016 2914 8563 8472 20. 428756 937524 129305 747925 165213 243121 ADDITION. . " - Add: Total. 63 + 59 + 17 + 64 + 23 + 36 = 87 + 84 + 68 + 82 + 72 + 28 = 42 + 35 + 49 + 39 + 18 + 65 = 96 + 42 + 28 + 50 + 25 + 32 = 54 + 96 + 16 + 26 + 16 + 44 = 83 + 73 + 94 + 45 + 25 + 46 = 28 + 18 + 31 + 13 + 23 + 32 = Total, Oral Work. 1. Ruth had 17 cents, and earned 8 cents more. How many cents had she then ? 2. 40 + 12 + 10 = ? 3. We had 62 books in our library and bought 18 more. How many have we now ? 4. Five pupils are absent from a class and 38 are pres- ent. How many pupils are there in the class ? 5. Edward had 29 cents in his bank and put in 6 more. How much money has he now ? 6. There are 42 pears in one box and 39 in another. How many in both boxes ? 7. Albert spent 8 cents for a ball, 4 cents for a pencil, and 12 cents for a writing-book. How much did he pay for all ? 8. Willard had 28 marbles in one bag and 12 in another. How many had he in both bags ? 9. A boy sold a pair of skates for $.75, which was $.25 less than they cost. What did they cost ? Add quickly : 16 19 24 14 32 29 17 9 8 7 12 8 11 18 8 ARITHMETIC. SUBTRACTION. Subtraction is the process of finding the difference be- tween two like numbers. The Minuend is the number from which we subtract. The Subtrahend is the number subtracted. The result in subtraction is called the Difference or Remainder. How can you find the minuend when the subtrahend and remainder are given ? (Illustrate.) How can you find the subtrahend when the minuend and difference are given ? (Illustrate.) 1. 2819 2. 8203 3. 4295 4. 7306 - 674 -1276 597 -1807 5. 2763 6. 37284 7. 36801 8. 18003 1289 - 9287 - 18463 921 9. From seventeen thousand sixteen, take nine thou- sand four hundred eighty-seven. 10. From seventy-two thousand three hundred eleven, take forty-six thousand nine hundred sixty-one. 11. Take eight thousand four, from thirty thousand. 12. Find the difference between sixteen thousand four hundred seventy-five, and twenty thousand seven hundred twenty-seven. Subtract : 1. 92874 2. - 8206 3. -38706 4. -26942 - 11392 94210 42840 98301 5. 38264 6. - 8291 7. 92593 8. 27075 29842 19327 87246 18092 SUBTRACTION. 9 1. The minuend is 7026 and the subtrahend 987. Find the difference. 2. Find the difference between 9284 and 13411. 3. $4216.56 - $1874.92 =? 4. $149.725 - $69.417 =? 5. Take sixteen thousand eight hundred seventeen, from twenty-four thousand five hundred forty-one. 6. The difference is 8037. The minuend is 19406. Find the subtrahend. 7. Take seventy-six dollars four cents two mills, from one hundred two dollars nine mills. 8. From sixty-five thousand three hundred sixteen, take twenty-four thousand forty. 9. Take $86.215 from $900.09. 10. What must be added to $.67 to make $3 ? 11. From a farm of 263 acres of land 97 acres were sold. How much was left ? 12. A man had $279, and spent $129.64. How much had he left ? Oral Work. 1. If 8 apples are taken from a basket containing 25, how many apples are left ? 2. James is 7 years younger than his brother, who is 21 years old. How old is James ? 3. If you have $1.25 and spend $.75, how much have you left ? 4. A farmer sold a horse for $125. If it cost him $140, how much did he lose ? 5. How much more than 32 is 45 ? 6. Fred has 32 marbles and Robert has 18. How many more has Fred than Robert ? 10 ARITHMETIC. 7. Find the difference between 36 and 45. 8. If you give a fifty-cent piece in payment for a book which cost 35 cents, how much change should you receive ? Subtract : 29 30 25 48 35 GO -12 -16 -13 .-15 ADDITION AND SUBTRACTION. Add, without copying : 1. 417 2. 1172 3. 1642 4. 183 28 2964 417 2057 135 42 305 125 2906 4281 2893 748 Subtract, without copying s 5. 1187 6. - 1375 7. -2156 8. 7073 -242 9100 4228 496 Add: 9. 2786 10. 5278 11. 43562 12. 22879 1947 4492 84601 43012 1838 8913 92873 92874 5287 6472 26461 69154 6496 7258 30725 82738 8035 5603 92837 56425 2956 8450 68154 78997 8375 7921 15210 37684 1698 2864 2835 1872 3263 1342 5832 5832 Subtract : 13. 10392 14. -28023 15. 7296 16. __ 73841 -3875 38641 -629 92040 INDICATED WOliK. 11 Oral Work. 1. A man travelled 36 miles one day and 13 the next. How many miles did he go in both days ? 2. Edith wrote 28 words in the morning and 17 in the afternoon. How many words did she write ? 3. John used 17 sheets of paper from a tablet containing 65 sheets. How many are left ? 4. James had 22 cents and earned 3 more. He spent 18 cents for a knife. How much had he left ? 5. What must be added to 16 + 12 to make 30 ? 6. Two trains leave Syracuse, and move in opposite directions. One goes 42 miles and the other 34. How far apart are they ? 7. Anna had $ 1. She spent $.42 for a book and $.08 for a tablet. How much had she left ? 8. Two boys skate in the same direction. One goes 57 yards and the other 49 yards. How far apart are they ? Indicated Work. Addition. NOTE. Pupils should be taught to indicate the opera- tion before finding result. Indicate the addition, by using sign, and give the result. 1. 4 and 5. 4. 5, 17, 26, and 39. 2. 36 and 45. 5. 16, 15, 18, and 26. 3. 36, 15, 16, and 38. 6. 29, 84, 72, and 56. 7. A lady found 3 roses on one bush, 5 on another, and .7 on another. How many roses did she find ? 8. James found 2 eggs in a hen's nest, 5 in another, and 6 in another. How many eggs did he find in all ? 9. The sum of 25 cents, 38 cents, and 18 cents. 10. The sum of 16, 18, and 56, added to the sum of 25, 19, and 34. 12 ARITHMETIC. Indicated Work. Subtraction. Indicate the subtraction and give the result. 1. 29 from 84. 3. $.74 from $1. 2. 63 minus 42. 4. $8.35 from $9.14. 5. Nineteen books are taken from a case containing 130 books. How many are left ? 6. A man spent $10.16 on Monday and $4.40 on Tues- day. How much more did he spend on Monday than on Tuesday ? 7. A boy had 36 marbles. He lost 5 one day and 11 the next. How many had he left ? 36 _ (6 + 11) = ? 8. If you have 60 cents and spend 18 cents for ribbon and 12 cents for thread, how much have you left ? MULTIPLICA TION. Multiplication is a short method of adding equal numbers. Add four 5's. ADDITION. MULTIPLICATION. 5 5 multiplicand. 5 _4 multiplier. 5 20 product. _5 20 The Multiplicand is one of the equal numbers added. The Multiplier is the number of equal numbers added. The result of multiplication is called the Product. The multiplier is never a concrete number. The multiplicand and product are like numbers. The product is the same in whatever order the numbers are multiplied. (Illustrate.) MULTIPLICATION. 13 NOTE. Teach how to multiply by 10, 100, 1000. Multiply : 1. 36 X 10 4. 28 X 100 7. 100 X 10 2. 100 X 16 5. 328 X 1000 8. 100 X 100 3. 236 X 100 6. 1000 X 512 9. 1000 X 1000 Multiply : 1. 872 2. 943 3. 206 4. 985 5. 876 16 24 37 59 84 6. 467 7. 317 8. 2936 9. 416 10. 487 238 504 28 834 236 Oral Work. 1. 536 X 10 6. 289 X 100 2. 10 X 694 7. 1000 X 36 3. 786 X 10 8. 1000 X 586 4. ' 100 X 395 9. 369 X 100 5. 1864 X 100 10. 289 X 1000 283 X 300 == ? Annex two naughts and multiply by 3. Write answers only : 1. 286 X 20 6. 384 X 2000 2. 25 X 20 7. 5000 X 382 3. 400 X 253 8. 9000 X 35 4. 500 X 286 9. 2811 X 10 5. 29 X 900 10. 27 X 700 Multiplication. 1. 2087 X 42 5. 746 X 835 2. 186 X 294 6. 3006 X 48 3. 1238 X 400 7. 1283 X 680 4. 653 X 29 8. 967 X 105 14 ARITHMETIC. 1. If there are 196 pounds in a barrel of flour, how many pounds in 240 barrels ? 2. Find the product, the multiplicand being three thou- sand five hundred seventy-eight and the multiplier four hundred six. 3. How many bushels of apples will 4 orchards produce, if there are 84 trees in an orchard, and the average yield 10 bushels to a tree ? 4. $126.34 X 28 = ? 5. Fifteen men earn $1.25 each in a day. What will their combined wages amount to in a week ? 6. Multiply six hundred nine by forty-eight. 7. $720.036 X 14 = ? 8. Seventy-four yards of carpet at $1.08 a yard, will cost what ? 9. Multiply nine dollars six cents two mills, by 17. 10. Find the cost of 472 bushels of apples at $1.20 a bushel. Oral Work. 1. How many pens in 2 boxes, if there are 144 pens in a box? 2. How many dimes in 27 dollars, if there are 10 dimes in a dollar ? 3. How many trees in 6 rows, if there are 15 trees in a row ? 4. How many quarters in 42 dollars, there being 4 quarters in a dollar ? 5. How many pounds in 3 sacks of flour, if there are 49 pounds in a sack ? 6. How many desks in three schoolrooms, there being 9 rows in each room and 7 desks in a row ? DIVISION. 15 DIVISION. Division is the process of finding how many times one number is contained in another. The number divided is the Dividend. The number by which the dividend is divided is the Divisor. The result of division is called the Quotient. When the divisor is not exactly contained in the divi- dend, the part of the dividend that is left is called the Remainder. (Illustrate.) The remainder and dividend are always like numbers. (Illustrate.) When the divisor is an abstract number, the dividend and quotient are like numbers. (Illustrate.) When the dividend and divisor are concrete, the quotient is abstract. (Illustrate.) The sign of division is -5-, and, when placed between two numbers, signifies that the first is to be divided by the second. As 12 ~- 3 = 4. Division is also indicated by writing the dividend above, and the divisor below, a dividing line. Thus, ^ = 4. Find the quotients : 1. 18^6 3. if 5. 63-7-9 7. ^ 2. 14 -T- 2 4. \& 6. 28 -T- 2 8. 3^ Find the quotients at sight : 1. 5 in 15 2. 8 in 32 3. 9 in 72 4. 11 in 99 5. 12 in 132 6. 5 in 60 7. 11 in 88 8. 10 in 100 9. 11 in 77 10. 12 in 84 11. 75 -i- 5 12. 56-5-8 13. 21--7 14. 63 ^- 7 15. 96 -4- 12 16 ARITHMETIC. SHORT DIVISION. 6)1974 Six is not contained in 1 (thousand), hence 329 we find how many times 6 is contained in 19 (hundreds), which is 3 (hundreds) times, with a remainder of 1 (hundred). Write the 3 hundreds under the hundreds of the dividend. Change the remainder 1 (hundred) to tens, and add it to the seven (tens), making 17 (tens). Dividing this by 6, we have 2 (tens) for a quotient, which is placed under the tens in the dividend, and 5 (tens) for a remain- der. Changing this remainder to ones, and adding the 4 (ones), we have 54 (ones), which, divided by 6, gives 9 (ones), which is placed under the ones in the dividend. The quotient is 329. Prove by multiplying the quotient by the 'divisor ; if it equals the dividend, the work is right. Divide by short division : 1. 2555-4-5 10. ^jpjL 19. 30005 5 2. 2436-4-4 11. **f^ 20. 288012 12 3. 2845 -r- 5 12. *af*a. 21. 300010 10 4. 4503-7-3 13. lajia 22. 99011-11 5. 2045 -r- 5 14. AAJAA 23. 33264-11 6. 2835-7-7 15. J-LflAi 24. 29280 12 7. 4986-4-9 16. lifaa 25. 36550 10 8. 2009-7-7 17. S^AJL 26. 28692 9 9. 3504 -T- 8 18. S-&J4JL 27. 333333-4-11 8)3989 After dividing, there is a remainder of 5, 498 f which must be divided by 8. This we can X 8 on iy indicate thus, . The quotient is 498|. To prove, multiply the integral part of the *" quotient by the divisor, and add the remainder. QQQQ If it equals the dividend, the work is right. LONG DIVISION. it Divide : 1. 35621-4-7 3. 31493^6 5. 82956 -r- 10 2. 42963-5-6 4. 50725-4-3 6. 93043-4- 7 To divide by 10, 100, or 1000. 1|QO)369|00 1|00)283|25 1|000)36|954 369 283^ 36 T Vo 4 o Cut off from the right of the dividend as many figures as there are naughts upon the right of the divisor. 1. 39826-4- 100 5. 26935 -j- 100 9. 39482-4-1000 2. 28964 -f- 1000 6. 26320 -4- 10 10. 2896 -4- 1000 3. 389000 -4- 1000 7. 61934 -- 1000 11. 99208 -4- 100 4. 42657 -4- 10 8. 40030 -4- 100 12. 500300 -4- 10 LONG DIVISION. Divide 1234 by 15. 82^ Quotient. 15)1234 SOLUTION. 15 is not contained in 12 (hundreds), so we must include the tens. 15 is contained in 123 (tens) 8 (tens) times. Multiplying 8 (tens) by the divisor, we have 120 (tens), and subtracting we have 3 (tens), which, changed to ones and added to the 4 (ones) makes 34 (ones). 15 is contained in 34 (ones), 2 (ones) times. Multiplying 2 (ones) by the divisor, we have 30 (ones), which, subtracted from 34 (ones), leaves a remainder of 4 (ones). 4 (ones) divided by 15 is -^, which is placed as part of the quotient. The entire quo- tient is 82yV Divide : 1. 14)392 2. 17)485 3. 16)507 4. 13)296 5. 18)1124 6. 15)793 7. 22)943 8. 19)1364 9. 24)3726 10. 27)851 11. 21)1240 12. 32)3964 18 ARITHMETIC. 13. 1289 -r- 16 27. 5630-^-62 14. 5683 -r- 37 28. 1289-^-73 15. 5436 -;- 18 29. 16428-7-84 16. 1286 -r- 39 30. 12582 -,-58 17. 2983-42 31. 1384-^-75 18. 1008-^-25 32. 6381 -r- 86 19. 9436-47 33. 12946 -T- 24 20. 8059-^-36 34. 9273 -r- 91 21. 1583 -v- 46 35. 7754 -f- 78 22. 4109-^51 36. 10846 -T- 45 23. 2695 -;- 57 37. 21431-^36 24. 3874^49 38. 27473 -T- 97 25. 9003 -T- 25 39. 35702 -f- 53 26. 5914 -T- 59 40. 40060 -f- 60 41. Divide seven thousand, four hundred sixteen, by twenty-seven. 42. A ship sailed 159 miles in 13 days. How far did it- sail in one day. 43. $96.64. was divided equally among 16 men. How much did each receive ? 44. How many pounds of butter at $.18 a pound can be bought for $8.82 ? 45. How many times is 19 contained in 4845. 46. Divide $137.34 by 18. 47. Divide $62.826 by 74. 48. A farmer wishes to put 9216 bushels of oats into bins holding 16 bushels each. How many bins can he fill ? 49. Divide six thousand, seven hundred and sixty-seven dollars, thirty-nine cents, seven mills, by sixty-seven. INDICATED WOKK. 19 Oral Work. 1. At $.12 each, how many books can be bought for $.72? 2. Divide $1 equally among 5 boys. 3. How many dozen are 84 oranges ? 4. When 8 pounds of butter cost $2, how much is it a pound ? 5. Four boys together earned $4.16. What should each receive if they all earned the same amount ? 6 56 -r- 14 = ? 60 -r- 15 = ? 75 -^ 25 = ? 7. A class of 25 scholars wrote 250 words. How many words did each scholar write ? 8. Among how many men can $45 be divided, if each man receives $15 ? 9. There are 108 pencils in 9 packages. How many pencils in a package ? Indicated Work. 1. James had 18 marbles and found 5 more. He after- wards lost 8 marbles. How many were left ? 2. A boy deposited $1 in the savings-bank on the 1st of the month, 75/ on the 10th, and $2.50 on the 25th. On the 30th he drew out $2.75. How much was left in the bank? 3. John had $5.18 in the bank. He drew out $1.50 on Monday, and $.75 on Tuesday. How much had he left ? $5.18 ($1.50 + $.75) = ? Teach use of parentheses. 4. The sum of 5 and 9 taken from 16. (Problem.) 5. The sum of 56 and 39 taken from 188. (Problem.) 6. The sum of 36, 15, and 87 taken from the sum of 39, 752, and 38. (Problem.) 20 ARITHMETIC. Review Work. 1. 46835 29728 42841 56307 92013 52464 27835 12928 52642 83754 Subtract : 4. 142864 - 92831 2. 62423 3. 36429 18730 72015 29640 49263 19283 52700 76274 29378 < 53359 42064 84206 19283 19827 87261 52640 36426 64385 92075 5. 92053 - 72935 6. -65842 98375 Write results only : 8. -5624 9287 7. 7329 -4526 9. 1356 2462 10. 2526 1847 Multiply : 11. 2846 X59 12. 3874 X 78 Write results only : 15. 2834 16. 595 X 6 X 700 Divide : 19. 2783 -f- 49 20. 12064^62 21. 13011 -T- 13 22. 12094^56 13. 9256 14. 6247 X 206 X 125 17. 1293 18. 463 X 8 X 90 23. 70943-f-29 24. 5964 ^ 87 25. 29635 -T- 98 26. 39305^70 INDICATED wo^lM^: ..,:> 21 Indicated Work. NOTE. The pupil should be required to read his indi- cated work. 1. 4 + 3x2 = ? 3. 4x3 + 2 = ? 2. (4 + 3) X 2 = ? 4. 4 X (3 + 2) = ? Teach that operations indicated by the signs of multi- plication and division should be performed first, unless the parenthesis is used. Indicate the operations, and find the result. 5. 4 increased by the product of 3 and 2. 6. The sum of 4 and 3, multiplied by 2. 7. The product of 4 and 3, increased by 2. 8. 4 multiplied by the sum of 3 and 2. 9. 84 less the product of 5 and 6. 10. The difference between 72 and 42, multiplied by 4. 11. 144 subtracted from the product of 18 and 18. 12. 6 multiplied by the difference between 11 and 6. 13. Henry has 58/ and John 63/. Charles has 4 times as much as both. How much has Charles ? 14. Lucy and Mary went shopping. Lucy bought lace for 55/, and Mary 5 yds. of ribbon at 15/ a yard. What was the amount of both purchases ? 15. A livery man bought 4 bu. of oats at $.40 a bushel, and a load of hay for $12. What did the hay and oats cost? 16. Henry had 75/. He bought a tablet for 10/, and paid 4/ apiece for 6 pencils. How much had he left ? 75/ - (10/ + 6 X 4/). 17. Horace travels 50 miles and Joseph 3 times as far. How much farther does Joseph travel than Horace ? 22 ARITHMETIC. FRACTIONS. Oral Work. One of the two equal parts of an apple is called one half of an apple. One half is written . One of the two equal parts of a number is of the number. 1. How many are of 4 oranges ? of 8 cows ? 2. of $10 are how many dollars ? One of the three equal parts of an apple is of it. One of the three equal parts of a number is of the number. 3. How many are of 6 men ? of 12 men ? 4. What is J of an apple ? Illustrate. 5. What is of any number ? 6. What is i of 8 ? i of 12 ? J of 16 ? One or more of the equal parts of a whole is a Fraction. Divide a number by 3 to get of it. Divide a number by 4 to get ^ of it. How do you get J of a number ? ? f ? J ? 7. Which is the greater part of an apple, or J ? Illustrate. 8. Which is the greater part of 12, or ? How much greater ? 9. When you have found J of a number, how do you find f ? f ? f ? 10. How many are f of 16 marbles ? Of 20 apples ? 11. How many are J of 12 ? | of 12 ? 12. James had 1 dollar, and Clara | of a dollar. Which had the more money ? Illustrate. 13. 1 is equal to I; ; |; T V 14. 2 is equal to J ; | ; | ; |. FRACTIONS. 23 Written Work. 1. How much is J of 8 ? 16 ? 24 ? 32 ? 40 ? 2. Find f of 16. of 24. | of 40. 3. Find \ of 7, 14, 21, 28, 35. 4. Find f of 7, 14, 21, 28, 35. 5. How much is J of 15 ? of 21 ? 6. Find ^ of 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 ? 7. Which is greater, T L of 40 or of 30 ? How much ? 8. Which is greater, ^ of 100 or J of 40 ? How much ? Oral Work. 1. How many are 3 men and 5 men ? 4 horses and 6 horses ? 2. How much are 3 sevenths and 2 sevenths ? 3. How much are and 4- + = . 6. i + J = ? 7- T% + T 1 5 = 11. *-?=? 12. - = ? 8. 9- 10. 13. 14. 15. 16. Written Work. 1. James had $J and George $J. How many cents had both? 2. Mary paid of a dollar for a book, f of a dollar for a hat, and J of a dollar for gloves. How many eighths of a dollar did she spend in all ? 3. How much is 2g + J ? 3 + | ? 6ft + A ? 4. 3} 5. 5-ft 6. IT*,- 7. 15f 24 ARITHMETIC. 8. A lady paid $1| for a book and $4J for a hat. How much did both cost ? 9. How much is 7 J ? 4| | ? 2^ ^ ? 10. 6f 11. 8$ 12. 15$ 13. -3ft 4^ 6f Oral Work. 1. (One-half of an apple) and (one-half of an apple) make how many apples ? 2. How many halves in a whole apple? In a whole dollar ? In any whole thing ? 3 * hlUf + half ^ halves' 2 + 2 = * 4. Two halves of an apple = ? 5 = ? 7_.? J=? $ = ? o lllll t * 3 ~r 3 i $ 5- 9. How many thirds in a whole apple ? 12. How many thirds in 4 ? How many halves in 3 ? 13. 9 = J. 10 = . Written Work. 1. JU 6. 9 3 ^r_ __ 9 2. J^ = ? 4. 1^ = ? Add: 9. 4 10. 7f 11. 8} 12. 5g 5 6 5f 7| 6| 2J- 4f 4J 13. How many yards in 3 pieces of silk containing respectively 14| yd., 19| yd., and 24 J yd. ? FEDERAL MONEY. 25 FEDERAL MONEY. 10 mills (m.) = 1 cent (ct.). 10 cents = 1 dime (di.). 10 dimes = 1 dollar ($). 10 dollars = 1 eagle (E.). 20 dollars = 1 double eagle. Oral Work. 1. How many dimes in 6 dollars ? 2. 4 eagles are how many dollars ? 3. A man had $80 in eagles. How many eagles had he ? 4. How many mills in 15 cents ? 5. In 5 dimes how many cents ? 6. A dollar, a quarter, a dime, and a nickel are how many cents ? 7. How many dollars are there in 130 dimes ? 8. How many dollars are there in 1300 cents ? 9. Divide 4 eagles among 5 men. How much would each receive ? 10. How many books at 50/ each can be bought for $2 ? Written Work. 1. Reduce 2 dollars, 5 cents, 7 mills to mills. 2. J of 3 double eagles are how many cents ? 3. Change 2971 mills to higher denominations. 4. 2000 dimes are how many dollars ? 5. How many cents are there in $17J ? 6. From 16 cents take 9 mills. 7. If you had a quarter of an eagle and spent $1.75 how much would you have left ? 8. 270 cents are how many dimes ? 9. How many five-cent pieces in a box containing $20 ? 26 ARITHMETIC. DRY MEASURE. 2 pints (pt.) = 1 quart (qt.). 8 qt. =1 peck (pk.). 4 pk. = 1 bushel (bu.). 36 bu. = 1 chaldron (chal.). Oral Work. 1. How many pecks in 9 bushels ? 2. In 22 pints how many quarts ? 3. A bushel of apples at $.25 a peck will cost what ? 4. How many pints are in 6 qt. 1 pt.? 5. What will a peck of chestnuts bring if sold at 10/ a quart ? 6. How many times can a quart cup be filled from a bushel of walnuts ? 7. How many quarts are in 9 pecks ? Written Work. 1. Eeduce 15 pk. 7 qt. to pints ? 2. How many quarts are in 846 pints ? 3. Henry gathered 16 bu. 2 qt. of walnuts, and sold them at f . 08 a quart. What did he receive for them ? 4. In 8136 pt. how many bushels, etc.? 5. What will 6 bu. 2 pk. of cranberries be worth at f .75 a peck ? 6. How many quarts are there in 7 bu. 3 pk. 7 qt.? 7. A peck of peaches were sold from a crate containing a bushel. How many quarts were left ? 8. How many bushels are there in 576 pints ? 9. 14 bushels of oats will last a horse how many weeks if he is fed 8 qt. a day ? 10. What will 32 pk. of walnuts bring at f .05 a pint ? LIQUID MEASURE. 27 LIQUID MEASURE. 4 gills (gi.) = 1 pint (pt.). 2 pt. = 1 quart (qt.). 4 qt. =1 gallon (gal.). 31i gal. = 1 barrel (bbl.). 2 bbl. = 1 hogshead (hhd.). 2 hhd. = 1 pipe (pi.). 2 pi. =1 tun. Oral Work. 1. How many pint cups can be filled from 8 quarts ? 2. A quart of milk was taken from a five-gallon pan How much was left ? 3. How many gallons in a hogshead ? 4. | of a gallon are how many quarts ? 5. How many hogsheads will 8 barrels of oil fill ? 6. A family uses 2 qt. of milk a day. What will the milk bill amount to in a week at the rate of 4/ a quart ? 7. How many quarts are there in 7 gal. 3 qt. ? Written Work. 1. Reduce 1 bbl. to gills. 2. 2% gallons of kerosene at 2? a pint will cost what ? 3. Paid $1.60 for cider at $.05 a quart. How many gallons were purchased ? 4. Change 711 gills to higher denominations. 5. How many 2-quart cups can be filled from a barrel of sirup ? 6. Change 7 gal. 2 qt. 1 pt. to gills. 7. In 276 pints, how many gallons ? 8. A merchant paid $10 for a barrel of molasses, and retailed it at $.40 a gallon. What was his gain ? 28 ARITHMETIC. 9. A half-barrel of vinegar was sold at 2/ a quart. What was received for it ? 10. How many gallons in 16 quarts ? 11. 13 gallons have been sold from a barrel. How many quarts are left? 12. At 7/ a quart what will 15 gal. of vinegar cost ? AVOIRDUPOIS WEIGHT. 16 drams (dr.) = 1 ounce (oz.). 16 oz. = 1 pound (lb.). 25 lb. = 1 quarter (qr.). 4 qr. =1 hundred weight (cwt.). 20 cwt. = 1 ton (T.). 2000 lb. = 1 ton. Oral Work. 1. How many ounces are in 4 pounds ? 2. In 32 drams, how many ounces ? 3. What will a pound of candy cost at $.02 an ounce ? 4. How many pounds in 4 tons ? 5. In 6000 lb. how many tons ? 6. How many pounds are in a hundred weight ? 7. Paid $.48 a pound for candy. How much is that an ounce ? 8. How many 3-pound packages can be made from 75 pounds of coffee ? Written Work. 1. How many ounces are there in 25 lb. ? 2. In 1\ tons, how many pounds ? 3. How many bullets weighing 2 oz. each can be made from 32 lb. 4 oz. of lead ? 4. Eeduce 18 lb. 7 oz. to drams. AVOIRDUPOIS WEIGHT. 29 5. What will 2 Ib. dried peaches cost at $.32 per pound ? 6. In 1364 drams, how many ounces ? 7. Five boys share equally 1 Ib. 9 oz. of candy. What is each boy's share ? 8. How many 4-ounce bags can I fill from a box hold- ing 7 Ib. of candy ? 9. What is the difference in ounces between a hundred weight and 97 pounds ? 10. What will 7f Ib. of indigo cost at $ .03 an ounce ? 11. Keduce 18 Ib. 4 oz. to drams. 12. A hundred weight is what part of a ton ? LINEAR MEASURE. 12 inches (in.) = 1 foot (ft.) 3 ft. =1 yard (yd.) 5% yd. = 1 rod (rd.) Oral Work. 1. How many inches long is a yard stick ? 2. Reduce 2 ft. 4 in. to inches. 3. 24 feet are how many yards ? 4. How many feet around a picture 2 ft. long and 1 ft. wide ? 5. How many inches in a half-yard ? 6. At $.16 a yard what will 1 yd. of ribbon cost ? 7. 6 f t. + 2 ft. + 4 ft. are how many yards ? 8. How many feet in 108 inches ? Written Work. 1. A rod is how many feet ? 2. In 124 inches, how many yards, etc. ? 3. Reduce 14 ft. 7 in. to inches. 30 ARITHMETIC. 4. Change 1848 in. to feet. 5. How many feet in 16 yards ? In 118 yards ? 6. What will be the cost of 16 yd. of wire at 2/ a foot ? 7. How many yards around a room 18 ft. long and 15 ft. wide ? 8. At $.36 a yard what will 18 inches of cord cost ? 9. A pole 119 ft. high is broken off 27 ft. from the bottom. How many yards are broken off ? 10. How many inches remain ? 11. What will it cost to build 12 yards of fence at $.42 a foot ? 12. A picture is 72 inches long and 36 inches wide. How many feet in the frame around it ? ADDITION AND SUBTRACTION OF DENOMINATE NUMBERS. Add : 7 bu. 1 pk. 3 qt. 6 bbl. 7 gal. 1 qt. 10 1 2 5 12 1 412 461 1 yd. 2 ft. 6 in. 4 bbl. 7 gal. 4 qt. 1 pt. 21 62 1 123 3 511 Find the sum : 1. 3 bu. 2 pk. 2 qt. 1 pt. ; 4 bu. 5 pk. 3 qt. ; 7 bu. 1 pk. 4 qt. 1 pt. 2. 7 Ib. 8 oz. 6 dr. ; 4 Ib. 11 oz. 5 dr. ; 2 Ib. 4 dr. 3. Ibbl. 14 gal. 2 qt. Ipt.; 2 bbl. 5 gal. 3qt.; 7 gal. 3 qt. 4. 4 yd. 2 ft. 7 in. ; 5 yd. 1 ft, 6 in. ; 2 yd. 11 in. 5. 16 Ib. 10 oz. 3 dr. ; 9 Ib. 7 oz. ; 5 Ib. 14 dr. COMPOUND ADDITION AND SUBTRACTION. 31 Subtract : 1. 16 Ib. 8 oz. 3 dr. 2. 7 yd. 3 ft. 9 in. 542 2 5 7 3. 16 da. 5 hr. 36 min. 4. 27 gal. 3 qt. 1 pt. 2 gi. 5 7 18 _18 213 5. From 6 dol. 7 di. 2 ct. 3 m. take 4 dol. 5 di. 8 ct. 1 m. 6. Find the difference between 4 Ib. 8 oz. 3 dr. and 2 Ib. 5 oz. 7 dr. 7. From a can containing 7 gal. 2 qt. 1 pt. of milk, 4 gal. 3 qt. were sold. How much was left ? 8. 6 ft. 11 in. were cut from a pole 18 ft. 9 in. long. How long was the pole then ? 9. Into a barrel were put 16 gal. 3 qt. 1 pt. of oil one day, 3 gal. 3 qt. the next, and 7 gal. 1 pt. the next. How much oil was there in the barrel ? 10. 9 Ib. 8 oz. of candy were sold from a box containing 14 Ib. How much was left ? 11. A 4-gallon pail and a quart bottle were filled from an 8-gallon cask of cider. How much was left ? 12. How much twine is there in 3 balls if there are 16 yd. 1 ft. in one, 17 yd. 2 ft. 8 in. in another, and 9 yd. 2 ft. 11 in. in the third ? Oral Work. 1. From a 4-gallon jug 2 qt. were taken. How much was left ? 2. 8 oz. were sold from a box of candy holding 5 Ib. How much was left ? 3. A boy cut 2 ft. from a piece of twine 4 yd. long How much was left ? 4. From 5 Ib. 7 oz. take 2 Ib. 5 oz. 32 ARITHMETIC. GENERAL REVIEW PROBLEMS. 1. At $1.50 a yard, what will 64 yd. of carpet cost ? 2. A man earns $65 a month, and spends $31. How long will it take him to save $238 ? 3. What is | of 2080 pounds ? 4. Add 1, , and f . 5. Write in figures seven hundred dollars six mills. 6. Two boys skate in the same direction. One goes 65 yd. and the other 3 times as far. How far apart are they then ? (Illustrate.) 7. Find the quotient, using 2734 and 15. 8. How many pints of walnuts can be sold from 5 bu. 2pk. ? 9. Multiply six thousand four hundred seventy-three by sixty-nine. 10. Charles has $42. James has 2 times as much. How much money have both ? 11. Find the difference between of 714 and $ of 1456. 12. Florence bought 6 dozen eggs at $.32 a dozen and 2 dozen oranges at $.02 apiece. What did all cost ? 13. A man bought a horse for $110 and a harness for $30. He paid J as much as both cost for a wagon. What did the wagon cost ? 14. At $.18 a peck what will 6 bu. of apples cost? 15. Find the product of 6 X 7 X 3 X 9. 16. A farmer bought goods to the amount of $6. He paid | of it in oats. How much money did he have to pay? 17. James paid $19.75 from a roll of bills containing $40. How much was left ? KEVIEW. 33 18. A lot which cost $125 was sold at a gain of of its cost. What was received for it ? 19. From 180 take f of itself. 20. 288 Ib. of coffee will fill how many 12-pound bags ? 21. At 27/ a pound what will three 30-pound jars of butter cost ? 22. If 5 dozen of eggs cost $1.20, what will % a dozen cost? 23. Take 2099 from 7809. 24. Find the sum of T 5 r of 132 and } J of 132. 25. 2 Ib. 4 oz. of candy at $.03 an ounce will cost what ? 26. A lady who had $100 bought a shawl for $30, and paid $2 a yard for 18 yd. of silk. How much had she left ? 27. Add $16.29 ; $2.71 ; $.49 ; $238 ; $16.12 ; and $42. 28. How many feet around a square table that is 18 in. on a side ? (Illustrate.) 29. In what year was a man born if his age is now 38 ? 30. A boy earns $4.20 a week. What will he earn in 10 days ? 31. What is the cost of 9 books at $J each and dozen pictures at $2 each ? 32. If 9 dozen of oranges cost $1.53, what will 12 dozen cost? 33. $96 $.45 = ? 34. 249 X 74 = ? 35. Add 726, 9463, 95, 18263, 9295, 38640, 1872, and 575. 36. At $3| a ton what will 3$ tons of coal cost ? 37. The dividend is 31232. The divisor is 32. Find the quotient. 84 ARITHMETIC. 38. If milk is 4/ a quart, how many gallons can be pur- chased for $1.92 ? 39. If six tons of hay cost $84, how many tons will $252 buy ? 40. How many pints in 18 bu.? 41. If a dozen pencils cost $.10 what will 3 boxes, each containing 6 dozen, cost ? 42. 8 boys each received $|. How many dollars did it take? 43. From J of the sum of 297 and 313 take of their difference. 44. If you paid $8 for a coat, $2j for a hat, and $5 for boots, what did all cost ? 45. Divide 37958 by 91. 46. Mr. Sherwood paid $1250 for a lot, built a house for $4275, and a barn for $1025. He sold it all for $9000. What was his profit ? 47. Add 813, 9257, 8564J, 267, and 9043. 48. A man divided $400 equally among his 5 sons. One spent g of his share for books. What did the books cost? 49. Add of $.63, i of $1, of $48, and of $.42. 50. Eeduce 1 hhd. to gills. 51. Take 2837 from 19364. 52. Add eleven thousand three hundred sixty-nine ; four thousand thirty-nine ; seventy-eight thousand four hundred fifteen ; four hundred eighty thousand five hundred seventy- three ; twelve thousand sixty-four. 53. When milk is 5/ a quart, what will it cost a family for 13 weeks if 14 qt. a week are used ? 54. How many thirds in 15 ? EEVIEW. 35 55. A room is 28 ft. wide and 32 ft. long. How many yards around it ? (Illustrate.) 56. Paid $34 for 8 tons of coal. What was it a ton ? 57. A man paid $6 apiece for 18 sheep, and had $78 left in his pocket. How much had he at first ? 58. How many pages are there in 17 books of 267 pages each ? 59. Divide 2083 by 19. 60. 14812 -^ 47 = ? 61. Divide three thousand eight hundred seventy-four by twenty-six. 62. | of 75 gal. of molasses were sold at $.18 a quart. How much was received for it ? 63. Add 213, 8237, 5164, 291, 9168, 3792, and 847. 64. Divide $325 equally among 13 men. What would each receive ? 65. From 7980 take 4518. 66. The minuend is 12138, the subtrahend 1497. Find the difference. 67. Add $11.08, $213, $9.643, $.876, $538.275, and 68. 137 oranges were sold from a box containing 225. What would they bring at $.04 each ? The rest were sold at $.05 apiece. What did they bring ? 69. Express in words 3130019. 70. How many pints in 75 gal. 3 qt. ? 71. Two men are 327 miles apart. They travel towards each other, one going 95 miles and the other 73. How far apart are they then ? 72. Take six thousand eight hundred forty-two from eleven thousand two. 36 ARITHMETIC. 73. How many dimes in 17 dollars ? 74. The multiplicand is 573 and the multiplier 38. Find the product. 75. A man who earned $1050 a year, spent | of it for rent. What did his rent amount to in 5 years ? 76. Mr. Jones has 945 sheep in one lot and J as many in another. How many sheep in both lots ? 77. Take $.38 from $38. 78. Write a number of three groups. 79. Write a number of two groups. Find the difference between the two numbers. 80. Add $17, $9, $6f , $14, $8J, and $.18. 81. John has 26 marbles. Fred has 16 more than twice as many. How many marbles have both ? 82. 4216 -r- 15 = ? 83. How many years since America was discovered in 1492? 84. At $24 a ton what will 263 tons of hay cost ? 85. How many pounds in 218 ounces ? 86. A barrel of flour weighs 196 pounds. How many pounds in 3 J barrels ? 87. A fruit dealer bought 5 dozen boxes of berries for $4.75 and sold them for $.09 a box. What did he gain ? 88. Bought 9 yd. of cloth at $1.75 a yard and paid $13.50 for making and trimmings of dress. What did it cost ? 89. Two cities are 540 miles apart. How fast must a train run to make the distance in 18 hours ? 90. A clerk's salary is $900 a year. He spends f of it. How much does he save in a month ? 91. } of the oranges in a box containing 300 were BEVIES divided equally among 5 persons. How many did each receive, and how many were left in the box? 92. A merchant who had 72 barrels of sugar sold | of them to one customer, to another, and f to another. How many barrels had he left ? 93. How many lemons in 3 loads, 28 boxes on each load and 100 lemons in a box ? 94. Two men travel from the same place in opposite directions. One goes 18 miles a day and the other 27. How far apart are they at the end of 4 days ? (Illustrate.) 95. A boy wishes to buy a bicycle for $60. He has $42, and earns $.75 a week for 6 weeks. How much does he still need ? 96. 8 men were paid $96 for 6 days' work. How much was that a day for one man ? 97. Find of $122. 98. What will 10 eggs cost at $.24 a dozen ? 99. 21432 - 947 = ? 100. If you had $7J, and spent $5J, how much would you have left ? 101. A farmer had a horse worth $265, and he exchanged it for a yoke of oxen and two cows. The oxen he sold for $118, and one cow for $75, the other for $54. What was his loss ? 102. A man paid $21 for a stove and as much for a chair. What did both cost ? 103. Multiply 627 by 94. 104. The minuend is 1362, the difference 584. What is the subtrahend ? 105. How many quarts of oats can be put in 35 peck measures ? 38 ARITHMETIC. 106. Express in figures two million, three hundred sev- enteen thousand, five. 107. 642 + 89 + 97 + 7362 + 745 + 469 + 1286 = ? 108. $3.12 -f- 4 = ? (Problem.) 109. Write the present year in figures. In words. In Roman. 110. What will 3 bu. of peanuts cost at $ .05 a pint. 111. 4829 -T- 16 = ? 112. J of $2440 = ? 113. What must be added to 816 to make 1003 ? 114. A man receives $658 rent a year for one store, $497 for another, and $276 for a third. What does he receive in rents for three years ? 115. From the sum of 1216 and 1890 take their difference. 116. A piano worth $162 was sold for f of its value. What was received for it ? 117. The product of two numbers is 216. One of the numbers is 18. What is the other ? 118. 317 x 56 = ? 119. What is 56 in the preceding example ? 120. How long would a bin of oats last 17 horses if it would last one horse 391 days ? 121. 27934 + 54 = ? 122. If 8 men earn $96 a week, what will 6 men earn ? 123. How many bushel measures can be filled from 640 pt. of oats ? 124. Find the product of 1826 and 89. 125. 1912 - 498 - 156 = ? 126. At 5/ an oz. what will 4 Ib. 5 ounce cloves cost? REVIEW. 39 127. Dr. Stokes sold his horse for $195, and lost $45. What did the horse cost ? 128. Find f of $2142. 129. A grocer bought 9 bu. of apples for $9, and sold them at the rate of $.30 a peck. What did he gain ? 130. How much more than J of 280 is f of 1535 ? 131. 9 gal. 3 qt. of milk will fill how many quart cups ? 132. If $3 will buy 7 books, what will a dozen cost ? 133. 10 bu. of cherries at 6^ a quart will cost what ? 134. $2.87 + $32.90 + $.15 + $921.83 + $16.74 = ? 135. To 384 add | of it. 136. of 18 ft. of cord are cut on. How many yards are left ? 137. How much money would a lady have left if she had a five-dollar bill and spent J of it ? 138. Paid $45.90 for 9 tons of coal. What was it a ton ? 139. How much can a boy save in 13 weeks if he saves 5 cents a day ? 140. How many feet around a table 1 yard square ? 141. What will a bushel of walnuts cost if 2 bu. cost $4 ? 142. Frank had $30.18, Rose had half as much, and James twice as much. How much money had the three ? 143. 6943 -v- 76 = ? 144. The sum of two numbers is 7315. One of the num- bers is 5817. What is the other ? 145. A boy ate J of a pie. What part of it was left ? 146. A man who had $5 spent of it for a book, and the rest for a* hat. What did the hat cost ? 147. What will $J + ${ + $$ + $J amount to ? 40 ARITHMETIC. 148. How many oranges in 18 dozen ? 149. Add 2|, 6J, 7J, 9, and 17J. 150. Find the product of 4129 and 37. 151. At 2/ an ounce what will 1 Ib. candy cost ? 152. Write in words 92073460. 153. How many pieces of wood 3 in. long will a 45-foot pole make ? 154. Bought nuts for $.06 a quart. What did 7 bu. cost ? 155. (916 + 159 - 187 - 213) X = ? 156. If the divisor is 178 and the quotient 123, what is the dividend ? 157. From } take ft- 158. A boy bought a pair of skates for $3, and sold them for $2J. What did he lose ? 159. 8 times the cost of a book is $22. What did the book cost ? 160. Find ft of $132. 161. If i a yard of cloth costs $2, what will 5 yd. cost ? 162. How many cents in 6 eagles ? 163. A gallon, 2 quarts, and a pint are how many pints ? 164. ^ = ? f J = ? f f = ? 165. From the sum of 8137 and 12291 take 1986. 166. Add 173, 8164, 97, 7931, 8, 9014, and 178. 167. of the oil in a barrel was sold. How many quarts were left ? 168. What part of 3 pk. is 12 qt. ? 169. Multiply $18.065 by 14. 170. A man who had $16| spent $9|. How much had he left ? 171. 384 pt. are how many gallons ? ADDITION. 41 Add: 1. 4287 2. 7236 3. 62345 5965 1834 18376 8473 3529 29046 4287 9287 92043 3069 5078 87536 4218 9217 42365 9246 3465 17393 3287 2974 24867 7249 3826 63587 2756 9375 20624 8461 5468 85626 3097 2083 13346 Add: 4. $173.25 5. $ 836.427 6. $7000.42 92.47 1521.500 83.92 9.38 936.008 6214.836 407.25 42.53 807.015 92.16 1006.279 78.26 57.84 83.442 9287.426 9.26 64.297 383.527 Subtract : 1. 27134 2. 18307 3. 5645 9862 11409 17013 4. 2365 5. 8034 6. 127421 9271 7296 86456 7. From 9864 take 3286. 8. 10016 - 9284 = ? Add: 9. 42J 10. 63j 11. 131 12. 42 ARITHMETIC. 1. 27938 -57 . 9 2. 62941 -83 == 9 3. 7209 -18 = 9 4. 8312 -83 _ 9 5. 64205 -42 o 6. 17483 -56 _ 9 7. 29006 -29 = 9 8. 18736 -54 __ 9 9. 56803 -92 = 9 10. 9063 -50 9 11. 6284 59 = ? 12. 92307 36 = ? 13. 53124 72 = ? 14. 96503 87 = ? 15. 600412 29 = ? 16. 37561 - 46 = ? 17. 25964 - 38 = ? 18. 40136 24 = ? 19. 50601 -f- 97 = ? 20. 80319 -g- 75 = ? Indicated Work. (Indicate the operation, and give result.) 1. James had $.75 and spent $.32 for a book and $.18 for paper. How much had he left ? 2. A man who owed $500 paid $275 and then borrowed $132 more. How much did he owe then ? 3. Laura bought 3 yd. of ribbon, paying $.18 a yard. How much change would she receive from a dollar ? 4. John skated 72 yards and Henry 3 times as far. How far apart were they if they went in the same di- rection ? 5. John had $27 in the bank and drew out $4.25. He then deposited $18. How much money had he in the bank? 6. Julia had $.75 and her sister 3 times as much. How much money had both ? 7. A merchant has 3 pieces of cloth, each containing 24 yards. He sells 18 yd. one day and 41 the next. How much has he left ? 8. Jennie bought a book for $.36 and gloves for $.75. How much change would she receive from a five-dollar bill? REVIEW. Oral Work. 1. Our school term is 20 weeks long. If we have two holidays, how many days do we attend school ? 2. f of 12 is what part of 16 ? 3. A pint of water is poured into a gallon measure, and the measure is then filled with milk. How much milk is in it ? 4. If 3 oranges cost 10 cents, how many can I buy for $.20? 5. There are 23 girls and 27 boys present. How many scholars are in the school-room ? 6. Fred paid $90 for a bicycle and sold it for $75. How much did he lose ? 7. How many oranges at $.04 each can you get for 2 dimes ? 8. 3 hats at $5 each and 5 umbrellas at $3 each will cost what ? 9. f of a pie is eaten. What part is left ? 10. How many yards in 27 ft.? 11. At 5/ each what will 40 tablets cost ? 12. Ten eggs out of 5 dozen are bad. How many are good ? 13. From 3 take 1. 14. From a cistern containing 25 bbl. of water 9 bbl. leaked out. How much was left ? 15. A lady bought 8 Ib. of meat at $.12 a pound, and gave the butcher a dollar bill. How much change did she receive ? 16. Take f of 12 from 30. 17. 3 books at $.50 each would cost what ? 18. How many feet in 36 inches ? 44 ARITHMETIC. 1. At $19.50 a ton find the cost of 12000 Ib. of hay. 2. In 68 oz. how many pounds ? 3. A cistern holds 14 bbl. 3 gal. 2 qt. of water. If 7 bbl. 3 qt. are taken out, how much is left ? 4. The subtrahend is 63281, the difference is 41263. Find the minuend. 5. Find the cost of a hogshead of molasses at $.15 a quart. 6. 10080^32. 7. 2J + } + 5 + 6J + 8} = ? 8. How many drams in 3 Ib. 12 oz. 9 dr. ? 9. $16.755 + $1.20 + $.45 + $.008 + $7.015 + $.602 + $25 = ? 10. How much fence would be needed to inclose a lot 72 ft. by 58 ft. ? 11. How many hogsheads will it take to hold 10710 gal. of molasses ? 12. Which is the greater amount of money, ^ of $350 or } of $630.27 ? How much greater ? 13. 125J - 86J = ? 14. How many yards of tape will it take to go around a table 3 ft. long and 5 ft. wide ? 15. If you pay $.25 apiece, how many books can you get for $9 ? 16. At $.16 a gallon, how many gallons of vinegar will pay for 6 Ib. of butter at $.32 a pound ? 17. Write and add all the numbers between 7 and 50 that end in the figure 8. 18. How many times can 16 be subtracted from 1328 ? 19. At 8/ a pound what will 4 Ib. of meat cost ? 20. What will 20 bu. of tomatoes bring at $.40 a bushel ? MISCELLANEOUS. 45 21. How many pounds in 526 ounces ? 22. A horse and $125 were given for a carriage valued at $270. What was the horse worth ? 23. Draw a square, and call each side 15 ft. How many inches half way around the square ? How many yards around the other half ? 24. If 6 men can do some work in 12 days, how long will it take one man to do it ? How long will it take 8 men? 25. 28 horses cost $6944. What is the cost of one ? 26. How many inches in 4 yd. 2 ft. 9 in.? 27. Reduce a jf 4 to a whole number. 28. 7 bbl. were pumped from a cistern holding 41 bbl. 16 gal. How much was left ? 29. At $.28 a pound, what will be the cost of 16 J pounds of coffee ? 30. If 4 men earn $64 in a week, how much can 13 men earn in the same time ? 31. 36741 -r- 92 = ? 32. The sum of two numbers is 8013. One of the num- bers is 2416. What is the other ? 33. If 8 tons of coal cost $42, what will 1 ton cost ? 34. In how many weeks will a family use 196 qt. of milk, if 2 qt. a day are used ? 35. How many yards of ribbon at $.24 a yard can be bought for $2.88 ? 36. How many quarts in of a barrel ? 37. 7 bu. of nuts at 6/ a quart will cost what ? 38. From 9}f take 6 T \. 39. Add three hundred seventeen thousand four hundred six ; nine million twenty-three thousand eight hundred 46 ARITHMETIC. fifteen ; forty-five thousand four hundred eighty-six ; six hundred nine thousand thirty. 40. A man carried in 3 tons of coal in a basket holding 40 pounds. How many basketfuls did he carry ? 41. How many times would you fill a quart cup to empty a barrel ? 42. If 17 Ib. of butter cost $5.95, what will 1 pound cost ? 43. A man sold a cow for $72. If f of this was profit, what did the cow cost him ? 44. If J a pound of tea costs $.26, what will 17 Ib. cost ? 45. What number multiplied by 72 gives 11232 ? 46. Take .f of 112 from f of 5000. 47. At $4.80 a dozen, what will 9 handkerchiefs cost ? 48. Reduce 1608 mills to higher denominations. 49. How much will a man earn in a week if he earns $2 J a day ? 50. A farmer sold f of 126 bu. of potatoes at $.90 a bushel. What did he receive for them ? 51. A house cost $4200, the barn ^ as much. What did both cost ? 52. How many yards of border will it take for the walls of a room 17 ft. square ? 53. How many pounds of butter at $.32 a pound can be bought for $160.64 ? 54. A man sold 2 bu. of peanuts by the half-pint. How many sales did he make ? 55. 262194 -+- 89 = ? 56. From $200 take $.082. 57. How long will a tub of butter containing 42 Ib. last a family, if 8 oz. a day are used ? MISCELLANEOUS. 47 58. 16f + 18 + 42J - 60 = ? 59. A man earned $90 a month, and saved in 3 months $55.20. How much did he spend each month ? 60. Add the odd numbers between 10 and 20 and multi- ply the sum by 75. 61. Find the difference between Jf and f . 62. A lady had 3 spools of thread with 50 yd. on each spool. She used 1 J spools. How many yards were left ? 63. At $4.50 a ton what will 19 tons of coal cost ? 64. Write the largest number containing two groups. 65. What number must be added to 1083 to make 4000 ? 66. A barrel of cider at $.05 a quart will cost what ? 67. Paid J of $640 for a horse, and J- of what was left for a harness. What did both cost ? 68. 2921 - 189 - 1716 = ? 69. Take the sum of $4.42 and $6.20 from the differ- ence between $19 and $2.88. 70. If a bushel of apples costs $|, what will 21 bu. cost ? 71. A boy had $J, $J, and 3 dimes. He spent J of it. How much had he left ? 72. Bought | of a yard of plush at $3.52 a yard, 4 yd. of ribbon at $.35 a yard, and had $2.28 left in money. How much had I at first ? 73. of a barrel of molasses was sold at $.12 a quart. What was received for it ? 74. How many times will a wheel 9 ft. in circumference revolve in going 15 yards ? * 75. Find the quotient of 32857 -r- 36. 76. A man paid $400 for a team of horses, and sold them at $187 apiece. What did he lose ? 77. From the sum of $27 and $.27 take their difference. 48 ARITHMETIC. 78. At f .06 a quart how many gallons of milk can I get for $1.44 ? 79. 3 qt. of walnuts cost $.24. What will 3 pk. cost ? 80. How many 3-ounce bags can be filled from 7| Ib. of candy ? 81. If 2989 is taken from the sum of 1397 and 4209, how much is left ? Add: 82. 12694 38563 72596 80357 94261 80357 29826 56837 43659 38426 12564 37842 Add: " 85. 12964 83756 23841 92084 64297 13826 24927 12562 17384 92644 75806 92109 74318 83. 86. 72846 38137 56421 74363 82146 37356 12945 73486 15942 66735 70859 29642 37642 92845 17376 69524 83126 54216 83709 52846 75912 86402 91929 65307 84. 87. 35832 15737 26840 12756 34265 19084 27653 12642 98731 70735 23853 73759 56384 83792 47275 74903 31296 96478 53129 41379 83567 42861 27272 93856 64925 MISCELLANEOUS. 49 88. 126430 -r- 54 = ? 89. 72426 -T- 87 = ? 90. How many 2-foot rules can be cut from a pole 8 yd. long ? ! 91. Paid | of $21 for a hat and of what was left for a cloak. What did both cost ? 92. Add nineteen dollars eighteen cents ; two hundred seventy dollars three cents four mills ; sixty-five dollars ; four dollars eighteen cents nine mills ; eight hundred two dollars thirteen cents seven mills. 93. How many 2-quart pails could be filled from a hogshead ? 94. If 18 tables cost $110.34, what is that apiece ? 95. Charles had J of $16 and spent | of it. What did he spend ? 96. A horse was fed 8 qt. of oats a day for 12 days ? How much was left from 11 bu.? 97. In 304 gills how many gallons, etc.? 98. 3 boys picked 7 bu. of nuts, and sold them at $.09 a quart. They divided the money equally. What was the share of each ? 99. How long will 4 tons 6 cwt. 1 qr. 15 Ib. of hay last if 40 Ib. are used daily ? 100. What will 17f Ib. of crackers cost at $.12 a pound ? 101. At $.06 a yard what will the cord around a table 3 ft. long and 2 ft. wide cost ? 102. How many 24-inch steps would you take in walking 32 yd.? 103. -4f A = ? 104. When lemons are $.24 a dozen, what will 17 baskets, each containing 48 lemons, cost ? 50 ARITHMETIC. 105. If a steamer goes 10 miles an hour, how many days will it take to go 3120 miles ? 106. A druggist sold alcohol at $.10 a pint. If he paid $35 for a cask containing 48 gallons, what was his gain ? 107. 13867 is the product and 49 the multiplier. What is the multiplicand ? 108. How many more pints are there in 18 bu. than in 18 gal.? 109. What number added to 23002 will make 31101 ? 110. Find the sum of $18.36, $57.043, $117, $26.136, $.18, $14, and $.009. 111. At $.05 a quart, how many quarts of peanuts can be bought for $15.40 ? 112. A man left $7000 to his wife and three children. The children received $1275 each. What did the wife receive ? 113. 79430 is the minuend and 36412 the subtrahend. Find the remainder. 114. A lady started on a journey with fifty dollars. She paid $16.62 for railroad fare, and stayed 3 weeks, paying $10.50 a week for board. How much had she left? 115. A barrel of vinegar at $.28 a gallon will cost what ? 116. If a laborer receives $.17| an hour, what will he receive for 12 hours' work ? 117. A farmer sold 2 bu. 3 pk. 4 qt. of clover seed at $.05 a quart. What did he receive for it ? 118. A merchant gained $295 by selling some goods for $7640. What did the goods cost him ? 119. If 6 bu. of apples cost $6.48, what will a peck cost ? 120. I paid | of $.40 for 3 lemons. How much was that apiece ? MISCELLAKEOUS. 51 121. From a piece of cloth containing 21 yd., 8 yd. were cut at one time and 4i at another. How many yards were left? 122. 4293 -4- 47 = ? 123. If 23 is the divisor and 147 the quotient, what is the dividend ? 124. A lot cost $2600 and a house 2 times as much. What did both cost ? 125. Bought 9 Ib. butter at $ a pound. How much change would I receive from $3 ? 126. Add 6134, 74927, 84, 8739, 952, 5734, and 1286. 127. 2873 X 94 = ? 128. On a card there are 4 dozen buttons. How many are there on 16 cards ? 129. 84 sheep were sold for $300. If they cost $3 apiece, how much was gained by the sale ? 130. 28375 ~ 15 = ? 131. How long would it take a boy earning $1.25 a day to earn $120 ? 132. Allowing 75 Ib. to a bushel, how many ounces will 42 bu. of corn weigh ? 133. What is the average daily attendance at a school if there are 916 scholars present on Monday, 938 on Tues- day, 934 on Wednesday, 927 on Thursday, and 915 on Friday ? 134. A milkman sells 17 customers each a pint of milk daily, 23 customers take a quart each, 47 take 2 qt. each, and 8 take 3 qt. each. How much milk does he sell in a week ? 135. 7 bushels of nuts are sold at $.04 a half-pint. What did they bring ? 52 ARITHMETIC. 136. A man having 5 Ib. 7 oz. of candy sold 3 Ib. 12 oz. How much had he left ? 137. A man steps 24 inches at each step. How many yards will he go if he takes 1279 steps ? 138. What will 8 months' rent amount to at $390 per year ? 139. 48 Ib. of camphor costing $.29 a pound was sold at $.03 an ounce. How much was gained ? 140. A man buys 27 horses for $3564, and sells them at a gain of $270. What does he receive apiece for them ? 141. What is the difference between 16 X $3.25 and 9 x $4.015 ? 142. How many pints in a cask containing 27 gal. 3 qt. 1 pt.? 143. Two persons together earned $236. One had $32 more than half of it. How much had the other ? 144. 119287 + 18 = ? 145. At $10 a week what will a man's board bill amount to in 17 1 weeks ? 146. What is the distance in yards around a table 5 ft. long and 3 ft. wide ? Illustrate. 147. Two persons are 1000 miles apart. They travel towards each other for 6 days, one going 42 miles a day and the other 37. How far apart are they then ? 148. A farmer sold 18| bu. of raspberries, 37 bu. of blackberries, and 56f bu. of strawberries. How many bushels of berries did he sell ? 149. At $2.75 a day, what will it cost a man to board 3 weeks ? 150. A hogshead of maple sirup at $.30 a quart will cost what ? CHAPTEE II. FACTORS AND MULTIPLES. 15 is the product of 3 and 5. 15 is a multiple of 3 and 5. 3 and 5 are factors of 15. A Factor of a number is an integer that will exactly divide it. An Integer is a whole number. 3 is a factor of 15, because it is contained in 15 without a remainder. What are the factors of 18 ? of 20 ? of 24 ? of 36 ? A number that has no factors except itself and 1 is a Prime Number. 17 is a prime number. Name several prime numbers. A number that has other factors besides itself and 1 is a Composite Number. 16 is a composite number. Name others. A prime number used as a factor is a Prime Factor. What are the prime factors of 18? of 20? of 24? of 36? Numbers are Prime to each other when they have no common factor or divisor. 4 and 9 are prime to each other, but are not prime numbers. Name other numbers prime to each other. 63 54 ARITHMETIC. A Multiple of a number is a number that exactly con- tains it. 5, 10, and 15 are multiples of 5. Give a multiple of 3 ; of 4 ; of 6 ; of 7. Give several multiples of 2. Give three factors of 18. What number is a factor of both 9 and 15 ? A number that is a factor of two or more numbers is a Common Divisor of them. Name a common divisor of 18 and 24. Name all the common divisors of 18 and 24. What is the greatest common divisor of 18 and 24 ? What is the greatest common divisor of 24 and 36 ? 30 is a multiple of what number ? What number is a multiple of both 2 and 5 ? . Give other numbers that are multiples of 2 and 5. A number that is a multiple of two or more numbers is a Common Multiple of them. What is a common multiple of 4 and 6 ? of 3 and 4 ? What is a common divisor of 24 and 36 ? What is the greatest common divisor of : 1. 20 and 30 4. 10, 15, and 20 2. 18 and 27 5. 7, 14, and 21 3. 25 and 40 6. 12, 18, and 24 Write the prime numbers from 1 to 25. Write the composite numbers from 1 to 25. Name a common multiple of : 1. 2 and 7 3. 2 and 6 5. 2, 3, and 4 2. 3 and 4 4. 5 and 4 6. 3, 4, and 5 Name three common multiples of 2, 3, and 4. FACTORS AND MULTIPLES. 55 What is the least common multiple of 3, 4, and 5 ? Is 24 a common multiple of 4 and 6 ? Give a common multiple of 4 and 6 that is larger than 24. Smaller than 24. What is the least common multiple of 4 and 6 ? Of 3 and 6 ? What is the least common multiple of 18, 27, and 30 ? Divide by any prime number 2 ) 18, 27, 30 that is exactly contained in any 3 ) 9, 27, 15 two or more of the numbers, 3)3, 9, 5 and bring down the quotients 1, 3, 5 and undivided numbers. Divide 2x3x3x3x5 = 270 again as before, and so continue Ans. until there is no number that will exactly divide the quo- tients and undivided numbers. The product of the divisors, quotients, and undivided numbers will be the least common multiple. The least common multiple of 18, 27, and 30 is how much less than their product ? Find the least common multiple of : 1. 5, 16, and 20 4. 3, 5, 9, and 10 2. 9, 14, and 12 5. 6, 9, 12, and 18 3. 4, 12, and 16 6. 2, 14, 7, 9, and 3 7. From the product of the numbers in each of the above examples, subtract their least common multiple. Add: 8. 3264 9. 29832 10. 98462 987 394 231 26942 99 9825 7298 4 259 369 6943 72491 56 ARITHMETIC. COMMON FRACTIONS. A common fraction is written with two numbers, one above the other, with a line between them : thus, f . The number above the line is called the Numerator. 4 is the numerator. The number below the line is called the Denominator. 5 is the denominator. The denominator shows the number of equal parts into which the unit is divided. The numerator shows how many of these equal parts are taken. (Illustrate.) The numerator and denominator are called the Terms of a Fraction. A Proper Fraction is a fraction whose value is less than 1. Its numerator is less than its denominator, as j. An Improper Fraction is a fraction whose value is 1 or more than !. Its numerator is equal to, or greater than, its denominator. | is an improper fraction because its value is 1. J is an improper fraction because its value is more than 1. An integer may be written in fractional form by giving it 1 for a denominator. A Mixed Number is an integer and fraction written together. 3J and oj are mixed numbers. Reduction of Fractions is the process of changing the form without changing the value. Multiplying both terms of a fraction by the same number does not change its value. (Illustrate.) Dividing both terms of a fraction by the same number does not change its value. (Illustrate.) To reduce a fraction to higher terms, multiply both terms by the same number. REDUCTION OF FRACTIONS. 57 Eeduce j to 60ths, g* } 2 2 = g. (To find the number by which you multiply, divide the required denominator by the given denominator.) In the above illustration the required denominator, 60, is 12 times the given denominator, 5 ; therefore the required numerator must be 12 times the given numerator. Which is greater, f or Jf ? Why ? Change the following fractions to higher terms : Mental Work. 1. \ to 4ths. 2. to lOths. 3. f to 12ths. 4. 5. 6. 7. 15. to 16ths. \ to 48ths. to ? 8 - f = If- 9. f = if 10. f = A- 11- i? 12. J, to 12ths. to 24ths. 13. f , 14. J, J to 28ths. , to 40ths. \ to higher terms. 16. How many 20ths in J 17. How many 66ths in 18. f equal how many 63d s ? 19. Mary bought J of a yard of velvet. How many 32ds of a yard did she buy ? 20. Change ^ to another form. 21. Change f to a fraction having 40 for a denominator. Written Work. 1. to 42ds. 5. f | to 120ths. 9. f f to 500ths. 2. f to 64ths. 6. ff to 150ths. 10. f J to 1680ths. 3. ^to75ths. 7. ftol44ths. 11. {% to 522ds. 4. iJ to 96ths. 8. H to 128ths. 12. Jf f to 9375ths. 58 ARITHMETIC. 13. {$ to 1330ths. 17. |, |, J, }i to 96ths. 14. ^ to 43121sts. 18. f, 3^, ^ to 250ths. 15. }$ to 1393ds. 19. ^, J, TW to lOOOths. 16. ,&V to 988ths. 20. f, }|, f * to 625ths. 21. Reduce |-| to a fraction having another form. 22. How many 328ths in ? 23. There are how many 840ths in f, jj, $ ? 24. Reduce ^ T to higher terms. To reduce a fraction to its lowest terms : Divide both terms by any common factor ; divide the result in the same way, and so continue until the terms are prime to each other. If the terms are large, divide by their greatest common divisor. Change the following to their lowest terms : Mental Work. 1. 6. H 11. |J 2- t 7. ft 12. Jf 3. H 8. || 13. f J 4. 4 9. |f 14. S| 5. } 10. ft 15. rffr Written Work. I- *f 8. ^ft 15. TVft 22. 2. T% 9- Ml 16- Hi 23. J 3. H 10. Jft 17. m 24. 4. H 11. Jft 18. ^ 25. 5. || 12. if} 19. /A 26. 6- f* 13. ^% 20. Ht 27. J 7. Tftft I 4 - *Wfr 21. ^ftV 28. i 29. How many thirds in |J ? 30. Express in simplest form 98 divided by 392. REDUCTION OF FRACTIONS. 59 31. Change -ff/^ to a fraction whose denominator is five. 32. Express in simplest form the quotient of 288 divided by 504. 33. What is the simplest form of |f f ? To reduce a mixed number to an improper fraction. 5| apples equal how many 4ths ? In 1 apple there are fou * ths ? i n 5 apples there are 5 X 4 20 . 20 , 3 23 __ 93 I fourths fourths ~ = * ' fourths fourths ' fourths ' fourths fourths Rule. Multiply the integer by the denominator of the fraction ; to the product add the numerator, and write the result over the denominator. Reduce the following to improper fractions : Mental Work. 1. 1J 8. 2f 15. 8ft 22. 12} 2. 4 9. 3f 16. 9J 23. 10^ 3. 3| 10. 4J 17. 10$ 24. llf 4. 5J 11. 5J 18. llf 25. 13J 5. 3J 12. 6f 19. 12J 26. 14^ 6. 2f 13. 8J 20. 10^ 27. 13| 7. 4J 14. TfV 21. 11{ 28. How many eighths in 13 ? Change 239 to the form of a fraction. 29. How many fifths in 32 ? Eeduce 25 to eighths. 30. In 15^ weeks how many sevenths of a week ? 31. What improper fraction will equal 13 J ? Written Work. 32. 25 34. 37^ 36. 270$ 38. 33. 9^ 35. 49^ 37. 19ft 39. 60 ARITHMETIC. 40. 174 45. 25^ 50. 29ft 55. 191^ 41. 25? 46. 59^ 51. 149? 56. 203^ 42. 15^ 47. 67^ 52. 128f 57. 98H 43. 23ft 48. 89f$ 53. 137 T 7 s 58. 87|J 44. 40J 49. 131? 54. 238H 59. 138W 1 60. In 237 how many ninths ? 61. Reduce 147 to elevenths. 62. How many sixteenths in 250 ? 63. Change 128^ to twenty-sixths. 64. In $730 how many nineteenths of a dollar ? 65. Change 297 to a fraction whose denominator shall be 81. 66. Express in another form, 142^. 67. There are how many 121 sts in 401 T J T ? To reduce an improper fraction to an integer or mixed number. foi ^ ths apples = how many whole apples ? In 1 apple there are ^^; in ^^ apples there are as many apples as ^^ is contained times in ^^- = 5 apples. 23 _ K, fourths ~~ ** Rule. Divide the numerator by the denominator. Change the following to integers or mixed numbers : Mental Work. 1. 1 7. 13. -V 3 - 19. 2. i 8. v- 14. ff 20. 3. f 9. 15. V 21. f 4 i 10. V 16. lf 22. ft 5. \ a 11. V 17. A|* 23. ff 6. i 12. V >8. ft 24. Vi REDUCTION OF FRACTIONS. 61 * 25. In z^ 2 - of a gallon how many gallons ? 26. In -Z-J 1 of a dollar how many dollars ? 27. f changed to sixths equals what mixed number ? Written Work. 1. JLJI 3. \^ 5. sff. 7. Hi* 9- W 2- W 4 - W 6. W W 10 - H a 11. *# 14. *f$3. 17. * 20. A|^a. 12. affi 15. *jp 18. W- 21. iff^ 13. V/ 16 - W 19 - -Tff- 22 ' *$%$* 23. How many pecks in Q-fja. pecks ? 24. Reduce f f& to a mixed number. 25. In f J^ miles how many miles ? 26. Express in another form, 2^*. 27. What is the quotient of 38540 -r- 197 ? 28. Prove the above by multiplication. A Fractional Unit is one of the equal parts of a whole unit. The fractional unit of is ; of J is J. Like Fractions are fractions that have the same fractional unit; J, f, |, f. Fractions having unlike fractional units can be changed to those having like fractional units ; %, , J = T 6 5 , ^, ^. NOTE. Fractions having a common denominator are like fractions. (Illustrate.) Fractions have a Common Denominator when their denom- inators are alike. The common denominator of two or more fractions is a common multiple of their denominators. The least common denominator of two or more fractions is the least common multiple of their denominators. 62 ARITHMETIC. To reduce fractions to a common denominator : Reduce each of the fractions to its lowest terms. Mul- tiply the denominators together for a common denominator. Divide the common denominator by the denominator of each fraction and multiply both terms of each fraction by the quotient. (Illustrate.) Mental Work. Change the following to fractions having a common denominator : I- *, i 4- t, } 7. i 4, i 2. *,} 6. },f 8. f, i, } 3. J,| 6. J,i 9. |,i,i Written Work. 1- i,i, 4. $, *,3 7. |,i,4 2- f J,i 5. f,f,| 8. f,J,f 3- 2, i, 6. i,|, | 9. Ai* NOTE. To reduce fractions to their least common de- nominator, find the least common multiple of the denomi- nators, and proceed as above. Change the following to fractions having a least common denominator : Mental Work. 1. i, 1 4. f A ;J 7. *> T 7 ^ 2. , | 5. t. 4> T 7 5 8. |, f, 3- i, | 6. *, f, 5 1 ! 9. * ? l ? i Written Work. 1. i, K'i 5. if, 1, A 9. i t, f , A 2- f, t.t 6. i, f, M 10. i A> 1? f 3. ft, A. 4 7. 9, i A> f 11. i? f > 3? T ? 5 4. J, 1,1 8. i, ii I, A 12. fc i i> H ADDITION OF FRACTIONS. 63 In how many ways may fractions be reduced? (Illus- trate.) ADDITION OF FRACTIONS. Rule. If the fractions are not like fractions, reduce them to a common denominator ; add their numerators, and write the sum over the common denominator. If neces- sary, reduce the result to lowest terms. If an improper fraction, reduce to an integer or mixed number. To add mixed numbers, add the integers and fractions separately, and unite the results. (Illustrate.) Mental Work. Add the following fractions : 1. i, i 6. f, 11. 2J,4, 1 2- ,* 7. |,i 12. i,,^ 3. , i 8. , 13. H, 2|, 4. 4. f, 9. f , J, 3 14. 3J, If, 5 5. J, i 10. i, J, H 15- ,3*,4 16. 2i,3j,2i. 17. A man paid $| for a book, $J for an inkstand, and $ J for writing paper. How much did he spend ? 18. Mary has $ ; her mother gave her $3. How much has she ? Written Work. Addition : 1. f A>i 6. i,},i,A 11. f, f i T 7 T, f 2. i> 4> "i 7 ^ 7- *, A ^ i 12. i, *,i, f 3. > f T 7 ^ 8 - > i ? T^ ? A 13. i> 2,|, 6 4. fi/l 9- i,*,*,i 14. i, i 2 i? T ? TT 5. A*,i 10. j, f, 6, | IB. 4|,f,J,/,. 64 ARITHMETIC. 16. t,*, A>* 21. 6|, 81, 5|, 7| 17. 6|, 8, f , i 22. 9J, 5f , 9, if 18. 3, , }, f 23. j, I, f, f 19. I, 4, , If 24. 5J, 7|, 9J, 45 20. f, J, A> T 8 * 25. 14?, 9f , 10 26. A man travels 25 mi. on Monday, 37g mi. on Tues- day, on Wednesday as many miles as on Monday and Tuesday. How many miles does he travel in three days ? 27. A farmer has 271 bu. of potatoes in one bin, 133f bu. in another, 47 T \ bu. in another. How many bushels has he ? 28. Mr. Brown has $130^ ; his wife $25| more than he has ; his son $78/ ? , and his daughter $5 more than his son. How much have all ? 29. How many yards of cloth will I have, if I buy 123J yd., 76| yd., and 58 yd. ? 30. 6J yd. of cloth are required for a coat, 3| yd. for trousers, and J yd. for a vest. How many yards are required ? 31. Find what your mother spends if she pays $8 T \ for your coat, $9J for your dress, $4 for your hat, $2j for your shoes, and $lj for your gloves. 32. How much land in a farm of five fields ; the first contains 26}^ acres, the second 50J acres, the third 41f acres, the fourth 69|- acres, and the fifth 52 acres ? 33. ? - 28if = 98. ? - 104 T 6 F = 99J. 34. After selling 197| bu. of potatoes, a farmer has 238J bu. left. How many bushels had he to sell ? 35. A lady paid $25^ for china, $12| for cooking uten- sils, $2| for tin-ware, and $25i for a stove. What was the amount of her bill ? SUBTRACTION OF FRACTIONS. 65 SUBTRACTION OF FRACTIONS. Rule. If the fractions are not like fractions, reduce them to a common denominator and write the difference of their numerators over the common denominator. (Illustrate.) To subtract mixed numbers, subtract integers and frac- tions separately. From 12 Take _9| 3J Ans. From llf = lOf f cannot be subtracted from f ; take Take 5f = 5f one of the 11 units, change to 7ths, Oral Work. 5f and add the f , making lOf . Subtraction : 1. |-i 5. A- f 9. 7 - i 13. 4 -14 2. f - i 6. 1- f 10. 8- 14. 6* -4 3- 1 -i 7. T 9 <7 - J 11. 12- H 15. 11 -8 4. i-i 8. 1| _ 3 12. 22-44 16. 191 9 ? 17. J. Edmund Y. Cooke of Cleveland had $150, and paid $21 J for printing. How much was left? 18. 9 _ ? = 8J J - ? = i 7 - ? = 2J 17 ? = f 19. Thomas Phillips bought a pony for $35. He paid $14| cash. What does he still owe ? 20. Minuend 75, subtrahend 24i. Find the difference. 21. What is the difference between 11 and J ? 22. Given 26f for minuend, 19 for difference ; what is the subtrahend ? 23. 8J- - 7 = ? 9^ T - 2^ = ? 12 J - | = ? 13 - 66 ARITHMETIC. Written Work. 1. S-J 6. 15 ~2J 11. H- 2. i-J 7. 8J-4J 12. |-4 3. | - i &. 10 f 13. ^ - f 4. 6 - J 9. 12J -5J 14. H-l 5. 3f | 10. 8 4J 15. if 16. Take from |. From take ^ T . 17. From the sum of $ and J take their difference. 18. What must I add to 9| to make 20 & ? 19. The sum of two fractions is || ; one of the fractions is : what is the other ? 20. John had 125J bu. of apples and sold 98 bu. How many bushels had he left ? 21. 91^ - 38$ = ? Take 291| from 378J, 22. From 23J tons of coal 13| tons were sold. How many tons remained ? 23. A dressmaker cut of a yard of velvet from a piece containing | of a yard ; how many sixteenths of a yard were left ? DENOMINATE NUMBERS. A Denomination is the name of a unit of weight or measure. In the number 4 pints, 1 pint is the unit and pint is the denomination. A Denominate Number is a number that has one or more denominations ; as, 6 pecks, 4 qt. 2 pt. A number having more than one denomination is a Compound Number; as, 4 qt. 2 pt. What is a concrete number ? Give several concrete numbers that are not denominate numbers. LINEAR MEASURE. 67 LINEAR MEASURE. 12 inches (in.) 1 foot (ft.) 3 feet 1 yard (yd.) 5% yards 1 rod (rd.) 40 rods 1 furlong (fur.) 8 furlongs 1 mile (mi.) 320 rods 1 mile 5280 feet 1 mile Oral Work. 1. John has a fish pole 4 yd. long. How many inches long is it ? 2. How many rods in 5 miles ? 7 miles ? 1 miles ? 3. How many feet in % mile ? 3 miles ? 1 miles ? 4. Stepping two feet at a step, how many steps will be taken in walking 2 miles ? 5. 40 rd. are what part of a mile ? 6. 2640 ft. are what part of a mile ? Written Work. 1. Eeduce 164808 in. to miles, etc. 2. Change 8 mi. 27 rd. 5 yd. 2 ft. to inches. 3. In 12024 in. how many rods, etc. ? 4. 9 mi. 3 fur. 5 rd. 3 yd. 1 ft. make how many feet ? 6. The diameter of the earth is 7912 miles ; how many feet long is it ? 6. The Atlantic cable between Newfoundland and Ire- land is 2200 mi. long. If it cost 10 cents a foot, what was the whole cost ? 7. At 2 cents a mile, what is the railroad fare from Syracuse to N. Y. City, the distance being 1599840 feet ? 8. Long Measure is used in measuring what ? 68 ARITHMETIC. Indicated Work. Division. NOTE. Division is indicated by the sign -f- , or by writ- ing the dividend over the divisor with a line between them ; thus, 16-7-4 may be written *-. 1. 8 + 4-j-2 = ? 2. (8 + 4) -f- 2 = ? 3. Which is the dividend and which the divisor in example 1 ? In example 2 ? 4. Indicate the division by the other method in exam- ples 1 and 2. Indicate and give results : 5. 20 increased by the quotient of 25 divided by 5. 6. The sum of 20 and 25, divided by 5. 7. The difference between 18 and 12, divided by 2. 8. The product of 11 and 6, divided by 3. 9. 150 plus the quotient of 16 divided by 4. 10. If 5 apples cost 20/, what is the cost of 1 apple ? What is the cost of 7 apples ? 11. Divide the product of 11 and 8 by 4. 12. If 5 horses cost $750, what will 8 horses cost ? 13. The product of three factors is 88. One of the fac- tors is 11. What is the product of the other two ? 14. What number multiplied by 16 gives 400 ? 15. How many yards of cloth at $5 a yard will pay for 15 cords of wood at $4.00 a cord ? 16. How many eight-quart cans will hold 16 gallons ? 17. Divide the product of 126 and 15 by 189. 18. Harry had 25/, and his uncle gave him 50/. He spent 35/ and divided the remainder equally among 4 poor boys. How much did each boy get ? REVIEW. Add from right to left and vertically 124 + 5789 + 45876 + 2356 + 18534 + 127302 + 103497 + 92345 + 246138 + 876 + 8953 + 68350 + 91203 + 75638 + 9021 + 716584 + 36492 + 312325 + 980705 + 10347 + 253614 + 875612 + 63826 + 531743 + Add without copying : 1. 3804| 2. 2756 31 945 5667 17589^ 69 124579 785673 103805 98764 362958 873286 651321 936432 1032867 943758 3. 437601f 605712 4. 308675 4756 3854621 2170 613521 25043 7394583 1438 6716579 148394 29367 5671432 76227$ 832299 Subtract : 1. 2478 954 7. 12348 9876 10. 35214 9876 2. 5876 1259 5. 1234f 985$ 8. 15943J 798 11. 3. 6. 24678 9899 8765J 928 9. 1325432 980993H Subtract without copying : 13. 236 14. 3214 1982 867 12. 15. 12546J 8231 923 70 ARITHMETIC. Written Work. 1. Add forty-three dollars ; seven dollars, twenty cents ; nineteen cents ; twenty-three dollars, three mills ; forty cents, six mills ; and eighty-five cents. 2. From one hundred nine dollars, subtract eight dollars, eighty cents, eight mills. 3. $840.268 + $23.01 + $14296.30 + $560.005 + $4.90 + $.009 = ? 4. Reduce $5284.40 to mills. $9456.03 to cents. 5. Change 15004 cents to dollars. 6524 mills to dollars. 6. How many half-dimes in 2 eagles ? 7. Paid for building a house $3170.75, for painting $240.37, for furniture $1605.40, for carpets $340|. How much money did I pay out ? 8. $6.375 X 98 = ? $8500 + 125 = ? $41.25 ^ 33 = ? 9. At of a dollar a pound, what cost 376 Ib. of maple sugar ? 10. ^y of a dollar equals how many cents ? 11. 25 cents are what part of a dollar ? 12. What will one spoon cost, if 2 dozen cost $47.76 ? 13. 134 + 39f - 35J = ? 958J - 194J = ? 14. 1,244 + 29,567 + 900,580 + 39 + 150 + 6 + 3,487,639 = ? 15. Multiplicand 246, the product 854661 ; what is the multiplier ? 16. A freight train ran 399 mi. one week ; 11 times as far lacking 246 mi. the second week ; how many miles did it run ? 17. At $.87| each what cost 12 doz. handkerchiefs ? 18. How many pints in 479 barrels ? REVIEW. 71 19. A yoke of oxen cost $137. What will 19 oxen cost at the same rate ? 20. 21 yd. of muslin cost $72.58. How much is yard worth ? Oral Work. 1. of 32 - f x 12 = ? | X 120 + | of 56 = ? 2. Divide the sum of 6 and 12 by their difference. 3. 9 X 6 divided by the product of 2 and 9. 4. How many times is 7 + 2 contained in 60 + 12 ? 5. Subtrahend 24, remainder 49 ; find the minuend. 6. Take the sum of 6 and 9 from their product. 7. What is the product of 12 and 96 ? 8. Divisor 7, quotient 9, remainder 5; what is the dividend ? 9. How many times is of 16 contained in 112 ? 10. The product of two numbers is 210, one of the numbers is 70 ; find the other. 11. A boy earned 39 cents, and his father gave him 21 cents. At 15 ct. each how many doves can he buy ? 12. Take from 20. From 18 take 17$. 13. At $ J a peck, what will 2 bu. of potatoes cost ? 14. One pint of molasses costs 20 ct. ; what will 1 gal. cost? 15. $18 X 9 =? 22 X ? = 154. 640 -*- ? = 128. 16. How many feet around a table 9 ft. long and 3 ft. wide? 17. Cloves are 7 ct. an ounce. What will 4-J- oz. cost ? 18. What are the factors of 26 ? 39 ? 42 ? 63 ? 19. Give 4 different multiples of 6. 20. Give three different factors of 40. 72 ARITHMETIC. Written Work. 1. How many pints, etc., in 1543 gallons ? 2. In 14 gal. 3 qt. 2 gills, how many gills ? 3. Keduce 52 gal. 1 pt. 3 gi. to gills. 4. Keduce 2024 gi. to hogsheads, etc. 5. How many quarts in 4 hogsheads ? 6. Change 92351 qt. to hogsheads, etc. 7. How many gills, etc., in 947 gallons ? 8. Keduce to gills and add the results : 14 gal. 1 pt. ; 18 gal. 3 qt. 1J pt. 9. How many mills in $6.62 ? $.37 ? $89 ? 10. 1500 cents equal how many dimes ? 11. 2932488 -v- ? = 40720. of 1245 -=- 73 = ? 12. I work 12 weeks at $14 a week ; how much wheat at $1.68 a bushel will pay my wages ? (Indicate.) 13. How many hours in this year ? 14. ft of 319 - |f of 96 = ? 69438 X 131 = ? 15. How many times must I fill a quart-cup to make 2 barrels ? 16. 64 gal. of kerosene oil cost $5.12. What will one quart cost ? 17. Change 3 hhd. to gills. 18. In 1 hhd. 1 bbl. 10 gal. how many quarts ? 19. At $.06 a pint, how many gallons of molasses can be bought for $7.68 ? 20. Add 12l, 34, 104^-, and 7J. 21. 240 - 103 T V = ? 512} - 90^ = ? 22. Give 4 factors of 340. Of 198. Of 160. 23. What is the quotient of 24 times 56 divided by 7 times 24 ? (Indicate.) 3ITTJ REVIEW. ^^^iUFOjJjiX 73 Written Work. 1. In 4 bu. 3 pk. 7 qt. 1 pt. how many pints ? 2. In 2099 pt. how many bushels, etc. 3. Change 82 pints to bushels, etc. 4. How many quarts in 2 bu. 2 pk. 1 qt. ? 5. 32 chaldrons equal how many pecks ? 6. Reduce 14 bu. 6 qt. to pints. 7. Reduce to pints and add the results : 7 qt. ; 14 bu. ; 2 pk. ; 138 qt. 8. How many pints of peanuts can a pedler sell, if he has 29 bu. ? 9. At 5 ct. a half-pint, how much will he receive for his peanuts ? 10. How many dollars in 250 dimes, 16 quarter-dollars, and 6 double eagles ? 11. At $1.40 a bushel, what cost 2 qt. of oats ? 12. Find the cost of 7 bbl. of vinegar at five cents a quart. 13. Write in figures seventy-four million, three thousand, two hundred eight, and divide by 59. 14. 1984 + 936 + 12305 + 894873 + 9973 - 2968 = ? 15. 2345 X 239 -4- 121 = ? 118 T 7 ^ + 29| + 240J = ? 16. 49638 X 498 - 29631 = ? 345962 -^ 196 = ? 17. 6, 15, and 4 are factors of what number ? 18. From | of 720 take f of 287. 19. Write 1895 in Roman Notation. 20. Find 3 factors of 120. Of 78. Of 196. 21. Dividend 2975, divisor 85 ; find the other factor. 22. 93600 -i- ? = 390. 190348 X 107 = ? 23. Find the greatest common divisor of 99 and 72. 74 ARITHMETIC. Written Work. 1. Eeduce 2 cwt. 15 Ib. 1 oz. to ounces. 2. In 2 T. 15 cwt. 40 Ib. how many pounds ? 3. In 41570 Ib. how many tons, etc. ? 4. In 2936 cwt. how many ounces, etc. ? 5. How many ounces in 1 T. 12 cwt. 70 Ib. ? 6. Change 3 tons to pounds. 7. Reduce 10 Ib. 2 oz. 14 dr. to drams. 8. How many two-ounce weights can be made out of 100 pounds of brass ? 9. What cost 26 cwt. of ham at 9 ct. a pound ? 10. One horse eats 6 qt. of oats a day. How long will 12 bu. last a span of horses ? 11. Oranges are worth 3 ct. each. How many dozen can be bought for $3.60 ? 12. 648 + 2903 + 18 + 16730 + 24985 + 7039875 + 39= ? 13. 15 men dig a cellar in 12 days. How long will it take 2 men ? 14. Dividend 346030, divisor 243. Find the quotient. 15. A railroad train travels at the rate of a mile a minute. How far will it go in a day ? 16. 1649 and 137 are factors of what number? 17. 9837650 -r- 293 = ? 248 X 95 +- 36 = ? 18. At 6J ct. a pound, what cost 4 cwt. of sugar ? 19. f of 168 + ? = 2496. 8J + lOf - 1J = ? 20. 9854 x 937 = ? 298 T V - 39 T \ = ? 21. How many quarter-acres in 96 acres ? 22. $748.30 was spent for coal at $5.75 a ton. How many tons were purchased ? REVIEW. 75 Written Work. 1. In 6 rd. 2 yd. 2 ft. how many feet ? 2. How many yards, etc., in 239 inches ? 3. Eeduce 330 ft. to rods. 4. How many inches in 32 rd. 14 feet ? 5. Reduce 24960 in. to rods, etc. 6. In 4976 fur. how many miles, etc. ? 7. Change 6| mi. to feet. 8. How many feet in 230 rods ? 9. In 4 mi. 3 rd. 1 yd. 2 ft. how many inches ? 10. Eeduce 54950 in. to furlongs, etc. 11. Reduce 9860 in. to rods, etc. 12. How many yards around a school-lot that is 12 rd. square ? (Illustrate.) 13. In 176 yd. how many rods ? 14. J of a mile from your house is your school ; how many feet is it ? 15. How many rods in 44 yards ? 16. In 620 rd. how many feet ? 17. 10 yd. of ribbon cost $3.50. What will 18 in. cost ? 18. If a foot of land costs $4.98, how much will 1 rd. cost? 19. What will be the cost of 18 ft. of wire at $.24 a yard? 20. How many feet around a lot 40 by 38 feet ? 21. 1210 -r- 5i = ? 315 ^ 31 = ? 198 -f- 16 = ? 1896 X 29J = ? 22. 316 X 14 x 139 =? 3758930 -r- 127 = ? T \ of 6282 = ? 23. 12987 + 10401 + 1605839 + 93 + 70936 = ? 76 ARITHMETIC. Sight Work. 1. Give two factors of 99. Of 121. Of 63. Of 91. 2. What number added to 8 X 4 will make 60 ? 3. Minuend 94, remainder 39 ; find the subtrahend. 4. 3 yd. of ribbon cost 72 ct. How much will one-half yard cost ? 5. 7 and 14 are factors of what number ? 6. Product 84, multiplier 7 ; find the other factor. 7. Jenny earned $.50 on Monday and 30 ct. on each of the following days in that week. She had left $1.25 ; how much did she spend ? 8. In $9.50 how many mills ? 9. From 36 subtract of itself. 10. 8x15-4x10 = ? 20x3 + 4x6=? 11. How many times must I take 3 to make 7 ? 12. 9 + 8 + 20 + 17-19 = ? 13. Give two equal factors of 49. 14. How many times of 16 is 9 X 8 ? 15. James, William, and John hire a boat for $3.20. James pays 75 ct., William 85 ct. How much does John pay? 16. What will 6 bu. of potatoes cost at 20 ct. a peck ? 17. 36 - (6 + 3) = ? (8 + 6x3) -5-7=? (30 + 40) -7X2=? 18. 4 qt. of oil can be bought for 32 ct. How many quarts can I buy for $1.60 ? 19. How many quarts, etc. in 39 pecks ? 20. 2 + 3J - H = ? ^f 10 - = ? 241 f = ? 21. 126 X 7 = ? 980 X 11 = ? 1908 -r- 12 = ? 22. 139-18^ = ? 27 -6|=? ^of!56=? DRILL EXERCISES. 77 Rapid Mental Work. Add: 1. 4 + i 5 . 4 + * 12 + A 13 . 4 + w 2. i + 4 6. J + A 10. 1 -f-T 1 ! 14 4 + TV 3. 4 + | 7. 4 + 4 11. J. 4-iV 15 4 + 4 4. 4 + i 8, T*r + lv 12. A -l-i 16 \ + A Subtract : 17. i- | 21, . *-4 25. J - -iV 29 - A IT 18. J- 4 22, *,-* 26. \ - -A 30 4 -TV 19. i - 4 23, *-iV 27. J- - -A 31 4 -TV 20. i i 24, 4-4 28. J - -Vr 32 4 -TV Written Work, 33. 121 X 21 48. 7822 X 11 34. 178 X 51 49. 8739 X 12 35. 1234 X 31 50. 9876 X 15 36. 1579 X 61 51. 2347 X 13 37. 2345 X 41 52. 7823 X 16 38. 298 X 71 53. 1578 X 14 39. 304 X 91 54. 6352 X 17 40. 1235 X 101 55. 2796 X 19 41. 2479 X 121 56. 3498 X 18 42. 1783 X 111 57. 8793 X 102 43. 243 X 81 58. 25796 X 103 44. 225 X 131 59. 25879 X 107 45. 1256 X 141 60. 12576 X 104 46. 897 X 151 61. 283546 X 109 47. 976 X 161 62. 85063 X 110 63. 8793 + 7632 + 98764 + 79321 . 64. 3579 + 25680 + 543872 + 895470. 78 ARITHMETIC. Written Work. 1. In 4 wk. 1 da. 2 h. how many minutes ? 2. Change 305840 sec. to years, etc. 3. If a clock loses 4 sec. an hour, how much time will it lose in 3 days ? 4. How much time will a person waste in a year, if he wastes 30 min. every day ? 5. If I cut 30 oranges into sixths, how many pieces have I ? 6. If 10 yd. of cloth cost $35, what will 3 yards cost ? 7. If of a ton of coal costs $2.00, what will 5 tons cost ? 8. A man bought a carriage for $54 ; spent $20 for repairs, then sold it so as to gain $19.00. What did he receive ? 9. Divide the product of 12, 23, 14, and 13 by the difference between 160 and 39. 10. A tree 68 ft. high was broken off by the wind ; the broken part was 49 ft. How high was the remaining portion ? 11. How many gallons, etc. in 2495 pints ? 12. In 470 qt. how many bushels, etc. ? 13. How many feet in 324 rods ? 14. If you go to bed at half-past seven, and rise at six o'clock, how many hours do you sleep ? 15. Bought a bushel of nuts at $2 and sold them at 5 ct. a pint. What did I gain ? 16. How many feet in 18 rods ? 17. In 840 da. how many weeks ? 18. What will 12 fans cost if seven cost $27.86 ? INDICATED WORK. 79 * INDICATED WORK. Cancellation. Indicate the division of : 1. 6 apples by 3 apples. 2. 6 times 1 apple by 3 times 1 apple. 3. 8 twos by 4 twos. 4. 8 times 2 by 4 times 2. 5. Twenty-four 7's by six 7's. 6. 24 times 7 by 6 times 7. 7. 18 x 6 by 9 x 6. 8. 40 X 11 by 8 X 11. When the same factor appears in both dividend and divisor, the quotient will be the same if that factor is cut out, or cancelled, from both. 9. How many times is 6 X 5 contained in 17 X 5 ? SOLUTION. First indicate the division, thus: 17 X 5 As the factor 5 appears in both dividend and 6x5 divisor, it should be cut out, or cancelled, thus : 10. Prove this work by the long method. 11. Change examples 2, 4, 6, and 8 to read like example 9, and find results by cancellation. 12. Divide 3x5x7x2 by 3 X 5 X 11. What factors appear in both dividend and divisor ? After these are cancelled, what is left in the dividend ? In the divisor ? 13. Divide the product of 5 times 7 times 17 by the product of 11 times 5 times 7. 14. Divide 6 x 11 X 16 by 6 X 11 X 13. 15. Divide 46 X 58 X 90 by 46 x 58 X 90. 16. Divide 24 X 36 X 17 by 24 x 36 X 13. 80 ARITHMETIC. Written Work. 1. At the rate of $.36 a dozen, what will 11 lemons cost? 2. How many years in 19 centuries 3. At J of a dollar a pound, what costs a box of tea weighing 35 pounds ? 4. How many inches in 24 yd. ? 5. 960345 -r- 296 = ? 649638 X 498 - 38731 = ? 6. 3406 X 107 -*- 368 = ? 9085763 ^ 193 = ? 7. A man has a lot 24 rd. long and 20 rd. wide. How many rods around of it, if divided by a fence parallel to the long sides ? How many rods around of it, when divided by a fence parallel with the short sides ? NOTE. Pupils should draw diagrams before solving. 8. Bought 6 bu. of corn for $4.20 ; sold 5 bu. for $4.20. How much did I gain on what I sold ? 9. A merchant, by selling tea at $.50 a pound, loses $.225. What did it cost him ? 10. A building worth $25675 is insured for J of its value. For what amount is it insured ? 11. A wheel 3 ft. 6 in. in circumference will turn how many times in going a mile ? 12. Multiply together the even numbers between 3 and 41. The prime numbers. 13. Find the product of 12547 and 546. Express the product in words. 14. An acre of land produces 25 bu. of oats. How many quarts will 18 acres produce ? 15. $6903.07 + $9564.305 + $12018 + $96138.42 + $.03 = ? 16. (24J + 18*) - (19J + 14*) = ? 68 - 15 = ? MISCELLANEOUS. 81 Written Work. 1. Write a number of three periods. 2. Divide the number in Ex. 1 by 79. 3. 1246 -f 329073 + 907 + 3647809 + 7069875 = ? 4. | of 960 - ft of 288 = ? 96542 x 937 = ? 5. At $.05 a half-pint what will a barrel of oil cost ? 6. Product $24875, multiplier 95 ; find the multipli- cand. 7. Divide the sum of 194703 and 191893 by their difference. 8. 695430 -f- 127 = ? 1073825 -f- 201 = ? 372984 -^ 37 = ? 9. 124J + 29| - 37J = ? 16J - 4J + 391* = ? 10. How many quarter-dollars in $50 ? 11. Paid $16 for coal, $12 J for wood ; how much change did I receive from a fifty-dollar bill ? 12. Take 29 X 942 from 65 X 875. 24893 X 297 = ? 13. Subtrahend 98560, remainder 340875 ; find the min- uend. 14. The sum of six numbers is 9,580,765; five of the numbers are 2,480; 396,407; 24,999; 38,950; and 37. What is the sixth? 15. $26,403 - $9,544.307 = ? / 5 of 32,240 x 79 = ? 16. How many yards of border shall I need for a room 16 ft. long and 14 ft. wide ? 17. How much will the above border cost at $.35 a yard ? 18. What is the greatest common divisor of 35 and 63 ? 19. At 75 ct. a bushel, how many bushels of apples will pay for 8 chairs valued at $1.25 each ? (Indicate.) 20. 780932 -T- 113 = ? 1532467 -5- 76 = ? 1038916 -*- 103 = ? 82 ARITHMETIC. Sight Work. 1. 10 x 10 35 -5- 7 = ? 6x4 + 50-f-5 = ? 8 X 9 X 2 - 50 = ? 2. How many half-pints in 4 bushels ? 3. What will 1 gal. of oysters cost at $.20 a pint ? 4. At 88 ct. a bushel, what will -J- pk. of peas cost ? 6. How many days in this and next month ? 6. A boy sells 8 qt. of peanuts in a day. How many bushels will he sell in 12 days ? 7. How many feet in 156 inches ? 8. In $25 how many 5-cent pieces ? 9. 5 men can build a cellar wall in 16 da. How long will it take 10 men ? 10. If 2 apples cost 3 ct., how many can you buy for 30 cents ? 11. What will 8 bananas cost if 2 dozen cost 48 cents ? 12. What is my laundry-bill if I have 18 pieces at $.60 a dozen ? 13. Give four factors of 120. 14. $5.60 x 10 -*- 40 = ? j + ? = f f . 4* - ? = 1J. 15. 72 gal. of milk will fill how many quart-cans ? 16. If you have fifteen minutes recess daily, how many hours will you have in 2 weeks ? 17. Change 5 dimes, 10 half-dimes, and J of a dollar to cents. 18. Find two factors of 121. Of 143. Of 49. 19. W T hat is the sum of the numbers in the above example ? 20. Dividend 14 X 9, divisor 7 ; what is the quotient ? 21. Find the greatest common divisor of 21 and 28. 22. 6 X 3 - 72 -f- 12 = ? 18 + 40 + 39 - 14 = ? INDICATED WORK, 83 INDICATED WORK. Cancellation. 1. Divide 8 X 18 x 15 x 7 by 4 x 6 x 11 X 9. 225 Since 4 and 6 are factors of 8 and $ X Iff X Iff X 7 13 respectively, they may be cancelled X X 11 X from both dividend and divisor, leaving * the factors 2 and 3 in the dividend. Cancelling the factor 3 from 9 in the divisor and 15 in the dividend, leaves 3 in the divisor and 5 in the dividend. Cancel the factor 3 again from both dividend and divisor. The product of the uncancelled factors in the dividend is 70, and in the divisor is 11. 70 -r- 11 = 6 T \. 2. Divide 36 X 27 X 49 X 38 X 50 by 70 x 18 X 15. First find result by long method, then by cancellation, and see how much time is gained by cancelling. 3. Divide the product of 28 X 38 X 48 by the product of 14 x 19 X 24 x 2 x2. 4. (26 X 5 x 54) -5- (13 X 5 X 6) = ? 5. What is the quotient of 36 X 48 X 16 divided by 27 X 24 x 8 ? Indicate and find result by cancellation. 6. How many bushels of apples at 50/ a bushel must be given for 15 Ib. of tea at 40/ a pound ? 7. If 5 horses cost $600, what will 8 horses cost ? 8. How many yards of cloth can be bought for $40, when 60 yd. cost $120 ? Add from right to left : 9. 3694 + 8976 + 38421 + 25982 = ? Multiply without copying : 10. 5894 X 9 = ? 68463 X 7. 3862 x 8. Divide by short division : 11. 3694 by 6. 7852 by 10. 6985 by 9. 84 ARITHMETIC. Add: 1. 450135 + 69089 = 680506 + 35025 = 66077 + 3498 = 940461 + 38690 = 76274 + 85176 = Subtract : 3. 133- 75 4. 140- 84 6. 122- 73 6. 112- 46 7. 740-280 8. 500-219 Multiply : 21. 149 X 23 22. 147 X 122 23. 146 X 147 24. 134 X 125 25. 147 X 431 26. 143 X 332 Divide : 39. 1392 -i-24 40. 2084 -r- 26 41. 1864-^29 42. 1272 -f- 27 43. 5176 -T- 19 2. 85785 + 643066 48740 + 678459 5624 + 575535 56025 + 390154 315270 + 421280 9. 810-560 10. 121 - 65 11. 146- 61 12. 134- 78 13. 269-134 14. 620-370 27. 238 X 235 28. 249 X 225 29. 248 X 238 30. 225 X 237 31. 235 X 246 32. 227 X 235 44. 21515 -f- 97 45. 10414-^84 46. 9170 -T- 75 47. 6101 -^ 51 48. 18017^36 Perform the indicated operations : 54. + 57. 21J ' 55. | + | 58. 19J 56. J + | 59. 14| + 15. 376 - 16. 437 - 17. 658 - 18. 1124- 19. 1112 - 20. 1237- 33. 1827 X 34. 7520 x 35. 8984 X 36. 6575 x 37. 3825 X 38. 2765 X 49. 50. 51. 43379 75405 210562 52. 1695832 53. 2601879 - 16| + 293 247 312 189 156 169 309 207 329 535 491 513 -f- 123 -f- 345 -^ 624 -^-3024 -^4068 MISCELLANEOUS. 85 1. Add twenty-five million, five hundred seventy-eight thousand, three hundred seven ; forty thousand, nine ; two hundred seventy-six; one million, two thousand, and one. 2. The product is $713184, the multiplicand $2346. What is the other term ? 3. 234 x 146 - 197 X 95 = ? 897038 -f- 123 = ? 4. How is the dividend obtained when the quotient, divisor, and remainder are given ? (Illustrate.) 5. Dividend 36 X 914 ; divisor 6 X 56. Find the quotient. 6. Reduce to other forms y^s ; *?^ ; 248f; Jf ; afji. 7. 18J + 24ft - ? = 24 & 16031 x 907 = ? . 8. 197 and 128 are the factors of what number ? 9. Multiply 987 by itself. 10. $78563 - $1735.841 = how much ? 11. A purse worth % $3 contains the largest 9 U. S. coins. What are purse and contents worth ? 12. What number added to 17 x 39 will make 408 X 706? 13. Divide 34768 by 16, and the quotient by 13. 14. Bought 50 Ib. sugar at $.05 a pound, giving in return 6 doz. eggs at $.40 a dozen, and the remainder in cash ? How much was the cash ? 15. Write two numbers, one odd and the other even, that are divisible by 99. , Add: 16. 51463 17. 21784 18. 245678 172903 158304 31235 278631 197428 8765 3895746 160895 432 6387595 598606 9872 86 ARITHMETIC. 1. How many days in the winter months ? 2. 6 X 998 -r- 39 = ? 648751032 + 297 = ? 3. In 735270 in., how many miles, etc. ? 4. A lady uses 5 yd. of ribbon for rosettes. If she uses 1 8 in. for each rosette, how many will she have ? 5. 1613 to 27ths. 129J + 38ft + 194 = ? 6. 248 fr to 66ths. J j-J to 20ths. 7 . 9437 + 207839 + 51697 + 666098 + 9017832 + 60012 = ? 8. 67048164 -5- ? = 9876. 24 x ? = 95160. 9. 5148 is a multiple of two numbers. 66 is one, what is the other ? 10. How many miles in 1943040 feet ? 11. 1600060 -5- (11 x 2 x 7) = ? 6816600 -g- (3 x 7 X 5) = ? 12. 7150990 -f- 77 = ? 2 x 4595200 -4- ? = 71800. 13. The school lot is 12 rd. front ; how many feet is it ? 14. How many half-pints in 1^ barrels ? 15. What is the total weight of 6 bales of hay weighing 8| cwt., 5ft cwt., 4 cwt., 7J cwt., 6J cwt., and 5 cwt. ? 16. What is the difference between ^ of 880 and J of 9680? 17. Quotient 560, dividend 218400. Find the divisor. 18. At $.06 a pint, how much do I receive for 8 bu. 6 quarts of berries ? 19. What will 13 Ib. of spice cost at 2 ct. a half- ounce ? 20. 5} + 9f 8| = ? 23. 17J + ? = 21. 89 J - 78f + 39 = ? 24. ? + 29} = 22. 18 T \ + 39 - 10J = ? 25. 249 T ^ - 12 = ? MISCELLANEOUS. 87 Denominate Numbers. Add: 1. 3 mi. 7 fur. 5 rd. 5. 3 gal. 2 qt. 1 pt. 2 gi. 5 mi. 1 fur. 6 rd. 7 gal. 5 qt. 2 pt. 3 gi. 2. 3 yd. 2 ft. 5 in. 6. 3 da. 12 hr. 45 min. 1 yd. 1 ft. 10 in. 9 da. 16 hr. 40 min. 3. 14 Ib. 7 oz. 7. 3 rd. 1 yd. 1 ft. 6 in. 7 Ib. 10 oz. 5 rd. 1 yd. 2 ft. 8 in. 45 Ib. 11 oz. 17 rd. 2 yd. 1 ft. 7 in. 4. 3 pk. 5 qt. 1 pt. 8. 7 cwt. 30 Ib. 8 oz. 1 pk. 7 qt. 1 pt. 9 cwt. 50 Ib. 9 oz. 5 pk. 4 qt. 3 cwt. 40 Ib. 10 oz. Subtract : 1. 5 wk. 6 da. 19 hr. 6. 125 cwt. 8 qr. 5 Ib. 2 wk. 3 da. 17 hr. 85 cwt. 9 qr. 3 Ib. 2. 29 gal. 3 qt. 1 pt. 7. 7 mi. 30 rd. 9 yd. 14 gal. 2 qt. 1 pt. 4 mi. 17 rd. 8 yd. 3. 9 bu. 6 pk. 7 qt. 8. 35 Ib. 10 oz. 5 dr. 5 bu. 7 pk. 3 qt. IT Ib. 12 oz. 8 dr. 4. 7 T. 3 cwt. 23 Ib. 9. 25 bu. 3 pk. 3 T. 5 cwt. 16 Ib. 15 bu. 5 pk. 1 pt. 5. 15 yd. 2 ft. 7 in. 10. 8 bbl. 31 gal. 2 qt. 5 yd. 6 ft. 2 in. 7 bbl. 9 gal. 3 qt. 11. What is a denomination ? 12. What is a denominate number ? 88 ARITHMETIC. Sight Work. Reduce to whole or mixed numbers : i- > > 1 F > a i*> f J, W> W W> ^F- 2. gallon equals how many pints ? 3. How many inches in f t. + J ft. + } foot ? 4- A = i, = A, = T V * + 3J + 4 = ? 5. I bought J of a yard of ribbon for 1 hat, and f for another. How many inches of ribbon did I buy ? 6. Sold of a pound of coffee to one customer, and j of a pound to another. How much was sold ? 7. What part of an hour is 50 minutes ? 8. 121 is a multiple of what two numbers ? 9. Change $24.38 to cents. 10. In 25 dimes and 4 mills, how many mills ? 11. $8.75 equals how many 5-cent pieces ? 12. In 4 bu. how many pints ? 13. Sold 8 bu. of nuts at 2 ct. a pint. How much money did I receive ? 14. At 6 ct. a pint, how many gallons of molasses can be bought for $3.84 ? 15. What will 8 erasers cost at $1.56 a dozen ? 16. If one clothespin cost % cent, what will 3 doz. cost ? 17. 8 X 9 - 2 X 6 =? 295 X 7 =? 9588 -- 12 =? 18. What will one-half bushel of sweet potatoes cost at $.37 a peck ? 19. At 15 ct. a quart, what will 12 qt. of strawberries cost? 20. How many times must I take 4 to make 18 ? 21. 6J + 5 = ? 19| X 12 = ? 30J - 10 = ? 22. 849 + 3 + 7 + 5 + 10 + 12 + 7 + 4 + 11 = ? MISCELLANEOUS. 89 Written Work. 1. Mary had $ 39J ; she spent $27f . How much was left? 2. Paid $190f on a bill, and I still owe $105f . What was the entire bill ? 3. Eeduce 24 chal. 2 bu. 2 pk. 7f qt. to pints. 4. In 528, how many ninths ? 5. Express in figures MDCCCLXXXIY. 6. 1235407 - (9456 + 8578) X 79 = ? 7. How many weights, weighing a quarter of a pound each, can be made from 128 Ib. of iron ? 8. 7056 pk. will fill how many barrels, each barrel holding 3 bushels ? 9. How many 3-inch pieces can be cut from 48 ft. of wire ? 10. Write 4956 in Roman notation. 11. Express in its simplest form the quotient of 1340 -j- 2660. 12. J of a hogshead of oil was sold. What did the remainder bring at 7 ct. a quart ? 13. Of what number are 5, 7, 3, and 9 factors ? 14. Eeduce 99 mi. to feet. 15. Express in other forms 17} ; *f jj*; T g|jj ; 293^. 16. 125 x 5 - ? = 100. 295873 -5- 73 = ? 17. How many quart-boxes will hold 79 bu. 1 pk. 3 qt. of grapes ? 18. How many times can you fill a bushel measure from 15872 pints of beans ? 19. 36} + 19J - 27^ = ? 20^ + 29 - 16J = ? 20. 107^ + 16? - 29 = ? 92 - 16| + 3| = ? 21. A half-pint bottle can be filled how many times from 724 gal. 2 quarts ? 90 ARITHMETIC. Add: 1. 71 2. 26^ 3. 132$ 171J 35J 607J 37| 59^ irif i 14 182 4. 119g 6. 195 6. 124J 108| 47J 156J QO 1 *Ji 91ii 132| 307 198 39| 7. 8. j + 43g + 72| 9. 66$ + 5J + 89| + 37| 10. 71J + 32| + 83J + 11. 94} + 19J + 67| + 12. 96/j + 42^ + 71 + 66 J 13. 49J + 23i + 81J + 16^ 14. 8g + 16i + 34| + 301 J 15. 14} + 19$ + 115J + 334| 16. 93J + 27^ + 103J + 1215 17. 905| + 814 + 73| + 39| 18. 2307| + 121 % + 114* + 79 19. 13J + 14/j + 15| + 16J 20. 19i + 18ft + 17} + 18| 21. 118J + 147J + 132| + 60ft 22. 90J + 701} + | + 27 23. 10-^ 24. 113} + 99 + 26 + 25. 460 + 909ft + 54f + 28}. MISCELLANEOUS. 91 Written Work. 1. A thermometer showed at noon 74^ degrees above zero. At 6 o'clock P.M. it showed 66| degrees. How much had the mercury fallen ? 2. A grocer buys 12 bu. of sweet potatoes for $14. He sells them at 50 ct. a peck. Does he gain or lose, and how much ? 3. Skates bought for $3 were sold for $2. How much was lost ? 4. Gave of $1500 to my brother and ^ to my sister. How much have I left ? 5. 8| + 11? - 6f = ? 94J - 30f + 19i = ? 6. #16| + $90^ + $120 + $345 + $98 = ? 7. $7012.06 + $8072.02 + $7865 = ? 8. A man carries 3 tons of coal in a basket holding 50 Ib. How many times does he fill the basket ? 9. From a barrel of oil 7J gal. leaked. How many quarts remained ? 10. If 2 bu. of nuts cost $6.40, what costs a half-pint ? 11. } of 1099 - 191 = ? ^ of 2232 x 903 = ? 12. How many hours in one half-day ? 13. In the winter season how many days ? 14. Reduce 24 rd. 3| ft. to inches. 15. Bought 175^ bu. of potatoes at one time, 97 bu. at another, then sold 89| bushels. How many bushels were left? 16. Tom steps 2 ft. at each step. How many rods will he travel in walking 99 steps ? 17. How many half-yard pieces can be cut from 114 ft of ribbon ? 18. 27 + 39 - 13J = ? 640-301^ + 69! = ? 92 ARITHMETIC. Written Work. 1. William J. Fitzgerald made $75 J on Monday; on Tuesday $17 more than on Monday ; on Wednesday as much as on Monday and Tuesday together. How much did he make in all ? 2. 4832 X 673 + ft of 3175 = ? 3. 587654 + 345 = ? 1956 X 49 + 36 = ? 4. A merchant buys 30^ Ib. of tea at one time, 25% Ib. at another, and 39J Ib. at another. He sells 17J Ib. to one customer, and 25J Ib. to another. What is left ? 5. A man earns $1500 a year. How much does he earn a day, if he works 313 days ? 6. What cost 4400 Ib. of coal at $4.60 a ton ? 7. A house cost $7800, a barn ft as much. What did both cost ? 8. A plot of grass is 3 rd. square. How many feet around it ? 9. 747 + 138J + 47J + + 130^ + 40 = ? 10. Pineapples are worth $.25 apiece. What will 8^ dozen cost ? 11. How long is the fence that extends around a lot 14 ^ rods long and 165 ft. wide ? 12. What number must be multiplied by 2463 to make 5666369 ? 13. 11967 X 290 are factors of what multiple ? 14. Quotient 4308, divisor 28701. Find the dividend. 15. If I spend $| for candy and $ T 3 ?J for fruit, what part of a dollar do I spend ? 16. If I take 5280 from the sum of 3782 and 7848, divide the remainder by 50, and multiply the quotient by 19, what is the product ? MISCELLANEOUS. 93 1. I pay $6 for the use of $100 for one year. How much do I pay for the use of $300 ? 2. One acre of corn yields 80 bu., another 2| times as much, a third 195 T 5 T bu. How many bushels do the three acres yield ? 3. At the rate of 9 yd. for $4.50, how many yards can be bought for $120 ? (Indicate.) 4. 2480 - 987J = ? 148J - 96| = ? 5. 297i - 39f = ? 285 + 95f = ? 6. How much change will you receive from a 10-dollar bill, if you buy 2 Ib. of tea at $.50, 18 Ib. of sugar at 5J ct., and 5 Ib. of beef at 18 cents ? 7. 18 - 13J + 195J + 13J = ? 907 - 138}i = ? 8. 2468^ - 348^ = ? 248^ - 83 = ? 9. 1694 x 395 - 2956 = ? 2958346 -=- 297 = ? 10. 3J yd. of ribbon are required for one hat, and 12| yd. for a dress. How much will it cost at $J a yard ? 11. John read of an hour, and was idle of an hour. How many minutes was he idle ? 12. What will 3 melons cost, if of a melon costs 15 cents ? 13. Add 105, 294, 135$, and 2997|. 14. Subtract 899J from 2349f . 15. What is gained on 250 Ib. of sugar, bought at 4 cents, and sold at 5J cents ? 16. If you study 5^ hours a day in school, and Ij hours at home, how many hours will you study in 40 weeks ? 17. What part of a dollar is 60 cents ? 18. In 8 J yards, how many inches ? 19. How many feet around a lot 4 rd. square ? In division, when is the quotient 1 ? 94 ARITHMETIC. Sight Work. 1. Reduce to whole or mixed numbers : II, , W, a l ft , II, W, W, W, -4*. 2. J gallon equals how many quarts ? 3. How many inches in ft. + ft. + J ft. ? 4. What will 2 bu. of onions cost, if a peck costs $.25 ? 5. Paid 10 ct. for a paper of pins, 8 ct. for some braid, and $.35 for a handkerchief. How much change do I receive from one dollar ? Give answers: . 6. of 45. 9. ft of 121. 12. }J of 600. 7. J of 60. 10. ^ of 108. 13. }f of 136. 8. | of 96. 11. Jg of 300. 14. |J of 200- 15. 9 X 12. 18. 15 X 10$. 21. 29J X 11. 16. 11 X 9J. 19. 19 X 6. 22. 35 X 8. 17. 13 X 8|. 20. 30 X 18. 23. 127 X 6. 24. How many pints in 33 gallons ? 25. 2930 - 128 = ? 6451 - 2314 = ? 9900 - 3647 = ? 26. 4J + 7J - 6| = ? l}f + 9J = ? 29J - 11 = ? 27. 108 is a multiple of what two numbers ? 28. From JJ take J. Take J from l. 29. 50 X 15 = ? 34. 600 ^ ? = 40 30. 28 X ? = 196 35. 11 X 330 = ? 31. 18 X ? = 198 36. 9 X 219 = ? 32. ? X 18 = 216 37. 8 X 304 = ? 33. ? X 27 = 297 38. Julia will be 13 years old on Aug. 2d next. In what year was she born ? 39. What is the cost of 72 eggs at 22 ct. per dozen ? MISCELLANEOUS. 95 Add: 1. 6f, 5J, 2f 4. 3, 4J, 8|, 4|. 2. 4f, 2i, 8^. 5. 8f, 2*, 8, 121&. 3. f , |, J, J. 6. 4J, 20, 64, 71*. Subtract : 7. 64JJ - 57. 10. 100J - 70|J. 8. 18JJ-7|. 11. 67A-58f- 9. 90j| 39f 12. 120J 39 T 7 T . 13. 9801 is a multiple of what two numbers ? 14. What is the smallest number that can be divided by 4, 6, 8, and 12 ? 15. From a piece of cloth containing 22 yd., 5| yd. were sold at one time, and 7 j yd. at another. How many yards were left ? 16. How much heavier is a cheese weighing 80| Ib. than one which weighs 29|J Ib ? 17. 136 18. 324 19. 603 20. 958 X 39f X 27 X 124 X 784 21. Find answers : -2479682 22. -9637821 23. -1503765 20083410 70526934 7420524 24. How many hours in two weeks ? 25. $2155.73 + $429.93 + $6873.68 + $793.94 + $6347.20 = ? 26. How many rods in 1004 yards ? 27. In 1584 ft. how many rods ? 28. Eeduce 2646 gal. to barrels. 29. Eeduce 539 gal. to gills. 30. How much does 97182 exceed 17324? 31. 250 -f- 12 = ? 32. 945 -f- 31. 33. 770 -5- 5 = ? 96 ARITHMETIC. Bead and add : 1. $1.70 2. $205.003 3. $2357.12 4.18 75.08 4378.355 .005 123.80 1238.638 5.673 785.155 1049.287 4. $14516.25 6. $3120.05 6. $7049.01 3847.00 18.92 419.87 356.73 97148.42 1007.89 3578.06 14.92 9890.04 75873.97 2793.78 874.84 8003.89 401.79 6914.79 4478.68 91.90 479.14 2.85 77891.64 8.54 93.99 783.43 319.99 536.72 58.19 64.32 .85 901.90 2.69 23.03 372.24 .75 7. $90774.81 8. $438.08 9. $75256.35 874.94 54.56 207.08 7098.41 9389.85 94.35 484.01 946.87 745.38 30.19 7.83 6975.43 78999.02 91.65 98374.58 4938.57 348.64 8783.94 26.78 9383.53 843.54 7.43 13.62 45.84 3825.96 98.64 935.78 346.29 2834.25 33.49 2863.46 536.16 215.17 3984.05 75.12 15.18 32.53 826.19 226.23 18.96 25.22 86.98 REVIEW. I J . ..- ^ 97 Add: 1. 8764 2. 18491 3. 98534 4. 983 2940 5864 36720 7354 381 6798 8406 34797 64169 48 42398 9048 495832 57629 4675 68576 1036749 29513 ' 75281 473865 5. 2019 + 8521 + 204 x 29 = ? (Multiply first.) 6. Express in other forms ^ 7 5 , 295|J, *f a, g|. 7. If 8 persons use a barrel of flour in 27 days, how long will it last 4 people ? (Indicate.) 8. How many dollars must I spend daily to use an in- come of $800 a year ? 9. 1534768 -f- 216 = ? of 2520 x 91 = ? 10. Sold a cargo of salt for $7497, thereby gaining $629. What did the salt cost ? 11. How many bushels of oats at $.40 must be given for 1600 bu. of wheat at $.75 a bushel ? (Indicate.) 12. The sum of two numbers is 247, the smaller number being 216. What is the product of the two numbers ? 13. Quotient 137, divisor 293, find the dividend. 14. What number added to 93,248 makes 978,365 ? 15. What number has for its factors 4, 9, 6, 7, 20 ? 16. If I give $141 for a piece of cloth containing 47 yd., for how much must I sell it to gain $2 on a yard ? 17. What number must I multiply by 135 that the pro- duct may be 505710 ? 18. How many days, etc., in 6954 minutes ? 19. 184J + 295* + 24 + 348 + 9634 = ? 20. 629J + 3068 + 247f + 96 + 2963 = ? 21. What is an integer ? 98 ARITHMETIC. 1. 274 + 8946 + 389014 + 376 + 39845 + 3756340 = ? 2. At 85 ct. a pound, what cost 89 Ib. of tea ? 3. What will 24 doz. of eggs cost at 3 ct. each ? 4. Dividend 972800, quotient 208 ; find the divisor. 5. 8, 9, 12, and 7 are factors of what multiple ? 6. Working 8 hours a day, how much will a man earn during this month at 15 ct. an hour ? 7. 3465 -f- (36 + 12) = ? 9871 X 119 X = ? 8. 529 X + 1554239 -f- 216 = ? 9. Bought 325 loads of barley, each load containing 50 bu., at 98 ct. a bushel. What did the barley cost ? 10. How many fourths of a pound in 396 pounds ? 11. If 25 acres of land cost $2450, what will acre cost ? 12. What is the value of 12 bales of cotton, containing 425 Ib. each, at 56 ct. a pound ? 13. One paper-cutter costs 37 ct. ; what will 12 dozen cost? 14. How many steps of 2 ft. each will a man take in going 3^ miles ? 15. A man sells 2| gross, 5j gross, 14JJ gross, and 10^ gross of lead pencils at 36 ct. a dozen. How much does he receive for all ? 16. 28 yd., 6^ yd., 40| yd., and 39 yd. of silk were sold. How many yards were sold ? 17. To the sum of 19} and 34f add their difference. 18. If 2 qt. of oil cost 10 ct., what will 2 gal. 2 qt. cost ? Divide : 19. 3869 -v- 18 22. 289352 -f- 9G 20. 2836 -r- 27 23. 368421 ~ 125 21. 3953 -+ 82 24. 28932 -^ 77 REVIEW. 99 Subtract : 1. 17} 2. 25^ 3. 37f -18f 4. 54| -15} 5. 113 T 7 F 9. 63| 6. 130$ 7. 59 -49} 11. 580 -98^ 8. 575 - 1931 10. 76} -Mi 12. 73f 13. 100} -33^ 14. 473 -357 15. 1000 -_899| 16. 1370 -349^ 17. 576^ -176| 18. 888 -239} 19. 1532 20. 131^ 21. 295$ -186| 22. 1038} -192j 23. 123J -108} 24. 234J -127} 25. 221} -109$ 29. 200f\> -176| 26. 238J -129J 30. 325} -105J 27. 233J -107} 31. 493tV 28. 277^ -159J 32. 601A -198|_ 33. 322 1 -1751 34. 347J -98f 35. 193^ 36. 246J -198f 37. 900} 38. 1006J- -290} 39. 2901f -190J 40. 2063 -249} 7. 36 12. 27 17. 54 8. 18 13. 40 18. 64 9. 44 14. 50 19. 72 10. 45 15. 56 20. 84 11. 48 16. 63 21. 96 100 ARITHMETIC. Multiply by 25 : 1. 16 ; 12 ; 24 ; 36 ; 28 ; 72 ; 48 ; 96. Mental Work. Give three factors of : 2. 12 3. 16 4. 20 5. 24 6. 32 Written. Give least common multiple of : 22. 2, 3, and 4. 25. 5, 3, and 6. 23. 3, 2, and 5. 26. 4, 8, and 12. 24. 2, 4, and 6. 27. 5, 10, and 16. Reduce to whole or mixed numbers : 28. V- 31. V- 34 - ft 37. 29. ^a 32. y 35. ^ 38. 30. V 33. V 36. iffA 39. Mental. Reduce to improper fractions : 40. 7 44. 7^ 48. 12^ 52. 41. 4f 45. 8 T 9 T 49. 13$ 53. 18f 42. 5$ 46. 9| 50. 17f 54. 124J 43. 6| 47. 12 T V 51. 9|f 55. 303J Divide without copying : 56. 120 -:- 30 60. 720 -f- 90 64. 840 -r- 120 57. 360 -j- 40 61. 960 -h 80 65. 960 ^ 240 58. 480-=- 60 62. 320 --40 66. 720-^360 59. 640 -T- 80 63. 1440 -f- 120 67. 5200 ^- 130 REVIEW. 101 Written Work. 1. A man buys 5 chairs at $9.75 each; he sells them at a profit of 50 ct. each. What does he get for the chairs ? 2. I buy some lace at $3.05 a yard. What will 18 yd. cost? 3. $12987.032 - $3498.107 = ? 39870 x 273 = ? 4. A barrel of flour weighs 196 Ib. What will 2 bbl. cost at 3 ct. a pound ? 5. $892.06 equal how many mills ? 6. 1387500 cents equal how many dimes ? 7. At 4 ct. a spool, what cost 3126 spools of thread? 8. Reduce 3 T. 15 cwt. 34 Ib. 6 oz. to ounces. 9. At $1.60 for 4 Ib. of cinnamon, what will 1 oz. cost ? Add: 10. 18J, 17f 30f 13. 248^, 26f, 950. 11. 28 T V, 14, 290. 14. 190^, 370, 29J. 12. 304J, 9 T 7 T , 95. 15. 87|, 31J, 40J. 16. 2963 x 27 = ? 19. 3078506 -r- 191 = ? 17. 3070 X 109 = ? 20. 2693705 + 205 = ? 18. 729580 x 308 = ? 21. 97548900 -r- 113 = ? 22. A field is 8 rd. long and 40 feet wide. How many feet around it ? 23. How many steps 30 in. long will a man take in walking 1 miles ? 24. In 132 ft. how many rods ? Inches ? 25. S. T. Smith bought furniture for $98J, $240f, J, $2096, and $306 T V What was his bill? 26. 8 x 25 28. 180 X 25 27. 25 X 6 29. 200 X 25 102 ARITHMETIC. 1. 13f 4- 19 J - 13 = ? 4. 204 - 38} + 16 = ? 2. 121 + 20$ - 30 = ? 5. 95J - 40| +391 = ? 3. 96$ + 13$ + 50J = ? 6. 75 - 31 + 647$ = ? Reduce to lowest terms or to higher terms : 7. Jft 10. |f 13. flfr 16. ^ 19. ft 8. H 11- Si 14 H 17- Jft 20 H* 9. H 12. if 15. f& 18. 21- ft* 22. In 3J hhd. how many quarts ? I 23. How many gills in 15 gal. 1 qt. 2 pints ? 24. Reduce 1010 pints to gallons, etc. 25. How much will a grocer make on 16 bu. of potatoes, if he buys them at 75 ct. a bushel, and sells them at 22 ct. a peck ? 26. In 3 da. 59 min. how many minutes ? 27. How many half -pint glasses of jelly can I fill from 21 gallons ? 28. How many feet long is a fence extending around a field 12 rd. long and 8 rd. wide ? 29. Draw a square ; call each side 13 ft. How many inches half-way around it ? How many yards around the other half ? 30. At 6 ct. a pound, what cost 150 Ib. of sugar ? 31. In 328563 gills, how many hogsheads, etc. ? 32. A dog runs ^ mi. If he leaps 2j feet each time, how many leaps does he make ? 33. | of 960408 - 243 = ? 985073 X 209 = ? 34. J of 86400 = ? 6370012 -s- 194 = ? 35. $6401.37 + $19587.01J + $29563.01^ = ? 36. 38420 -r- 250 = ? 37. 1289626 -5- 352 = ? MISCELLANEOUS. 103 1. 1504 mills equal how many dollars ? 2. How many yards in 2 miles ? 3. At $f a yard, what cost 16 yards of muslin ? 4. What will it cost to dig a ditch Ij miles long, at 75 ct. a rod? 5. How many feet of moulding will be required for a room 16 ft. long and 14 feet wide ? (Illustrate.) 6. John steps 1 \ ft. at each step ; how many steps will he take in walking 66 rods ? 7. George's kite-string was 120 yd. long ; he lost 180 ft. of it. How many inches remained ? 8. At $.06J a spool, what cost 3126 spools of silk ? 9. How many quarter-dollars in $19 ? 10. In 210 half -dimes, how many dollars ? 11. From a board 22 ft. long, a carpenter sawed 3 ft. 5 in. How much remained ? 12. How many inches in \ of 12 yards ? 13. What will it cost to fence a farm J of a mile square, at 95 ct. a foot ? 14. 32840 X 490 = ? 1093457 X 47 = ? 15. 19 + 13} + 27 = ? 18. 270} + 9| + 2^ = ? 16. 31| - 10J + 30^ T = ? 19. 190 - 38^ + 1J = ? 17. 240J - 91| + 987 = ? 20. 19J + 2 + 360 = ? 21. A tree is 4 ft. 10 in. high. How many inches must it grow to measure 73 feet in height ? 22. At 18 ct. each, what will Mrs. Gale pay for 12759 salt-barrels ? 23. If the blackboard is 5 yd. long and 6 ft. 3 in. wide, how many yards around it ? 24. How many feet in 4 rods ? 104 ARITHMETIC. Compound Numbers. 1. From 9 rd. 1 ft. take 2 rd. 2 ft. 2. Find the difference between 5 yd. 2 ft. 4 in. and 16 rd. 3. Take 12 yd. 2 ft. from 15 yd. 1 ft. 4. What taken from 25 gal. 1 qt. leaves 6 gal. 3 qt. ? 5. 50 min. 30 sec. 16 min. 45 sec. = ? 6. Find the sum of 187 T., 116 T. 1314 lb., 147 lb., 53 T. 711 lb., 21 T. 35 lb. 7. Add 93 lb. 15 oz., 19 lb. 6 oz., 28 lb., 32 lb. 22 oz., 6 lb. 9 oz., 21 oz. 8. 2 gal. 2 qt. 1 pt. + 16 gal. 3 qt. 1 pt. + 80 gal. 1 qt. + 25 qt. 1 pt. = ? 9. How many hours, minutes, and seconds in 4 hr. 12 min. 22 sec., 15 hr. 26 min. 29 sec., 45 min. 15 sec., 35 se^. ? Subtract : 1. 35 da. 2 hr. 1 min. 3. 17 gal. 2 pt. 5 da. 5 hr. 5 min. 8 gal. 3 qt. 1 pt. 2. 120 bu. 4. 100 mi. 15 rd. 45 bu. 2 pk. 5 qt. 84 mi. 24 rd. Find results : 1. 18 lb. 14 oz. 12 dr. 3. 18 gal. 3 qt. 1 pt. __ 9 _ 9 37 lb. 3 oz. 14 gal. 1 qt. 1 pt. 2. 15 yd. 3 ft. 6 in. 4. 25 bu. 2 qt. 1 pt. 7 yd. 2 ft. 7 in. + 14 bu. 2 qt. 5. 193 gal. 2 qt. 1 pt. + ? = 230 gal. 3 qt. 6. 2 hhd. 2 gal. 2 gi. - ? = 1 hhd. 1 gi. REVIEW. 105 1. What cost 16 chests of tea, each weighing 52 Ib. at ct. a pound ? 2. A single rail is two rd. long; how many rails will be required to lay a mile of railroad ? 3. From $10,009 take $9.073. 648730 X 207 = ? 4. Sold a horse for $145.27, which is $37.69 more than it cost ; what did it cost me ? 5. How many barrels, etc., in 24680 gills ? 6. John King bought a pair of horses for $480, a har- ness for $80.50, and a carriage for $150 less than he paid for both horses and harness. What was the cost of the carriage ? 7. 6J + 9| + 18J + 120 + 960f = ? 8. A of 990 X f of 560 = ? 2963 - 981^ = ? . 9. What cost a pound of beef, if 894 Ib. cost $80.46 ? 10. A farmer sells 120 bu. of wheat at $1.12 a bushel, for which he receives 27 bbl. of flour. What does the flour cost him a barrel ? (Indicate.) 11. 36 Ib. of tea cost $23.40. What cost 5 pounds ? 12. 63 X ? = 93840. 99 X 33847 = ? 13. Add 120 dollars, 85 dimes, 9 cents, 7 mills, 44 hak- dimes, and one eagle. 14. A man bought horses for $556 a pair ; how many horses can he buy for $20016 ? 15. At $7 a ton, what cost 500 Ib. of hay ? 16. A house cost $11622, which is 13 times as much as the lot cost. What did the lot cost ? 17. Find 4 factors of 1270, 1960, 340, 230. 18. In 2 mi. 5 rd., how many inches ? 19. Find the sum, difference, and product of 294 and 79. 106 ARITHMETIC. GENERAL REVIEW PROBLEMS. Sight Work. 1. 3| + 44 + 8 = ? 24 + 6J - 1J = ? 2. Take 3J + 9 + 3 from 44. 3. How many pounds of hay in 7 tons ? 4. How much fringe will be needed to go around a rug 9J ft. by 3 ft. ? 5. What will the fringe cost at 20 ct. a yard ? 6. Nutmeg is worth $.04^ an ounce. What will j of a pound cost ? 7. From f of 63 take of 42. 8. Divide 3 by 14. 11 by 21. 16 by 30. 9. A boy picked 2 bu. of cherries ; he sold J of them at 5 ot. a quart. What did he receive for them ? 10. William owes $42f for his pony. He saves $2^ a week. If he pays what he saves in a month, what does he still owe for the pony ? 11. Change to simplest form J, ^ T , $$, f f, ^fr. 12. -i-2 = ? 13. 33 -s- 5J = ? 22 -*- 3J = ? 21 -^ 4J = ? 14. What is a fraction ? Factor ? Product ? 15. 168 is a multiple of what two numbers ? 16. From 12J take 9f . 120 - 13J = ? 17. 12 X 13 - 9 = ? 120 -v- 5 X 7 = ? 18. 4j + gj. _ 6 = ? 27 + 18J- + 40J = ? 19. Give four factors of 96, 48, 56, 32. 20. 8 X 9J = ? 10 X 30f = ? | of 1080 = ? 21. 6| X 24 = ? 8i X 35 = ? 11 X ? = 154. 22. How many feet in 6 rods ? MISCELLANEOUS. 107 1. How many cents in 4 double eagles ? 2. How many half-dimes ought I to receive in exchange for 1 eagles ? 3. Eeduce to cents and add : 4 eagles, 30 dollars, 5 half- dollars, 3 quarter-dollars, 12 dimes, and 60 half-dimes. 4. How many cents in $642j ? 5. J of 270 + J- of 1960 f of 560 = ? 6. 18J- + 13J + 19 = ? 10. 131 - 10 + 49 = ? 7. 204f-160^+140^=? 11. 20 + 39J + 12 = ? 8. 9034 - 1041^ + | = ? 12. 130 - 29 - 13J = ? 14. In 1568 oz. how many pounds ? 15. How many half-ounce weights can be made from 98 Ib. of brass ? 16. What cost 58 cwt. of pork at 11 cents a pound ? 17. For $98.44, how many pounds of butter can be bought at 27 ct. a pound ? 18. In a carload of potatoes were 330 bushels. I bought J of them at 37 ct. a bushel. What did they cost me ? 19. At 38 ct. a pound, what cost 97 Ib. of coffee ? 20. 701 X 391 - 44506 = ? 21. A boy having 600 marbles, gave away of them, lost J of them, and sold the remainder at the rate of 2 for a cent. How much money did he receive ? 22. 1135 + 209J + 28 + 129$ - 201 = ? Find quotients : 23. 28326-^-58. 24. 39823 -f- 73. 25. 98463-^215. 26. 38848 -j- 79. 27. 33333-^99. 108 ARITHMETIC. Subtract : 1. 256375 2. 189730 3. 245793 78597 - 98751 - 104832 4. 23121 5. 9675 6. 243967 ? _ 9 _ 9 8573 7356 145578 7. - 57869743 8. -20503742 9. 250606 490132001 910602053 - 172739 10. 149682356 11. 1200000*0 12. - 143654 - 37035874 990909 1950000 13. 70000000 14. -15357580 15. 340302056 69570649 987543123 259686765 16. 843 17. 3927 18. 732685 2692 83528 6128 74863 9 624689 738342 23687 9 9 94649 749309 486903 38652 9708 2161037 380241 2905790 19. 2435693 - 856487 = ? 20. 8754321 - ? = 1235697 21. ? - 258721 = 6492130 22. 48325 - ? = 45993 23. ? - 87543 = 123473 24. 594789-7 = 239896 25. Subtrahend 84798, remainder 109878, find the minuend. MISCELLANEOUS. 109 Written Work. 1. Change 3960 in. to rods, etc. 2. | of a barrel of vinegar leaked out. How many pint bottles can be filled from what remained ? 3. Write in Arabic notation one million, one hundred six thousand, five hundred nineteen. 4. From 37 bu. how many pints of walnuts can be sold ? Find the cost at 8 ct. a pint. 5. I step 24 in. in one step. How many steps do I take in going around the school lot, if it is 198 by 132 feet ? 6. Paid $7360 for 46 horses ; what was the cost of one horse ? 7. Find the product of 3496 and 4857. 8. 16/ T + 13} + 391 = ? 241J + 39 - 10H = ? 9. Paid $7260 for a house and lot ; the house cost $ of it. What did the lot cost ? 10. 19 + lOjj- + 16 - 29ft = ? 11. | + J + J + ft + 247 = ? 29703 X 43 = ? 12. Write a mixed number, and change it to another form. 13. Find the cost of 3 tons of hay at ct. for one pound. 14. | equals how many 270ths ? 15. What part of 500 is 200 ? 16. Subtrahend 341^, remainder 250$ . Find the miss- ing term. 17. Reduce 45649 oz. to higher denominations. Divide : 18. 3869846 by 2832. 19. 2863521 by 983. 20. 3612152 by 76. 110 ARITHMETIC. Multiply : 1. 378 X 35 10. 4029 X 867 X 59 2. 367 X 24 11. 4705 X 395 X 87 3. 434 X 52 12. 521 X 742 x 835 4. 805 X 45 13. 471 X 235 X 749 5. 1234 x 14 14. 905 X 403 X 607 6. 2135 X 127 15. 632 x 320 X 478 7. 3598 X 103 16. 876 X 324 X 903 8. 28453 X 897 17. 1286 X 521 x 632 9. 1925 X 5076 18. 2010 X 837 X 156 19. Multiplicand 1875, multiplier 906 ; what is the pro- duct? 20. Product 468750, multiplier 625 ; what is the multi- plicand ? 21. Multiple 66516, one factor 172; what is the other factor ? Divide : 22. 4352 -f- 32 29. 210522 -r- 1248 23. 5438 -f- 51 30. 847916 -r- 2016 24. 8877 -T-. 77 31. 869827 -?- 5068 25. 20492 -f- 168 32. 851724 -5- 5028 26. 42642 -r- 194 33. 1778136 -v- 1923 27. 84774 -j- 246 34. 3855810 -r- 2135 28. 147380 -f- 690 35. 11080268 -r- 756 36. Multiple 611604, one factor 756; find the other factor. 4 37. Divisor 354, dividend 95580 ; find the quotient. 38. Divisor 812, quotient 1243 ; what is the dividend ? 39. Dividend 87654, divisor 203 ; find the quotient. 40. Dividend 75432, quotient 123 ; what is the divisor ? REVIEW. Ill Solve at sight as many as possible : 1. Reduce 94 to a fraction whose denominator is 7. 2. Change 247 to the form of a fraction. 3. Express in its simplest form the quotient of 56 divided by 63. 4. From | take f . From 1J take T V 5. 11 - } = ? - A = ? 9J - 2| = ? 6. Subtrahend 8^, remainder 15 J ; what is the minuend? 7. (20 - 16J) + (24| - 13|) = ? 9^ - 7| = ? 8. A man earns $7-J a week; his expenses are $4f. How much does he save ? 9. What must be added to the sum of $ and 7| to make 20 ? 10. of my money is in bills, in silver, and the rest in gold. What part is in gold ? 11. Mary has f 10 in 50-cent pieces ; how many has she ? 12. pound of tea cost 37 ct. ; how much will 2 pounds cost? 13. At 16 ct. a pound, how much will 3| lb. of meat cost ? 14. A of 50 + T 7 * of 60 == ? 29587 X 107 = ? 15. 6J + 8| + 9 = ? 22. | - ? = lOf 16. 80 - 37f + 10 = ? 23. 84 - 3l = ? 17. 29^ - 11 + 6 = ? 24. 6J - ? + 3 = 7. 18. ? - 7J = 9|. 25. 9J + 2 + 1J = ? 19. ? + 13 = 20. 26. 2| - + is = ? 20. 17 ? = 4. 27. 24 + 30 - 20J = ? 21. 25 - ? = 17^. 28. 16 _ 13 + 26 = ? 29. Product 187, multiplier 11 ; find the multiplicand. 30. Product 136, multiplicand 7 ; find multiplier. 31. ? x 12 = 1728. 112 ARITHMETIC. 1. What will 15 bu. 2 pk. 2 J qt. of nuts bring, at 12 ct. a quart ? 2. What number is that to which, if you add 18|, the sum will be 97? ? 3. A boy sold his book for $ 3J, which was $ less than he paid. What did it cost him? 4. What part of 49 is 27 ? Of 88 is 44 ? 6. How many packages of 8 oz. each can be made from 100 cwt. of cinnamon ? 7. At 18 ct. a pound for roast-beef, what will 5 Ib. 7 oz. cost? 8. What cost 349 Ib. of sugar at 5 ct. for one pound ? 9. 19| + 15| + 17f = ? 2960 x 91 = ? 10. At 12 ct. a pound, what cost 18| Ib. of cheese ? 11. When J of a year has passed, how many days re- main ? How many hours ? 12. At 5 ct. a pint, what will a barrel of vinegar cost ? 13. What will 625 oysters cost at one dollar for 100 oysters ? 14. Change 1260 pt. to barrels, and find the cost at $22 a barrel ? 15. 4 hours of a day have passed ; how many minutes remain ? 16. 2990 is a multiple of what 3 numbers ? 17. I buy 1 bu. of pears for $2 and sell them so as to gain $2.20. How much does one peck bring ? 18. 6 X 998 -f- 39 = ? 450^ - 340f = ? 19. 25796 -*- 137 = ? 20. 3576 X 1247 = ? REVIEW. 113 2. 518JJ - 309f J. 5. 802 T V - 128if . 3. 532 T \ - 227. 6. 132f + 719 j. 7. How many pounds of butter at 33 ct. a pound can be bought for 55 Ib. of tea at 78 ct. a pound ? (Indicate.) 8. A pedler bought 491 yd. of cloth at $.81 a yard. He used 29 yd., and sold the rest for $.95. How much did he gain ? 9. A man exchanged 159 cords of wood, at $5 a cord, for a horse valued at $144, and the balance in sheep at $3 each. How many sheep did he get ? 10. What cost 9 hats, if 5 cost $12.25 ? 11. To what number must three times 27 be added to make 120 ? 12. Find the cost of 11 Ib. 4 oz. of tea at 48 cents for one pound ? 13. A flag-pole 180 ft. high broke off way up. How many yards broke off ? 14. 200 + 143}f - 39}f = ? 15. (24} - 12|) + (if + f) - ? 16. 1958 - 496H + 97 = ? 17. If a man's pulse beats 72 times in a minute, how many times will it beat in 2 hours ? 18. A miller buys 12890 pounds of wheat at the rate of 90 cents for 60 pounds. How much does he pay for it ? 19. If 36 men can do a piece of work in 84 days, how long will it take 42 men to do it ? (Indicate.) 20. 81} + 13i - 27i = ? 21. 100 - 69^ + 38| = ? 22. 120i - 75^ + 65| = ? 23. 300 - 18J - 59J = ? 114 ARITHMETIC. 1. Define fraction ; improper fraction, and give examples. 2. MjVi 11 equals what mixed number ? 3. Eeduce }Jf to its lowest terms. 4. Change , f , f , to fractions having a least common denominator. 5. Change 16 to fourths. 5 to twelfths. 6. From seven million, sixty-three thousand, eight hun- dred six, take six million, forty-five thousand and thirty- seven, and multiply the result by three hundred seventeen. 7. Mrs. A. L. Jones bought a sewing-machine for $65. She paid $20 cash. How many months will it take her to pay the rest, if she pays $5 a month ? 8. Mr. Bradley Phillips sold 5 lots at $864 each, then purchased a house and lot for $2563; how much did he have left ? 9. What part of 2 bu. is 1 bu. 4 quarts ? 10. Lulu Lane paid $20 for 15 piano lessons, and $35 for 20 violin lessons. What did each lesson cost ? 11. Add f , A, 6f, 17JJ. 31* - 27f + 17H = ? 12. The quotient is 151, the divisor 364, the remainder 39. Find the dividend. 13. When you change J to g do you change the value ? How do you affect the number of parts ? How do you affect the size of the parts ? 14. Eeduce 13 bu. 2 pk. 7 qt. 1 pt. to pints. 15. How many feet in 9| miles ? 16. Miss Marie Hunt paid $38.50 for 20 reading lessons. How much did she pay for 2 lessons ? 17. 103 X 2894 = ? 18. 39 - 13f + 15 = ? 19. 384526 -5- 169 = ? REVIEW. 115 Mental Work. Add: 1. 52 + 65 8. 950 + 790 15. 31 + 22 + 34 2. 44 + 77 9. 190 + 320 16. 20 + 30 + 81 3. 26 + 82 10. 180 + 450 17. 10 + 43 + 62 4. 31 + 93 11. 120 + 680 18. 14 + 21 + 16 5. 57 + 95 12. 360 + 590 19. 42 + 24 + 40 6. 68 + 73 13. 780 + 270 20. 49 + 17 + 20 7. 93 + 38 14. 240 + 660 21. 37 + 25 + 61 Subtract : 22. 65- 18 29. 650 290 36. 231 165 23. 74- 49 30. 410 320 37. 723 184 24. 82 - 60 31. 330 250 38. 656 378 25. 93- 27 32. 720 160 39. 964 197 26. 63- 19 33. 224 35 40. 179 145 27. 55 - 37 34. 421 187 41. 683 362 28. 49 - 23 35. 542 439 42. 731 719 Multiply : 43. 89 x 2 50. 210 X 8 57. 96 X 25 44. 56 X 3 51. 320 X 9 58. 25 X 84 45. 94 x 4 52. 360 X 3 59. 18 X 25 46. 72 x 5 53. 470 X 2 60. 36 X 25 47. 62 x 7 54. 1200 X 6 61. 25 X 48 48. 42 X 8 55. 2500 X 3 62. 25 X 33 49. 81 X 9 56. 2100 X 4 63. 320 X 25 Divide : 64. 175- 5 71. 240 -f- 12 78. 1680 -^_ 210 65. 192- 4 72. 520 -f- 13 79. 2250 -- 250 66. 196- 6 73. 600 -i- 15 80. 1890 -i- 270 67. 189- 3 74. 700 -i. 14 81. 1160 -r- 290 68. 406- 7 75. 640 -f- 16 82. 1900 -f- 380 69. 174- 3 76. 680 -i- 17 83. 1560 _i- 260 70. 244 4 77. 720 -f- 18 84. 2080 -f. 520 116 ARITHMETIC. Test Work. 1. Copy and add : 695, 378, 26, 409, 825, 6. 2. 482 X 743 -T- 216 - 647 = ? 3- (i + H)-(A + *)- 4. Five boxes of oranges contained 156 oranges each, and were sold at $3 per hundred. Find the amount of money received. 5. A piece of cloth was sold for $112.70 at $2.45 per yard. How many yards did the piece contain ? 6. From 7$ yards of cloth 5^ were sold. How much remained ? 7. Add J, I, J, A, 3, 7, ft, 4. 8. How many hours in this month ? 9. How do you reduce dollars to cents ? 10. How do you reduce mills to dollars ? 11. Define multiplication, quotient, product. 12. Change 6 miles to rods. 13. When do you use short division and when long division ? 14. What do you do when the partial dividend will not contain the divisor ? 15. How many pounds of butter at 15 cents a pound will pay for 9 boxes of berries at $.25 a box, and 25 pounds of sugar at 9 cents a pound ? 16. Write in figures one million, six thousand five. CHAPTER III. BILLS AND ACCOUNTS. 1. A receipted bill. MR. JOHN D. WILSON SYRACUSE, Jan. 1, 1895. Bought of H. E. BARRETT. 16 yd. Silk @ $1.50 $24 00 6 yd. Irish Linen " .60 3 60 4 pair Hose " .50 2 00 1 Cane Seat Rocker 4 00 9 yd. Lace " .50 4 50 $38 10 Received Payment, H. E. BARRETT. Mr. John D. Wilson is the debtor, and H. E. Barrett the creditor. The bill is a written statement of Mr. Wilson's account. It is a receipted bill because the creditor, Mr. Barrett, has acknowledged in writing that he has received payment. Study the form of the bill, and tell what things are necessary in every receipted bill. If Mr. Wilson had paid the money to Mr. Barrett's clerk or collector, how could it have been receipted ? 117 118 ARITHMETIC. 2. Mr. John Q. Adams buys of D. McCarthy & Co. : 4 pounds of coffee at 27 cents a pound, 18 pounds of sugar at 5 cents a pound, 5 gallons of molasses at 60 cents a gallon, 16 pounds of rice at 8J cents a pound. Make out the bill. 3. James Smith, farmer, sold Richard Dunn, grocer, the following : 6 barrels of potatoes at $1.80 a bbl., 2 tons of hay at $16 a ton, 3 cords of wood at $4 a cord, 360 pounds of butter at 24/ a pound. Make a receipted bill. 4. Syracuse, Dec. 5, 1894. Edward Smith sold B. M. Watson 65 yd. Brussels carpet at $1.25 ; 24 yd. oil cloth @ 35/ ; one dozen pair of merino hose @ $3.50 ; 2 willow chairs @ $4.50. Make bill, find the footing, and properly receipt it. 5. Make out a bill of groceries. Foot it, and receipt it, with F. H. Mead as creditor and Wm. H. Scott as debtor. To THE TEACHER. See that the prevailing prices are used, and that the quantities are consistent. 6. J. H. Acker bought of II. A. Strong of San Francisco, the following articles: 15 bbl. flour @ $8.00; 6 tubs of butter, 120 pounds in a tub, at 24 cents a pound ; 5 barrels of beef, 200 Ib. to the barrel, at 5/ ; 25 sacks of flour @ 95/; 450 pounds of ham at 11 J/. Make out bill, and receipt it. 7. Make out a bill of hardware, another of groceries, and another of dry goods. 8. Make out a bill of goods bought at a shoe store ; at a stationer's ; at a furniture store. INDICATED WORK. 119 Indicate the operations and find the results : 1. The sum of 26 and 50, added to their difference. 2. The sum of 26 and 50, multiplied by their difference. 3. The sum of 26 and 50, divided by their difference. 4. The sum of 26 and 50, less their difference. 5. Divide the product of 9 times 112 times 16, by the product of 28 times 18. 6. The sum of two numbers is 6943. The greater is 587. What is the less ? 7. At the rate of 8 pencils for 48 cents, how many cents will 10 pencils cost ? 8. At the rate of 8 pencils for 48 cents, how many pencils can be bought for $1.02 ? 9. If a man is 52 years of age now, how old was he 26 years ago ? How old will he be in 26 years ? 10. How many dozen eggs at 24 cents a dozen, can be bought for $7.20 ? 11. If 16 horses cost $1280, what will 19 horses cost ? 12. From a flock of 130 sheep, 56 were sold, and after- ward 48 were purchased ; how many sheep in the flock ? 13. How many barrels of flour, at $5.80 a barrel, will pay for 850 pounds of maple sugar at 15/ a pound ? 14. If I have $36, and pay all but $16 for cloth at $1.50 a yard, how many yards can I buy ? 15. John had $3, Henry 4 times as much, and Charles J as much as Henry. How much had all ? 16. If I have $18, and buy 2 tons of coal at $4.50 a ton, and 2 cords of wood at $2.50 a cord, how much money will I have left ? 17. John had $76J ; he deposited $75 in the bank, and then earned $28*. How much money had he ? 120 ARITHMETIC. ADDITION AND SUBTRACTION OF FRACTIONS. 1. From a cask of oil containing 42 gal., a merchant sells 9 gal., 15} gal., and 7 gal. How much remains in the cask ? 2. If 7^ feet are subtracted from a rod, how many feet will remain ? 3. A farmer brought to market 3J bushels of raspber- ries, 2} bushels of blackberries, and 3j bushels of currants. How many bushels of berries did he bring to market ? Add: 4. 324J 5. 695? 6. 1694J 7. 2698^, 593} 284$ 2436J 2963J 282^ 989j 9843f 2956? MULTIPLICATION OF FRACTIONS. Oral Work. 1 apple multiplied by 4 = 4 apples. 3 apples multiplied by 4 = 12 apples. J multiplied by 4 = J. f multiplied by 4 = |. | multiplied by 3 = T 6 T . 1. 5 times T \ = ? 5 times T 2 T = i^. 2. 6 times 1 = |J. 4 times f\ = ? 3. 6 times ^ = 1^. ^ multiplied by 6 = ? 4. At $} each, what will 8 books cost ? 5. If a horse eat f of a bushel of oats a week, how many bushels will he eat in 10 weeks ? 6. Multiply $5i by 2 ; by 3 ; by 4. 7. Multiply these fractions by 3, and give products in simplest form : f, f, f, f, f, , f, 2, 3J. MULTIPLICATION ^F^^A^TiS^'S. l 121 8. Multiply the fractions in Ex. 7 tJy^SjDy 6 ; by 10 ; and give products in simplest form. 9. J of 6 = ? f of 6 = ? 10. of a number is how many times J of it ? 11. If a pound of tea costs 48 cents, what will of a pound cost ? What will f of a pound cost ? 12. | of 12 = ? SOLUTION. J of 12 = * = 3. f of 12 = 3 X 3 = 9. ^4ras. 13. f of 7 = ? SOLUTION. J of 7 = J. | of 7 = 5 times J = ^ = 4$. J?*s. NOTE. | o/ 7 is the same as f times 7, and the product is the same as 7 times f . 14. f of 5 = ? ft of 4 = ? | of 3 = ? of 6 = ? 15. $10 multiplied by | = ? ^ X 10 = ? 16. f times 8 = ? 4xf = ? f X 4 = ? 17. How much is % of J of an inch ? (Illustrate.) 18. Illustrate that \ of of an apple is J of an apple. 19. Multiply f by J ; f by ; by 4 ; J by f 20. How much is J of } ? f of f ? J of f ? J of ft = ? 21. A man owned f of a farm, and sold of his share. What part of the farm did he sell ? 22. James had $|, and John 1 as much. How much had both? 23. If a pound of tea costs $|, what will Ib. cost ? 24. | X = ? 3X5 = ? T % X J = ? 25. f X J = ? jXi = ? JX| = ? 26. Find of | ^ of J ; of ft ; of ft. 27. Find f of 6 ; of 9 ; of 12 ; of 15. 122 ARITHMETIC. 28. } of | = ? (Problem.) 29. T \ of = ? Written Work. 1. | of | = ? SOLUTION. } of f = 2 X & of f . i of J = i. ^ of | = 2 X i = | i = i- Rule. Keduce integers and mixed numbers to im- proper fractions. Multiply the numerators together for the numerator of the product, and the denominators for the denominator of the product. Cancel when possible. Two or more fractions joined by (of) form a Compound Fraction. The word of is equivalent to the sign of multiplication. 1. Multiply & by 6. 5. 87 times J = ? 2. Multiply ig by 10. 6. 45 times }f = ? (Problem.) 3. Multiply { by 17. 7. 35 times ^ = ? 4. Multiply JJ by 16. 8. 18 times f = ? (Problem.) 9. Multiply 18| by 8, first reducing the mixed number to an improper fraction. 10. Multiply 18f by 8 without reducing the mixed num- ber to an improper fraction. First multiply the fraction, then the integer, by 8, and add the products. Thus : 18J SOLUTION. 8 8 times } = 2 ? * = 6 6 8 times 18 = 144 144 150 150 11. 16| multiplied by 10. 14. Multiply 27|$ by 38. 12. 126 1 multiplied by 9. 15. Multiply 35f by 46. 13. 326 multiplied by 5. 16. 9 times 2}f = ? MULTIPLICATION OF FRACTIONS. 123 17. At 37 J cents a pound, what will 16 Ib. of tea cost ? 18. Find the cost of 17 barrels of flour at $7| a barrel ? 19. | of 160 = ? 23. Multiply 195 by |f . 20. | times 160 = ? 24. Multiply 326 by f J. 21. 160 times f = ? 25. Multiply 584 by -%. 22. J of 126 = ? 26. Multiply 1680 by if. 27. Multiply 47 by 6}. 47 SOLUTION. f of 47 = 14 J _6| 6 times 47 = 102 14J Il6| Ans. 102 116J 28. Multiply 65 by 7f . 30. 4 J times 48 = ? 29. Multiply 125 by 14f . 31. 10/ times 385 = ? 32. At f 16 a ton, what will 17} tons of hay cost ? Multiply : 33. 12| by 30. 36. 24 Jf by 48. 34. 36 T \ by 60. (Problem.) 37. 28f by 9. (Problem.) 35. 41 f \by20. 38. 3269|by5. (Problem.) 39. What will 65} bushels of potatoes cost at 40 cents a bushel ? 40. If a train travels 58 f miles in one hour, how far can it travel in 24 hours ? Find the product : 41. if X fV? 44 - J X i? X T \ 47. 16| X 6| 42. fo x 11 45. J j X J X 6J 48. 5j X 5j 43. 25 X if 46. 5g X 7J X 4| 49. } X ff 50. What is the cost of 28| yards of broadcloth at $2f a yard ? 51. What is the cost of 4} cords of wood at $4} a cord ? 124 ARITHMETIC. Change the following compound fractions to simple fractions : 52. $ of T 6 T of || of 14. 53. of f of 21-f . 54. What is the cost of 35 Ib. of butter at 37^ cents a pound, 28 bushels of wheat at 87 cents a bushel, and 250 bushels of oats at 22J- cents a bushel ? DIVISION OF FRACTIONS. Oral Work. 1. J is contained in 1 how many times ? SOLUTION. th ? rd is contained in 3 times. 2. is contained in 1 how many times ? SOLUTION. Since is contained in 1 three times, is contained in 1 one-half as many times, or J of 3 times, which is | times. Ans. 3. f is contained in $ how many times ? SOLUTION. is contained in one J times. It is con- tained in just as many times as it is in 1. Since is contained times in 1, in it is contained J of | times, which is f times. Ans. Thus we see that j -f- is equivalent to f multiplied by f . 4. By what do we multiply when we divide by | ? | ? 5? i? i? 4 ? 5- 1-1 = ? i-f = ? f-i=? 6. How many times is J contained in 1 ? in ? in ? in J ? in f ? 7. Write problems, using the fractions in Ex. 5. 8. How many pens can be bought for 3 cents if one pen costs a cent ? 9. If tea costs $| a pound, how much tea can be bought for $1 ? for $1 ? for f f ? DIVISION OF FRACTIONS. 125 Written Work. NOTE. To divide by a fraction is to multiply by that fraction inverted. Rule. Multiply the dividend by the divisor inverted. Reduce integers and mixed numbers to improper factions, and cancel when possible. Find the quotients : L i-i-f 4. |f -2 7. 51-2J 2. H-s-f 5. 21 -f 8. 10 -f 3. y 5 _i_ 3 6 . 16 -T- 4f 9. 5^ -^ 8f f -T- } may be written |. Such an expression is called a Complex Fraction, and is used simply to indicate division. 10. Change Examples 1 to 9 to the form of complex fractions. 11. Eead each of them. A Complex Fraction is a fraction having a fraction for one or both of its terms. 12. Eead the following, and change to simple fractions. 4| 18| | 10 4f 2' *' r t' 7f 13. 22| -^ 7 = ? SOLUTION. 7 is contained in 22|, 3 times, with a re- 7)22| mainder of If or J ; this divided by 7 = ^ or J. 3J The quotient is 3J. When not convenient to di- vide this way, proceed according to rule. Prove by other method. 14. 16| - 5 = ? 16. 45? ~ 6 = ? (Problem.) 15. 30f -;- 7 = ? (Problem.) 17. 33J ^ 9 = ? 18. 264 ^ 4 = ? 19. 9826^ -T- 6 = ? (Problem.) 126 ARITHMETIC. 20. 17 -r- 4| = ? 4 17 This may be divided according to rule, but 3 _3 the division is sometimes shortened as follows : 14 ) 51 Reduce both dividend and divisor to thirds. 3A 4 = and 17 = J * is contained in as many times as 14 is contained in 51, which is 3^ times. 21. 34 -f- 2g = ? (Problem.) 24. 99 -r- 15J = ? 22. 16 ^ 5J = ? 25. 3694 -5- 2f = ? 23. 126 -r- 3| = ? (Problem.) 26. 85 -r- 4^ = ? 27. If 16 bushels of apples cost $8|, what will one bushel cost ? 28. A sum of $2369 J was divided equally among 5 heirs. What was each one's share ? 29. How much is wheat a bushel when 15 bushels cost 30. When | of a pound of tea costs $$, what is the price per pound ? 31. The product of two numbers is 16|. One of the numbers is 7. What is the other ? 32. At $1 per yard, how many yards of cloth can be purchased for $11 ? 33. In how many days can a horse eat 6 bushels of oats, if he eats | of a bushel a day ? 34. If it takes 2 yd. of ribbon to trim a lady's hat, how many hats can be trimmed with 10 yards ? 35. If 4 apples cost 6 cents, what will 1 apple cost ? 3 apples ? 11 apples ? 36. If 5 oranges cost 16| cents, what will 7 oranges cost ? 37. 6 X ? = 14| 40. 5}| X ? = 6 38. 5| X ? = 2J 41. 3i X ? = 15J 39. 31 X ? = 12 42. X ? = 1J REVIEW. 127 43. If | of an acre of land is worth $250, what is 1 acre worth ? 44. What will 5 doz. eggs cost, when f of a dozen cost 15 cents ? 45. At $5 J a barrel, how many barrels of flour can be bought for $75 ? Written Work. Find the sum : i. i + i+i * + * + * + ! 2. t + A + 7 - J + f + t + 3- + | + T 5 * * + * + * + 4- + A + t 9 ' + t+T 7 s + 6- ! + + 10 - A + i + i + H 11. 34| 12. 165J 13. 364| 29H 284f 19f Find the difference and prove by addition : 1. i - | 6. 15J - 12/ ff 11. 16f - llf 2. || - } 7. 18J - 16J 12. 28| - A 3 - ft - J 8 - 28 ^ - 13 I 13 - 184 - 4. J 9. 65| 18 J 14. 5. -3 10. 43 1 - 289 15. Find the value of : I- J + l-i + i 5. 4i 2-? + J-t + * 6. 12| - 5^ + 7 A 3- T 4 ir + T 8 5 - A - A 7. 6 + 4* - 1| + * -- + 8 - 9-4 T 3 - + 3?- 128 ARITHMETIC. PARTS OF A DOLLAR. 6J cents = $ T V 37 cents == $$ 8 cents = $ T ^ 50 cents = $ 12| cents = $J 62 cents = $f 16 cents = $J 66| cents = f f 25 cents = $ 75 cents = $J 33J cents = $ 87 cents = f J What is the cost of 33 books at 16/ each ? 33 books cost 33 times 16/, or 33 times $J = $ $5.50. ^47i5. Oral Work. Multiply : 1. 12 cents by 16. 7. 37 cents by 8. 2. 16| cents by 12. 8. 50 cents by 15. 3. 25 cents by 20. 9. 62 cents by 8. 4. 33 cents by 27. 10. 66| cents by 9. 5. 6J cents by 16. 11. 75 cents by 4. 6. 8J cents by 24. 12. 87 cents by 8. 13. What is the cost of : 16 Ib. of bacon @ 12/ a pound ? 16 balls @ 50/ each ? 36 yd. of ribbon @ 33J/ a yard ? 36 Ib. of candy* @ 25/ a pound ? 8 Ib. of tea @ 62/ a pound ? Written Work. Make out a bill, and supply place, date, and names. 14. 66 Ib. of pork at 12 cents a pound. 48 Ib. of veal at 16 cents a pound. 65 boxes strawberries at 25 cents a box. 15 yd. flannel at 33^ cents a yard. 80 Ib. honey at 25 cents a pound. 48 Ib. tea at 50 cents a pound. PARTS OF A DOLLAR. 129 1. Find the cost of : 16 Ib. of tea @ 62J/. 2 yd. flannel @ 66f /. 84 gal. molasses @ 75/. 16 doz. oranges @ 37 J/. 50 Ib. butter @ 25/. 2. At 25/ a pound, how many pounds of butter can be bought for $8 ? As many pounds as 25/ or $8 -f- $1 = 32 pounds. Ans. Oral Work. Divide : 3. $3 by 33J/. 4. $5 by 25/. 5. $2 by 8J/. 6. $9byl2i/. 7. $1 by 6i/. 13. At 25 cents apiece, how many hats can be bought for $6? 14. At 25 cents a pound, how many pounds of cheese can be bought for $5 ? 15. At 16| cents a dozen, how many dozen eggs can be bought for $4 ? 16. How many pounds of beef can be bought for $4, at $.16| a pound ? 17. At 33 \fl a yard, how many yards of linen can be bought for $10 ? 18. How many penknives can be bought for $6, at 33J cents apiece ? 19. 24 x $.12^ = ? 20. $24 ^- $.12J = ? is contained times in 8. f!0by50/. 9. $6 by 33J/. 10. $4byl2j/. 11. $2by25/. 12. $3 by 130 ARITHMETIC. Written Work. 1. At $.75 a bushel, how many bushels of corn can be bought for $48 ? 2. At 62J/ a bushel, how many bushels of oats can be bought for $45 ? 3. At 66 cents each, how many knives can be bought for $7J ? 4. When potatoes are 25 cents a bushel, how many bushels can be bought for $16 ? 5. When butter is 33J/ a pound, how many pounds can be bought for $68 ? 6. At 87J cents each, how many arithmetics can be bought for $80 ? What is the cost of : 7. 6 doz. lemons at 37 cents a dozen ? 8. 96 Ib. of candy at 25/ a pound ? 9. 356 pounds of butter at 32J/ a pound ? 10. 234 bu. of wheat at 84| cents a bushel ? 11. 46 yards of oil cloth at 36J cents a yard ? 12. 96 doz. eggs at 16/ a dozen ? 13. At 37J cents each, how many articles can be bought for $80 ? for $63 ? for $33 ? 14. At $| a day, how much will a boy earn in a week ? in 26 days ? in 300 days ? 15. Make out two bills, using fractional parts of a dollar for prices. Supply names, place, date, etc. Divide : 16. $12i by 25 cents. 20. $18.33^ by 33J cents. 17. $16.75 by 33J cents. 21. $15.89 by 12J cents. 18. $14.53 by 16| cents. 22. $18.66f by 33J cents. 19. $65.14J by 50 cents. 23. $400.15 by 12J cents. QUESTIONS OF RELATION. 131 THE THREE QUESTIONS OF RELATION. (a) 3 X 4 = 12 ; (b) therefore 12 +- 4 = 3 ; (c) and 12 -f- 3 = 4.. These statements placed in the form of questions would be as follows : (a) 3 times 4 = what ? Ans. 12. (b) 12 is how many times 4 ? Ans. 3 times. (c) 12 is 3 times what ? Ans. 4. 1. Form the three questions from the following state- ment, 5 X 6 = 30. Question a. % of 8 equals what ? Ans. 4. Question b. 4 is J of what ? Ans. 8. Question c. 4 is what part of 8 ? Ans. J. SOLUTION. The product of two numbers divided by one of them equals the other. Since J X 8 == 4, then, 4 -f- = 8, and, 4 -i- 8 = . 2. J of 24 = what ? With the product and one of the given numbers, form a question like question b ; like question c. 3. 16 is 4 of what? SOLUTION. It is evident that 16 is the product of two numbers, and ^ is one of them. Therefore, 16 -=- J = 48. Ans. 4. What part of 48 is 16 ? SOLUTION. It is evident that 16 is the product of two numbers, and 48 is one of them. Therefore, 16 -s- 48 = J = i, Ans. 132 ARITHMETIC. NOTE. The foregoing treatment of this class of ques- tions will be found exceedingly helpful in Percentage and all its applications. Each of these three typical questions forms the basis of the other two. When one of these is given, the pupils should be required to give both the others until very familiar with them, and when a problem is found, involving one of the Three Questions, pupils should be able to state the right question at sight. When the pupil can state the question, the correct solu- tion is assured, for the question is the key that unlocks the problem. Solve each of the following, and tell whether it is under question a, b, or c, and state the other two questions. Mental Work. 5. | of 12 is what ? 11. 4 is J of what ? 6. 8 is what part of 24 ? 12. What part of 10 is 9 ? 7. What, part of 18 is 6 ? 13. f of 6 = what ? 8. 15 is | of what ? 14. 9 is of what ? 9. How much is of 9 ? 15. What part of 8 is 1 ? 10. What part of 9 is 3 ? 16. 4 is what part of 5 ? Written Work. To THE PUPIL. Think what the question is before attempting to solve the problem. 17. James had 56 marbles, and John f as many. How many had John ? 18. What is the question in Example 17 ? 19. A man sold f of his farm, and had 50 acres left. How many acres were there in the entire farm ? (Ques- tion.) How many acres did he sell ? (Question.) 20. A boy had 15 cents, and spent 6 cents for candy. What part of his money did he spend ? (Question.) What part was left ? (Question.) BEVIEW OF FRACTIONS. 133 21. What is the cost of a ton of hay, when f of a ton is worth $9 ? (Question.) 22. | of a basket of eggs were sold for $6.24. At the same price, what did the entire basket sell for ? (Question.) 23. If I can build a wall in 8 days, what part of the wall can I build in 5 days ? (Question.) 24. James spent $12, which was f of his money. How much had he at first ? How much had he left ? 25. In my pasture are 49 sheep, which is T 7 5 of my entire flock. How many sheep have I ? (Question.) What part of my sheep were not in the pasture ? How many sheep were not in the pasture ? 26. John runs 360 yards, and Henry as far. How far does Henry run ? (Question.) 27. Henry ran 300 yards, which is $ as far as John ran. How far did John run ? (Question.) 28. If Dick has $5 and Eobert #15, what part of Robert's money is Dick's ? (Question.) What fraction of Dick's is Robert's ? (Question.) 29. If a barrel can be filled by a pipe in 40 minutes, what part of it can be filled in 35 minutes ? (Question.) 30. What part of 15 is 8 ? (Problem.) 31. 50 is what part of 150 ? (Problem.) 32. 50 is | of what ? (Problem.) 33. What is VV of 34 ? (Problem.) 34. What part of is f ? 35. What part of f is J ? REVIEW OF FRACTIONS. Written Work. 134 ARITHMETIC. 2. Reduce to lowest terms : || ; 3. Eeduce to integers or fractions : 8 T \ ; 9f ; 15J. 4. Reduce to mixed numbers : f j ; ^ ; ^ ; *?f&. 5- f + f = ? A + M + 41 = ? 6. 6| + 7 A = ? 29J + 17 = ? 7. Add : 206|, 509 T %, 1056|, 109 T V a I - 1 - ? f - t\ = ? ii - } = ? I - A = 9. 142 _ 14J = ? 17J - 15| = ? 183 - 124f = ? 10. j of 16 = ? 5 x 11 = ? f X i XJi X }f 11. 2064 X 3} = ? 2693f X 4 = ? 2j x 4 x 3J = ? 12. J -5- 6 = ? g^4 = ? ^ T ^-9 = ? }| ^- 6 = ? 13. 16 -f- 5 = ? 14 -f- i = ? 15 ^- J = ? 14. 216| - 5 = ? 3046g -4- 2 = ? 321^ ^ 18 = ? 15. 5 _,. 3 = ? 3 j. ^ = ? 3} -j- 2| = ? 42 -r- 5J = ? 16. Find the cost of four pieces of cloth, measuring 23 yd., 25| yd., 22J yd., and 21J yd., at $2 a yard ? 17. Charles has $15| in the bank. How much more must he earn before he has $50 ? 18. From the sum of 2J and 31J, take the difference of 5J and 4J. (Indicate.) 19. Which is greater, the product of | and J, or the dif- ference ? How much ? 20. How many inches in j of a yard ? (Question.) 21. A man drew from the bank $68, which was | of his entire deposit. How much had he at first ? (Question.) 22. A sum of money was divided between John and James. John received f of it, and James $300. How much did John receive ? 23. A merchant paid $272 for flour, at $9f a barrel. How many barrels did he buy ? REVIEW OF FRACTIONS. 135 24. Subtract 2 from both terms of the fraction J. Do you increase or diminish the fraction, and how much ? Add 2 to each term, and find the difference. 25. What must be added to f + If + J to make 5 ? 26. What number multiplied by 25 J will make 264 ? 27. A lady gave $J for ribbon, $1 for pins, $| for velvet, and $-J for lining. What change did she receive from a five-dollar bill ? 28. What will 4| Ib. of raisins cost, at 15/ a pound ? 29. From 25 yd. of cloth, a tailor used If yd. for making a vest, 2 yd. for a pair of pantaloons, 3 yd. for a coat, and 4J yd. for an overcoat. How much cloth was left ? 30. I owned } of a farm and sold of my share. What part of the farm did I sell, and what part did I retain ? 31. How many rods in 3642 feet ? 32. If 4 quarts of grain are given for grinding a bushel, what part is the cost for grinding ? (Question.) 33. Henry had $78 in the bank. He withdrew $15 at one time and $38 at another time. What part of his money did he withdraw ? (Question.) What part re- mained ? (Question.) 34. A man having 150 sheep sold of them and kept the remainder. How many were sold ? How many were kept? 35. A man owning $ of a factory sold of his share. What part of the factory did he sell ? 36. What part of 6 bushels 1 peck is 4 bushels 2 pecks ? 37. A man sold 50 sheep, which was f of his whole flock. How many sheep had he at first ? 38. At $4| a yard, what will 5 yards of velvet cost ? 136 ARITHMETIC. 39. 4 yards of ribbon cost 35/; what was the price per yard ? 40. At $3 a ream, what will of a ream of paper cost? 41. If 20f Ib. of sugar cost 80 cents, what will 50 Ib. cost ? (Use cancellation.) 42. If 20| Ib. of sugar cost 80/, how many pounds can be bought for $2.00 ? 43. I bought 55 yd. of cloth at $g a yard, and sold it at $f a yard. What was the profit ? 44. If an acre of land is worth $150, what part of an acre will $50 buy ? 45. If a teacup holds J of a pint, how many cupfuls in a quart ? A gallon ? 46. If | of my money is $93, how much is | of it ? (1st Question.) (2d Question.) 47. You sell to John Grouse the following goods : 48 Ib. of butter @ 29/. 24J Ib. of coffee @ 45/. gal. of oil @ 12|/. Ib. of tea @ 37|/. Make out a bill and receipt it. 48. How many feet of wire will be required to fence a lot 145J ft. deep, and having 55^ ft. front, the fence being four wires high ? (Illustrate.) 49. How long will it take a boy to pay for a $75 bicycle, if he can pay $f a day ? 50. f of a class are boys ; there are 30 girls. How many pupils in the class ? 51. If 2| Ib. of tea cost $1.00, how much will 3J Ib. cost? REVIEW OF FRACTIONS. 137 52. From a hogshead of molasses ^ leaked out; what part of the molasses remained ? (Question.) How many gallons remained ? 53. At $1 a pound, what is the cost of % pound of tea ? 54. Find the product of , f , 5f , and 2. 55. From 369 take 25f . 56. Find the sum of of 44, } of 16, and j of 24. 57. Find the product of J, , f , and |. 58. What is a compound fraction ? 59. A paymaster pays each of his men $*. How much does he pay 22 men ? 60. If a press can print T 5 r of a paper in a second, how many papers can it turn out in a minute ? 61. If the fare for a ride on the steamer Ontario is $$, what will it cost 21 persons to take the ride ? 62. A family eat 2 Ib. of meat a day. How much meat will they eat in a week ? 63. The distance from Harry's home to school is j of a mile. How many miles does he walk in a week in going to and from school ? 64. Each person at a church contributed, on an average, $g. There were 295 persons present. What was the amount of the collection ? 65. If Mary can pick 4 quarts of berries a day, how many quarts can she pick in a week ? 66. Mr. Lynn, the shoemaker, charges $ for making a pair of shoes. How much will he charge for making a dozen pair ? Indicate and cancel : 1. What is the value of J of my lot, if T 7 ^ of it is worth $84 ? 138 ARITHMETIC. 2. If 3 yards of cloth are worth $5f, what are 5| yards worth ? 3. If 3 bushels of grain are worth $4|, how many bushels can I buy for $26 ? 4. How many pounds of coffee at 37J/ a pound can be bought for $24f ? 5. If 3 boxes of lemons cost $6|, what will 32 boxes cost? 6. If 3 boxes of oranges can be bought for $6J, how many boxes can be bought for $72 ? 7. A boy earned $5j at the rate of 37/ a day. How many days did he work ? 8. 7 cords of wood at $4 a cord will pay for how many tons of coal at $3 a ton ? 9. | of a ton of coal at $4 a ton will pay for what part of a cord of wood at $5 a cord ? 10. How many peaches at !/ each can be exchanged for 50 apples at cent each ? 11. J is what part of f of f ? 12. | is f of what number ? 13. The product of two numbers is 2| ; one of them is f . What is the other ? SQUARE MEASURE. An Angle is the difference in direction of two lines that meet. (Illustrate.) A Right Angle is the angle of a Square. (Illustrate.) A Surface is that which has length and breadth, but not thickness, as a page in a book, the top of a table. A Rectangle is a surface having four sides and four right angles. A Square is a rectangle having equal sides. SQUARE MEASURE. 139 Draw two different kinds of rectangles, and tell their difference. Point out some surfaces in your room that are rectangles, and not squares. Are there any perfect squares in the room ? A square inch is a square, 1 inch long and 1 inch wide. Draw a square inch. A square foot is a square, 1 foot long and 1 foot wide. Draw a square foot upon the blackboard. See that the sides are exactly 1 foot long, and the angles right angles. What is a square yard ? Illustrate upon the blackboard. Separate it into square feet. What is a square rod ? Let two pupils come early in the morning and mark off a square rod on the floor, and let others measure and see if it is correct. What is a Square Mile ? The Area of a surface is the number of square units that it contains. What is the square unit in this figure ? How many square inches in it ? SOLUTION. - - There are six square inches in a row, and there are 4 rows, therefore 4 times 6 square inches = 24 sq. inches. Ans. NOTE. In finding areas, as in all multiplication, the multiplier is abstract, and the unit of the product must be the same as the unit of the multiplicand. Oral Work. 1. How many square feet in a surface 6 ft. long and 1 ft. wide ? 6 IN. 140 ARITHMETIC. 2. In a surface 6 ft. long and 2 ft. wide ? (Illustrate.) 3. How many square feet in a platform 5 ft. long and 4 ft. wide. (Illustrate.) 4. What is the area of a mirror 3 ft. long and 2 ft. wide ? 5. Draw a rectangle 4 inches long by 2 inches wide, and divide it into square inches. 6. Draw a 2-inch square and divide it into square inches. 7. What is the difference between a 2-inch square and 2 square inches ? (Illustrate.) 8. How many square feet does a 5-foot square contain ? Show the difference between a 5-f t. square and 5 square feet. 9. If the top of a table is 5 ft. long and 4 ft. wide, what is its area ? 10. If the area of a floor is 20 square yards, and the length 5 yards, what is the width ? The length and breadth of a rectangle are called its Dimensions. Principle 1. Area is the product of length multiplied by breadth. The numbers representing length and breadth must be like numbers. Principle 2. Area divided by either dimension gives the other. 11. Give a problem, to find area, when length and breadth are known. 12. From that problem give another, to find length, when area and width are known. 13. Give another, to find width, when area and length are known. 14. How long is a path whose area is 48 square feet, and whose width is 4 feet ? 15. How wide is a pavement that contains 56 sq. yards and is 8 yards long ? SQUARE MEASURE. 141 TABLE. 144 square inches = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30i square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile or section (sq. mi., sec.) Written Work. 1. How many square inches in 3 sq. feet ? 2. How many square yards in 54 sq. feet ? 3. In 12^ sq. ft. how many sq. inches ? 4. How many acres in 480 square rods ? 5. In 7212 sq. in., how many sq. feet ? Reduce to square inches : 6. 4 sq. ft. 15 sq. in. 7. 8 sq. ft. 110 sq. in. 8. 2 sq. yd. 4 sq. ft. 78 sq. in. 9. 25 sq. yd. 6 sq. ft. 12. 2 sq. ft. 10. 6 sq. yd. 12 sq. in. 13. J sq. yd. 11. 5J sq. yd. 14. sq. yd. sq. ft. 15. 1 sq. rd. 1 sq. yd. 1 sq. ft. 1 sq. in. Reduce to higher denominations : 16. 5680 sq. in. 19. 7200 sq. in. 17. 38204 sq. in. 20. 450 sq. rd. 18. 3296 sq. in. 21. 264321 sq. ft. 22. In 30 A. 14 sq. rd. how many square rods ? 23. Reduce 34232 sq. rods to higher denominations. 24. Reduce 14 sq. rd. 18 sq. yd. 15 sq. ft. 82 sq. in. to inches. 25. What part of a sq. yard is a sq. ft. ? (Question.) 26. 2 sq. ft. is what part of 2 sq. yd. ? (Question.) 27. Reduce 120 sq. rd. 4 sq. yd. to sq. yards. 142 ARITHMETIC. 28. In 40 acres, how many square rods ? square yards ? 29. In of an acre how many sq. rods ? sq. yards ? 30. How many square feet in the top of a table 5 ft. long and four feet wide ? How many square inches ? 31. What is the area of a basement 36 ft. by 18^ ft. ? 32. At 22/ a square foot, what is the cost of a flag walk 50 ft. X 6 ft, ? 33. How wide is a floor whose area is 300 sq. ft., and whose length is 20 ft. ? 34. 5 flag-stones, each 5 ft. by 6 ft., will cover how much surface ? 35. In a room are 4 doors, each 7 ft. 6 in. by 2 ft. 8 in. How many square feet in the surface of the four doors ? How many square inches ? 36. A pasture containing 4 acres is 20 rods wide. How long is it? 37. A city lob containing 8400 sq. ft. is 140 ft. deep. What is the frontage ? (Illustrate.) 38. How many acres in a field mile square ? 39. Is there any difference between a mile square and a square mile ? (Illustrate.) 40. A farmer had a section of land. He sold of it to one man and J to another. How many acres had he left ? What part of the farm is left ? 41. How many sq. yards in a ceiling that is 15 ft. square. (Illustrate.) 42. From | of a sq. mile of land, I sold a farm 80 rods long by 40 rods wide. How many acres have I left ? 43. A garden is 300 ft. long and 200 ft. wide. How many flower-beds in J of it, if each bed covers 100 sq. ft. ? 44. How many square rods in a field 20 rods square ? SQUARE MEASURE. 143 45. How many square yards of carpeting will cover a floor 5 yd. long and 4 yd. wide ? 46. A school yard of 120 sq. rods is 8 rods wide. How long is it ? 47. How many square inches in a board 15 ft. long and 12 inches wide ? (Illustrate.) 48. A field 160 by 50 rods contains how many acres ? 49. How many square yards in a strip of wall paper 28 yd. long and 2 ft. wide ? 60. How many square feet of boards will it take to make a tight board fence 150 ft. long and 4| ft. high ? t 51. How many 3-inch squares can be cut from a strip of pasteboard 3 ft. long and 18 inches wide ? 52. A blackboard is 16 ft. long and 4 ft. 4 in. wide. What is its area in square feet ? 53. How many inch squares in a piece of paper 10 X 8 inches ? How many half-inch squares ? 54. At 4/ a sq. ft., how much will 10 cherry boards cost, each board being 12 ft. long and 1 ft. 3 in. wide ? 55. How many square feet in a roof that is 30 ft. long and 20 ft. wide on each of the two sides ? 56. How many rods long must a pasture be that is 20 rd. wide to contain 2 acres ? 57. The floor of a bedroom is 9 feet wide, and is cov- ered by 12 sq. yards of carpet. How long is it ? 58. What will be the cost of 16 feet of wire cable at 12 / a foot ? t 69. How many blocks 3 inches square can be cut from a board 9 X 12 inches ? 60. Eeduce 1 sq. rod to square feet. 61. Eeduce 2798 sq. inches to higher denominations. 144 ARITHMETIC. SHORT MULTIPLICATION. To multiply when the unit figure of the multiplier is 1. 3825 X 31 Multiply by 3, write the product under 11475 the multiplicand one place to the left, 118575 and add. Multiply and prove : 1. 6984 X 51 4. 2863 x 71 2. 3584 x 21 5. 564 x HI 3. 5832 X 41 6. 2852 x 121 A\%en the tens figure of the multiplier is 1. 3862 x 15 Multiply by 5, write the product under the 19310 multiplicand one place to the right, and 57930 add. Multiply and prove : 7. 2836 X 14 11. 2864 x 12 8. 38987 X 17 12. 3984 X 13 9. 2863 x 18 13. 3549 X 19 10. 3684 x 16 14. 896 X 15 15. At $21 a ton, what is the cost of 285 tons of hay ? 16. At 18 cents each, what will be the cost of 286 blank books ? 17. How far can a train run in 648 hours at the rate of 51 miles an hour ? 18. A wholesale grocer purchased 3984 dozen eggs at a dozen. What was the cost ? 19. If a common soldier receives $14 a month, what must be paid 2856 soldiers for one month's service ? 20. At $31 a head, what will be the cost of 356 cattle ? 21. 384 x 18 X 41 = ? PABTS OF 100 AND 1000. 145 PARTS OF 100 AND 1000. Written Work. Short Multiplication. 1. What part of 100 is 12J ? 25 ? 33| ? 2. What part of 1000 is 125 ? 250 ? 333 ? 3. How much is 1 of 100 ? \ of 1000 ? 4. How much is of 100 ? J of 1000 ? 5. How much is of 100 ? J of 1000 ? 6. 25 times 24 = ? EXPLANATION. 25 times 24 is J as much as 100 times 24. 1-00 times 24 is 2400. \ of 2400 is 600. Ans. To multiply by 25, annex two naughts, and take one- fourth of the result. 7. Tell how to multiply by 12, 33J, 125, 250, 333J. 8. Multiply first by 25, then by 250 : 8, 20, 28, 72, 96, 248, 1696. 9. Multiply first by 12, then by 125 : 16, 24, 48, 96, 144, 264, 1448. Rapid Work, Without Pencil. 10. 56 X 25 15. 25 X 444 20. 42 X 33J 11. 12* X 32 16. 25 X 368 21. 80 X 12 i 12. 36 X 33J 17. 33J X 993 22. 25 X 68 13. 96 X 12* 18. 33J X 846 23. 76 X 25 14. 168 X 12 i 19. 72 X 12* 24. 168 X 12* Written Work. 25. 250 x 44 29. 333 J X 42 33. 1863 X 333 J 26. 27 X 333^ 30. 250 X 648 34. 250 X 7676 27. 125 X 168 31. 125 X 648 35. 125 X 1672 28. 125 X 64 32. 426 X 333 J 36. 9684 X 250 146 ARITHMETIC. Oral Review 1. If 5 bbl. of flour cost $25, what will 8 bbl. cost ? 2. If 5 bbl. of flour cost $25, how many barrels can be bought for $60 ? 3. How many inches long is a 10-foot pole ? 4. What is the cost of 2 doz. lead pencils at 4/ each ? 5. What is T \ of 84 ? 6. What will 5 oranges cost, if a dozen cost 36 cents ? 7. What number added to 4 will equal 9 ? 8. A boy had 48 marbles. He lost 8 in the morning, 7 at recess, and 6 at noon. In the afternoon he found 7 and bought 12 new ones. How many marbles had he at the close of the day ? 9. If 4 men can do a piece of work in 9 days, in how many days can 6 men do it ? 10. If a boy can earn $30 in 6 weeks, how much can he earn in 10 weeks ? 11. What are the prime factors of 48, 72, and 96 ? 12. In } of an apple how many eighths ? 13. Which is larger, or |, and how much ? 14. ^ is how many times ^ ? 15. What is the difference between and ? 16. If one man eats Ib. of meat at a meal, how many pounds will 12 men eat ? 17. If a barrel of apples cost $2 J, what will 4 bbl. cost ? 18. How much is 4 times 5^ ? 19. If a barrel of flour costs $5J, what will 6 barrels cost? 20. 2i is of what number ? 21. 3f is of what number ? MISCELLANEOUS REVIEW. 147 Write in figures : 1. Two hundred ninety-six thousand, forty. 2. Three million, three thousand, three. 3. Two hundred six million, five hundred five thousand. 4. One hundred million, one hundred thousand, one. 5. Eight million, eight. 6. Write the smallest number that can be written with six figures. The largest. 7. In reading numbers, what name is given to the first right hand group ? the second ? the third ? 8. What number is made up of eight 8's written hori- zontally ? Counting from the right, what is the value of the sec- ond 8 ? the fourth 8 ? the seventh 8 ? Add these three values, and express their sum in words. 9. What is the simple value of any figure ? The local value ? Write in figures : 10. Five hundred fifty-five million, five hundred fifty-five thousand, five hundred fifty-five. 11. Two hundred million, two hundred thousand, two. Eead the following : 12. 206,391,401 18. 350,053,350 13. 960,004,003 19. 999,999,999 14. 11,311,114 20. 298,432,624 15. 239,239,239 21. 83,083,083 16. 1,001,001 22. 8,000,008 17. 21,000,021 23. 316,048,006 148 ARITHMETIC. Add: 4,329 2. 216,824 3. 16,826 26,263 315,962 194,295 34,046 16,822 999,999 99,004 1,981 698,423 21,982 124,306 482,394 821,366 281,016 100,004 98,989 192,498 216,429 26,032 29,260 918,492 98,482 234,234 215,498 132,014 16,215 888,888 4. Subtract the 1st sum from the 3d sum in the above examples. 5- If a clerk earns $1000 a year, how much will he have left after paying his board at the rate of $20 a month ? 6. How much will a girl spend for car fare in a year, if she pays two fares every day except Sundays ? Multiply : 7. 8926 8. 58231 9. 46829 10. 99849 298 8753 389 7689 11. What is the sum of the four products in the above examples ? The four multipliers ? 12. Subtract the sum of the multipliers from the sum of the multiplicands. Add, without copying : 13. $ 742,155.74 14. $8,362.04 429.39 36.09 6,873.68 2,040.03 397.49 295.98 1,956,374.20 3,699.84 284,326.19 1,582.67 MISCELLANEOUS. 149 Multiply without copying : 1. 580092 2. 698468 3. 6940059 9 8 7 Subtract without copying : 4. 28796 5. 36095 6. From 361181 from 398162 from 263069 29999 7. Divide without copying : 86929 by 8 ; by 9 ; by 10 ; by 11 ; by 12. 8. The multiplicand is 3894, the multiplier 12. What is the product ? Prove in two ways. 9. The dividend is 98486, the divisor 94, what is the quotient ? Prove. 10. The dividend is 3846, the divisor 34, the quotient 113, and the remainder 14. Prove. 11. The divisor is 171, the quotient 22, and the remainder , 80. What is the dividend ? 12. The multiplier is 396, and the product is 5148, what is the multiplicand ? 13. The minuend is 9842, the subtrahend 2989. What is the difference ? Prove. 14. The subtrahend is 2869, the remainder 86594. What is the minuend ? 15. The minuend is 64682, the remainder 64497. What is the subtrahend ? Which is the greater, and how much ? 16. 48 X 364 or 63 X 295. 17. 8694 - 356 or 3695 + 4782. 18. 8695 16 or 1000 - 555. 19. Write in Roman, 36,984 ; 28,695 ; 186,005. 20. 189763-4-329. 21. 286976 -f- 57. 150 ARITHMETIC. Indicated Work. Indicate, and solve by cancellation : 1. How many boxes, each holding of a quart, can be filled from 16 quarts of berries ? 2. How many books at f | apiece can be bought for $12 ? 3. If a man earn f of a dollar an hour, in how many hours can he earn $ 15 ? 4. A laborer can build a wall -f? of a rod long in an hour. How many such walls can he build in 3 days of 10 hours each ? 5. At $ I a yard, how many yards of cloth can be bought for Parts of a Dollar. 1. If coffee is worth 25/ a pound, what will be the cost of 3 J pounds ? 2. When tea is worth 37/ a pound, how much tea can be bought for f of a dollar ? 3. At f | a pound, what will 2 Ib. of tea cost ? 4. How many books can be bought for $3f, when one book can be bought for 62 cents ? 5. At 12 JX a quart, how many quarts of berries can be bought for $16f ? Add, without copying : 6. 36984 7. $ 384.62 8. 2683 29.36 95426 2943.99 943 269.46 29 98.43 354 269.62 2983 8936.15 362 26.18 513 827.25 MISCELLANEOUS. 151 1. A man divided his estate worth $ 8694 equally among his 5 children. How much did each receive ? 2. The salary of the President of the United States is $50,000 a year. How much is that a day ? 3. How many yards of linen at 25/ a yard, must be given for 35 bushels of potatoes at 50/ a bushel ? Solve by cancellation. 4. If 10 men earn $2600 in a month, how much can 28 men earn in the same time ? 5. A man earns $50 a week and spends $29 a week ; what can he save in 20 weeks ? 6. I bought 50 sheep for $200, but 10 of them died. At what price must I sell each of the rest so as to lose no money ? 7. What must I ask apiece in order to gain $40 on the sheep that are left ? 8. From the sum of $684.96 and $283.94, take their difference. 9. How many 2-foot steps will a boy take in walking around a square lot 140 feet on a side ? 10. Find the cost of a horse at $250, a carriage at $195, a harness at $38.75, and a robe at $15.35. 11. How much does 28,983 exceed 14,299 ? 12. Write 3,649, multiply it by 12, divide the product by 11, add 482 to the quotient, and from the sum subtract 2872. What is the remainder ? (Indicate.) 13. A grocer bought 50 bushels of potatoes for 48 cents a bushel, and sold them for 64^ a bushel. What was his profit ? 14. If 28 Ib. of butter cost $8.40, what will 350 Ib. cost ? 15. The divisor is 75, the quotient 184. What is the dividend ? 152 ARITHMETIC. Add up, and from right to left : 1. 369 + 2869 + 2864 + 369 = ? 9842 + 3984 + 983 + 25 =? 6894 + 2693 + 4292 + 15 = ? 726 + 526 + 39847 + 324 = ? 32 + 2864 + 122 + 1592 = ? 16 + 294 + 9821 + 296 =? 954 + 36 + 296 + 324 = ? 2989 + 2962 + 3281 + 1692 = ? 287 + 392 + 293 + 264 = ? 21 3050 9834 = ? Find the cost of the following : 2. 5 Ib. sugar @ 7/. 7 Ib. beef @ 14/. 3 bus. oats (3 37/. 9 Ib. coffee @ 35/. 4 doz. eggs @ 24/. 6 gal. kerosene @ 8^. | doz. oranges @ 48/. 4 qt. berries @ 25/. 3. What will 3 bu. and 2 pecks of corn cost, at 45/ a bushel ? 4. What is the cost of 8 Ib. 4 oz. of honey at 16/ a pound ? Of 7} Ib. ? 5. What will 4 bu. apples cost, at 18/ a peck ? 6. What will a pound of candy cost, at 5/ an ounce ? 7. What is the cost of 2 bu. of berries, at 7/ a quart ? 8. How many bushels of potatoes, at 45/ a bushel, will pay for 28 sheep, at $2.94 each ? 9. At 8/ a yard, how many yards of ribbon can be bought for $4.00 ? MISCELLANEOUS. 153 1. What is a fraction ? An integer ? A fractional unit ? An improper fraction ? A mixed number ? 2. Find the sum of f, f , f , and . 5. What is the common denominator of two or more fractions ? 6. What is the least common denominator of two or more fractions ? Reduce to lowest terms : 7- If, iff, , Hi, Hi*, itt- Change the following to 48ths : Reduce to integers or mixed numbers : Reduce to improper fractions : 10. 16, 5$, 268$, 16, 54^, 78J. Add: 11. 28J 12. 6 13. 14J 14. 16{ Ji _8i_ Find the difference : 15. 16* 16. Give results : 20. | of 56. (Problem.) 22. } X 5 X A X }} X 21. f of 144. 23. J of } of f of T ^ of T 154 ARITHMETIC. 1. The sum of two fractions is }| ; one of them is ^. What is the other ? 2. Multiply jj by f . Which is greater, the product or the multiplicand, and how much ? 3. The quotient is f, the dividend $. What is the divisor ? 4. How do you find divisor when quotient and dividend are known ? How do you find multiplicand when product and multiplier are known ? 5. When the sum of two numbers and one of them are known, how do you find the other ? 6. Give an example similar to Example 1. Example 2. Example 3. Find the result. 7- ?-ft = 11. f + i + ? = 4i 8. $ -r- ? = 2 12. ? + ll = 18J 9- X ? = A 13. ? - / 9 = ^ . 10. I - ? = fr 14. ? X ? = i 15. A coal dealer sold 3f tons of coal to one man, and 7$ tons to another. How much did he sell to both ? 16. If a load of hay was bough: for $ll and sold for $16|, what was the profit ? 17. A teamster feeds his horses J a bushel of oats in the morning, bu. at noon, and f bu. at night. How much does he feed daily ? 18. John gave $f for skates, $ for straps, and $| for a sled. How much did all cost ? 19. A merchant sold f of a bushel of seeds to one farmer, 5j to another, and 8 bushels to another. How many bushels were sold in all ? 20. Write a problem in addition of fractions, using a mixed number, a fraction, and an integer. Solve it. MISC ELL ANEOUS. 155 1. Multiply together 3580 and 15, and divide the prod- uct by 24 times 75. (Cancel.) 2. How many loads of apples, each containing 22 bushels, at 50/ a bushel, will pay for 132 Ib. of butter, at 25/ a pound. (Cancel.) 3. Divide two hundred sixty-seven million, one hundred five, by twenty-three thousand, seventeen. 4. Which is worth the more, 24 J tons of hay at $14 a ton, or 68 tons of coal at $4f a ton ? 5. The product of three numbers is 360 ; two of them are 12 and 6. What is the other ? (Indicate and cancel.) 6. 14 x 18 X ? = 12600. (Factor, indicate, and cancel.) 7. When the product of three numbers is known, and the product of two of them, how do you find the third ? 8. Keduce 15^ miles to feet. 9. How many feet in a mile ? How many rods ? 10. Eeduce 5694 oz. to higher denominations. 11. How many pounds in 3 tons of coal ? , 12. Find the cost of 5 Ib. 8 oz. of coffee at 35/ a pound. 13. Eleven ounces is what part of a pound ? (Question.) 14. Find the cost of 6 Ib. 11 oz. of candy at 38/ a pound. 15. At $.05 an ounce, what is the cost of 3j pounds, of cloves ? 16. Eeduce 3694 pints to higher denominations. 17. How many pints will a 5-gallon cask hold ? 18. 3 pints is what part of a gallon ? 19. A farmer sold to a merchant 2 firkins of butter of 56 Ib. each, at 24^/ a pound, and bought 4 boxes of tea weighing 35 Ib. each, at 40/ a pound. Did the merchant owe the farmer, or did the farmer owe the merchant, and how much ? 156 ARITHMETIC. Oral Work. PARTS OF A DOLLAR. Find the cost of : 1. 8 boxes of berries at 12/ a box. 2. 24 Ib. of coffee at 25/ a pound. 3. 36 doz. eggs at 16/ a dozen. 4. 32 pounds of sugar at 6J/ a pound. 5. 36 bars of soap at 8J/ a bar. Make a bill of example 1-5, and supply names and date. 6. How much is f of $1.00 ? f of $1.00 ? of $1.00 ? J of $1.00 ? | of $1.00 ? J of $1.00 ? Find the cost of : 7. 60 Ib. of coffee at 33J/ a pound. 8. 8 yards linen at 37 J/ a yard. 9. 40 bushels of corn at 62/ a bushel. 10. 15 yards of flannel at 66|/ a yard. 11. 8 bushels of wheat at 75/ a bushel. 12. 8 hats at 87J/ each. QUESTIONS OF RELATION. 13. A foot is what part of J yard ? 14. A boy gave his brother 8 marbles, which was f of all 'he had. How many had he ? 15. In a school there are 20 boys, f of the pupils are girls. How many pupils in the school ? 16. A pint is what part of a gallon ? 17. A lineal foot is what part of a lineal yard ? 18. A square foot is what part of a square yard ? 19. f of my sheep are in one pasture, and the remainder, 12 in number, are in another. How many in both pastures ? 20. What part of f is f ? MISCELLANEOUS. 157 1. How many steps of 25 inches each must a person take in walking 15 miles ? 2. Paid $565.85 for 6500 pounds of honey, and sold it at 12^ cents a pound. What was the whole gain ? 3. Paid $189 for flour at $6| a barrel. How many barrels did I buy ? 4. The area of Illinois is 55,405 square miles, and of "New York 49,170 square miles. How much larger than New York is Illinois ? 5. The difference between two numbers is 4,031, the greater number is 8,456. What is the smaller ? 6. The multiplicand is 6,412 and the multiplier 29. What is the product ? 7. The divistfr is 48, the quotient 514, and the remain- der 13. What is the dividend ? 8. What is the cost of 8700 Ib. of hay at $11 a ton ? 9. The product of three numbers is 42,512. One of the numbers is 164, and another 15. What is the third number ? 10. If 216 cabinets cost $211,320, what will 340 cost ? 11. When hay is $25 a ton, how many barrels of flour, at $9 a barrel, will exactly pay for 45 tons of hay? (Indicate.) 12. Reduce f } to lowest terms. 13. Find the sum of f , f , and 4^. 14. If 9J yards cut from a roll of cloth leaves 34f yards, what is the length of the roll ? 15. What is the value of 3| times of |f ? 16. Find the number that must be added to f of ^ to make | of ^. 17. What fraction must f be divided by to give 3jr ? 158 ARITHMETIC. 1. Change ^ to a fraction having 84 for its denominator. 2. How many times is ^ + i + i contained + + ? 3. What fraction of 5 bushels is ^ of 3 bushels ? 4. Reduce f of ^\ of ^ of 8 j to a simple fraction. 5. Sold a horse for $105f, which was J of its cost. What was its cost ? (Question.) 6. What cost 22 tons of coal at $3 J a ton ? 7. Bought 3J bu. of corn for $4|, what was the price ? 8. When $1.25 is paid for of a cord of wood, what is the cost of one cord ? Of 12f cords ? 9. If 30 men can build a wall in 20 days, in how many days can $ as many men build it ? 10. If | of a store is worth $6000, what is the whole store worth ? 11. Find the sum of $13.50, $2.63, $5000, $.32, $6.13. 12. What is the cost of 60 Ib. of crackers at $.14 a pound, and 142 loaves of bread at $.08 a loaf ? 13. What will 2134 pair of boots cost at $1.75 a pair ? 14. What will 212 bu. of oats cost at $1.37 a bushel ? 15. What will 75 yd. of cloth cost at 66f / a yard ? 16. At $5.12 a dozen, how many dozen knives can be bought for $30.75 ? 17. At $.87 each, what must be paid for 24 books ? 18. How many rods of fence are required to enclose a lot 25 rd. wide and twice as long ? (Diagram.) 19. At $2 a foot, what will it cost to build a stone wall 24 rods long ? 20. What is the distance in inches around a room 22 yards square ? 21. What cost 27 in. of ribbon at 60 cents a yard ? 22. Define a rectangle. A square. Area. MISCELLANEOUS. 159 1. What part of an acre is 130 sq. rd. ? 75 sq. rd. ? 2. Bought nuts for $2 a bushel and sold them for 10 cents a quart. What did I gain on 5 bushels ? 3. I burn a pint of kerosene every night. What will a 4 weeks 7 supply cost me at 18 cents a gallon ? 4. If 200 Ib. of flour can be packed into a barrel, how many barrels will be needed to pack 4 T. 10 cwt. ? 5. What cost 3 Ib. 8 oz. of butter at 32/ a pound ? 6. What is the difference between 25 long tons, and 28 common tons ? 7. Explain the difference in the use of a long ton and a common ton. 8. A coal dealer bought 10 car loads of coal, each car containing 14 long tons. How many short tons did he buy ? (Indicate and cancel.) 9. How many hours in f of a week ? 10. How many dozen pint bottles will be needed to hold 5 gal. 3 qt. of milk ? 11. A grocer buys a barrel of oil containing 40 gal. for $40, and sells it at $.30 a pint. What does he gain ? 12. Reduce 12 A. 144 sq. rd. to square feet. 13. Reduce 60 gal. 3 qt. 1 pt. to pints. 14. Eeduce 48 miles to feet. 15. Change -$y of a mile to feet. ' 16. What part of 2 bu. is 1 bu. 5 qt. ? (Question.) 17. A rectangular lot containing 2648 sq. ft. has a front- age of 262 feet. What is the other dimension ? 18. If the top of your desk is 2j ft. long and 18 in. wide, how many yards in the four sides ? 19. It is l miles to the post-office. How many miles must I travel to go the post-office and back four times ? 160 ARITHMETIC. 1. What costs a mile of wire at 3 cents a foot ? 2. At 5 cents a quart, what will be my milk bill for May and June if I take 3 quarts daily ? 3. 2 of 48 is ^ T of what number ? (Question.) 4. A farm is -J mile square. How many acres does it contain ? 5. How long must a field be, that is 40 rd. wide, to contain 5 acres? 6. What will 8 Ib. 15 oz. of sugar cost at If a pound ? 7. In a school, T 7 y of the pupils are girls ; there are 90 boys. How many pupils in the school ? 8. What cost 18 dozen blank-books at 33 J/ each ? 9. How many 3-inch squares in a square foot ? 10. How many yards of carpet J yd. wide will be re- quired for a room 18 ft. square ? 11. The area of a sidewalk a mile long, is just one acre ; how wide is it ? (Indicate and cancel.) 12. If an iron rail is 33 ft. long, how many rails will be required to lay a double-track railroad 10 miles long? 13. I have $570 in my safe, which is f of what I have in the bank. How much money have I ? 14. How many square yards in a room 28 ft. long and fifteen feet wide ? 15. How many square yards in a piece of calico 48 inches wide and 48 yards long ? 16. I have a piece of land 200 ft. deep, and fronting 250 ft. on the street. I cut it into 50-foot lots, and fence each lot. What will my fencing cost, at $5 a rod ? 17. How many times will a wheel 5 ft. in circumference turn in going 26 miles ? 18. How many square feet in a room 4 yd. wide and 20 ft. long? MISCELLANEOUS. 161 1. How many acres in a street 50 feet wide and f of a mile long ? 2. How many square yards of silk in 400 feet of 4-in. ribbon ? 3. From an acre of land I sold 10 square rods and also a piece 10 rods square. How much have I left ? 4. A field containing 6 acres is 600 ft. long. What is the distance around it ? 5. From the product of 3* and 4J take half their difference. 6. If 7 barrels of flour cost $47, what will 21 barrels cost ? (Indicate and cancel.) 7. Write 1567849 in words ; in Roman Notation. 8. What is the difference between 32 x 534 and 63 X 542? 9. How many more times is 18 contained in 216 than it is in 72 ? 10. Multiply 462 by 12, and write the product in figures and words. 11. What number must be added to 365 to make 840 ? 12. What number taken from 8,473 will leave 2,416 ? 13. 7,532 less what number will give 457 ? 14. How many hours are there in January ? 15. The product of two factors is 216, one of the factors is 12 ; what is the other ? 16. Two men, who are 1550 miles apart, travel towards each other, one 36 miles a day, the other 40 miles a day. In how many days will they meet ? 17. Divide 246,752 by 41, and prove your work. 18. What is the difference between a figure and a number ? 162 ARITHMETIC. 1. If 9578 is the remainder and 709 the subtrahend, what is the minuend ? 2. Divisor is 46, quotient is 52, remainder is 7. Find dividend. 3. How many pounds of sugar at 9 cents a pound, can you get for 18 dozen eggs at 25 cents a dozen ? 4. Multiply 752 by 18 ; add 234 to the product ; subtract 562 from the sum ; and divide the remainder by 24. 5. Divide the product of the sum and difference of 222 and 66 by 72. (Indicate.) 6. Write in figures, MDXLI ; XCXCIX. 7. Bought 18 Ib. of steak at 24/, 4J doz. eggs at 36/, 3 qt. of molasses at 18/, and a bushel of potatoes for Make out in the form of a bill, and receipt it. 8. How many 5-cent pieces in $840 ? 9. 6020 X 40 X 12 = 18 X ? 10. Eeduce to lowest terms : ffc 11. Change to an improper fraction 12. Multiply 175f by 12 ; 124 by 6| ; 6| by 12|. 13. Eeduce to whole or mixed numbers : ^ff A ; ^ly 11 . 14. Divide the product of 4J and 6| by 5. 15. Add : 12|, 16|, 24f , 40J. 16. What is the difference between 84 J and 42| ? 17. If a man travels 230 miles in 5j days, how far can he travel in 3J days ? (Indicate and cancel.) 18. If f of a farm is worth $7000, what is the whole farm worth ? 19. Multiply f of 12| by 36J. (Cancel.) 20. Divide 27| by f of 8J. 12. A boy paid $4 for shoes and f 9f for a coat. How much more did the coat cost than the shoes ? MISCELLANEOUS. 163 1. A stone wall 5 ft. high encloses a field 175 feet long and 53 ft. wide. How long is the fence ? 2. A farmer kept his sheep in two pastures ; one-half of his flock was in one, and 53 in the other. How many had he ? 3. A fourth of a hogshead of oil leaks out. How many pint bottles can be filled from the remainder ? 4. Give four factors of 48. 5. What is the difference between 5 square feet and 5 feet square ? (Illustrate.) 6. How many square yards in the surface of a desk 26 feet long and 18 inches wide ? 7. Reduce 2013 gills to higher denominations. 8. How many bricks, 8 in. by 4 in. by 2 in., laid on the side, will be required to pave a walk 60 ft. long and 5 ft. wide ? (Indicate and cancel.) 9. Make a receipted bill of the following, sold to yourself: 5J yd. of cloth at $1.75; 2 pair of shoes at 10. How many square feet of sidewalk 8 ft. wide will be needed to make a walk in front of a lot 8 rd. wide ? 11. How many lots of 20 sq. rd. each will 15J acres of land make ? 12. How many feet in a mile ? in f of a mile ? in 17| miles ? 13. Find the product : X | X f X j- X 3 X 4. 14. Change f to a fraction having 153 for a denominator. 15. The sum of two numbers is 47 f , and one of them is 18$. What is the other ? 16. Divide 14| by f of 19. 17. What part of two yards is two feet ? 164 ARITHMETIC. Addition. The following table shows the number of the boys and girls registered, and the average attendance in the schools of Syracuse for the year 1893. Add and make the sums of the totals alike. NUMBER REGISTERED. AVERAGE DAILY ATTENDANCE. SCHOOLS. m 1 d 00 3 3 & O I 1 O "o ri High .... 373 558 290 435 Salina .... 273 261 215 207 Jefferson . . . 168 140 . 142 118 Genesee . . . 263 197 , t 210 158 Prescott . . . 358 329 287 257 f Seymour . 420 502 325 388 . Putnam . . . 402 457 . 332 364 . Irving .... 225 198 . 172 158 . Delaware . . . 287 307 221 228 . Porter. . . . 485 515 365 375 Gere .... 196 198 158 160 . Danforth . . . 195 207 163 159 Brighton . . . 241 292 182 213 . Rose .... 9 9 . 8 7 . Clinton . . . 289 270 . 228 218 Vine .... 109 122 . 82 89 Grace .... 209 179 . 173 145 Merrick . . . 177 184 152 153 Madison . . . 389 409 315 311 Bassett . . . 92 81 78 72 Frazer .... 223 243 183 207 Townsend . . 190 199 170 173 . Franklin . . . 570 514 . 469 445 May .... 289 302 219 254 Montgomery 324 260 286 225 . Adams . . . 163 168 144 149 Total . . . Total, PARTS OF A DOLLAR. 165 Oral Work. PARTS OF A DOLLAR. 1. At 6 J cents apiece, how many pencils can be bought for i of a dollar ? for $ J ? for $4 ? 2. At 8j/ a box, how many boxes of berries can be bought for $1 ? for $ \ ? for 75 cents ? for $2 ? 3. At 12^ cents each, how many tablets can be bought for $3 ? for $ \ ? 4. At 16f/ a dozen, how many dozen eggs can be bought for $5 ? for $? 5. How many pounds of coffee at 25/ a pound, can be bought for $3.00 ? for $ ? 6. How many straw hats can be bought for $2.00 at 33J cents each ? 7. At 37J/ a pound, how many pounds of coffee can be bought for $3.00 ? for $J ? 8. How many days must James work, at 50/ a day, to earn $5.00 ? QUESTIONS OF RELATION. 1. A man deposited in the bank f of his money which was $3000. How much had he ? 2. What part of a yard is one-half a foot ? 3. What part of the week is passed at midnight on Wednesday ? 4. John had $12. He spent $4 for a coat. What part of his money remains ? 5. A gentleman divided 50 cents among 5 children. To four of them he gave f of the money. How much did the fifth receive ? 6. At midnight on the 16th of June, what part of the month has passed ? 166 ARITHMETIC. 1. If J of of a piece of land is worth $5000, what is the whole piece worth ? 2. Change ^, ^, ^ to lowest terms. 3. If a man can travel 15f miles in one day, how far can he travel in 5 J days ? 4. A man received $91 J for working 30 J days. How much did he earn a day ? 5. What number is that from which, if 2Sf is taken, the remainder will be 40 J ? 6. Find the cost of f. of 130 acres of land at ^ o f $81 an acre. (Indicate and cancel.) 7. What number multiplied by $ will give 4f ? 8. In 5g acres, how many building lots of $ of an acre each? 9. What part of 10 miles is of 12 miles ? 10. f of a farm is worth $8000. What is the whole farm worth ? 11. If 3 be added to both terms of the fraction f , will its value be increased or diminished, and how much ? 12. A girl sold a pair of skates for $1^, which was f of their cost. How much did she lose ? 13. Multiply 25-| by f . 14. How many acres in of a mile of street 66 ft. wide ? 15. If 3} yd. of gingham cost 22 cents, how many yards can be bought for 75 cents ? 16. What part of a ton is 175 Ib. ? (Question.) 17. How many sheep, at $8 each, can be exchanged for 12 horses at $250 each ? (Indicate and cancel.) 18. Divide by |. 19. If | of my salary is $1200, what is my salary ? MISCELLANEOUS. 167 1. Fifteen men do a piece of work in 8 days. How long will it take 8 men to do it ? 2. Divisor is 3f, quotient is 4f . Find the dividend. 3. If If yd. of silk cost $llj, what will one yard cost ? 4. If a roll of carpet containing 80 yards is worth $160, what is f of a yard worth ? 5. How many quarts of nuts at 8^ cents a quart, will it take to buy 5 J bu. of apples at $} a bushel ? 6. From a hhd. of molasses containing 56 gal. 2 qt., there were sold 25 gal. 1 pt. How many quarts remained ? 7. Change 29 gal. 2 qt. 1 pt. to gills. 8. How many rods of fence will be required to enclose a rectangular field 95 yd. long and 60 yd. wide ? 9. A farmer divides his farm of 425 acres into fields of 12 acres each. How many fields has he ? 10. If 2f yd. of ribbon cost 45 cts., what will 3 yd. cost? 11. How many yards in 5 pieces of cloth, each contain- ing 19 yards ? 12. The product is 960, the multiplier is 52. Find the multiplicand. 13. The width of a room is f of its length. How many square feet in the floor, if the width is 18 ft. ? 14. ,J of what number equals 200 ? 15. How many square yards in the four walls and ceil- ing of a room 20 ft. long, 15 ft. wide, and 12 ft. high ? 16. A lady had $36.84. She spent J of it for a cloak and ^ for a pair of shoes. How much was left, and what fraction of the entire sum was it ? 17. How many yards of carpet, 1 yd. wide, will be needed for a room 24 ft. by 18 ft. ? 168 ARITHMETIC. Written Work 1- J X | 8. fX | 3. |X J 4- H x A 6. X A 16. 38946f X 9 17. 25986JI X 8 18. 39825 X 6 19. 298f J X 4 20. 39846^ X 25 6. 32-j X 4 7. 14J X 16 8. 135 X 18 9. 75J X 15 18^x20 10. 11. 16 X 4J 12. 18 X 8-J 13. 46 X 3f 14. 129 x 15f 15. 161 X 25 T 8 7 21. 36982 x 5| 22. 4986 X 16f 23. 28761 X 18 24. 39840 X 16H 25. 26358 X 4| Reduce the following compound fractions to simple fractions : 26. |of Jof A 29. fcofiJofH 27. | of JJ of if 30. of | of | 28. fof JJofff 31. f off of A Divide, and prove by multiplication : 1. M - A 6. 35| --5 11. 26 -*- 2 i 2. '** 7. 26| -f-7 12. 13 -If 3. ft-* A 8. ISA -r-5 13. 16 -^- 2 i 4. if - T\ 9. 15f -j-12 14. 84 -5^ 5. t-M 10. 121| -f-6 15. 69 -7| 16. 2834^ -J- 6 20. 21983 ~=~ i 17. 3984 --5 21. 39562 -T- ^ r 18. 2693f -8 22. 17523 -r- I ! 19. 2345f -^ 6 23. 28884 -5- 1 b 24. i of 1 -*- f Of f 27. T X 1 X T 6 T -5- f X6 yY 25. f of i- |of A 28. 1 x ixi i ^~ A x H 26. } of li-H |of| 29. HX, li-*- 1 IX A X 2 REVIEW. 169 1. Find the cost of 15 doz. collars at $.16| each. 2. Paid $24 for cuffs at 16 f/ a pair. How many dozen pairs were bought ? 3. If one bushel of wheat is worth $1.37^, how many bushels can be bought for $198 ? 4. Divide $3,542,305 by 864. 5. Divide of 24 qt. of nuts among 20 boys. 6. If 4 qt. of nuts cost 25 cents, what will half a bushel cost? (Indicate and cancel.) 7. If I of a load of hay is worth $7, what will 2 loads be worth ? (Indicate and cancel.) 8. 2J X 3 = ? 2J -r- 3 = ? 9. At 80/ a lb., what will 5 oz. of tea cost ? 10. How many sq. yards in the surface of a table 12 ft. by 5 ft. ? 11. of 16 is of what number ? 12. 108 is J of $ of what number ? 13. If one man can do a piece of work in 12 days, how long will it take 5 men to do it ? 14. A field is 825 ft. long and 396 ft. wide. How many rods of fence will be required to enclose it ? 15. How many bushels of oats at $f a bushel, will pay for g of a barrel of flour at $4.75 a barrel ? 16. How many arithmetics at 37J/ each can be pur- chased for $384 ? 17. How many pounds of butter at 25/ a pound can be bought for $8.50 ? at 37/ ? 18. How many dozen eggs at 12 \p a dozen can be bought for $3 ? for $11 ? for $3.75 ? 19. At $.62^ a yard, how many yards of cloth can be bought for $31.50 ? 170 ARITHMETIC. 1. What are unlike fractions ? How are they added ? 2. Write three unlike fractions and add them. 3. Write two unlike fractions and find their difference. 4. Write a compound fraction having four simple frac- tions and change it to a simple fraction. 5. Write a complex fraction with an integer for the numerator and a fraction for the denominator, and reduce it to its simplest form. 6. Write three other forms of complex fractions and change them to simple fractions. 7. Can you write still other forms of complex fractions ? 8. James had } of an apple and gave J of of it to his brother. What part did he give to his brother ? What part remained ? 9 --, ? ' 2* f 13J- 4 10. How many days in of a common year ? 11. How much cloth will be required for 3 doz. vests allowing f yd. for each ? 12. What will one orange cost if 3 oranges cost 5/ ? 13. How many days must a boy work to earn $5^ if he earns $| a day ? 14. 168 is f of what number ? 15. of 12 are how many times 3 ? 16. of 12 is of what number ? 17. I bought 3 tons of coal, and used f of a ton one month and T 9 ^ of a ton the next month. How much was left? 18. A, B, and C own a melon ; A owns J, B f , and C the rest. What part does C own ? 19. 3 pints is what part of 3 gallons ? (Question.) REVIEW. ^gA- CALiFO 171 1. What is a factor ? a prime number ? a multiple ? 2. What are the prime factors of : 72; 64; 48; 32; 80? 3. Write all the prime numbers between 10 and 40. 4. Write the composite numbers between 59 and 71, and find their prime factors. 5. What are the prime factors of 484 ? 6. What is the greatest common divisor of 25 and 150 ? 7. 12 X9X6X 20 --2x3x3x4x5. 8. 75 X 33 x 2 x 125 -- 16 X 150. 9. What number is a multiple of 5 ? 10. Write four numbers that are multiples of 6. Are there any more ? What is the least multiple of 6 ? 11. What is the least common multiple of 18, 28, and 30 ? 12. Find the sum of the prime numbers from to 30. From 30 to 60. 13. Divide 504 by 82, using prime factors and cancel- lation. Prove the work. 14. Divide 28 X 56 X 15 by 6 X 5 x 14. (Cancel.) 15. At 7/ a pound, how many pounds of sugar can be bought for $60.55 ? 16. A man bought 700 bushels of wheat at $.95 a bushel. He sold 256 bushels at $1.10 a bushel, 45 bushels at $1.15 a bushel, and the remainder for $395. How much was his gain ? 17. At the rate of 6 pencils for 48 cents, how many pen- cils can be bought for 32/ ? (Indicate.) 18. pt. is what part of f pt. ? 19. If gold is worth $18.60 an oz., what is the value of a cup that weighs 6 oz. 5 pwt. ? 20. (5 + 3) X 4 minus 5 + 3x4 = ? 172 ARITHMETIC. 1. What number other than 71 will evenly divide 8662 ? 2. Why do you change fractions to a common denom- inator before adding them ? 3. Keduce T %, ? \, }f to 205ths. 4. Edwin had $25j. He spent $J for skates, $1J for gloves, and $6 J for a suit of clothes. How much was left ? 5. 13J is divisor, 1080 is quotient, 12J remainder. Find the dividend. 6. If 6 spoons cost f 9|, what will 15 spoons cost ? (* + %) X M 14J - 2J 8. 8 is what part of 13 ? 9. 76 -f- 228 = ? 4J -r- ? = 10. 10. ^ T of a hhd. leaked out. How many 2 pint cans can be filled from the remainder ? 11. ^ of a flock of sheep was sold and 27 were left. How many in the flock at first ? 12. How many square inches in an 8 ft. square ? 13. Find the area in square yards of a lawn 35 by 28 ft. 14. How many lots 10 rd. deep and 5 rd. wide can be sold from a 5 A. field ? 15. J. F. Worclen bought of Dey Bros, the following: 18 yd. lace at 16|/ a yard, 4 dozen ties at $1.50 a dozen, 3 dozen handkerchiefs at $.25 each. Make out the bill, and receipt it ? 16. 4003 X 1000 -f- 201 = ? 17. Bought 25 Ib. of butter at 25 cents a pound. I gave in part payment 35 Ib. of cheese at 12 cents a pound, and paid the remainder in cash. What did I pay in cash ? REVIEW. 173 1. How many cheeses of 45 Ib. each, at 12^ cents a pound, will pay for 15 bbl. of apples, each containing 3 bu. at 87J ct. a bushel ? (Indicate.) 2. 13| + 14|i - 4J = ? 3. Multiply seven thousand and five by three hundred five. 4. Add 210 ; 15,262,986 ; 15,438 ; then multiply \ of the sum by 130. 5. From 125| yd. of cambric were sold 87 T \ yd. What remained ? 6. 8754321 +- 873 = ? 9853021 + 901 = ? 7. How many half-pint bottles will be needed to hold 2J gal. of catsup ? 8. Reduce 21240 pt. to bushels, etc. 9. A train moves at the rate of a mile in 2 min. In how many hours will it go 330 miles ? 10. A man having 420J miles to go has traveled 193f mi. How many miles has he still to go ? 11. How long will 20 bu. of oats last a horse if he is fed 8 qt. a day ? 12. Divide $9340526 equally among 105 people. 13. At 3 ct. a gill, what will 560 gal. cost ? 14. 16 J + lOg- + 120 + 96 + 38^ = ? 15. Remainder 56|, minuend 379 ; what is the subtra- hend? 16. Subtrahend 297^ ; remainder 194 ; find the minu- end. 17. U648 + 9718 + 23149 + 7116 + 73999 + 1476 = ? 18. At $1.75 a sq. rod, find the cost of 6 acres of land. 19. (52 + 35) X 6 plus 52 + 35 X 6 = ? 20. 16 + 3 x5 minus 31 = ? 174 ARITHMETIC. 1. Write an improper fraction, and change it to a mixed number. 2. Find the product of f , |f , f, f . 3. Divide the largest of the above fractions by the smallest. 4. Buy 4 bu. 2 pk. of nuts at $1.60 a bu. 5. How many barrels of flour will 19236 Ib. make ? 6. A boy spent f of his money, and had $1.50 left; how much had he at first ? (Question.) 7. I bought 8 cwt. 10 Ib. of sugar for $81, and sold it at 12/ a pound. What was my gain ? 8. How many square feet in a floor 20 ft. 9 in. long and 17 ft. 6 in. wide ? 9. If a lot is 198 ft. long, and 132 ft. wide, how many rods around it ? 10. A lot contains 3 A. and is 20 rd. wide ; how long is it? 11. Change 8f to 25ths. 12. Find the product of 7 j- and 9J. 13. The product of two factors is 25 J, one factor is 8 ; what is the other ? 14. I bought 7 bu. of apples at $.67 a bu. How much change should I have left from a two-dollar bill ? 15. Multiply 2i by j- and divide the product by Ij. 16. John had $3.60 left after spending | of his money. How much had he at first ? (Question.) 17. Find the area of a floor 18J ft. long and 4 yd. wide. 18. How much will 8 bu. 3 pk. of potatoes cost at $.20 a peck ? 19. 9f + 7|.+ 6A = ? 20. Name two equal factors of 121. CHAPTER IV. PLASTERING AND PAINTING. Plastering and painting are usually done by the square yard. Deductions are frequently made for openings. Windows and doors are openings. 1. A room is 18 ft. long, 12 ft. wide, and 8 ft. high. How many square yards in the walls and ceiling ? In problems of this kind first make a diagram represent- ing the four walls in a line. The ceiling may be represented as extend- ,!? FT.__ ^ ing from a side wall or an end wall. Draw this diagram (enlarged) on stiff paper, then cut and fold the pattern to rep- resent the walls and ceiling of the room. o z 18 FT. 12 FT. 18 FT. 12 FT. SOLUTION. The length of the four walls is 2 X (18 ft. + 12 ft.) = 60 ft. ; Area of 4 walls, 60 ft, by 10 ft. = 600 sq. ft. Area of ceiling, 18 ft. by 12 ft. = 216 sq. ft. Area of four walls and ceiling = 816 sq. ft. = 77f sq. yd. 2. If in the room mentioned above there are two doors, 3 ft. by 7 ft., and 4 windows 3 ft. by 6 ft., how many square yards remain in the four walls and ceiling ? Represent these doors and windows on the diagram. 175 176 ARITHMETIC. To THE TEACHER. See that the pupil draws all dia- grams to scale that is, in proper relative proportions. 3. How much will it cost at 25 cents a square yard to plaster the room mentioned in Example 1, full deductions being made for openings ? 4. What will it cost at 33J cents a square yard to plas- ter the walls and ceiling of a room 24 ft. long, 18 ft. wide, and 9 ft. high ? The room has 3 doors 3 ft. by 7 ft., and 6 windows 3 ft. by 6 ft. One-half of the openings deducted. (Diagram.) 5. How many square yards of plastering in a room 40 ft. long, 30 ft. wide, and 12 ft. high, deducting 80 sq. ft. for openings ? 6. At 35/ a sq. yard, what will it cost to plaster a room 18 ft. square and 9 ft. high, having 3 doors 7 ft. by 3 ft., and 5 windows 6 ft. by 3 ft., no allowance for openings ? 7. A lot 100 ft. by 200 ft. is enclosed by a tight board fence 5 ft. high. At 6j/ a square yard, what will it cost to paint both sides of it ? 8. What will it cost to paint the outside of a freight car 3 coats, at 7 cents a sq. yard for each coat, the car being 25 ft. long, 9 ft. high, and 8 ft. wide ? 9. Write an original problem for plastering a room. Make a diagram of it, and solve. 10. Write an original problem for painting. Make dia- gram, and solve. 11. The room in Example 6 has a wainscoting 3 ft. high. How many square yards of wainscoting ? 12. Plaster a room having the following dimensions : length 40 ft., width 28 ft., height 16 ft. Deduct for 12 windows, 7 ft. by 3 ft., and 4 doors 9 ft. by 3 ft. 6 in. What is the cost at 25 cents a square yard ? CARPETING ROOMS. 177 13. Paint the outside of a cubical box, 3 ft. X 3 ft. X 3 ft. with 3 coats of paint. What is the cost at 6 cents a square yard for each coat ? 14. How many square feet of boards necessary to con- struct a box 4 ft. long, 3 ft. wide, and 2 ft. high, with cover ? 15. How many flag-stones 5 by 5 ft. will be required for a sidewalk covering 275 sq. feet ? 16. At 18/ a square yard, what will it cost to plaster the ceiling and the four walls of a room that is 18 ft. long, 15 ft. wide, and 9 ft. high ? 17. At 30/ a square yard, what will be the cost of ce- menting the walls and floor of a basement 40 ft. long, 30 ft. wide, and 8 ft, high? 18. A cistern is 12 ft. square and 9 ft. deep. What will it cost to cement it at 12 / a sq. yard ? 19. What will be the cost of turfing a lawn 50 ft. by 38 ft. at 9/ a square yard ? 20. What is a rectangle ? CARPETING ROOMS. In carpeting rooms, it frequently occurs that more car- peting must be purchased than is needed to cover the floor. This waste is often necessary to make the patterns match. A part of one outside strip must sometimes be turned under. All this waste must be paid for by the purchaser. If borders are used, there is a waste of % the breadth of the border in mitering. (Illustrate.) 1. How many sq. yards in the surface of a floor 18 ft. long and 15 ft. wide ? (Diagram.) 2. How many lineal yards of carpet 1 yd. wide must be purchased to cover it ? 178 ARITHMETIC. 3. How many lineal yards of carpet yd. wide, will be needed to cover it ? How long and how wide is each lineal yard ? What part of a square yard will it cover ? Ans. sq. yd. How many times is J sq. yard contained in 30 sq. yd. ? 4. How many yards of carpet } yd. wide will be required for a hall 18 ft. wide and 30 ft. long, the strips running the long way of the hall ? How many yards must be pur- chased if the strips run the short way ? Make a diagram showing the strips or breadths. How many lineal yards in each strip ? How many strips ? SOLUTION. Since one strip covers a space yd. wide, it will take as many strips to cover a space 6 yd. wide, as yd. is contained times in 6 yd. = 8 strips. 8 strips each 10 yd. long = 80 lineal yards. 5. At $1.25 a yard, what will it cost to carpet a hall 11 J ft. wide and 28 ft. long, the carpet being 1 yard wide. How many yards long is each strip ? How many strips ? 6. How much would the carpet cost if there were a waste of 1 J yd. for matching ? NOTE. The latter of the two methods indicated above is the more practical. . 7. How many yards of ingrain carpet 1 yd. wide will be required for a floor 17 ft. wide and 20 ft. long, strips run- ning lengthwise ? (Diagram.) How many yards long is each strip ? How many strips would be needed if the room were 15 ft. wide ? How many if it were 18 ft. wide ? It being 17 ft. wide, how many strips must be purchased ? How much of the last strip is to be turned under ? CARPETING ROOMS. 179 8. How many yards of carpet f yd. wide will be re- quired to cover a floor 25 ft. square ? 9. Write an original problem for carpeting a room. Draw a diagram, and solve it. 10. At $1.15 a yard, what will be the cost of carpeting a room 8 yd. long and 5 yd. wide, with carpeting J yd. wide, allowing If yd. for waste, and $3 for making and laying ? 11. A sidewalk 10 ft. wide is to be laid on the 4 sides of a lot 100 ft. square. At 25/ a sq. foot, what will be the cost ? The walk will be 110 ft. long on each of the four sides, as shown by the dia- gram. The entire length of the walk is 4 X 110 ft. = 440 ft. ; this multiplied by the width of the walk will give its area. 12. What would be the cost of a 5-foot walk laid on the inside of the lot, close to the sides ? 13. A man builds a house 45 by 35 ft., on a lot 60 by 150 ft. How many sq. yards has he left for a garden ? 14. What would it cost to build a tight fence 5 ft. high, around the above lot, at 25/ a square yard ? 15. What is the cost of a farm 80 rods square, at $50 an acre ? 16. How long is a rectangular field that contains 50 acres and is 80 rods wide ? i 110 FT. 100 FT. 1 8 JJOOL JJ OU 180 ARITHMETIC. PAPERING WALLS. A roll of wall paper is commonly 8 yards long and 18 inches wide. A double roll is 16 yards long. Borders are sold by the lineal yard. 1. A room is 18 ft. by 15 ft., and 9 ft. high ; making no deduction for openings, how many square yards in the four walls and ceilings ? (Diagram.) 2. How many sq. yards will one roll cover ? 3. How many rolls will be needed to paper the room ? 4. If the room has 2 doors 3 ft. by 8J ft., and 4 win- dows 3 ft. by 6 ft., how many sq. yards of surface are to be covered ? 5. How many rolls will cover it ? 6. At 33 \fl a roll, what will be the cost of papering ? 7. At 50/ a lineal yard, what will the border cost ? 8. A hall is 12 ft. wide, 38 ft. long, and 10 ft. high. The area of the openings is 117 sq. feet. (Diagram.) The paper costs 38/ a roll. The border costs 66|/ a yard. Cost of putting on: 2 men 1J days each at $2 a day. Full deductions made for openings. Find the entire cost and make bill. 9. What will be the cost of carpeting the hall men- tioned above, with carpet f yd. wide at $1.50 a yard ? 10. A sitting-room is 20 ft. by 17 ft., and 10 ft. high. It has 3 doors 2 ft. 6 in. by 7 ft., and 4 windows 3i ft. by 6 ft. How many rolls of paper will cover the walls and ceiling, deducting \ the area of the openings ? 11. At $1.15 a yard, what will it cost to carpet the above sitting-room with carpet 1 yd. wide ? BEVIEW. 181 Review. 1. How many square yards in a lot 16 rods long and 10 rods wide ? 2. How many lineal feet of fence will enclose this lot ? 3. A farmer divided his farm, containing 150 acres, 3 roods of land, among his 4 sons and 2 daughters equally. How much did each receive ? 4. How many lots of 20 sq. rods each will J of a square mile make ? 5. How many rods of fence will enclose a lot J of a mile square? 6. How many 4-in. squares may be cut from a square foot of paper ? 7. What part of an acre in a field 30 rods by 4 rods ? 8. How many square feet of glass in 6 windows of 8 panes each, each pane being 14 by 12 inches ? 9. What will it cost to pave a street 2^ miles long and 2 rods wide, at $12 per square rod ? 10. How many yards of oil-cloth 8 ft. wide will cover a floor 32 by 22 ft., if the strips run lengthwise, allowing 1 yard for waste ? 11. How many yards of crash 18 inches wide will cover a floor 18 ft. square, allowing If yards for waste ? 12. How many sq. yards to be plastered in the ceiling and walls of a room 26 by 20 ft. and 12 ft. high, there being 3 windows 6 ft. by 2 ft. 8 in., and 2 doors 8 by 3 ft. ? 13. How many square feet of boards in a tight fence 10 rods long and 6 ft. high ? 14. How many sq. rods in a garden 231 ft. long and 165 ft. wide ? 15. What will it cost, at 25/ a sq. foot, to construct a slate blackboard 34 ft. 3 in. long and 4 ft. 6 in. wide ? 182 ARITHMETIC. 1. How many square feet in 6 boards each 9 ft. by 9 in. ? 2. How many farms of 16 acres can be made from half a square mile of land ? 3. A lot contains l acres, and is 60 rods long. How wide is it ? 4. How many square inches in a board 8 ft. long and 9 in. wide ? 5. At $5.80 per square rod, what will of an acre of land cost ? 6. Find the square contents of a floor 21 J ft. long and 15J ft. wide. 7. Find the number of planks, 12 ft. long and 9 in. wide, required to lay a floor 22 by 16 ft. 8. How many rolls of paper 24 ft. long and Ij ft. wide will be needed for the walls of a room 28 by 24 ft. and 9 J ft. high, no allowance for openings ? 9. What will the paper cost at $.37 J per roll ? 10. How many yards of oil-cloth f of a yard wide will this room require if the strips run lengthwise, and there is a waste of 1 yards for matching ? 11. What will the oil-cloth cost at $.25 per yard? 12. How many yards of carpet of a yard wide will be required for a room 24 ft. long and 18 ft. wide, if the strips run lengthwise, and there is a waste of 9 in. for matching in each strip ? 13. A ceiling 25 X 30 ft. contains how many square yards ? 14. What is the difference between a surface of 6 square inches and one 6 inches square ? 15. What is the cost of a rectangular field 40 rd. wide and 92 rd. long, at $63 an acre ? 16. How many square inches in of a sq. yard ? REVIEW. 183 1. How many square yards to be plastered in a room 22 ft. long, 20 ft. wide, 10 ft. high, the wainscoting being 3 ft. high., deducting 100 sq. ft. for openings ? 2. How many bricks, 8x4 inches, will pave a walk 200 ft. long and 1 rod Avide ? 3. What will be the cost of 8 slate blackboards 30 ft. by 3 ft. 4 in., at 25 cents a square foot ? 4. How much will it cost to lath and plaster the four walls and ceiling of a room 20 ft. long, 16 ft. wide, and 10 ft. high, at 25 cents a square yard ? No allowance for openings. 5. What will be the cost of carpeting a room 15f ft. long and 12 ft. wide, with carpet f yd. wide, at a dollar a yard ? the strips to run from side to side. 6. How many single rolls of paper will be required to paper the walls of a hall 80 ft. long, 40 ft. wide, and 15 ft. high ? the paper to be 18 inches wide, and no allowance for openings. 7. At $1.50 a yard, find the cost of a carpet 27 inches wide, for a room 15 ft. by 12 ft., if the strips run length- wise, and there is a waste of of a yard on each strip for matching ? 8. A room 27 ft. long and 21 ft. wide is to be carpeted with carpet a yard wide, at $1.25 a lineal yard. Would the cost be different if the carpet were J yd. wide at $1.00 a yard, and how much ? 9. How much linoleum will be required to* cover a room 19 ft, square, and what will it cost at 87 J/ a square yard ? 10. How many yards of carpet f yd. wide will cover a room 18 ft. long and 15 ft. wide, if the strips run the long way ? the short way ? 184 ARITHMETIC. Review. Addition. The following table shows the amounts paid for teach- ers' salaries in the Syracuse schools for the year 1893. Add both ways, and* make the sums of the two totals agree. SCHOOLS. SENIOK. JUNIOR. PBIMAKY. TOTAL. Salina . . . Jefferson . Genesee . Prescott . . Seymour . . Putnam . . Irving . $2 7 972.13 500.00 4,469.54 5,447.63 4,038.51 $1,800.38 1,300.00 2,349.38 1,400.00 2,060.25 2,589.38 2,282.76 $2,110.50 2,001.13 2,574.13 1,892.38 2,962.13 2,431.64 2,533.63 . . . Delaware . . Porter . . . Gere . . . Danforth . . 425.00 4,483.63 2,293.00 2,322.89 1,849.69 2,165.01 2,&54.86 3,919.50 1,842.78 2,230.95 Brighton . . Rose 3,446.56 1,115.82 1,749.63 425 00 Clinton . . 2,774.39 2 856 20 Vine 1 799 00 Grace . 1,409.01 3 160 71 Merrick 1,685.12 2 319.69 Madison . . Bassett . . Frazer . . . Townsend Franklin . . May . . . Montgomery . Adams . 4,785.14 3,352.51 850.00 2,493.13 1,043.25 2,320.20 2,348.26 2,846.25 2,125.63 2,542.89 1,297.00 2,249.00 934.57 2,472.78 2,875.00 4,704.40 2,728.64 4,413.45 2 246 25 ... Asylum . . St. Vincent . 1,498.75 375.00 550.00 550.00 . . . Total PARTS OF 100 AND 1000. 185 SHORT MULTIPLICATION. Written Work. 1. What part of 100 is 37| ? 62 ? 66| ? 75 ? 87 ? 2. What part of 1000 is 375 ? 625 ? 666| ? 750 ? 875 ? How much is 3. | of 100 | of 1000 4. | of 100 | of 1000 5. of 100 of 1000 6. 2 of 100 J of 1000 7. J of 100 I of 1000 8. How much is 37 J times 56 ? EXPLANATION. 37| is f of 100, therefore 37 times 56 is | as much as 100 times 56. 100 times 56 is 5600. 700 ?. of WW = 2100. To multiply by 37, annex two naughts and take f . Tell how to multiply by 33^, 25, 16|, 8J, 25, 50. Tell how to multiply by 62, 66, 75, 87, 625, 666f . Find the products : 9. 48 X 37 19. 375 x 1128 10. 64 x 37J 20. 648 X 375 11. 72 x 37 21. 625 x 888 12. 88 X 62 22. 723 X 625 13. 96 X 62J 23. 272 x 666J 14. 66- x ise 24. 666| x 189 15. 66 X 284 25. 750 X 948 16. 288 X 75 26. 844 x 750 17. 75 X 87 27. 112 x 875 18. 96 X 87 28. 328 X 875 186 ARITHMETIC. Review. Indicated Work. Indicate, and find the result : 1. From the product of 8 and 5, diminished by the quo- tient of 56 divided by 8, take the sum of 11 and 3. 2. To 6 plus the product of 8 and 9, add the sum of 6 and 8 multiplied by 9. 3. From the sum of 5, 8, and 11 take the sum of 3, 6, and 4. 4. John earned 70/ a day, but was obliged to pay 15/ for lunch, and 10/ for carfare. How much could he save daily ? 5. A man bought a table for $20, a chair for $8, and a mattress for $6. How much change should he receive from a fifty-dollar bill ? 6. From the sum of 6, 5, and 4 take the difference be- tween 20 and 30. (6 + 5 + 4) - (30 - 20) 7. To the difference between 15 and 8, add the difference between 11 and 16. 8. From the sum of 11, 5, and 6 diminished by 4, take the difference between 60 and 56. Add and prove : 16 + 389 + 3684 + 38 = 19 + 763 + 986 + 49 = 28 + 982 + 428 + 76 = 76 + 786 + 5963 + 95 = 94 _|_ 594 + 3694 + 87 = 78 + 362 + 4896 + 63 = 96 + 28 + 985 + 99 = 89 + 9 + 8694 + 76 = 96 + 364 + 9899 + 94 = BEVIEW. 187 Mental Work. Find the sum : 1- * + t 5- + 2 9. f + f 2. i + t 6. 1J + 1J 10. f + i 3- J + | ? 2 + 5 11. f + 4 - T + T<y 8 - 4 + * A 12 - I + A Find the difference : 13. J 17. 1J - i 21. | -J- 14. i i 18. 2^ 1^ 22. I i 15. _ i 19. 3J - J 23. f T ^ 16. | 20. 1 1} 24. T \ ^ Find the product in simplest form : 25. 2 X f 29. i of f 33. 2 J X 4 26. 3 X 5 30. f of 34. 1J X 6 27. J X 4 31. | of ^ 35. 2| X 3 28. | X 3 32. of J 36. ! X i Find the quotients in simplest form : 37. j ^. 3 40. i -s- 2 43. j. -4- 5 38. j -f- 4 41. 2 -T- J 44. $ -5- 39. f H- 7 42. 3 -f- J 45. ^ ^ 46. What is a complex fraction ? 47. Change examples 43 and 44 to complex fractions. Simplify : 52. 3 3 49. - 51. | t 56. Define a compound fraction. 188' AKITHMETIC. Written Work. Find the sum : 1. 324| 2. 25 16* 136 297$ Subtract : 841 163ft If . 16ft 3. $ 14.15J 4. 7.03 112.19ft 7.29} 5. 384 16^ o OQ 7 5J 11. 253 6. 28} - 12g 9. 16J - 11J 12. 15 - 7. 163J - 129J 10. 28J - 4J 13. 11 - 28J 20J 19* 28J - 5J Find the product : 14. 126} X 6 17. 182 X 5J- 20. 28$ 15. 1586| X 3 18. 66 X 2j 21. 3| 16. 2894f X 7 19. 38 X 7| 22. 17 Find the quotients : 23. )1824 26. 6)1352^ 29. 28^ 24. 5^)1856 27. 7)1892$ 30. 15} 25. |)18934 28. 8)1624 31. 1^ Simplify : QO -*-*^^" QK 7? QQ O OA. - OO. OO. 7 6 % 33. ?i 36. ?4 39. i 41 4J 7J 34. A 37. 40. ^ 5} f 5 41. } of if of ft ~ ft of T \ of f . 42. Add } of 5i and f of 5 of J. X 7f X 5J- X 5J -i- 2| REVIEW. 189 1. Express by figures, three million, twenty-five thou- sand, three hundred eight. Separate into groups and read : 2. 36948006 5. 9984321 8. 3890000 3. 5000609 6. 300500200 9. 2503001 4. 2890006 7. 2983261 10. 2863259 11. Express the number of this year in Roman Notation. Read and add : 12. $369.842 13. $369.56 14. $ .39 15. $394.634 25.053 26.03 .384 28.52 695.06 9.565 .06 3.216 298.065 28.696 1.103 53.698 694.01 239.001 .582 21.034 986.959 598.091 .98 500.005 16. ReadLXXXI; CCCCXCI; DCLXXXIX. 17. 165 + 16 X 5 + ? = 395. 18. 384 + 64 -v- 8 - ? = 250. 19. (384 + 64) -r- 8 - 45 = ? 20. (384 + 64)^8x6-58x4=? 21. 384 + 64 -r- 8 X 6 - 58 X 4 = ? Rapid subtraction, without copying. 22. 500000 25. 568594 28. 3695.286 6852683 965989 56397.01 23. 389648 95999 24. 3000530 29856331 26. $ 3984.325 53681.20 29. $ 7787.10 2999.999 27. $3698 299.598 30. Multiply 589635 by 286 ; by 3950 ; by 90006. 31. Divide, without copying : 369081 by 9 ; by 10 ; by 11 ; by 12. Add the quotients. 190 ARITHMETIC. Indicate the operations and find results : 1. To the product of 5 and 6, add 5, and divide the sum by fourteen minus seven. 2. From 65 subtract the quotient of 60 divided by five, and add three. 3. Thirty plus twenty-five minus twenty-three multi- plied by three. 4. Six times nine, minus seven times seven. 5. 46 minus 16 plus 10, divided by three. 6. 35 plus 15 minus the quotient of 14 divided by 9. 7. 35 times 4, minus 14 plus 16. 8. Multiply the quotient of 56 divided by 14 by 12, and subtract 12 from the product. (56 -^ 14) X 12 - 12 ; or [(56 ~ 14) X 12] - 12 ; or f f X 12 - 12. I NOTE. Explain use of brackets, and horizontal vin- culum. 9. 3 fourths of 80, plus the sum of 25 and 16. Eead, and find results : 1. What is the value of (22 + 8 + 16) X (18 + 10 + 21) divided by (9 + 5 + 7) X (15 + 8). 2. (56 + 4 -v- 12) x (15 - 7) = ? 3. 25 - 13 + 10 -r- 10 = ? 4. (125 + 75) X 6 + 36 ~ 3 - 92 = ? 5. (28 - 7) X 6 + (92 + 7) -^ 9 - 86 + 10 -f- 12. 6. 2x4x8x13x7x16 divided by 26 X 14 X 8 = ? 7. Multiply by 6 the quotient of 72 divided by 12, and subtract 6 from the product. EEVIEW. 191 8. f = 9. 15 = 10. A = Mental Work. 1 . In f , how many twentieths ? 2. How many sixths in f of a mile ? Reduce 3. f to 9ths. 4. | to 24ths. 5. 5f to 6ths. 6. to 24ths. To lowest terms : 19- if 20. if 21. if 22. M To improper fractions : 35. 7 39. 5] 36. 9 40. 37. 14^ 41. 38. 2A 42. 11. IJtoSths. 12. 2J to 20ths. 13. 4 to 25ths. 14. 1 to 16ths. 15. if 16. if 17. if 18. 23. U 27. -|f 31. 24. |^ 28. Jft 32. 25. | 29. 4* 33. 26. 3ft 30. J-gg 34. 43. 12f 47. 12^ 44. 11^ 48. 313 45. 6^ 49. 10 T 9 - 46. 3| 50. 20| To' integers or mixed numbers : 51. 52. 53. 54. V V 55. 56. 57. 58. 59. |f 60. fft 61. Yr 62. M^ 63. 64. 65. 66. To a common denominator : 67. |, J 69. i,J 71. 68. |, | 70. i, f 72. 73. Add i and ; | and } ; i, , and J. 74. Multiply | by | ; f by f ; f by J. J, 192 ARITHMETIC. Oral Work. 1. If a man spends J of a dollar each day, how many dollars will he spend in 11 days ? 2. If a bushel of apples cost $1.24, what will a peck cost? 3. 5 is J of what number ? 4. 5 is | of what number ? 5. If it costs 37 cents a week for milk tickets, what will it cost for 8 weeks? 6. 4 is 3 times what number ? 7. 4 is J of what number ? 8. If 6 is | of some number, what is ? J ? 9. Add f and A; and f ; $ and ft? 10. 3 is | of what number ? 11. What is the difference between and J ? 12. How many eighths does it take to make one-half? How many GOths in J ? 13. How many times 12 are 3 times 8 ? 14. \ of 28 are how many times 2 ? 15. If 3 oranges cost lo/, what will 2 oranges cost ? 16. If 5 apples cost 2 cents, what will 1 apple cost ? What will 8 apples cost ? How many can I buy for 8 cents ? 17. A boy spent 6 cents, which was } of all he had. How much had he ? (Question.) 18. If 3 men do a piece of work in 5 days, how long will it take one man to do it ? 19. A boy who had \ of an apple gave away one-half of what he had. What part of the apple did he give away ? 20. i of f f of \ = ? 21. 9 is { of what number ? REVIEW. 193 Indicated Work. Indicate and cancel : 1. How many sq. yards of carpeting in a floor 24 ft. long and 18 ft. wide ? 2. How many acres in a field 140 rods long and 80 rods wide ? 3. What will be the cost of 1632 quarts of nuts at $2.50 a bushel ? NOTE. In cancellation, it is convenient to change cents to the fraction of a dollar, thus, 40/ = $f, 37/ == $f, $2.50 = $2J, etc. 4. At 6/ a quart, how many gallons of cider can be bought for $4.80 ? 5. 30 pounds of steel will make how many horse shoes, each weighing 6 ounces ? 6. In what time can a boy, at 60/ a day, earn as much as a man can earn in 40 days at $3.00 a day ? 7. What will it cost to paint 8 blackboards each 18 ft. long and 3 ft. wide, at 40/ a sq. yard ? 8. A man sold 15| cords of wood at $8^ a cord, and received payment in wheat at $J a bushel. How many bushels of wheat did he receive ? 9. If 2 yards of cloth cost $10, what will J of a yard cost? 10. If 8 bushels of wheat cost $6, how many bushels can be bought for $! ? 11. If 3J Ib. of tea cost $2J, what will f of a pound cost? 12. A garden is 3 rods wide and 10 rods long. How many 3 foot paces will a man take in going around it ? 13. How many boards 1 ft. wide and 12 ft. long will it take for a tight fence 5 ft. high and 50 ft. long ? 194 ARITHMETIC. CUBIC MEASURE. A Solid is anything that has length, breadth, and thick- ness. A Rectangular Solid is a body having six rectangular sides or faces. A Cube is a body having six square sides or faces. Every cube is a rectangular solid, but every rectangular solid is not a cube. A cubic inch is a cube 1 inch long, 1 inch wide, and 1 inch thick. A cubic foot is a cube 1 ft. long, 1 ft. wide, and 1 ft. thick. A cubic yard is a cube 1 yd. long, 1 yd. wide, and 1 yd. thick. The Volume or Solidity of a body is the number of cubic units that it contains. How many cubic inches in a block 4 inches long, 3 inches wide, and 2 inches thick ? The block is made up of two layers, each 1 in. thick. In the top layer there are 4 times 3 cu. inches. In the two layers there are 2 X (4 X 3 cu. in.) = 24 cu. inches. To find the volume or solid contents of a rect- angular solid, obtain the product of the length, breadth, and thickness. NOTE. The three dimensions must be of the same denomination. 3 IN. WIDE 10 CUBIC MEASURE. 195 1. How many cubic inches in a block 12 inches long, 6 inches wide, and 4 inches thick ? 2. What is the volume of a block of marble 6 ft. long, 5 ft. wide, and 4 ft. thick ? 3. What is the volume of 6 bricks, each being 8 by 4 by 2 inches ? 4. The solid contents of a block are 720 cubic feet. It is 10 ft. high and 9 ft. wide. How thick is it ? To find one dimension when the solidity and the other two dimensions are known, divide the solidity by the product of the two given dimensions. TABLE OF CUBIC MEASURE. 1728 cubic inches = 1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.). 16 cubic feet = 1 cord foot (cd. ft.). 8 cord ft. or 128 cu. ft. = 1 cord (cd.). 5. A cord of wood is 8 ft. long, 4 ft. wide, and 4 ft. high. How many cubic feet does it contain ? A cord foot is J of a cord. It is 4 ft. high, 4 ft. wide, and 1 ft. long. 8 of these placed together make a cord. NOTE. Illustrate, using 8 books of equal size. 6. Prove that there are 1728 cu. inches in a cu. foot. 7. Prove that there are 27 cu. ft. in a cubic yard. Keduce to cubic inches : 8. 11 cu. ft. 145 cu. inches. 11. 1 cord 8 cu. ft. 9. 38 cu. ft, 1046 cu. inches. 12. 3 cords 150 cu. in. 10. 4 cu. yards 15 cu. feet. 13. 1 cord foot. Reduce to higher denominations : 14. 498326 cu. in. 16. 369426 cu. in. 15. 269842 cu. in. 17. 846394 cu. in. 196 ARITHMETIC. Reduce to lower denominations : 18. J cu. yd. 20. 16 cu. yd. 5 cu. ft. 19. | cu. yd. 21. ] cu. yd. 22. 18 cu. yd. 3 cu. ft. 159 cu. in. 23. 15 cords 14 cd. ft. 24. J cu. ft. 25. 78 cu. yd. 26. 6 cu. yd. 6 cu. ft. 6 cu. in. 27. 14J cu. yd. 28. What part of a cu. yard is 2 cu. feet. (Question.) A cubic yard = 27 cu. ft. The question then becomes, What part of 27 cu. ft. is 2 cu. ft. ? 29. What part of a cubic foot is 144 cubic inches ? 30. What part of a cubic yard is 144 cubic inches ? 31. A cubic inch is what part of a cubic foot ? 32. 2 cord feet are what part of a cord ? 33. What will be the cost, at 20/ a cubic yard, to dig a cellar 45 ft. long, 36 ft. wide, and 5 J ft. deep ? 34. Reduce | of a cu. foot to a fraction of a cu. yard. 35. What is the cost of a block of granite 6 ft. 6 in. long, 4 ft. 3 in. wide, and 3 ft. thick, at $15.75 a cubic yard ? 36. What is the difference between 5 cubic inches and a cube 5 inches on each edge ? 37. How many cubic feet in a cube two feet on each edge ? How many cubic inches ? 38. What is the capacity in cubic feet of a dry-goods box 3 ft. by 2J ft. by 2 f t ? 39. How many 2-in. cubes can be cut from a block of marble 2 ft. long, 18 in. wide, and 15 in. thick ? CAPACITY. 197 Capacity of Cisterns. 1. A wine gallon fills 231 cubic inches of space. How many gallons will fill a cubic foot of space ? J^ T 8- = about 7| gallons. Ans. NOTE. It is sufficiently accurate in estimating large liquid capacities to reckon 1\ gal. to the cubic foot. 2. A cistern is 5 ft. square and 6 ft. deep. How many gallons of water will it contain ? Solve both ways and note the difference. 3. What is the capacity in barrels of the cistern in Ex. 2? 4. How many barrels of water in a cistern 7 ft. long, 4 ft. wide, and 5 ft. deep ? (Indicate and cancel.) 5. What is the capacity in barrels of a tank 3 ft. by 4 ft. and 4f ft. deep ? 6. How many gallons of water in a tank whose capacity is 150 cubic feet ? 7. How many cubic feet in a cistern that will hold 1500 gallons of water ? Capacity of Bins. 8. A bushel fills 2150.42 cubic inches of space, which is l cu. ft., nearly. NOTE. In estimating large quantities of grain, it is sufficiently accurate to reckon \\ cu. ft. to the bushel. 9. How many bushels of grain will a bin hold that is 6 ft. long, 5 ft. wide, and 4 ft. deep ? (Indicate and cancel.) Solve both ways and note the difference. 10. How many bushels in a bin that contains 150 cubic feet? 11. How many cubic feet in a bin that holds 150 bushels ? 1 98 ARITHMETIC. Kead and add : 1. 36,942,006 2. $69804.41 2,000,052 306.005 3,540,004 2904.046 81,569,298 198.36 1,005,000 2940.265 3. How many bushels of oats at 55/ a bushel can be bought for $52.25? 4. The divisor is 36846, the quotient 515, the remainder 1658. What is the dividend ? 5. Bought 356 bbl. of flour at $6.50 a barrel, and sold the whole for $2198.50. How much did I gain or lose ? 6. A newsboy buys 96 papers a day at 18/ a dozen, and sells them at 3/ each. How much money has he made at the end of a week if he has 16 old papers left ? 7. How many square yards in a pavement 348 ft. long and 6 ft. wide ? Write results without copying : 8. 89742 -- 9 12. 2189362 -=- 482 9. 168439 -f- 12 13. 3698452 -^ 1962 10. 486982 x 9 14. 2893004 -^- 1009 11. 290006 x 8 15. 3698421 + 2196 16. From 6990026 17. Take 36984 Take 980059 From 161103 18. Take 38948 From 94321 19. How many tiles 12 in. by 6 in. will it take to cover a hearth 3 ft. wide and 6 ft. long ? (Diagram.) 20. If an acre of wheat yields 30 bushels, how much will a field yield that is 25 rd. long and 45 rd. wide ? MISCELLANEOUS TABLES. 199 MISCELLANEOUS TABLES. 12 units or things, 1 dozen, (doz.). 12 dozen, 1 gross, (gro.). 20 units, 1 score. 196 lb., 1 barrel of flour. 200 lb., 1 bbl. of beef, pork, or fish. 280 lb., 1 barrel of salt. PAPER. 24 sheets, 1 quire. 20 quires, 1 ream. 2 reams, 1 bundle. 5 bundles, 1 bale. Oral Work. 1. How many are 3 gross of pencils ? 2. What is the cost of J gross of pens at 60/ a gross ? 3. How many are 3 score years ? 4. How many are 3 score and 10 years ? 5. How many are 5 dozen eggs ? 6 J dozen eggs ? 6. How many sheets in 3 quires of paper ? 7. How many quires in a ream ? 8. 84 eggs are how many dozen ? 9. How many reams in a bale of paper ? 10. How many pounds in J bbl. of flour ? 11. How many pounds in bbl. of salt ? Written Work. 12. How many dozen in 66 cucumbers ? 13. What is the cost of 6 quires of paper at $4 a ream ? 14. What will 8 gross of copy books cost at 10/ apiece ? 16. If paper is bought at $2 a ream and sold at 18/ a quire, what is the gain on 6 reams ? 200 ARITHMETIC. TIME. 60 seconds, 1 minute, (min.). 60 minutes, 1 hour, (h.). 24 hours, 1 day, (da.). 365 days, 1 common year, (y.). 366 days, 1 leap year. Also: 7 days, 1 week (wk.). 12 months (mo.), 1 year. 100 years, 1 century. The solar year is 365J days, nearly. For convenience 365 days is taken for a common year. The . day, in 4 years, amounts to another day, making every 4th year 366 days. This is called leap year. This extra day is added to February. Days in each month : January 31 days. July 31 days. February 28 or 29 days. August 31 days. March 31 days. September 30 days. April 30 days. October 31 days. May 31 days. November 30 days. June 30 days. December 31 days. Oral Work. 1. How many minutes in 2 hours ? ^ hour ? 2J hours ? 2. Reduce to hours : 180 minutes ; 120 minutes ; 30 minutes. 3. How many hours in a week ? 4. How many days in March and June ? 5. What 4 months have 30 days ? 6. How many have all the rest except February ? 7. What part of an hour is 45 minutes ? (Question.) TIME. 201 Written Work. Reduce to lower denominations : 1. 15 min. 29 sec. 5. 3 wk. 4 da. 6 min. 2. 16 da. 16 hr. 6. f of a day. 3. 15 da. 6 hr. 7. of a day. 4. 3 wk. 11 hr. 8. 5 weeks. Reduce to higher denominations : 9. 1565 sec. 13. 359642 sec. 10. 3246 min. 14. 29843 hr. 11. 3694 h. 15. 3000 min. 12. 56984 sec. 16. 300000 hr. 17. lleduce J of an hour to the fraction of a day ? (Question.) 18. What part of a week is J of a minute ? (Question.) 19. How many weeks in July, August, and September ? 20. If a boy walk J mile in 10 minutes, how long will it take him to walk 16 miles ? 21. How many days from the 4th of May to the 4th of July? 22. How many hours in the month of August ? The Winter months are December, January, and February. The Spring months are March, April, and May. The Summer months are June, July, and August. The Autumn months are September, October, November. 23. Which is the shortest season ? Why ? 24. Is it the shortest in leap year ? 25. How many more days in this year ? 26. If a man earn \fl a minute, what will he earn in a day of 8 hours ? 27. If a clock loses \ a second every day, how much will it lose in the summer months ? 202 ARITHMETIC. Reduce to lower denominations : 1. 48 Ib. 12 oz. to ounces. 2. 15 cwt. 90 Ibs. to pounds. 3. 15 Ib. 7 oz. 14 dr. to drams. 4. 3 gal. 3 qt. 1 pt. to pints. 6. 4 bbl. 4 gal. to pints. 6. 4 qt. 3 pt. 3 gi. to gills. 7. 4 bu. 3 pk. 6 qt. to quarts. 8. 2 bu. 2 pk. 2 qt. 1 pt. to pints. 9. 16 bu. 4 qt. to pints. 10. ton to pounds. 11. J hhd. to quarts. 12. 6J gal. to gills. 13. J bushel to quarts. Eeduce to higher denominations : 14. 2698 dr. to pounds, etc. 21. 2180 pt. to bushels, etc. 15. 4694 oz. to pounds, etc. 22. 3694 pk. to bushels. 16. 4924 Ib. to tons, etc. 23. 3 oz. is what part of 1 Ib.? 17. 2946 pt. to hhd., etc. 24. 4 pt. is what part of 1 gal.? 18. 3694 gi. to gal., etc. 25. 4 qt. is what part of 1 bu.? 19. 3314 pints to gal., etc. 26. J pk. is what part of 2 pk.? 20. 734 qt. to bushels. 27. 3 pk. is what part of 3 bu.? 28. How many times can a 2-quart pail be filled from a half -bushel of chestnuts ? 29. A man bought sweet potatoes at $2 a bushel, and sold them at 70/ a pk. What did he gain on 3 bu. ? 30. If vinegar retails at 1|/ a pint, what will a barrel of vinegar sell for ? 31. From a barrel of wine f was sold in one day, and \ in another. How many quarts are left ? CUBIC MEASURE. 203 1. How high is a room that is 24^ ft. long, 20 ft. wide, and contains 4410 cubic feet ? 2. How long a building that is 10 ft. wide and 8 ft. high will it require to hold 10 cords ? 3. A pile of wood 4 ft. long. 4 ft. wide, and 4 ft. high, will cost how much at $4.50 a cord ? 4. How many cubic inches of space in a box 10 in. long, 6 in. wide, and 4 in. high ? 5. What part of a cubic yard is the space in this box ? 6. The volume of a rectangular solid is 1200 cu. ft. It is 20 ft. long and 4 ft. high. How wide is it ? 7. A piece of marble 16 ft. long, 12 ft. wide, and 9 ft. thick can be sawed into how many blocks of a cubic foot each? 8. What will it cost to make an embankment contain- ing 54270 cu. ft. of earth at $.15 a cubic yard ? 9. At $6.40 a cord, hqw much money will pay for a pile of wood 80 ft. long, 32 ft. wide, and 12 ft. high ? 10. Find the number of cubic inches in a piece of ice 3 ft. long, 2 J ft. wide, and 18 inches thick ? 11. How many cords of wood may be piled in a shed 62 ft. long, 20 ft. wide, and 12 ft. high ? 12. What is the difference between 6 cubic inches and a 6-inch cube ? 13. If 300 cu. ft. of breathing space is required for a pupil, how many pupils should be seated in a schoolroom 30 ft. square and 10 ft. high ? 14. How many cubic yards of earth will be removed in digging a cellar 36 ft. square and 6 ft. deep ? 15. How many books 9 in. by 6 in. by 1^ in. can be packed into a chest 4 ft. by 2 ft. by 1^ ft. ? (Indicate and cancel.) 204 ARITHMETIC. 16. How many cubic feet of snow on 18 square feet of ground, the snow being 9 inches deep ? 17. How many bricks, 8x4x2 inches, will it take to build a wall 16 x 8 x 1 J ft ? (Cancel.) 18. How many cubic inches in a stick of timber 1 foot square at the ends and 8 ft. long ? 19. How many cubic feet of air in a room 16 ft. 8 in. long, 12 ft. 9 in. wide, and 8 ft. 4 in. high ? 20. How many 2-in. cubes may be made from a piece of wood 6 by 4 by 2 ft.? 21. How many loads of earth, each load containing a cubic yard, in an excavation 30 ft. long, 26 ft. wide, and 14 feet deep ? 22. How many cubic inches in 3 barrels, there being 231 cubic inches in a gallon ? 23. A cubic yard of earth makes a load. How many loads must be excavated for a cellar 32 ft. long, 30 ft. wide, and 6J ft. deep ? 24. A brick is 8 in. long, 4 in. wide, and 2 in. thick. How many will it take to build a wall 30 ft. long, 20 ft. wide, and 1 J ft. thick, using no mortar ? 25. A vat is 4 ft. long, 3 ft. wide, and 6 ft. 9 in. deep. How many gallons of oil will it hold ? 26. How many bricks, 8 by 4 by 2 in., will be needed to build a wall 16 ft. long, 4 ft. high, and 1 ft. thick ? (Use cancellation.) 27. How many blocks 8 in. long, 6 in. wide, and 1 in. thick, can be packed into a box 2 ft. by 1J ft. by lj ft ? (Use cancellation.) 28. A pile of wood is 32 ft. long, 12 ft. high, and 8 ft. wide. How much will it cost at $3.37^ a cord ? 29. A cord foot is what part of a cord ? MEASUREMENTS. 205 30. How many cubic feet of water will a trough hold that is 6 ft. long, 2 ft. wide, and 18 in. deep ? 31. How many cubic feet of air in a room 18 by 15 by 10 ft. ? 32. How many cords of wood in a pile 32 ft. long, 8 ft. high, and 4 ft. wide ? (Use cancellation.) 33. There are 231 cubic inches in a gallon. How many gallons will a tank hold that is 6 ft. by 6 ft. and 2 ft. deep? 34. A and B are 1,000 miles apart and travel towards each other, A at the rate of 90 miles a day and B at the rate of 80 miles a day. In how many days will they meet ? 35. A square mile of land may be cut up into how many 80-acre farms ? (Diagram.) 36. In the above problem what will be the dimensions of each farm ? 37. An ordinary brick is 8 in. by 4 in. by 2 in. How many sq. inches of paper will exactly cover it ? 38. Make a diagram on thick paper, cut, and fold into the form of a brick. 39. A chest is 4 ft. long, 2 ft. wide, and 2 ft. high. How many yards of cloth will be required to cover all sides except the bottom, allowing ^ yd. for laps. (Diagram.) 40. Which will hold the more water, a can 6 in. by 4 in. by 3 in., or a can in the form of a 4-inch cube ? 41. How many bricks, 8 by 4 by 2 in., will fill 1 cubic foot of space ? (Factor and cancel.) 42. How many bricks in a pile 10 ft. long, 8 ft. wide, and 9 ft. high ? 43. At $.87^ a bushel, what will be the cost of the wheat that can be contained in a bin 15 ft. long, 12 ft. wide, and 4 ft. high ? 206 ARITHMETIC. Questions of Relation. Oral Work. What part of 1. 7 is 5 ? 6. 14 is 8 ? 11. 16 is 4 ? 2. 11 is 6 ? 7. 25 is 15 ? 12. 28 is 14 ? 3. 13 is 8 ? 8. 36 is 24 ? 13. 15 is 7 ? 4. 16 is 15? 9. 27 is 18? 14. 30 is 8 ? 5. 21 is 11 ? 10. 19 is 18 ? 15. 42 is 6 ? Form the other questions from the first five of the above. Of what? 1. 12 is J 4. 6 is I 7. 25 is g 10. 15 is J 2. 16 is 5. 18 is 8. 36 is f 11. 9 is ft 3. 11 is | 6. 14 is } 9. 24 is f 12. 7 is J Write problems, using the first five of the above. How much is 1. f of 25 ? 4. | of 60 ? 7. f of 9 ? 10. ^ of 28 ? 2. I of 36 ? 5. } of 24 ? 8. | of 45 ? 11. f of 35 ? 3. | of 48 ? 6. f of 14 ? 9. ft of 16 ? 12. f of 42 ? From the first five of the above form the other questions. Written Work. 1. What part of 15 is 3J ? 2. What part of 15f is 6 ? 3. f is what part of f ? 4. 2| is what part of 5J ? 5. 381 is T 9 o of what ? 6. is f of what ? 7. 224 is |f of what ? 8. 2 1 is f of what ? 9. How much is T 9 7 of REVIEW. 207 Written Work. Add both ways. Prove by adding sums. 1. 684 + 396 + 28 + 934 + 287 399 + 763 + 198+ 26+ 32 39 + 63 + 984 + 699 + 721 897 + 673 + 987 + 38 + 489 394 + 998 + 483 + 976 + 694 21 + 8 + 64 + 16 + 114 371 + 973 + 163 + 814 + 748 29 + 28 + 63 + 982 + 36 Multiplication. 2. 395 x 382 7. 3983 X 4 i 12. 369 X 6 3. 786 X 25 8. 26984 X 3 13. 263J X 15 4. 3985 X 683 9. 2634 X 6* 14. 394| X 14 5. 2963 X 29 10. 2835 X 4 5 15. 784f X 14 6. 876 X 250 11. 3892 X 5J 16. 298| X 8 Subtraction. 17. 36983 4999 18. 2984 46321 19. 3685f _398| 20. 264f 5983^ Division. 21. 3985 -5- 185 24. 269 -f- 36| 27. 29| -f- 22. 3553 -T- 13| 25. 2896 -f- 15 28. 38^ -^- 15f 23. 7899 -j- 15f 26. 9894 ^ 95f 29. 36J -;- 17| Factor and cancel : 30. Divide the product of 25 and 96 by the product of 15, 8, and 12. 31. Divide 17280 by 1440. 32. What is the quotient of 21 x 28 X 36 divided by 27 X 14 X 6? 33. Divide 16 X 14 X 12 X 7 X 4 by 32 x 28. 208 ARITHMETIC. Written Review. 1. How much will J Ib. of coffee cost if 4 Ib. cost 80/? 2. A family use 5 pt. of milk daily. How many quarts will they use in the month of March ? 3. How much will 2 dozen bananas cost at the rate of 2 for 3 cents ? 3 for 2 cents ? 4. What will peck of apples cost, if 2 bushels cost $1.50 ? 5. How many hours in the month of February, 1896 ? 1867? 6. If I buy 10 yd. of cloth at $2 per yard, and pay for it in wool at $ a pound, how many pounds of wool will it require ? (Use cancellation.) 7. If wood is worth $4 a cord and coal $5 a ton, how many cords of wood will pay for 16 tons of coal ? (Use cancellation.) 8. What will be the cost of 5 Ib. 6 oz. of steak at 16/ a pound ? 9. My lawn is 120 ft. long and. 60 ft. wide. If my lawn mower cuts a strip 16 inches wide, how many strips must I cut to mow the entire lawn, mowing from side to side ? from end to end ? (Diagram.) How many miles do I walk in mowing the lawn 15 times ? 10. 40,000 is what part of half a million ? (Question.) 11. How many cubic yards of air in a schoolroom 36 ft. by 36 ft. and 12 ft. high ? 12. If there are 45 children in the room, how many cubic yards for each child ? 13. How much do I gain on 3 bushels of berries bought at $2 a bushel and sold at 11 / a quart ? 14. How many sq. inches in 3 sq. yards ? 15. How many inches in 3| rods ? FRACTIONS. 209 Oral Work. 1. If a pie is cut into 8 equal parts, and 5 of those parts are divided among 10 boys, what part of the pie will each boy receive ? 2. If a boy can ride a bicycle 18 miles in an hour, how far can he ride in 20 minutes ? 3. If I can buy a pound of raisins for ll/, what part of a pound can I buy for 8/ ? 4. If coffee is 37/ a pound, how many pounds can be bought for $15 ? 5. When coffee is 37/ a pound, what will 40 pounds cost? 6. Tf two apples are divided among three boys, what part of an apple will each receive ? 7. 18 is T 3 ^ of what number ? 8. If 3 boys together earn 5 dollars, what will be each boy's portion ? 9. How much is of five ? 10. Which is greater of 4 or j of 1 ? 11. How much is $ of 5 ? J of 11 ? f of 6 ? 12. Among how many boys can I divide 5 oranges giving each boy J of an orange ? 13. Reduce to whole or mixed numbers, ^ ; -y- ; - 1 /- ; \ 6 ; V; ; 14. Reduce to improper fractions, 7 ; 14J ; 5; 8; 8|. 15. Reduce 2J to 8ths. 16. How many bushels in 7 pecks ? 17. How many feet in 28 inches ? 18. How many gallons in 10 quarts ? 19. What part of a gross is a dozen ? 210 ARITHMETIC. Written Work. 1. How many loads of earth must be taken from a cellar 32 ft. long, 27 ft. wide, and 7 ft. deep ? 2. Find the cost of 14 cwt. of hay at $12 a ton. 3. When is a fraction in its lowest terms ? 4. A room is 18 J- ft. long, 15 ft. wide, and 9 ft. high. How many square yards of plaster are needed for the walls and ceiling ? 5. Cover the floor of above room with carpet } yards wide. 6. How many yards of lining J yd. wide, will it take to line a quilt 2j yd. square ? 7. A silver pitcher weighs 3 Ib. 2 oz. What will it cost at $.15 a pwt. 8. How many gallons of water in a cistern 8 X 5 X 7 ft. ? 9. Add: 9857, 7783, 8939, 2791, 9367, 4683, 8479, 9847. 10. [(496 X 37) - 6453] -^ 87 = ? 11. Eeduce f of T \ of of fa of ft of }| to a simple fraction. 12. What is a prime number ? 13. What is a composite number ? 14. What is the difference between 5f + 8| and 3^ + 4f? 15. How much will a barrel of cider cost at 3 \fi a pint ? 16. Bought 4765 laths at $1.98 per M. What was the cost? 17. Find the cost of 3 gross of buttons at 33 J/ a doz. 18. Paid $3.50 for a bushel of chestnuts. What must I ask per quart to gain $1.30 ? MEASUREMENTS. 211 1. How many cubic feet in a solid block measuring 2 yards on each edge ? 2. An acre of land costs $420. How much will a square rod cost ? 3. In j how many 20ths ? 4. What is the value of 38 Ib. 8 oz. pure gold at $1.04 a pennyweight ? 5. How many yards of carpet 27 in. wide will cover a floor 12 by 14 ft. ? 6. How many cubes 3 X 4 x 2 in. can be packed into a box 8 X 4 X 6 ft. ? 7. 10 sq. rd. are what part of an acre ? 8. A room 40 ft. square contains 20000 cu. ft. of air. How high is it ? 9. From f of 4J take of 3|. 10. How many days in the first 5 months of a leap year ? 11. What are the dimensions of a 5 ft. cube in inches ? 12. When f of a year has passed, how many days remain ? 13. Are all the factors of 24 found in 56 ? 14. Find the factors of 108. 7201 x 207 = ? 144 16. Find the cubic inches in a 3J ft. cube ? 17. The capacity of a room 12 ft. square is 1368 cubic feet. What is its height ? 18. How many gallons of water will a cistern 11 X 7 X 6 ft. hold ? (Cancel.) 19. How many inches in all the edges of a 2 ft. cube ? 20. How many gallons in a tank, 4 ft. 6 in. long, 2 ft. 8 in. wide, and 1 ft. 6 in. deep ? 212 ARITHMETIC. 1. Find the area of one face of a cube 2 ft. 9 in. long. 2. What is meant by debtor ? Creditor ? 3. Make the following into a bill and receipt it, 2 Ib. sugar at 5/ ; 9 Ib. beef at 8/ ; 2 Ib. coffee at 37/ ; 1 Ib. tea at 50/ ; 2 J Ib. cheese at 13/. 4. What are the prime factors of 36 ? 5. What are the factors of any product ? 6. How much hay at $16 a ton will $4 buy ? 7. How many feet of fence will enclose a lot 80 rd. square ? 8. of a hogshead of vinegar leaked away. How many quarts remained ? 9. If a bushel of chestnuts is bought for $4 and sold at 12/ a quart, what is the loss ? 10. 5280 ft. = ? 366 da. = ? 2240 Ib. = ? 49 Ib. = ? 128 cu. ft. = ? 29 da. = ? 11. (j of 6i) - (i of 3|) = ? 12. If I use 14 from a great gross of buttons, how many buttons will be left ? 13. A room is 40 ft. square and 12 ft. high. How many sq. yards of plastering in the walls and ceiling, making no allowance for openings ? 14. What would it cost to carpet the above room with carpet 1 yd. wide at $1 J a yard ? 15. I use 3J cu. feet of ice a day. How many cubic feet Will I use during the spring and summer months ? 16. If 4 tons of coal cost $17, what will 3 cwt. cost ? 17. How many cubic inches in a cubic block whose edge measures four feet ? 18. Change 3J| to 200ths. DECIMALS. 213 DECIMAL FRACTIONS. A Common Fraction is generally expressed by placing the denominator below the numerator. A Decimal Fraction has a denominator, but the latter is not generally written. The denominator of a common fraction may be any number. The denominator of a decimal fraction must be 10, or 100, or 1000, etc. NOTE. The word decimal comes from the Latin word decem, ten, and so the denominator of every decimal frac- tion is 10, or 10 X 10, or 10 X 10 X 10, etc. A Decimal Fraction, or Decimal, is a fraction whose unit is divided into tenths, hundredths, thousandths, etc. A Decimal is always written at the right of a period (.), called the Decimal Point. We use the decimal point in writing U.S. money to separate the dollars from the cents and mills. 5 dollars and 28 cents is written $5.28. But 28 cents is ffo of a dollar; hundredths therefore are written like cents, with two decimal figures. NOTE. Any figure at the right of the decimal point is a decimal figure. Tenths are written like dimes, with one decimal figure. fV = -3 ; T 8 u = - 8 5 yV = - 7 - Hundredths are written with two decimal figures, ^ = 28; flfr = .39; T {fo = .06. Thousandths are written with three decimal figures, = -025. Name the denominators : 1. .6 3. .105 5. .05 2. .17 4. .006 6. .225 214 ARITHMETIC. Change to decimal form : 1 vU* 6 * ^Ar 11 ?& 16 2 *U* 7 -vU* 12 i i 17 o .J'jiJL Q --A- 1^-2- 18 4. _6_ Q 6 14.500 1Q * TO *' 1015 " TOOO i ^ < 5. y\^ 10. y-J-g-g 15. y^ 20. Change to common fractions, and read : 21. .36 26. .485 31. 5.6 22. .7 27. .016 32. 5.06 23. .125 28. .16 33. 5.006 24. 12.2 29. .06 34. 5.600 25. 6.25 30. .6 35. 5.060 Write, first as common fractions, then as decimals : 36. Six tenths. 37. Seventy-five hundredths. 38. One hundred twenty-five thousandths. 39. Sixteen, and forty-eight hundredths. 40. Twelve, and four tenths. 41. Six tenths. 42. Six hundredths. 43. Six thousandths. 44. How many decimal figures are required to express thousandths ? hundredths ? tenths ? 45. Read the numerators only in Examples 36 to 43. Write the following as decimals, and read the numerator of each, then the denominator of each. 46. Two hundred eighty-two thousandths. 47. Fifty-six hundredths. 48. Seven tenths. 49. Six hundred thousandths. DECIMALS. 215 Oral Work. 1. What part of 10 units is 1 unit ? 2. What part of 1 ten is 1 unit ? 3. What part of 2 hundreds is 2 tens ? 4. In the number 555, what is the value of the first 5 at the right ? the second 5 ? the third 5 ? 5. Upon what does the value of any figure depend ? 6. The value of the first five is what part of the value of the second five ? (Question.) 7. -fa is what part of 2 units ? 8. In the number 5.5 the value of the right-hand 5 is what part of the value of the left-hand 5 ? (Question.) 9. Write five and five-tenths decimally. 10. .5 is what part of 5 ? 11. In the decimal .555 what is the value of the first 5 to the right ? the second 5 ? the third 5 ? 12. How much greater is the value of the third 5 than the second 5 ? (Question.) 13. How much greater is the second 5 than the first 5 ? In the number 555.555 we have found that the value of each 5 is ^ as much as the next 5 to the left. Also that the value of any 5 is 10 times as great as the value of the next 5 to the right. In decimals as in integers, any figure removed one place to the right is diminished tenfold, and when removed one place to the left it is increased tenfold. 14. In the number 32.6 what would be the value of the 2 if it were removed one place to the right ? One place to the left ? Read: 15. .222 17. 22.2 16. 2.22 18. 222. 216 ARITHMETIC. Integers and Decimals. I ii !l i 1 1 fl * i i i I l!j lis HI HI ^ WHH WHO HKH H W S 4, 243, 684 .268, 504 The number is read 4 million, 243 thousand, 684 and 268 thousand 504 millionths. To read decimals : For the numerator, read the decimal as an integer. For the denominator, give the denomination of the last figure. NOTE. The denominator of a decimal is always named, but seldom written. In reading an integer and a decimal, read " and " where the decimal point occurs. What is the denominator when the decimal has one fig- ure ? two figures ? three figures ? four figures ? five figures ? six figures ? seven figures ? Eead: 1. .368 7. 28.3005 13. .4983695 2. .894 8. .28962 14. 4.98369 3. .5328 9. 15.60534 15. 49.8369 4. .2053 10. 37.00537 16. 498.369 5. 25.623 11. 25.203602 17. .000400 6. 7.0063 12. 38.000006 18. .0004 19. In Example 11 remove the point two places to the right and read. Four places to the right. One place to the left. 20. Kead denominators only, in the last five Examples. DECIMALS. 217 To write a decimal : Write the numerator ; prefix naughts when necessary to express the denominator ; place the point at the left. How many decimal figures are required to express tenths ? hundredths ? thousandths ? ten-thousandths ? hundred- thousandths ? millionths ? ten-millionths ? Write decimally : 1. Eight tenths. 2. 29 hundredths. 3. Sixteen, and 284 thousandths. 4. 4584 ten-thousandths. 5. Twenty-five hundredths. 6. Twenty-five thousandths. 7. Twenty-five .ten-thousandths. 8. Twenty-five hundred-thousandths. 9. Twenty-five millionths. 10. 1650, and 464 thousandths. 11. One thousand one, and 36 hundred-thousandths. 12. Sixteen, and six thousandths. 13. Seven hundred eighty-four millionths 14. Twelve hundred-thousandths. 15. Seventy-five ten-thousandths. 16. Seven hundred fifty ten-thousandths. 17. Seven thousand five hundred ten-thousandths. 18. 324 millionths. 19. 7685 hundred-thousandths. 20. One million three hundred six thousand four hundred two, and three hundred five thousand four hundred ten ten-millionths. 21. Three hundred-thousandths. 218 ARITHMETIC. Oral Work. 1. In the three decimals, .4, .40, .400, is there any differ- ence in value ? 2. What is the effect when a naught is annexed to a decimal ? 3. What is the effect when a naught is annexed to an integer ? Give an example. 4. In the decimals, .4, .04, .004, is there any difference in value? 5. What is the effect when a naught is prefixed to a decimal ? two naughts ? Give examples. 6. What is the effect when a naught is prefixed to an integer ? Naughts annexed to decimals do not change their values. (Prove this.) For each naught prefixed to a decimal the value is dimin- ished tenfold. (Prove.) The denominator of a decimal when expressed is always 1 with as many naughts as there are figures in the numera- tor. (Prove.) REDUCTION OF DECIMALS. Decimals may be reduced to a common denominator by annexing naughts sufficient to give the same number of decimal figures to all the decimals. Eeduce to a common denominator : 7. .5, .365, and .4689. 8. .18963, .5, 7.84, .16005. 9. .28, 3.5, .00005, .256. 10. .5, .05, .005, .0005, .00005. 11. .0463, .03, .1, .100010. 12. .38, 1.16, .4, 78.592. DECIMALS. 219 To reduce a decimal to a common fraction. 1. Eeduce .375 to a common fraction. SOLUTION. .375 as a common fraction is -- 5 -375 = Rule. Write the numerator, omitting the point, sup- ply the denominator, and reduce to lowest terms. Eeduce to common fractions : 2. .25 8. .125 14. .368 3. .35 9. .875 15. 16.75 4. .75 10. .375 16. .00125 5. .64 11. .455 17. .054 6. .52 12. .025 18. .0250 7. .38 13. .561 19. .01375 20. Eeduce .37^ to a common fraction. SOLUTION. - .37* = ^ = = = Eeduce to common fractions : 21. .12 24. 22. .62J 25. .03| 28. .66 23. .06 26. .25| 29. .36J Eeduce to mixed numbers : 30. 16.25; 2.33 J ; 34.75. To reduce a common fraction to a decimal. Eeduce J to a decimal. J = 3 times . 3 = (3.0) 30 tenths. of 3.0 = (.7) 7 tenths and 2 tenths remainder. 2 tenths = 20 hundredths. of .20 = .05. Hence J = .7 + .05 = .75. Rule. Annex decimal naughts to the numerator, and divide by the denominator. Point off in the quotient as many decimal places as there are naughts annexed. 220 ARITHMETIC. The division will not always be exact. In such cases, write the remainder over the divisor as a common fraction, or place the sign (+) after the decimal to show that the result is incomplete. Thus, | = .142f, or .142 +. Reduce to decimals : 1- I 6- i n - i* 16. 2. | 7- } 12. / s 17- ft 3- $ 8. 13. J 18. / 5 4. J 9- A 14. {2 w- i* 6- ft io- 51 15. 3A 20. 2^ Add : .35, 4.375, 28.3065. .35 RULE. Write the numbers so that the deci- 4.375 mal points stand in a column. Add as in inte- 28.3065 gers, and place the point in the sum directly under the points above. Add: 2. 3.25 3. 4.5 4. .004 7.163 .168 4.1 15.0032 2.12 16.1563 5. .175 + 1.75 + 17.5 + 175. 6. 145 + 14.5 + 1.45 + .145 + .0145. 7. 3.2 + 14.0063 + .006 + 25,384 + .1. 8. .8 + .446 + 59.3 + 2.575 + 1.0056 + .3. 9. 1.45 + 2.365 + 96 + .96 + 15.863 + 4.3 + .0004. 10. 446 + 44.6 + 37562 + 9 + .8 + .321 + .16. 11. 21.0005 + .3842 + .1 + .005 + 3.6 + .158. 12. 1.0006 + 2001.1 + .003 + 5.5 + 11.1111. DECIMALS. 221 SUBTRACTION. Rule. Write the numbers so that the decimal point of the subtrahend stands directly under the decimal point of the minuend, subtract as in integers, and place the point directly under the points above. NOTE. It is sometimes convenient to give the decimals the same denominator, by annexing decimal naughts. Subtract : 1. 24.3 2. 2.86 3. 4. 4. 2.46 4.5 1.325 1.15 .005 Find remainders : 5. 7 - .15 10. 29.325 - 15.14 6. 1 - .004 11. 3.852 - .125 7. 13 - 2.1 12. 1.1111 - .0011 8. 3.256 - 1.05 13. 500 - .05 9. 256.1 - 1.256 14. 25.3894 - 15.005 15. From twenty-eight, and twenty-five thousandths take fourteen, and twenty-five hundredths. 16. From one tenth take one thousandth. 17. Which is the greater, fifty thousandths or five hundredths ? 18. Take one thousandth from one thousand. 19. From 5 hundred take 5 hundredths. MUL TIPLICA TION. Oral Work. How much is 2 times .3 ? 3 times .3 ? 4 times .3 ? 7 times .02 = ? 12 times .06 = ? 12 times $.12 =? AXA = T**5 .1 X.I = .01. -3 X .05 = .015. .3X.3=.09. 222 ARITHMETIC. T ^ x T 3 ff = rMff How many naughts in the denom- inator of the product ? How many naughts in the denominators of both factors ? Every decimal has a corresponding common fraction, and for each naught in the denominator of the common frac- tion, there is a decimal figure in the decimal. .05 X .3 = .015 How many decimal figures in both factors ? Rule. Multiply as in integers, and point off from the right of the product as many decimal figures as there are decimal figures in both factors. NOTE. If there are not figures enough, prefix naughts. Naughts at the right of a decimal have no value, and may be omitted. 2.8 1.25 .005 25 X 8 .6 .03 .06 22.4 .750 .00015 1.50 Find the products : 1. .18 X .15 8. 13.3 X 1.3 2. 1.0005 X .2 9. 100 X .01 3. 2.5 X .06 10. 100.56 X .0005 4. 56 X .005 11. 25.32 x 1.05 5. .005 X 1.6 12. 2.84 x 2 6. 25.05 X 1.15 13. 3.28 X 12J 7. 2.863 X 100 14. 1.111 X 1000 To multiply by 10, 100, 1000, etc. 1.265 1. Multiply 1.265 by 100. 100 Remove the point one place to the right for 126.500 every naught in the multiplier. Do not write the multiplier. DECIMALS. 223 Oral Work. Multiply : 2. 3.84 by 10 7. .3 by 100 3. 12.63 by 100 8. 1.869 by 100 4. 1.5555 by 1000 9. 32.856 by 1000 5. 1.358 by 1Q 10. 138.56 by 1000 6. .25 by 1000 11. 11.11 by 100 DIVISION. Since in multiplication there are as many decimals fig- ures in the product as in both factors, in division the quotient will have as many decimal figures as the number of decimal figures in the dividend exceeds those in the divisor. .5) 12.685 Since there are three decimal figures in the 25.37 dividend and one in the divisor, there must be two in the quotient. Prove by multiplying dividend by quotient. In all cases divide as in integers, then place the decimal point. Divide 399.552 by 192. 2.081 192) 399.552 When the divisor is an integer, place the point in the quotient directly over the point in the dividend, in long division (directly 1^ under in short division). 192 Prove by multiplying the divisor by the quotient. 4) 12 If we multiply both dividend and divisor by 3 the same number, the quotient will not be changed. Multiply both 4 and 12 by 2; by 6; by any number. What is the effect in each case ? Divide 28.78884 by 1.25. 224 ARITHMETIC. 23.031 + When the divisor contains decimal 1.25')28.78 884 figures, move the point in both divisor and dividend as many places to the right as there are decimal figures in the divisor, then place the point in the quotient as if the divisor were an ~jcj 4 integer. 125 NOTE 1. The new point in both 9 dividend and divisor may be placed on a line with the tops of the figures, and the original point may stand, to preserve the reading of the decimals. NOTE 2. In the above example, the moving of the point two places to the right in both dividend and divisor is equivalent to multiplying each by 100. NOTE 3. If the quotient does not have a sufficient number of figures, prefix naughts. NOTE 4. Before commencing to divide see that there are at least as many decimal places in the dividend as in the divisor. NOTE 5. If there is a remainder, after all the figures of the dividend are used, annex decimal naughts and con- tinue the division. NOTE 6. It is not usually necessary to have more than four decimal figures in the quotient. Divide .125 by .5 Divide 1.25 by .5 Divide 12.5 by .5 Divide at sight : 1. 3)3.33 4. .03).333 7. .5)25 5. .003).333 8. .05).25 6. 5^)2.5 .07).28 DECIMALS. 225 18. 376 - 19. 376 - 20. 37.6 21. 3.76 .6 .06 .6 - .06 22. 1.875 -f- .005 23. 15.55 ^ .1 24. 1 + .1 25. .1 -T- 1 Find quotients, and prove : 10. 3.57 -T- .7 11. .488 -*- .12 12. 16.55 -4- .05 13. 13.13 -^ 1.3 14. 1.111 -- .01 15. 5555 ^- .5 16. .875 -f- .05 17. 73.5 ~ 1.05 Divide : 26. Twenty-five ten-thousandths by 25 hundredths. 27. 4678 hundred-thousandths by 9 ten-thousandths. 28. 3582 ten-thousandths by 3 hundredths. To divide by 10, 100, 1000, etc., remove the decimal point one place to the left for each naught in the divisor. Divide at sight : 33. 100)189.36 34. 100).189 35. 100)148.369 29. 10)365.8 30. 10}5_ 31. 10)148.963 32. 10)115.55 Eead and add : 1. 32.065 6.006 4.25 .032 .25 25.01 111.11 2. 36.9486 2583.04 4.9602 15.15 100.001 56.56 141.141 226 ARITHMETIC. Review of Decimals. 1. Tell where to place the decimal point in any product. 2. In any quotient. 3. In any remainder. 4. In any sum. 5. From 1 take 1 millionth. 6. Add 1 tenth, 1 hundredth, and 1 thousandth. 7. Find the product of 1 multiplied by .15. 8. Multiply at sight : 36.984 by 1000. 9. Divide at sight : 159.83 by 1000. Change to decimals : 10. i, f, *, 16}, 255. Change to common fractions : 11. .28, .38, .375, 15.125. 12. If John earns $8 in a week, how much can he earn in 7.5 weeks ? 13. If a barrel of flour costs $5.25, how many barrels can be bought for $105 ? 14. How many rods of fence around a rectangular field 10.018 rd. long, and 6.2892 rd. wide ? 15. Eeduce 5.75 rods to feet. 16. How many rods in 247.5 feet ? . 17. What is the effect when a decimal figure is removed one place to the left ? to the right ? 18. What is the effect when an integral figure is removed one place to the left ? One place to the right ? 19. At 15^ a peck, how many pecks of pop-corn can be bought for $3.75 ? 20. What is the cost of 28.78 yd. of cloth at $3.15 a yard? DECIMALS. 227 Written Work. Short Methods. 1. Multiply 2.222 by 200. Remove the point two places to the right, and multiply by 2 ; thus : 222.2 X 2 = 444.4. Am. Multiply and prove : 2. 6.84 X 500 5. 3.553 X 800 3. 28.96 X 2000 6. 4.583 X 4000 4. 1.876 X 80 7. 156.2 X 200 8. Divide 444.4 by 200. Eemove the point two places to the left, and divide by 2; thus: 4.444 _=_ 2 = 2.222. Ans. Divide and prove : 9. 369.5 -f- 500 15. 2.835 -=- 700 10. 36.66 -f- 60 16. 35.46 -f- 40 11. 36.48 -4- 8000 17. 808 -r- 700 12. .159 -r- 900 18. 2585 -r- 5000 13. 172.8 -f- 120 19. 3.672 -f- 6000 14. 1.44 + 1200 20. 25.55 -5- 50 21. How much is j of 324 ? 22. How much is .75 of 324 ? Oral Work. Questions of Relation. 23. .5 of 16 equals what ? 24. 8 is what decimal part of 16 ? 25. 8 is .5 of what ? 26. How much is .06 of $200 ? 27. $12 is what decimal part of $200 ? 28. Five cents is what part of $5 ? Express the answer as a decimal fraction. 228 ARITHMETIC. MISCELLANEOUS REVIEW. 1. How much must be paid for lathing and plastering the walls and ceiling of a room 36 ft. long and 20 ft. wide at 25 ct. a square yard ? 2. How many cubic yards of earth were removed in digging an excavation for a cellar 28 ft. 9 in. long, 18 ft. 8 in. wide, and 5 ft. 6 in. deep ? 3. How many yards of carpet 27 inches wide will cover a room 22 ft. 6 in. long, and 18 ft. wide. 4. How deep is a lot 100 ft. wide that contains 20450 sq. ft. ? 5. How many square inches in the surface of a 12-inch cube? 6. In a pile of wood 64 ft. long, 24 ft. wide, and 8 ft. high, how many cords ? 7. A block of marble is 3 ft. long, 20 in. wide, and 18 in. thick. How many 3-inch cubes can be cut from it ? (Solve by calculation.) 8. A man purchased 4 pieces of sheeting, each contain- ing 35 yd., at 10 cents a yard, and pays for it in eggs at 15 cents a dozen. How many dozen eggs does it require ? 9. Divide the product of 8, 7, 5, 3, 6 by the product of 15, 4, 7. 10. What is a Eectangle ? 11. Define a Proper Fraction. 12. Which is greater, and how much, | of 24, or of 30 ? 13. If tea is worth $ a pound, how much can be bought for $J ? (Question.) 14. What is the cost of 40.5 tons of coal when .3 of a ton costs $4.48 ? 15. John had 48 marbles and lost .25 of them. How many did he lose ? REVIEW. 229 1. If f of an acre of land produces 20 bushels of grain, how many bushels will 4 acres produce ? 2. How many yards of cloth at $3.75 a yard can be purchased for $288.75? 3. 869.44 X 2000 = ? 4. 869.44 -r- 2000 = ? 5. How many dozen eggs at 12 cents a dozen will pay for 96 bushels of corn at 60 cents a bushel ? 6. A coal-dealer received 12 car-loads of coal. Each car contained 18000 pounds. How many tons did he receive ? 7. What is the cost of 3142 pounds of plaster at $3.75 a ton? 8. If a man's income is $1800 a year, and his expenses are $4.50 a day, how much can he save in a year ? 9. In how many years will a man lay up $1000 if he earns $75 a month, and his expenses amount to $35 a month ? 10. A boy sells oranges at the rate of 12 for 36 cents. How much does he receive for 48 oranges ? for 53 oranges ? for 38 oranges ? 11. The divisor is 3.875 and the quotient 46.5, what is the dividend ? 12. The quotient is 206.48 and the dividend 268.432, what is the divisor? 13. How many 16ths in | ? 14. When it costs $1.50 to travel 75 miles on the railroad, how much will it cost to travel 375 miles ? 15. $12 is .06 of how many dollars ? 16. What decimal part of a bushel is a peck ? 17. What part of a yard is 1.5 feet ? 230 ARITHMETIC. 1. Find the product of 248 and 763. 2. Find a quotient, using 6954 and 218. Find another. 3. The sum of four numbers is 12006 ; 6492, 3005, and 943 are three of the numbers. Find the other. 4. A field 213 rd. long and 180 rd. wide contains how many sq. rd. ? How many acres ? 5. Multiply f of 432 by $\. i + A-s-* = ? (* + A) * = ? 7. What is a numerator ? What is a fraction ? 8. Find the area of the floor in a room 15 ft. long and 12 ft. wide. 9. Write three factors of 72. 10. Find the value of 15 X 75 x 81 6 X 9 X 3 X 25 11. How many cubic inches in a marble block 6 ft. long, 4 ft. wide, and 2 ft. 9 in. thick ? 12. If 6 cwt. of coal cost $1.80, what will a ton cost ? 13. How many boards 12 ft. long, and 6 inches wide, will make a floor 24 X 18 ft. ? (Cancel.) 14. A pile of wood 4 ft. long, 4 ft. wide, and 4 ft. high, will cost how much, at $6.25 a cord ? 15. 429 has three factors. If 11 is one of them, what are the other two ? 16. A field is 30.65 rd. wide, and nine times as long as it is wide. Find its area in sq. rd. 17. How many thousandths in 2 ? 18. 2.0705 -T- 2.05 = ? Write your answer in words. 19. Write a bill of the following sale to James P. Brown : 12J tons of coal at $6.25; 18 bushels of corn at $.75; 478 ft. of lumber at $17 NUMBER. White's TWO Years With Numbers. Number Lessons for second and third year pupils. 40 cts. AtWQOd'S Complete Graded Arithmetic. Present a carefully graded course in arithmetic, to begin with the fourth year and continue through the eighth year. Part I. 200 pages. Cloth. 40 cts. Part II. 382 pages. Half leather. 75 cts. Walsh's Mathematics for Common Schools. Special features of this work are its division into half-yearly chapters instead of the arrangement by topics ; the omission, as far as possible, of rules and definitions ; the great number and variety of the problems; the use of the equation in solution of arithmetical problems; and the introduction of the elements of algebra and geometry. Parti. 218 pages. 35 cts. Part. II. 252 pages. 40 cts. Part III. 365 pages. Half leather. 75 cts. Sutton and Kimbrough's Pupils' Series of Arithmethics. PRIMARY BOOK. Embraces the four fundamental operations in all their simple relations. 80 pages. Boards. 22 cts. INTERMEDIATE BOOK. 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