IN MEMORIAM FLORIAN CAJORI -,-', A rf \ 3 and 5 ' (. 2 and 6, 4 and 4. ADDITION. Uniting the parts. How many are 7 soldiers and 1 soldier? 1 s and 7 S's ? 3 s's and 5 s's ? 6 s's and 2 s's ? 4 birds and 4 birds? 5 b's and 3 b's? 2 b's and 6 b's? What are the two parts of 8? What do they make when united or put together? How many are 5 marbles and 3 marbles? 4 ill's and 4 ill's? 6 cakes and 2 cakes? 7 buds and 1 bud? How manv is 1 more than ti 7 ? 3 more than 5 ? 4 more than 4 ? Ike has 5 peaches, and Jim has three peaches more than Ike; how many peaches has Jim? Ben jumped 6 feet and Tom jumped 2 feet further than Ben; how far did Tom jump? The Number Eight. 29 Copy and read : 1 + 7 = 8; 3 + 5 = 8; 5 + 3 = 8; 7_|_1 == 8 ; 2 + 6 = 8; 4 + 4 = 8; 6 + 2 = 8; 8 + = 8. SUBTRACTION. Taking away one part. How many soldiers would be left in the squad if 1 soldier were taken away ? If two soldiers were taken away? If 3 soldiers? If 4 soldiers? 5 soldiers? 6 soldiers ? 7 soldiers ? 8 soldiers ? What are the two parts of 8? When one of them is taken from 8, what is left? I wish to put 8 birds in two cages so as to have 3 birds in one cage, how many birds will be in the other cage? How can I put 8 hogs in two pens so as to have 2 hogs in one of the pens? Moses and Joseph ate 8 biscuits together ; if Joseph ate 4 biscuits, how many did Moses eat? Which is the more, 8 or 5 ? How many more ? Copy and read : 8 2 = 6; 8 3=5; 8 6 = 2. MULTIPLICATION. Uniting equal parts. Make 8 marks. How many times did you make 1 / *' mark? 8 times 1 mark are how many? Are 2 marks a part of 8 marks ? If you make 2 marks 4 times, thus : //, //, I It 111 how many marks will there be in all? How man} 7 , then, are 4 times 2 marks? 4 times 2 sol- diers? Are 4 marks a part of 8 marks? If you make 4 marks 2 times, thus: ////, ////, how many marks will there be in all ? How many, then, are 2 times 4 marks? 2 times 4 soldiers? Ike has 4 marbles and John has 2 times as many as . Ike ; how many marbles has John ? Harry has 4 times 2 dimes and Jane has 2 times 4 dimes which has the more ? Copy and read : 8x1 -8; 4x2=^8; 2 X 4 = = 8. 30 Intermediate Arithmetic. DIVISION. Measuring by a part. Does 8 contain its parts ? Does the squad of 8 soldiers contain 1 soldier 8 times? Suppose 2 soldiers were taken away ; then 2 soldiers more were taken away ; then 2 more; then 2 more; would there be any soldiers left? How many times can 2 soldiers be taken away? How many times, then, do 8 soldiers contain 2 soldiers? If 4 soldiers were taken away, and then 4 soldiers more, would any soldiers remain ? How many times can 4 soldiers be taken away? How many times, then, do 8 soldiers contain 4 soldiers? OBJECT EXERCISES. Take 8 blocks ; put them on a table and call them soldiers. What stands' for soldier ? Ans. S. Now separate the soldiers into two parts every way you can, calling the results thus : 1 s and 7 s's are 8 s's, etc. Now take off 1 soldier at a time, calling the results thus : 1 s from 8 s's leaves 7 s's ; 2 s's from 8 s's leave 6 s's, etc. Now put on 1 soldier at a time, calling the results thus : 1 time 1 s is 1 s ; 2 times 1 s are 2 s's, etc. Now take off 1 s at a time, calling the results thus : 1 s contains 1 s, 1 time; 2 s's contain 1 s, 2 times, etc. Next put on 2 soldiers at a time, calling the results thus : once 2 s's is 2 s's ; 2 times 2 s's are 4 s's, etc. Next take off 2 soldiers at a time, calling the results thus : 2 s's contain 2 s's, 1 time ; 4 s's contain 2 s's, 2 2 times, etc. Now put on 4 soldiers at a time, etc., and then take off 4 s's at a time, etc. The Number Nine. 31 Learn and recite : THE TABLE OF EIGHT. and 8 are 8 from 1 and 7 are 8 1 from 2 and 6 are 8 2 from 3 and 5 are 8 3 from 4 and 4 are 8 4 from 5 and 3 are 8 5 from 6 and 2 are 8 6 from 7 and 1 are 8 7 from 8 and are 8 8 from 1 time 8 is 8 1 in 2 times 4 are 8 2 in 4 times 2 are 8 4 in 8 leaves 8 8 leaves 7 8 leaves 6 8 leaves 5 8 leaves 4 8 leaves 3 8 leaves 2 8 leaves 1 8 leaves 8, 8 times 8, 4 times 8, 2 times LESSON IX. ABOUT THE NUMBER NINE. Is 9 one more than 8 ? Hold up 9 fingers. Make 9 steps and count them as you step. clclilclciJJclclci How many a's are in the above row ? Count and see if there are 8 a's and la? 7 a's and 2 a's? 6 a's and 3 a's? 5 a's and 4 a's? What are the two parts of 9? An*. / ij an( J *' 7 and 2 < (> and 3, 5 and 4. 32 Intermediate Arithmetic. ADDITION . Uniting the parts. Count 9. How many are 8 men and 1 man ? 7 m's and 2 ill's? 6 cats and 3 cats? 5 c's and 4 c's? What are the two parts of 9? When united do they make 9 ? Is 1 more than 8 the same as 2 more than 7? Is 3 more than 6 the same as 4 more than 5? How many are 4 and 5 ? 5 and 4 ? 6 and 3 ? 3 and 6? 7 and 2? 2 and 7? Five eggs in one nest and 4 eggs in another; how many eggs in all ? 7 tulips and 2 tulips are how many ? 6 daisies in one cluster and 3 in another ; how many daisies in both clusters ? Copy and read: 1+8 = 9; 2+7=9; 3 + 6 = 9; 5 + 4 = 9. S U BTR ACTION . Taking aivay one part. Count 9 backward. How many are 2 ducks less than 9 ducks? 4 doves less than 9 doves? 6 rats less than 9 rats? What are the two parts of 9? When one of them is taken from 9, what is left? 9 birds were on a limb, but three of them have flown, how many birds are left? The old hen has 9 chicks; 6 of the chicks are on one side of the fence, how many are on the other side? A boy had 9 marbles but lost 5 of them, how many marbles had he left? If 7 chairs are 1 part of a set of 9 chairs, how many are in the other part? 7 from 9 leaves how many ? Copy and read: 9 1 = 8; 92 = 7; 9 3 = 6; 9 4 = 5; 9-5=4. MULTIPLICATION.- Uniting equal parts. Make 9 aV. How many times did you make la? How many, then, are 9 times l*a? 9 times 1 horse? Object Exercises. 33 9 times 1 boy? Are 3 a's a part of 9 a's? If you make 3 a's 3 times, thus : aaa, aaa, aaa, how many a's will there be in all? How many, then, are three times 3? 3 times 3 cows? 3 times 3 cats? Show me 3 times 3 fingers. Show me 3 times 3 planks. Little Hal is 3 years old, and Mattie is three times as old as Hal; how old is Mattie? Are 9 times 1 dollar more than 3 times 3 dollars? Copy and read : 9X1 = 9; 3X3 = 9. DIVISION .Measuring by a part. Does 9 a contain its parts? Is 1 a one of the parts? How many a's does a row of 9 a's contain ? Make 9 a's. Erase 3 a's at a time, until all are gone. How many times did you erase? How many times, then, are 3 a's contained in 9 a's? Is a yard -stick 3 feet long? How many times is a yard -stick contained in 9 feet? What is a yard -stick for? Ans. For measuring. Can I measure a pole, 9 feet long, by it? How many measures would it take? How many does 7 lack of containing one 9 times? How many does 5 lack ? 6 ? 8 ? 3 ? How many do 8 cups lack of containing 3 cups 3 times? How many do 4 cups lack? Copy and read: 9-^-1=9; 9^3 = 3; 9 -=- 9 = 1. OBJECT EXERCISES. Take 9 blocks, put them on the table and call them horses. What stands for horses? Ans. li. Now separate the horses into two parts every way you can, calling the results thus : 1 li and 8 li's are 9 h's, etc. N. I. 3. 34 Intermediate Arithmetic. Next take off 1 horse at a time, calling the results thus: 1 horse from 9 horses leaves 8 horses; 2 li's from 9 h's leave 7 h's, etc. Next put on one horse at a time, calling the results thus: 1 time 1 h is 1 h; 2 times 1 h are 2 h's, etc. Next take off 1 horse at a time, calling the results thus: 1 h contains 1 h, 1 time; 2 h's contain 1 h, 2 times, etc. Next put on 3 horses at a time, calling the results thus : 1 time 3 h's is 3 h's ; 2 times 3 h's are 6 h's, etc. Now take off 3 h's at a time, calling the results thus : 3 h's contain 3 h's, 1 time ; 6 h's contain 3 h's, 2 times, etc. NOTE. The teacher may repeat the object exercises, giving the blocks or pebbles such names as will interest and amuse the pupils. Learn and recite- THE TABLE O,F NINE. and 9 are 9 from 9 leaves 9 1 and 8 are 9 1 from 9 leaves 8 2 and 7 are 9 2 from 9 leaves 7 3 and 6 are 9 3 from 9 leaves 6 4 and 5 are 9 4 from 9 leaves 5 5 and 4 are 9 5 from 9 leaves 4 6 and 3 are 9 6 from 9 leaves 3 7 and 2 are 9 7 from 9 leaves 2 8 and 1 are 9 8 from 9 leaves 1 9 and are 9 9 from 9 leaves 1 time 9 is 9 1 in 9, 9 times 3 times 3 are 9 3 in 9, 3 times The Number Ten. 35 LESSON X. ABOUT THE NUMBER TEN. What lesson is this? What is it about? 10 is the next number after what? Count all your fingers. How many have you ? How many frogs in the pic- ture? How many are on the log? How many are on the ground ? If another frog gets on the log, how many would then be on the log and ground ? How many if 2 frogs more get on the log? How many if 3 frogs more get on the log? 4 frogs more? 5 frogs? Is one part of 10 frogs on the log and the other part on the ground ? ( 1 and 9, 3 and 7, What are the two parts of 10? Ans. < 2 and 8, 4 and 6, 15 and 5. ADD IT ION. Uniting the parts. How many are 7 frogs and 3 frogs? 8 f's and 2 f's? 9 f's and 1 f ? 4 men and 6 men ? 5 ill's and 5 in's ? 3 birds and 7 birds? 2 b's and 8 b's? What are the two parts of 10? How many do they make when put together? How many are 5 marbles and 5 marbles? 7 eggs and 3 eggs? 6 cakes and 4 cakes? Charles is 7 years old and Fannie is 3 years older; how old is Fannie? Mary has 6 roses and Jane has 4 roses; how many roses have both girls? Copy and read : 4 -f 6 = 10; 8 -f 2 = 10; 3 -f 7 = 10. 36 Intermediate Arithmetic. SUBTRACTION. Taking away one part. If 10 frogs were on a log, and 1 frog should leap off, how many frogs would be left? How many would be left if 2 frogs should leap off? If 3 frogs should leap off? If 4 frogs? 5 frogs? 6 frogs? 7 frogs? 8 frogs? 9 frogs? 30 frogs? What are the two parts of 10? When one of them is taken from 10, what is left? How many are 8 and 2 ? How many, then, is 10 less 2 ? There are 10 frogs in all ; if 4 frogs are on the log, how many are on the ground ? Ann had 10 little birds, but the old cat ate 3 of them ; how many birds has Ann now ? 10 pinks in one bed, and 6 pinks hi another bed ; how many more pinks in one than in the other? 10 hogs were in the garden, but John turned 4 hogs out; how many re- main in the garden ? Copy and read: 10 3 = 7; 10 5 = 5; 10 8 = 2. MULTIPLICATION. Uniting equal parts. Count 10. Make 10 c's. How many times did you make 1 c? How many, then, are 10 times 1 c? 10 times 1 horse? Are 2 c's a part of 10 c's? If you make 2 c's 5 times, thus : CC, CC, CC, cc, CC ; how many c's will there be in all? How many, then, are 5 times 2 c's ? 5 times 2 cups ? 5 times 2 caps ? If you make 5 c's 2 times, how many c's will there be in all? How many, then, are 2 times 5 c's? 2 times 5 cats? Are 5 times 2 dollars the same as 2 times 5 dollars? Little Ella is 5 years old, and her brother is 2 times as old as she; how old is her brother? Copy and read : 10 X 1 = 10 ; 5 X 2 = -- 10 ; 2 X 5 = 10. Object Exer rises * 37 DIVISION. Measuriny by u o 5 fl < s < c ^ ^ WT^ ^ , i- j^ / O G C WHO WHO WHO WHO WHO r-j ;-J r^ ^H ,-i r~* r* * r* , . _ Places of units i2^2 S^32 ^^^ ^^-^ eooii-i Figures 360,504,725,913,876. Periods I Numbers Fifth. Fourth. Third. Second. First. \ Names Trillions. Billions. Millions. Thousands. Ones. EXERCISES IN NUMERATION. 34. i. Read the number represented by 37000401064. OPERATION. Separate the number into periods, thus : 37,000, 401,064. The 4th period is billions, the 3d is millions, the 2d is thousands, the 1st is units ; hence, the number is 37 billion million 401 thousand 64 units, or thirty-seven billion four hundred one thousand sixty-four. RULE. I. Begin at the right and separate the number into periods. II. Then begin at the left and read each successive period as if it stood alone, giving each its name except the period of units. N. I.-4. 50 Intermediate Arithmetic. In this manner read 2. 43564. 8. 3. 75031. 9. 4. 132140. 10. 5. 5720307. 11. 6. 4006009. 12. 7. 7205806. 13. 54311237. 801603709. 4321780651. 123456789. 123456654321. 230405060708090. EXERCISES IN NOTATION. 35. 1. Express in figures fifty-three billion sixty-five million three hundred seven. OPERATION. Since billions, the highest, number named, occupy the fourth period, there will be four periods in the number. Now, beginning at the left, we fill each of the periods with the given numbers of billions, millions, thousands, and units, as if each stood alone, and obtain 53,065,000,307. KULE I. Consider from the greatest number named the necessary number of periods. II. Begin at the left and fill each of the successive pe- riods as if it stood alone. NOTE. There must be three figures in every period, except the one at the left, which may have one, two, or three. All vacant orders and periods must be filled with ciphers. In this manner express in figures : 2. Eighteen thousand five hundred thirty-six. 3. Thirty-two thousand eight. 4. Forty-seven thousand two hundred. 5. Two hundred forty thousand five hundred one. Notation and Numeration. 51 6. Six million five thousand forty-seven. 7. Nine million twenty-three thousand thirty-one. 8. Twenty-nine million four hundred twelve thou- sand five. 9. One hundred seven million eleven thousand one hundred four. 10. Seven hundred thirty million six hundred nine thousand three hundred ninety-two. 11. Thirteen billion thirteen million thirteen thou- sand thirteen. 12. Two hundred forty-five billion one hundred seven million fifty nine thousand eight hundred seventy. NOTATION OF DOLLARS AND CENTS. 36. The Sign of Dollars is 8, which is read, dollars. Thus, $12 is read 12 dollars. 37. The Sign of Cents is c, or cts., which is read, cents. Thus, 23c., or 23 cts., is read 23 cents. 38. Dollars and cents may be written as one number by placing a point (.) between them. Thus, 25 dollars 34 cents, is written $25.34. 39. Since it takes 100 cents to make a dollar, cents always occupy two places at the right of the point. Hence, when the number of cents is less than 10, a cipher must he written between it and the point. Thus, 5c. is written $.05 ; and 3 dollars 8 cents is written $3.08. Neither the sign ($) nor the point (.) should be omitted. 52 Intermediate Arithmetic. 40. Exercises in Numeration of Dollars and Cents. Read : 1. 85.35. 2. $7.40. 3. 810.09. 4. $.06. 5. $17.13. 6. $33.07. 7. $ .34. 8. $1.01. 9. $3140.43. 10. $504.67. 11. $5008.03. 12. $.04. 41. Exercises in Notation of Dollars and Cents. Ex- press in figures and signs : 1. Thirteen dollars fifteen cents. 2. Forty dollars fifty cents. 3. Forty-three dollars seven cents. 4. One hundred dollars twenty cents. 5. Sixty dollars ten cents. 6. Thirty-five cents. 7. Nine cents. 8. Eighty-four dollars six cents. 9. Ninety-nine dollars twelve cents. 10. Fifty-four cents. 42. A Scale in Arithmetic is the relation between the successive orders of units. In the Arabic system of notation, the scale is ten ; that is, the value of the unit in any order is ten times as great as the unit in the next lower order ; hence, it is called the Decimal Scale, from the Latin word decem, meaning ten. NOTATION OF OBJECTS. 43. In this work objects are frequently denoted by the first letters of their names. Thus, an arm, an ap- ple, etc., is denoted by a; a box, a bin, a boy, etc., is denoted by b. Notation and Numeration. 53 Conversely, an a may be regarded as denoting an apple, or an ax, etc.; li as denoting an hour, a hand, etc. If a letter appears more than once in the same example, each one denotes the same thing. Thus: 2 b and 1 b are 3 b, may be read, 2 boys and 1 boy are 3 boys ; or 2 boxes and 1 box are 3 boxes. ABBREVIATIONS. 44. The colon (:) is employed to denote " the follow- ing ," or "as follows" and to signify that the term or phrase preceding it is to be prefixed to each of the phrases following it. Thus, "from 5 take : 3, 4;" means "from 5 take 3; from 5 take 4." QUESTIONS FOR REVIEW. 45. What is: 1. A unit? 2. A number? 8. The unit of a num- ber ? 4. An abstract number ? 5. A concrete number ? 6. Nota- tion? 7. Numeration? 8. A scale? Name: 1. The figures. 2. The digits. What is the greatest number that can be expressed by: 1. One figure? 2. Two figures? 3. Three figures? 4. Four figures? How many figures in : 1. One place? 2. One period? How are places and periods numbered ? Repeat the table of : 1. Tens. 2. Thousands. 3. Places. 4. Periods. What is the rule for: 1. Reading numbers? 2. Writing num- bers? What stands for: 1. Dollars? 2. Cents? How arc dollars and cents written as one number? In this book: What often stands for an object? What does a colon ( : ) denote? A DDITION. INDUCTIVE EXERCISES. 48. 1. Count 7. Am. 1, 2, 3, 4, 5, 6, 7. 2. Count $7. ^m-. 81, $2, $3, $4, 3. Iii counting, what are the next 3 numbers above $4? Ans. $5, $6, $7. 4. If I count $4 and then count S3 more, how much will I have counted in all? Ans. $7. 5. In counting, what are the next 4 numbers above 6 hats? Ans. 7 hats, 8 hats, 9 hats, 10 hats. 6. If I count 6 hats and then count 4 more, how many hats will I have counted in all? Ans. 10 hats. In counting, what is the : 7. 1st number above 8? 8. 2d number above 9 ? 9. 3d number above $5 ? 10. 4th number above 8 hours? 11. 5th number above 4 pecks? How many are : 7. 8 and 1 ? 8. 9 and 2? 9. 85 and $3 ? 10. 8 hours and 4 hours ? 11. 4 pecks and 5 pecks? 12. James makes 7 steps and then 4 steps more; how many steps does he make in all? 13. If you have 5 apples and I give you 3 more, how many apples will you then have? 14. Seven peaches are on one tree and 6 on another tree ; how many peaches on both trees ? 15. John paid 8 cents for apples and 4 cents for pears; how much did he spend? (54) Addition. 55 How many are : 16. $6 and $1. 17. 7 men and 2 men? 18. 5 pints and 3 pints. How many are: 16. $1 added to 17. 2 men added to 7 men? 18. 3 pints added to 5 pints? DEFINITIONS. 47. Like Numbers are those which have units of the same kind, as 7 and 13; 3 pounds and 11 pounds; 6 tens and 20 tens. 48. Unlike Numbers are those which have units of different kinds, as 5 yards and 17 pints ; 3 days and 25 feet. 49. Addition is the process of uniting two or more like numbers into one, 50. The numbers added are called the parts, and the number obtained by adding, the sum or amount. Thus, 7 balls and 3 balls, when united, make a group of 10 balls. Here 7 balls and 3 balls are the parts, and 10 balls the sum or amount. 51. A Sign is a symbol used to indicate an operation or relation. 52. The Sign of Addition is -f- > which is read : and or plus ; plus means more. When -}- stands between two numbers, it indicates that they are to be added. 53. The Sign of Equality is = , which is read : are or equals. 54. The Sign of Interrogation is ? , which is read : what or how many. It signifies that the answer is to be found, and when found, belongs in the place occupied by the sign. 56 Intermediate Arithmetic. Thus, 9 + 5 = 14 is read: 9 and 5 are 14, or 9 plus 5 equals 14. Beginners should adopt the first reading. Again, 8 -f 3 = ? is read: 8 and 3 are how many. EXERCISES. 55. Read : 1. 8 +5 13. 2. 9 -1-6 =15. 3. 5 +4 =? 4. 6c. -f 4c. = lOc. 56. Express by signs 5. $7 + $10 = $17. 6. $6 + $3 =? 7. 7 men -f 4 men = 11 men. 8. 5 pins + 6 pins = ? 1. 3 and 5 are 8. 2. 2 and 7 arc what ? 3. 3 -f- 4 equals what ? 4. 3c. and lOc. are 13c. 5 . 6 balls added to 3 balls are 9 balls. 6. How many are 3 and 12? 7. The sum of 6 and 2 is 8. 8. $4 and $8 are how many ? 57. PRINCIPLE. Only like numbers can be added. 58. The Complemental Parts of a number are the num- bers whose sum equals that number. Thus: the complemental parts of 7 are 3 and 4, or 1, 2, 3 and 1. 59. By Addition we find a number when its comple- mental parts are given. Complemental will frequently be denoted by the letter c. Thus, the c parts of 8 are 5 and 3, or 6 and 2. SUGGESTION TO TEACHERS. In all the examples in Addition, the pupil should be required to point out the c parts and the ivliole. 60. The following table should be thoroughly com- mitted to memory : Addition. 57 ADDITION TABLE. 1. 2. 3. 4. 5. 1 + 1= 2 2+1= 3 3+1= 4 4+1= 5 5+1= 6 1-1-2= 3 2+2= 4 3 + 2= 5 4+2= 6 5+2= 7 1+3= 4 2+3= 5 3+3= 6 4+3= 7 5+3= 8 l-f4= 5 2+4= 6 3+4= 7 4+4= 8 5+4= 9 14-5= 6 2+5= 7 3+5= 8 4+5= 9 5+5=10 1+6= 7 2+6= 8 3+6= 9 4+6=10 5+6=11 1+7= 8 2+7= 9 3+7=10 4+7=11 5+7=12 1+8= 9 2 + 8=10 3+8=11 4+8=12 5+8=13 1+9=10 2+9=11 3+9=12 4+9=13 5+9=14 6. 7. 8. 9. 10. 6 + 1 7 7+1= 8 8+1= 9 9+110 10+1 11 W | J. f J- Vy l J. l_ J. 6+2= 8 7+2= 9 8+2=10 9+2=11 10+2=12 6+3= 9 7+3=10 8+3=11 9+3=12 10+3=13 6+410 7-4-411 8+4=12 9+4=13 10+4=14 \-r | 1 ^^^^ J_ \_T 9 | I -L J. 6+5=11 7+5=12 8+5=13 9+5=14 10+5=15 6+6=12 7+6=13 8+6=14 9+6=15 10+6=16 6+7=13 7+7=14 8+7=15 9+7=16 10+7=17 6+8=14 7+8=15 8+8=16 9+8=17 10+8=18 6+9=15 7+9=16 8+9=17 9+9=18 10+9=19 NOTE. Pupils should read the tables thus: 1 and 1 are 2, 1 and 2 are 3, 2 and 1 are 3, 1 and 4 are 5, 4 and 1 are 5, etc. The table is expressed in signs to familiarize the pupil with their use and meaning. DRILL EXERCISES. 61. The following, read vertically, are the two com- pie mental parts of the number written under them : 58 Intermedia fe Arithmetic. 2 32 43 543 654 7654 8765 98765 12 12 123 123 1234 1234 12345 6 8 9 9 9 98 78 18 17 16 15 98 987 987 9876 67 567 4j>_6 3456 13 14 12 10 9876 2345 11 SUGGESTIONS TO TEACHERS. These parts and their sum should be thoroughly committed to memory, so that they will be recog- nized at a glance. For this purpose the following is recommended : Procure two boards of convenient di- mensions and dress them smoothly, so that one will slide freely on the other. Mark, number, and place j them as shown in the dia- gram, and require the class in daily drill, to name the sum of the opposite num- bers. Then shift the posi- tion of the upper board, and proceed as before, until all the combinations have been reached. These boards, especially with the addition of two or three similar ones, will afford abundant drill exercises in Addition, Subtraction, and Multiplication. We shall subsequently refer to them as the Combination Boards. See Art. 104. COMBINATION BOARDS. MENTAL EXERCISES. 62. i. Frank is 10 years old, and his sister is 3 years older; how old is his sister? Addition. 59 2. John has 6 pears, and William has 5 more than John ; how many pears has William ? 3. Susan has 7 roses and Lucy has 3 roses; how many roses have both girls? 4. Thomas killed 6 squirrels on Monday, and 8 squir- rels on Friday ; how many squirrels did he kill on both days? 5. 7 rabbits are in the garden, and 4 rabbits are in the yard ; how many rabbits in all ? 6. 1 hen has 8 chicks, and another hen has 5 chicks; how many chicks have both hens? 7. A boy worked 9 hours on Tuesday, and 6 hours on Wednesday ; how many hours did he work on both days? 8. John caught 5 fishes, and Ben caught 8 fishes; how many were caught by both boys? 9. Moses is 9 years old now; how old will he be 7 years from now ? 10. Emma has 8 tulips, and Ann has 8 tulips more than Emma ; how many tulips has Ann ? 11. A man traveled 6 miles, and then went 3 miles further; how far did he travel in all? 12. Henry spent 10 cents for oranges, and 6 cents for apples ; how much did he spend in all ? 13. How many are 2 and 2? 3 and 4? 4 and 5? 6 and 6? 14. 7 and 7? 8 and 8? 9 and 9? 5 and 6? 7 and 3? 8 and 4? 15. 9 and 2? 5 and 7? 6 and 8? 7 and 9? 2 and 5? 1 and 8? 16. 6 and 4? 5 and 5? 8 and 2? 5 and 3? 9 and 9? 6 and 7? 17. What is the number whose complemental parts are 2 and 3? 3 and 4? 4 and 5? 5 and 6? 9 and 60 Intermediate Arithmetic. 4? 7 and 8? 1, 3, and 4? 2, 3, and 7? 3, 5, and 8? 1, 2, 3, and 4? 18. How many are 5 and 6? 15 and 6? 25 and 6? 35 and 6? 45 and 6? 55 and 6? 65 and 6? 75 and 6? 85 and 6? 95 and 6? 19. How many are 8 and 7? 18 and 7? 28 and 7? 38 and 7 ? 48 and 7 ? 58 and 7 ? 68 and 7 ? 78 and 7? 88 and 7? 98 and 7? 20. How many are 9 and 8? 19 and 8? 29 and 8? 39 and 8? etc., to 107. 21. How many are 4 and 9? 14 and 9? 24 and 9? etc., to 103. 22. How many are 7 and 5? 17 and 5? 27 and 5? etc., to 102. 23. How many are 4 and 10? 14 and 10? 24 and 10? etc., to 104." 24. How many are 4 and 5? 5 more than that? 5 more than that? 5 more than that? 5 more than that? 25. How many are 7 and 3? 3 more than that? 3 more than that? 3 more than that? etc., to 22. 26. How many are 8 and 4? 4 more than that? etc., to 32. 27. How many are 9 a's and 6 a's? 6 a's more than that? etc., to 33 a's. 28. How many are 6 c's and 7 c's? 7 c's more than that? etc., to 41 c's. 29. How many are 2 ii's and 8 ii's? 8 ii's more than that? etc., to 32 n's. 30. How many are 3 cows and 6 cows? 7 tens and 4 tens? 9 fives and 3 fives? 5 thirds and 2 thirds? 8 tenths and 4 tenths? Addition. 61 EXERCISES IN MAKING PROBLEMS. 63. i. Make a problem of 5 b + 7 b = ? Ans. 5 bats and 7 bats are how many ? Or, Mary has 5 beads and Susan has 7 beads; how many beads have both girls ? 2. Make a problem of 6 111 + 8 ill : Ans. 6 men and 8 men are how many ? Or, there are 6 marbles in one pile, and 8 marbles in another pile; how many marbles in both piles? Make problems of the following, and then give the answers : 6. 36a + 5a= =? 7. 54d + 9d = ? 8. lOp + 6p + 5 p 3. 7 c + 8 c=? 4. 6 f +9 f =? 5. 10 h + 5 h = ? SLATE EXERCISES. 64. Copy and add the following, placing the answer under the line : 4 9 5 7 8 4 7 10 34 46 57 76475989389 3 5 3 5 6 5 3 8 42 34 16 2 4 4 3 2 5 9 2 5 4 4 7 8 9 7 2 4 7 7 3 6 5 3 1 2 3 4 3 4 3 4 5 2 7 9 7 5 6 7 8 1 6 7 1 2 2 5 5 6 3 8 9 7 8 4 5 8 7 5 6 7 4 8 10 8 62 Intermediate Arithmetic. 7 3 9 6 5 7 4 3 5 8 5 8 3 4 8 9 2 3 6 9 7 5 9 5 3 7 9 5 8 6 9 10 3 8 10 5 4 12 15 17 27 30 20 36 17 The following exercises may be performed, first on the slate, and then mentally : MENTAL EXERCISES. 65. i. Name every 10th number from: to 100: 5 to 105; 2 to 102; 7 to 107; 9 to 109; 3 to 103: 8 to 108; 4 to 104; 1 to 101; 6 to 106. 2. Count 8 on to: 7; 15; 21; 34; 46; 52; 68; 72; 89; 97. 3. What is the sum of: 23 and 7? 37 and 5 ? 48 and 3 ? 59 and 4 ? 66 and 8 ? 72 and 9 ? 84 and 7 ? 99 and 3? 45 and 5 ? 76 and 7? 93 and 5 ? 87 arid 8? 63 and 8? 4. 16 + 7 = =? 35 + 6=? 43 + 8=? 74 + 3 = ? 98 + 7 = ? 57 + 9=? 5. To every 10th number from 5 to 95 add: 7; 3; 8. Add or count : 6. By 2's from to 18. 7. By 3's from to 30. 8. By 4's from to 40. 9. By 5's from to 50. Name : 10. By 6's from to 60. 11. By 7's from to 70. 12. By 8's from 'to 80. 13. By 9's from to 90. 14. Every 3d number from 2 to 32. 15. Every 4th number from 3 to 39. Addition. 63 16. Every 5th number from 4 to 104. 17. Every 6th number from 5 to 65. 18. Every 7th number from 6 to 76. 19. Every 8th number from 7 to 87. 20. Every 9th number from 8 to 71. 21. What pairs of digits, when added, will make: 4? 5? 6? 7? 8? 9? 10? 11? 12? 13? 14? 15? 16? Find the sum of: 22. 5, 7, 9, 3. 23. 2, 4, 6, 8. 24. 7, 1, 9, 3, 5. 25. 10, 6, 8, 4, 7. 26. $7, $8, $9. 27. 6c, 7c, 3c, 9c. 28. 5 hats, 4 hats, 7 hats. 29. 16 m, 10 m, 9 m, 7 m. 38. 45 + 9 + 7- 39. How many 30. $50, $10, $9, $7, $3, 31. 6 + 8 + 3=? 32. 9 + 5+ 6 = ? 33. 10 + 6 + 7 + 3 = ? 34. 23 + 7 + 6 + 5 = ? 35. 7 + 3+9 = ? 36. 9 + 7 + 8 = ? 37. 34 + 10+6 + 5 = ? Q 9 are 35 and 20? OPERATION. 3 tens and 2 tens = 5 tens or 50. 50 + 5 -- 55. 40. There are 45 boys and 50 girls in school. How many pupils in all? 41. If Charles reads 78 pages one day and 60 pages the next day, how many pages will he read in both days? 42. A drover bought 67 sheep from one man and 30 from another, how many sheep did he buy of both ? 43. How many are 63 and 20? 78 and 40? 37 and 70? 45 and 30? 97 and 40? 50 and 23? 70 and 85? 90 and 44 ? 80 and 56 ? 60 and 84 ? 44. How many are 49 and 37? OPERATION. 49 + 30+7 = 79 + 7 = 86. 64 Intermediate Arithmetic. 45. John gave 25 cents for a slate and 42 cents for a book; what did both cost? 46. A pole is 43 feet in the air, 19 feet in the earth, and .18 feet in the water. How long is the pole? 47. A lad, having spent 43 cents, finds he has 59 cents left. How much had he at first? 48. How many are 43 and 22? 64 and 53? 47 and 35 ? 84 and 21 ? 74 and 56? 73 and 91 ? 53 and 41 ? 75 and 92 ? 39 and 24 ? 43 and 87 ? 49. How many are 2 tens and 12 ones? OPERATION. 12 ones=l ten 2 ones. 2 tens and 1 ten 2 ones are 3 tens 2 ones == 32. 50. How many are 3 tens and 25 ones? Ans. 5 tens 5 ones = 55. 51. How many are: 4 tens and 17 ones? 6 tens and 34 ones? 8 tens and 73 ones? WRITTEN EXERCISES. 66. 1. Find the sum of 4864, 785 and 693. OPERATION. EXPLANATION. Write the numbers or parts, so that units of the same order stand in the same col- ' 5 umn ones under ones, tens under tens etc. ; and draw a line beneath them. 6342 Adding, from the bottom, the column on the right, we get 12 (=1 ten and 2 ones) ; write the two below the line for the ones of the required sum. Adding the 1 ten with the tens of the given parts, which is the next column, we get 24 (=2 hundreds and 4 tens) ; write the 4 below the line for the tens of the required sum. Adding the 2 hundreds to the hundreds of the given parts, which are the numbers of the 3d column, we get 23 (=2 thousands and 3 hundreds) ; write the three in the place of hundreds in the sum, and carry the 2 to the next column of thousands, which, added to 4, gives 6 thousands. Hence the sum is 6342. To prove the work we begin at the top and add down. Addition. 65 2. 45364 -|- 8965 + 786 + 9374 -f 47 = ? OPERATION. 45364 67. In adding it is best to use only the follow- 8965 ing wording: 7, 11, 17, 22, 10 (emphasize 6, and 786 write it down while pronouncing it), carry 2 ; 6, 9374 13, 21, 27, 33, carry 3 ; 6, 13, 22, 25, carry 2 ; 11, 1<), 47 24, carry 2 ; 6. "6453G From the preceding examples we derive the following RULE. I. Write the parts so that like orders of units shall stand under each other. II. Begin at the right, add each column separately, write the units 7 figure of the sum under the column added, and carry the tens, if any, to the next column. PROOF. Perform the addition in the reverse direction, from top to bottom, and if the results agree the work is probably correct. In this manner add and prove : (3) 87 49 64 (6) 37 (7) 65 (8) 48 Ans. 96 183 70 89 98 34 84 (9) 95 (10) 21 (11) 52 (12) 78 (13) 80 (14) 67 36 54 76 89 63 27 48 79 81 67 94 57 Ans. 179 (15) 939 827 705 693 Ans. 3164 N. I. 5. (16) (17) (18) (19) (20) 818 729 60S 590 486 706 695 587 405 705 694 581 446 967 628 582 434 993 879 530 66 Intermediate Arithmetic. (21) (22) (23) (24) (25) 38025 75631 9998 642 43384 9467 8467 74635 9753 965 7098 983 672 85671 8741 Ans. 54590 68. Table of distances on the Mississippi River, com- piled in integers of miles, from the surveys of the Mis- sissippi River Commission : Jetties to New Orleans . . 96 N. O. to Donaldsville, La. . 78 Donald, to Plaquemine, La. 32 Plaq. to Baton Rouge, La. . 20 B. R. to Bayou Sara, La. . 34 B. S. to Mouth Red R., La. 35 Mouth R.R. to Natchez, Miss. 64 Natchez to St. Joseph, La. . 52 St. J. to Vicksburg, Miss. . 49 Vicks. to La. State Line . . 47 La. S. L. to Greenville, Miss. 44 Green, to Arkansas City, Ark. 40 A. C. to mouth Ark. R., Ark. 37 Mou. Ark. R. to Helena, Ark. 95 Helena to Memphis, Tenn. 76 Memp. to Fort Pillow, Tenn. 58 Fort P. to Columbus, Ky. . 151 Columbus to Cairo, 111. . . 21 How far is it from New Orleans by river : 26. To Baton Rouge? Ans. 130 miles. 27. To Natchez? Ans. 263 " 28. To Vicksburg? Ans. 364 " 29. To Memphis? Ans. 733 " 30. How far is it from the mouth of Red River to the mouth of Arkansas River? To the Jetties? Ans. 363 miles ; 295 miles. 31. How far will a man travel who takes a boat at Baton Rouge, La., and goes to Helena, Ark. Ans. ? 32. How far is it from the Jetties to Memphis? Cairo? Ans. ? 33. A farmer sold 4 bales of cotton; the first weighed 463 pounds, the second 458 pounds, the third 417 pounds, and the fourth 513 pounds; what was the whole weight? Ans. 1851 pounds. Addition. 67 34. An army is composed of 34379 infantry, 8625 cavalry, and 1792 artillery-men ; how many men in the army? Ans. 44796 men. 35. A butcher bought 6 oxen which weighed 1345 pounds, 1623 pounds, 978 pounds, 1174 pounds, 819 pounds, and 1796 pounds ; what was the total weight ? Ans. ? Find the sum of: 36. 2564, 34875, 16374, 985, 76. Ans. 54874. 37. 14200 yards, 672 yards, 1265 yards, 3789 yards. Ans. 19926 yards. 38. 340 acres, 281 acres, 57 acres, 426 acres, 5 acres. Ans. 1109 acres. 39. 25 days, 460 days, 191 days, 763 days, 1084 days. An*. ? NOTE. When numbers of dollars and cents are to be added, they must be written so that the points stand under each other. 40. $36.27 + 85.96 + $1208. -f $120.40 + $75.00 + $.94 = ? Ans. $1446.57. 41. $50.04 + $7.80 + $102.10 + $15.08 + $208.00 + $3.43 = ? Ans. $386.45. 42. $304.00 + $75.75 -f $12.05 + $27.54 + 85.81 + $63.02 = ? Am. ? 43. What is the amount of $45.63, $3.68, $37.45, $93.07, $2.84, $175.50, and $430.12? Ans. $788.29. 44. Mr. Hicks owes Mr. Johnson $130.50, Mr. Jackson $475.12, Mr. Turner $980, and Mr. Wafer $17.64; how much does he owe them all? Ans. $1603.26. 45. A farmer sold a horse for $109.50, four bales of cotton for $197.85, three hogsheads of sugar for $239, and a load of corn for $13.25; what did he receive for all? Ans. ? 46. North America contains 8,593,000 square miles, 68 Intermediate Arithmetic. South America 7,362,000 square miles, Europe 3,825,000 square miles, Asia 17,300,000 square miles, and Africa 11,557,000 square miles; how many square miles are in these five countries? Ans. 48,637,000 square miles. 69. When the parts are equal, the whole or sum may be indicated by inclosing one of the equal parts in a parenthesis, and writing the number of parts before it. Thus, 5 (7) indicates 5 sevens, or 7-|-7-|-7-f7-{-7 = 35. EXERCISES. i. 4 (357)=? OPERATION. 357 357 357 357 1428 6(8435.10) =? Ans. ? 7 (374 hogs) = ? Ans. ? 4(25681bs.) =? Ans. ? 10 (504 cents) = ? Am. ? 2. 5(645) =? Ans. 3225. 6. 3. 7(807) =? ,4m. 5649. 7. 4. 3(2768)=? Ans. 8304. 8. 5. 9(215) =? Ans. 1935. 9. 10. A father gave each of his five sons $125; how much did he give them all? Ans. $625. 11. What is the total weight of 3 bales of cotton if each bale weighs 410 pounds ? Ans. 1230 pounds. 12. James has 5 boxes, and in each box there are 175 chestnuts ; how many chestnuts has he in all ? Ans. ? 13. What will 6 horses cost if each horse cost $150? Ans. ? 70. PARALLEL PROBLEMS. NOTE. In these and in subsequent parallel problems the men- tal, denoted by in, involve the same principles as the succeed- ing written problem or problems, and are intended for two pur- poses, viz : Addition. 69 1. To supply the place of mental arithmetic ; hence, they should be solved mentally and recited orally. 2. To furnish indirectly an explanation of the principles and terms embraced in the parallel written problems, so that the pupil, by proper diligence, may comprehend and solve the latter unaided. I." 1 What is the 9th number above 12? 2. What is the 837th number above 968? Ans. 1805. 3. m I sold a hog for $13, which lacked $5 of being as much as the hog cost me ; what did the hog cost ? 4. By selling a farm for $1305 I lost $960.50; what did the farm cost me? Ans. $2265.50. 5.m James spent $8 for pants, $9 for a coat, and $5 for a vest; how much did he spend in all? 6. A merchant invests $3275 in a house, $9760.23 in merchandise, $2987.53 in improvements, and $1624.35 in clerks' hire ; what is his total investment ? Ans. $17647.11. 7. m John gave his mother 12 pears, his father 7 pears, his sister 6 pears, and had 5 pears left ; how many had he at first? 8. A farmer paid $4875.60 for a farm, $1782 for stock, $2416.98 for supplies, and had $3026.05 left; how much money had he at first? Ans. $12100.63. 9. William paid $7 for a hog, $9 for a calf, $5 for a sheep, and sold them so as to make $6; what did he receive for all? 10. A speculator bought a drove of horses for $3764.15, a drove of cattle for $2017.55, a drove of sheep for $620, and sold them at a profit of $784.85 ; what did he receive for all? Ans. $7186.55. ll. m James has 3 marbles, John 5 more than James, and Moses 6 more than John ; how many have all ? 12. Mr. Taylor has 735 sheep, Mr. Jackson has 842 70 Intermediate Arithmetic. more than Mr. Taylor, and Mr. Hulse 634 more than Mr. Jackson ; how many have the three men ? Ans. 4523 sheep. I3. m Six years ago Peter was 9 years old; how old will he be 8 years from now? 14. A man married 17 years since, at which time he was 24 years old; how old will he be 22 years hence? Ans. 63 years. 15. Add by 9's from to 45. 16. Add or count by 278's from to 1390. Ans. 0, 278, 556, etc. 71. QUESTIONS FOR REVIEW. What are: 1. Like numbers? 2. Unlike numbers? What is Addition? What is the sign of Addition? What is denoted : 1. By the sign = ? 2. By the sign (?) ? What are the complemental parts of a number ? What do we find by Addition ? What stands for complemental ? What is the : 1. Principle of Addition ? 2. Rule for Addition? How may we prove Addition ? What is meant by parallel problems? Ans. Problems which involve the same principles. SUBTRACTION. INDUCTIVE EXERCISES. 72. l. Count 7 backward. Ans. 7, 6, 5, 4, 3, 2, 1. 2. Count $7 backward. Ans. $7, $6, $5, 84, $3, $2, $1. 3. How do we count backward? 4. In counting backward, what are the next three numbers below $7? 5. How many are left when $1 is taken 3 times from $7? 6. In counting backward, what are the next 4 num- bers below 10 hats? Ans. 9 h's, 8 h's, 7 li's, 6 h's. 7. How many are left when 4 hats are taken from 10 hats? Ans. 6 hats. In counting backward, what is: 8. The 1st number below 9? 9. The 3d number below 12 ? 10. The 3d number below $8? 11. The 4th number below 12 c? How many are: 8. 9 less 1? 9. 12 less 3? 10. $8 less 83? 11. 12c'sless4c's? 12. If I make 7 marks ///////, and rub out 3 of them, how many will be left? 13. John had 12 balls and gave James 5; how many balls did John have left? 14. Eight peaches are on a tree, if 5 are taken off, how many will be left? (71) 72 Intermediate Arithmetic. How many are: 15. $7 less $1? 16. 9 men less 2 men? 17. 8 pints less 3 pints? Subtract : 15. $1 from $7. 16. 2 men from 9 men. 17. 3 pints from 8 pints. DEFINITIONS. 73. Subtraction is the process of taking from a number a given number of like units. 74. The number to be diminished is called the Minu- end, the number by which it is diminished, the Subtra- hend, and the result the difference or remainder. Thus : $3 taken from $7 leaves $4. Here, $7 is the minuend, $3 the subtrahend, and $4 the remainder. 75. The sign of Subtraction is , which is read : less or minus. When stands between two numbers it indi- cates that the one after it is to be taken from the one before it. Thus : 8 3 5 is read : 8 less 3 is 5, or 8 minus 3 equals 5. Beginners should adopt the first reading, or : 3 from 8 leaves 5. Again, 10 4 = ? is read: 10 less 4 is how many? or 4 from 10 leaves how many ? 76. Read: EXERCISES 1. 9 in's 6 m's = 3 m's. 2. 7d's 4d's=3 d's. 3. 12 boys --7 boys=? 77. Express by signs: 1. $14 minus $5 is $9. 2. $12 less $7 is $5. 3. 7 plus 5 minus 3 is 9. 4. 6 + 5 3 = 8. 5. 7 + 8--6--4 = 6. 6 . 10 6 + 53 = 6. 4. 16 less 7 is how many? 5. 3 from 11 leaves 8. 6. 6 from 14 leaves 8. Subtraction. 73 In each of the preceding examples point out the minuend, subtrahend, and remainder. Thus, in Ex. 3, 7 plus 5 is the minuend, 3 is the subtrahend, and 9 the remainder. 78. PRINCIPLE. Only like numbers can be subtracted. RELATION OF SUBTRACTION TO ADDITION. 79. 4 and 3 are how many? 4 and what number are 7 ? What number and 3 are 7 ? 4 -f ? = 7 ? Ans. 3 ; because 7 4=3. ? + 3=7? Ans. 4; because 7 3 = 4. In a similar manner answer the following: 9 + ?= 12. ? + 7 = 15. 5 cents +? = 11 cents. 8 + ? = 14. 7 + 3 = 10. ? + 6 pecks = 9 pecks. PRINCIPLES. 1. Subtraction is the reverse of Addition. 2. By Subtraction we find one of the complemental parts of a number, when the number and the other part are given. Thus : if 6 is one of the c parts of 11, the other part is 11 6, or 5. SUGGESTIONS TO TEACHERS. In all the examples and problems in Subtraction, the pupil should be required to point out the c parts and the whole. Thus, in 11 - - 5 == 6, 5 and 6 are the parts, and 11 is the whole. Again, in the problem : James had $15, but lost $8, how much had he left? $8 and $7 are the parts, and $15 the whole. 80. Since Subtraction is the reverse of Addition, by reversing the table of the latter, we get the 74 Intermediate Arithmetic. SUBTRACTION TABLE. 1 2 3 4 5 1--1==0 2 2 = 3 3 = 4- -4 = 5 5 = 2--l==l 3 2 = 1 4 3 = 1 5--4 = l 6 5 = 1 3--l==2 4 2 = 2 5 3 = 2 6- -4 = 2 7 5 = 2 4 1 = 3 5 2 = 3 6 3 = 3 7--4 = 3 8 5 = 3 5 1-4 6 2 = 4 7 3 = 4 8--4 = 4 9 5 = 4 6 1 = = 5 7 2 = 5 8 3 = 5 9 --4 = 5 10 5 = 5 7--I==G 8 2 = 6 9-3 = 6 10--4 = 6 11 5 = 6 8 1 = 7 9 2 = 7 10 3 = 7 ll--4 = 7 12 5 = 7 9 1 = 8 10 2 = 8 11 3 = 8 12 --4 = 8 13 5 = 8 10 1 = 9 11 2 = 9 12 3 = 9 13--4 = 9 14 5=9 6 7 8 9 10 6 6 = 77 = 8 8 = 9 9 = 10 10 = 7 6 = 1 8 7 = 1 9 8 = 1 10 9 = 1 11- -10 = 1 8 6 = 2 9 7 = 2 10 8 = 2 ll--9 = 2 12 10 = 2 9 6 = 3 10 7 = 3 11 8 = 3 12 9 = 3 13 10 = 3 106 = 4 11 7 = 4 12 8 = 4 13 9 = 4 14 10 = 4 ll--6 = 5 12 7 = 5 13 8 = 5 14 9 = 5 15 10 = 5 12 6 = 6 13 7 = 6 14 8 = 6 15 9 = 6 16 10 = 6 13 6=7 14 7 = 7 15 8 = 7 16 9 = 7 17--10 = 7 14 6 = 8 15 7 = 8 16 8 = 8 17 9 = 8 18 --10 = 8 15 6 = 9 16 7==9 17 8 = 9 18 9 = 9 19 --10 = 9 NOTE. Pupils should read the tables thus : 1 from 1 leaves none, 1 from 2 leaves 1 , 1 from 3 leaves 2, etc. The table is ex- pressed in signs to familiarize the pupil with their use and mean- ing. DRILL EXERCISES. 81. In these exercises each figure is to be subtracted from the number that stands above the group. The exercises should be written on the board, and used in class drill daily, until every pupil can call all the re- sults instantly. Subtraction. 7"> 2345 6 7 8 1 12 123 1324 13542 153246 lfi:V)437 18 17 16 15 14 13 12 9 89 798 6897 58796 486597 3856479 9 10 11 18347562 183659742 63752849 MENTAL EXERCISES. 82. 1. Susan is 14 years old, and her brother is 5 years younger; how old is her brother? 2. William has 11 pears, and John has 6 less than William ; how many pears has John ? 3. Susan and Lucy together have 10 roses ; if 7 of them are Susan's, how many has Lucy? 4. Thomas killed 8 sqirrels on Tuesday, and 14 on Tuesday and Wednesday together ; how many did he kill on Wednesday ? 5. There are 13 rabbits in the garden and yard; if 6 rabbits are in the yard, how many are in the garden ? 6. Two hens have 13 chicks together; if one hen has 8 chicks, how many chicks has the other? 7. A boy worked 15 hours in two days ; if he worked 6 hours in one day, how many did he work the other? 8. Ben caught 17 fishes and Moses 9 fishes ; how many more fishes did Ben catch than Moses? 9. 7 years from now Henry will be 15 years old ; how old is he now ? 10. Emma has 14 tulips and has 7 tulips more than Ann ; how many tulips has Ann ? 11. A man traveled 9 miles; how far would he have traveled if he had gone 5 miles less? 76 Intermediate Arithmetic. 12. Harry spent 16 cents for oranges and 8 cents less for apples; how much did he spend for apples? 13. How many is: 13 less 6? 17 less 8? 8 less 5? 14. 10 less 2? 10 less 5? 9 less 8? 7 less 5? 16 less 8? 15. 16 less 7? 14 less 6? 12 less 7? 11 less 9? 10 less 6? 16. 10 less 3? 11 less 5? 18 less 9? 16 less 9? 14 less 7? 17. If 7 is one of the C parts of 15, what is the other? 18. If 9 is one of the c parts of 13, what is the other? 19. How many is: 104 less 6? 94 less 6? 84 less 6? 74 less 6 ? 64 less 6 ? 54 less 6 ? 44 less 6 ? 34 less 6 ? 24 less 6 ? 14 less 6 ? 20. How many is : 103 less 7 ? 93 less 7 ? 83 less 7 ? 73 less 7 ? 63 less 7 ? 53 less 7 ? 43 less 7 ? 33 less 7 ? 23 less 7? 13 less 7? 21. How many is: 105 less 9? 95 less 9? etc., to 6. 22. How many is: 101 less 10? 91 less 10? etc., to 1. 23. How many is: 53 less 8? 8 less than that? 8 less than that? 8 less than that? 8 less than that? 24. How many is: 43 less 5? 5 less than that? etc., to 3. 25. How many is: 65 less 9? 9 less than that? etc., to 2? 26. How many is 33 diminished by 7 four times? 27. How many is 30 less 5, less 5, less 5, less 6? 28. 17 a's 9 a's = ? 25 b's - 6 b's = ? 32 e's 5 c's? 40 n's 10 n's=? EXERCISES IN MAKING PROBLEMS. 83. 1. Make a problem of: 11 S--5 s = = ? Ans. 5 spoons from 11 spoons leaves how many? Subtraction. 77 Or, James had 11 strings, but gave 5 strings to his sister; how many did lie have left? 2. Make a problem of: 13y--6y==? Problem: Mike is 13 years old, and Henry is 6 years old ; how much older is Mike than Henry? Make problems of the following, giving the answers to each : 6. 23 a - 6 a -- ? 7. 32 b 7b =? 8. 65 in 9 m = =? 3. 9 c - -2c =? 4. 11 r - -8r =? 5. 17 b 9b=? 9. 7a-f6a--5a ? Ans. If from the sum of 7 apples and 6 apples I take 5 apples, how many apples will be left? 10. 10 b + 6 b 3 b = ? 11. 12 a + 9 c 6 c =? 12. 24 g; +8 g +3 g 6 g = ? 13. 15 in -f- 6 ni - 4 in - - 5 in = = ? MENTAL EXERCISES. 84. l. Name every tenth number from: 100 to 0: 105 to 5 ; 102 to 2 ; 107 to 7 ; 109 to 9 ; 103 to 3 ; 108 to 8 ; 104 to 4 ; 101 to 1 ; 106 to 6. 2. Take 8 from: 15; 23; 29; 42; 54; 61; 76; 80; 97; 105. 3. What is the difference between 23 and 7? 42 and 5? 51 and 3? 63 and 4? 74 and 8? 81 and 9? 91 and 7 ? 102 and 3 ? 50 and 5 ? 83 and 7 ? 98 and 5? 95 and 8? 4. 23- -7:=? 41 --6 = =? 51- -8==? 77 - - 3 = ? 105 7 = ? 66 9 = ? 5. To every 10th number from 105 to 5 subtract 7; 3; 8. 78 Intermediate Arithmetic. Subtract : 6. By 2's from 18 to 0. 7. IJy 8s from 30 to 0. 8. IJy 4's from 40 to 0. 9. IJy 5's from 50 to 0. Name : 10. By 6 f s from 60 to 0. 11. By 7's from 70 to 0. 12. By 8's from 80 to 0. 13. By 9's from 90 to 0. 14. Every 3d number from 32 to 2. 15. Every 4th number from 39 to 3. 16. Every 5th number from 104 to 4. 17. Every 6th number from 65 to 5. 18. Every 7th number from 76 to 6. 19. Every 8th number from 87 to 7. 20. Every 9th number from 71 to 8. 21.5 7 4 = ? 25. 85 +88 84 = ? 22. 9+65=? 26. 89 + 87 -f S6 $3 = ? 23. 7 + 8 + 6 4 = ? had In- It/ft? 37. Edward had ?<5-">, and aft< r s}M-ndin) f O--t Subtracting like numbers, we have 3 hunds. t>> . tens 1 one ==371. 3/1 Hence, the RULE. I. Write the subtrahend under the minuend, pluc- iny unit* titiflt-r unit*, tens under ten*. ef>\ II. Begin at the right, subtract each figure of the siib- trahend from the figure above it. and write the difference below. From : 2. 375 take 124. Ans. 251. 3. ns; take 505. Ans. 182. 4. 9Js:J take 823. An*. 160. 5. 841 take 31. Ans. 810. 6. 9324 take 113. Ans. 0211. 7. 7379 take 7163. Ans. 216. 8. 2:V.)4<; take 1814. Ans. 22132. 9. 68438 take ^25. Ans. 68213. 10. 7 How many yards of cloth will it take, at 5 cents a yard, to amount to 37295 cents? Ans. 7459 yards. 21. A merchant spent $33224 for hats, paying, on an average $4 apiece; how many hats did he buy? Ans. 8306 hats. 22. A farmer received $1950 for a lot of land which he sold at the rate of $6 per acre; how many acres did he sell? Ans. ? 23. 4308 chestnuts were divided equally among 6 boys; how many chestnuts did each boy receive? Ans. ? Division. 115 24. There are 12320 yards in 7 miles ; how many yards in one mile? Ans. 1760. 25. I counted the legs of all the horses in a drove, and found that there were 476 ; how many horses were in the drove? Ans. ? LONG DIVISION. 136. When the divisor is greater than 12 we write down all the figures employed, and call the operation Long Division. 137. CASE I. When the quotient is not greater than 9. 1. Divide 91 by 21. EXPLANATION. - - The first figure of the dividend is 9, and that of the divisor is 2 ; 2 in 9, 4 times. Place the 4 on the right, mul- tiply it by 21, subtract the product 84 from 91, which gives 7 remainder. 2. Divide 442 by 75. The first part of the dividend that con- tains the first figure (7) of the divisor is 44; 7 in 44, 6 times. Place the 6 on the right, multipl y it by 75, and since the result, 450, is larger than 442, 6 is too large. Hence, we take the next less number (5), put it in place of 6, multiply it by 75, and since the result, 375, is less than 442, 5 is the correct quotient, and the answer is : quo. 5 rein. 07. 3. Divide 337 by 35. Since 33 is less than 35, we say 3 in 33, 11 times. But in dividing in this manner we can never get a greater quotient than 9 ; hence, instead of 11 we write 9 on the right, multiply it by 35, and, since the result, 315, is less than 337, 9 is the correct quotient. OPERATION. 21)91(4 84 OPERATION. 75)442(6 450 75)442(5 375 67 35)337(9 315 22 116 Intermediate Arithmetic. From the preceding work we derive the RULE. I. Divide the first figure or figures of the divi- dend by the first figure of the divisor ; place the result on the right and call it the trial quotient. II. Multiply the divisor by the trial quotient and place the product under the dividend; if it is larger than the dividend, the trial quotient is too large and must be dimin- ished; if it is smaller, subtract it from the dividend, and if the remainder is less than the divisor the work is correct ; if greater, the trial quotient is too small and must be in- creased. PROOF. The same as in short division. Divide: 4. 5. 6. 7. 8. 9. 16. 17. 18. Divide 517604 a by 89325 a. R&ni. 71069 a. 138. CASE II. When the quotient is more than 9. 1. Divide 9156 by 21. OPERATION. EXPLANATION. 21 in 91, by CASE I, 21)9156(436 goes 4 times and rem. 7. Place 4 on the right and annex 5 'to 7, making 75. 75 By CASE I, 21 in 75, 3 times and rem. gg 12. Place 3 on the right and annex 6 to 12, and we find by CASE I, 21 in 126 goes 6 times, which place on the right. 126 117 by 23. Ans.Q.5,R.2. 10. 300 by 45. Rem. 30, 400 by 76. Rem. 20. 11. 967 by 98. Rem. 85. 311 by 88. Rem. 47. 12. 573 by 75. Rem. 48. 728 by 93. Rem. 77. 13. 805 by 237. Rem. 94. 643 by 75. Quo. 8. 14. 933 by 465. Rem. 3. 340 by 49. Quo. 6. 15. 1080 by 135. Rem. 0. Divide 43657 by 8705. Rem. 132. Divide $34637 by $9604. Rem. $5825. llivision. 117 2. Divide 876000 by 125. OPERATION. By CASE I, 125 in 87G, 7 times and 125)876000(7008 rem. 1. Place 7 on the right and an- 75 nex 0, the next figure of the dividend to 1 , making 10. Now 125 in 10, times and rem. 10. Place on the right and annex the next figure to 10, making 100. Now 125 in 100, times and 100 rem. Place on the right and annex the next figure to 100, making 1000. By CASE I, 125 in 1000, 8 times, which we place on the right. From the foregoing examples and operations we de- rive the following RULE. I. Write the divisor on the left of the dividend, with a line between them, and draw a line on the right. II. Find how many times the divisor is contained in the least number of the left hand figures of the dividend that will contain it, and place the quotient on the right. III. Multiply the divisor by this quotient figure, subtract the product from the figures of the dividend used, and to the remainder annex the next figure of the dividend. IV. Divide as before, and continue the operation until all the figures of the dividend have be-en brought down. V. When one of the partial dividends is less than the di- visor, write for the next figure of the quotient, and bring down the next figure of the dividend. PROOF. Add the remainder to the product of the divisor and quotient; the result should be equal to the dividend. NOTE. When there is a remainder after all the figures of the dividend have been brought down and divided, it may either be set off by itself, QT it may be written over the divisor and annexed to the quotient. 118 Intermediate Arithmetic. Divide : 3. 625 by 25. Ans. 25. 4. 759 by 33. Ans. 23. 5. 864 by 36. Ans. 24. 6. 882 bv 42. Ans. 21. 8. 9. 10. 11. 12. 1778 by 14. Ans. 127. 2169 by 18. Rem. 9. 3639 by 27. Rem. 21. 7540 by 59. Rem. 47. 35645 by 215. Rem. 170. Quo. 138. Rem. ? Ans. 143fff. Ans. Ans. Ans. 4ff Ans. 864 by 36. 882 by 42. 7. 270 by 18. Ans. 15. 13. 58650 by 425. 14. 98629 by 687. 15. 75863 by 3421. 16. 10000 by 2749. 17. $132 by $27. 18. $457 by 56. 19. How many hours will it take a railway-train to go 800 miles at the rate of 32 miles an hour? Ans. 25 hours. 20. What is the weight of a bale of cotton if 25 bales weigh 11400 pounds? Ans. 456 pounds. 21. How many hogsheads of sugar will it take to weigh 17400 pounds, if 1 hogshead weighs 1450 pounds? Ans. 12. 22. The average price of a drove of horses is $127, and the price of the whole drove is $13335 ; how many horses are in the drove? Ans. 105 horses. 23. AVith $19608, how many cows can I buy at $43 a head? Ans. 456. 24. How many bales, each weighing 475 pounds, can be made of 93100 pounds of cotton ? Ans. 196 bales. 25. AVilliam can haul 1248 pebbles in his wagon; how many trips will he have to make to haul off 91104 pebbles? Ans. 73. 26. If the distance around a wheel is 56 inches, how many times will the wheel turn over ir^ going a dis- tance of 7504 inches? Ans. 134. Division. 119 27. At what price per head must I sell 148 sheep to receive $1036? Ans. $7. 28. The salary of the President of the United States is $50000 a year; how much is that a day, there being 365 days in 1 year? Ans. 8136fff 29. There are 56 pounds in a bushel of corn ; how many bushels in 12345 pounds? Ans. 220ff. 139. Make problems of the following, taking the terms in order, and giving the answer to each : 30. 26992 h H- 482 h = ? si. 3212 m + 868 m -+- 34 m =? 32. 4120 d--520 d-r-150 = ? 33. 703 p X 8 204p-r-125 p==? CONTRACTIONS IN DIVISION. 140. CASE I. When the divisor is 10, 100, 1000, etc. 1. Divide 1625 by 100. Dividing according to the rule of Long OPFR \TION Division, we obtain the quotient 16 and remainder 25. 100)1625(16 Now we observe that this answer could 100 have been obtained by simply cutting off the last two figures (25) of the dividend for a remainder, and taking the balance of the dividend for a quotient. Hence, to divide by 10, 100, etc., we have the RULE. Cut off from the right of the dividend as many figures as tJiere are ciphers at the right of the divisor ; the remaining figures of the dividend ivill be the quotient, and those cut off on the right will be the remainder. 2. Divide 375 by 10. Ans. 37, Rem. 5. 120 Intermediate Arithmetic. 3. Divide 4316 by 100. Ans. 43, Rem. 16. 4. Divide 60524 by 1000. Ans. 60, Rcm. 524. 5. How much is 1 tenth of 43 ? 1 hundredth of 471 ? 6. There are 10 dimes in one dollar ; how many dol- lars in 40 dimes? 260 dimes? 500 dimes? 7. A farmer having $3254, bought horses at $100 each ; how many horses did he buy, and how many dollars had he left? 8. A dealer has 1895 cigars, and wishes to put them in boxes of 100 cigars each ; how many boxes does he need, and how many cigars will he have left over? 141. CASE II. When the divisor is any number with ciphers annexed. 1. Divide 73153 by 2700. Dividing by the rule of Long Division, QPERATION. we obtain the quotient 27 and remainder 27,00)731,53(27 253, which result could have been ob- ^A tained thus : Cut off the two O's of the divisor and the last two figures of the dividend ; divide the remaining figures of the dividend 253 (731) by the remaining figures of the di- visor (27), and to the remainder (2) annex the two figures cut off (53) for the true remainder. Hence, we have the RULE. Cut off the ciphers from the divisor, and also cut off the same number of figures from the right of the dividend; divide the remaining figures of the dividend by the remaining figures of the divisor, and to the remainder, if any, annex the figures cut off' from the dividend for a true remainder. 2. Divide 37657 by 50. Ans. 753, Rem. 7. 3. Divide 43787 by 600. Ans. 72, Rcm, 587. 4. Divide 35016 by 700. Ana. 50, Rem. 16. Division . 12 1 5. Divide 63242 by 3500. Ana. 18, R&m. 242. 6. Divide 71831 by 6400. Ans. 7. Divide 93045 by 17000. Ans. 5 T 8. Divide 184973 by 23000. Ans. 9. Divide 846 by 40. Ans. ? 10. Divide 7593 by 900. Ans. ? 11. Divide 23956 by 3700. Ans. ? 12. If 40 barrels of molasses cost $480, what is the price of 1 barrel? Ans. $12. 13. A former sold 600 acres of land for $7800; how much was that per acre? -4ns. $13. 14. A merchant sold 8000 yards of cloth for 184000 cents ; how much was that per yard ? Ans. 23 cents. 15. John and Henry gather 6275 chincapins, and de- sire to put them in sacks containing 290 chincapins each ; how many sacks do they need, and how many chincapins will be left over? MENTAL EXERCISES. An important class of problems involving Multiplication and Division. 142. i. If 4 yards of cloth cost 20 cents, what will 7 yards cost at the same rate?* OPERATION. EXPLANATION. Since 4 yards cost 20 cents 4)20 we divide 20 by 4 to get the cost of 1 yard, F which gives 5 cents. Now, since 1 yard cost _ 5 cents, we multiply 5 by 7 to get the cost of 7 yards. 35 cents. 2. If 5 yards of cloth cost 40 cents, what will 8 yards cost? Ans. 64 c. *The words "at the same rate" are supposed to follow several of the following exercises. 122 Intermediate Arithmetic. 3. If 4 apples cost 12 cents, what will 9 apples cost? 4. If 6 peaches cost 24 cents, what will 10 peaches cost? Ans. 40 c. 5. If 7 melons cost 70 cents, what will 9 melons cost? 6. If 9 hats cost $27, what will 7 hats cost? Ans. $21. 7. If 8 books cost $16, what will 11 books cost? 8. If 3 tens cost $15, what will 7 tens cost? Ans. $35. 9. If 2 threes cost $24, what will 5 threes cost? 10. If 5 fourths cost S30, what will 3 fourths cost? 11. If 4 11 cost 24 cents, what will 9 u cost? Ans. ? 12. If 4 boys kill 8 squirrels, how many will 5 boys kill? * . Ana. ? 13. If 5 cats catch 15 rats, how many will 7 cats catch ? Ans. ? 14. If 6 boys eat 18 biscuits, how many will 12 boys eat? Ans. ? 15. If 5 girls have 40 fingers, how many fingers have 7 girls? Ans. ? 16. If 7 horses have 28 feet, how many feet have 12 horses ? Ans. ? 17. If 5 gallons = =20 quarts, then 8 gallons = ? Ans. 32 quarts. 18. If 4 quarts = 8 pints, than 11 quarts =? 19. If 6 dimes = 60 cents, then 13 dimes = ? 20. If $9 = 90 dimes, then $6 = ? Ans. 60 dimes. WRITTEN EXERCISES. 1. If 18 acres cost $270, what will 20 acres cost? Ans. $300. 2. If 25 tables cost $400, what will 12 tables cost? Ans. $192. 3. If 31 cows cost $465. what will 60 cows cost? Ans. $900. Division. 123 4. If 43 readers cost 688 cents, what will 25 readers cost? Ans. 400 cents. 5. If 752 sheep cost $4512, what will 137 sheep cost? Ans. $822. 6. If 20 bushels=640 quarts, then 17 buxhels - = ? Ans. 544 quarts. 7. If 15 yards = 540 inches, then 23 yards = ? -4ns. 828 inches. 8. If 75 ones = 675 ninths, then 63 ones = ? Ans. 567 ninths. 143. PARALLEL PROBLEMS. 1. What number multiplied by 12 will make 108? 2. One of the factors of 4375 is 175, what is the c factor? Ans. 25. 3. m If 9 boys together catch 72 fishes, how many does each boy catch on an average? 4. A field of 57 acres produced 1539 bushels of wheat, what was the average product of an acre? Ans. 27 bushels. 5. If 11 vards of cloth cost 132 cents, what is the %' cost of 1 yard? 6. A man sold 59 acres of land for $1062 ; what did he receive per acre? 7. What number is contained in 68 9 times and 5 over? 8. A man had $937, and, after paying some labor- ers at the rate of $25 apiece, had 812 left; how many laborers were there? Ans. 37. 9. m How often is 13 a less 5 a contained in 56 a? 10. A clerk's yearly salary is $1500 and his expenses 8945; in how many years can he lay up $4440? Ans. 8 years. 124 Intermediate Arithmetic. ll. m A boy sold a merchant 4 oranges at 6 cents apiece, and 9 apples at 4 cents apiece, and took his pay in cigars at 5 cents a piece ; how many cigars did he get? Ans. 12 cigars. 12. A farmer sold 27 cords of wood at $5 a cord, and 47 hogs at $7 apiece, and took in exchange flour at $8 a barrel; how many barrels did he get? Ans. 58 barrels. 13. How many minutes will it take 2 boys to re- move 80 rails if one boy removes 6 rails, and the other 4 rails, every minute? 14. Two railway trains are 540 miles apart, and travel towards each other at the rates of 25 miles and 20 miles per hour ; how many hours before they will meet? Ans. 12 hours. 15. If a horse walks 35 miles in 7 hours, how far will he walk in 11 hours at the same rate? 16. If a railway train goes 304 miles in 16 hours, how far will it travel in 21 hours at the same rate? Ans. 399 miles. 17. How much is ^ of the sum of 12 and 8? 18. What is the average width of a field which is 332 yards wide at one end, and 478 yards wide at the other? Ans. 405 yards. 19. How much is ^ of the sum of 11, 9, and 4? 20. A farmer killed 3 hogs ; one weighed 165 pounds, another 173 pounds, and the third 181 pounds; what was the average weight of the three hogs? Ans. 173 pounds. 21. m Henry started on a journey of 62 miles, and traveled at the rate of 5 miles a day for 4 days ; how many days will it take him to complete the journey if he goes at the rate of 6 miles a day? 22. A laborer engaged to remove 4703 bricks. After Division-. 125 working 24 hours, removing 125 bricks each hour, how many hours will it take him to complete the work if he removes 131 bricks per hour? Ans. 13. 23. m Frank sold 10 oranges at 6 cents apiece, and with the money bought apples at 5 cents apiece ; how many apples did he get? 24. A man sold 144 horses at $135 apiece, and in- vested the money in land at $18 per acre; how many acres did he get? Ans. 1080 acres. 25. A boy spent 42 cents for cakes, and 1-sixth as much for nuts ; what are the c parts of what he spent in all? How much did he spend? 26. A farmer has 306 acres in one field, and 1-eigh- teenth as much in another field ; how much has he in both fields? Ans. 323 acres. 144. QUESTIONS FOR REVIEW. What is: 1. Division? 2. The dividend? 3. The divisor? 4. The qnotient ? 5. The sign of division ? What is denoted by the sign -=- ? Name the three principles of division. Can division be ef- fected by subtraction? What is the relation of division to multiplication? What do we find by division ? What is meant by : 1. Short division? 2. Long division? Give the rule for each. How do we find: 1. The half of a number? 2. The third of a number? 3. The fourth f 4. The fifth? etc. DIVISORS AND MULTIPLES. 145. A Divisor of a number is one of its factors.* Thus, 4 is a divisor of 12, since 4X 3= 12. Also, 1, 2, 3, 4, 6, and 12 are divisors of 12, since each is contained in 12 an exact number of times. 1. 1 is a divisor of every number. 2. Every number is a divisor of itself. Any number is exactly divisible : 3. By 2, if its last figure is divisible by 2. 4. By 4, if its last two figures are divisible by 4. 5. By 8, if its last three figures are divisible by 8. 6. By 3 or 9, if the sum of its figures is divisible by 3 or 9. 7. By 5, if it ends in 5 or 0. 8. By 6, if it is divisible by 2 and 3. MENTAL EXERCISES. How many times is : 1. 1 contained in 11? 1 contained in 143? 2. 13 contained in 13? 97 contained in 97? Which of the numbers 2, 3, 4, 5, 6, 8, 9, and 10 are divisors of : 250? 3056? 4581? 1722? 45460? 761? 23202? 45128? 37301? 371820? "The terms numbers, divisors, multiples, and factors, as here used, de- note integers. (126) Divisors and Multiples. 127 146. A Common Divisor of two or more numbers is any number which will exactly divide all of them. Thus, 2 and 4 are common divisors of 8 and 12, since they di- vide each of them without a remainder. EXERCISES. 1. What are the common divisors of 6, 12, and 24? Ans. 2, 3, and 6. 2. What are the common divisors of 8, 16, and 24? Aiis. 2, 4, and 8. 3. What are the common divisors of 12, 18, and 24? Ans. 2, 3, and 6. 4. What are the common divisors of 36, 54, and 72 ? Ans. 2, 3, 6, 9, and 18. 5. What are the common divisors of 24, 36, and 48? Ans. 2, 3, 4, 6, and 12. 147. The Greatest Common Divisor of two or more numbers, denoted by G. C. D., is the greatest number that will exactly divide each of them. Answer these five questions by referring to the pre- ceding exercises : What is the G. C. D. of 6, 12, 24 ? Of 8, 16, 24? Of 12, 18, 24? Of 36, 54, 72? Of 24, 36, 48? 148. PRINCIPLE. The c factors of the G. C. D. of several numbers are all the factors common to all. Thus, the c factors of 12 are 2, 2, 3, the c factors of 18 are 2, 3, 3. Now, from each we may take the factor 2, then the factor 3, and as no equal factors remain, 2 and 3 are the c factors of the G. C. D. of 12 and 18. 128 Intermediate Arithmetic. 149. What is the G. C. D. of 18, 27, and 36? EXPLANATION. First divide 18, 27, and OPERATION. 36 by any number that will divide each of 3)18 27. 36. them, as 3. Then divide the quotients 6, o\~c Q 7i7 9, and 12, by any number that will exactly divide each of them, as 3. Now there is 2. 3. 4. no number except 1 that will divide the quotients 2, 3, 4. Hence, 3 and 3, the two divisors, are the c fac- tors of the G. C. D. ; that is, the G. C. D. is 9. WRITTEN EXERCISES. 1. What is the G. C. D. of 12 and 30? Am. 6. 2. What is the G. C. D. of 24, 30, and 54 ? Ans. 6. 3. What is the G. C. D. of 35, 56, and 70? Ans. 7. 4. What is the G. C. D. of 18, 36, and 72? Ans. 18. 5. What is the G. C. D. of 48, 72, and 144? Ans. 24. 6. What is the G. C. D. of 32, 48, 64, and 160? Ans. 16. 7. What is the greatest common divisor of 60, 90, 150, and 210? Ans. 30. 150. A Prime Number is a number which has no di- visors except itself and 1. Thus, 1, 2, 3, 5, 7, etc , are prime numbers. 151, A Composite Number is a number which has other divisors than itself and 1. Thus, 4, 6, 8, 9, 10, etc., are composite numbers. Is 13 a prime or a composite number? Ans. A prime number, as it has no divisors except 1 and 13. Is 21 a prime or a composite number? Ans. A composite number, as it can be divided by 3 and also by 7. Divisors and Multiples. 129 152. Prime Factors are factors which are prime num- bers. Thus, 2, 2, and 3 are the prime factors of 12. WRITTEN EXERCISES. 153. 1. Write all the prime numbers between 1 and 25 ; 25 and 50 ; 50 and 75 ; 75 and 100. 2. What are the prime factors of 56? OPERATION. EXPLANATION. We divide the number by any prime factor ; then divide the quotient by any 2)28 prime factor, etc., until the quotient 1 is obtained. 2)14 The several divisors are the prime factors re- LSJL J..U-1*^ ldV> L*yi O 1 t- . -_ quired. Hence, the answer is 2, 2, 2 and 7. ^ ' What are the prime factors of: 3. 50? Ans. 2, 5. 5. 4. 60 ? Ans. 2, 2, 3, 5. 5. 108? Ans. 2, 2, 3, 3, 2. 6. 640? Ans. 5, seven 2's. 7. 455 ? Ans. 5, 7, 13. 8. 680? Ans. ? 9. 1155? Ans. ? 10. 7800? Ans. ? 11. 2310? Ans. ? 12. 4290? Ans. ? 154. A Multiple of a number is a number which con- tains it an exact number of times. Thus, 24 contains 6 exactly 4 times, hence 24 is a multiple of 0. Is 16 a multiple of 8? 25 a multiple of 5? 27 of 7? 36 of 9? 42 of 6? 54 of 9? 63 of 7? 64 of 10? 72 of 12? 155. A Common Multiple of two or more numbers is a number which is exactly divisible by each of them. Thus, 12 is a common multiple of 2, 3, 4, and ; 18 of 2, 3, 6, and 9. N. i. 9. 130 Intermediate Arithmetic. EXERCISES. 1. Name three common multiples of 2, 3, and 6. Am. 6, 12, and 18. 2. Name three common multiples of 4, 5, and 10. Ans. 20, 40, 60. 3. Name three common multiples of 3, 4, 12. Ans. 12, 24, 36. 4. Name three common multiples of 7 and 2; 5 and 8; 3, 7, and 2; 5 and 12; 3, 5, and 10; 2, 4, and 8. 156. The Least Common Multiple of two or more num- bers, denoted by L. C. M., is the smallest number which these numbers will exactly divide. Answer these questions by referring to the preceding exercises: What is the L. C. M. of 2, 3, and 6? Of 4, 5, and 10? Of 3, 4, and 12? What is the L. C. M. of 5 and 6? 6 and 7? 6 and 8? 5 and 9? 7 and 10? 2, 3, and 5? 157. PRINCIPLE. The c factors of the L. C. M. of two or more numbers are all the prime factors of each. Thus, the prime factors of 12 are 2, 2, and 3. 30 are 2, 3, and 5. " " 50 are 2, 5, and 5. Taking out the facto. 2 from each, then the factor 3 from the first and second, then the factor 5 from the second and third, there are left, 2 in the first, and 5 in the third. Hence, the c factors of the L. C. M. of 12, 30, and 50 are 2, 3, 5, 2, and 5. WRITTEN EXERCISES. 158. What is the L. C. M. of 12, 15, and 25? EXPLANATION. Divide any two or more of the numbers by any common prime factor, as 3, and bring down with the quo- Divisors and Multiples. 131 tients 4 and 5 such numbers (25) as do not OPERATION. contain the divisor. Again, divide out by 5, o\io 15 o r as it is a prime factor common to 5 and 25, and bring down the 4. Now there is no 5) 4 5 25 factor except 1 that will divide two of the 415 numbers 4, 1, and 5. Hence, the two di- visors 3 and 5, and the quotients 4 and 5 are the c factors of the L. C. M. ; that is, the L. C. M. is 3 X 5 X 4 X 5 = = 300, as it contains all the prime factors of 12, 15, and 25. Find the L. C. M. of: 2. 6 and 12 ; 9 and 15 ; 20 and 25. Ans. 12 ; 45 ; 100. 3. 12 and 20 ; 16 and 24 ; 35 and 42. Ans. 60; 48 ; 210. 4. 2, 4, 5, and 12 ; 3, 7, 9, and 14. Ans. 60; 126. 5. 5, 8, 10, and 12; 7, 9, 12, and 18. Ans. 120 ; 252. 6. 5, 6, 10, and 15; 6, 12, 15, and 20. Ans. 30; 60. 7. 15, 20, 30, and 40; 16, 20, 32, and 40. Ans. 120; 160. 8. 25, 36, 50, and 72; 48, 60, 96, and 120. Ans. 1800 ; 480. 9. 5, 7, 11, and 15 ; 3, 7, 13, and 39. Ans. 1155 ; 273. 10. 4, 5, 6, 10, 12, 15, 20, and 30. Ans. 60. 159. QUESTIONS FOR REVIEW. "What is: 1. A divisor of a number? 2. A common divisor of two or more numbers? 3. The G. C. D. of two or more numbers? Give an example of each. What is: 1. A prime number? 2. A composite number ? 3. A prime factor? 4. A multiple of a number? 5. A common mul- tiple of two or more numbers ? 6. The L. C. M. of two or more numbers? Give an example of each. What is the principle of : 1. The G. C. D. ? 2. The L. C. M. ? When is a number exactly divisible by : 2 ? 3? 4? 5? G? 8? 9? 10? COMMON FRACTIONS. INDUCTIVE EXERCISES. 160. When an apple, an orange, a number, or a bar of soap is divided into two equal parts, what is each part called? Hew is 1-half written? Ans. %. W How do we get ^ of a thing ? Ans. Divide it into two equal parts, and take one of the parts. What is i of a pile of 2 books ? of 82 ? J of $4 ? i of $16? When an apple, an orange, a number, or a bar of soap is divided into three ^ ^_ ^ equal parts, what is each part called? What are two of the parts called? How are 1-third and 2-thirds written? Ans. % and -f. How do we get -|- of a thing ? Ans. Divide it into three equal parts and take one of the parts. What is i of a pile of 3 books ? \ of $3 ? i of $6 ? 1 of $12? i of 24c.? \ of 60 bushels? o o o How do you get f of a thing? Ans. Divide it into three equal parts and take one of the parts 2 times. (132) Common Fractions. 133 What is 1 of a pile of 3 books? | of $3? J of $12? | of 27c.? When an apple, an orange, a number or a bar of soap is divided into four equal parts, what is each part called? What are two of the parts called? What are three of the parts called? How are 1-fourth, 2-fourths, and 3-fourths written? Ans. J, f , and f . How do we get ^ of a thing? Ans. Divide it into four equal parts, and take one of the parts. What is \ of a pile of 4 books? J of $4? J of $20? \ of 40c. ? How do we get f of a thing ? Ans. Divide it into four equal parts and take 1 part 2 times. How much is | of a pile of 4 books ? } ot 4 ? f of 16? | of 32? How do we get f of a thing? Ans. Divide it into four equal parts, and take 1 part 3 times. How much is f of a pile of 4 books ? j of 4 ? f of 12? f of 40? DEFINITIONS. 161. Fractional Parts are parts obtained by dividing any thing, or a unit, into any number of equal parts. Thus : halves, thirds, fourths, fifths, sevenths, tenths, etc., are fractional parts. 162. A Fractional Unit is one of the equal fractional parts into which a thing is divided. 134 Intermediate Arithmetic. Thus- I 1 ' half ' !-third, 1-fourth, 1-fifth, 1-tenth, 1-twelfth, j ill \ \ TO T5 > are fractional units. 163. A Fraction is a fractional unit taken one or more times. C l-fourtb, 2-fifths, 3-sevenths, 7-tenths, ~\ Thus :] 1 X i 2 X i 3 X \ 7 X T V [ are fractions. 11 371 V. t 3 7 T7 164. The Terms of a fraction are the two numbers used to express it. 165. The Denominator is that term which names the parts expressed by the fraction. It is written below the horizontal line. 166. The Numerator is that term which numbers the parts expressed by the fraction. It is written above the horizontal line. Thus : the terms of the fraction are 5 and 6 ; the denomina- tor is 6, and the numerator is 5. The fraction means : 1. 5 of the parts when 1 unit has been divided into 6 equal parts. 2. The fractional unit 1-sixth, or , taken 5 times. 3. The quotient of 5 divided by 6. Read each of the following fractions, name the terms, the numerator, the denominator, the fractional unit, the number of fractional units, and give the three meanings of each fraction : I; I; ; A; j ; 5 ninths; 7 twelfths. 167. A Proper Fraction is one of which the numera- tor is less than the denominator ; as -f-, f , f , etc. 168. An Improper Fraction is one of which the numer- ator equals or exceeds the denominator ; as f , \-, ^, etc. Common Fractions. 135 169. A Mixed Number is an expression consisting of a whole number and a fraction ; as 2|, 5J, 12 J, etc. 170. The Value of a fraction is the quotient of the numerator divided by the denominator. The value of f is 2 ; of - 1 / is 4 ; of *f- is 5. When the numerator equals the denominator the value of the fraction equals 1 ; as f, f , ||, etc. The value of a proper fraction is less than 1, as its numerator is less than its denominator. The value of an improper fraction is equal to or exceeds 1, as its numerator equals or exceeds its denominator. 171. A Compound Fraction is a fraction of a fraction ; as i of i | of f of T 5 T . MENTAL EXERCISES. 172. l. How many halves () in one (1) ? Why? 2. What is the w r orth of one orange, if 1 half of it is worth 5 cents ? 6c. ? lOc. ? 3. How many thirds (-J) in one (1) ? Why ? 4. What is the worth of one bale of cotton, if 1 third of it is worth $10? 812? $20? 5. How m^nj fifths (i) in one (1) ? Why? 6. What is the length of a pole, if 1 fifth of it is 3 feet long? 7 feet long? 10 feet? 7. How many sevenths (-J-) in one (1) ? Why ? 8. What is the weight of a rock if 1 seventh of it weighs 5 pounds? 9 pounds? 9. How many tenths in one (1) ? Why ? 10. How many marbles in a box, if 1 tenth of them is 6 marbles? 9 marbles? 25 marbles? 11. What is the number whose half is 1 ? Whose third is 2 ? Whose fourth is 5 ? Whose sixth is 2 ? Whose ninth is 3? Whose tenth is 7? 136 Intermediate Arithmetic. 12. How much is ^-fourths (f) of 28? ANALYSIS. I fourth of 28 is 7 ; 3 fourths of 28 is 3 times 7 = = 21, Ans. 13. How much is 2 ^refe of 12? f of 15? f of 30? 14. How much is 3 fourths of 16? f of 20? f of 40? 15. How much is 4 fifths of 20? f of 30? f of $50? 16. How much is 5 sixths of 18? 17. How much is 7 ninths of 18? 18. How much is 5 twelfths of 24? |of36? 19. If one acre of land cost $12, what is the cost of fof24? |of60c.? f of 36? | of 45c.? of an acre ? -J- of an acre ? f of an acre ? 20. If one bushel of potatoes cost 30 cents, what is the cost of -J- of a bushel ? f of a bushel ? ^ of a bushel ? 21. 5 sixths of a number is 10, what is the number? ANALYSIS. If 5 sixths is 10, I sixth is $ of 10, or 2 ; hence, the number is 6 times 2 -= 12, Ans. 22. What is the number of which 5 sixths is 20? f is 20? 4- is 16? f is 10? A is 21? ( O 1 A 23. What will a melon cost if f of it cost 20 cents? If | of it cost 18 cents? If f of it cost 35 cents? FUNDAMENTAL PRINCIPLES. 173. CASE I. To multiply and divide fractional units by whole numbers. Into how many parts is this bar of soap divided ? What is 1 part called? Is it a fractional unit? If each of the three parts is cut into two parts, how many parts will there be in all? What is 1 part called? Is it a fractional unit Common Fractions. 137 Is 1 part of the first bar equal to 2 times 1 part of the second bar? What does this show ? Ans. That 2 times ^ =- J. What else does it show ? Ans. That 1 half of ^ = 1. Hence, 1. To multiply a fractional unit by a number, we may divide the denominator by that number. 2. To divide a fractional unit by a number, we multiply the denominator by that number. MENTAL EXERCISES. How much is : 1. 2 times ? Ans. 2. 4 times ? Ans. 3. 5 times y 1 ^? Ans. How much is : 10. 1 half of i? 11. 1 fourth of ' 12. 1 s*atf^ of J? 13. 1 fifth of 4-? 4' 4, 2 X Jl 12 ? "2- 5 7 X 2T ? 6 11 X filf ? 75X1 8. 8) ; \ fortieth f 9 3 X 1 fifteenth? -4ns. Ans. 14. 15. 16 17. 1 tenth -^ 1 third -- 1 ninth -r- 174. CASE II. To change the fractional unit. Into how many equal parts is this bar of soap divided? What is 1 part called? 2 parts ? Is J a fractional unit? Here is an equal bar ; in- to how many parts is it divided? What is 1 part called ? 2 parts ? etc. Is a fractional unit? Is 1 part of the first bar, or the second, or f ? equal to two parts of 138 Intermediate Arithmetic. Are 2 parts of the first bar equal to 4 parts of the second ? What does this show ? Ans. That f = = f, and $==-. Hence, 1. Multiplying both terms of a fraction by the same number does not alter the value of the fraction. 2. Dividing both terms of a fraction by the same num- ber does not alter the value of the fraction. EXPLANATION. Can 24 be written thus: 6 fours f Thus: 12 twos? Are 6 fours equal to 12 twos? Why? Ans. The unit two is half of the unit four, but is taken twice as often, since 12 is twice 6; hence, they are equal. Are 3 fifths, or f , equal to 6 tenths, or T % ? Why ? Ans. The unit 1 tenth is half of the unit 1 fifth, but is taken twice as often, since 6 is twice 3 ; hence, f -f^. Why, then, is the value of a fraction not changed by multiplying both terms by the same number ? Ans. Because it decreases the fractional unit in the same ratio that it increases the number of times it is taken. Why is the value of a fraction not changed by divid- ing both terms by the same number? Ans. Because it increases the fractional unit in the same ratio that it decreases the number of times it is taken. REDUCTION OF FRACTIONS. 175. Reduction of a Fraction consists in changing its o o terms without altering its value. 176. CASE I. To reduce a fraction to its lowest terms. A fraction is in the lowest terms when no number greater than 1 will exactly divide its numerator and denominator. Common Fractions. 139 1. Reduce f to its lowest terms. OPERATION. f)\ 3 15 ) 4"9" ~ " ^TF EXPLANATION. Dividing both terras of f-$ by 2, we get -$. Now dividing both terms 5)13. 3 of Jo- by 5, we obtain . In the second operation we divide both 2d OPERATION. terms by their G. C. D., 10. 10) ^ = Hence the, RULE. Divide both terms of the fraction by any number greater than 1 that will exactly divide them, and continue the operation as long as possible. Or, Divide both terms by their G. C. D. WRITTEN EXERCISES. Reduce to lowest terms: ol4212445 75 A^o 232 9 9 * To i "5 "6 5 T65 635 1 2"o' -./*. -5, g-, 3, I Q 84 112 81 6313 5 AW* 343 9 9 P 1 T9"65 T4~"05 TO^S"? "815 TTO' /1/Wf. y, 5-, 4^, A. 2_0_7 j_2_8 39 J.9.A _5 2_8_ A^Q J.3. ? 3 9 1_2. * 99"5 885 1435 2255 17712' t<5< 115 11? ' 23* 177. CASE II. To reduce a whole number to a frac- tion. 1. Reduce 4 to a fraction whose denominator is 3. EXPLANATION. We write 4 thus: f, then OPERATION. multiply both terms by 3, and obtain - 1 -/-. r-L 2 -. Hence, the RULE. Multiply the whole number by the given denom- inator and place the product over the denominator. WRITTEN EXERCISES. Reduce : 2. 5 to a fraction whose denominator is 3. Ans. ^-. 3. 7 to a fraction whose denominator is 8. Ans. 140 Intermediate Arithmetic. 4. 9 to a fraction whose denominator is 6. Ans. ^-. 5. 11 to a fraction whose denominator is 5. Ans. ? Reduce : 6. 7 to fifths. Ans. - 3 /. 7 8 to thirds. Ans. --. 8. 15 to fourths. Ans. ? 9. 20 to tenths. Ans. ? 178. CASE III. To reduce a fraction to higher terms. 1. Reduce & to a fraction whose denominator is 15. o EXPLANATION. -Write f, and on its right place 15 with a line above it. Now say 5 in OPERATION. 15 3 times. 3X3 = 9, which place over the f = =73- 15. Hence, the RULE. Divide the denominator of the required fraction by the denominator of the given fraction, multiply the quo- tient by the numerator, and place the product over the larger denominator. WRITTEN EXERCISES. Reduce : 2. | to a fraction whose denom. is 24. Ans. -Jf. 3. f to a fraction whose denom. is 42. Ans. ff. 4. _z_ to a fraction whose denom. is 66. Ans. ? 5. | to sixteenths. Ans. ff. | 7. f to fortieths. Ans. ? 6. f to forty-fifths. Ans. f f. | 8: T 7 ^ to sixtieths. Ans. ? 179. CASE IV. To reduce a mixed number to an im- proper fraction. 1. Reduce 5| to an improper fraction. EXPLANATION. AVe multiply 5 by 3, add OPERATION. 2 to the product, and place the sum 17 over 5 X 3 = 15 the denominator. 15 -f~ 2=17, ^. Common Fractions. 141 ANALYSIS. 5f Hence, the 5 ones and 2 thirds, 15 thirds and 2 thirds = 17 thirds. RULE. Multiply the -whole number by the denominator, add the numerator to the product, and place the sum over the denominator. WRITTEN EXERCISES. Reduce to improper fractions : 2. 4J. 3. 7|. 4. 5f. 5(~* O \ ** w"fT* 6. lOf. 7 O3 I t/"?. O 3 ' _3_0 4 ' 28 8. 9. 15f. 10. J -3' ;s >4' j 14 - 6 - 3 I 15. 45/y. 50 Ans. Ans. 11. ISf. 12. 17^. Ans. 13. 21f. Ans. 16. 17. 60^. Ans. 18. 1352jj. Ans. 19. 24444. Ans. 180. CASE V. To reduce an improper fraction to a whole or mixed number. l. Reduce ^- to a mixed number. EXPLANATION. We divide the numerator by the denominator. OPERATION. 3)10(3i ANALYSIS. ^ -10 thirds = Q thirds and 1 Mird = 3 and 1 tfiiYd == 3}. 2. Reduce 2 ^ to a mixed number. ANALYSIS. - 2 ? 2 - 22 fourths = 2Qfoiirt)is and 2fourths=5 and |. OPERATION. i22( 20 2 - - 1 Hence, the RULE. Divide the numerator by the denominator, and re- duce the fractional remainder, if any, to its lowest terms. 142 Intermediate Arithmetic. WRITTEN EXERCISES. Reduce to whole or mixed numbers : 3. -V-- 4. -y-. 5. ^ 5 -. ft ^ - 7 48 I . ~ a". . 9. . 7-. Ans. . 54. 8. - 6 ^. Ana. 16. , 13. W. Ans. 131. rr 1 O o Ans. 84. 9. 10. ^. ^4ns. 11. i-jp. ^4m\ 12. 78 T2- Ans. 14. W. Ans. 40 15. ^. ^4m-. ? 16. ^/. ^4m. 5^. 17. - 6 F 2 f. -4na. 12 J. 181. CASE VI.- -To reduce a compound fraction to a simple one. 1. Reduce -f of J to a simple fraction. ANALYSIS. By Art. 173, | of ^ = = T5) then f OPERATION. of i will be 2 times 1 fifteenth, or T 2 ^, and f of f 2. y i _8_ will be 4 ^m^s 2 fifteenths, or T \, ^4ns. Hence, RULE. Multiply the numerators together for the numer- ator of the answer, and the denominators together for the de- nominator. EXERCISES. 2. Reduce f of f to a simple fraction. Ans. J| = -fa. 3. Change f of -f$ to a simple fraction. ylns. ^. 4. Reduce 2 thirds of 3 tenths to a simple fraction. ^4n.s. 1 ^/f/^. 5. Change 4 ^/^/is of 2 sevenths to a simple fraction. 6. What simple fraction is equal to f of f of f ? Ans. y\. 7. What simple fraction is equal to f of -f of -}-f. 8. Reduce f of 7J of 5 to a simple fraction. Common Fractions. 143 EXPLANATION. Reduce 1\ to an im- proper fraction (- 2 /) ; also the whole number 5, by putting 1 under it fora fX-^-Xf = ; W r : ~- denominator, and proceed as before. 9. Reduce f of 3^ of \ to a simple fraction. Ans. ^. 10. Reduce J of -f- of 9^ to a simple fraction. Ans. f. 11. What is the value of f of f of l|f? Ans. 1. 12. What is the value of J of T 3 T of 16J? Ans. 3. 182. Cancellation. Instead of multiplying the numer- ators, then the denominators, and then reducing to lowest terms, the same result may be obtained by first striking out or cancelling all factors common to the numerator and denominator. By this process the work is often materially shortened. 13. Reduce f of -f of f to a simple fraction. EXPLANATION. First, there is a 3 OPERATION. in both numerator and denominator, q 9 9 one over 5, and the other under 2 ; we o f _ o f _ _ cancel these by making a mark across ft ft 7 7 them. Next, we cancel the two 5's in the same manner. Cancelling a number is dividing by it ; hence, 1 is supposed to take the place of the number canceled ; that is, ^121 2 of - of - means - of - of -. Hence the answer is -. 7 1 1 7 7 14. Reduce f of f of ^ to a simple fraction. EXPLANATION. - Write 3X3 OPERATION. in the place of 9, 3 X 5 in the 9 a o \/ K place of 1-5 ; then cancel as in the - of - of = 1. preceding example. 4 15. Reduce f of -J of -J-f- to a simple fraction. Ans. f. Reduce to a simple fraction : 16. | of | of J of 6. Aiis. 3. 144 Intermediate Arithmetic. 17. f of f of f of &$ of f . Ans. 2. 18. f of T 2 T of f of 7J of T V Ans. . 19. f of f of J of T 6 r of 3f of 8| of i An*. I. 183. CASE VII. To reduce fractions to a common de- nominator. Fractions have a common denominator when their denominators are the same : as f and -f , y\- and -fa. 1. Reduce |- and J to a common denominator. Multiplying both terms of -|- by 3, the denominator of the other fraction, we have i = - |. Now multiplying both terms of \ by 2, the denominator of the other fraction, w T e have i = = f. -Hence, in the place of ^ and | we have f and |, and these have a common denominator. Reduce to a common denominator : 2. | and -f. ^4ns. f-|- and 1 I AJ t-7 JJ ana TS-- 2- Reduce to their least common denominators : 3 .9 8 - ' 24 A 2. 0. 6 ' " 24 JL 14. 12 ' "24 25, 26. 27. 28. i o o T\ r\ O "35 4"5 ana 6"' 2. 4. 5. 35 95 65 1 3 5 7 45 85 165 32' _3 _5_ _7 < 175 125 13"5 J. N. 10. 4?1S. Ans. Ans. Ans. 8 125 12 185 8 9 12") 8 185 12 "" 18 "6"o"5 2S " 10 12' 15 185 10 3T5 28 "605 7 18' 27 fir- 146 Intermediate Arithmetic. ADDITION OF FRACTIONS. INDUCTIVE EXERCISES. 185. PARALLELISMS. WHOLE NUMBERS 1. Add: 4 threes and 7 threes. Ans. 11 threes. 2. Add: 8 threes and 9 fours. These are unlike, and must be reduced to like units. 8 threes = 2 twelves. 9 fours = 3 twelves. Now adding the like num- bers, we get 5 twelves. FRACTIONS. 1, Add: 4 thirds and 7 thirds. Ans. 11 thirds. 2. Add: 8 thirds and 9 fourths. These are unlike, and must be reduced to like units. 8 thirds = 32 twelfths. 9 fourths = 27 twelfths. Now adding the like num- bers, we get 59 twelfths. PRINCIPLE. To add two numbers, whether whole or frac- tional, they must be reduced, if not already so, to like units. 186. CASE I. When the denominators, or fractional units, are alike. 1. Add together f, -f-, and . EXPLANATION. Since the numbe added have the same unit, viz : 1 seventh, we OPERATION. f = 3 sevenths. EXPLANATION. Since the numbers to be 2 __ o seventh'* add as in whole numbers, and obtain 6 sev- Z_m enths, or f. Hence, 6 sevenths^. RULE. Add the numerators and place the sum over the common denominator. Addition of Fractions. 147 What is the sum of: 2. f , f , | ? ylns. f = 3. 4. 5. 346? Am ? Ans. l- 7 - A TT' 9. $t> S-V -, H ? Ans. OPERATION. 9 187. CASE II. When the denominators, or fractional units, are unlike. l. Add together f and f. EXPLANATION. Since the numbers to be added, viz : 3 fourths and 2 thirds, are unlike, they cannot be added in their present form. Reducing them to a common denominator, or unit, by Art. 183, we obtain ^- 2 - and T \, the sum of which, by Case I, is j| = lj\. Hence, RULE. Reduce the fractions to a common denominator, and proceed as in Case L NOTE. In addition and subtraction, the fractions should be written under each other after the manner of whole numbers. 3 f : 2 8 - -jnr 1 7 What is the sum of and and and 7. -fi and 8. I- and 4. 5. 6. 3 6" 5 8" f? f? A? Ans. Aiis. Ans. Ans. Ans. f f and 1? ^4ns 3.1 _ = 1 i.i f f and 2 3 ? Ans. 113. 24* A- 9. 2 3) 3 T> and 6. ~5 ? Ans. 2 f-J- 45 42" 93 T2- *f 10. 11. 12. 1, 5 6") 5 6) and and and 1 "8 8 &" ? ? ? Ans. Ans. 037 ^8*' 2 95 O 3 1 3 Z 5"OT' 11 1 3. 1! 8 "Q"l and TT .? Ans. 24 M- 14. Find the sum of 7| and 9J. EXPLANATION. When there are mixed numbers, we add the fractions first, and then add their sum to the sum of the whole num- bers. Adding T 9 2 and T 8 2, we get H = l T 6 s ; put down the T \ and carry 1 to be added to the sum of the whole numbers, we get 17y\. OPERATION. 8 72 _ 7 f t~ - *J Q3 - Q 9 TT- 148 Intermediate Arithmetic. 15. Add together 12J and 15J. Ans. 27f. 16. Add together 23f, 18f, and 32|. ^?is. 74ji. 17 A man paid $13J for a pair of pants, $17f for a coat, and $5f for a vest. What did he pay for all? 18. One boy weighs 64f pounds, another boy 56f, pounds, and the third boy 49^ pounds. What is the total weight of the three boys? -4ns. 170JJ pounds. 19. A man planted 120f acres in corn, 75 J acres in cotton, 32^ acres in wheat, and 15fV acres in oats. How many acres did he have in cultivation? Ans. 243 acres. SUBTRACTION OP FRACTIONS. INDUCTIVE EXERCISES. 188. PARALLELISMS. WHOLE NUMBERS. 1. From 4 fives take 2 fives. Ans. 2 fives. 2. From 15 fours take 8 fives. These are unlike, and must be reduced to like units. 15 fours = 3 twenties. 8 fives = 2 twenties. Now subtracting like num- bers, we get 1 twenty. FRACTIONS. 1. From 4 fifths take 2 fifths. Ans. 2 fifths. 2. From 15 fourths take 8 fifths. These are unlike, and must be reduced to like units. 15 fourths - - 75 twentieths, 8 fifths = 32 twentieths. Now subtracting like num- bers, we get 43 twentieths. Subtraction of Fractions. 149 PRINCIPLE. --To subtract one number from another, whether whole or fractional, they must be reduced, if not already so, to like units. 189. CASE I. When the denominators or fractional units are alike. 1. Subtract T % from ^. ANALYSIS. Since the numbers to be subtracted, viz : 3 tenths and 9 tenths, have the same unit, 1 tenth, we subtract as in whole numbers, and obtain 6 tenths, or T % = f , Ans. Hence, the * RULE. Take the less numerator from the greater, and place the difference over the common denominator. 2. What is the difference between f and f ? 3. What is the difference between ^ and How much more is : 7. -- than -? Ans. ? 8. 1^ than T \? Ans. ? 9. A! than 4-1- ? Ans. ? 4. T 7 -g than y^? Ans. 5. if than T 7 2-? Ans. f. 6. i| than A ? Ans. \. 1 '.) LI) O -i 190. CASE II. When the denominators or fractional units are unlike. l. From f take %. EXPLANATION. Since 3 fifths and 1 Aa?/are un- OPERATION. like, they can not be subtracted in their present 3 form. Reducing them to a common denominator, i by Art. 183, we obtain 6 tenths and 5 tenths, the difference between which is 1 tenth, or y 1 ^. yV Ans. Hence, the RULE. Reduce the fractions to a common denominator, and proceed as in Case I. 150 Intermediate Arithmetic. Find the difference between : 2. f and f. 3. f and }. 4. -J and J. 5. f and f. 6. T 9 T and TV 7. y 9 ^ and f 8. | and i 2 T . 16. From 8f take 5|. EXPLANATION. We first reduce the fractions to a common denominator, then take their differ- ence, and unite it to the difference of the whole numbers. Thus, -^ from T 9 j leaves y 1 ^ ; 5 from 8 leaves 3 ; now uniting the 3 and y 1 ^, we get 3^2. 5 ^ 9. 2" and 24' Ans. II 3 28' 10. 3. and T3~' Ans. H- 1 11. 5 TT and 2. Ans. 23 99 T8-- 12. 10 13 and 3 8' Ans. ? yW 13. 1 4 and 3 Ans. ? 14. 2 and 10 43' Ans. ? ? 15. H and A- Ans. ? OPERATION. 8! = 8A 8 From : 17. 12f take 7|. 18. take 18J. Ans. 19. 351 take 22f. 20. 27 take 23 . Ans. 4. 21. 451 take 71. Ans. 38M. 24' 22. 1641 take 73f Ans.t 23. 195ttakel26 T V Ans.t 24. 200^ take 85 A-. Ans. ? 25. A rope was 48J feet long, but 17| feet were cut off; how long was the rope then? 26. From 8 take I. EXPLANATION. 8 is equal to 7 and one, or, re- ducing one to ninths, 7 and f . Hence, we write 8 under the form of 7f, and subtract f as in the preceding examples. How much more is : OPERATION. 8 = 7* 5 5 9" 9 71. 27. 12 than J? Ans. llf 28. 13 than -f^ Ans. 12 T 6 T . 29. 14 than f ? Ans. 13|. 30. 43 than ^ Ans. 42 T %. 31. 64than T 3 T ? 32. 75 than ^? 33. 84 than T y? 34. 125 than -i? Ans. ? Ans. ? -4ns. ? Ans. ? Subtraction of Fractions. 151 35. From 7J take 5f. EXPLANATION. We reduce the fractions to a OPERATION. common denominator, and as we can not subtract 7-g- = 7^2 A from T \, we take one, or }? from the 7, and add ^3 - - c; 9 i_i &/ x *-^ 4. 12 it to ' T 4 2> making ||. Then we say, T \ from || leaves T 7 2, and 5 from 6 (7 less one) leaves 1. lyV What is the value of: 36. 8 3f? 37. 16f 12f ? 38. 43J 18f? ,4ns. 24^. 39. 68|- -49J? And. 40. lOO- - 41. 146f 86}f: 42. A man had $5J and paid for a knife $f ; how much did he have left? Ans. $4^. 43. One melon weighs 25^ pounds, and another weighs 17 T \ pounds ; how much heavier is one than the other? Ans. 7f pounds. 44. Frank and John went fishing ; Frank walked IS^V miles and John 2g\ miles ; how much further did Frank walk than John? Ans. 15f|f miles. 45. A farmer sold If acres from a field containing 37f acres ; how many acres had he left in the field ? Ans. 35|-f acres. 46. One bale of cotton weighs 463^ pounds, and an- other bale weighs 17f pounds less ; what does the lighter bale weigh ? Ans. ? 191. In the following examples the whole and one or more parts are given, and the c part required. See Art. 89. 47. The whole is 385J days, and one part 67^ days; what is the other or c part? Ans. 317j^ days. 48. A man had $1673f and spent $356f ; how many dollars had he left? Ans. $1316|J. 152 Intermediate Arithmetic. 49. A man had S540J- and spent $271f for sheep, and 8180J for hogs; how much money did he have left? Ans. $88 T V. 50. A flag-pole, standing in the water, is 100 feet long; 72 J feet of its length are above the water, 'and 12f feet are in the mud below the water ; how deep is the water? Ans. 14fJ ft. 51. Two boys, Charles and Henry, are 400 feet apart; if Charles goes towards Henry 167f feet, and Henry goes towards Charles 207f feet, how far apart will they then be? Ans. 24^ ft- MULTIPLICATION OP FRACTIONS. INDUCTIVE EXERCISES. 192. PARALLELISMS. WHOLE NUMBERS. 1. Multiply : 5 threes by 4 threes. Since 5 X 4 20, and three X three = nine, 5 threes X 4 threes 20 nines. 2. Multiply: 7 threes by 5 fours. Since 7 X 5 35, and three X four = twelve, 7 threes X 5 fours = 35 twelves. FRACTIONS. 1. Multiply: 5 thirds by 4 thirds. Since 5 X 4 20, and 1 third X 1 third = 1 ninth, 5 thirds X 4 thirds = 20 ninths. 2. Multiply : 7 thirds by 5 fourths. Since 7 X 5 = 35, and 1 third X 1 fourth = 1 twelfth, 7 thirds X 5 fourths = 35 twelfths. Multiplication of Fractions. 153 PRINCIPLE. To multiply two abstract numbers, whether whole or fractional, we may multiply the numbers regarded as units together for a new unit, and the numerators, or numeral factors, together for a new numerator. 193. CASE I. --To multiply a fraction by a whole number. 1. Multiply f by 3. EXPLANATION. -- The multiplicand is 5 1st. OPERATION. sixths and the multiplier 3. 3 times 5 sixths 5^0 . i s _ - 91 i - .. tf X O - -g- - Z T . are lo sixths, or -^ = 2$. Instead of multiplying the factor 5 by 3, we may multiply the unit factor 1 sixth by ,5 ,. i *v i i. i* 2d. OPERATION. 3. By Art. 173, 3 times 1 sixth is 1 half. Hence, 5 sixths X 3 = 5 halves, or f = = 2$. f X 3 = f = 2. Hence, the RULE. Multiply the numerator by the whole number, or divide the denominator by the ivhole number when it can be done without a remainder. EXERCISES. Multiply : . 2. | by 3. Ans. 1| 3. |- by 4. Ans. 8- H by 8. 9. || by 7. Ans. 2J. 10. ff by 11. 11. f by 12. Ans. ? 12. ^ by 10. 13. if by 4. ^- by 6. ,4ns. 5. f by 7. Ans. 4. 6. -5^- by 8. Ans. 1%. 7. JJ- by 5. -4ns. ? 14. If 1 gallon of syrup cost f of a dollar, what will 6 gallons cost? Ans. $3f. 15. If 1 bushel of potatoes cost f of a dollar, what will 8 bushels cost? Ans. ? 16. If 1 basket holds f of a bushel of corn, how many bushels will 8 baskets hold? Ans. 6f bushels. 154 Intermediate Arithmetic. 17. If 1 bucket holds -f- of a gallon of water, how many gallons will 5 buckets hold ? Ans. ? 18. Multiply 7} by 5. We may reduce 7| to an improper frac- OPERATION. tion, and multiply as in the preceding exer- 3 cises. It is generally better, however, to ^ multiply thus : 5 X f = = = = 3 !- 5 X 7 == 35. _A Now, adding 3| to 35, we have 38f. 35 NOTE. Since 8]- X 5 = 5 X 8}, it is imma- terial which we regard as the multiplier. 38f What is the value of: 19. 6-| X 7? Ans. 46|. 20. 7|X 3? Ans. 22^ 21. 12f X 9? Ans. 115^ 22. 10} X 8? Ans. i 23. 7fX20? Ans. 24. 9X 18|? Ans. 168. 25. 12 X 13f? Ans. 166. 26. 20 X llf? Ans - ? 27. 48x30yV? Ans. ? 28. 72 X 1234? Ans. ? 29. What will 32 gallons of brandy cost at $lj- per gallon? Ans. $36. 30. What will IS-j^ barrels of apples cost at $3 per barrel? * Ans. $45^. 31. What will 556 pounds of cotton amount to at 8f cents a pound ? Ans. 4865 cents. What will be the cost of: 32. 654 pounds of cotton at 7| c. a pound ? Ans. 5014 c. 33. 255 pounds of sugar at 9f c. a pound? Ans. 2448 c. 34. 876 yards of prints at 5} c. a yard ? Ans. 5037 c. 35. 1260 bushels of corn at 64| c. a bushel ? Ans. 81480 c. 36. 570 yards of silk at $1| a yard? Ans. $798. 194. CASE II. To multiply a fraction by a fraction. 1. Multiply f by J. Multiplication of Fractions. 155 EXPLANATION. Multiplication means tak- OPERATION. ing one number as many times as there are 5sx_l--5X2-.io "7 / ^ U TTxTT ^T* units in another. In f there are only f of a unit. Hence, we are required to take f f of 1 time. Now, 1 time f is f , hence f of 1 time f is f of f = |-J. Hence, the RULE. Multiply the numerators together for a numera- tor, and the denominators together for a denominator. Or, Regard X as meaning of, and proceed as in the re- duction of compound fractions to simple ones. NOTE. Mixed numbers must be reduced to improper fractions. Multiply : 2. | by Ans. 3. i by f Ans. 4. f by f. Ans. J, 5 - - 7J by 8f 5. ^n. 42. . 9. 6f by 31. Ans. 10. 7f by 2f. Ans. 20^. 11. 4^ by 2J. 6. 4J by |. What will be the cost of: 12. 12 yards of cloth at 8 c. a yard? Ans. 96 c. 13. 4 barrels of cider at $3 a barrel? Ans. $13. 14. 25| pounds of coffee at 12 c. a pound t Ans. 308 c. 15. If barrels of flour at S5f a barrel? Ans. $10ff. 16. 9f bushels of corn at 62J c. a bushel? Ans. 570^ c. 17. 240y\ acres of land at $25f an acre? Ans. $6156ff. What is the cost of: 18. 16 pounds of cheese at 8| c. a pound? Ans. 136 c. 19. 15| yards of cambric at 15 c. per yard? Ans. 235 c. 20. 11 cords of wood at $3J per cord? Ans. $38|. 21. 15 \ yards of broadcloth at $3|^- a yard? Ans.$57^%. 22. 15f yards of ribbon at 40 c. per yard? Ans. 630 c. 23. 8 J yards of silk at $^ per yard? Ans. 84f. 24. 348 pounds of cotton at 7| c. per pound ? 156 Intermediate Arithmetic. DIVISION OF FRACTIONS. INDUCTIVE EXERCISES. 195. PARALLELISMS. WHOLE NUMBERS. 1. Divide 15 fours by 3 Jours. Ans. 15-r-3=5. 2. Divide 15 fours by 8 Jives. Reducing to like units : 15 fours = 3 twenties, 8 fives = 2 twenties. Now, 3 twenties -r- 2 twenties = 3 -f- 2 U. 5. FRACTIONS. 1. Divide 15 fourths by 3 fourths. Ans. 15 -j- 3 2. Divide 15 fourths by 8 fifths. Reducing to like units : 15 fourths =- 75 twentieths, 8 fifths = 32 twentieths, Now, 75 twentieths -?- 32 twentieths = 75-^32 = 211 PRINCIPLE. To find how often one number is contained in another, whether whole or fractional, they must be reduced, if not already so, to like units. 196. CASE I. To divide a fraction by a whole number. 1. Divide * by 5. 1st OPERATION. EXPLANATION. The dividend is 20 thirds, the divisor 5. Now, 20 thirds -f- 5 = 4 thirds == f = = 1$. Instead of dividing the factor 20 by 5, we may divide the unit 1 20 _^ F; _ _ 20. 4 11 ~3 ' T 5 U ~ ~ 3 * third by 5. By Art. 173, 1 third -*- 5 = = 1 fifteenth. Hence, 20 thirds -4- 5 == 20 fifteenths == f = f = 1$. o P; 4 11 Q O -K- J.-5-. 2d OPERATION. Division of Fractions. 157 RULE. Divide the numerator or multiply the denominator by the whole number. EXERCISES. Divide : 2. f by 2. Ans. J. 8 - if by 6. 3. f by 2. Ans. 4. f by 6. Ans. 5. 12 fifths by 4. ^4ws. f. 6. 15 halves by 5. ^4?is. 1|-. 7. ft by 9. Ans. 1 o / 3 8' 4 2T- /; 4 1 /^ j /~v rt 9. ^ by 8. ^Lns. 2|. 10. f by 9. Ans. -fa. 11 1 VT7 19 /J 1} <3 ? 1> TS" D J 1Z - -fins, i 12. ff by 25. Ans. ? 13. 7 ninths by 4. J.TIS. -j 7 g-. 14. If 4 tops cost $f, what will 1 top cost? Ans. %%. 15. If 3 melons cost $f, what will 1 melon cost? Ans. $-f-. 16. If 5 spellers cost S^-J, what will 1 speller cost? Ans. $ T 2 T . CASE II.- -To divide 1 by a fraction. INDUCTIVE EXERCISES. 197. Divide a bar of soap into three equal parts, thus : How often are 2 parts con- ^^^ tained in 3 parts f Ans. f . What stands for 2 parts? What stands for 3 parts? Ans. 1. How often, then, is | contained in 1 ? Ans. f . How often is f contained in 1 ? Ans. |. Why ? Ans. Because f - = 3 fourths, and 1 = 4 fourths, and 4 fourths -j- 3 fourths = ^. How often is f contained in 1 ? Ans. % times, because 5 sevenths is contained in 7 sevenths -J 198. The Reciprocal of a Fraction is the result of in- terchanging the places of its terms. The reciprocal of | is f ; of f, |; of f ; of 4, -J- ; of 2 or J, f ; etc. 158 Intermediate Arithmetic. Inverting a fraction is taking its reciprocal. 199. From the preceding articles we derive the RULE. To find how often a fraction is contained in one, or 1, we take its reciprocal, or invert it. EXERCISES. How often is : 1. f contained in 1? Ans. f times. 2. ^ contained in 1 ? Ans. {% times. 3. f contained in 1 ? Ans. f times. 4. f contained in 1 ? Ans. 2 times. 5. Is | equal to i? Why? Ans. Because each is contained in 1 2 times. 6. Is T \ equal to ^? Why? 7. Is -fa equal to ^? Why? 8. Is 3^ equal to J|? Why? How many : 9. | will it take to make 1 ? Ans. 4. 10. T \ will it take to make 1? Ans. 5. 11. -iV will ^ take to make 1? Ans. 4J. 200. CASE III. To divide a whole number or a frac- tion by a fraction. 1. Divide 5 by f. EXPLANATION. By Case II. f is contained OPERATION. in 1 | times ; hence, in 5 it is contained 5 JL v & 1 5 71 l /^ 2 ~ 2 ' ' '2 times f or *- 7|. 2. Divide f by f. EXPLANATION. By Case II, f is contained OPERATION. in 1 1 times ; hence, in f it is contained f 8vl--H-- \ 01 X 5---ZTF- times or=l. Division of Fractions. 159 Hence, the RULE. Invert the divisor and proceed as in multiplica- tion of fractions. NOTE. This rule is also applicable to Case 1. EXERCISES. Divide : 3. 18 by 2 Ans. 27. 10. H by 2"' Ans. 3. 4. 20 by i- Ans. 80. 11. 3 i by 2^ Ans. 5. 5. 6 by C Ans. 10. 12. 5J by 3 Ans. 7. 6. 15 by 5 8"' Ans. 24. 13. o i by I- Ans. 3. 7. I by f- Ans. 32 2T- 14. 71 i 2 bv *t 4- Ans. ? 8. 1 by 3 Ans. 5 6* 15. 5 i by 3|. Ans. l\ 9. 3 T by 2 Ans. 9 16. 6J by If- Ans. 3 i 17. Divide 13J by 7. 18. Divide 24| by 2f. 19. Divide 15f by 3f. 20. How many f make 7J? 21. How many If make 14|? 22. How many $f make SIOJ? 23. How many Sf make $f ? What will 1 yard of cloth cost : 24. If 4 yards cost 12 cents? 25. If 4 yards cost 12|- cents? 26. If 4|- yards cost 12 cents? 27. If 4| yards cost 12^- cents? 28. If 3J yards cost 13J cents? 29. If 5J yards cost 31|- cento? 30. If 3J- yards cost $| ? 31. If 51 yards cost $f ? 32. If 3| yards cost $2f ? 33. If 5 yards cost 10. Ans. 12. Ans. 2c 4c. Ans. ? -4ns. -4ns. ? Ans. ? 160 Intermediate Arithmetic. WRITTEN EXERCISES. 201. An important class of problems invoking Multiplica- tion and Division of Fractions. (See Art. 142.) * 1. If If yards of cloth cost $lf, what will 2i yards cost? EXPLANATION, Since If yards cost OPERATION. $1|, we divide If by If to get the j _._ j 2 - - n. cost of 1 yard, which gives $ft. Now ^ v 91 ill o ju since 1 yard costs $|f, we multiply 20 A ^ 5 - - 100 - 4 ioTT- f ^ by 2i to get the cost of 2i yards, and obtain v>2jYo> Ans. 2. If 2J- yards of cloth cost $3f, what will If yards cost? Ans. $2-J. 3. If f yard of cloth cost 21 cents, what will 2-f yards cost? Ans. 60 cents. 4. If 2 yards of cloth cost $lf, what will 2J yards cost? Ans. $1^-. 5. If 3^ yards of cloth cost $4f, what will 2J yards cost? Ans. S3. 6. If f yard of cloth cost 3J cents, what will 13 yards cost? Ans. ? How much will : 7. 12 yards of cloth cost if 3|- yards cost 21 cents? Ans. 72 cents. 8. 15^- bushels of corn cost if 3J bushels cost $lf? Ans. $6.51. 9. 7^ gallons of syrup cost if 2J gallons cost Ans. 10. 13^- pounds of beef cost, if ^ pounds cost H c. ? = : '- Before the pupils begin these exercises, let them solve six or eight of the examples under Art. 142. Division of Fractions. 161 11. How much will 25^ acres of land cost, if 2|- acres cost $26 ? Ans. $304. 12. How much will 10} barrels of flour cost, if l^ barrels cost $8f ? . Ans. $86. 13. A farmer bought 6J pounds of nails for 25 c., and desires to get 8J pounds more at the same rate; how much will they cost? Ans. 32 c. 14. A farmer's price for 250^- acres of land is $2505 ; what is his price for 175} acres? Ans. $1757 J-. 202. PARALLEL PROBLEMS. 1. Reduce to lowest terms T 6 , -J-f, 2. Reduce to lowest terms Iff, T VoV Ans. f, -f. 3. m Change f to twelfths ; } to twentieths. 4. Change -JJ to one hundred sixty -eighths. Ans. -}-j-f. 5. What is the number whose seventh is 12? 6. What is the number whose thirty-sixth is 95? Ans. 3420. 7. Six-sevenths of a number is 12 ; what is the number ? 8. ff of a number is 138; what is the number? Ans. 282. 9. What is the weight of a beef, if } of it weighs 600 pounds ? 10. What is the weight of a beef, if ff of it weighs 629 pounds? Ans. 697 pounds. 11. If f of an acre of land cost $12, what will one acre cost? 12. If |f of a lot is valued at $3225, what is the price of the whole lot ? Ans. $3750. 13. If 3 fourths of a pound of sugar cost 9 cents, what will 1 fourth of a pound cost? N. I. 11. 162 Intermediate Arithmetic. 14, If f-f of a load of cotton cost $175, what would of the load cost? Ans. $7. 15. Reduce ^ 8 - to a mixed number. Ans. 31f. 6.m What is the sum of -J- and -^? ^ and ^? 17. What is the sum of -f and ff ? What is the number whose c parts are : 18 m 5 and |? 7 and J? 9f and 3^? 6J and 5|? 19. 18fand36fV? 481f and 196f ? Ans. 55J, 678i|. 20. A merchant has two pieces of prints, one con- tains 23f yards and the other 48f yards ; how many yards in both pieces ? Ans. 72^J- yards. 21. m What is the difference between ^ and ^? 7 and f ? 22. What is the difference between |f and -^J ? Ans. f-^. If one of the parts of: 23. m 8 is 1J, what is the c part? 24. 12 is 7|, what is the c part? Ans. 4|. 25. $8J is $5, what is the c part? 26. A man had $125f and spent $83 J ; how much had he left ? Ans. $42^-. 27. From 134J gallons of water there were drawn off 117f gallons; how many gallons were left? Ans. 16^. 28. If two of the parts of 8 are 2 and 3J, what is the c part? 29. If two of the parts of 165f are 17^ and 88f, what is the c part? Ans. 59^-. 30. Three boys together weigh 210J pounds. The first boy weighs 71| pounds, and the second 69| pounds ; what does the third boy weigh ? Ans. 69j^ pounds. 31.*^ What will be the cost of 9 apples at 2 c. apiece? Of 12 peaches at f c. apiece ? Of 8 gallons of syrup at $| a gallon ? Of 9 bushels of corn at If a bushel ? 32. What will 568 pounds of cotton cost at 9| c. per pound? Ans. 5538 c. Division of Fractions. 163 33. What will 18J pounds of butter cost at 18f c. per pound? Ans. 351^ c. 34. What is the number whose c factors are 5J and 5|? Ans. 27. 35. What will be the cost of 1 yard of cloth if 4 yards cost $|f ? If 4 yards cost $f ? 36. What will be the cost of 1 yard of cloth if 3^- yards cost 25-f c. ? Ans. 7^ c. 37. If a man travels 4 miles in 1 hour, how far will he go in 3| hours? 38. If a horse travels 6^ miles in 1 hour, how far will he go in 5^ hours? Ans. 33J miles. 203. QUESTIONS FOR REVIEW. What is: 1. A fractional unit? 2. A fraction? 3. The terms? 4. The denominator ? 5. The numerator ? 6. A proper fraction ? 7. An improper fraction ? 8. A mixed number ? 9. The value of a fraction? 10. A compound fraction? 11. The reciprocal of a fraction ? How may a fractional unit be : 1. Multiplied by a whole num- ber ? 2. Divided by a whole number ? Why is the value of a fraction not changed by : 1. Multi- plying both terms by the same number ? 2. Dividing both terms by the same number? What is reduction of fractions ? Give the rule for reducing : 1. A fraction to its lowest terms. 2. A whole number to a fraction. 3. A fraction to higher terms. 4. A mixed number to an improper fraction. 5. An improper fraction to a whole or mixed number. 6. Compound fractions to simple ones. 7. Fractions to a com- mon denominator. 8. Fractions to their least common denomin- ator. What is the principle of: 1. Addition? 2. Subtraction? 3. Multiplication? 4. Division? What is the rule for: 1. Addition? 2. Subtraction? 3. Mul- tiplication ? 4. Division ? How do we find how often a fraction is contained in 1 ? In di- vision of fractions, why do we invert the divisor ? DECIMAL FRACTIONS. INDUCTIVE EXERCISES. 204. If an orange, an apple, a number, or a bar of soap be divided into ten equal parts, what is one of the parts called? Ans. 1 tenth. What are two of the parts called? Three of the parts? Four? Five? If, now, each of these tenths be divided into ten equal parts, what is one of the parts called? Ans. 1 hun- dredth. What are two of the parts called? Three of the parts? Four? Five? If, now, each of. these hundredths be divided into ten equal parts, what is one of the parts called ? Ans. \ thousandth. In the number 327, what is the unit of 7 ? Ans. one. See Art. 31. What is the unit of 2? Of 3? Is the unit of 2 one-tenth of the unit of 3? Is the unit of 7 one-tenth of the unit of 2? If, now, we write other figures after 7, thus: 327568, will the unit of 5 be 1 tenth of the unit of 7? Will the unit of 6 be 1 tenth of the unit of 5? Will the unit of 8 be 1 tenth of the unit of 6 ? Placing a point ( . ) after 7, thus : 327.568, indicates that its unit is one j what, then, is the unit of 5 ? Of 6? Of 8? (164) Decimal Fractions. 165 1. What, then, is denoted by 327.568? Ans. 327 and 5 tenths 6 hundredths 8 thousandths. Or, 327 and > + T o + T. 4 3 2 3? Ans. 514 3. 4. Scruples in 12 Ib. 7 dr. ? 4ns. 3477 scr. 5. Pounds, etc., in 99 dr.? Ans. 1 Ib. 3 dr. 6. Ounces, etc., in 167 scr. ? Ans. 6 oz. 7 dr. 2 scr. 7. Drams, etc., in 583 gr. ? Ans. 9 dr. 2 scr. 3 gr. 8. Pounds, etc., in 564307 grains? Ans. ? 9. Grains in 5 Ib. 5 dr. 5 gr. ? Ans. ? MEASURES OP MONEY. 267. I. UNITED STATES MONEY. NOTE. For table and exercises under this head, see United States Money, page 174. II. ENGLISH MONEY. 268. English or Sterling money is the currency of Great Britain. The denominations, or measures, are the pound (.), the shilling (s.), the penny (d.), and the farthing (far.). TABLE. 4 far. = 1 d. 12 d. = 1 s. 20 s. = 1 . NOTE. A florin = 2 s. ; a guinea = 21 s. ; and 1 ==$4.84. Measures of Monet/. 193 MENTAL EXERCISES. 269. How many : 1. Farthings in 3d.? 5 d. ? | d. ? i d. ? 2. Pence in 20 far. ? 30 far. ? 2s.? 6s.? s. ? 3. Shillings in 24 d. ? 72 d.? 3 ? 1 ? ? 4. Which is the lowest denomination? The next? etc. 5. Which is the highest denomination? The next? etc. WRITTEN EXERCISES. 270. Reduce: 1. 5 . 4 s. 10 d. to pence. Ans. 1258 d. 2. 16 s. 5 d. 3 far. to farthings. Ans. 791 far. 3. 7s. 1 far. to farthings. Ans. 337 far. 4. 251 d. to pounds. Ans. 1 . 11 d. 5. 100 far. to shillings. Ans. 2 s. 1 d. 6. 793 s. to pounds. Ans. 39 . 13 s. 7. 10 . 3 d. to farthings. Ans. ? 8. 53675 far. to pounds. Ans. ? 271. 111. FRENCH MONEY. TABLE. 10 milliemes (mi-lame) = 1 centime. 100 centiemes (son-teem') = 1 franc. NOTE. One franc is equal to $.186 U. S. money. MEASURE OF TIME. 272. The Units, or Measures, used in measuring time, are the century (C.), the year (yr.), the month (mo.), the week (wk.), the day (d. or da.), the hour (hr. or h.), the minute (m.), and the second (sec. or s.X N. I. 13. 194 Intermediate Arithmetic. TABLE. 60 s. = 1 m. 60 m. = 1 h. 24 h, = 1 da. 7 da. 1 wk. 365 da. = 1 yr. The Solar Year is exactly 365 da. 5 hr. 48 m. 49.7 sec., or 365^ days nearly. In four years this fraction amounts nearly to one day. To provide for this excess one day is added to the month of February every fourth year, which is called Leap Year (L. yr.). Every year, except those ending with two O's, that is exactly divisible by 4 is a L. yr. ; as 1844, 1856, 1884. Every year ending with two O's that is exactly divisible by 400 is a L. yr. ; as 1600, 2000, 2400. Every year which is not so divisible is a common year ; as 1847, 1855, 1900, 1800. A common year consists of 365 days, a leap year of 366 days, and a century of 100 successive years. The Civil Year is divided into twelve Calendar months, thus : January (Jan.) 1st mo.. .31 da. February (Feb.) 2d mo.. .28 da. March (Mar.) 3d mo... 31 da. April (Apr.) 4th mo. ..30 da. May (May) 5th mo.. .31 da. June (June) 6th mo.. .30 da. July (July) 7th mo. 31 da. August (Aug.) 8th mo. 31 da. September (Sep.) 9th mo. 30 da. October (Oct.) 10th mo. 31 da. November (Nov.) llth mo. 30 da. December (Dec.) 12th mo. 31 da. MENTAL EXERCISES. 273. How many: 1. Seconds in 2 m.? 5m.? | m. ? J m. ? ^m.? 2. Minutes in 180 s. ? 90s.? 3 hr.? J hr. ? hr.? 3. Hours in 120 m. ? 600 m. ? 2 da. ? \ da. ? \ da. ? 4. Days in 48 hr. ? 36 hr. ? 240 hr. ? 2 wk. ? 5 wk.? 5. Is 1824 a common or a leap year? 1838? 1874? 1855? 1900? 1700? 1600? 1950? 2200? 2800? 3000? Circular Measure. 195 WRITTEN EXERCISES. 274. How many: 1. Days in 32 common years? Ans. 11680 da. 2. Days in 32 leap years? Ans. 11712 da. 3. Hours in 5 yr. 120 da. 15 hr. ? Ans. 46695 hr. 4. Hours in 10 L. yr. 106 da, 17 hr. ? Ans. 90401 hr. 5. Minutes in 3 wk. 5 da. 10 hr. 12 m. ? Ans. ? 6. Weeks, etc., in 583 hr. ? Ans. 3 wk. 3 da. 7 hr. 7. Years, etc., in 45375204 m. ? Ans. 86 yr. 120 da. 13 hr. 24 m. 8. Days, etc., in 1000000 sec. ? Ans. ? 9. How many days are in the century beginning with the year 1801 and ending with the year 1900? Ans. 36524 da. SUGGESTION. Multiply 365 da. by 100, and to the product add as many days as there are L. yr. CIRCULAR MEASURE. 275. The measures used in meas- uring angles and the arcs of cir- cles are the circle (cir.), the degree (), the minute ('), and the sec ond CO- TABLE. 60"= = 1'. 60'= 1. 360= 1 cir. EXERCISES. 276. How many : 1. Seconds in 7 15' 25"? 2. Minutes in 1 cir. ? 3. Minutes in 3 cir. 150 15'? CIRCLE. Ans. 26125". Ans. 21600'. Ans. 73815'. 196 Intermediate Arithmetic, 4. Seconds in 16 50"? Ans. 57650". 5. Degrees, etc., in 7453"? An*. 2 4' 13". 6. Circles, etc., in 584375'? Ana. 27 cir. 19 35'. 7. Circles, etc., in 73564807"? Ana. ? 8. Seconds in 10 cir. 5 35' 45"? Ans. ? PAPER MEASURE. 277. The measures used in measuring paper are the bale (b.), the bundle (bun.), the ream (rm.), the quire (qr,), and the sheet (sht). TABLE. 24 sht. = l qr. 20 qr. =1 rm. 2 rm. =1 bun. 5 bun. = 1 b. MISCELLANEOUS TABLE. 12 units = 1 dozen. 12 dozen = 1 gross. 20 units 1 score. 4 inches = 1 hand. 6 feet = 1 fathom. 8 furlongs = 1 mile. See, also, Art. 113. THE OLD FRENCH MEASURE. 278. The old French Linear and Land Measure is still partly used in Louisiana, and in other French settlements of the United States. TABLE. 12 lines = 1 inch. 12 inches = 1 foot. 6 feet = 1 toise. 32 toises = 1 arpent. 1024 sq. toises 1 sq. arpent. The French foot equals 12.79 English inches. The arpent is the old French name for acre, and is equal to about jj- of an English acre. Exercises in Reduction. 197 EXERCISES IN REDUCTION. 279. Reduce: 1. 5 R). 8 oz. 11 pwt. to grains. Ans. 32904 gr. 2. 13 bu. 5 pk. 6 qt. to quarts. Ans. 462 qts. 3. 2 mi. 45 ch. to yards. Ans. 4510 yd. 4. 5 ch. 3 yd. 2 ft. to inches. Ans. 4092 in. 5. 6 yr. 25 da. 6 hr. to minutes. Ans. 3189960 m. 6. 12 L. yr. 18 da. to hours. Ans. 105840 hr. 7. 25 cu. yd. 15 cu. ft. to cubic feet. Ans. 690 cu. ft. 8. 16 sq. rd. 12 sq. yd. 8 sq. ft. to sq. ft. 9. 50 pk. 2 qt. to pints. Ans. 804 pt. 10. 18 bar. 10 gal. 2 qt. 1 pt. to pints. Ans. 4621 pt. 11. 13 25' to seconds. Ans. 48300". 12. 12 . 5 s. 11 d. to pence. Ans. 2951 d. 13. 5 hhd. 15 gal. 1 pt. to gills. Ans. 10564 gi. 14. 15 A. 3 R. 20 P. to poles. Ans. 2540 P. 15. $2, 5 d. 6 c. to cents. Ans. 256c. 16. 7 hunds. 5 tens, 3 ones to ones. Ans. 753 ones. 17. 2 lb. 3 3, 4 3, 2 9 to scruples. Ans. 662 . 18. $43.75 to mills. Ans. 47750 m. 19. 5 t. 7 cwt. 74 Ib. to pounds. Ans. 10774 Ib. 20. 7 b. 1 bun. 1 rm. to quires. Ans. 1460 qr. Reduce : 21. 3779 in. to rods. Ans. 18 rd. 5 yd. 2 ft. 11 in. 22. 12500 m. to days. Ans. 8 d. 16 h. 2 m. 23. 4392 P. to acres. Ans. 27 A. 1 R. 32 P. 24. 24352 far. to ., etc. Ans. 25 . 7 s. 4 d. 25. 47643 cu. in. to cu. yds Ann. I cu. yd. 987 cu. in. 26. 1075 gi. to gallons. AIM. 33 gal. 2 qt. 3 gi. 27. 953 9 to pounds. Ans. 3 Ib. 3 , 5 3, 2 9. 28. 895 pt. to bushels. Ans. 13 bu. 3 pk. 7 qt. 1 pt. 29. 8433 qrs. to reams. Ans. 421 rm. 13 qr. 30. 24563 sq. in. to sq. yds. 198 Intermediate Arithmetic. COMPOUND ADDITION. 280. i. Add together 5 yd. 2 ft. 9 in. ; 6 yd. 1 ft. 7 in. and 4 yd. 2 ft. 4 in. Ans. 17 yd. 8 in. EXPLANATION. Since only like numbers can be OPERATION. added, we write inches under inches, feet under yd. ft. in. feet, etc. Adding the column of inches, we get 529 20 in., which, divided by 12, gives 1 ft. 8 in. Set Q ^ the 8 in. under the column of in., and carry the 1 ft. to the column of ft. ; adding this column, we . get 6 ft., which equals 2 yd. and ft. ; writing 17 under the column of ft., and carrying 2 to the col- umn of yd., we have 17 yds. Hence, the RULE. I. Write the numbers to be added so that those of the same unit may be in the same column. II. Add each column, beginning at the right, divide the sum by the number of units of the column added which equals one of the next higher, set the remainder under that column, and carry the quotient to be added to the next. 2. What is the sum of 9 . 16 s. 8 d., and 10 . 12 s. 7 d. ? Ans. 20 . 9 s. 3 d. 3. What is the sum of 7 . 13 s. 6 d., 2 . 17 s. 9 d., 3 . 8 s. 3 d., 9 . 11 s. 8 d. ? Ans. 23 . 11 s. 2 d. 4. What is the sum of 4 bu. 3 pk. 1 qt., 7 bu. 2 pk. 3 qt., 1 bu. 1 pk. 7 qt., and 8 bu.? Ans. 21 bu. 3 pk. 3 qt. 5. What is the sum of 5 Ib. 7 oz. 10 dr., 7 Ib. 11 oz., 8 dr., 12 Ib. 5 dr., 13 Ib., 3 Ib. 6 oz. 3 dr? Ana. 41 Ib. 9 oz. 10 dr. COMPOUND SUBTRACTION. 281. 1. From 7 Ib. 5 oz. 9 pwt. 7 gr. take 3 Ib. 4 oz. 12 pwt. 4 gr. Ans. 4 Ib. 17 pwt. 3 gr. Compound Subtraction. 199 EXPLANATION. - Since only like numbers can be .subtracted, we write gr. under gr.. pwt. , I)-. oz. pwt. gr. under pwt.. etc. I" ~ Q y Beginning at the right, we subtract 4 gr. from 7 gr. and get 3 gr., which we write ^ under the column of gr. 4 17 3 Since 12 pwt. is larger than 9 pwt., we take 1 oz. from the 5 oz., leaving 4 oz., and add it, or 20 pwt., to 9 pwt., making 2!) pwt. 12 pwt. from 29 pwt. leave 17 pwt. ; which we write under the pwt. Since 1 oz. was taken from 5 oz., we subtract 4 oz. from 4 oz. and get oz., which we write under oz. 3 Ib. from 7 Ib. leaves 4 Ib., which we write under Ib. Hence (see Art. 87), the RULE. I. Write the less number under the greater so that those of the same kind shall be in the same column. II. Begin at the right and subtract each term from the one above it, if the latter is the greater, and place the dif- ference under the numbers subtracted. III. // any term is greater than the one above it, add to the one above the number of units of that column which equals one of the next higher, from the sum subtract the lower term, ivrite the remainder below, and carry one to the next term to be subtracted, and so on with all the columns. 2. From 45 A. 2 R. 17 P. take 19 A. 3 R. 36 P. Ans. 25 A. 2 R. 21 P. 3. From 65 cu. yd. 20 cu. ft. 1252 cu. in. take 55 cu. yd. 26 cu. ft. 956 cu. in. An*. 9 cu. yd. 21 cu. ft. 296 cu. in. 4. From 85 bu. 2 pk. take 45 bu. 1 pk. 6 qt. Ans. 40 bu. 2 qt. 5. From 5 yr. take 3 yr. 9 mo. Ans. 1 yr. 3 mo. 6. From 12 Ib. 3 3, 1 3, take 5 Ib. 7 .5, 5 3, 2 B. Ans. 6 Ib. 7 3, 3 5, 1 200 Intermediate Arithmetic. COMPOUND MULTIPLICATION 282. Multiply 3 5 s. 9 d. by 7. Ans. 23 3 d. EXPLANATION. We write the multiplier under OPEBATION. the term on the right; multiply each term as > Si d> in simple numbers, setting down and carrying as 59 in compound addition. Thus, 7 X 9 d. - - 63 d., which divided by 12 gives 5 s. 3 d. Write the 3 below, and carry 5s. 7 X 5 s. - - 35 s. and 5 s. = 23 3 40 s., which divided by 20 gives 2 s. Write the below and carry 2 . 7 X 3 == 21 and 2 == 23 . Hence, the RULE. Multiply each term of the multiplicand, beginning at the right, by the multiplier ; divide the product by the number of units of the term multiplied which equals one of the next higher; set the remainder under that term, and carry the quotient to be added to the next product. 2. Multiply 22 A. 3 R. 35 P. by 6. Ans. 137 A. 3 R. 10 P. 3. Multiply 3 Ib. 4 oz. dr. 2 scr. by 4. Ans. 13 Ib. 4 oz. 2 dr. 2 scr. 4. Multiply 13 bu. 2 pk. 1 pt. by 15. Ans. 202 bu. 2 pk. 7 qt. 1 pt. 5. Multiply 5 Ib. 3 oz. 13 dr. by 7. Ans. 36 Ib. 10 oz. 11 dr. 6. How many bushels in 9 bins, each containing 120 bu. 3 pk. 3 qt.? An*. 1087 bu. 2 pk. 3 qt. COMPOUND DIVISION. 283. 1. Divide 44 bu. 3 pk. 3 qt. by 7. Ans. 6 bu. 1 qt. 5 pk. Parallel Problems. 201 EXPLANATION. Dividing 44 bu. by 7 we OPERATION. get bu. and 2 bn. over. Write the below, 1)U pk t reduce the 2 bu. to pk., and to it add 3 pk., 7^44 which gives 11 pk. 11 pk. -f-7 - = l pk. and 4 fi. 1 f pk. over. Write the 1 below, reduce the 4 pk. to qt., and to it add 3 qt, making 35 qt., which divided by 7 gives 5 qt. ; write the 5 below. Hence, the RULE. I. Place the divisor on the left of the dividend, and divide the left-hand term by it, writing the quotient under that term. II. Reduce the remainder, if any, to the next lower unit, adding in like terms of the dividend, if any, and divide the sum by the divisor ; and so on, for the other terms. 2. Divide 24 Ib. 7 oz. 8 pwt. by 2. Ans. 12 Ib. 3 oz. 14 pwt. 3. Divide 10 hhd. 46 gal. 1 qt. 1 pt. by 7. Ans. 1 hhd. 33 gal. 2 qt. 1 pt. 4. Divide 11 6 s. 3 d. by 5. Ans. 2 5 s. 3 d. 5. Divide 36 Ib. 10 oz. 9 dr. by 5. Ans. ? 284. PARALLEL PROBLEMS. l. What will 4^ quarts of plums cost at 6 cents a quart? At 3 cents a pint? At 5 cents a pint? 2. What will 5 hogsheads of molasses cost at 37 cents a gallon? Ans. $116.55' 3. m If 1 bushel of potatoes cost 80 cents, what is the price of 1 peck? 4. If 12 bushels of potatoes cost $6.48, what will be the value of 1 peck? Ans. 13^ c. 5. A boy picked 3 quarts of cherries, and sold them at the rate of 5 cents a pint ; how much did he receive ? 6. A farmer gathered 24 bu. 3 pk. of peaches, and 202 Intermediate Arithmetic. sold them at the rate of 3 cents a quart; how much did he receive? Ans. $23.76. 7. m A boy bought half a bushel of chestnuts for 30 c., and sold them at 20 c. a peck; how much did he make? 8. A man bought 1 seventh of a hogshead of sugar for $12.60, and retailed it at 20 cents a pint ; how much did he make? Ans. $1.80. 9. How many gills are there in f of a gallon? 10. How many pints are there in -}J of a hogshead? Ans. 399 pt. ll. m How many hours in f of a day? In T 7 ^ of a day? 12. How many minutes in f of a day? Ans. 1200 mi. 13. How far will a man travel in 2 hours, if he goes 5 chains in 1 minute? 14. How far will a man travel in 3 weeks at the rate of 5 miles an hour? Ans. 2520 mi. 15. What will f of a ream of paper cost at 20 cents a quire? At 30 cents a quire? 16. What will | of a bale of paper cost at 1^ cents a sheet? Ans. $28.80. I7. m How long will it take a man to travel 100 miles if he goes 5 miles an hour? 18. How long would it take a horse, traveling at the rate of 8 miles an hour, to go around the' world, a distance of 25000 miles? Ans. 130 da. 5 hr. 19. How long would it take a locomotive to go from the earth to the moon, a distance of 240000 miles, trav- eling at the rate of 25 miles an hour ? Ans. 1 yr. 35 da. 20. "i How many times will a wheel 8 feet in circum- ference turn over in going 120 yards? 21. How often will a wheel 12 feet in circumference turn over in going 5 miles? Ans. 2200. 22. m If a buggy, whose wheels are 12 feet in circum- Parallel Problems. 203 ference, goes 400 yards in 10 minutes, how often do the wheels turn over in 1 minute? 23. If a locomotive, whose wheels are 15 feet in cir- cumference, runs at the rate of 45 miles an hour, at what rate do the wheels revolve per minute? Am. 264. 24. m What will it cost to build a fence 2 miles long at SI a chain ? 25. What will 36 miles of telegraph wire cost at 75 cents a rod ? Ans. $4320. 26.f" If a lad makes 5 steps in walking a rod, how many steps will he make in going 3 chains? 27. If 20 rails are required to build a fence 1 rod long, how many rails will it take to inclose a field ^ of a mile long and ^ of a mile wide? Ans. 2400 rails. 285. QUESTIONS FOR REVIEW. What is a: 1. Simple number? 2. Compound number? State what each of the following measures are used for, name the meas- ures, and repeat the Table : 1. Linear Measure; 2. Square Meas- ure ; 3. Solid or Cubic Measure ; 4. Liquid Measure ; 5. Dry Meas- ure ; 6. Troy Weight ; 7. A voirdupois Weight ; 8. Apothecaries' Weight; 9. English Money; 10. Measure of Time; 11. Circular Measure ; 12. Paper Measure. What is the rule for: 1. Reduction Descending? 2. Reduction Ascending ? How many: 1. Cubic inches in a gallon? 2. Cubic inches in a bushel ? 3. Dollars in 1 ? 4. Cents in 1 franc ? 5. Days in a leap-year ? 6. Acres in an arpent ? Which are the: 1. Leap-years? 2. Common years? Name the number of days in each month. IMPORTANT APPLICATIONS. 286. CASE I. To find the time between two dates. 1. What length of time elapsed from July 10, 1843, to Jan. 4, 1845 ? EXPLANATION. "Write the latter or greater date for the minuend, and the earlier for the OPERATION. subtrahend, giving the month its number in- yr mo da stead of the name. Thus, since Jan. is the 1845 1 4 1st month, we write 1 under mo. and under 1843 7 JQ it write 7, as July is the 7th month. Now subtract as in compound subtraction, allow- 5 ing 12 mo. to the yr. and 30 da. to the mo. Find the time from : 2. May 12, 1848, to June 1, 1860. Ans. 12 yr. 19 da. 3. June 1, 1861, to Oct. 20, 1872. Ans. 11 yr. 4 mo. 19 da. 4. June 15, 1846, to Jan. 10, 1848. Ans. 1 yr. 6 mo. 25 da. 5. April 20, 1868, to Aug. 1, 1869. Ans. 1 yr. 3 mo. 11 da. 6. Henry was born March 9, 1868, and Harry Sept. 15, 1875 ; how much older is Henry than Harry ? Ans. 7 yr. 6 mo. 6 da. 7. A note dated July 15, 1879, was paid May 21, 1882; how long did it run? Ans. 2 yr. 10 mo. 6 da. To what age did the following live : 8. Washington, born Feb. 22, 1732; died Dec. 14, 1799? (201) Important Applications. 205 9. Jefferson, born April 2, 1743; died July 4, 1S20? 10. Lincoln, born Feb. 12, 1S09; died April' 15, 18G5? 11. Calhoun, born March 18, 17*2; died March 31, 1850? 12. Webster, born Jan. 18, 1782; died Oct. 24, 1852? 13. Clay, born April 12, 1777; died June 29, 1852? 287. CASE II. To find the area of rectangular sur- faces. A rectangular surface is a surface in the shape of a sheet of paper, or of an ordinary square cornered gar- den of four sides. The figure represents a rectangular surface 5 in. long and 3 in. wide, and evidently contains 3x5= 15 square inches. 15 sq. in. is called its area. Hence, the RULE. Multiply the length by the width, expressed in like units. 1. What is the area of a floor 18 ft. long and 12 ft. wide? Ans. 216 sq. ft., or 24 sq. yd. 2. What is the area of a rectangular garden 88 yd. long and 55 yd. wide? Ans. 4840 sq. yd., or 1 A. 3. What is the area of a rectangular field 35 ch. long and 24 ch. wide? Ans. 840 sq. ch., or 84 A. 4. How many acres in a meadow 17 ch. long and 12 ch. wide? Ans. 20.4 A. 5. How many acres in a field 60 rd. long and 44 rd. wide? Ans. 16 A. 6. How many yards of carpeting 3 ft. wide will it take to cover a floor 20 ft. long and 15 ft. wide ? 206 In ter mediate Arithmetic. We divide the number of sq. ft., 300, OPERATION. by the width of the carpet, 3 ft., which 20 X 15 = 300 gives the length of the carpet, 100 ft., 300 -=-3 100 and this divided by 3 gives 33 yd. 1QQ ft. -r- 3 = 33J yd. 7. How many yards of carpeting 4 ft. wide will it take to cover a floor 22 ft. long and 18 ft. wide? Ans. 33 yd. 288, CASE III. To find the volume of a rectangular - solid. A rectangular solid is a solid in the shape of an or- dinary goods-box, or of a room. The figure represents a rectan- gular solid 5 ft. long, 4 ft. high, 3 ft. wide, and evidently contains 5 x 4 X 3 = 60 cu. ft. 60 cu. ft. is called its volume. Hence, RULE. Multiply the length, width, and height together. NOTE. The dimensions must be expressed in terms of the same measure. 1. What is the volume of a rectangular box 6 ft. long, 3 ft. wide, and 5 ft. high ? Ans. 90 cu. ft. 2. What is the volume of a room 20 ft. long, 18 ft. wide, and 12 ft. high? Ans. 4320 cu. ft., or 160 cu. yd. 3. How many cubic feet in a marble slab 60 in. long, 24 in. wide, and 6 in. thick ? Ans. 5 cu. ft. 4. How many cubic yards of coal in a pile 18 ft. long. 10 ft. wide, and 7 ft. high ? Atis. 46| cu. yd. 5. How many cubic feet of corn in a crib 20 ft. long, and 12 ft. wide, the corn being 5^ feet deep in the crib ? Ans. 1320 cu. ft, Important Applications. 207 6. How many cords of wood in a pile 20 ft, long, 12 ft. wide, and 6 ft. high? Ans. Hi cords. OPERATION. * OX 2X6 ^ How many cords in a pile of wood : 7. 24 ft. long, 8 ft. wide, and 4 ft. high ? Ans. 6 c. 8. 18 ft. long, 6 ft. wide, and 8 ft. high ? Ans. 6f c. 289. CASE IV.-- To find the number of gallons in a rectangular box or tank. 1. How many gallons of water will a tank hold which is 6 ft. long, 5 ft. wide, and 8 ft. high? OPERATION. 6X5X8X7^ = 1800 gal. Ans. EXPLANATION. Since there are 231 cu. in. in one gallon, and 1728 cu. in. in one cubic foot, 1 cu. ft.=-y^ 2 T 8 - gal. or about 7% gal. Hence, the RULE. Multiply the number of cu. ft. by 7J. 2. How man}'- gallons of molasses will a box hold which is 4 ft. long, 2 ft. wide, and 9 in. (f ft.) high? Ans. 45 gal. 3. A box is 3 ft. long, 2 ft. wide, 8 in. (f ft.) high, and is full of honey ; how much honey is in the box ? Ans. 30 gal. What is the capacity of a tank which is : 4. 6 ft. long, 4 ft. wide, and 10 ft. high ? Ans. 1800 gal. 5. 4 ft. long, 4 ft. wide, and 9 ft. high? Ans. 1080 gal. 6. 5 ft. square at the bottom and 7 ft. high ? Ans. 1312J gal. 208 Intermediate Arithmetic. 7. 3 ft. square at the bottom and 6 ft. high? Ans. 405 gal. How much milk will a box hold which is : 8. 1| ft. long, 1-J- ft. wide, and J ft. high? Ans. 7-J- gal. 9. If ft. long, 1 ft. wide, and 6 in. high? Ans. 7\ gal. 10. 9 in. long, 8 in. wide, and 4 in. deep? Ans. 1J gal. 290. CASE V. To find the number of bushels in a rectangular box, bin, or granary. 1. How many bushels of wheat will a box hold which is 5 ft. long, 4 feet wide, and 3 ft. deep? OPERATION. 5X4X3X.8 = 48.0 = 48 bu. EXPLANATION. Since there are 2150.4 cu. in. in one bushel, and 1728 cu. in. in one cubic foot, then 1 cu. ft. = 2T^?bu. about .8 bu. Hence, the RULE. Multiply the number of cu. ft. by .8. How many bushels will a bin hold : 2. Which is 6 ft. long, 5 ft. wide, and 2| ft. high? Ans. 60 bu. 3. Which is 6 ft. long, 4| ft. wide, and 2| ft deep? Ans. 57f bu. 4. Which is 5-J ft. long, 3f ft. wide, and 2 ft. deep? Ans. 40 bu. 5. Which is 7J- ft. long, 2-J feet wide, and 3 ft. deep? Ans. 53^- bu. 291. CASE VI. To find the number of board feet in boards or planks. A Board Foot is used in measuring boards, planks, and sawed timber generally. It is 1 foot long, 1 foot wide, and 1 inch thick, and contains 144 cu. in. Important Applications. 209 l. How many board feet in a plank 13 ft. long, 18 in. wide, and 2 in. thick? Am. 39 B. ft. OPERATION. 13 X 18 X 2 12 Hence, the = 39. .-- Multiply the length in feet by the width and thickness expressed in inches, and divide the product by 12. NOTE. If a board or plank is less than 1 in. thick, it is dis- regarded; that is, the calculation is made as if it were 1 in. thick. 2. What is the number of feet in a board 15 ft. long, 9 in. wide, and ^ in. thick. Ans. 11 J B. ft. 3. How many feet in 12 planks, each 7 ft. long, 8 in. wide, and 1|- in. thick? Ans. 84 B. ft. 4. How many feet in 9 boards, each 11 ft. long, 6 in. wide, and f in. thick? Ans. 49 \ B. ft. 5. How many feet in 36 plank, each 10 ft. long, 11 in. wide, and 1^ in. thick? Ans. 440 B. ft. 292. PARALLEL PROBLEMS. l.m Albert was born Feb. 17, 1868, and Robert Feb. 17, 1875; how much older is Albert than Robert? 2. The Brooklyn Suspension Bridge was commenced Jan. 3, 1870, and was opened for travel July 4, 1882 ; how long was it in building? Ans. 12 yr. 6 m. 1 da. 3. m How many acres in a field 8 ch. long and 5 ch. wide ? 4. How many acres in a field 45 rd. long, and 38 rd. wide? Ans. 10 A. 2 R. 30 P. N. L 14. 210 Intermediate Arithmetic. 5. m If carpeting is 4 ft. wide, how many yards will it take to cover a stage 12 ft. wide, and 5J ft. deep? 6. If carpeting is 5 ft. 6 in. wide, how many yards will be required to cover a floor 16^- ft. long, and 15^ ft. wide? Ans. 15 J yd. 7. m How many marble slabs, 3 ft. long and 2 ft. wide, are necessary to pave a walk 150 ft. long, and 4 ft. wide? 8. How many bricks, 8 in. long and 4 in. wide, are required to construct a pavement 285 ft. long, and 7 ft. 4 in wide? Ans. 9405. 9. What will it cost to cut a ditch 60 ft. long, 3 ft. wide, and 2 ft. deep, at $J per cu. yd? 10. A street is 600 ft. long, and 55 ft. wide; \vhat will be the cost of elevating it 3 ft., at 60c. per cu. yd.? Ans. $2200. 11. "i A tin box 3 ft. long, 2 ft. wide, and 1 ft, high, is full of honey; what is the honey worth at 50c. per gallon ? 12. A cistern 6 ft. square at the bottom, and 10 ft. high, is 2 thirds full of water ; how long will it supply a family that uses 45 gallons of water per day ? 13. m A man sells corn at $^ per bushel ; what should be his price for a box of corn 5 ft long, 3 ft. wide, and 2 ft. deep? 14. Corn weighs 56 pounds per bushel; what is the weight of the corn in a granary 9 ft. long, 5 ft. wide, and 6 ft. high, the granary being 3 fourths full ? 15. If lumber is worth $1-J- per hundred board feet, what is the value of 20 planks, each 10 ft. long, 6 in. wide, and 1 in. thick? 16. If lumber is worth $18.50 per thousand board feet, what is the valne of 36 plank, each 16 ft. long, 8 in. wide, 1 in. thick? Ans. $7.104. p ERCENTAGE 293. The term per cent means by the hundred. Thus : 3 per cent means 3 hundredths, or yf-Q, or .03; 5 per cent means 5 hundredths, or yf-g-, or .05 ; 13 per cent means 13 hundredths, or j 1 ^, or .13. 294. The process of calculating by hundredths is called Percentage. INDUCTIVE EXERCISES. 295. 1. A man had 100 sheep, but lost 7 of them ; what part of his sheep did he lose? Ans. yJo-, or 7 per cent. 2. A man had 50 sheep, but lost 3 of them ; what part of his sheep did he lose? Ans. fo = jfa j or 6 per cent. 3. A man had 25 hogs, but lost 7 of them ; what part of his hogs did he lose? Ans. 7 7 T = T 2 A = 28 per cent. 1 O i v v -L 4. A man had $200, and gained $13 more ; what part of his money was his gain ? Ans. -fo = -f^ - - 6^ per cent. 5. What per cent of a number is ^ of it? Ans. ^ = y^ = = 50 per cent. 6. What per cent of a number is : -i of it? Ans. 331 i of it ? Ans. 25. i of it ? Ans. 20. 1 of it? | of it? i of it ? i of it ? T V of it ? A of it? 2*0 of it? A of it? ^r of it? (211) 212 Intermediate Arithmetic. What fractional part of a number is 20 per cent of it? A 20 7. What fractional part of a number is : 25 per cent of it? Ans. ^. 50 per cent of it? Ans. \. 75 per cent of it? Ans. f. per cent of it ? 100 per cent of it? 12^ per cent of it? DEFINITIONS. 296, Rate is a given allowance. 297, When rate is expressed by the hundred, the number of hundredths taken or allowed is the Rate Per Cent. 298, Percentage is the result obtained by taking a cer- tain per cent of any given number. 299. The Base is the number on which the percentage is computed. Per cent is indicated by %. Thus, 6% is read : 6 per cent. 300. CASE I. The Base and Rate % given to find the Percentage. EXERCISES. 301. 1. What is 6% of 150 sheep? ANALYSIS. 6% of 150 sheep is T f ^ of 150 sheep, which is 9 sheep. Hence, the RULE. Multiply the base by the rate expressed decimally. How much is : 2. 2 of450bu. ? Ans. 9bu. 3. 15 of 240 c.? Am. 36 c. 4. Sty of$120?^7is. 810.20. 5. 20% of $185.70? Ans. ? 6. 50% of $124.37? Ans. ? 7. 331^ of 183 lb.? Ans. ? Percenfaf/e. 213 How many dollars are made by selling : 8. A cow which cost $18, so as to gain 20% ? Ans. $3.60. 9. A horse which cost $135, so as to gain 15^ ? Ans. $20.25. 10. A wagon which cost $84, at an advance of 12-J-^ ? Ans. 310J. How many dollars are lost by selling : 11. A lot which cost $2345, so as to lose 9% ? Ans. $211.05. 12. A carriage which cost $245, so as to lose 8% ? Ans. $19.60. 13. A saddle which cost $14, at a discount of 1\% ? Ans. ? 302. CASE II. --The Base and Percentage given to find the Rate %. EXERCISES. 303. 1. What per cent of 16 is 4? ANALYSIS. 4 is T \ of 16. -& = \ = -fifa = 25 per cent. Hence, the RULE. Multiply the number which is the percentage by 100, and divide the product by the base. What per cent of: 2. 500 is 40? Ans. 3. 825 is 50? Ans. 4. $637 is $38.22? ,4ns. 6%. 5. $300 is $37.50? Ans.12^%. 6. 7. 40 is 2 16s.? Ans. 8. 50 gal. is 2 gal. 3 qt.lAns. ? 9. 20yd. is 5 ft. 10 in.? Ans.? 10. A man bought a cow for $40, and sold her so as to gain $10; what per cent did he make? Ans. 15%. 11. A man bought a cow for $50, and sold her at a loss of $10; what per cent did he lose? Ans. 20%. 214 Intermediate Arithmetic. COMMISSION. 304. Commission is a sum of money paid to an agt for buying or selling goods or other property. It is a percentage of the amount invested or collected. MENTAL EXERCISES. 305. How much does an agent get for his services who collects : 1. $500, and charges 2 per cent? Ans. $10. 2. $650, and charges 4 per cent? Ans. ? 3. $200, and charges 2^ per cent? Ans. ? 4. $325, and charges 8 per cent? Ans. ? How much does an agent receive who sells goods to the amount of: 5. $450, at 2% commission? Ans. 6. $125, at 8% commission? Ans. ? 7. $50, at 3J% commission? Ans. ? WRITTEN EXERCISES. 306. 1. A commission merchant sold wheat for $1900; what was his commission at 2f per cent? Ans. $52.25. 2. A commission merchant invested $1250 for another party ; what was his commision at If per cent ? Ans. $20. What does a commission merchant receive for his services who sells : 3. 5 horses at $150 each, at 2^ per cent commission? Ans. $18.75. Profit and Loss. 215 4. 12 bales of cotton, averaging 450 pounds to the bale, at 8 cents a pound, commission being If per cent ? Ans. $7.20. 5. hlids. of sugar, averaging 1450 pounds to the lihd., at 5 cents a pound, commission being If per cent? An*. $7.61}, 6. 120 bu. apples @ $1.50, 020 Ib. butter @ $.25, 1250 Ib. bacon @ 14 c., commission being \\% ? Ans. $8.92^-. PROFIT AND Loss. 307. Profit and Loss denote the gain or loss in busi- ness transactions. They are computed by percentage. 308. The cost is the base; the per cent of gain or loss, the rate ; the gain or loss, the percentage. In case of a gain, the selling price is the amount, and in case of a loss, the selling price is the difference. 309. PARALLEL PROBLEMS. l . A man bought a gun for $8 and sold it for $10 ; what per cent did he make? Ans. NOTE. The gain was $10 $8 =$2, and the cost $8, and by Art. 303, $2 are 25$, of $8. What per cent does a man make who buys : 2.m A saddle for $10, and sells it for $12? Ans. 3. A book for $42, and sells it for $63? Ans. 4.m A table for $3, and sells it for $6? Ans. ? 5. A cow for $120, and sells her for $220? Ans. 8 6.m A horse for $60, and sells him for $90? Ana. ? 216 Intermediate Arithmetic. 7. A man bought a coat for $10, and sold it for what per cent did he lose? Ans. 20^. NOTE. The loss was $10 $8 ==$2, and the cost $10, and by Art. 303, $2 are 20% of $10. What per cent does a man lose who buys : 8.m A saddle for $12, and sells it for $10? Ans. 9. A secretary for $62, and sells it for $40? 10. A table for $6, and sells it for $3? Ans. 50%. 11. A cow for $25, and sells her for $23? Ans. 8 ft. 12. A horse for $80, and sells him for $60? Ans. ? 13. A horse for $80.50, and sells him for $62? Ans. ? 14. A farm for $320, and sells it for $280? Ans. Vl What will a merchant receive for an article which : 15. m Cost $10, if sold so as to gain 20% ? Ans. $12. 16. Cost $122, if sold at a profit of 25% ? Ans. $152.50. 17. Cost $24, if sold at a gain of 5% ? Ans. ? 18. Cost $282, if sold at an advance of 15% ? Ans. ? 19. Cost $8, if sold so as to lose 50% ? Ans. $4. 20. Cost $53, if sold at a loss of 22% ? Ans. $41.34. 21. m Cost $25, if sold at a loss of 12 Jg ? Ans. ? 22. Cost $182, if sold at a discount of Yl\ ? Ans. ? 310. QUESTIONS FOR REVIEW. What is: 1. Rate? 2. Rate per cent? 3. Percentage? 4. The Base? How is per cent indicated ? How do we find : 1. The percentage, when the base and rate