I$mmmmmmmmmmimmmmmim m i m u i ii iiii iii i i i iii Mm iiiit iiymn M wiww iiiwtimi^w wB^^ AN INTRODUCTORY ARITHMETIC wiw i ia i iwiimiin; SENSENIG^ARDERSOH jBft^Buiiirr ^^fim iso DEI No. ^ H 5[|)c Sinocnig-^nberson Scrica of ^IritlimEticB mTRODUCTORY ARITHMETIC BY DAYID M. SE^N^SEKIG, M.S. KOBEET F. ANDERSOJS^, A.M. INSTRUCTORS IN MATHEMATICS, STATE NORMAL SCHOOL, WEST CHESTER, PENNSYLVANIA SILVER, BURDETT AKD COMPAlSrY NEW YORK BOSTON CHICAGO aAio3 S4- Ete Srngenifi'^ntiergon Series of arithmetics By DAVID M. SENSENIG, M.S., and ROBERT F. ANDERSON, A.M., Instructors in Mathematics, State Normal School, "West Chester, Pennsylvania. AN INTRODUCTORY ARITHMETIC. A first book in arithmetic, inductive in method and carefully graded, developing the subjects by natural steps. It gives the pupil an intelligent understanding of processes and abundant practice in operations. 262 pp. ESSENTIALS OF ARITHMETIC. A compre- hensive and practical book for grammar grades, giving thorough instruction and drill in fundamental processes and much information that is useful in business transac- tions. The treatment of mensuration leads up to algebra and geometry. 343 pp. 60 cents. THE NEW COMPLETE ARITHMETIC for High Schools, Academies, and Normal Schools. A prac- tical text-book in which, besides the usual topics, special attention is given to business papers, with photographic reproductions of business forms, to property and life assurance, and mensuration. 437 pp. 90 cents. SILVER, BURDETT AND COMPANY New York Boston Chicago eOUCATION DCTf I Copyrighi, 1903, By Silver, Burdett and Company
1
How many goats and 6 goats are 13 goats ?
How many rabbits and 5 rabbits are 13 rabbits ?
How many owls and 7 owls are 14 i ##^#####«0 i
owls ? 14
llow many bats and 5 bats are 14 bats ?
How many minks and G minks are 14 minks ?
18 WHOLE NUMBERS AND FRACTIONAL PARTS.
How many flags and 6 flags are 15
flags ?
15
IIow many hoops and 8 hoops are 15 hoops ?
How many wheels and 8 wheels arc
10 wheels ?
16
How many spokes and 7 spokes are IG spokes ?
How many trout and 8 trout are 17 | —
trout ?
How many frogs and 7 frogs are 17 frogs
17
How many shingles and 8 shingles r— -
are 18 shingles ?
How many nails and 9 nails are 18 nails ?
How many wheels and 9 wheels are i
18
19 wheels ?
19
26. Exercise.
Copy, writing the proper figures in place of dots :
. + 9 = 10
► + 9 = 12
. + 8 = 14
. + 5 = 10
. + 6 = 12
. + 9 = 15
. + 8 = 10
+ 8=12
. + 8 = 15
. + 6 = 10
+ 7 = 12
+ 9 = 16
. + 7 = 10
+ 9 = 13
+ 8= 16
. + 9 = 11
+ 7 = 13
• + 9 = 17
. + 6 = 11
+ 8 = 13
+ 9 = 18
. +8 = 11
. + 9 = 14
+ 10 = 18
. + 7 = 11
+ 7 = 14
+ 9 = 19
SUBTRACTION.
19
27. Oral Exercise.
Bead Exercise 26, supplying tlie missing numbers.
28. Oral Exercise.
Read, supplying the missing numbers :
10 horses less 1 horse are — horses.
10 cows less 6 cows are — cows.
10 calves less 8 calves are — calves.
10 mules less 2 mules are — mules.
10 sheep less 7 sheep are — sheep.
10 goats less 4 goats are — goats.
10 pigs less 9 pigs are — pigs.
10 birds less 5 birds are — birds.
10 geese less 3 geese are — geese.
11 ducks less 2 ducks are — ducks.
11 carts less 5 carts are — carts.
11 coats less 8 coats are — coats.
11 vests less 3 vests are — vests.
11 hats less 6 hats are — hats.
11 caps less 4 caps are — caps.
11 collars less 7 collars are — collars.
11 buttons less 9 buttons are — buttons,
12 pins less 3 pins are — pins.
12 needles less 6 needles are — needles.
12 skates less 8 skates are — skates.
12 plows less 5 plows are — plows.
12 wheels less 4 wheels are — wheels.
12 rings less 7 rings are — rings.
12 cuffs less 9 cuffs are — cuffs.
• ••••
• ••••
• •
20 WHOLE NUMBERS AND FRACTIONAL PARTS.
13 matches less 4 matches are — matches.
13 lamps less 6 lamps are — lamps.
13 cars less 8 cars are — cars.
13 boys less 5 boys are — boys.
13 girls less 7 girls are — girls.
13 women less 9 women are — women
14 dogs less 5 dogs are — dogs.
14 cats less 8 cats are — cats.
14 mice less 6 mice are — mice.
14 birds less 9 birds are — birds.
14 flowers less 7 flowers are — flowers,
15 houses less 6 houses are — houses.
15 boards less 8 boards are — boards.
15 shingles less 9 shingles are — shingles.
15 bricks less 7 bricks are — bricks.
16 slates less 7 slates are — slates.
16 pencils less 9 pencils are — pencils.
16 buds less 8 buds are — buds.
17 twigs less 8 twigs are — twigs.
17 trees less 9 trees are — trees.
18 shrubs less 9 shrubs are — shrubs.
• ••••
• ••••
• ••
• ••••
• ••••
• ••••
• ••••
• ••••
• ••••
• ••
•••
• ••••
• ••••
• ••
• •••
29. Exercise.
Copy, writing the proper figures in place of dots :
10 - 1 = . 10 - 7 = • 11 - 9 =
10-9=- 10-4=. 11-3 =
10-2=. 10-6=. 11-8 =
10-8=. 10-5=. 11-4 =
10-3=. 11-2=. 11-7 =
ADDITION
21
11-5 =
13 - 9 =
15 - 6 = •
11 - 6 = <
13 - 5 =
15 - 9 = «
12-3= ^
13 - 8 = <
15 - 7 = .
12 - 9 = .
13 - 6 = .
15 - 8 = '
12-4= .
13 - 7 =
16 — 7= .
12 - 8 = .
14 - 5 = .
16 - 9 = «
12 - 5 = .
14 - 9 =
16 - 8 = .
12-7= .
14 - 6 = .
17-8= .
12-6 =
14 - 8 = «
17-9= .
13 - 4 = .
14 - 7 = •
18 - 9 = .
30. Oral Exercise.
Eead Exercise 29, supplying the missing numbers.
31. If we count 3 to 4, we have 7.
Counting 3 to 4 is called adding 3 to 4.
32. Oral Exercise.
2 4 13
9
To each of the numbers above the line:
/. Add 1. Thus, 2 and 1 are 3, 4 and 1 are 5, and so on.
2. Add 2. 4. Add 4. 6. Add 6. 8. Add 8.
3. Add 3. 5. Add 5. 7. Add 7. 9. Add 9.
33. Oral Exercise.
To the teacher. — Teach Roman notation to XIX. Teach United
States coin to one dollar.
/. Charles had 2 goldfish and his uncle gave him 3
more ; how many had he then ?
22 WHOLE NUMBERS AND FRACTIONAL PARTS.
2. Sara found 3 eggs in one nest and 4 in another ; how
many did she find in all ?
3. Mr. Rice has 2 sons and 4 daughters ; how many
children has he ?
4. A farmer had 4 sheep and bought 5 more ; how
many had he then ?
5. James is 2 years older than Mary ; if Mary is 6
years old, how old is James ?
6. How many lambs had a farmer at first, if he had 5
left after selling 6 ?
7. Jane spelled 6 words correctly and missed 4 ; how
many was she given to spell ?
8. How many horses has a man, if he keeps 3 to drive
and 6 to work ?
9. Harry has walked 4 miles toward home and has 7
miles yet to go ; how far from home was he when he
started ?
10. How many animals did a blacksmith shoe to-day, if
he shod 7 horses and 5 mules ?
//. How many boys were there at first in a class, if 7 re-
mained after 3 were promoted ?
12. How many policemen must be on duty so that when
4 go home 8 remain ?
10 cents make 1 dime.
13. 1 cent and how many cents make a dime ?
14. A dime is how much more than a 5-cent piece ?
15. A dime and a cent make how much ?
16. A dime and a 5-cent piece make how much ?
ADDITION AND SUBTRACTION. 23
17. A dime, a cent, and a 5-cent piece make how
much ?
18. A dime is worth how many 5-cent pieces ?
19. A cent and a 5-cent piece are together worth how
much less than a dime ?
34. Exercise.
To the teacher. — Let the pupil solve problems in Exercises orally
when possible.
/. There are 3 persons in a boat that will carry 8 per-
sons ; how many more may get in ?
2. I found 6 eggs ; how many more must I get to have
12 ?
7 days mahe 1 week.
3. Name the days of the week.
4. Name the first day of the week ; the third ; the
second ; the fourth ; the sixth ; the fifth ; the seventh.
5. Last week father worked 4 days ; how many days
was he idle ?
6. A man owns 6 horses ; how many more must he buy
to have 14 ?
7. Samuel has spent 2 days with his cousin ; how long
has he yet to stay, if his visit is to last a week ?
8. Mother put up 17 cans of fruit to-day ; 8 of them
were pears and the others peaches ; how many were
peaches ?
9. I had to write 16 words ; I have written 7 of them ;
how many have I yet to write ?
24 WHOLE NUMBERS AND FRACTIONAL PARTS.
10. My uncle wishes to have 12 cows ; he now has 5 ;
how many more must he get ?
11. K boy paid 10 cents for a loaf of bread and a pound
of sugar. If the sugar cost 6 cents, what did the loaf of
bread cost ?
12. How many days of the week are left after Wednes-
day has passed ?
35. If we take 3 from 7, we have 4.
Taking 3 from 7 is called subtracting 3 from 7.
36. Oral Exercise.
Subtract :
/. 2 from
2 4 3 O 9 5 8 10 7 11
Thus, 2 from 2 leaves 0, 2 from 4 leaves 2, and so on,
2. 3 from
3 6 4 9 5 8 12 lO 7 11
3. 4 from
7 5 6 4 12 8 11 9 13 10
4. 5 from
8 5 O 12 14 11 9 13 lO 7
6. 6 from
8 10 6 9 12 7 14 11 13 15
6. 7 from
8 lO 9 7 12 14 11 16 13 15
7. 8 from
10 12 14 16 8 9 11 13 16 17
8. 9 from
11 13 9 12 16 10 14 17 15 18
ADDITION AND SUBTRACTION. 25
37. Oral Exercise.
/. Mary saw 8 stars, but a cloud now covers 3 of them ;
how many can she still see ?
2. There were 7 houses in a row, but a fire burned 3 of
them ; how many were then left ?
3. A gardener planted 9 trees, but 4 of them died ; how
many lived ?
4. A farmer had 11 colts, but sold 5 of them ; how
many had he left ?
5. 9 boys were examined ; if 5 of them failed, how
many passed ?
6. Amos had 10 problems to solve ; if he has solved 3,
how many has he yet to solve ?
7. There were 12 women in a parlor, but 4 of them
went away ; how many remained ?
8. A woman bought 12 eggs, and cooked 6 of them for
supper ; how many had she left ?
12 is called a dozen (doz.).
9. A man bought a dozen oranges and ate 3 of them ;
how many had he left ?
10. A lady bought a dozen ears of corn and cooked 8 of
them for dinner ; how many had she left ?
//. James planted 15 ferns, but only 9 of them lived ;
how many died ?
12. A boy had 10 firecrackers and shot off 6 of them ;
how many had he then left ?
26 WHOLE NUMBERS AND FRACTIONAL PARTS.
33. Oral Exercise.
/. A farmer had 9 lambs ; if he sold all but 2, how
many did he sell ?
2. A newsboy bought 15 morning papers and sold all
but 6 ; how many did he sell ?
3. Jessie bought a pound of sugar and gave the store-
keeper 10 cents. He gave her 4 cents change ; how much
did he charge for the sugar ?
4. There are 4 collars left in a box that had contained
a dozen ; how many were taken out ?
5. 14 persons came into the dining room for dinner ;
there is room for only 9 ; how many must wait ?
6. Frank has 6 cents left out of 14 cents that his father
gave him ; he spent the rest for a tablet ; how much did
it cost ?
7. Chester had 13 marbles, but he lost all but 4 ; how
many did he lose ^
8. How many shad did a man sell, if he had a dozen
and sold all but 3 ?
9. Last spring I planted 12 shade trees ; they all died
but 4 ; how many died ?
10. K boy had 11 pigeons and sold all but 4 ; how many
did he sell ?
//. Frank had a dozen plums ; he kept 4 for his sister
and ate the others ; how many did lie eat ?
12. I have 5 cents yet to get in order to have 14 cents ;
how much money have I ?
ADDITION AND SUBTRACTION. 27
39. Oral Exercise.
/. 6 pupils are absent from a class and 9 are present ;
how many belong to this class ?
2. A man was sent to prison 5 years ago and he has yet
7, years to serve ; how many years must he serve in all ?
3. There were 12 knives on a table, but only 9 forks ;
how many more knives than forks were there ?
4. A farmer had 8 sheep left after selling 7 ; how many
had he at first ?
5. There were 16 tramps in a workhouse, but 7 of them
were set free ; how many were held ?
6. How many girls were in a class at first, if 7 remained
after 6 were promoted ?
7. How many cents must I put in my money bank to
have 14 cents in it, if it already contains 9 cents ?
8. I have put 4 cans of corn into a box that will hold
a dozen cans ; how many more will it hold ?
9. There are 14 sheep and lambs in a field ; how many
of them are lambs, if there are 6 sheep ?
10. How many men must my father hire to have 14, if
he already has 7 ?
//. After a fire had burned 8 stores in a block, there
were 9 left ; how many were there before the fire ?
12. A girl bought a pound of sugar for 6 cents and re-
ceived 4 cents change ; how much money did she give the
grocer ?
28 WHOLE NUMBERS AND FRACTIONAL PARTS.
Forming, Writing, and Reading Numbers from 20
to lOO; Addition and Subtraction.
40. Two tens make twenty , written 20.
Twenty
Two tens and one make twenty-one, written 21.
Twenty-one
Two tens and two make tiventy-two, written 22.
Twenty-two
Two tens and three make twenty-three, writ-
ten 23.
ij
Twenty-three
1!"
Twenty and four make twenty-four, written 24.
Twenty-four
Twenty and five make twenty -five, written 25.
Twenty-five
Twenty and six make twenty-six, written 26/
Twenty and seven make tiventy-seven,
written 27.
Twenty-six
TTll
Twenty-seven
NUMBERS FROM 20 TO 100.
29
Twenty and eight make twenty -eight,
written 28. ^ ^
Twenty-eight
Twenty and nine make twenty -nine, ffll ffl 1 1 1 1 j 1 1
written 29. li 11 1 I I 1 1 1 J
Twenty-nine
41. Oral Exercise.
Beginning at 1, count to 29.
Read these numbers : 20, 21, 22, 23, 24, 25, 26, 27, 28, 29.
Tell the number of tens and ones in each of the above
numbers.
How many ones are there in 2 tens and 4 ? 2 tens and 6 ?
Three Tens
42. Three tens make thirty, written 30.
Three tens and one make thirty-one, writ-
ten 31.
43. Oral Exercise.
Three tens and two make what number ?
Three tens and three make what number ?
Three tens and four make what number ?
Three tens and five make what number ?
Three tens and six make what number ?
Three tens and seven make what number ?
Three tens and eight make what number ?
Three tens and nine make what number ?
Thirty-one
30 WHOLE NUMBERS AND FRACTIONAL PARTS.
Beginning at 30, count to 30.
Read these numbers : 30, 31, 32, 33, 34, 35, 36, 37, 38, 39.
Tell the number of tens and ones in each of the above
numbers.
How many ones are there in 3 tens and 1 ? 3 tens and 6 ?
Four Teng
1?^
44. Four tens imxkQ forty, written 40.
-1
i
Forty
45. Oral Exercise.
Four tens and one make what number ?
Beginning at forty, name the numbers that are formed
by forty and one ; forty and two ; forty and three ; forty
and four ; forty and five ; forty and six ; forty and seven;
forty and eight ; forty and nine.
Read : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49.
Beginning at forty, count to forty-nine.
46. Five tens make Jifii/, written 50.
Six tens make sixty, written 60.
Seven tens make seventy, written 70.
Eight tens make eighty, writ-
ten 80.
Five Tens
itj
ui
Fifty
Six Tena
Eighty
NUMBERS FROM 20 TO 100.
31
Nine Tens
Nine tens make ninety, written
90.
Ninety
Ten Tens
Ten tens combined make one hundred,
written 100.
One Hundred
47. Oral Exercise.
/. Name the numbers from fifty to sixty.
2. Name the numbers from sixty to seventy.
3. Name the numbers from seventy to eighty.
4. Name the numbers from eighty to ninety.
5. Name the numbers from ninety to one hundred.
6. Pvead : 50, 51, 52, 53, 54, 55, 56, 57, 58, 59.
7. Read : 60, 61, 62, 63, 64, 65, 66, 67, 68, 69.
8. Read : 70, 71, 72, 73, 74, 75, 76, 77, 78, 79.
9. Real! : 80, 81, 82, 83, 84, 85, 86, 87, 88, 89.
10. Read : 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.
48. Exercise.
Write in fiofures the numbers :
/. From one to ten.
2. From ten to twenty.
3. From twenty to thirty.
4. From thirty to forty.
5. From forty to fifty.
6. From fifty to sixty.
7. From sixty to seventy.
8. From seventy to eighty.
9. From eighty to ninety.
10. From ninety to one hun-
dred.
5.
32
9.
51
13.
44
17.
72
21. 91
6.
34
10.
57
14.
46
18.
70
22. 19
7.
36
11.
20
15.
64
19.
15
23. 92
8.
39
12.
33
16.
69
20.
88
24. 87
32 WHOLE NUMBERS AND FRACTIONAL PARTS.
49. Oral Exercise.
Read :
/. 22
2. 25
5. 28
4. 30
50. Oral Exercise.
/. Count from 10 to 20.
2. Count backwards from 20 to 10.
3. Count from 20 to 30.
4. Count backwards from 30 to 20.
5. Count from 30 to. 40.
6. Count backwards from 50 to 40.
7. Count from 50 to 60.
8. Count backwards from 70 to 60.
9. Count from 70 to 80.
10. Count backwards from 90 to 80.
51. In numbers of two figures, the right-hand
figure is called the Units' figure, and the left-hand
figure the Tens' figure.
52. Oral Exercise.
/. In 87 name the tens' figure ; the units' figure.
2. In 63 how many tens are there, and how many
units ?
3. In 90 how many tens are there, and how many units ?
4. How many figures are needed to write a number
mailc uj) of 3 tens and 2 units ?
ADDITION AND SUBTRACTION. 33
5. Name the number made up of 6 tens and 2 units.
6. What is the largest number that can be written with
one figure ? With two figures ?
7. What two numbers can be written with the figures
8 and 9 ?
To the teacher. Teach Roman notation to C. See p. 79.
53. Oral Exercise.
/. Add each of the numbers 2, 5, 3, 6, 4, 7, 9, 8, 10 to
10 20 30 40 50 60 70 80 90
Name results only. Thus, 12, 15, 13, and so on ; then 22, 25,
23, and so on ; etc.
2. Add each of the numbers 2, 5, 3, 6, 4, 8, 7 to
11 21 31 41 51 61 71 81 91
3. Add each of the numbers 2, 5, 3, 6, 4, 7 to
12 22 32 42 52 62 72 82 92
4. Add each of the numbers 2, 5, 3, G, 4 to
13 23 33 43 53 63 73 83 93
5. Add each of the numbers 2, 5, 4, 3 to
14 24 34 44 54 64 74 84 V\
6. Add each of the numbers 2, 4, 3 to
15 25 35 45 5^ 65 75 85 95
7. Add each of the numbers 3 and 2 to
16 26 36 46 56 66 76 86 96
8. Add 2 to
17 27 37 47 57 67 77 87 97
8
34 WHOLE NUMBERS AND FRACTIONAL PARTS.
64. Oral Exercise.
Note. The sign ? denotes how many?
/. 74.3_|_4_5=? Answer thus : 7, 10, 14, 0.
2. 3+4 + 2=? 7. 5 + 4 + G + 3=?
3. 7+5 + 6=? 8. 7 + 4 + 3-5=?
4. 2 + 5-1 = ? 9. 9-4 + 6-7=?
5. 14 - 9 + 7 = ? /(?. 14 - 9 + 8 - 6 = ?
ff. 18 - 9 - 4 = ? //. 18 - 9 - 3 - 4 = ?
55. Exercise.
/. I bought a tablet for 4 cents, a pencil for 5 cents,
and a sponge for 6 cents ; how much did I pay for all ?
2. There are 4 chairs at each side of a table and 1 at
each end ; how many are there in all ?
3. A bicycle, a cart, and a wagon have, in all, how
many wheels ?
4. From a box of 1 dozen collars I sold 6 to one man
and 3 to another ; how many remain ?
5. A man owed 12 dollars. lie paid 4 dollars at one
time and 3 dollars at another ; how much does he still owe ?
6. A boy added three numbers, and the answer was 10.
If two of the numbers were 3 and 5, what was the third ?
7. 2 sheep, 4 cows, 2 horses, and 2 mules make how
many head of stock ?
8. A laborer had 5 dollars left of his month's wages
after paying 4 dollars house rent and 9 dollars store bill ;
what were his wages ?
9. Out of a dozen eggs mother used 2 for breakfast, 4
for dinner, and 4 for supper ; liow many were left ?
ADDITION AND SUBTRACTION. 35
10. 6 women, 5 girls, 3 men, and 2 boys are how many
persons ?
//.I have 4 brothers and 3 sisters ; how many children
are there in our family ?
12. Mother sent me to the store for a 5-cent spool of
thread, a 4-cent loaf of bread, and a 2-cent box of matches.
How much change should I bring back to her out of a
dime and a 5-cent piece ?
66. Oral Exercise.
/. Add each of the numbers 2, 5, 3, 6, 4, 8, 7, 9 to
19 29 39 49 59 69 79 89
Name results only. Thus, 31, 24, 23, and so on ; then 31, 34,
33, and so on ; etc.
2. Add each of the numbers 2, 5, 3, 0, 4, 8, 7, 9 to
18 38 58 28 48 68 88 78
3. Add each of the numbers 5, 3, 6, 4, 8, 7, 9 to
17 37 57 27 47 67 87 77
4. Add each of the numbers 5, 8, 6, 9, 7, 4 to
16 36 56 26 46 . 66 86 76
5. Add each of the numbers 5, 8, G, 7, 9 to
15 35 55 25 45 65 85 75
6. Add each of the numbers G, 8, 7, 9 to
14 34 54 24 44 64 84 74
7. Add each of the numbers 8, 7, 9 to
13 33 53 23 43 63 83 73
8. Add each of the numbers 8 and 9 to
12 32 52 22 42 62 82 72
9. Add 9 to
11 31 51 21 41 61 81 71
36 WHOLE NUMBERS AND FRACTIONAL PARTS.
Forming, Writing, and Reading Numbers from 1 GO
to lOOO; Addition and Subtraction.
57. Two hundred is written 200 ; three hundred,
300 ; nine hundred, 900.
58. One hundred and one make one hundred one,
written 101.
One hundred and two make one hundred two,
written 102.
One hundred and ten make one hundred ten,
written 110.
Two hundred and forty make two hundred forty,
written 240.
Six hundred and seventy-two make six hundred
seventy-two, written 672.
Nine hundred and ninety-nine make nine hundred
ninety-nine, written 999.
59. In numbers of three figures the first from the
right is called the Units' figure, the second the Tens*
figure, and the third the Hundreds* figure.
60. Exercise.
Write in figures :
/. Four hundred ; seven hundred.
2. Five liundred ; eight hundred.
5. Six hundred ; nine hundred.
4. One hundred three ; one liundred eleven.
5. Two liundred twenty ; three hundred ninety-nine.
6. Four hundred four ; five hundred sixty-eight.
NUMBERS FROM 100 TO 1000.
37
7. Six hundred sixteen ; seven hundred seventy-seven.
8. Nine hundred ninety-nine ; four hundred forty-four.
61. Oral Exercise.
Read :
/. 103.
8. 213.
15.
900.
22.
527.
29.
149.
2. 112.
9. 400.
16.
901.
23.
752.
30.
222.
3. 126.
10. 407.
17.
910.
24.
257.
31.
806.
4. 129.
//. 510.
18.
888.
25.
600.
32.
419.
5. 131.
12. 111.
19.
808.
26.
999.
33.
344.
6. 209.
13. 222.
20.
919.
27.
909.
34.
707.
7. 211.
/4. 876.
21.
764.
28.
526.
35.
818.
62. Oral Exercise.
/. Count from 100 to 110.
2. Count from 330 to 340.
3. Count from 489 to 500.
4. Count from 789 to 911.
63. Oral Exercise.
/. In 765 how many units ?
hundreds ?
2. In 860 how many units ?
hundreds ?
3. In 409 how many units ?
hundreds ?
5. Count backwards from
210 to 200.
6. Count backwards from
370 to 360.
7. Count backwards from
910 to 890.
8. Count backwards from
701 to 689.
how many tens ? how many
how many tens ? how many
how many tens ? how many
38 WHOLE NUMBERS AND FRACTIONAL PARTS.
4. llow iiijuiy figuroK iiiu; ii(3(3dod to write u number made
14) of 3 hundreds, 2 tens, and 5 units ?
5. How many figures are needed to write a number made
up of 4 hundreds and 2 tens ?
6. How many figures are needed to write a number made
up of 2 hundreds and 3 units ?
7. Name tlie number made up of 2 hundreds and 2 tens.
8. What is the largest number that can be written with
three figures ? What is the smallest ?
To the teacher. — Teach Roman notation to M. See p. 79.
64. The process of taking two or more numbers
together is Addition.
65. The answer in addition is called the Sum.
66. Add 427 and 532.
2 units and 7 units are 9 units.
427
3 tens and 2 tens are 5 tens.
- — 5 hundreds and 4 hundreds are 9 hundreds.
Sum 959 „
The sum is 9o9.
Tn practice^ say 2 and 7 are 9 ; S and 2 are 5 ; 5 and ^ are 9.
67. Exercise.
Copy and add :
/. 28 3. 43 5. 50 7. 25
JU 36^ 32 43
21
2. 52 4. 05 6. 43 8. 10
^ 34 j^ 52
31
ADDITION AND SUBTRACTION. 39
9. 80
13. 235
17. 222
21. 222
11
723
333
333
7
444
10. 53
14. 327
18. 423
22. 708
23
652
322
260
13
31
//. 42
15. 326
/5. 321
23. 421
35
362
432
243
22
146
324
12. 14
/ff. 342
20. 321
24. 423
32
635
372
332
23
305
143
68. The process of taking one number from another
is Subtraction.
69. The answer in subtraction is called the Differ-
ence.
70. Subtract 252 from 594.
2 units from 4 units leaves 2 units.
Minuend 594 5 tens from 9 tens leaves 4 tens.
Subtrahend 252 2 hundreds from 5 hundreds leaves 3 hun-
Difference 342 ^reds.
The difference is 342.
In practice, say 2 from 4 haves 2 ; 5 from 9 leaves J^ ; 2 from 5
leaves 3.
40 WHOLE NUMBERS AND FRACTIONAL PARTS.
71. Exercise.
Copy and subtract :
/. 57
7. 38
13. 456
19. 704
23
17
234
203
2. 8G
8. 26
U. 397
20. 420
52
4
175
310
3. 65
9. 48
/5. 658
21. 517
32
33
423
306
4. 79
10. 99
16. 965
22. 427
_45
66
532
120
5. 59
//. 87
n. 878
25. 575
16
75
532
475
6. 97
12. 74
/S. 757
24. 379
6
33
243
375
72. Add 76 and 38.
8 units and 6 units are 14 units, or 1 ten and 4 units.
76 Write 4 in units* place and add 1 to the tens. 1 ten,
38 3 tens, and 7 tens are 11 tens, or 1 hundred and 1
114 ten. Write 1 in tens' place and 1 in hundreds' place.
The sum is 114.
In 'practice, say 8 and 6 are 14; 1, 3 and 7 are 11.
73 Exercise.
Copy and add :
/. 43
2. 53
3. 78
4. 73
48
27
51
36
ADDITION AND SUBTRACTION. 41
5. 48
/3. 89
2/. 82
29. 93
85
97
20
27
19
34
55
6. 79
14. 98
22. 58
30. 75
66
43
67
89
38
32
56
7. 37
/5. 39
23. 27
31. 185
87
93
96
237
14
8. 54
16. 37
24. 89
52. 476
98
99
9
238
17
9. 76
17. 54
25. 37
35. 567
59
77
29
196
41
23
10. 27
/5. 87
26. 17
34. 438
98
16
29
299
51
93
//. 89
19. 48
27. 99
35. 347
7
37
9
468
55
83
20
f2. 58
20. 98
28. 54
36. 298
^
32
35
386
29
27
58
42 WHOLE NUMBERS AND FRACTIONAL PARTS.
37. 275
40. 74
43. 127
46. 175
378
186
346
428
319
416
38. G54
41. 237
44. 486
47. 209
196
179
201
127
302
348
39. 289
42. 37
45. 174
45. 498
69
864
297
126
488
299
74. Subtract 37 from 65.
7 units cannot be taken from 5 units. Take 1 ten
from the G tens, leaving 5 tens. Add this 1 ten,
Mm. 65 . , . . ^ . ^ .
o 1. oi*y or 10 units, to 5 units, making 15 units. 7 units
^.^ — from 15 units leaves 8 units. 8 tens from 5 tens
Diff. 28
leaves 2 tens.
The difference is 28.
In practice, say 7 from 15 leaves 8; 3 from 5 leaves 2.
76. Exercise.
Copy and subtract :
/. 37
5. 56
9. 70
13. 27
19
89
18
18
2. 42
6. 74
10. 62
14. 43
29
36
9
19
3. 45
7. 40
//. 53
n. 67
26
29
18
19
4. 81
8. 91
12. 61
16. 46
38
82
14
4
ADDITION AND SUBTRACTION. 43
17. 52
23. 34
29.
70
35. 436
28
15
27
187
18. 70
24. 90
30.
52
36. 491
36
44
36
238
19. 32
25. 21
31.
186
37. 572
27
9
28
387
20. 50
2^. 41
32.
192
38. 500
41
29
49
283
21. 93
27. 56
33.
228
39. 406
84
17
182
308
22. 77
28. 92
34.
374
40. 702
58
13
182
184
76. Exercise.
Find the missing numbers :
/. 8+9+6:rro 3. 2 + 3+ 4 =
16 + 32 4- 27 = o 20 + 30 4- 40 =
24 + 41 + 44 = . 200 + 300 + 400 =
+ • =
2 4- 3 4- 4
4. Add down ; subtract across.
2. 11 4- 27 + 38 = . . 186 - 27 = •
9 4- 18 4- 60 = . 220 - 185 = .
36 4- 7 4- 8 =^ 407 - 200 =^
• 4~ •4" • -— • • — • — — •
44 WHOLE NUMBERS AND FRACTIONAL PARTS.
77. Exercise.
/. There are 30 days in June, 31 days in July, and 31
days in August. How many days are there in these 3
months ?
30 is the number of days in June.
31 " " *' " " " July.
31 '* " ** " '' '' August.
T " '' ** *' " " the 3 months.
2. A drover sold in one day 36 cows, 42 hogs, and 21
sheep ; how many head of stock did he sell ?
5. A woman sold a dealer 28 pounds of poultry, 16
pounds of butter, and 36 pounds of lard ; how many
pounds of produce did she sell ?
4. A laborer worked 26 days in January, 23 days in
February, and 25 days in March ; how many days did he
work during these months ?
5. I spent at a grocer's to-day 36 cents for coffee, 24 cents
for sugar, and 78 cents for meat ; how much did I spend
in all ?
Note. Ct. stands for cent or cents.
36 ct. was the amount spent for coffee.
24 ct. " '* '* " " sugar.
78 ct. " '' " " " meat.
? '' '' '' " " all.
e. A clerk had $25 of liis month's salary left nftor pay-
ing a $20 store bill and $16 for house rent ; wliul was liis
month's salary ?
ADDITION AND SUBTRACTION. 45
7. After 28 yards were sold from a piece of bunting,
there remained 32 yards ; how many yards were there in
the piece at first ?
8. A farmer had 36 bushels of wheat and used 19
bushels of it for seed ; how many bushels had he left ?
36 is the number of bushels the fanner had.
19 " *' " " " he used for seed.
T '' " " " '' " had left.
9. A girl invited 40 of her friends to a picnic, but 12
of them failed to come ; how many came ?
10. There are 88 houses on a certain street ; if 28 of
them are on one side of the street, how many are on the
other side ?
If. A huckster had 80 shad ; if he sold all of them but
18, how many did he sell ?
12. A field of 68 acres was divided into two parts ; if
the first part has in it 49 acres, what is the size of the
second part ?
68 acres is the size of tlie field divided,
49 " " " " " " first part.
? " '' " " " " second part.
13. How much older is my father than I am, if his age
is 70 years and mine 24 years ?
14. I sold 32 yards from a piece of cheese cloth contain-
ing 60 yards ; how much remained ?
15. A dealer bought 3 pieces of printed lawn containing
50 yards, 15 yards, and 36 yards ; how much did he buy
in all ?
46 WHOLE NUMBERS AND FRACTIONAL PARTS.
16. K flagpole 86 feet long had 37 feet broken from the
top ; how long was the piece that was left standing ?
17. A. contractor has employed 2G Italians, 14 Hun-
garians, and 22 Polanders ; how many laborers has he
employed ?
18. A boy had 18 papers left after selling 24 morning
papers and 21 evening papers ; how many had he to
begin with ?
19. 52 girls were given a problem ; if 19 of them had
the wrong answer, how* many solved the problem cor-
rectly ?
20. A drover sold a horse for 95 dollars ; after he had
paid a debt of 27 dollars out of this sum, how much had
he left ? I
21. K boy weighed 101 pounds and has since gained 19
pounds ; how much does he now weigh ?
22. There were 28 men, 7G women, 62 boys and 78
girls at a Sunday-school picnic ; how many persons were
there ?
23. There are 68 apple trees in an orchard and 18 more
pear trees than apple trees ; how many trees are there
of both kinds ?
24. A man's wages are 42 dollars a month and his son's
18 dollars less ; how mucli do both earn in a month ?
25. A drover had 264 sheep and bought 187 more ; how
many had he then ?
26. Find the sum of three hundred fifty, two hundred '
twenty-five, and one hundred eighty.
MULTIPLICATION, DIVISION, MENSURATION. 47
27. A has 275 dollars and B 189 dollars ; how much
would each have if A were to give B 186 dollars ?
28. I owed 278 dollars, but paid him 129 dollars ;
how much do I still owe him ?
29. A miller had 736 bushels of wheat but sold 478
bushels of it, and later bought 127 bushels ; how many
bushels had he then ?
30. Add six hundred eighty-nine and two hundred six-
teen, and take four hundred four from the sum.
Multiplication, Division, and Mensuration.
78. Oral ExercisCo
Count by 2's :
/. From 2 to 20 ; from 20 to 2.
Thus, 2, 4, 6, etc. ; then 20, 18, 16, etc.
2. From 1 to 21 ; from 21 to 1.
Count by 3's :
5. From 3 to 30 ; from 30 to 3.
4. From 1 to 31 ; from 31 to 1.
5. From 2 to 32 ; from 32 to 2.
79. Oral Exercise.
How many are 1 and 1 ?
1 and 1 are how many times 1? #+• = ••
How many are 2 times 1 ?
What are the two parts of 2 ?
48 WHOLE NUMBERS AND FRACTIONAL PARTS.
What are the two parts of 3? •# + • = •••
llow many are 2 and 2 ?
2 and 2 are how many times 2? ••-{-•# = ###•
JIow many are 2 times 2 ?
What are the two parts of 4 here shown ?
Are these parts alike ?
When the parts of a number are alike they are said to
he equal.
Wliat is one of the two equal parts of 2? •• = #-4-#
How many ones are there in 2 ?
What is one of the two equal •••• = ##-f-##
parts of 4 ?
How many twos (2's) are there in 4 ?
How many are 2 times 3 ?
• • •
What is one of the two equal parts of 6 ? • • •
Each of the ttvo equal parts of a 7iumber is called one
half of it.
One half is written •}.
What is one half of 6 ?
llow many 3's are there in 6 ?
llow many are 2 times 4 ? • • • •
AVhat is one half of 8 ?• • • • #
How many 4's are there in 8 ?
How many are 2 times 5 ? • • • • •
What is i of 10 ? • # • • •
How many 5's are there in 10 ?
MULTIPLICATION, DIVISION, MENSURATION. 49
;Io\v many are 2 times 6? ••••••
What is i of 12 ? ••••••
g_ How many 6^s are there in 12 ?
How many are 2 times 7? •••••••
What is ^ of 14 ? •••••••
How many 7's are there in 14 ?
How many are 2 times 8? ••••••••
What is i of 16? ••••••••
How many 8's are there is 10 ?
How many are 2 times 9? •••••••••
What is i of 18? •••••••••
, How many 9's are there in 18 ?
80. Oral Exercise.
How many are 3 times 1 .''
i What is one of the tliree equal parts of 3 ?
Bach of the three equal parts of a fiumber is called
I one third of it.
One third is written ■^.
[ What is i of 3 ?
How many I's are there in 3 ?
How many are 3 times 2 ? • •
What is i of 6 ? J J
How many 2's are there in 6 ?
4
60 WHOLE NUMBERS AND FRACTIONAL PARTS.
IIow many are 3 times 3 ? • • •
What is i of ? • • •
How many 3's are there in 9 ?
How many are 3 times 4 ? • • • •
What is i of 12 ? * • • • •
# • • •
How many 4's are there in 12 ?
How many are 3 times 5 ?
What is i of 15 ?
How many 5's are there in 15 ?
How many are 3 times G ?
What is i of 18 ?
How many G's are there in 18 ?
How many are 3 times 7 ?
What is ^ of 21 ?
How many 7's are tliere in 21 ?
How many are 3 times 8 ?
What is i of 24 ?
How many 8's are there in 24 ?
How many are 3 times 9 ?
What is 4 of 27 ?
How many 9'8 are there in 27 ?
MULTIPLICATION, DIVISION, MENSURATION. 51
81. The sign x is read times.
82. Exercise.
Copy, supplying the missing numbers :
1 + 1 =
2x1 =
2 + 2 =
2x2 =
3 + 3 =
2x3 =
4 + 4 =
2x4 =
5 + 5 =
2x5 =
G + G -
2 x G =
7 + 7
2x7
8 + 8
2x8
9 + 9
2x9
83. Exercise.
Copy, supplying the missing numbers, and learn
2x1 =
2x2 =
2x3 =
TABLE.
2x4 =
2x5 =
2 X G =
2x7 =
2x8 =
2x9 =
84. Exercise.
Copy, supplying the missing numbers
2x1+1=
2x4 + 4 =
2x7 + 7= .
3x1 =
3x4=.
3x7=.
2x2+2=
2x5 + 5= «
2x8 + 8= .
3x2 =
3x5 =
3x8=.
2x3 + 3= .
2 X G + 6 = .
2x9+9=.
3x3=.
3x6=.
3x9=.
62 WHOLE NUMBERS AND FRACTIONAL PARTS.
85. Exercise.
Copy, supplying the missing numbers, and learn :
TABLE.
3x1=. 3x4=. 3x7=.
3x2=. 3x5=. 3x8=.
3x3=. 3x6=. 3x9=.
86. Exercise.
Copy, supplying the missing numbers :
/.
2
X
2= .
19.
. X
1 = 2
37.
3
X
. =9
2.
3
X
8= .
20.
2 X
.= 10
38.
3
X
. =27
3.
2
X
7= .
21.
3 X
7= .
39.
•
X
4= 12
4.
3
X
3= .
22.
. X
1 = 3
40.
•
X
2=4
6.
2
X
.= 6
23.
2 X
1 = .
41.
3
X
. =3
6.
2
X
. = 4
24.
3 X
.= 21
42.
•
X
6= 18
7.
.
X
9 = 27
25.
2 X
4= .
43.
2
X
. = IG
8.
3
X
5 = .
26.
. X
9 = 18
44.
•
X
3 = 9
9.
•
X
8 = 24
27.
2 X
3= .
45.
3
X
. = 15
10.
•
X
2 = 4
28.
2 X
6 = .
46.
.
X
7 = 21
11.
3
X
6 = .
29.
• X
5 = 10
47.
•
X
5 = 15
12.
2
X
.= 8
30.
2 X
.= 14
48.
3
X
2= .
13.
2
X
5 = .
31.
3 X
.= 24
49.
2
X
9= .
14.
2
X
.= 2
32.
2 X
8= .
50.
3
X
• = 12
15.
2
X
.= 18
33.
3 X
9= .
51.
•
X
7 = 14
16.
2
X
.= 12
34.
. X
1 = 2
62.
3
X
1 = .
17.
3
X
.= 6
35.
. X
G = 12
53.
3
X
. = 18
18.
3
X
4= .
36.
• X
2 = 6
64.
•
X
8 = 16
MULTIPLICATION, DIVISION, MENSURATION. 63
87. Oral Exercise.
/. What do 2 ])osUil cards cost ?
2. What do 2 two-cent stamps cost ?
1 pint.
1 quart.
There are tivo pints (pt.) m a quart (qt.).
3. How many pints are there in 2 qnarts ?
4. How many pints are there in 3 quarts ?
5. What do 3 one-cent stamps cost ?
6. How many feet have 2 horses ?
7. How many horses are needed to make 2 six-horse
teams ?
8r How many horseshoes will it take to shoe 3 horses
all around ?
Measure a yard-stick with a foot-rule to find that :
There are three feet (ft.) in a yard {yd.).
9. How many feet are there in 2 yards ?
70. How much fare will two persons pay, if each pays
5 cents ?
//. How many days are there in two weeks ?
12. How many feet are there in 3 yards ?
13. How many days are there in 3 weeks ?
54 WHOLE NUMBERS AND FRACTIONAL PARTS.
1 quart.
1 gallon.
There are four quarts in a gallon {gal.),
14. How many quarts are there in 2 gallons ?
15. How many trees are there in two rows, each con-
taining 8 trees ?
16. How many quarts are there in 3 gallons ?
17. Three persons each pay 5 cents car fare ; how much
do they all pay together ?
18. How many panes of glass are needed for 3 windows,
if 8 panes are needed for each window ?
19. I can hold out 9 pounds in each hand ; what weight
can I hold out with both hands ?
20. How many persons are there at 3 tables, if 6 per-
sons are at each table ?
21. A grass-plot is enclosed by 3 rows of hedge, each 9
yards long ; how many yards of hedge are there in all ?
Find the cost of :
22. 2 tablets at 6 cents each.
23. 3 lemons at 3 cents each.
MULTIPLICATION, DIVISION, MENSURATION. 55
24. 3 pounds of sugar at 6 cents a pound.
25. 2 sheep at 4 dollars apiece.
88. Exercise.
Copy, writing in each case the proper figure in place
of X :
/. i of 2 = X
7. 1 of 14 = X
13.
1 of 12 = X
2. i of 4: = X
8. ^ of 16 = X
14.
1 of 15 =3 X
5. i of 6 = X
9. i of 18 == X
15.
i of 18 =: X
4. ^ of 8 = X
10. ^ot 3 = X
16.
i of 21 = X
5. 1 of 10 = X
11. ^ of 6 = X
17.
J of 24 = X
6. i of 12 = X
/2. 1 of 9 = X
18.
i of 27 = X
89. Oral Exercise.
Read Exercise 88, supplying the missing numbers.
90. Exercise.
Copy, supplying the missing numbers :
13. 15 = — fives.
14. 27 = — nines.
15. 14 = — sevens.
16. 18 = — sixes.
17. 16 = — eights.
18. 24:=:— eights.
91. Oral Exercise.
Read Exercise 90, supplying the missing numbers.
92. Exercise.
/. i of 8 pints are — pints, or — quarts.
2. J of a dozen are how many ?
3. Divide 6 cents equally between 2 boys. What part
/.
2 = — ones.
7.
9 = — threes.
2.
4 = — twos.
8.
12 — — fours.
3.
3 = — ones.
9.
12 =: — sixes.
4.
6 = — threes.
10.
8 = — fours.
6.
6 = — twos.
11.
21 = — sevens.
6.
10 = — fives.
12.
18 = — nines.
66 WHOLE NUMBERS AND FRACTIONAL PARTS.
of the money does each receive ? How much does each
receive ?
4. If 2 quarts of syrup cost 12 cents, 1 quart will cost ^
of — cents, or — cents.
5. How much oil does a stove burn in 1 hour, if it
burns 4 quarts iu 2 hours ?
6. If I buy 3 pints of ice cream for 3 persons, how mucli
will that allow for each person ?
7. When 6 quarts of milk fill 3 cans of the same size,
how much does each can hold ?
8. 18 pails were tied in 2 bundles of the same size ; how
many were put in a bundle ?
9. I wish to have 18 pounds of sugar put up in 3 pack-
ages of the same weight ; how much must be put in each
package ?
10. 24 children are formed into 3 classes of the same
size ; how many are there in a class ?
//. Make and answer a question about J of 18 cents.
12. Mary spent 15 cents for 3 pounds of sugar. Ask a
question about this, and give the answer.
13. A boy had 12 cents. He spent ^ of it for fire-
crackers and J of it for ice cream ; how much did he
spend for both ?
14. A girl who earns 9 dollars and spends \ of it for
board has how much left ?
93. Exercise.
/. If one lemon costs 3 cents, for G cents — lemons can
be bought.
MULTIPLICATION, DIVISION, MENSURATION. 57
2. How many 3-foot ropes can be cut from a rope 9 feet
long ?
3. How many times can a gallon measure be filled from
12 quarts of milk ?
4. How many 4-horse teams can be formed from 8 horses ?
5. How many 8-foot ropes can be cut from a rope 16
feet long ?
6. How many packages of oatmeal, each containing 4
pounds, must I buy to have 12 pounds ?
7. 2 quarts of raspberries at 8 cents a quart, with 5
cents, will pay for — sacks of salt at 3 cents a sack.
8. Last week I worked 3 days and was paid 6 dollars ;
how much was that a day ?
9. In 12 pints there are — quarts.
10. 15 cents will pay for — car rides at 5 cents each.
//. How many tables must be set for 24 persons, if 8
persons can be seated at a table ?
12. How many are there in half a dozen ?
13. How many are there in one third of a dozen ?
14. 2 pecks of potatoes at 8 cents a peck are worth —
pounds of flour at 4 cents a pound.
15. If I paid 10 cents for pencils at 5 cents apiece,
how many did I buy ?
94. Oral Exercise.
Count by 4's :
/. From 4 to 40. 4. From 2 to 42.
2. From 40 to 4. 5. From 3 to 43.
3. From 1 to 41. 6. From 43 to 3.
58 WHOLE NUMBERS AND FRACTIONAL PARTS.
Count by 5's :
7. From 5 to 50.
8. From 50 to 5.
9. From 1 to 51.
10. From 2 to 52.
//. From 3 to 53.
12. From 4 to 54.
95. Exercise.
Copy, supplying the missing numbers :
3x1 + 1.=
3x4+4=
3x7 + 7= .
4x1=.
4x4 =
4x7=.
3x2 + 2=. .
3x5+5=
3x8 + 8= .
4x2=.
4x5 =
4x8=.
3x3 + 3= .
3x6 + 6= .
3x9 + 0= .
4x3=.
4x6 =
4x9=.
96. Exercise.
Copy, supplying the missing numbers and learn :
TABLE.
4x1=. 4x4=. 4x7
4x2=. 4x5=. 4x8
4x3=. 4x6=. 4x9
97. Exercise.
Copy, supplying the missing num^bers :
4x1+1=. 4x4+4=. 4x7+7
5x1=. 5x4=. 5x7
4x2+2=. 4x5+5=. 4x8+8
5x2=. 6x5=. 5x8
4x3+3=. 4x6+6=. 4x9+9
5x3=. 5x6=. 5x9
MULTIPLICATION, DIVISION, MENSURATION. 59
98. Exercise.
Copy^ supplying the missing numbers and learn :
TABLE. ^
5xl=» 5x4=» 5x7=»
5x2=- 5x5=» 5x8=-
5x3=- 5x6=« 6x9=-
99. Exercise.
Copy, supplying the missing numbers :
/.
4 X
1 = .
19.
5 X . = 15
37.
4
X
.= 8
2.
4 X
. = 12
20.
. X 5 = 25
38.
.
X
4= 16
3.
4 X
6= .
21.
4x2=.
39.
5
X
4= .
4.
4 X
.= 32
22.
. X 6 = 24
40.
5
X
5 = .
5.
5 X
'= 20
23.
4 X 5 = •
41.
.
X
2 = 8
6.
4 X
4= .
24.
. X 1 =5
42.
5
X
.= 25
7.
5 X
. = 35
25.
. X 7 = 28
43.
•
X
5 = 20
8.
5 X
7 = .
26.
5 X . = 30
44.
4
X
7= .
9.
• X
2 = 10
27.
4 X . = 20
45.
4
X
.= 36
10.
5 X
3= .
28.
. X 6 = 30
46.
4
X
9 = .
11.
4 X
. = 4
29.
5x0=.
47.
•
X
9 = 45
12.
• X
9 = 36
30.
. X 3 = 15
48.
5
X
.= 45
13.
• X
8 = 40
31.
4x3=.
49.
5
X
6= .
14.
4 X
. =28
32.
5 X . = 10
50.
4
X
.= 16
15.
• X
7 = 35
33.
. X 1 = 4
51.
•
X
4 = 20
16.
4 X
8= .
34.
5 X . = 5
52.
4
X
. = 24
17.
5 X
. = 40
35.
5x8=.
53.
5
X
2= .
18.
5 X
1= .
36.
. X 3 = 12
54.
•
X
8 = 32
60 WHOLE NUMBERS AND FRACTIONAL PARTS.
iOO. Exercise.
/. In one week there are — days, and in 4 weeks there
are 4 times — days, or — days.
2. ^n one yard there are — feet, and in 5 yards there
are 5 times — feet, or — feet.
5. In one gallon there are — quarts, and in 4 gallons
there are 4 times — quarts, or — quarts.
4. In one quart there are — pints, and in 5 quarts there
are 5 times — pints, or — pints.
Q
1 quart. 1 peck. 1 bushel.
There are 8 quarts (qt.) in a peck (pk.).
There are Jf. jjecks in a hnshel [by.).
5. How many quarts are there in 4 pecks ? in 3 pecks ?
6. How many quarts are there in 2 pk. ? in 6 pk. ?
7. How many pecks are there in 4 bushels ? in 5 bushels?
8. How many pecks are there in 2 bu. ? in 3 bu. ?
5. 4 X 2 and how many make a dozen ?
10. A dime and how many cents will pay for 3 pounds
of sugar at 6 cents a pound ?
//. How much change should a lady receive who
bought 3 pounds of soap at 4 cents a pound and gave the
clerk a dime and a 5-cent piece ?
MULTIPLICATION, DIVISION, MENSURATION. 61
Find the cost of :
12. 2 pencils at 5 cents each.
13. 4 pounds of sugar at 5 cents each.
14. 5 oranges at 2 cents each.
15. 4 eye shades at 8 cents each.
16. 4: yards of ribbon at 6 cents a yard.
17. 6 packs of firecrackers at 5 cents a pack.
18. 5 loaves of bread at 6 cents a loaf.
19. 4 quarts of milk at 7 cents a quart.
20. 5 papers of pins at 5 cents a paper.
101. Each of the four equal jmrts of a number is
called one fourth (i) of it.
102. Each of the five equal parts of a number is
called one fifth (^) of it.
103. Exercise.
Copy, writing in each case the proper figure in place
of X :
/. i of 4 = X 7. i of 28 = X /3. J of 20 -- X
2. J of 8 = X S. i of 32 = X /4. ^ of 25 =r X
5. i of 12 = X 5. i of 36 = X 15. ^ of 30 = x
4. i of 16 = X /O. i of 5 = X 16. i of 35 = X
5. \ of 20 = X //. i of 10 == X 17. ^ of 40 = X
6. i of 24 = X 12. ^ of 15 == X 18. ^ of 45 = x
104. Oral Exercise.
Bead Exercise 103, supplying the missing numbers.
62 WHOLE NUMBERS AND FRACTIONAL PARTS.
105. Exercise.
Copy, supplying the missing numbers :
1. 4 — — ones. 7. 24 = — sixes. 13. 30 = — sixes.
2. 10 = — twos. 8. 30 = — nines. 14. 6 = — ones.
3. 8 = — twos. 5. 20 == — fives. /5. 45 1= — nines.
4. 15 - —fives. 10. 12 = —threes. 16. 32 = —eights.
5. 16 = — fours. //. 25= — fives. 17. 35 = — sevens.
6. 20 = — fours. 12. 28 = — sevens. 18. 40 = — eights.
106. Oral Exercise.
Head Exercise 105,supplying the missing numbers.
107. Exercise.
/. If I pay 20 dollars for 4 weeks' board, how much is
that a week ?
2. I wish to cut 40 yards of carpet into 6 equal strips ;
how long must I make each strip ?
3. 10 pigeons will make — pair of pigeons.
4. How many quart bottles can be filled from 8 pints ?
5. How many 5-cent boxes of indigo can be bought for
20 cents ?
6. A school of 24 boys was divided into 4 equal classes ;
how many boys does each class contain ?
7. How many are 24 less 8 divided by 4 ?
8. How many times must I reach into a box containing
eggs to count out 2 dozen, if I take out 6 each time ?
9. In 10 quarts there are — pocks.
10. How many pieces, each 3 feet long, can I cut from
15 feet of twine ?
MULTIPLICATION, DIVISION, MENSURATION. 63
11. K painter received 18 dollars for painting a barn ;
if he worked 6 days, how much did he earn each day ?
12. IIow many 5-cent combs can be bought for 40 cents ?
13. How many gallon jugs can be filled with 20 quarts
of vinegar ?
14. Allowing a pound of meat for 2 persons, how much
meat should be bought for a dinner at which 10 persons
are to be served ?
15. How many pounds of sugar at 4 cents a pound can be
bought in exchange for 3 pounds of lard at 8 cents a pound ?
20 is called a score.
16. How many are there in i of a dozen ? In i of a
score ?
108. One of the two
equal parts of 1 is i ; or
ioflisi
One of the three equal
parts of 1 is i ; or ^ of 1
is i.
J of 2 is i plus -J, or f,
read two thirds.
109. Find J of 5.
Answer thus : | of 5 is 2 and 1 over ; i of 1 is |.
The answer is two and one half ; written 2|.
110. Oral Exercise.
Find:
/. J of 3. 3. J of 9. 5. I of 11. 7. i of 17.
2. J of 7. 4. i of 13. 6. ^ of 15. 8. J of 19.
64 WHOLE NUMBERS AND FRACTIONAL PARTS.
111. Find i of 17.
Answer thus : i of 17 is 5, and 2 over ; J of 2 is |.
Tlie answer is 5J.
112. Oral Exercise.
Find :
/. i of 5. 5. 4 of 4. 9. J of 10. 13. i of 13.
2. J of 8. 6. I of 7. 10. i of 22. 14. J of 25.
3. \ of 11. 7. i of IG. //. i of 23. 15. J of 26.
4. J of 14. 8. i of 19. /2. i of 28. /ff. J of 29.
113. Exercise.
/. J of 3 pints is — pints.
2. Find J of 7 feet.
5. Divide 8 yards of ribbon equally among 3 girls.
4. If a man earned 3 dollars in 2 days, in 1 day he
earned J of — dollars, or — dollars.
5. A lady bought 3 quarts of plums for 25 cents ; how
much were they a quart ?
6. At 7 cents a quart, what will a pint of milk cost ?
7. An iron rod 9 feet long was divided into 2 equal
parts ; find the length of each part.
8. When 3 loaves of bread cost 10 cents, how much
does one loaf cost ?
9. A family that uses a pound of tea in 3 weeks, uses
how much per week ?
10. When 3 quarts of oysters fill 2 cans of equal size,
how much does each can hold ?
//. How many days are there in lialf a week ?
MULTIPLICATION, DIVISION, MENSURATION. 65
114. How many 2's in 7 ?
3^ twos
(2) (2) (2) (lof2)
How many 3's in 11 ?
33-threes
11 = 3f threes.
(3) (3) (3) (|-of3)
116. Exercise.
Supply the missing numbers, using lines to illustrate :
/. 5 = — twos. 4. 7 = — threes. 7. 11 = — twos.
2. 4: = — threes. 5. 3 = — twos. 8. 10 = — threes.
5. 9 = — twos. 6. 8 — — threes. 9. 14 = — threes.
116. How many 2's in 7 ?
Answer thus: There are 3 twos in 7 and 1 over.
This 1 over is ^ of another 2.
Therefore, the number of 2's in 7 is 3i, read
three and one half.
117. Oral Exercise.
How many :
/. 2's in 3 ? 3. 2's in 13 ? 5. 2's in 17 ? 7. 2's in 19 ?
2. 3's in 5 ? 4. 3's in 20 ? 6. 3's in 16 ? 8. 3's in 25 ?
5
66 WHOLE NUMBERS AND FRACTIONAL PARTS.
118. Exercise.
/. In 7 pints there are — quarts — pint.
2. In 11 feet there are — yards — feet.
3. How many 2-cent stamps can be bought for 15 cents,
and what change will there be ?
4. 17 ducks will make how many pair, and leave how
many over ?
5. A liveryman hired out 21 horses, 9 of them for
driving singly and the others for double teams ; how
many double teams were there ?
6. How many sheets of drawing paper at 3 cents each
can be bought for 10 cents, and how much change will
there be ?
7. If one yard of braid costs 2 cents, for 9 cents as many
yards can be bought as — cents is contained times in —
cents, or 4^.
8. At 2 cents a pound, how many pounds of putty can
be bought for 7 cents ?
9. A man who earns 2 dollars a day must work how
many days to earn 11 dollars ?
10. How many bags of grain has a farmer, if he has 28
bushels in 3-bushel bags ?
119. Exercise.
/. What is each of the two equal parts of
1 called ?
2. How many halves are there in 1 ?
3. Read : ^ ; |.
MULTIPLICATION, DIVISION, MENSURATION. 67
4. What is each of the four equal parts of
1 called ?
5. What are two of the four equal parts of
1 called ?
6. Read: i;f; };*.
i of 2 := J + J, or f , or J.
7. Show that J of 3 = }.
120. Find \ of 14.
Answer thus : i of 14 is 3 and 2 over ; i of 2 is |, or \,
The answer is 3i.
121. Oral Exercise.
Find :
/. \ of 5. 4. \ of 9. 7. \ of 25.
2. \ of 7. 5. \ of 11. 8. \ of 27.
5. J of 6. ff. J of 10. 9. J of 22.
10. \ of 31.
//. }of 33.
12. iof34.
122. How many 4's are there in 14 ?
**** **** **** ^** 14 = 3^ fours.
4 4 4 iof4
123. Exercise.
Supply the missing numbers, making drawings to illus-
trate :
/. 9 = — fours.
^. 7 = — fours.
5. 6 = — fours.
4. 10 = — fours.
5. 13 = — fours.
6. 22 = — fours.
7. 15 = — fours.
8. 23 = — fours.
9. 25 = — fours.
68 WHOLE NUMBERS AND FRACTIONAL PARTS.
124. Exercise.
/. At 4 cents a yard, how many yards of lace can I
buy for a dime, and what change will there be ?
2. In 19 quarts there are — gallons — quarts.
3. At 7 dollars a bushel, what is a peck of clover seed
worth ?
4. When a gallon of oil costs 9 cents, how much is it a
quart ?
5. What is each of the five equal parts of 1 called ?
6. Read: i; |;|; t;|.
7. Find ^ of each number in the outer ring :
Suggestion, Solve as in S 120.
8. Show that there are 2| fives in 12.
9. 2 pounds of lard at 9 cents a pound will buy how
many yards of lining at 4 cents a yard ?
10. A boy had 22 cents in 5-cent and one-cent pieces.
He had 7 one-cent pieces ; how many 5-cent pieces had he?
//. How many 5-ccnt packages of fire-crackers can a
boy get for a pair of pigeons worth 28 cents ? How much
money will he have left?
12. A vessel that holds 9 quarts has in it 1 quart of
MULTIPLICATION, DIVISION, MENSURATION. 69
water ; liow iiuiiiy gallons of water can still be ])oured
into it ?
13. I have four pieces of money, together worth 12 cents.
Two of them are one -cent pieces ; what are the other
two ?
125. 3 X 23 = 23 + 23 + 23 = G9. The answer may
be found by a shorter method by making use of the Table
on p. 120, thus :
^3 3 X 23 = 3 X (3 units + 2 tens).
_£. 3x3 units = 9 units, which we write in units' place.
69 3x2 tens = 6 tens, which we write in tens' place.
The answer is 69.
In practice, say 3 times S are 9 ; 3 times 2 are 6.
126. The process of taking a number as many
times as there are units in anotlier number is called
Multiplication. The answer in multiplication is called
the Product.
127. Multiply 21 by 2.
Multiplicand 21
Multiplier 2
Product 42
128. Exercise.
Copy and multiply, and state which number is the mul-
tiplicand, which the multiplier, and which the product :
70 WHOLE NUMBERS AND FRACTIONAL PARTS.
/. 42
7.
30
13.
41
19.
201
25.
203
2
3
5
4
3
2. 33
8.
21
14.
120
20.
212
2ff.
121
3
3
2
4
4
3. 20
9.
21
15.
203
21.
200
27.
101
2
4
2
4
5
4. 40
10.
32
16.
101
22.
112
28.
323
2
4
3
4
3
5. 13
11.
40
17.
132
23.
102
29.
213
3
4
3
4
3
6. 32
12.
31
18.
332
24.
111
30.
222
3
5
3
5
4
129. One of the 3 equal parts of 39 may be found thus :
i of 39 = i of (3 tens + 9 units).
i of 3 tens = 1 ten, which we write in tens' place.
■^*^ 3 of 9 units = 3 units, which we write in units' place.
The answer is 13.
In practice, my ^ of d is 1 ; ^ of d is S.
130. Finding one of the equal parts of a numbei
is called Division. The answer in division is called
the Quotient.
131. The sign of division is -^, read divided by.
Thus, 24 -s- 3 is read ^4 divided by 3.
Note. 24-1-3 may also be expressed 3)24.
MULTIPLICATION, DIVISION, MENSURATION. 71
132. Divide 28 by 2.
Divisor 2)28 Dividend
Quotient 14
133. Exercise.
Copy and divide, and state which number is the divi-
dend, which the divisor, and which the quotient:
/. 2)42 7. 3)60 13. 4)80 19. 2)406 25. 4)444
2. 2)64 8. 3)33 14. 4)40 20. 3)333 26. 4 )480
5. 2)86 5. 3)69 /5. 5)^ 21. 3)369 27. 4 )408
4. 2)40 /O. 4)^ /^. 5)55 22. 3)960 25. 5 )555
5. 3)36 //. 4)44 /7. 2)222 23. 3)609 25. 5)505
5. 3)96 /2. 4)_48 /5 2 )264 24. 3)^06 50. 5)550
134. Multiply 45 by 4.
45 4 X 45 = 4 X (5 units + 4 tens).
4 4x5 units = 20 units, or 3 tens.
180 Write in units' place, and add 2 tens to the next
product.
4x4 tens = 16 tens, which added to the 2 tens makes 18 tens,
or 1 hundred and 8 tens.
Write 8 in tens' phxce and 1 in hundreds' place.
The product is 180.
In practice^ say If, times 5 are 20 ; Jf, times Jf, are 16, and 2 are 18.
72 WHOLE NUMBERS AND FRACTIONAL PARTS.
135.
Exercise.
Copy and multiply :
/. 37
7. 34
13.
66
19. 46
25.
192
2
3
4
5
5
2. 39
5. 87
14.
89
20. 55
26.
287
2
3
4
5
3
3. 48
9. 64
15.
94
2/. 136
27.
498
2
3
4
2
2
4. 56
10. 95
16.
19
22. 176
28.
185
2
3
5
3
5
5. 75
//. 23
17.
28
23. 135
29.
225
2
4
_5
3
4
^. 29
12. 57
18.
37
24. 186
30.
199
_3
4
5
4
4
136.
Exercise.
/. How many pails are there in 4 bundles, if each
bundle contains 52 pails ?
52 is the number of pails in 1 bundle.
4
? is the number of pails in 4 bundles.
2. Mow many are there in 4 score ?
MULTIPLICATION, DIVISION, MENSURATION. 73
3. How many oranges are there in 3 boxes each con-
taining 36 oranges ?
4. A bushel of oats weighs 32 pounds ; find the weight
of 3 bushels of oats.
82 pounds is the weight of 1 bushel of oats.
3
? pounds is the weight of 3 bushels of oats.
5. A bushel of wheat weighs 60 pounds ; find the weight
of 5 bushels of wheat.
6. How much will a clerk save in 4 months, if he saves
$26 a month ?
There are 24 Jiours {hr.) in one day {da.).
7. How many hours are there in 4 days ?
8. A barrel of flour weighs 196 pounds; find the weight
of 4 barrels of flour.
9. Find the cost of 3 pounds of butter at 23 cents a
pound.
10. Find the cost of 4 pounds of ham at 16 cents a
pound.
//. K there are 24 sheets of paper in a box, how many
sheets are there in 4 boxes of the same size ?
12. Find the cost of 3 hundred shad at $25 per hundred.
137. Divide 312 by 4.
4 ) 312 i of 812 = i of (31 tens + 2 units).
78 i of 31 tens = 7 tens, and 8 tens over.
3 tens + 2 units = 32 units.
\ of 32 units = 8 units. The quotient is 78.
In practice, say i ^ 31 isl and 8 over ; i ^ 82 is 8.
74 WHOLE NUMBERS AND FRACTIONAL PARTS.
138. Exercise
Copy and divide :
/. 2)34
/2.
4)64
23.
7)91
34. 3)252
2. 2)76
13.
4)72
24.
8)96
35. 3)405
3. 2)92
14.
4)76
25.
7)98
36. 4)164
4. 2)58
15.
4)92
26.
2)124
37. 4)876
5. 2)98
16.
5)65
27.
2)136
38. 4)172
6. 3)42
17.
5)75
28.
2)154
39. 4)952
7. 3)48
18.
5)85
29.
2)306
40. 4)708
8. 3)54
19.
5)95
30.
2)178
41. 5)515
9. 3)72
20.
6)72
31.
3)153
42. 5)585
10. 3)84
21.
6)84
32.
3)657
43. 5)195
//. 4)56
22.
6)90
33.
3)174
44. 5)705
139. Exercise.
/. A dealer bought 4 dozen eggs for 84 cents ; how
much was that a dozen ?
84 cents was the cost of 4 dozen.
4) 84 cents
? cents was the cost of 1 dozen.
2. IIow much was paid a week to a workman who
received $75 for 5 weeks' work ?
MULTIPLICATION, DIVISION, MENSURATION. 75
3. A piece of carpet measuring 72 yards was cut into
4 strips of equal lengtli ; find the length of each strip.
4. 56 chairs are to be arranged in 4 rows with the same
number in each row ; how many must be put in a row ?
56 is the number to be put in 4 rows.
4)56
? is the number that must be put in a row.
5. 144 handkerchiefs were put up in 3 boxes, the same
number being in each box ; how many were put in a box ?
6. 5 bushels of feed weighed 225 pounds ; what did each
bushel weigh ?
7. A farmer sold 2 cows, the first for 35 dollars and the
second for 43 dollars ; what was the average selling price ?
35 dollars was the selling price of the first.
43 " " " " '' second.
~78 "■ " " " " both.
2778 "•
? was the average selling price.
8. A merchant bought 2 hams ; the first weighed 23
pounds and the second 25 pounds ; what was their average
weight ?
140. Finding hov7 many times 3 feet is contained
in 12 feet is measuring 12 feet by 3 feet.
141. The process of measuring one number by
another is Mensuration.
142. The sign of mensuration is :, read measured
hy.
Thus, 12 : 3 is read 12 measured by 3,
76 WHOLE NUMBERS AND FRACTIONAL PARTS.
143. The number of 4'8 in 12 is 3 ; that is, 12:4 = 3.
But 12^4 = 3.
Therefore, 12:4 = 12^4. Therefore,
To measure one number hy another^ divide the
first hy the second,
144. Oral Exercise.
Measure :
/. 12 by 4. Answer thus : 12 : 4 = 12 -i- 4, or 3.
2. 14 by 2. 6. 32 by 4. 10. 28 by 4. /4. 24 by 4.
5. 24 by 8. 7. 30 by 5. //. 40 by 5. 15. 35 by 5.
4. 18 by 2. 8. Ti by 3. 12. 32 by 4. 16. 18 by 3.
5. 27 by 3. 9. 36 by 4. 13. 21 by 3. 17. 25 by 5.
145. 2 feet is contained in 14 feet as many times
as 2 is contained in 14 ; that is,
14 feet : 2 feet = 14 : 2, or 14 -^ 2. Therefore,
To measure one amount hy another^ divide the
number of units in the first by the number of units
in the second,
146. Oral Exercise.
Measure :
/. 14 feet by 2 feet.
Answer tlms : 14 feet : 2 feet = 14 -*- 2 = 7.
2. 16 cents by 2 cents. 7. 45 dollars by 5 dollars.
3. 24 yards by 3 yards. 8. 28 men by 4 men.
4. 20 quarts by 4 quarts. 9. 30 feet by 5 foot.
5. 25 days by 5 days. 10. 20 pounds by 4 pounds.
B. 36 quarts by 4 quarts. //. 32 gallons by 4 gallons.
MULTIPLICATION, DIVISION, MENSURATION. 77
147. Exercise.
/. A man bought 144 colored crayons in boxes contain-
ing four pieces each ; how many boxes did he buy ?
144 was the number of pieces bought.
4 was the number of pieces in each oox.
4) -144
? was the number of boxes bought.
2. How many 2-cent stamps can be bought for 60
cents ?
3. How many gallon jugs can be filled with 64 quarts
of vinegar ?
A. How many 3-bushel bags are needed to hold 168
bushels of grain ?
5. How many horses can be shod all around with 264
horseshoes ?
6. How many bushels are there in 72 pecks ?
7. How many 5-cent pieces will make 75 cents ?
8. 144 dozen eggs are to be packed in crates, each hold-
ing 4 dozen ; how many crates are needed ?
9. At 5 cents a pound, how many pounds of sugar can
be bought for 90 cents ?
10. How many yards long is a rope that is 426 feet
long ?
//. If a family use 2 quarts of milk a day, in how many
days will they have used 5 gallons of milk ?
12. In how many days will a boy save 60 cents, if he
saves 5 cents a day ?
78 WHOLE NUMBERS AND FRACTIONAL PARTS.
Forming, Writing, and Reading Numbers from
lOOO to lOOOO.
148. Ten hundred make one thousand^ written
1000.
Two thousand is written 2000 ; three thousand,
3000.
One thousand one is written 1001.
One thousand ten is written 1010.
Five thousand two hundred is written 5200.
Four thousand seven hundred twenty is written
4720.
Nine thousand nine hundred ninety-nine is
written 9999.
149. Exercise.
Write in figures:
/. Three thousand. 7. Seven thousand.
2. Four thousand. 8. Eight thousand.
3. Five thousand. 9. Nine thousand.
4. Six thousand. 10. One thousand two.
5. Two thousand twenty. //. Six thousand five hundred.
6. Four thousand forty. 12. Nine thousand fifteen.
13. Seven thousand six hundred eighty-four.
14. Nine thousand four hundred forty.
15. Eight thousand six hundred four.
16. Seven thousand eight hundred sixty-six.
17. Nine thousaiid eiglit hundred twenty.
18. Four tliousand nine hundred ninety-nine.
19. One thousand one hundred one.
ROMAN NOTATION. 79
160. Exei
else.
Read,
and write in words :
/. 400C.
5. 1200.
9.
1009.
13.
8777.
2. 7000.
6. 5021.
10.
9090.
14.
G666.
3. 7002.
7. 9888.
11.
5007.
15.
2754.
4. 1011.
8. 7232.
12.
8201.
16.
3215.
Roman Notation.
151. Capital letters are often used instead of
figures in numbering chapters and sections of books,
pages of prefaces and introductions, divisions on
clock dials, etc.
This method of writing numbers is called the
Roman System.
I=:l Y = 5 X = 10 L = 50 C = 100
D = 500 M = 1000.
Note. The number for which a letter stands \b called the
value of the letter.
152. If a letter is repeated its lvalue is repeated.
Thus, II = 2 ; III = 3 ; xx = 20 ; cc = 200 ; mmm = 3000.
153. If a letter is written before one of greater
7)alue^ the difference of tlielr values is expressed.
Thus, IV = 4 ; ix = 9 ; xl = 40 ; xc = 90.
154. If a letter or a combination of letters is
written after a letter of greater value, the sum of
the values is expressed.
Thus, VI =r 6 ; xiii = 13 ; lv = 55 ; lix = 59.
80 WHOLE NUMBERS AND FitACTIONAL PARTS.
155. A dash placed over a letter or a combina-
tion of letters multiplies its value by 1000.
Thus, V = 5000 ; vli = 7000 ; Tx = 9000.
156. Units, tens, hundreds, and thousands only
are written according to the principles given above.
TABLE.
Units.
Tens.
Hundreds.
Thousands.
I = 1
X
=
10
C
= 100
M
= 1000
II =2
XX
=
20
CC
= 200
MM
= 3000
III = 3
XXX
=
80
CCC
= 300
MMM
= 3000
IV = 4
XL
=
40
CD
= 400
IV
= 4000
V =5
L
=
50
D
= 500
V
= 5000
yi = 6
LX
=
60
DC
= 600
vT
= COOO
VII = 7
LXX
=
70
DCC
= 700
vli
= 7000
VIII = 8
LXXX
=
80
DCCC
= 800
VIII
= 8000
IX =9
xc
=
90
CM
= 900
ix
- 9000
157. In writing numbers between those in the
above table, hundreds are written after thousands,
tens after hundreds, and units after tens.
Thus, 15G7 is written mdlxvii ; 2053, mmliii : 4508, fvDViii ;
3600, MMMDC.
Note, hit = 4 ; xxxx = 40 ; cccc = 400, are also iu use.
158. Exercise.
Write by the Roman method :
/. The nine simple numbers. 3. The nine hundreds.
2. The nine tens. 4. The first nine thousands.
The numbers between
5. 10 and 20. 8. 40 ami 50. //. 70 and 80.
6. 20 and 30. 9. 50 and 60. 12. 80 and 90.
7. 30 and 40. 10. 60 and 70. 13. 1)0 and 100.
UNITED STATES MONEY.
81
The following numbers :
U. 101. 17. UO.
20.
400.
'23.
1899.
15. 120. 18. 149.
21.
6G6.
24.
1 90G.
16. 150. 19. 194.
22.
924.
25.
1910.
Write in figures the following :
26. IX.
27. XIV.
28. XIX.
29. XXXIV.
30. XXXVIII.
31. XLIX.
32. LXIV.
33. XCIX.
34. CLX.
55. CLXII.
55. CCCLXV.
37. CDIX.
55. CDLXIV.
39. DCCCV.
40. MCM.
4/. MCMV.
42. MCMIX.
43. MCMXIX.
44. MDCCCXCIX.
45. VII.
46. VCCIV.
United States Money.
159. In the money of the United States,
10 cents = 1 dime.
10 dimes = 1 dollar.
100 cents = 1 dollar.
160. The dollar sign is $. It is placed before the
number of dollars to be expressed.
Thus, 5 dollars is expressed $5.
161. If a sum of money consists of dollars and
cents, a point called the Decimal Point is written
between the number of dollars and the number of
cents. If the number of cents is less than ten, a
6
82 WHOLE NUMBERS AND FRACTIONAL PARTS.
cipher is written between the decimal point and the
number of cents.
Thus, S dollars 25 cents is expressed $3.25, and Jf. dollars 8
cents is expressed $1^.08.
162. The abbreviation for the word cent or cents
is ct. or ^.
Thus, 25 cents may be expressed 25 ct. or 25i^.
163. A sum of money less than a dollar, such as
36 cents, may also be expressed $0.36.
164. Oral Exercise.
Eead:
/. $2. 3. $15.25.
5. $0.62.
7. $88.66.
2. $3.05. 4. $37.50.
6. $0.07.
8. $ 0.12^.
165. Exercise.
Write, using the dollar sign :
/. 6 dollars. 4. 7 dollars 5 cents.
2. 75 dollars 25 cents. 5. 87 cents.
3. 11 dollars 15 cents. 6. 9 cents.
166. Exercise.
/. How many cents are there in $3 ? $5 ? $12 ?
2. Change $4.36 to cents.
$4.36 = 436 ct.
5. How many cents are there in $2.25 ? $4.36 ? $14.36 ?
MAKING CHANGE. 83
4. Change to cents :
$6.84; $9.32; $15.36; $28.45.
5. Change 600 ct. to dollars.
600 ct. = $6.00, or $6.
6. Change to dollars :
500 ct. ; 900 ct.; 1500 ct. ; 2800 ct.
7. Change 324 ct. to dollars and cents.
324 ct. =$3.24.
8. Change to dollars and cents :
539 ct.; 608 ct.; 3125 ct.; 6175 ct.
Making Change.
167. A man bought a fork for 03^/ and gave the store-
keeper a dollar bill ; how will the storekeeper make the
change, using as few pieces of money as possible ?
The storekeeper will give the man 2 cents, a dime, and
a quarter-dollar. He may count out the change, saying,
63 ct., 65 ct., 75 ct., 1 dollar.
168. Oral Exercise.
Make the change in the following examples, using as
few pieces of money in each case as possible ; and state
how the change may be counted out :
Sales. Money Paid.
/. 17^'. Quarter-dollar.
2. 32^. Half-dollar.
3. 63r/. Dollar.
84 WHOLE NUMBERS AND FRACTIONAL PARTS.
4. 62^'. Half-dollar and quarter-dollar.
5. 12^'. A dime and a 5-cent piece.
6. 33^'. A quarter dollar and a dime.
7. $1.15 ^2-bill.
8. $2.12 $2-bill and a $l-bill.
9. $3.28 $5-bill.
10. 9^. $2-bill.
//. $28.37. $20-bill and a $10-bill.
Addition and Subtraction.
169. Add:
3745 In practice we ma}' say :
2G34 3, 7, 13. Write 2 in units' place and add 1 to the
1853 tens.
8232 Then 1, 6, 9, 13. Write 3 in tens' place and add
1 to the huncl reels.
Then 1, 9, 15, 23. Write 2 in the hundreds' place and add 2 to
the tliousands.
Then 2, 3, 5, 8. Write 8 in the thousands' place.
The sum is 8233.
170. Exercise.
Copy and add :
/. 345 3. 378
5. 765
7. 5482
9. 3482
870 745
080
2708
2351
2. 421 4. 987 6. 2768 8. 5832 10. 7683
899 876 1289 3168 2297
ADDITION AND SUBTRACTION.
85
r/. 3275
13.
2845
15. 3754
17. 329
19.
328
876
738
2832
896
798
49
897
1709
539
854
654
29
r2. 832
14.
2597
16. 763
18. 327
20.
377
2375
3685
854
83
479
1783
1763
963
785
901
78
637
507
Find the missing numbers by adding down and across
21. 234 + 567 + 728 =
109 + 106 + 34 =
48 + 729 4- 138 =
•+• + • =
22. 728 + 925 + 324 =
75 + 109 + 7 =
768 + 325 + 785 =:
+
23. 308 + 78 + 230 =
106 + 7 + 98 =
99 + 126 4- 3 =
+
+ • =
24. 1024 + 276 + 325
106 +72+78
48 + 9 + 3
+ . =
Find the missing numbers by multiplying across and
adding down ;
25. 3 X 228 = . 26. A: X 164 = . 27. 5 x 75 =
3 X 456 = . 4 X 108 = . 5 x 110 =^
3 X 325 = . 4 X 378 = . 5 x 306 =
3x184=. 4x76=. 5x 9 =
3 X
4 X . =
5 X
86 WHOLE NUMBERS AND FRACTIONAL PARTS.
171. Subtract :
111 ])iactice be^in at units and subtract thus :
8214
8 from 14 leaves 6.
1 7«3o
3 from 10 leaves 7.
7 from 11 leaves 4.
1 from 7 leaves 6. The difference is 6476.
172. Exercise.
Copy and subtract :
/. 832 6. 1000 //. 7001 16. 7200 2f. 3020
574 _389 2098 _99 1982
2. 722 7. 1070 12. 3010 17. 3G72 22. 7008
448 179 2111 908 1999
3. 320 8. 2080 13. 5002 18. 7202 23. G320
107 1979 4983 5193 2198
4. 250 9. 3420 14. 5000 /5. 8201 24. 3507
148 2837 3092 7192 2198
6. 700 10. 9000 /5. 8030 :?0. 3380 25. 4320
188 1807 1929 1099 1738
173. Find the sum of 113.25, 110.48, and 10.70.
113.25 Write the sums, placing dollars under dollars and
10.48 cents under cents. Add as in ordinary numbers,
Q' «6 and place the decimal point before the last two
$24.49 figures to separate dollars from cents.
ADDITION AND SUBTRACTION. S7
174. Exercise.
Find the
sum :
1.
2.
3.
4.
% 3.25
$ 8.48
$12.38
$24.06
5.63
7.35
15.09
33.45
7.42
0.67
32.26
8.37
12.56
26.89
31.56
15.00
5. $7.25 4- $6.48 + $15.23 + $28.16.
6. $9.32 + $17 + $ 0.64 + $36.99.
7. $0.56 + $1.38 + $10.49 + $ 0.18 + $14.38.
176. From $27 take $14.27.
Write dollars under dollars and cents under
$27 00
cents. Subtract as in ordinary numbers, and place
the decimal point before the last two fissures to
separate dollars from cents.
176. Exercise.
Find the difference :
/. $4.26 2. $15.42 5. $146.38 4. $96.45
3.83 9.83 27.94 0.87
5. $34.10 - $19.46 7. $82.56 - 37.49 9. $10 - $ 6.48
6. $ 1.48 - $ 0.79 5. $ 5 - 3.69 10. $50 - $42.65
177. Exercise.
/. A man bought a lot of land for $350 and built on it
a house that cost $1875 ; how much did both cost him ?
88 WHOLE NUMBERS AND FRACTIONAL PARTS-
2. James had $2.75 and earned $3.75 more ; liow much
had he then ?
5. In a certain town there are 35 78 males and 4423
females ; how many people has the town ?
4. Take 11.25 from 15.00.
5. I had a 1000-mile ticket, but have used 695 miles
of it ; how many miles may I yet ride on it ?
6. A farmer^s crop for a certain year was 268 bushels of
wheat, 308 bushels of oats, 564 bushels of corn, and 78
bushels of rye ; how much grain did he raise that year ?
7. Frank and John together have $4.85. If Frank has
$2.90, how much has John ?
8. A wagon loaded with coal weighed 5268 pounds. If
the wagon weighed 1825 pounds, what was the weight of
the coal ?
9. A farmer bought a plow for $8.75 and gave in pay-
ment a $20-bill ; how much change should he receive ?
10. A farmer was paid $175.20 for his wheat and $106.30
for his potatoes ; how much was he paid for both ?
//. The number of ladies at a normal school duriiiij a
certain year was 539 and the number of men 221 ; how
many were there of both ? By how many did the former
exceed the latter ?
12. My brother earned $1.75 on Monday, $1.60 on
Tuesday, $1.85 on Wednesday, $1.35 on Thursday, and
$1.75 on Friday ; how much did he earn in tlio 5 days ?
13. I owed B $100 and gave him a cow worth $45.50
and the balance in cash ; how much cash did I give
him ?
MULTIPLICATION, DIVISION, MENSURATION. 89
14. Find the sum of one thousand four hundred eight,
three thousand forty-eight, and six hundred seventy-six.
15. Find the sum of all numbers that are greater than
1298 and less than 1304.
16. The following table gives the number of bushels
of grain bought by a dealer in one week ; fdl in the
totals :
Grain.
Mon.
Taes.
Wed.
Thurs.
Fri.
Sat.
Total.
Wheat
235
431
500
27
300
307
480
45
201
408
578
38
406
560
401
46
327
456
492
62
195
231
324
14
Corn
Oats
Rye
Total
Multiplication, Division, and Mensuration.
178. Oral Exercise.
Count by 6's :
/. From 6 to 60.
2. From 60 to 6.
3. From 1 to 61.
4. From 2 to 62.
5. From 3 to 63.
6. From 4 to 64.
7. From 5 to 65.
8. From 65 to 5.
Count by 7's :
9. From 7 to 70.
10. From 70 to 7.
//. From 1 to 71.
12. From 2 to 72.
13. From 3 to 73.
14. From 4 to 74.
15. From 5 to 75.
16. From 6 to 76.
90 WHOLE NUMBERS AND FRACTIONAL PARTS.
179. Exercise.
Copy, supplying the missing numbers :
5x1 + 1= .
5x4+4=
5x7 + 7= .
G X 1 = . -
6x4 =
6x7=.
5x2 + 2= .
5x5+5=
5x8 + 8= .
6x2=.
6 X 5= «
6x8=.
5x3+3=-
5x6+6=
5x9 + 9= .
6x3=.
6x6=,
6x9=.
180. Exercise.
Copy, supplying
the missing numbers and learn :
TABLE.
6x1=.
6x4=.
6x7=.
6x2=.
6x5=.
6x8=-
6x3=.
6x6=.
6x9=.
181. Exercise.
Copy, supplying the missing numbers
6x1+1=.
6x4+4=
6x7+7=.
7x1=.
7x4 =
7x7=.
6x2 + 2= .
6x5+5=
6x8 + 8= .
7x2=.
7x5=.
7x8=.
G X 3 + 3 = .
6x6+6=
6x9 + 9= -
7x3=.
7x6=.
7x9=.
182. Exercise.
Copy, supplying the missing numbers and learn :
7x1=. 7x4=. 7x7 =
7x2=. 7x5=. 7x8=
7x3=. 7x6=.. 7x9=
MULTIPLICATION, DIVISION, MENSURATION. 91
183. Oral Exercise.
Eead, supplying
the missing numbers :
/. 6 X 3 = .
19.
•
X 6 = 42
37.
• X
1 = 6
2. 7 X 7 = .
20.
6
X 2= .
38.
6 X
4= .
5. 7 X . = 49
21.
•
X 4 = 24
39.
7 X
. =42
4. 7 X 9 = .
22.
7
X . =21
40.
7 X
. =63
5. 7x4=.
23.
6
X 5= .
41.
6 X
. = 12
ff. 6 X 8 = .
24.
•
X 2 = 14
42.
• X
7 = 42
7. 7 x . = 28
25.
.
X 6 = 36
43.
6 X
6 = .
8. . X 7 = 49
26.
.
X 8 = 56
44.
6 X
. =24
5. 6 X 7 = .
27.
6
X 1 = .
45.
7 X
7= .
10. . X 2 = 12
28.
6
X . =42
46.
6 X
. =30
//. 6 X . = 36
29.
6
X 9= .
47.
6 X
. =6
/2. 7 X 5 = .
30.
6
X . = 54
48.
7 X
• = 35
/5. 7 X 6 = .
31.
.
X 9 = 54
49.
• X
3 = 18
14. . X 8 = 48
32.
•
X 9 = 63
50.
• X
5 = 35
15. . X 4 = 28
33.
.
X 1 = 7
51.
7 X
2 = .
/e. 7 X 2 = .
34.
6
X . = 18
52.
• X
3 = 21
17. . X 5 = 30
35.
7
X 1 = .
53.
6 X
. =48
/5. 7 X . = 56
36.
7
X 8= .
54.
7 X
3= .
184. Exercise. '
1. Find the cost of these groceries together :
6 pounds of sugar at 5 cents a pound.
2 pounds of crackers at 5 cents a pound.
2 quarts of vinegar at 4 cents a quart.
2. In a room there are 7 rows of desks and in each row
8 desks ; how many desks are there in all ?
5. A man who sleeps 9 hours a day sleeps how long each
week?
92 WHOLE NUMBERS AND FRACTIONAL PARTS.
4. A farmer bought 6 lambs at 4 dollars apiece and sold
them at 5 dollars apiece ; how much did he make on each
lamb ? How much on the 6 lambs ?
5. In 6 bushels there are — pecks.
6. In 7 pecks there are — quarts.
7. Mr. Brown gave 3 boxes of strawberries at 8 cents a
box for syrup at 6 cents a quart ; how many quarts did he
receive ?
8. A lady spends 6 cents a day for milk ; how much is
that a week ?
9. Show by dots that 6 x 3 — 18.
10. \ dozen pigeons will make how many pair ?
//. How many bushels will 7 two-bushel bags hold ?
12. How many pencils are there in 7 boxes, if each box
contains 6 pencils ?
185. Each of the six equal parts of a number is
called one sixth {\) of it.
188. Each of the seven equal parts of a number is
called one seventh (|) of it.
187. Exercise.
Copy, writing the
proper figures in
place of x^:
/. ^of 6 = x
7. J of 42 = X
13. \ of 28 =
X
2. i of 12 := x
8. \ of 48 = X
U. ^ of 35 =
X
5. \ of 18 = X
9. i of 54 = X
15. ^ of 42 =
X
4. J of 24 = X
10. \oi 7 = X
16. 1 of 49 =
X
5. \ of 30 = X
//. |of 14 = X
n. \ of 50 =
X
6. i of 36 = X
12. |of 21 =x
18. \ of 63 =
X
MULTIPLICATION, DIVISION, MENSURATION. 93
188. Oral Exercise.
Read Exercise 187, supplying the missing numbers.
189. Exercise.
Copy, supplying the missing numbers :
/. 12 = — twos. 7. 30 = — fives. 13. 63 = — nines.
2. 7 = — ones. 8. 24 = — fours. 14. 4'^ = — sevens.
3. 14 = — twos. 9. 48 = — eights. 15. 35 = — fives.
4. 18 = — threes. 10. 42 — — sixes. 16. 36 = — sixes.
5. 6 = — ones. //. 28 = — fours. /7. 4C = — sevens.
6. 21 = — threes. 12. 54 = — nines. 18. 56 — — eights.
190. Oral Exercise.
Read Exercise 189, supplying the missing numbers.
191. Exercise.
/. Divide 12 cards equally among 6 boys. What part
of them will each boy receive ? How many will each boy
receive ?
2. How many yards of carpet are there in a piece 21
feet long ?
3. 48 girls are marching 6 in a row ; how many rows
are there ?
4. How many 5-cent pieces should I get in exchange
for 30 cents ?
5. 3 gallons of syrup will fill how many 2-quart jars ?
6. How long will 3 pecks of oats last a horse that eats
4 quarts a day ?
94 WHOLE NUMBERS AND FRACTIONAL PARTS.
7. In 28 days there are — weeks.
8. At 4 for a cent, how much must I pay for 28
rivets ?
9. How many yards of muslin at 8 cents a yard can be
bought for 8 quarts of plums at G cents a quart ?
10. A boy earned 54 dollars in six months ; how much
was he paid a month ?
11. A dozen will make how many pair ?
12. How many working days are there in a week ?
13. A man who earns 3 dollars each working day earns
how much in a week ?
14. 6 pounds of lard at 8 cents a pound, and how many
cents, will pay for a 50-cent bucket ?
192. Oral Exercise.
Count by 8's :
/. From 8 to 80.
2. From 80 to 8.
3. From 1 to 81.
4. From 2 to 82.
5. From 3 to 83.
6. From 4 to 84.
7. From 5 to 85.
8. From 6 to 86.
9. From 7 to 87.
10. From 8 to 88.
Count by 9's :
//. From 9 to 90.
12. From 90 to 9.
13. From 1 to 91.
14. From 2 to 92.
15. From 3 to 93.
re. From 4 to 94.
17. From 5 to 95.
18. From 6 to 96.
19. From 7 to 97.
20. From 8 to 98.
MULTIPLICATION, DIVISION, MENSURATION. 95
193. Exercise.
Copy, supplying
the missing numbers
7x1 + 1=.
7x4+4=.
7x7+7=.
8 X 1 r= .
8x4=.
8x7= •
7x2 + 2= .
7x5 + 5= .
■7x8 + 8= .
8x2=.
8x5=.
8x8=.
7x3 + 3= .
7x6 + 6= .
7x9 + 9= .
8x3=.
-8x6=.
8x9=.
194. Exercise.
Copy, supplying
the missing numbers,
TABLE.
and learn :
8x1=.
8x4=.
8x7=.
8x2=.
8x5=.
8x8=.
8x3=.
8x6=.
8x9=.
195. Exercise.
Copy, supplying
the missing numbers
8x1 + 1= .
8x4 + 4= .
8x7 + 7= .
9x1=.
9x4=.
9x7=.
8x2 + 2= .
8x5 + 5= .
8x8 + 8= .
9x2=.
9x5=.
9x8=.
8x3 + 3= .
8x6 + 6= .
8x9 + 9= .
9x3=.
9x6=.
9x9=.
196. Exercise.
Copy, supplying
the missing numbers,
TABLE.
and learn :
9x1=.
9x4=.
9x7=.
9x2=.
9x5=.
9x8=.
9x3=.
9x6=.
9x9=.
96 WHOLE NUMBERS AND FRACTIONAL PARTS.
197. Oral Exercise.
Head, supplying the missing numbers :
/.
8
X
2= .
19.
9
X
.= 63
37.
.
X
1 = 8
2.
.
X
7 = G3
20.
•
X
9 = 81
38.
.
X
5 = 45
3.
•
X
1 = 9
21.
9
X
.= 36
39.
9
X
.= 72
4.
8
X
8= .
22.
9
X
5 = .
40.
8
X
7= .
5.
9
X
7= .
23.
8
X
. = 56
41.
9
X
9= .
6.
.
X
9 = 72
24.
9
X
3= .
42.
•
X
6 = 54
7.
•
X
8 = 72
25.
•
X
2 = 18
43.
.
X
4 = 36
8.
8
X
6= .
26.
9
X
.= 9
44.
9
X
.= 27
9.
9
X
8= .
27.
9
X
1 = .
45.
9
X
G = .
10.
8
X
.= 72
28.
•
X
2 = 16
46.
•
X
6 = 48
11.
8
X
. = 32
29.
8
X
.= 40
47.
9
X
.= 18
12.
8
X
.= IG
30.
8
X
. = 24
48.
8
X
.= 8
13.
8
X
5= .
31.
8
X
1 = •
49.
9
X
.= 54
14.
•
X
8 = G4
32.
.
X
4 = 36
50.
•
X
7 = 56
15.
9
X
4= .
33.
8
X
4 .
51.
9
X
. = 81
16.
8
X
9= .
34.
8
X
3= .
52.
•
X
3 = 27
17.
.
X
5 = 40
35.
9
X
.= 45
53.
•
X
3 = 24
18.
8
X
.= 64
36.
9
X
2= .
54.
8
X
. = 48
198. Exercise.
/. In 8 gallons there are — quarts.
2. In 8 weeks there are — days.
3. In 8 pecks there are — quarts.
4. 8 X 2 + . = 9 X 2.
5. 9 three-bushel bags will hold how much wheat ?
6. 8 two-cent stamps will cost how much ?
7. At 4 dollars a hundred, what will 9 hundred cabbage
plants cost ?
MULTIPLICATION, DIVISION, MENSURATION. 97
8. What should be paid for 9 dozen herrings at 8 cents
a dozen ?
9. What should be paid for 8 hundred rails at 6 dollars
a hundred ?
10. How many feet long is a fishing line 8 yards long ?
/ /. How many persons are seated in a dining-room at
6 tables, if there are 8 persons at each table ?
12. A drover sold 8 pair of mules and one more ; how
many did he sell ?
13. After a farmer sold 8 bags of wheat each holding 3
bushels, he had 5 bushels left ; how many had he at first ?
14. In 1 bushel there are — pecks, and in 1 peck there
are — quarts ; then in 1 bushel there are — times —
quarts, or — quarts.
15. Show that there are 8 pints in a gallon.
199. Each of the 8 equal parts of a number is one
eighth (|) of it.
200. Each of the 9 equal parts of a number is one
ninth (i) of it.
201. Exercise.
Copy, writing the proper figures in place of x :
/. i of 8 = X
7. ^ of 56 = X
13. i of 36 = X
2. J of IG = X
8. '\ of 64 = X
14. ^ of 45 = X
3. 1 of 24 = X
9. \ of 72 = X
15. ^ of 54 = X
4. 4 of 32 = X
10. \Qi 9 = X
16. ^ of 63 = X
5. i of 40 = X
//. -iof 18 = X
n. ^ of 72 = X
6. -^ of 48 == X
7
12. i of 27 = X
18. i of 81 = X
98 WHOLE NUMBERS AND FRACTIONAL PARTS.
202. Oral Exercise.
Read Exercise 201, supplying the missing numbers.
203. Exercise.
Copy, supplying the missing numbers :
/. 8 = — ones. 7. 27 — — threes. 13. 72 = — nines.
2. 18 = — twos. 8. 56 = — sevens. 14. 04 = — oights.
3. 24 = — threes. 9. 45 = — fives. 15. 40 = — fives.
4. 32 = —fours. 10. 36 = —fours. 16. 63 = —sevens.
5. 9 = — ones. //. 48 = — sixes. 17. 72 = — eights,
ff. 16 = — twos. 12. 54 = — sixes. 18. 81 = — nines.
204. Oral Exercise.
Read Exercise 203, supplying the missing numbers.
205. Exercise.
/. 18 cents will buy — 2-cent stamps.
2. How many 8-foot logs can be cut from a log 40 feet
long ?
3. Make and solve a problem about ^ of 16 boys.
4. 800 posts cost 56 dollars ; how much was that for
each hundred posts.
5. My brother and I bought 9 dozen tomato plants for
72 cents ; how much was that a dozen ? How much
should he pay if he took 6 dozen ?
6. A log 42 feet long will make a 10-foot log and how
many 4-foot logs ?
7. A grocer put 40 pounds of sugar in 8 sacks of the
same size ; how much did he put in each sack ?
MULTIPLICATION, DIVISION, MENSURATION. 99
8. A farmer had 20 pigs. lie kept 4 of them and sold
the rest ; how many pair did he sell ?
9. How many 5-dollar bills should I get for 45 silver
dollars ?
10. A farmer needs 24 pounds of twine to bind his
wheat ; how many 3-pound balls must he buy ?
11. A woman sold a dealer chickens at 8 cents a pound ;
he paid her 72 cents for them ; how many pounds did she
sell him ?
12. I wish to plant 64 trees in 8 rows ; how many must
I put in a row ?
2 .r «
206. Find f of 9.
T0f9
Find f of 15.
i of 15 is 5, and f of 15 is 2 times 5 or 10.
207. Oral Exercise.
Find:
/. f of 6. 5. f of 12. P. ^ f of 28. 13. f of 20.
2. f of 3. 6. f of 21. 10. J of 32. 14. f of 30.
5. I of 9. 7. J of 12.* //. jof24. /5. I of 45.
4. f of 15. S. I of 16. 12. f of 10. 16. I of 25.
208. Exercise.
/. J of 27 ct. is — ct. ; and | of 27 ct. is 2 times — ct.,
or — ct.
2. What is I of 6 pounds ?
100 WHOLE NUMBERS AND FRACTIONAL PARTS.
3. What is I of 12 quarts ?
4. J of 16 bushels is — bushels, and } of 16 bushels is
3 times — bushels, or — bushels.
5. Find J of 12 oranges.
6. Find f of 10 cents.
7. If 18 cents was the cost of 3 pounds of sugar, J of
18 cents, or — cents, was the cost of 1 pound, and 2 times
— cents, or — cents, was the cost of 2 pounds.
8. How much should be paid for 2 pencils, if 3 pencils
cost 15 cents ?
9. A baker charged me 12 cents for 3 loaves of bread ;
what should he charge me for 2 loaves ?
10. If 3 horses are fed 24 quarts of oats a day, how
much does that allow for 2 horses ?
11.1 owed a 16-dollar store bill and paid J of it ; how
much did I pay ?
12. Martha is 20 years old and May is J as old ; how
old is May ?
13. A boy paid 10 cents for a tablet, and J as much for
a ruler ; how much did he pay for both ?
14. If 5 loaves of bread cost 30 cents, 1 loaf costs J of
30 cents, or — cents ; and 4 loaves cost 4 times — cents,
or — cents.
15. Wliat should I be paid for 3 days' work, if I was
paid 15 dollars for 5 days' work ?
16. A boy was given 25 words to spell ; he missed | of
them ; how many did he spell correctly ?
17. There are 21 dots in three rows, each containing
the same number ; how many dots are there in 2 rows ?
MULTIPLICATION, DIVISION, MENSURATION. 101
18. A grocer charged me 9 cents for 3 lemons, bat one
of them was bad ; how much should he have charged me
for the good lemons ?
19. I just paid $30 for 3 months' rent ; I now wish to
pay 2 months' rent in advance ; how much should I pay ?
20. If 4 persons use 8 pounds of meat a week, how much
should 5 persons use ?
209. Exercise.
Copy and multiply :
/. 27 6. 19 //. 18 16. 13 21. 128 26. 123
6 7 8 9 8 8
2. 39 7. 28 12. 26 17. 24 22. 109 27. 109
_6 _7 _8 _9 7 8
3. 58 8. 36 13. 74 18. 37 23. 138 28. 108
_6 ^ _8 _9 7 9
4. 66 9. 54 14. 93 19. 55 24. 145 29. 117
_6 _7 _8 __9 6 8
5. 79 10. 70 15. 50 20. 86 25. 139 30. 122
6 7 8 9 6 8
210. Multiply $2.38 by 6.
$2.38 Multiply as in ordinary numbers, and place the
Q decimal point before the last two figures of the
$14 28 pro^^uct to separate dollars from cents.
'io'S WHOL^' NUMBERS AND FRACTIONAL PARTS.
211. Exercise.
Multiply :
/. $4.28 3. $0.56
4 9
5. $48.07
8
7. $27.08
8
2. $8.72 4. $138.40
7 6
6. $94.18
9
8. $125.15
5
9. $3.25 by 5. //. $21.38 by 8. /5. $38.90 by 7.
10. $19.20 by G. 12. $75.03 by 9. 14. $1025.03 by 8.
212. Exercise.
Copy and fill in the missing amounts in the following:
/. The cost of :
6 plum trees at $1.25 each = $,
8 peach trees at $0. 80 each =
7 pear trees at $1.10 each =
9 apple trees at $0.75 each =
Total =i
The cost of :
4 kegs of nails at $2.75 each = $. . .
6 bushels of clover seed at
$5.75 each = ...
7 tons of bran at $18. (;0 each = ...
8 tons of coal at $5.75 each = . . .
Total =$...
MULTIPLICATION, DIVISION, MENSURATION. 103
3. The cost of :
9 sheep at 16.75 each =
5 horses at $125 per head. —
8 cows at $40.50 per head. =
8 hogs at $8.75 per head. . =
Total =
213. Exercise.
/. Find the cost of 6 yards of cloth at 15 cents a yard.
2. A car that makes 24 trips a day makes how many
trips a week ?
3. How many hours are there in a week ?
4. A gardener planted 8 rows of trees ; he planted 16
trees in each row ; how many trees did he plant in all ?
5. A farmer sold 8 loads of wheat ; there were 25 bushels
in each load ; how many busiiels were there in all ?
6. A bushel of potatoes weighs 60 pounds ; find the
weight of 8 bushels of potatoes.
7. A bushel of corn weighs 56 pounds ; find the weight
of 6 bags of corn, each holding 3 bushels.
8. How many eggs will 9 crates hold, if each crate holds
6 dozen ?
9. How many persons can be seated in 8 rows of chairs,
if there are 38 chairs in each row ?
10. How many persons can be seated in 6 cars, if there
are 36 seats in each car and 2 persons can sit on each seat?
//. A dealer sold to each of 8 men 125 barrels of flour
and has 50 barrels left ; how many barrels had he at first ?
104 WHOLE NUMBERS AND FRACTIONAL PARTS.
12. My lot cost $125, and my house 6 times as much ;
how much did they both cost ?
214. Oral Exercise.
20 30 40 50 60 70 80 90
Multiply each of the numbers above the line
/. By 2.
Thus, 3 times 20 are 40 ; 2 times 30 are GO ; etc.
2. By 3.
4.
By 5. 6.
By
7.
8. By 9.
3. By 4.
5.
By 6. 7.
By
8.
215. Oral Exercise.
Multiply :
/. 25 by 3.
Thus, 3 times 20
are
60 and 3 times 5 are 15 ;
60 and 15 are 75.
2. 32 by 2.
9. 43 by 5.
16.
02 by 7.
3. 48 by 2.
10. 38 by 5.
17.
84 by 8.
4. 34 by 3.
//. QQ by 5.
18.
15 by 8.
5. 25 by 3.
12. 72 by 6.
19.
30 by 8.
6. 43 by 4.
13. 43 by 0.
20.
02 by 9.
7. 50 by 4.
14. 25 by 6.
21.
2b by 8.
8. QQ by 4.
15. 75 by 6.
22.
30 by 8.
216. Exercise.
/. AVhat is the cost of 2 handkerchiefs at 18 cents each ?
2. What is the cost of 3 hatchets at 30 cents each ?
3. How many boxes of rivets are there in 6 packages,
each package containing a dozen boxes ?
4. How many yards of picture cord are there in 4 coils,
each containing 25 yards ?
MULTIPLICATION, DIVISION, MENSURATION. 105
5. A lady bought 2 yards of gingham at 18 cents a
yard; how much change should she receive from 50
cents ?
There are 16 oimces {oz.) in a pound {lb.) avoirdupois.
6. How many ounces are there in 5 pounds avoirdupois ?
7. How many ounces are there in 3 pounds 2 ounces ?
8. 4 strips of carpet, each 8 yards long, are needed to
carpet a certain floor ; how many yards are required ?
9. Find the cost of 3 yards of muslin at 12 cents a yard,
and 3 yards of flannel at 20 cents a yard.
10. How much should a trader pay for 2 dozen eggs at
15 cents a dozen and 3 pounds of butter at 20 cents a
pound ?
//. How many are 4 score and 10 ?
12. How many are a half dozen dozen ?
13. How much should a bushel of potatoes cost at 15
cents a peck ?
14. How much is received for a gallon of cream at 16
cents a quart ?
15. If a peck of clover seed weighs 15 pounds, what
does a bushel weigh ?
16. The distance between two towns is 12 miles ; how far
docs a motorman ride each day, if he makes 4 round trips
on a trolley line connecting these towns ?
217. 3 rows of 4 dots each contain #^00
as many dots as 4 columns of 3 dots # • # ^
each, which shows that 3x4 = 4x3. #•••
106 WHOLE NUMBERS AND FRACTIONAL PARTS.
218. Oral Exercise.
Multiply :
/. 2 by 20. Thus, 20 times 2 equals 2 times 20, or 40.
2. 2 by 25. 6. 3 by 60. 10. 4 by 60. 14. 6 by 48.
3. 2 by 40. 7. 3 by 25. //. 5 by 25. 15. 7 by 14.
4. 2 by 26. 8. 3 by 42. 12. 5 by 50. 16. 8 by 24.
5. 2 by 54. 9. 3 by 18. 13. by 36. 17. 9 by 31.
219. Exercise.
/. A boy sold 24 newspapers at 2 cents each ; how much
did he receive for them ?
Since he sold one newspaper for 2 cents, he sold 24 for 24 times
2 cents, which equals 2 times 24 cents, or — cents.
2. Find the cost of 20 two-cent postage stamps.
3. A trolley car conductor collected on one trip 5 cents
from each of 18 persons ; how much did he collect ?
4. How many persons are seated at 16 tables, if there
are 4 persons at each table ?
5. How many days are there in 21 weeks ?
6. How much does it cost to ride 28 miles on a railroad,
if the fare is 3 cents a mile ?
7. If school is in session 6 hours a day for 20 days of a
month, how many hours is it in session in a month ?
8. Find the gain on a dozen tablets bought for 50 cents
and sold at 6 cents each.
220. Find the cost of 126 yards of cloth at 5^ a yard.
$1.26 126 X 5^ =the cost.
5 But 126 X 5^ = 5 X 126^, or 5 x |1.26.
rrri Therefore, 5 x $1.26, or $6.30 = the cost.
In practice, multiply by the smaller mtn^>er.
MULTIPLICATION, DIVISION, MENSURATION. 107
221. Exercise.
Find the cost of :
/. 126 bushels of seed at 15 per bushel.
2. 228 pounds of soap at 5^ per pound.
3. 476 pounds of sugar at 6^ per pound.
4. 208 quarts of milk at 6^* per quart.
5. 165 barrels of apples at $4 per barrel.
6. 125 yards of carpet at $2 per yard.
7. 288 fruit trees at $1 each.
8. 385 gallons of oil at 90 per gallon.
9. A bushel of chestnuts at 8^ a quart.
10. 75 cans of corn at 8^ each.
222. Exercise.
Copy and divide :
/. Q)Q^
10. 7)84
19. 6)954
25. 8)624
2. 6)60
//. 8)96
20. 6)750
29. 8)920
3. 7)77
' 12. 7)98
21. 7)147
50. 8)704
4. 7)70
13. 6)102
22. 7)791
31. 9)819
5. 8)88
14. 9)108
23. 7)812
52. 9)765
6. 6)84
15. 8)104
24. 7)903
55. 9)648
7. 6)90
16. 6)366
25. 7)814
34. 9)531
8. 6)96
17. 6)672
26. 8)248
55. 9)396
9. 7)91
18. 6)804
27. 8)896
36. 9)612
223. Divide $18.75 by 5.
5 "i 118 75 Divide as in ordinary numbers, and place
"T IT the decimal before the last two figures to sepa-
rate dollars from cents.
108 WHOLE NUMBERS AND FRACTIONAL PARTS.
224. Exercise.
Divide :
/. 2 )118.72 5. y )$35.14 9. $178.50 by 5
2. 3 )$31.05 6. 9 )1193.05 10. $374.24 by 4
3. 6 )115.48 7. 5 )1100.10 //. $127.00 by 5
4. 8 )133.60 8. 6 )$127.08 12. $38.07 by 9
225. Exercise.
Find the cost of 1 pound of each, if :
/. 6 lb. of coffee cost $2.10.
2. 8 lb. of dried beef cost $2.00.
5. 5 lb. of ham cost $0.80.
4. 4 lb. of tea cost $2.80.
5. 7 lb. of butter cost $1.96.
6. 9 lb. of cheese cost $1.44.
7. 3 lb. of yarn cost $1.05.
8. 8 lb. of prunes cost $1.20.
226. Exercise.
/. A lady bought 6 yards of cloth for 96 cents ; how
much was it a yard ?
2. A dealer sold 6 horses for $900 ; he received the
same sum for each ; how much did he receive for each ?
3. 7 men caught 819 shad and shared them equally ;
how many did each receive ?
4. A man earned $84 in 6 weeks and his son earned $66 ;
liow much did both earn ? How much did both earn in
a week ?
MULTIPLICATION, DIVISION, MENSURATION. 109
5. A man rode 644 miles on Tuesday and \ as far on
Wednesday ; how far did he ride in the two days ?
6. In 9 class-rooms there are 405 desks ; if each room
contains the same number of desks, how many are there
in each room ?
7. A dealer had 500 dressed chickens ; he sold 68 of
them and shipped the others in 6 barrels of equal size ;
how many did he put in each barrel ?
8. Divide 818 - 106 by 8.
9. I owed $718, and paid $32; I wish to pay the
rest in 6 equal parts ; how much must I pay in each
part ? *
10. Make and solve a problem about 672 trees that were
sold during 6 days.
//. Divide 386 + 428 - 128 by 7.
12. Divide $648 among 3 persons, giving the first
person ^ of the money, the second ^ of it, and the third
the remainder.
227. Exercise.
Measure :
/. 96 by 6. 5. 402 by 6. 9. 801 by 9.
2. 108 by 9. 6. 833 by 7. 10. 726 by 6.
3. 272 by 8. 7. 504 by 9. //. 749 by 7.
4. 896 by 7. 8. 376 by 8. 12. 816 by 8.
228. Measure $12.05 by $0.05.
$13.05 : $0.05 = 12050 : 5^ = 1205 -- 5, or 241.
110 WHOLE NUMBERS AND FRACTIONAL PARTS.
229. Exercise.
Measure :
/. $4.24 by $0.04. 7. $12.72 by 6^.
2. $5 by 5^'. 8. $0.96 by 4^.
3. $18.36 by $0.09. 9. $12.50 by 5^.
4. $12.32 by $0.08. /O. $12.25 by 5^.
5. $50 by $0.05. //. $81 by 9^.
6. $27.99 by $0.09. 12, $120 by 80.
230. Exercise.
Find how many can be bought :
/. 2-cent postage stamps for $1.72.
2. 5-cent tablets for $11.65.
3. 3-cent pencils for $0.84.
4. 5-ceut loaves of bread for $2.40.
5. 4-cent boxes of matches for $0.60.
6. 8-cent spools of thread for $1.12.
7. 8-cent note books for $3.44.
8. 5-cent spools of thread for $1.
231. Exercise.
/. llow many pecks are there in 216 quarts ?
2. How many weeks are there in 364 days ?
3. How many trains of 9 cars each can be made up
from 189 cars ?
4. How many weeks will it take to get $344 by saving
$8 a week ?
5. Find the cost of 108 marbles, if 9 are bought for a
cent.
MULTIPLICATION, DIVISION, MENSURATION. 1 1 1
6. How many 8-cent stamps can you buy for 96 cents ?
7. How long will it take a boat to sail 224 miles, if it
sails 8 miles an hour ?
8. If coal costs $6 a ton, how many tons can be bought
for $216 ?
9. How many groups of 6 marbles each can be made
with 150 marbles ?
10. A dealer bought 8 dozen hats in boxes of 6 hats
each ; how many boxes were there ?
//. A dealer bought 750 cigars, and after selling 324 of
them, put the rest in packages of 6 each ; how many
packages had he ?
12. If the distance around a wheel is 6 feet, how many
times will it turn in going 882 feet ?
13. 144 trees can be planted in rows of 6 trees each, 8
trees each, or 9 trees each ; how many rows of each kind
would they make ?
232. Exercise.
/. What is each of the three equal parts
of 1 called ?
2. How many thirds are there in 1 ?
r 3. Read : i ; f ; |.
4. What is each of the six equal parts of
1 called ?
5. What are 2 of the six equal parts of 1
called ?
112 WHOLE NUMBERS AND FRACTIONAL PARTS.
6. Kead : J ; ^ ; f ; S ; I ; f
7. In J there are how many sixths ?
8. In f there are how many sixtlis ?
9. In J there are how many sixths ?
10. Show that ^ of 2 is |, or f
//. Show that ^ of 3 is J, or J.
12. Show that ^ of 4 is ^, or f.
/3. Show that J of 5 is |,
233. Find ^ of 26.
Answer thus : ^ of 26 is 4, and 3 over ; ^ of 2 is |, or §.
The answer is 2^.
234. Oral Exercise.
Find :
/. J of 7. 4. I of 13. 7. J of 28. 10. \ of 34.
2. ^ of 8. 5. ^ of 16. S. J of 25. //. J of 45.
5. I of 9. 6. ^ of 20. 9. J of 40. 12. J of 50.
235. How many 6's are there in 16 ? 16 = 2| sixes.
Supply the missing numbers, using drawings to illus-
trate :
/. 8 = — sixes. 4. 15 = — sixes. 7. 28 = — sixes.
2. 9 = — sixes. 5. 17 = — sixes. 8. 33 = — sixes.
5. 14 = — sixes. 6. 20 = — sixes. 9. 16 = — sixes.
MULTIPLICATION, DIVISION, MENSURATION. 113
237. Exercise.
/. If G pounds of soap cost 25 cents, how much is the
soap a pound ?
2. A dealer has 50 handkerchiefs which he wishes to
put up in boxes of half a dozen each ; how many boxes
will there be, and how many handkerchiefs over ?
3. What is each of the seven equal parts of 1 called ?
4. Show that i of 2 is f .
5. Read : f ; f ; 4 : f ; f ; -I-
6. Show that there are 2f sevens in 16.
7. Find \ of each number in the outer ring of the left-
hand design.
8. Find how many 7's there are in each number in the
outer ring of the right-hand design.
9. 25 days are — weeks — days.
10. K man who earns 15 dollars in the G working days
of a week earns how much a day ?
//. January has 31 days ; this is how many weeks and
days ?
12. April has 30 days ; this is how many weeks and days?
8
114 WHOLE NUMBERS AND FRACTIONAL PARTS.
13. When board costs 10 dollars a week, how much is it
per day ?
14. How many pounds of sugar at 6 cents a pound can
be bought for 8 pounds of duck at 7 cents a pound ?
238. Oral Exercise.
\
1. What is each of the 8
equal parts of 1 called ?
2. How many eighths are
there in 1 ?
3. How many eighths are
there in J ? in J ?
4. How many eighths are there in J ?
5. Eead : I ; f ; I ; I ; S ; |.
6. How many fourths are there in f ? in ^ ? in | ?
7. How many halves are there in ^ ? in f ?
8. What is each of the 9 equal parts
of 1 called ?
9. How many ninths are there in 1 ?
10. How many ninths are there in \ ?
inf ?
//. Read: | ; 4 ; | ; | ; 3 ; 4 ; i ; i
12. How many thirds are there in ^ ? ^ ?
1
..............1
1
9
J !
1 f
j I
239. Exercise.
/. Show that i of 2 is |, or J.
2. Show that i of 4 is |, or J.
3. Show that ^ of 6 is |, or J.
4. Show that -^ of 3 is ^, or J.
5. Show that i of G is i or |.
MULTIPLICATION, DIVISION, MENSURATION. 115
240. Oral Exercise.
What is :
/. iof 5? 5. ^ of 30? //. iof 25 ?
2. ^ of 25 ? 7. ^ of 35 ? 12. 1 of 30 ?
5. i of 20 ? 8. \ of 42 ? /5. ^ of 38 ?
4. ^ of 22 ? 5. ^ of 50 ? /4. ^ of 40 ?
5. ^ of 18 ? 10. I of 12 ? 15. ^ of 53 ?
241. Exercise.
/. Show that there are 2 J eights in 20.
2. In 15 quarts there are — pecks — quarts.
3. In 25 quarts there are — pecks — quarts.
4. Show that there are 2f nines in 20.
5. Show that there are 2^ nines in 21.
6. If a peck of plums sold for 50 cents, how much were
they a quart ?
7. 25 cents will pay for how many pounds of rice at 8
cents a pound, and how many cents will be left over ?
8. 38 dots will make how many lines of 9 dots each,
and how many dots will be over?
242. Exercise.
/. How many days are there in G weeks ?
2. 30 quarts are — pecks — quarts.
5. 2 sets of casters, 4 in a set, cost 20 cents ; how much
did each caster cost ?
4. Find the gain on 40 papers bought at 1 ct. each and
sold at 2 ct. each.
5. Since 16 ounces make a pound, how many pounds
will 2 cakes of soap weigh, if each cake weighs 8 ounces ?
116 WHOLE NUMBERS AND FRACTIONAL PARTS.
6. How old is a man who was 40 years old 16 years ago ?
7. A trolley car that makes 2 round trips an hour over
a road 3 miles long, runs how far each hour ?
8. Supply the missing numbers :
1 — t — t = t = t = ff = f — ¥ = t"
9. Since — pints make a quart, and — quarts make a
gallon, there are — times — pints, or — pints, in a gallon.
10. At 8 cents a pint, what is the selling price of a gal-
lon of cream ?
11. A. dime, a 5-cent piece, and a one-cent piece are
together worth how much less than 2 dimes ?
12. How many crates does a market man require to pack
54 dozen eggs, if he puts 6 dozen in a crate ?
13. — pints make a gallon; how many gallons are
there in 64 pints ?
243. Oral Exercise.
Find :
/. f of 12. 6. f of 35. //. 4 of 70. 16. | of 48.
2. I of 18. 7. f of 28. 12. | of 16. 17. I of 80.
3. I of 30. 8. 4 of 21. 13. I of 32. 18. f of 27.
4. I of 24. 9. 4 of 42. 14. | of 40. 19. ^ of 36.
5. f of 14. 10. f of 49. 15. I of 24. 20. \ of 18.
244. Exercise.
/. If times a number is 30, the number is what part
of 30 ? What is the number ?
2. If times a number is 24, the number is — of 24, or
— ; and 5 times the number is — times — , or — .
MULTIPLICATION, DIVISION, MENSURATION. 117
3. If 7 times a number is 56, what is 6 times the number?
4. ^^ of a score are how many ?
5. A farmer had 63 sheep and sold ^ of them ; how many
had he left ?
6. I of a week is f of — days, or — days.
7. A turkey weighing 13 pounds, live weight, weighed
only I as much when dressed ; how much did it lose in the
dressing ?
8. 7 cans of lard of equal size weighed 70 pounds ; how
much did 2 of them weigh ?
9. If 6 dozen bananas cost 60 cents, what should 3 dozen
cost?
10. If 6 hundred rails sold for 48 dollars, 2 hundred
should sell for — dollars.
//. I have 42 tablets in 7 packages of equal size ; how
many tablets are there in 3 of these packages ?
12. A miner earned $30 a month and his son ^V ^^
much ; how much did both earn a month ?
13. If a turkey weighing 9 pounds sold for 90 cents,
what should one weighing 8 pounds sell for ?
14. In taking 8 steps a man walked 24 feet ; how far
had he walked when he had taken 6 steps ?
15. A father gave each of his 3 sons ^ of a farm of 56
acres ; how many acres had he left ?
16. If 8 times a number is 32, what is 7 times the
number ?
17. A of 90 cents is — cents, or — dimes.
18. A carpenter bought 18 hinges for 9 doors ; how many
will he use to hang 4 doors ?
1 1 8 WHOLE NUMBERS AND FRACTIONAL PARTS.
246. Exercise — Miscellaneous Problems.
/. Write the letters used in the Roman system of nota-
tion and the number for which each stands.
2. Express by the Roman system :
19; 44; 406; 1909.
3. Express in figures :
XCIV; MCD; MCMVI; CMI ; VIV.
4. In $7486, the 7 expresses how many dollars ? the
4 ? the 8 ?
5. 87 + 168 + 326 + 308 = ?
6. 327 + 109 + 426 + ? = 1000.
7. 1487 - 896 = ?
8. 2939 - ? = 1000 ?
9. ? - 526 = 879 ?
10. K dealer bought seventeen hundred forty bushels
of wheat and sold eleven hundred ninety bushels of it ;
how much had he left ?
//• What number is 101 greater than 999 ?
12. A. man bought a house costing $7250 and paid
$4750 on it ; how much does he yet owe ?
13. A man bought a vest for $2.75 and a pair of shoes
for $3.50 ; how much change should he receive from a
$10-bill ?
14. From twenty, dollars and seventy-five cents take the
sum of four dollars and twenty cents and six dollars and
eighty-five cents.
15. A farmer bought a horse for $160 and sold him so
MISCELLANEOUS PROBLEMS. 119
as to gain ^ of this sum ; find the selling price of the
horse.
16. There were 1685 soldiers in a camp, but f of them
were sent away ; how many remained ?
17. Which is nearer 1000 and by how much, 899 or
1101?
18. From 4275 take 36 hundred.
19. How old was my father 25 years ago, if he is 74
years old now ?
20. If the first problem on a page is the 128th and the
last the 153d, how many problems are there on the page ?
21. Add down ; add across :
448 + 327 + 1320 =
79 + 108+ 768 =
175 + 927 + 7856 =
8 + 128 + 96 =
. + . + . =
22. Find the missing amounts by adding down and
subtracting across. See that the answer to (7) is the sum
of the amounts found for column (c).
(a)
(b)
(c)
(1)
$6750.30 -
$1007.50 =
p
(2)
35.35 -
20.95 =
?
(3)
180.00 -
178.25 =
?
(4)
3738.00 -
987.95 =
p
(5)
3753.60 -
8.85 =
p
(6)
37.85 -
5
13.70 =
?
(7)
?
?
120 WHOLE NUMBERS AND FRACTIONAL PARTS.
246. The following table, called the Multiplica-
tion Table, shows the results obtained by taking
each of the numbers from 1 to 12, inclusive, from 1
to 12 times.
multiplication table.
1 X
1= 1
2x1=2
3x1=3
4x1=4
1 X
2= 2
2x2=4
3x2=6
4x2=8
1 X
3=3
2x3=6
3x3=9
4 X 3 = 12
1 X
4= 4
2x4=8
3 X 4 = 12
4 X 4 = 16
1 X
5= 5
2 X 5 = 10
3 X 5 = 15
4 X 5 = 20
1 X
6= 6
2 X 6 = 12
3 X 6 = 18
4 X 6 = 24
1 X
7= 7
2 X 7 = 14
3 X 7 = 21
4 X 7 = 28
1 X
8= 8
2 X 8 = 16
3 X 8 = 24
4 X 8 = 32
1 X
9= 9
2 X 9 = 18
3 X 9 = 27
4 X 9 = 36
1 X
10 = 10
2 X 10 = 20
3 X 10 = 30
4 X 10 = 40
1 X
11 = 11
2 X 11 = 22
3 X 11 = 33
4 X 11 = 44
1 X
12 = 12
2 X 12 = 24
3 X 12 = 36
4 X 12 = 48
5 X
1 = 5
0x1=6
7x1=7
8x1=8
5 X
2 = 10
6 X 2 = 12
7 X 2 = 14
8 X 2 = 16
5 X
3 = 15
6 X 3 = 18
7 X 3 = 21
8 X 3 = 24
5 X
4 = 20
6 X 4 = 24
7 X 4 = 28
8 X 4 = 32
5 X
5 = 25
6 X 5 = 30
7 X 5 = 35
8 X 5 = 40
5 X
6=30
6 X 6 = 36
7 X 6 = 42
8 X 6 = 48
5 X
7 = 35
6 X 7 = 42
7 X 7 = 49
8x 7 = 56
5 X
8 = 40
6 X 8 = 48
7 X 8 = 56
8 X 8 = 64
5 X
9 = 45
6 X 9 = 54
7 X 9 = 63
8 X 9 = 72
5 X
10 = 50
6 X 10 = 60
7 X 10 = 70
8 X 10 = 80
5 X
11 = 55
6 X 11 = 66
7 X 11 = 77
8 X 11 = 88
5 X
12 = 60
6 X 12 = 72
7 X 12 = 84
8 X 12 = 96
9 y
1= 9
10 X 1 = 10
11 X 1 = 11
12 X 1 = 12
9 X
2= 18
10 X 2 = 20
11 X 2 = 22
12 X 2= 24
9 X
3= 27
10 X 3 = 30
11 X 3= 83
12 X 3 = 36
9 X
4= 36
10 X 4 = 40
11 X 4= 44
12 X 4 = 48
9 X
5= 45
10 X 5= 50
11 X 5= 55
12 X 5 = 60
9 X
6= 54
10 X 6 = 60
11 X 6= 66
12 X 6= 72
9 X
7= 63
10 X 7 = 70
1 1 X 7 = 77
12 X 7 = 84
9 X
8= 72
10 X 8 = to
11 X 8= 88
12 X 8 = 96
9 X
9= 81
10 X 9 - 90
1 1 X 9 ^ 99
12 X 9 = 108
9 X
10= 90
10 X 10= 100
11 X 10= 110
12 X 10 = 120
9 X
U = 99
10 X 11 ■--- 110
11 X 11 = 121
12 X 11 = 132
9 X
12 = 108
10 X 13 = 120
11 X 12 = 132
12 X 12 = 144
MULTIPLICATION, DIVISION, MENSURATION. 121
247. 10 X 5 = 50. From this we see that
Annexing a cipher to a number multiplies it
by 10.
248. Oral Exercise.
Read, supplying the missing numbers :
/. 10 X 7 =
2. 10 X 9 =
5. 10 X 13 =
4. 10 X 17 =
5. 10 X 27 =
6. 10 X 35 =
7. 10 X 42 =
8. 10 X 54 =
9. 10 X 61 =
10. 10 X 75 =
//. 10 X 29 =
12. 10 X 80 =:
13. 10 X 225 =
14. 10 X 201 =
15. 10 X 278 =
16. 10 X 370 =
17. 10 X 300 =
18. 10 X 990 =
249. 100 X 5 = 5 X 100 = 500. From this we see
I that
Annexing two cipJiers to a number multiplies it
by 100.
250. Oral Exercise.
Read, supplying the missing numbers :
/. 100 X 7 =
2. 100 X 9 =
3. 100 X 2 =
4. 100 X 6 =
5. 100 X 21 =
6. 100 X 15 =
7. 100 X 35 =
8. 100 X 20 =
9. 100 X 48 =
10. 100 X 99 =
//. 100 X 70 =
12. 100 X 10 =
13. 100 X 25 =
14. 100 X 72 =
15. 100 X 69 =
251. 1000 X ^ = ry X 1000 = 5000. From this we
see that
A nnexing three ciphers to a number multiplies it
by 1000.
122 WHOLE NUMBERS AND FRACTIONAL PARTS.
262. Oral Exercise.
Read, supplying the missing numbers :
/. 1000 X 2 = . 4. 1000 X 1 = . 7. 1000 x 8 = •
2. 1000 X 4 == . 5. 1000 x = . 8. 1000 x 3 = .
5. 1000 X 7 = • 6. 1000 X 5 = . 9. 1000 x 6 = •
253. Exercise.
/. 100 dimes are worth — cents.
2. At 15 cents a quart, how much should be received
for 10 quarts of chestnuts ?
3. Find the weight of 10 shad, if they average 8
pounds a pair.
4. What must be paid for 100 postal cards ?
5. A drover bought 100 head of steers at an average of
$35 a head ; how much did he pay for them ?
6. Find the weight of 1000 two-pound sacks of salt.
7. Find the cost of a 100-acre farm at $20 an acre.
8. How many ounces are there in 100 pounds avoirdu-
pois ?
There are 12 months (mo.) in a year (yr.).
9. 100 years is called a century ; how many months
are there in a century ?
10. K keg of nails weighs 100 pounds ; find the weight
of 15 kegs of nails.
SUG. 15 X 100 lb. = 100 X 151b.
//. There are 100 cents in a dollar ; liow many cents are
there in 5 dollars ?
MULTIPLICATION, DIVISION, MENSURATION. 123
254. Multiply 27 by 300.
27
Since 300 = 100 x 3,
300 X 27 = 100 X 3 X 27 = 100 x 81 = 8100.
8100
255. Exercise.
Find the product of :
/. 20 X 15. 5. 50 X 33. 9. 200 x 35.
2. 30 X 45. 6. 60 X 75. fO. 200 x 47.
3. 40 X 62. 7. 70 X 38. //. 200 x 35.
4. 40 X 60. 8. 70 X 80. 12. 300 x 28.
266. Exercise.
/. Find the cost of 40 sewing machines at $28 each.
2. Find the tuition paid by 45 students, if each pays
1200.
SuG. 45 X $200 = 200 x $45.
5. How many feet of boards does a man buy, if he buys
500 boards each 18 feet long ?
4. How much fertilizer must be bought for a 12-acre
field, if 200 pounds are required for an acre ?
6. A barrel of beef weighs 200 pounds ; find the weight
of a shipment of 17 barrels of beef.
6. How many yards are there in 12 pieces of bunting,
each containing 50 yards ?
7. There are 24 sheets in a quire, and 20 quires in
a ream ; how many sheets are there in a ream of paper ?
8. How much house rent does a man pay in 3 years,
if he pays $20 a month ?
124 WHOLE NUMBERS AND FRACTIONAL PARTS.
267. Multiply 374 by 24.
374
24
We first multiply 374 by 4, obtaining 1496 for the prod-
uct. We next multiply 374 by 2, obtaining 748 for the
1496 product ; and since the 2 is 3 tens, we write 748 so that
748 the 8 stands in tens' place. The sum of these products,
QQ7fi o^ the entire product, is 8976.
268. Exercise.
Copy and multiply :
/. 43 8. 72 15. 59 22. 342 29. 198
13 87 57 13 39
64
9. 79
f6. 86
23. 452
30. 276
26
77
76
14
42
3. 94
10. 81
17. 38
24. 209
31. 378
28
98
57
16
28
4. 85
11. 45
18. 92
25. 390
32. 309
52
56
21
21
27
95
12. 99
19. 47
26. 409
33. 290
36
91
19
21
39
6. 96
13. 65
20. 51
27. 580
34. 409
49
37
99
22
26
7. 57
14. 48
21. 427
28. 214
55. 275
66
23
12
24
27
MULTIPLICATION, DIVISION, MENSURATION. 125
259. Exercise.
/. Find the weight of 25 bushels of oats, if they weigh
32 pounds to the bushel.
2. How many hours are there in May, which has 31
days ?
3. How far will a steamboat run in a day, if it runs 16
miles an hour ?
4. A barrel of flour weighs 196 pounds ; find the weight
of barrels of flour.
5. Find the cost of 25 thousand feet of boards at $32
per thousand feet.
6. Find the cost of 16 hundred fence posts at $18 per
hundred.
7. A man left $8975 to his wife and 4 children, the
children receiving $1250 each ; how much did the wife
receive ?
8. A dealer bought 28 grain drills at $62 each and sold
them at 1100 each ; find his gain.
SuG. $100 - $62, or $38 = the gain on each.
9. A farmer bought 15 steers at $35 a head, and after
feeding them for 6 months at a cost of $40 a montli,
sold them at an average of $58 a head ; find his
gain.
10. If a clerk's wages are $30 a month, find his wages
for a year.
/ /. 52 weeks and 1 day make a common year ; how many
days make a common year ?
12. A. farmer put 25 loads of lime on his lan^, each
126 WHOLE NUMBERS AND FRACTIONAL PARTS.
load averaging 75 bushels ; how much lime did he put on
his land ?
13. How many dozen eggs will 24 crates hold, if each
crate holds 72 dozen ?
260. Divide 435 by 4.
i of 435 = i of (4 hundred + 3 tens + 5 units).
4 ) 435 i of 4 hundred = 1 hundred.
108f i of 3 tens = tens, and 3 tens over.
3 tens + 5 units = 35 units.
I of 35 units = 8 units, and 3 units over,
i of 3 units = f of a unit. The quotient is 108f,
261. Exercise.
Copy and divide :
/. 2)37
10. 5)72
19. 7)93
25. 8)500
2. 2)51
//. 5)83
20. 7)99
29. 8)362
3. 3)40
12. 5)99
21. 7)130
50. 8)815
4. 3)47
13. 5)93
22. 7)152
5/. 8)854
5. 4)57
14. 6)79
23. 7)230
32. 8)963
6. 4)74
15. 6)87
24. 7)136
55. 9)200
7. 4)59
16. 6)92
25. 8)257
34. 9)309
8. 4)94
17. 6)89
26. 8)45^
55. 9)401
9. 5)67
18. 6)94
27. 8)005
5tf. 9)118
MULTIPLICATION, DIVISION, MENSURATION. 127
262. Exercise.
/. When handkerchiefs are sold at 2 for 25 cents, how
much are they apiece ?
2. If a turkey weighing 6 pounds sold for 75 cents, how
much was that a pound ?
3. In 2 rods there are 33 feet ; how many feet are there
in 1 rod ?
4. How much must be paid for a pint of varnish at 65
cents a quart ?
5. If 2 dressed hogs weigh 625 pounds, what is their
average weight ?
6. What was the price paid per acre, if a lot of 7 acres
cost $825 ?
7. What was the average weight of 2 steers, if one
weighed 875 pounds and the other 964 pounds ?
8. What is the average age of the five members of a
graduating class, if their ages are 17 years, 16 years, 18
years, 15 years, and 15 years, respectively ?
9. A farmer paid a boy $110 for 9 months' work ; what
were the monthly wages paid ?
10. A lady paid 50 cents for 4 yards of velvet trimming ;
how much was it a yard ?
//. Two cans of lard weighed 50 pounds apiece and a
third can weighed 48 pounds ; what was their average
weight ?
263. Divide 48 by 16.
^48 ^^^ ^^ ^^ "^ ^' ^"^^^ 16 X 3, or 3 X 16 r= 48.
128 WHOLE NUMBERS AND FRACTIONAL PARTS.
264. Exercise.
Divide :
/. 22 by 11. 18. G5 by 13. 35. 051 by 93.
2. 3G by 12. 19. 54 by 18. 36. 406 by 58.
5. 28 by 14. 20. 95 by 19. 37. 384 by 48.
4. 45 by 15. 21. 100 by 50. 38. 423 by 47.
5. GO by 15. 22. 12G by G8. 39. 308 by 44.
6. GG by 22. 23. 108 by 3G. 40. 558 by 62.
7. 60 by 20. 24. 129 by 43. 4/. 305 by 61.
8. 72 by 24. 25. 124 by 31. 42. 134 by 67.
9. 81 by 27. 26. 288 by 72. 43. 102 by 34.
/O. 64 by 16. 27. 324 by 81. 44. 126 by 42.
//. 72 by 18. 28. 260 by 52: 45. 171 by 57.
12. 84 by 21. 29. 470 by 94. 46. 192 by 64.
13. 92 by 23. 50. 174 by 87. 47. 207 by 69.
14. 78 by 13. 31. 184 by 92. 48. 176 by 88.
/5. 90 by 18. 32. 228 by 76. 49. 300 by 75.
16. 96 by 16. 33. 318 by 53. 50. 440 by 55.
17. 91 by 13. 34. 576 by 96. 51. 140 by 28.
266. Divide 795 by 15.
iV of 79 is 5, the first figure of the quotient,
15 ) 795 ( 53 Taking 15 x 5, or 75, from 79, wo have 4 re-
-45 maining.
45 Annexing 5, the next figure of the dividend, to
the remainder, we have 45.
^^0 of 45 ia 3,the second figure of the quotient.
Taking 15 x 3, or 45, from 45, we have notliing remaining.
The quotient is 53.
MULTIPLICATION, DIVISION, MENSURATION. 129
266. Exercise.
Divide :
/. 143 by 11. 10. 656 by 41. 19. 3770 by 65.
2. 276 by 12. //. 867 by 51. 20. 4420 by 85.
3. 429 by 13. 12. 528 by 22. 21. 1804 by 82.
4. 602 by 14. 13. 1664 by 26. 22. 1932 by 21.
5. 795 by 15. U. 2088 by 72. 23. 1098 by 18.
6. 294 by 21. 15. 1092 by 42. 24. 1736 by 28.
7. 465 by 31. 16. 1056 by 24. 25. 3752 by 56.
8. 782 by 23. 17. 2024 by 44. 26. 2538 by 54.
9. 800 by 32. 18. 2520 by 56. 27. 1961 by 53.
Measure :
28. 252 by 12. 30. 684 by 38. 32. 4000 by 32.
29. 286 by 22. 31. 1215 by 45. 33. 3591 by 24.
267. Exercise.
/. A man was paid a debt of $125 in 25 bills of equal
value ; what was the value of each bill ?
2. The population of Railway, N. J., was 7105 in 1890
and 7935 in 1900 ; what was the average gain per year for
the ten years ?
3. A teacher was employed at $810 for 9 months, but
he resigned at the end of 7 months ; how much had he
earned ?
4. If a farm of 84 acres is valued at $2100, what value
is placed on each acre ?
5. What is the average number of miles that a train
9
130 WHOLE NUMBERS AND FRACTIONAL PARTS.
must run per hour to reach a city 990 miles distant in 18
hours ?
6. A man's lot of 25 acres and 2 cows are valued at $685 ;
if the cows are valued at $30 apiece, what value is placed
on the land per acre ?
7. A man bought property valued at $1872; how much
must he pay a year, if he is given 12 years to pay for it
and is to pay the same amount each year ?
8. If 12 acres of wheat yielded 336 bushels, how much
should 25 acres yield at the same average ?
9. I bought 14 thousand feet of white pine boards for
$448 ; what did I pay per thousand feet ?
10. A drover bought 25 sheep for $100 and sold them
for $150 ; what was the average gain per head ?
263. Exercise.
/. How many trips did a farmer make to haul away 900
bushels of wheat, if he hauled 75 bushels at a load ?
900 bushels was the amount hauled.
. ^ 75 bushels was the amount hauled at one load.
^ 900 bu. : 75 bu. = 900 -f- 75, or • = the number of trips.
2. A clerk pays $25 a month rent for a house valued at
$3750 ; in how many months will the rent amount to the
value of the house ?
3. A dealer shipped 702 dozen eggs in crates holding 72
dozen each ; how many crates did he ship ?
4. llow many feet are there in 1728 inches ?
5. There are 32 quarts in a bushel ; how many bushels
are there in 608 quarts ?
MULTIPLICATION, DIVISION, MENSURATION. 131
6. How many cows at 125 a head can be bought for
$320, and what amount will be over ?
25 ) 320 ( 12 1320 is the amount to be spent.
25
$25 is the price paid for each cow.
f-Q 1320 : $25 = 320 -f- 25, or 12, is the number that
"2() can be bought, with $20 over.
7. Corn weighs 56 pounds to the bushel. A farmer's
crop when sold weighed 7125 pounds ; how many bushels
had he, and how many pounds over ?
8. At 15 cents a yard, how many yards of flannel can be
bought in exchange for 9 pounds of chickens at 10 cents
a pound ?
9. If the distance around a wheel is 12 feet, how many
times will the wheel turn in going 5280 feet, or 1 mile?
269. Since 10 x 4 = 40, 40 -^ 10 = 4, or 4|0.
Since 100 x 4 = 400, 400 ^ 100 = 4, or 4|00.
Since 1000 x 4 = 4000, 4000 -^ 1000 = 4, or 4|000.
Therefore,
Cutting one cipher from the right of a nwnriber
divides the number by 10, cutting two ciphers di-
vides it by 100, three ciphers by 1000^ and so on.
270. Oral Exercise.
Read, supplying the missing numbers:
/. 30-^10=« 6. 400 -^ 100= .//. 7000 -^ 1000 =
2. 60 -^• 10 = . 7. 200 -^ 100 = . 12. 8000 -^ 1000 =
5. 90 -^ 10 = o 5. 500 -^ 100 = . 13. 9000 ^ lOOO =
4. 500 -^ 10 = . 9. 4000 -^ 100 = • 14. 5000 -^ 1000 =
5. 320 -f- 10 = . 10. 3200
100 =: . /5. 8000' -^ 1000
132 WHOLE NUMBERS AND FRACTIONAL PARTS.
271. 1200 ^ 20 = 60.
Also, 120|0 -r 6|0, or 120 -^ 2 = 60.
Therefore,
Qutting off tl\,e same number of ciphers from the
right of two numbers does not change tfteir quotient.
272. Oral Exercise.
Read, supplying the missing numbers :
/. 40-^20=:. 5.150^30 = - 5. 3600 -^ 400 =•
2. 80 ^ 20 = • 6. 300 -r- 50 = • 10. 7200 -^ 600 = •
5. 90-^30=. 7.350^70=. //. 8000 -^ 4000 = •
4. 100 -T- 20 = . 5. 630 -^ 90 = . 12. 9000 ^ 3000 = •
Forming, Reading, and Writing Numbers Above
lOOOO; Fundamental Operations.
To the teacher. The study of numbers above 10000 may be de-
ferred, if desirable, and taken up at the close of Chap. II or
Chap. IIL
273. Thousands are counted up to 999 thousand.
274. A thousand thousand is called a million.
Millions are counted up to 999 million.
275. A thousand million is called a billion.
Billions are counted up to 999 billion.
276. When more than three figures are used to
write a whole number, they are separated by com-
mas into groups, or Periods, of three figures each.
NUMBERS ABOVE 10000. 133
277. Counting I'roni the right,
In the lirst j)eriod any number of units from 1 to
999 is found.
In the second period any number of thousands
from 1 to 999 is found.
In the third period any number of millions from
1 to 999 is found.
In tlie fourth period any number of billions from
1 to 999 is found.
278.
TABLE.
m
(D
1
g
m
3
^
B m
O
a
■■? §
a §
5
5J OQ
OD
'^ -a ^
-S S £
i
o g
'S
mdr
n-bi
lion
indr
Q-m:
llior
s
T3
§ i
= « •«
= /. Jof 1 =
293. Oral Exercise.
/. Line MN is divided I
M \-
into how many equal parts ? I
2. What is each part called ?
5. What are 2 parts called ? 3 parts ?
4. How many thirds are there in 1 ?
5. PQ is divided into
how many equal parts ?
6. What is each part called ?
7. What are 2 parts called ? 3 parts ? 6 parts ?
5. How many sixths are there in 1 ?
9. RS is divided into , , ,
R ' I I M I I
I h
how many equal parts ?
10. What is each part called? What are 2 parts called?
12 parts ? 3 parts ?
/ /. How many thirds are there in 2 ? How many sixths ?
10
146
FRACTIONS.
12. What is J of J ? i of J ?
13. How many sixths are there in J ? How many
twelfths ?
14. How many sixths are there in f ? How many
twelfths ?
/5, How many sixths are there in J ? How many
twelfths ?
16. How many thirds are there in IJ ? How many
sixths ?
n. How many thirds are there in If ? How many
sixths ?
18. How many twelfths are there in |^ ? in J ?
Read, supplying the missing numbers :
19.
1= ^
27.
n= i
35.
3 =*
43.
H= *
20.
1= t
28.
n= i
3e.
H=i
44.
iS=i
21.
1 = tV
29.
H = ij
37.
H = i
45.
V= i
22.
i = -6
30.
If = TV
38.
3 =i
46.
= 4
23.
1= -c
31.
■ft" —
39.
3 =j
47.
J = *
24.
i = TV
32.
1 = i
40.
i ofi= .
48.
i =rt
25.
t = TV
33.
A= 4
41.
Jofi= .
49.
=i
26.
t = TV
34.
H= 5
42.
n = i
SO.
« = *
294. One of the equal parts into which a unit is
divided is a Fractional Unit.
Thus, \, ^, ^, ^, -jij are fractional units.
295. A number made up of like fractional units is
a Fraction.
Thus, I, g, 5, 6 7 V *ire fractions.
REDUCTION. 147
296. When a fraction is written in figures, the
number below the line is called the Denominator and
the number above the line the Numerator. The
numerator and denominator together are called the
Terms of the fraction.
Thus, in the fraction f, 5 is the denominator, 4 the numerator,
and 4 and 5 togetlier are the terms.
297. A Proper Fraction is a fraction whose numera-
tor is less than its denominator ; as, |.
298. An Improper Fraction is a fraction whose nu-
merator is equal to or greater than its denominator ;
as, J and J.
299. An Integer is a whole number ; as, 7.
300. A Mixed Number is a number made up of an
integer and a fraction ; as, 2|.
Reduction.
301. -
I _L I J- I X I J- I ±1 X | -v--|--7--'
88^888 8T
Here we see that
Therefore,
Multiplying or dividing both terms of a fraction
by the same number does not change its value.
148
FRACTIONS.
303. Oral Exercise.
,
Bead, siipp
lying the missing
numbers :
'■i = i
7. f = 5V
13. ^ = i
'»• n = i
^- i = i
«• J = tV
u. I = t
'0- U = i
3. i = i
9- t = rt
^5- J = i
2'- « = i
^■i = i
'0- 1 - TT
^6- U = i
22- H = i
«■ * = IS
"• l = 5S
^7. H = t
'3. H = i
e- 1 = 51
'2- A = bV
/5. il = ^
2^- M = *
303. Exercise. ,
1 1
1 1
Show tliat :
1
mi 1
' 1
, 1
1 1
1 1
/. 2J = |.
Thus, '
1
1 1
1
1 ___J
2. 2 = |.
4. 2J = f .
6.
n = Y-
8. 3f = V^.
5. 3=.|.
5. 2i = i.
7.
n = V-
9- H = V-
304. Changing the form of a number without
changing its value is called Reduction.
305. Reduce 21 to thirds.
Since 1 equals 3 thirds, 2 equals 2 times 3 thirds, or 6 thirds,
and 2J equals 6 thirds plus 2 thirds, or 8 thirds.
306. Reducing 2 to J, or 2| to } is reducing a whole
or a mixed number to an improper fraction.
307. Oral Exercise.
Reduce :
/. 2 to halves. 5. 4 to halves.
2. 4 to thirds. 4. 3 to halves.
Reduce to improper fractions :
7. 2J. 10. 8f. 13. 7J.
8. SJ. //. 3i. 14. 6}.
9. 2f. 12. 3f. 15. 4J.
5.
2 to fifths.
6.
5 to thirds.
16.
4J.
17.
12J.
18.
131.
REDUCTION.
149
308. Reduce 12| to fifths.
1 s
12 =
12f =
5'
13 X f , or ^/ .
¥ + f , or %^.
309. Exercise.
Reduce to
/. 131.
2. 18|.
3. 161.
improper fractions :
8. GG§. /5.
5. 42f 16.
10. 20|. /7.
18|.
55|.
361.
22.
23.
24.
112^.
118f.
300f.
4. 14|.
//. 44f.
/5.
165.
25.
299|.
5. 37|.
6. ^2\.
7. 87J.
12. 30J.
/5. 48|.
14. 2G|.
19.
20.
21.
331.
14J. '
67J.
26.
27.
28.
187}.
575f
627J.
310. Exercise.
Show that : i - i
/. \ = %\. Thus, r-^^ ^—
1
1
~3
1.
T
i i
2. 1 = 3.
3. J = 3J.
^- J -If.
5. V = 2}.
6. ^ = 3i,
7. ^^=^.
1
8.
9.
9? = i|.
311. Changing | to 2, or | to 3J, is reducing an
improper fraction to a loliole or mixed number.
iof 2
312.
Here we see that f = J of 2, or 2 h- 3. Hence,
A fraction may he considered to denote that the
numerator is to he divided hy the denominator.
150 FRACTIONS.
313. Oral Exercise.
Reduce to a whole or mixed number :
/. J. Thus, I equals ^ of 7, or 2^.
2. f. 4. |. 6. ^. 8. Y- 10. V. 12- V-
5. |. 5. Y- 7. y. P. J^. //. J^. 13. Y.
314. A fraction is in its lowest terms when its
numerator and denominator have no common di-
visor.
Thus, f is in its lowest terms, since 3 and 4 have no common
divisor.
316. Reduce |^ to lowest terms :
Therefore, to reduce a fraction to its lowest
terms,
Divide out of both terms all common dimsors of
them; or^
Divide hotli terms by their greatest common
divisor.
316. Oral Exercise.
Reduce to lowest terms :
'• A-
5. f,.
9.
fS-
13.
if-
17.
*i-
2. «.
«■ U-
10.
«•
14.
fJ-
18.
«•
3- !i
7- e-
11.
If-
15.
fl-
19.
n-
4. if.
«■ !«•
12.
if-
16.
if
20.
tW-
MULTIPLICATION AND DIVISION.
151
317.
Multiplication and Division.
4
' ^ 1 I I -I--,—
T—r
Here we see that
Therefore,
To multiply a fraction hy a whole number^ mul-
tiply the numerator or divide the denominator by
the whole nuuiber.
Note. Divide the denominator when possible.
318. Oral Exercise.
Read, supplying the missing numbers :
/. 2 X 1 = <
13.
4
X
tV= •
2. 2 X i = .
14.
3
X
1 = •
5. 3 X f =
15.
3
X
1 = •
4. 2 X 21 =
16.
4
X
24 = .
5. 2 X 4i = .
17.
5
X
5A= •
tf. 3 X 9^ =
18.
5
X
3tV = •
7. 2 X 1 = .
19.
6
X
i - •
5. 3 X f =
20.
3
X
4 = •
i?. 5 X A -
21.
2
X
i = •
10. 2 x5t%= <
22.
G
X
ItV - •
//. 3 x3^ =:
23.
4
X
3f = .
12. 4 X 5y%=:
24.
4
X
n = •
152
FRACTIONS.
319.
"j— I r
I 8 ^ ^
Here we see that
I - 2 = J - Jl^; or, J - 2 = I - ,-f^.
Therefore,
To divide a fraction hy a whole number, divide
the numerator or multiply the denominator hy the
whole number.
Note. Divide tlie numerator when possible.
320. Oral Exercise.
Kead, supplying the missing numbers :
/.
1
-^ 2 = .
2.
1
-^ 3 = .
3.
f
-f- 3 = .
4.
i
-^ 4 = .
5.
4|
-^ 2 = .
6.
6|
-^ 3 = .
7.
1
-f- 3 = .
8.
A
-^ 5 = o
Note.
9- -h -
-10 =z .
17. ^ of 5i
= .
10. jt -
- 5 = .
/5. \ of IGJ
= •
//. 101 -
- 5 = .
' /5. 1 of 1
= .
12. laij -
- 3 = .
20. J of §
r= .
13. A ot f = .
2/. 1 of 5
= .
14. ^ of 1 = .
22. i of J
— •
/5. i of J =
. 23. \ of ISi
= .
/fi. i of H =
• 24. i of 10 jV
— •
i of * hav
e the sail
le meaning.
321. Oral Exercise — Miscellaneous Problems.
/. Name at sight the whole or mixed number equivalent
to each of the following :
MISCELLANEOUS PROBLEMS. 163
2. What part of a quart is a pint ?
3. A half peck is how many quarts ?
4. How many quarter pecks are there in a half peck ?
5. How many quarts are there in 3 pints ?
6. f of a foot is how many inches ?
7. How many half inches are there in a foot ?
8. How many half feet are there in a yard ?
9. What part of a dime is 1 cent ?
10. K woman used half a gallon of milk for breakfast
and one fourth of a gallon for dinner ; how many quarts
did she use for both meals ?
//. What will f of a pound of dried beef cost at 32
ct. per pound ?
12. If 4 pounds of coffee cost If, what part of a dollar
did 1 pound cost ? How many cents ?
13. If 3 pen trays cost $lf, what part of a dollar did
one cost ? How many cents ?
14. How many yards of ribbon are needed to make 5
bows, each | of a yard long ?
15. A. boy has four 2J-dollar gold pieces ; how much
money has he ?
16. If a $5-bill is exchanged for half dollars, how many
should be received ?
17. K rod contains 5|- yards ; how many yards are there
in 2 rods ?
18. How many yards are there in 3 rods ?
154 FRACTIONS.
Reduction.
322. Reduce f, |, and i to 12ths.
12 = 4 X 3, 3 X 4, or 2 X G.
I = ii^ = A-
i = Tx-l = ->^-
b 2x6 ' ^
323. Fractions that have the same number for a
denominator are said to have a Common Denominator.
324. Reducing f, |, and | tv) 12ths is reducing
fractions to a common denominator.
325. Fractions that have a common denominator
are called Similar Fractions.
Thus, A, i^if, JAiid [•}, haviug a common denominator, are similar
fractions.
326. Exercise.
Reduce :
/. I and f to 15ths. 7. f, |, and \ to 12ths.
2. I and f to 20ths. 8. |, J, and ^ to 30ths.
3. \ and | to 8ths. 5. J, f , and \ to 20ths.
4. -I and | to 18ths. 10. |, i, and f to 24ths.
5. 3% and ^ to lOOths. //. i, |, and ^ to 60ths.
6. f and ^ to 3Gths. 12. |, ^, and f to 72d8.
327. The Least Common Denominator (1. c. d.) of
two or mor(^ fractions in theii* lowest terms is the
least number tliat their denominators will divide:
Thus, the least common denominator of J, 5, and \ is 12, for 12
is the least number that 3, 4, and 6 will divide.
REDUCTION. 156
328. Reduce J, f, and } to similar fractions having
their least common denominator.
The 1. c. d. of i, f, and f is readily seen to be 12.
Now, 12 = 6 X 2, 4 X 3, or 3 X 4.
1 _ 6^ _ 6
■2 — 6x2 ~ ^''^'
1 = 3"^ = 1%.
329. Exercise.
Reduce to similar fractions having their least common
denominator :
/, f and f. 4. i and |. 7. J, i, and i.
2. J and f. 5. } and |. S. f, f, and |.
5. I and |. 6. f and Vo- 9. i, |, and |.
330. Reduce j\, f, and /^ to similar fractions having
their least common denominator.
When the 1. c. d. cannot readily be seen, as in the above
problem, it may be found thus :
12 = 2 X 2 X 3. The 1. c. d. must have in it the prime
g _ 2 X 2 X 2 factors 2, 2, and 3 in order that 12 may
divide it ; it must have in it the addi-
tional prime factor 2, in order tliat 8
may divide it, and the additional prime
2 X 3 X 5, or 120. factor 5 in order that 20 may divide it.
20 = 2 X 2 X 5.
1. c. d. = 2 X 2 X
Now, 120 -*- 13 = 10 ; 120 -^ 8 = 15 ; 120 -i- 20 = 6.
120 = 10 X 12, 15 X 8, or 6 X 20.
- 10x7 _ la
'12 ~ioVn- ■»2o-
K 15 X 5 7 5
16x8
— "120-
"s% — rIT^ — '^0'
166 FRACTIONS.
331. Exercise.
Keduce to similar fractions having their least common
denominator :
1' i\, nh and ^j.. 5. A, ^, and Jf- 9. |J, \l and iJ.
2. -I, -i^, and }|. ff. A, f §, and A- ^^- i I'ff. and .^
•?• ^S^ T% and f§. 7. ^\, jK, and ^'V- ^^- iz^ ^V and ^g^.
^- 2\> ih and iJ. 5. ^\, A, and g^^. /2. ^^, ^^g, and H.
Addition and Subtraction.
332. Find the sum of f, |, and f
1. c. d. = 40.
_ 6J