^Wi
i 1 I I
^^'^
IN MEMORIAM
FLORIAN CAJORl
INTERMEDIATE ARITHMETIC
BY
BRUCE M. WATSON
SUPERINTENDENT OF SCHOOLS, SPOKANE, WASH.
AND
CHARLES E. WHITE
PRINCIPAL OF FRANKLIN SCHOOL, SYRACUSE, N.Y.
D. C. HEATH & CO., PUBLISHERS
BOSTON NEW YORK CHICAGO
Copyright, 1907,
By D. C. Heath & Co.
1a2
INTEODUCTION
The Intermediate Arithmetic is divided into two parts, each
containing a full year's work. Throughout the book the pupil
is led to see, in each new topic, an extension and application of
principles previously learned. Fractions are treated as expres-
sions of division. The work in decimals is presented as an
extension of the decimal scale of notation to numbers smaller
than one. Percentage is treated as an application of decimals
— a familiar topic under a new name. There is no formal divi-
sion of percentage into " cases," but the pupil is led to apply
his knowledge of the relation of product and factors in deter-
mining the process to be employed in the solution of each indi-
vidual problem. All technical commercial terms are reserved
for later consideration.
The work in denominate numbers is confined to such prob-
lems as people of the present generation are likely to meet in
their daily vocations. Extended reductions and intricate
measurements are not required. Attention has been given to
the development of ideas of proportion, the real purpose of
the so-called " ratio exercises " found in some courses of study.
An effort has been made to shorten the course and simplify
the work, to a reasonable degree, by reducing each topic to as
few cases as possible, and by employing the simplest and most
generally applicable processes.
The study of arithmetic should furnish the child a means of
interpreting mathematically the world about him. It should
iii
iv INTRODUCTION
enable him to measure and relate the facts of geography,
history, and science. It should bear directly upon the vital
interests of the home and family. Care has been taken in the
selection of problems to meet these requirements, so far as
possible, with due regard to the mathematical content of the
exercises, which must always be the first consideration.
I
INTERMEDIATE ARITHMETIC
^
IIs^TERMEDIATE ARITHMETIC
PART ONE
NOTATION AND NUMERATION
1. That which tells how many is number; e.g. 11, 14 (books),
25 (cents).
2. One is a unit ; e.g, 1 dollar, 1 house, one.
Every number is made up of units. Three contains 3 units.
Twenty contains 20 units.
3. Expressing numbers in figures or letters is notation ; e.g,
7, 29, VII, XXIX.
4. Expressing numbers by means of figures is Arabic notation ;
e.g. 13, 4728, 23806.
1, 2, 3, 4, 5, 6, 7, 8, and 9 are called significant figures, because
each figure has a value.
The figure 0, called zero, or naught, has no value, but is used
to give the significant figures their proper places in a number.
5. Expressing numbers by means of letters is Roman notation :
e.g. VIII, CD, XCIV.
3
ARABIC NOTATION
ARABIC NOTATION
!
h "5
o
3
rt
o
Jz; 00 s H
< ^ o <2
■C = .- "D
£ ^ -
O TJ "V 3 -O
C = C C O C
= 3a>--3a>j=3
J^ X|-CQII-Sl|-HX|-3
46 5, 20 9, 315, 087
This number is read, four hundred sixty-five billion^ two huu"
dred 7iine million^ three hundred fifteen thousand^ eighty-seven.
A comma (,), sometimes called a separatrix, is used between
periods to aid in reading numbers.
7. Oral
1. Name the periods in this number. Name the places.
2. How many periods are there ? How many places ?
3. How many places are there in each period ?
4. How does the name of each period compare with the
name of its right-hand place ?
8. Written
Express in figures: .
1. Two hundred thousand, sixteen.
2. Eleven thousand, two.
NUMERATION 5
3. Four million, six hundred eight thousand, three hundred
seventy-five.
4. Twenty-five thousand, seven.
5. Nineteen thousand, seventeen.
6. Twenty-seven million, six hundred fifty.
7. Eighty million, six hundred nine thousand, four hundred
twenty-eight.
8. Six hundred twenty million, seventeen thousand, four
hundred seventy-seven.
9. Four hundred thirty-six thousand, fifty-one.
10. One hundred fifty-seven million, six hundred eight thou-
sand, four hundred seventy-seven.
11. Three billion, fifty-seven million, four hundred seventeen
thousand, sixty.
12. Write a number containing five places.
13. Write a number containing three periods. How many
places does it contain ?
14. Write a number containing three periods in Avhich the
thousands' period has no value.
NUMERATION
9. Naming the places of figures and reading numbers is numer-
ation. Thus, to numerate 43,008,160, we should say "Units,
tens, hundreds, thousands, ten-thousands, hundred-thousands,
millions, ten-millions — forty-three million eight thousand one
hundred sixty."
10. Numerate the numbers below :
1. 385 4. 315,129 7. 8,460,000
2. 1,421 5. 6,785,342 8. 423,000,501
3. 25,678 6. 35,000,730 9. 8,003,040,631
KOMAN NOTATIOIT
ROMAN NOTATION
11. The Roman notation uses seven capital letters to express
numbers, as follows :
I (1), V (5), X (10), L (50), C (100), D (500), M (1000).
12. These letters are combined according to the following
principles of Roman notation:
1. Placing a letter after one of greater value adds its value to
that of the greater,
2. Placing a letter before one of greater value subtracts its value
from that of the greater.
3. Placing a letter between two letters of greater value subtracts
its value from their sum,
4. Repeating a letter repeats its value,
5. Placing a bar over a letter multiplies the value of the letter
hy 1000.
ILLUSTRATIONS
13. 1. X = 10, V = 5, XV = 10 + 5 =15. Which prin-
ciple does this illustrate ?
2. V = 5, I = 1, IV = 5 — 1 = 4. Which principle does
this illustrate ?
3. C = 100, L = 50, X = 10, CXL = 100 + 50 - 10 = 140.
Which principle does this illustrate ?
4. X = 10, XXX = 10 + 10 + 10 = 30. Which principle
does this illustrate ?
5. D = 500. D = 500000. Which principle ?
6. CCCLX = 360. Which principle ?
7. MCM = 1900. Which principle ?
8. MDCLXVI = 1666. Which principle ?
9. Write a number to illustrate each principle.
ADDITION 7
10. Express in Roman notation all numbers from 1 to 100.
Note. — For many years the Roman notation was the one chiefly used in
Europe. The ancient Greeks also had a system of notation that employed
the letters of the Greek alphabet. Both of these systems were awkward
and were not easily used in making computations. The Arabic system of
notation, now employed by all the great nations of the world, was used first
in India, and afterward brought to Europe by the Arabs.
ADDITION
14. Addition is the process of uniting two or more numbers into
I one number ; e.g. 2 + 5=7.
15. The numbers added are addends; e.g, 3 + 10 = 13; 3
and 10 are the addends.
16. The result of addition is the sum ; e.g, 8 books and 7
books are 15 books; 15 is the sum.
17. The addends and the sum are called the terms of addition.
18. Oral
Add:
1. 31
5
71
7
63
2'
8."
56
61
6/
46
8
34
7
26
«
19
9
9
71
Note. — In adding a column, always look for combina-
tions of two or three figures whose sum may be taken as
an addend. Thus, in finding the sum in Example 1, say
9, 19, 26, 34, 46, 56, 63, 71.
Read the column downward, making similar combina-
tions.
8
ADDITION
Add:
2. 4 3.
9
4. 5
5. 6
6.
8
7. 8
8. 5
9. 8
5
4
9
6
7
6
7
2
6
6
6
7
9
5
6
4
5
2
4
6
2
2
4
6
2
7
3
8
9
3
3
7
3
5
5
2
1
7
2
3
4
8
2
7
6
9
4
8
3
4
4
5
7
1
6
4
9
3
5
6
3
8
9
7
814^8736
10. Add 25 ant? 47.
25 + 40 = m. Say 25, 65, 72.
65 + 7 = 72. Ans,
In a similar way find the sums indicated in exercises 11-25 :
11. 28 + 26 16. 62 + 28 21. 62 + 24
12. 42 + 75 17. 57 + 36 22. 65 + 34
13. 63 + 29 18. 72 + 29 23. 53 + 46
14. 27 + 38 ^ 19. 35 + 26 24. 64 + 44
15. 45 + 34 20. 44 + 38 25. 73 + 27
26. A drover bought 8 cows, 5 horses, and 10 sheep. How
many animals did he buy ?
27. Fred paid 10 dollars for a goat, and 12 dollars for a cart
How much did both cost him ?
28. In a certain class 28 pupils were present, 5 were absent
on account of sickness, and 4 were absent for other reasons.
How many pupils belong to the class?
29. A farmer sold 25 bushels of apples to one man, 10 to
another, and 8 to another. How many bushels did he sell?
ADDITIOI!^ 9
30. John had 40 cents in his bank. He added 8 cents on
Monday, and 10 cents on Wednesday. How much money had
he then in his bank ?
31. A man paid 6 dollars for paint, and 10 dollars for labor.
How much did he pay for both?
32. Bought sheep for 50 dollars, turkeys for 12 dollars, and
chickens for 8 dollars. How much did they cost?
33. A man in repairing his house paid 65 dollars for lumber,
8 dollars for paint, 1 dollar for nails, and 30 dollars for labor.
What was the cost of his repairs?
34. How many fish did Mr. A catch in four days if he
caught 12 the first day, 8 the second, 9 the third, and 7 the
fourth ?
35. A girl spent 5 cents for car fare, 4 cents for pencils, 8 cents
for paper, 10 cents for ribbon, 15 cents for lunch, and had 9 cents
left. How much money had she at first?
19. Written
Add, and test your work hy adding downward:
1.
28
2. 639
3. 1050
4. 126
5. $115.85
39
874
394
149
' 327.15
76
596
769
1260
495.27
42
421
564
1004
160.03
89
397
285
986 .
598.09
I§
269
784
24
784.06
97
7. 857
8. 283
9. $208.40
10. 1356.24
98
943
2075
32.03
35.09
79
268
298
26.07
2.15
68
207
963
18.94
30.05
40
976
859
236.29
5.16
87 '
888
876
28.15
304.29
10
)
ADDITION
11. 2673
12. 837
13.
628
14. 8063
846
2964
4307
259
1025
418
526
8264
92
3825
8279
1287
837
842
428
428
642
29
4273
3064
4983
561
746
42379
8698
29
394
6507
2789
387
786
93289
15. Three boys went fishing, and caught 16 perch, 19 pickerel,
and 8 black bass. How many fish did they catch in all ?
16. Two trains starting from the same place ran two days in
opposite directions. One ran 530 miles the first day and 525
miles the second, while the other ran 492 miles the first day
and 510 miles the second. How far apart were they at the end
of the two days? (Illustrate by a picture.)
17. A man bought coal for $5.60, wood for $3.45, and a
stove for $45. What was the whole cost?
18. There are 112 bushels of wheat in one bin, 175 in an-
other, and 234 in the third. How many bushels in all ?
19. There are 218 pages in my reader, 245 in my arithmetic,
195 in my geography, and 189 in my language book. How
many pages in the four books ?
20. The Bum of all the sides of a figure is its perimeter.
1. Find the perimeter of a figure whose sides are 39 inches,
45 inches, 28 inches, b^ inches, 75 inches, and 17 inches.
2. What is the perimeter of a seven-sided piece of land
whose sides are 209 feet, 683 feet, 129 feet, 463 feet, 928 feet,
93 feet, and 290 feet ? (Illustrate.)
SUBTRACTION 11
^^ SUBTRACTION
21. Subtraction is the process of finding the difference between
two numbers; e.g. 21— 1 = 14:; 13 cents — 5 cents = 8 cents.
22. The number from which we subtract is the minuend. The
number subtracted is the subtrahend. The result of subtraction
is the difference or remainder.
The difference is always the number that must be added to
the subtrahend to obtain the minuend ; e.g. 17 — 9 = 8. 17 is
the minuend, 9 is the subtrahend, and 8 is the difference or
remainder.
23. 77ie minuend, subtrahend, and remainder are called the
terms of subtraction.
24. Oral
From 83 take 57.
83 - 50 = 33. Say 83, 33, 26.
33 - 7 = 26. Ans.
In a similar way find the differences indicated in exercises 1-15.
1. 38-19
6.
72-26
11.
63-25
2. 27-18
7.
66-37
12..
,48-19
3. 42-15
8.
92-48
13.
51-27
4. 61-22
9.
87-39
14.
84-47
5. 81-36
10.
42-29
15.
75-39
16. Frank lives 12 blocks from school, and Henry 5 blocks
In the same direction. Their homes are how many blocks
apart ? How many blocks apart would they be if Henry lived
5 blocks from school in the opposite direction ? (Illustrate.)
17. Mary added two numbers, and the sum was 28. One
of the numbers was 16. What was the other ?
18. 20 is how much more than 11?
12 . SUBTRACTION
19. Tom has 20 marbles, and Edward 11. Tom has how
many more than Edward ?
20. If you pay 15 cents toward the purchase of a slate cost-
ing 20 cents, how much do you still owe ?
21. Nell had 21 chickens, but a dog killed 10 of them.
How many were left?
22. Lucy is 20 years old, and her sister is 6 years younger.
How old is her sister?
23. John had 1 20. He spent 1 10 for a coat and 1 3 for a
hat. How much had he left ?
24. A pole was 19 feet long. I cut off 4 feet at one time
and 6 feet at another. How many feet were left ?
25. What number must be subtracted from 43 to leave 25 ?
26. John's heart beat 78 times a minute when he was well,
but 130 times a minute during a severe illness. How much
faster did the heart beat during illness than in health?
25. Written
From 58500 take 26937.
58500 = 50000 + 7000 + 1400 + 90 + 10
26937 = 20000 + 6000+ 900 + 30+ 7
31563 = 30000 + 1000+ 500 + 60+ 3
In finding the difference, we write only this :
58500
26937
31563 and say, 7 from 10 = 3, 3 from 9 = 6, 9 from
14 = 5, 6 from 7 = 1, 2 from 5 = 3.
To test the work, add the subtrahend and remainder. If the minu-
end is obtained, the work is correct. Do not write the numbers again, but
make the test with the numbers as they stand.
In what other way may we test subtraction?
SUBTRACTION
13
Subtract and test results
1.
2819
674
5.
2763
1289
9.
92874
11392
13.
38264
29842
17.
12.15
1.12
21.
134.28
24.28
2.
10.
14.
82'03
1276
37284
9287
94210
8206
19327
8291
18. $35.28
17.05
22.
.21
27.13
3.
7.
4295
597
36801
18463
11.
42840
38706
15.
92593
87246
19.
$25.18
1.15
23.
$17.80
16.75
7306
1807
8.
18003
921
12.
98301
26942
16.
27075
18092
20.
$36.51
16.82
24.
$75.00
24.32
25. From seventeen thousand sixteen, take nine thousand
four hundred eighty-seven.
26. From seventy-two thousand three hundred eleven, take
forty-six thousand nine hundred sixty-one.
27. Take eight thousand four, from thirty thousand!
REVIEW AND PRACTICE
26. Oral
1. Read 359,016,007,138 ; $3,894,760.15; 1,010,101.
2. ReadCLIX; DCCXXXVI; MCMXIIL
3. Numerate 3,057,608.
4. Jennie bought a skein of Shetland floss for 10 cents, 3
skeins of embroidery silk for 12 cents, and a pair of knitting
needles for 10 cents. How much change should she receive
from a 50-cent piece?
14 REVIEW AND PRACTICE
5. The sum of 3 numbers is 100. Two of them are 29 and
37. What is the other?
6. Albert has earned 15 cents, 25 cents, and 17 cents toward
a pair of gloves that cost f 1. How much more money must
he obtain in order to pay for the gloves ?
7. Helen, Howard, and Henry wanted a canoe that cost $47.
They obtained f 19 by renting their row-boat, their mother con-
tributed $12, and their father the remainder of the price. How
much did their father give ?
8. Edith made some purchases, gave the clerk a dollar, and
received in change three cents, one nickel, two dimes, and a
quarter. What was the amount of her purchases?
27. Written
1. A farmer having 456 bushels of corn sold 84 bushels to
one man and 135 bushels to another. How many bushels did
he have left ?
2. A man started to walk 112 miles in three days. He
walked 32 miles the first day, and 41 miles the second. How
far must he walk the third day to complete the journey ?
3. I bought a cow for $42, another for $48, and a third for
$56. For how much should I sell them to gain $28?
4. A lady bought sugar for 65 cents, tea for 55 cents, molas-
ses for 72 cents, butter for 84 cents, starch for 25 cents, and
gave in payment a five-dollar bill. How much change should
she receive ?
5. The distance by rail from Galveston to San Antonio is
572 miles, from San Antonio to Tucson 932 miles, and from
Tucson to Los Angeles 501 miles. What is the distance by
rail from Galveston to Los Angeles ?
REVIEW AND PRACTICE 15
6. Two vessels start from points 850 miles apart and sail
toward each other. How far apart are they when one has
sailed 246 miles and the other 352 miles? (Illustrate.)
7. A man sold one horse for $145 and another for $182.
On the first he gained 1 23, and on the second 1 36. What was
the cost of both?
8. A boy bought apples for $A5 and pears for $.62, and
sold them all for $1.50. What was his profit?
9. John sold 62 newspapers, Frank 48, and Henry 27 less
than both of them. How many did Henry sell ?
10. A grocer sold butter for $45 and cheese for $62. On the
butter he lost $6 and on the cheese he gained $14. What was
the cost of both?
11. A farmer bought a barrel of flour for $6.35, sugar for
$2.15, coffee for $1.46, tea for $1.20, and gave in payment
$3.15 worth of butter and the remainder in cash. What did
he pay in money?
12. The sum of 52 and 64 is how much greater than the
difference between 124 and 69?
13. From a flock of 320 sheep 76 were sold at one time and
112 at another. How many remained ?
14. A man bought 148 bushels of potatoes from A, 216
bushels from B, 183 bushels from C, and afterwards sold all
but 137 bushels. How many bushels did he sell?
15. The sum of three numbers is 342. Two of the numbers
are 84 and 96. What is the third number?
16. John's father gave him $2.25, and his uncle gave him
$1.40. He earned enough besides so that he bought, with the
whole, a suit of clothes for $8. How much did he earn?
16 REVIEW AND PRACTICE
17. Claude took 987 steps in coming to school, Francis 865,
and Alice 398 less than the number taken by both the boys.
How many steps did all three take?
18. A ship loaded with iron sailed from Cleveland to a
port 332 miles west of that city. A car loaded with machinery
at Cleveland was taken to a city 619 miles east of Cleveland.
How far apart were the ship and the car when each had reached
the end of its trip? (Illustrate.)
19. The first Thanksgiving was in 1621 and the day has been
observed every year since. How many times has the day been
observed?
20. A father and his three sons earned $2461 in a year.
The first son earned f 676, the second $456, and the father
$1080. How much did the third son earn?
21. A train started from Chicago with 324 passengers. On
the way to St. Paul 185 passengers left the train, and 149 came
aboard. How many passengers were on the train when it
reached St. Paul?
22. A retail grocer bought at a wholesale grocery three bar-
rels of apples for $4.50, a box of lemons for $2,70, and three
barrels of flour for $ 12. 30. He handed the wholesale grocer one
gold piece and received 50 cents in change. What was the
value of the gold piece ?
23. During one week, a man put into the bank $687, drew
out $489, put in $348, drew oat $298, and then had $1386 left
in the bank. How much had he in the bank at first?
24. A farmer having 215 acres of land, used 21 acres for
corn, 36 for oats, 29 for barley, 18 for potatoes, 52 for meadow,
and the rest for pasture. How many acres were used for
pasture ?
MULTIPLICATION
17
MULTIPLICATION
28. Multiplication is taking one number as many times as there
are units in another ; e.g. 6 times 9 are 54.
29. The number multiplied is the multiplicand ; the number
by which we multiply is the multiplier ; the result of multiplica-
tion is the product; e.g. 12 times 20 are 240. 20 is the multi-
plicand^ 12 is the multiplier^ and 240 is the product. 20 and
12 are factors of 240.
30. The multiplier^ multiplicand^ and product are called the
terms of multiplication.
31. The multiplier and multiplicand are factors of the product.
The product is the same in whatever order the factors are
taken; e.g. 6 times 7 are 42, and 7 times 6 are 42; 3x5x4
are 60 and 4x3x5 are 60.
32.
ORAL EXERCISES
(Reviewing work of primary arithmetic)
3
4
5
2
6
1
8
7
12
10
11
9
7
12
9
11
5
10
4
1
3
6
2
8
1. Multiply every number in the upper line by each number
in the lower line.
2. How do we multiply a number by 10 ? By 100 ? By 1000 ?
3. Multiply 7 by 10 ; by 100 ; by 1000.
4. Multiply 34 by 10 ; by 100 ; by 1000.
5. Multiply 11 by 3 ; by 30; by 300.
6. Multiply 12 by 7 ; by 70 ; by 7000.
18 MULTIPLICATION
7. The product is a multiple of the multiplicand. Of what
other number is it a multiple ?
8. Of what number is 33 a multiple?
9. Name 5 divisors of 24.
10. Multiply 23 by 12.
11. Multiply 23 by 10, also by 2. Add the two products.
How does this result compare with the answer to question 10?
12. Multiply 30 by 5, then by 20. Add the two products.
The sum is how many times 30 ?
13. 90 times 48, added to 6 times 48, are how many times
48?
14. 7 times 786, 40 times 786, and 3 times 786, added, make
how many times 786?
Q-ive the products at sight :
15. 86 16. 307 17. 315 18. 73 19. 32
7 5 10 100 20
20. 14
21. 28
22. 25
23. 16
24. 28
30
200
300
700
1000
25. 75
26. 103
27. 212
28. 31
29. 205
2000
40
50
400
600
30. Edward feeds his horse 11 quarts of grain per day and
his chickens 3 quarts. He feeds his horse how much more in
30 days than he feeds his chickens in the same time ?
31. Find the cost of 50 five-cent postage stamps.
32. 11 and 11 are the factors of what number ?
33. Of what number are 3, 7, and 5 the factors ?
34. What must be paid for a dozen junior baseballs at 70
cents apiece ?
MULTIPLICATION
19
33. Written
Multiply 1283 by 967.
1283
967
8981
7698
11547
1240661
We multiply 1283 by 7, 60, and 900, and then add the
results, which are called partial products. The sum of
the partial products is the product required. We omit
ciphers at the right of the partial products after the first.
Read each partial product as if the ciphers were ex-
pressed.
Multiply $34.79 by 806.
$34.79
806
20874
27832
128040.74
1. 324x24
2. 296x39
3. 387x45
4. 263x56
5. 892x63
6. 728x75
7. 398x84
8. 987x98
9. 516x31
10. 798x43
11. 896x79
12. 598x36
13. 287x49
14. 799x99
Observe that the right-hand figure of each partial
product is written directly under the figure by which
we multiply to obtain it. Cents in either factor give
cents in the product.
15. 296x28
16. 694x39
17. 206x54
18. 128.15x28
19. 134.98x27
20. 119.84x46
21. $7.85x124
22. 128.75x15
23. $36.91x45
24. $126.93x87
25.
26.
27.
28.
$.17.85x48
$19.63x49
$75.10x97
$16.35x764
29. $280.52x236
30. $356.04x328
31. $987.62x475
32. $396.41x641
33. $806.04x879
34. 238x307
35. 5126x208
36. 934x9000
37. 1027x2005
38. 386x1080
39. 527x2300
40. 4008x7003
41. $29.05x108
42. 4040x8356
20 MULTIPLICATION
34. PROBLEMS
1. a. What is the perimeter of a square farm whose side is
309 rods ? 5. What is its area ?
2. a. If it costs 14.78 a day to support a certain family,
how much does it cost for a month of 31 days ? h. How much
does it cost for a month of 28 days ? e. How much does it
cost for a year ?
3. There are 2000 pounds in one ton. How many pounds
are there in 496 tons ? (Solve it in the shortest way.)
4. There are 24 hours in a day, 60 minutes in an hour, and
60 seconds in a minute, a. What is the number of seconds in
a day ? h. In a week ?
5. a. How many minutes are thef^ in the month of May?
h. In the month of February ? c. In the month of September ?
6. Alice hemmed 4 dozen handkerchiefs, each 12 inches
square. How many inches of hem did she make ?
7. A manufacturer put up 13 tons of cereal in sample
packages, each containing 1 ounce. How many packages did
he make ?
8. 25,079 crates of strawberries, each containing 36 quarts,
were shipped from the city of Oswego in one season. What
were they worth at 8 cents a quart ?
9. There are 12 things in a dozen, 12 dozen in a gross, and
12 gross in a great gross. How many pens in a case containing
307 great gross ?
10. What is the value of 347 barrels of shredded cocoanut,
each containing 105 pounds, at 15 cents a pound ?
DIVISION 21
DIVISION
35. Division is the process of finding one of two factors when
the other factor and the product are given; e.g. 35 is the prod-
uct of 5 and 7. When 35 and 5 are given, we divide 35 by
5 to obtain 7 ; when 35 and 7 are given, we divide 35 by 7 to
obtain 5.
36. The number divided is the dividend ; the number by which
we divide is the divisor, and the result of division is the quotient ;
e.g. 42 -^ 7 = 6. 42 is the dividend, 7 is the divisor, and 6 is
the quotient.
When the divisor is not exactly contained in the dividend,
the part of the dividertd that is left is called the remainder ; e.g.
59 -T- 8 = 7 quotient and 3 remainder.
37. The dividend, divisor, and quotient are called the terms of
division.
38. Oral
1. 6 oranges at 3 cents apiece cost 18 cents. Which of these
numbers is a product ? Which are factors ? When 18 and 6
are given, how can 3 be found ? When 18 and 3 are given,
how can 6 be found ? When 3 and 6 are given, by what opera-
tion can 18 be found ?
2. The area of a page of Henry's book is 35 square inches.
If the length is 7 inches, what must be the width? If the
width is 5 inches, what must be the length ? 35 is which term
in division ? 35 is what of 5 and 7 ?
3. There are 9 square feet in 1 square yard. How many
square yards are there in 108 square feet ? 108 is the product
of 9 and what other number ? 108 is which term in division ?
108 is what of 9 and 12 ?
22 DIVISION
4. Fred paid 54 cents for some sugar. The number of cents
that a pound cost is one factor of 54. What is the other fac-
tor ? What, besides 54 cents, must be given in order that we
may find the cost of one pound ? What, besides 54 cents, must
be given in order that we may find the number of pounds that
Fred bought ?
5. What is
a. The number of feet in 132 inches ?
h. The number of pecks in 72 quarts ?
c. The cost of a month's rent at $120 a year ?
d. The number of weeks in 77 days ?
e. The price of a lawn mower, when 15 lawn mowers cost
1150?
6. The first number in each line below is a factor of every
other number in the line. Find the factor not given of each
number;
15; 75; 355; 525; 405
84; 217; 280; 763; 497
110; 44; 121; 880; 2211
819; 945; 189; 360; 963
64; 328; 176; 728; 960
7. In the following statements tell which numbers are fac-
tors and which are products :
a. There are peaches in 7 baskets if each basket con-
tains 12 peaches.
h. 12 quarts of berries cost 96 cents.
c. Jerome's wages for 9 weeks at f a week amounted to
27 dollars.
d. 6 fountain pens at $3 apiece cost $ .
e. il will pay 20 car fares at apiece.
f. 72 cents will buy 12 pounds of sugar at cents a pound.
a.
5
h.
7
c.
11
d.
9,
e.
8
DIVISION
28
39. Written
1. ' Divide
3125
317)990625
951
396
317
792
634
1585
1585
990625 by 317.
317 is not contained in 9 or 99, but it is contained in
990 three times. This is 3 thousand because 990 is thou-
sands. The remainder is 39 thousand. Put 600 with it,
and 317 is contained in 396 hundred 1 hundred times, with
a remainder of 79 hundred. The remaining figures of the
quotient are obtained in a similar manner. When the
second figure (from the left) of the divisor is 1, 2, or 3, it
is well to use the first figure for a " guide figure " in ob-
taining each quotient figure. Thus, in this example, we
say, 3 in 9 = ? 3 in 3 = ? 3 in 7 = ? 3 in 15 = ?
Note. — Be careful to write the first figure in the quo-
tient directly above the right-hand figure of the part of
the dividend used in obtaining the first quotient figure.
2. Divide 10192 by 49.
208
49)10192
98
392
892
In finding the second quotient figure, we
see that 49 is not contained in 39 ; so we
place in tens' place in the quotient and
bring down 2 units. 49 is almost 50. We
may therefore take 5 for a guide figure.
3. Divide 8531 by 672
12|U
Quotient
672)8531
672
1811
1344
In this example the dividend does not
contain the divisor an exact number of
times, hence there is a remainder. The re-
mainder may be written over the divisor as
a part of the quotient ; thus, 12|f| quotient.
Note. — In dividing by any number not
larger than 12, short division should be
467 Remainder
used. That is, no work should be written except the dividend, divisor, and
quotient. In such examples the quotient may be written either above or
below the dividend according to convenience. If the dividend contains
cents, and the divisor is a whole number, the quotient also contains cents.
24 DIVISION
Solve examples 4-68, and test your work hy multiplying the
quotient hy the divisor and adding the remainder^ if there is one,
to obtain the dividend :
4.
4503 - 3
26.
28,692 - 9
48.
130,052 - 2
5.
2045 -^ 5
27.
333,333 ^ 11
49.
168,754 - 9
6.
2835 ^ 7
28.
35,621 ^ 7
50.
385,980 - 5
7.
4986 -^ 9
29.
42,963 ^ 6
51.
769,520 -V- 7
8.
2009 -^ 7
30.
50,725 -r- 3
52.
387,052 - 10
9.
3504 ^ 8
31.
82,956 ^ 10
53.
943,769 -r- 12
10.
.6 193a.
32.
93,043 ^ 7
54.
748,136 - 4
11.
^_±a
33.
65,407 -f- 5
55.
7,688 - 31
12.
iLa840
34.
39,842 - 9
56.
12,978 - 42
13.
12 3 M.
35.
27,392 - 8
57.
31,509 -f- 81
14.
5 8 3 4.5.
36.
63,594 - 9
58.
40,948 -^ 58
15.
11^
37.
31,493 -^ 6
59.
68,476 -h 68
16.
lA&M.
38.
25,324 - 5
60.
168,665 -H 427
17.
2 9 31A
39.
28,764 -f- 4
61.
190,855 -^ 931
18.
^|ij6.
40.
36,099 H- 9
62.
293,004 - 801
19.
30,005 ^
5
41.
14,412 - 12
63.
129,324 - 756
20.
288,012 -
12
42.
36,930 -f- 11
64.
3,247,654^79
21.
300,010 ^
10
43.
24,003 ^ 6
65.
294,490 = 98 X?
22.
99,011 -^
11
44.
30,502 - 8
66.
503x7=637,804
23.
33,264 -
11
45.
29,333 - 11
67.
58,487 has what
24.
29,280 -
12
46.
675,262 - 5
factor besides 143 ?
25.
36,550 -^
10
47.
349,872 - 8
68.
?x 215 = 66,220
REVIEW AND PRACTICE 26
REVIEW AND PRACTICE
40. Oral
1. A farmer exchanged 12 barrels of apples at |3 a barrel
for coal at i 4 a ton. How many tons of coal did he receive ?
2. In what time will a boy earn as much at f 3 a week, as a
man earns in 6 weeks at $8 a week ?
3. Nell is 3 years old and Will 5. Their sister's age is twice
the sum of their ages. What is the sister's age ?
4. How many gallons of milk will a family use in the month
of June if they use 2 quarts a day ?
5. Frank rides 6 miles an hour and Albert 9. a. How far
apart will they be in 6 hours if they start at the same time and
place and ride in the same direction ? h. If they ride in op-
posite directions ?
6. Grace bought 2 dozen lemons. She used | of them for
lemonade and gave away 6. How many remained?
7. Helen bought -^ of a yard of cambric. She used half a
yard in her dress and wasted ^ of a yard in cutting. How
much was left ?
8. A man owed $96. He made 5 payments of 1 12 each.
How much did he then owe ?
9. Luther has % 6 and Leon 3 times as much. How much
have both?
10. 118 plus 112 is how much less than 4 times $12?
11. The product is 125 and one of the factors is 5. What
is the other factor ?
12. The divisor is 11, the quotient 12, and the remainder 9.
What is the dividend ?
13. The dividend is 85, the quotient 9, and the remainder 4.
What is the divisor ?
26 REVIEW AND PRACTICE
14. How many days will a 12-gallon keg of water last 24
shipwrecked men if each man drinks a pint a day ?
15. If Frances can knit 21 stitches a minute, how many
stitches can she knit in half an hour ?
16. The product of 20 and 16 is how much less than 20 times
20?
41. Written
1. How many tons of coal at f 5 a ton will pay for 15 tons of
hay at $11 a ton?
2. A man started on a journey of 724 miles. After he had
traveled 12 hours at the rate of 32 miles an hour, how far was
he from his journey's end?
3. A farmer bought six sacks of flour at §1.25 a sack, 25
pounds of sugar at 6 cents a pound, and two pounds of coffee at
30 cents a pound. He paid for it in butter at 24 cents a pound.
How many pounds of butter were there ?
4. A man having 1738.58 in a bank drew out 1132.75 at
one time, f 175.50 at another, and $216 at another. How much
money then remained in the bank ?
5. If the divisor is 38, the quotient 209, and the remainder
23, what is the dividend?
6. The product of three numbers is 1260 and two of them
are 12 and 7. What is the third?
7. A grocer buys 88 gallons of molasses at $.56 a gallon.
For what price per gallon must he sell it in order to gain
112.32?
8. The dividend is 1821, the quotient 32, and the remainder
29. What is the divisor?
9. Two trains start at the same time from points 1216 miles
apart and travel toward each other, one at the rate of 35 miles
REVIEW AND PRACTICE 27
an hour, the other at the rate of 41 miles an hour. In how-
many hours will they meet?
10. A lady bought 12 yards of dress goods at $1.75 a yard,
8 yards of silesia at f.25 a yard, 2 pairs of gloves at $1.45 a
pair, 6 handkerchiefs at 8.25 apiece, and 3 yards of table linen
at 1.95 a yard. She paid 118.75. What did she still owe?
11. How many pounds of cheese at 14 cents a pound will
pay for 3 barrels of flour at $4.20 a barrel?
12. At what rate per hour must a train run to go as far in 9
hours as another train running 27 miles an hour can go in 12
hours?
13. A farmer paid $1125 for cows, horses, and farming tools,
and 8 times as much for a farm of 125 acres. What was the
price per acre?
14. A farmer had 46 acres of alfalfa. He cut three crops a
year. The first crop yielded If tons per acre, the second IJ
tons, and the third 1 ton.
a. How much was it worth at $8 a ton?
h. How many pounds of alfalfa did he obtain?
15. At a certain post office there were sold in one year 12,400
twenty-five-cent stamp books ; 4600 forty-nine-cent stamp
books ; 2250 ninety-seven-cent stamp books. How much was
received for all of them?
16. A merchant owed a debt amounting to $9892. He
made four payments of $1980 each. How much did he then
owe?
17. A river is 2174 miles long. A steamboat started at
the mouth of the river and traveled up stream for 6 days at the
rate of 149 miles a day. The boat was then how far from
the source of the river? (Illustrate.)
28
REVIEW AND PRACTICE
18. a, A rural free-delivery mail carrier on a certain route
is on duty 298 days in a year and rides 24 miles each day.
How far does he ride in a year ?
5. He starts at 8.15 a.m. and returns home at 3.15 P.M. every
day. How many hours does he spend on the road in a year ?
The number of pieces of mail delivered and collected by
him in one month was as follows:
Registered Letters
Common Letters .
Postal Cards . .
Newspapers . .
Circulars . . .
Packages . . .
Collected
3
877
297
27
26
The total number of pieces delivered was how much
greater than the num-
ber collected ?
d. If there were
189 families on this
route, what was the
average number of
pieces delivered to
each family?
e. If this carrier's
salary is §900 a year,
and it costs him 813
a month to keep his
horse, how much of
the salary is left to pay him for his year's work ?
/. The Post-office Department of the United States pays the
carrier's salary. He sells stamps to the amount of $31.50 per
REVIEW AND PRACTICE
29
1^^^
month, and sends the money to the Post-office Department.
The amount received from this mail route is how much less per
month than the cost of delivering the mail?
19. a. In the year 1905 the Syracuse post-office received
$411,630.95 for stamps, registering letters, writing money
orders, and other postal business.
The expense of carrying on the
post-office was f of this amount.
What was the expense of carry-
ing on the post-office ?
h. How much did the Post-office
Department gain on account of
this post-office ?
c. In this office 722 sacks and
pouches of mail were handled in
one day, each sack and pouch con-
taining an average of 154 pieces
of mail. How many pieces of mail were handled ?
d. At the same rate, how many pieces were handled in a
year ?
20.. a. Willis has 15 hens. They laid at the rate of 120
eggs apiece in one year. How many eggs were laid by all of
them?
h. The food for the hens cost $21. Willis sold the eggs at
an average price of 22 cents a dozen. How much more did he
receive for the eggs than he paid for the food for the hens ?
c. What was the profit from one hen ?
d. What would be the profit from 75 hens at the same rate ?
e. Willis has a rectangular hen park 30 ft. long and 15 ft.
wide. How many feet of wire netting will inclose it ?
21. From one hundred twenty-two thousand take eighty-
seven thousand ninety-four.
30 INDICATED WORK
INDICATED WORK
42. In problems requiring several operations, or steps, it is
well first to indicate the operations by means of signs; e.g.
5208 -T- 3 X 8 means that we are to divide 5208 by 3 and multi-
ply the quotient by 8.
The parenthesis ( ) is sometimes used to inclose certain
numbers or expressions which are to be taken together as one
thing ; e.g. 18 x (15 + 10) means that 18 is to be multiplied by
the sum of 15 and 10.
Operations indicated within a parenthesis should always be
performed first; e.g. 5208 -r- (3 x 8) means that we are to multi-
tiply 3 by 8 and divide 5208 by the product.
When the parenthesis is not used, indicated multiplication
and division should be performed before indicated addition and
subtraction; e.g. 125 + 3x18 — 46^23 means that we must
multiply 3 by 18, then divide 46 by 23, then add and subtract
results as indicated; thus, 125 + 3 x 18 — 46 ^ 23 = 125 + 54 —
2 = 177.
43. Perform the operations indicated:
1. 5208-f-3x8
2. 5208 H- (3 X 8)
3. 203x607015-596034
4. 203 X (607015 - 596034)
5. 487 + 598 + 645- (2030 -1435)
6. 9346 - (6342 + 347 + 89) +2349
7. 9346-6342 + 347 + 89 + 2349
8. 41983 - 87 X 103 + 47
9. (41983 - 87) X (103 + 47)
10. 2310 ^ 10 X 7 + 604 X 35
11. 2310 -i- (10 X 7) + 604 X 35
INDICATED WORK 81
12. 3056 + 9821-7x48-29
13. 3056 + (9821 -- 7) x (48 - 29)
14. (|1.25x6 + 25x|.06 + 2x$.30)-5-24. See example 3,
page 26.
15. $738.58 -(1132.75 + 1175.50 + 1216). See example 4,
page 26.
16. 1216 -^ (35 + 41). See example 9, page 26.
44. Indicate and find results:
1. 1.75 less the sum of $.32 and 1.18.
2. 1500 less $275, and th.e result added to 1 132.
3. $18 more than the difference between $27 and $425.
4. The product of 1125 and 8, divided by 125.
5. The sum of 18 yards and 41 yards taken away from 4
times 69 yards.
6. The sum of 498 and 747 divided by the difference be-
tween 2342 and 2425.
7. George earns 55 cents a day and Harry 79 cents. How
much do they both earn in the month of October, allowing for
4 Sundays ?
8. The product of 162 and 39 divided by the difference of
87 and 61.
9. Frank's earnings for the 6 days of the week were $.43,
$.59, $.62, $.79, $.38, $.48. How much more must he earn
before he can buy a $5 set of books ?
10. The quotient of 12,848 -^ 16 is how much less than the
number of hours in 1 year ?
11. A lady bought 10 yd. of silk at $1.10 a yard and 2 yd.
of silesia at 25^ a yard. How much change should she receive
from a 20-dollar bill ?
32 SPECIAL CASES UST MULTIPLICATION" AND DIVISION
SPECIAL CASES IN MULTIPLICATION AND DIVISION
PRINCIPLES
45. 1. Each removal of a figure one place to the left increases
its value tenfold; e.g. 5 = 5; 50 = 5 x 10; 500 = 50 x 10.
2. Each removal of a figure one place to the right diminishes
its value tenfold', e.g. 500-^10 = 50; 50 -j- 10 = 5.
46. Oral
1. What is the shortest way to multiply by 10 ? By 100 ?
By 1000? By 10000? By 1 with any number of ciphers
annexed ?
2.8x10 = ? 8x100 = ? 8x1000=? 8x10000 = ?
3. Cutting off a cipher from the right of a number is the
same as moving all the figures one place to the right. How
does it affect the value of the number ?
4. How many ciphers must be cut from the right of a
number to divide the number exactly by 100 ? By 1000 ?
By 10000?
5. Divide each of these numbers by 10 :
50; 600; 5290; 36000; 4500; 321560.
6. Divide each of these numbers by 100 :
300; 4500; 6000; 78000; 70000; 831000.
7. Divide 9600000 by 10000.
8. Multiply each of these numbers by 10, 100, and 1000 :
7; 61; 20; 310; 402; 910; 653; 1020.
47. Written 287
1. Multiply 287 by 3700. 3700
2009
86l
1061900 Product
SPECIAL CASES IN MULTIPLICATION AND DIVISION 3B
2. Divide 435600 by 1800.
242 Q uotient
18j9P)4356p
36
75
72
36
1800 = 18 X 100.
Therefore we divide by 100 and
then by 18.
How do we divide by 100 ?
3. Divide 83,645 by 13,000.
Q^%%%\ Quotient
78
6
Multiple/ and test hy dividing
4. 432 by 20,100
5. 69 by 38,000
6. 420 by 80,000
7. 1242 by 3020
8. 5003 by 960
Divide and test results:
14. 257,830 by 590
15. 4410 by 70
16. 34,376 by 100
17. 1,333,800 by 1900
18. 1,308,580 by 260
19. 572,400 by 3600
20. 42,978 by 300
21. 642,359 by 470
When we divide by 1000, there is a
remainder of 648. When we divide
by 13, there is a remainder of 5 in
thousands' place. 5000 + 648 = 5648,
whole remainder.
the product hy the multiplier:
9. 208 by 6500
10. 320 by 420
11. 86 by 12,000
12. 409 by 30,800
13. 6900 X 413
22. 8,205,900 by 4200
23. 367,298 by 1600
24. 368,700 by 3600
25. 496,789 by 420
26. 805,060 by 3090
27. 367,059 by 7800
28. 8,079,600 by 71,000
29. 4,380,700 by 3210
34
IDEAS OF PROPORTION
48. Oral
IDEAS OF PROPORTION
Fig. a.
Fig. B.
1. Figure B is how many times as large as figure A ?
2. If figure A is 1 inch long, how long is figure B ?
3. If A and B are pieces of cloth, and A is worth f 5, what
is B worth ?
4. If A is a piece of land containing 10 acres, what is B ?
5. If A is a piece of land worth $12, what is B worth ?
6. If B is worth |60, what is A worth ?
7. If A and B are blocks of wood and A weighs 9 pounds,
what does B weigh ?
8. If B weighs 39 ounces, what does A weigh ?
9. If A and B are fields, and A can be plowed in 4 days,
how long will it take to plow B ?
<5d ■ 66 66 66 66
10. a, 10 pears are how many times 2 pears ?
h. 2 pears are what part of 10 pears ?
c. If 2 pears cost 3 cents, 10 pears cost cents.
d. Two pears are worth cents, when 10 pears are worth
25 cents.
IDEAS OF PROPORTIOl^
36
11. a. One dollar is how many times one dime ?
h. One dime is what part of one dollar ?
c. If one dollar will buy 30 pencils, one dime will buy
pencils.
d. If one dollar will pay for 70 apples, one dime will pay
for apples.
e. Frank can earn one dollar in hours if he can earn
one dime in two hours.
/. If one silver dollar weighs one ounce, one silver dime
weighs ounce.
12. If John rides 9 miles in 2 hours, in what time can he
ride 27 miles at the same rate ?
Analysis : 27 miles are 3 times 9 miles. Therefore, if John rides 9 miles
in two hours, he can ride 27 miles in 3 times two hours, or hours.
Solve and analyze each of the following problems :
13. If a man's wages for 12 hours are 5 dollars, in how many
hours will he earn 120 ?
14. If 20 men can do a piece of work in 5 days, how long
will it take 10 men to do the same ?
15. When 75^ will buy 6 pineapples, how much should be
paid for 2 pineapples ?
FACTORS AND MULTIPLES
FACTORS AND MULTIPLES
49. One of the numbers that are multiplied to produce a num-
ber is a factor of that number ; e.g. 2, 3, and 5 are factors of 30
because 2 x 3 x 5 = 30.
50. A number that exactly contains another number is a mul-
tiple of that number ; e.g. 21 is a multiple of 7. It is also a
multiple of 3.
51. A number that is composed entirely of whole units is an
integer ; e.g. 7, 13, 200. Can you name a number that is not
an integer ?
52. A factor that is an integer is called an integral factor ; e.g.
8 is an integral factor of 56.
53. A number that is not the product of integral factors other
than itself and 1 is a prime number; e.g. 2, 3, 5, 7, 11, and 13.
54. A number that is the product of integral factors other than
itself and 1 is a composite number; e.g. 16, 24, 35, 1000.
55. A factor that is a prime number is a prime factor; e.g,
13 is a prime factor of 26.
A number that is exactly divisible by 2 is an even number;
e.g. 2, 4, 6, 8, 10.
A number that is not exactly divisible by 2 is an odd number ;
e.g. 1, 3, 5, 7, 11.
Note. — In finding the factors of a number it is customary to consider
only integral factors.
56. Oral
1. Give the factors of 21 ; 35 ; 49 ; 77 ; 26 ; 39 ; 34 ; 15 ; 91.
2. Name three factors of 30.
3. Name as many factors of 24 as you can.
PRIME FACTORS 37
4. Of what numbers are 7, 2, and 13 the prime factors ?
5. Name four multiples of 9.
6. 132 i^ the product of 12 and what other factor?
7. Name all the prime numbers smaller than 50.
8. 84 is the product of three factors. Two of them are 2
and 6. What is the other ?
9. Of what number are 2, 3, 5, and 7 the prime factors?
10. Give the prime factors of 15; 25; 21; 33; 77; 30;
42; 51.
11. 5, 2, and what other number are the prime factors of 70 ?
12. Give two factors of 30 that are not prime.
13. What even number is prime ?
57. Rule for finding whether a Number is Prime or Composite.
1. If the given number is odd^ divide it by 3.
2. if 3 gives a remainder^ divide the given number by 5.
3. Continue this process, using each prime number in order as
a divisor, until an exact divisor is found, or until the divisor
equals or exceeds the quotient. If no exact divisor is found until
the divisor used equals or exceeds the quotient, the number is prime.
Otherwise it is composite.
e.g. To find whether 71 is prime or composite,
3 )71 5 )71 7 )71 11)71
23 — 2 rem. 14 — 1 rem. 10 — 1 rem. 6 — 5 rem.
Since the divisor 11, is greater than the quotient 6, and no
exact divisor has been found, 71 must be prime.
Even numbers need not be tested ; for no even number, ex-
cept 2, is prime. Why ?
38 PRIME AND COMPOSITE NUMBERS
58. In finding the factors of a number, it is useful to remem-
ber that
a. A number is divisible by 2 if the figure in units' place is
even.
l. A number is divisible by 5 if the figure in units' place is
or 5.
59. Find whether each of these numbers is prime or composite :
1. 143 5. 211 9. 121 13. 231 17. 437-
2. 123 6. 221 10. 97 14. 161 18. 401
3. 324 7. 119 11. 213 15. 87 19. 593
4. 163 8. 208 12. 215 16. 78 20. 395
60. Written
1. Find the prime factors of 7020.
2
7020
By what kind of numbers do we divide? Why?
Which divisors do we use first?
What besides the divisors is a prime factor?
3 . 5 . 13 Prime factors, Ans.
2
3510
3
3
1755
585
3
195
5
65
2.
13
2.3.3
2.
I the prime factors of:
120 8. 45
14. 3381
20.
169
3.
42
9. 189
15. 667
21.
561
4.
6Q
10. 665
16. 310
22.
1001
5.
110
11. 429
17. 399
23.
1265
6.
105
12. 425
18. 1287
24.
682
7.
462
13. 414
19. 253
25.
729
CANCELLATION 39
CANCELLATION
61. Dividing both dividend and divisor by the same number
affects the quotient how ?
462 _ ;2 X ^ X 7 X ^ _ ^ ^ .
We might express this work as follows : dividing both divi-
dend and divisor by 2, then by 3, then by 11 :
7
77
m
^ = 7 Quotient
n
1
Taking out the same factor from both dividend and divisor is
cancellation.
62. Solve hy cancellation:
1. Divide 86 X 27 X 49 X 38 X 50 by 70 x 18 x 15.
2. (28x38x48)^(14x19x24x2x2)=?
3. (26 X 5 X 54) - (13 X 5 X 6) = ?
4. What is the quotient of 36 x 48 x 16 divided by 27 x
24 X 8 ?
5. Divide 5 x 45 x 7 x 20 by 49 x 5 x 4 x 9.
6. Divide 5 X 51 X 7 X 9 X 4 by 17 X 20 X 12 X 7 X 2.
7. Divide 25 x 2 x 72 x 14 by 6 x 9 x 120.
8. How many bushels of potatoes at 50 cents a bushel must
be given in exchange for 15 pounds of tea at 40 cents a pound ?
40 REVIEW AND PRACTICE
«o r\ 7 REVIEW AND PRACTICE
63. Oral
1. Name the letters used in Roman notation and give the
value of each.
In finding the sums i
and difEerei
ures first, thus :
36 + 46= ?
36 + 40 = 76
76+ 6 = 82 .4ns.
Say 36, 76, 82.
2. Mnd the sums
'
36 + 47
89 + 27
81 + 29
62 + 38
76 + 39
48 + 24
48 + 53
36 + 17
3. Find the differences :
28-19
41-14
31-13
62-28
43-16
97-58
93-
-27^
= ?
93 -
-20z
= 73
73-
-7 =
66 Ans.
Say
93, 73, 66.
82 + 69
78 + 36
38 + 78
29 + 92
29 + 33
26 + 35
42 + 71
42 + 99
31-14
45-36
75-37
109 - 87
62-19
203 - 174
81-45 76-59 58-29 311-82
4. Crive products at sight:
403 X 10 86 X 200
86 X 100 15 X 40
22 X 10,000 18 X 300
14 X 20 12 X 6000
5. Find results:
27,000 + 13,000 345,000 ^ 100
218 - 38 6250 -f- 10
550x100 435-^10
19
x40
16
x500
200
xl90
403
x8
8324
-f-100
2800
^400
1635-
4-200
REVIEW AND PRACTICE
41
6. Henry can row a boat 20 rods in a minute,
and Eva can row 15 rods in a minute. If Eva
is 60 rods ahead of Henry, in how many minutes
can he overtake her ?
7. a. How many strokes of a force pump are
required to fill -J of a tank that holds 200 gallons
of water, if a pint is pumped at each stroke ?
h. How long would it take at 20 strokes per
minute ?
64. Written
A Force Pump
Find sums and test your work. Can you do it in four min-
I T'V _ J 1 Jl -1 .
utes ? Do not copy addends.
a, 49 5.
235
c.
8749
d. 1346.25
392
419
3254
29.48
48
786
286
934.29
6759
592
39
98.65
24
839
458
813.78
864
496
3476
92.48
9837
318
239
9.62
481
745
8375
46.78
28
932
468
932.86
938
467
9628
^8.93
2. Subtract and test
;
a, 4352 h.
38290
c, 4001
d. 603040
1987
8199
102
13048
3. Divide and test:
a. 153825 by 25.
h. 49386
by
78. c.
12634 by 500.
d, 983,700 by 1500. e. 863,426 by 19,000. /. 163,801 by 690.
4. 2, 3, 5, 7, 11, 13, and 17 are the prime factors of what
number ?
42 LEAST COMMON MULTIPLE
5. What prime factor beside 19 and 11 has 8987?
6. Indicate the work and solve:
a. Divide by 37 the result obtained by adding 111 to the
product of 148 and 6090.
h. A merchant bought 345 pounds of wool of one man, 3067
pounds of another, 468 pounds of another, and 384 pounds of
another ; and sold ^ of it at 27 cents a pound. What did he
receive for the part sold ?
7. Make and solve a problem that might be indicated thus :
110.00 - (1.35 4-12.20 + 16.19 + 1.18).
8. Solve by cancellation,
(48 X 36 X 55 X 26) -^ (12 x 22 x 13).
LEAST COMMON MULTIPLE
65. Oral
1. 3 X 4 = ? 12 is what of 3? Of 4 ?
2. 2 X 6 = ? 12 is what of 2? Of 6 ?
3. Name all the numbers of which 12 is a multiple.
4. Define multiple.
66. A number that exactly contains two or more numbers is a
common multiple of those riumbers ; e.g. 12 is a common mul-
tiple of 2, 3, 4, and 6. 36 is also a common multiple of 2, 3,
4, and 6.
Can you name any other common multiple of 2, 3, 4,
and 6?
67. The smallest number that exactly contains two or more num-
bers is their least common multiple (L. C. M.); e.g. 18 is the
least common multiple of 3, 6, and 9. 36 is a common multi-
ple of 3, 6, and 9. Why is it not the least common multiple?
LEAST COMMON MULTIPLE 43
68.
Oral
Find the L. C. M. of.
1.
2 and 3
2.
2, 3, and 4
3.
4 and 6
4.
9 and 6
5.
10 and 6
6.
8 and 6
7.
5, 3, and 2
8.
1, 2, 6, and 4
9.
2, 3, and 9
10.
5, 4, and 2
11.
7, 4, and 2
12.
10, 5, and 4
13.
2, 4, 8, and 12
14.
4, 5, and 12
15.
7 and 8
16.
16 and 32
17.
2, 3, 6, and 5
18.
4, 9, 3, and 12
69. When the least common multiple is a large number,
the following direct method is employed in finding it.
Let it be required to find the L. C. M. of 12, 15, and 18.
12 = 2 X 2 X 3
15 = 3 X 5
18 = 2 X 3 X 3
What kind of factors have we found ? A number, in order
to contain 12, must have what prime factors? What prime
factors must it have in order to contain 15 ? 18 ?
A number that contains 12, 15, and 18 must have how many
factors 2 ? How many factors 3 ? How many factors 5 ?
What is the smallest number that has the factors 2, 2, 3, 3,
and 5 ? What, then, is the L. C. M. of 12, 15, and 18 ?
The prime factors may be easily found in this way:
2 112 15 18
^ — fi — T^ Q ^^ what kind of numbers do we divide ?
3 5 3 2 X 3 ><; 2 ;x; 5 X 3 = 180 L. C. M.
44 GREATEST COMMON DIVISOR
70. Find the L. C. M.:
1. 18, 27, 30 8. 15, 60, 140, 210 15. 10, 15, 6, 14
2. 9, 12, 18 9. 24, 42, 54, 360 16. 48, 20, 21
3. 16, 48, 60 10. 25, 20, 35, 40 17. 9, 36, 45, 63, 42
4. 21, 27, 36 11. 14, 21, 35, 45 18. 25, 15, 30, 50
5. 36, 40, 48 12. 24, 48, 96, 192 19. 13, 19
6. 18, 24, 36 13. 15, 18, 20, 60 20. 2, 3, 4, 5, 6
7. 15, 30, 21, 28 14. 16, 24, 40 21. 7, 8, 9, 10
GREATEST COMMON DIVISOR
71. A number that will exactly divide two or more numhers is a
common divisor of those numbers ; e.g, .5 is a common divisor of
30, 40, and 60.
72. The largest number that will exactly divide two or more
numbers is their greatest common divisor (G. C.T),)\ e.g. 10 is
the greatest common divisor of 30, 40, and 60.
Note. — A common divisor is sometimes called a common factor^ and the
greatest common divisor is sometimes called the highest common factor.
73. Numbers that have no common divisor are prime to each
other ; e.g. 13 and 15.
74. Oral
1. Find the G-. 0. D. of:
a. 6, 9, 12 e. 8, 24, 40 L 30, 45, 60
5. 10, 30, 35 /. 14, 28, 42 /. 18, 27, 36
c. 2, 10, 16 g. 33, 22, 77 h. 12, 24, 36, 48
d. 12, 30, 18 h. 21, 27, 39 I. 24, 32, 48
2. Name two numbers of which 7 is a common divisor.
3. Name three numbers of which 9 is a common divisor.
4. Name two numbers which are prime to each other.
GREATEST COMMON DIVISOR
46
5. What is the greatest number that will exactly divide 12,
30, and 36 ?
6. Name two numbers of which 11 is the G. C. D.
7. Tell which of these pairs of numbers are prime to each
other:
a. 12 and 7 b. 16 and 20 c. 19 and 21 d. 8 and 15
75. Written
1. Find the greatest common divisor of 336, 504, and 924.
336 = ;2x;2x2x2x^x /
504 = ;2x;2x2 x^x3x7
924 = ;2 X ;2 X ? x 7 X 11
2 X 2 x 3 X 7 = 84 G. 0. D.
Factoring the numbers and selecting the common prime factors, we find
them to be 2, 2, 3, and 7. Since all of them are factors of each of the given
numbers, their product, 84, is the greatest common divisor required.
The common prime factors may easily be found in this way:
2
336
504
924
2
168
252
462
3
84
126
231
7
28
42
77
4
6
11
2 • 2 • 3 • 7 Common prime factors.
Find the a. 0. D. :
2. 63,42
3. 90,105
4. 112, 168
5. 132,156
6. 40, 60, 80
8. 36, 48, 24 14. 63, 126, 189
9. 40, 56, 72
10. 18, 54, 32
11. 45, 60, 90
12. 36, 72, 81
15. 36, 81, 135
16. 91, 143, 156
17. 192, 400, 240
18. 168, 210, 308, 350
7. 64, 144, 560 13. 44, 121, 33 19. 1980, 945
46 REVIEW OF INTEGERS
20. Find the greatest number that will exactly divide 189,
378, and 504.
21. Find all the common prime factors of 360, 540, and 450.
22. Find the product of all the common prime factors of 108,
144 and 360.
23. Find a number that is prime to 210.
24. Name three numbers of which 11 is the greatest common
divisor.
REVIEW OF INTEGERS
76. Oral
1. (15-4)x(3 + 2) = ?
2. 15-4x3+2=?
3. Numerate 137,640,507,239.
4. Name the periods in the above number.
5. ReadDCXLIV.
6. What is the value of ^^?
7. 35 + 48 = ? (35 + 40 = 75; 75 + 8 = 83. Say 35, 75, 83.)
In the same way add :
a. 6S and 29; b. 58 and 15; e. 49 and 33; d. 67 and 24.
8. The product of two factors is 45. If one factor is 9,
what is the other? If one factor is 15, what is the other? If
one factor is 6, what is the other?
Which of the factors given in your answer is not an integral
factor?
9. The product of three factors is 108. Two of them are 4
and 3. What is the other?
REVIEW OF INTEGERS • 47
10. Find tlie difference by subtracting the tens first :
a. 64-25; 5.81-32; c. $.76 - $ .28; (^. 11.27 -1.79.
11. 24 X 20 = ? 30,700 -- 100 = ? 1200 -^ 200 = ?
22. 235 X 1000 = ? 208 X 10 = ? 4000 -5- 400 = ?
13. 1,500,000-^1500 = ?
14. Edward earned $3 one week and $6 the next. How
much was left after he had spent | of it?
15. The change for 1.24 from 11.00 is |.06 + |.70=$.76.
Say 6, 76.
Find the change from 11.00 for:
a. $.28 c. $.18 g. $.69 ^.$.52 2. $.79 A;. $.72
h. $.10 d. $.42 /. $.37 A. $.39 j. $.83 L $.35
16. a. One day is what part of a week? If a man pays $84
for a week's travelling expenses, they average how much per
day ? h. At the same rate, what would they be for 11 days ?
17. What part of $5 is $21? If $5 will buy 20 splint
baskets, how many such baskets will $ 2^ buy ?
18. How many times is $20 contained in $400?
19. Whatpart of $400is$20?
20. A cent is what part of a dime ? 7 cents are what part of
a dime ?
21. If a dime will pay for 20 steel hooks, how many such
hooks will 7 cents buy ?
22. 6 is what part of 12 ?
23. When a hardware merchant makes a profit of $1.44 on
12 window screens, what does he make on 6 of them ?
48 REVIEW OF INTEGERS
77. Written
Test and time yourself on the first eight examples.
1. Add :
2. Add:
3. Subtract: 4. Subtract
287
627
807,204 230,007
965
438
99,197 150,008
473
796
287 .
69
342
109
5. 2059x78
218
246
781
627
6. 786 X 205 -f- 210
968
749
280
3578
7. 302 X (4780 - 3874)
421
372
642
7986 .
8. 346793-^5700
568
8144
9. A park in the shape of a rectangle, 135 rods long and
48 rods wide, contains how many square rods of land ? How
many acres ?
10. A certain street is 6 rods wide. How long must it be to
contain 77 acres of land ?
11. A farmer has 20 cows and feeds each of them two quarts
of corn meal a day. How long will 100 bushels of corn meal
last them ? (Solve by cancellation.)
12. A man owed his grocer f 135. He paid | of the debt in
labor and the rest in cash. a. How much cash did he pay ?
h. How many days did he work, if he received $2 a day ?
13. In the year 1905, 1,027,421 immigrants came to this
country ; 317,000 of them settled in New York State, 222,300 in
Pennsylvania, 20,000 west of the Mississippi River, and the rest
FRACTIONS
49
in other parts of the country, a. How many settled in other
parts of the country ? h. The entire number was how many
times the number that settled west of the Mississippi ?
REVIEW OF FRACTIONS
(^Studied in the Primary Arithmetic)
M,
i
I I I I
I I I I I
N.
i
78. Oral
1. This figure shows that \ — ^-^. What else does it show ?
2. Make a figure to show that f = J.
3. Make a figure to show that f = ^.
4. Make a figure to show that |^ = f and ^ = f •
5
. What is the value of ^
' ^1 H
^?
f? J^:
> 1?
¥?
6
. Express in lowest terms:
I; f;
A; 1
' is> ri'
A
' 2^'
7.
* + i = ?
16.
1 + 1 =
?
25.
f2-
f=?
8.
* + i + i
= ?
17.
i + i =
?
26.
A +
J = ?
9.
f + j = ?
18.
|-i =
?
27.
tV +
t=?
10.
i-i = ?
19.
f + f =
?
28.
T>2 +
i = ?
11.
i + i = ?
20.
*-J =
?
29.
i-
i = ?
12.
i-i = ?
21.
J + f =
?
30.
i +
1 = ^
13.
i + i = ?
22.
J + i =
?
31.
i +
i=?
14.
i--i = ?
23.
A + i =
?
32.
1-
tV = ?
15.
i + i = ?
24.
l%-i =
?
33.
f-
i = ?
50 FRACTIONS
34. Harold bought a melon. He gave |- of it to Clarence and
^ of it to Howard. What part of the melon did he keep ?
35. A blotter is 1^ inches long and 3| inches wide. What
is the sum of the two sides ? Of the two ends ? What is the
perimeter ? Draw the blotter, full size.
3. 41Jj 4. 25^
268|_ _8^
7. 84| 8. 152f
91^ 801-1
79. Written
Add:
1. 12i
13i
2.
58i
25j
28|
S. 18J
6.
325|
5|
27f
41i
Subtract and test your work:
9. 43f
10.
85V^
29J
37|
13. 132f
14.
203
24A
161
11. 401f 12. 204-j^
123J 131
15. 83 16. 401 If
HA j5|_
17. What must be added to 5J to make 16| ?
18. What must be taken from 18^^^ to leave 6|?
19. Laurence bought a pencil 6lf inches long and cut it into
3 pieces, two of which were 3| and 2^1 inches long. How long
was the third piece? Prove the correctness of j^our answer by
drawing a line 6^| inches long and cutting off pieces 3| and 2^^
inches long.
20. Indicate by signs the work required for example 19.
FRACTIONS 61
FRACTIONS
80. One or more of the equal parts of a unit is a fraction; e.g.
i' |! f ; *A-
81. A fraction is always an expression of division. For ex-
ample, if 1 inch is divided into 8 equal parts, each part is -J of
an inch. If a line 7 inches long is divided into 8 equal parts,
one part is -J of an inch long. That is, 1 in. -v- 8 = J in., and
7 inches -i- 8 = |- inch.
Take your rule and draw a line 1 inch long. Divide it into
4 equal parts. How long is one part? Draw a line 3 inches
long. Divide it into 4 equal parts. Measure one of the parts.
3 inches -^ 4 = ?
Draw a line 5 inches long. Divide it into 8 equal parts.
Measure one of the parts. 5 inches -^-8 = ? 3h-7 = ?
9^11 = ?
82. The number above the line in a fraction is the numerator.
It is always a dividend. In the fractions ^, ^, ^, |j, the
numerators are 1, 7, 15, and 23.
83. 77ie number below the line in a fraction is the denominator.
It is always a divisor. In the fractions J, -|, ^^^ -||, the de-
nominators are 3, 9, 5, and 12.
84. The numerator and denominator are the terms of a frac-
tion; e.g. the terms of -^j are 7 and 11.
85. The value of a fraction is the quotient obtained by dividing
the numerator by the denominator.
REDUCTION OF FRACTIONS
86. Changing the form of a number without changing its value
'ds reduction ; e.g. 8 pt. = 4 qt.; $7 = 700 ct. ; 7 ft. = 2^ yd.;
^^=3; i| = f;| = A-
52 KEDUCTION OF FRACTIONS
REDUCTION TO LOWEST TERMS
87. A fraction is in its lowest terms when the numerator and
denominator are prime to each other ; e.g, |^, f^^ ||^.
88. Oral
1. Dividing both dividend and divisor by the same number
affects the quotient how ?
2. 1^ compares how with ^ ?
3. -^j compares how with | ? What did we do with the
terms of ^^ to obtain ^ ?
4. Show by these circles that
i=h
t = |-
x\ = h
^ = -1-
A=l-
iV = t-
fo = h
5. How are these fractions reduced to lowest terms ?
6. Reduce to lowest terms : |; |; |; |; |; -^; |; f; -fji
h f^; A; t'j; ^r' f^; ^; ^; A; if; if; if; if; if; A;
IXL. _8_. 8
12' 12' 1?-
89. Written
Reduce || to lowest terms. || = || = J 4^«. We divide
both terms by 2 and then by 3.
If we use the greatest common divisor (6), we shall need to divide only
once, thus f f = |.
Note. — We may often save time by remembering that an even number
will never exactly divide an odd number. Can you tell why?
REDUCTION OF FRACTIONS 63
22.
^1
23.
IH
24.
3%
25.
T%
26.
m
27.
m
28.
m
90. Reduce to lowest terms :
1- If 8- l¥? 15- ^s¥j
3. M -10. 5^ 17. 11^
4- M 11- ^J 18. ^
5. If 12. Ill 19. ^
6- M 13- iifs 20. iH
'• A% "• SWV 21. ^%
29. Express in lowest terms 230 -f- 345.
30. Express in lowest terms 98 divided by 392.
31. Express in lowest terms 437 -4- 2484.
32. Express in lowest terms the quotient of 288 divided
by 504.
33. What are the lowest terms of if J ?
REDUCTION OF IMPROPER FRACTIONS TO INTEGERS OR
MIXED NUMBERS
91. A fraction whose numerator is smaller than its denominator
is a proper fraction; e.g. f, ^^p \^, The value of a proper
fraction is always less than 1.
92. A fraction ivhose numerator equals or exceeds its denomi-
nator is an improper fraction, e.g. |^, |, ^|. The value of an
improper fraction compares how with 1 ? f
93. A numher that is composed of an integer and a fraction is
a mixed number; e.g. 5|, lOi 201-j\.
54 REDUCTIOlSr OF IMPROPER FRACTIONS
94. Oral
1. A boy has two half dollars. That is the same as how
many whole dollars ? Six half dollars equal how many whole
dollars ? How do you find it ?
2. Eleven half dollars make how many dollars and how
many halves over ? How do you find it ? Write it.
3. How many quarters make a dollar ?
4. How many dollars are there in 8 quarters ? 40 quarters ?
5. Fifteen quarters make how many dollars and how many
quarters over ? Write it. What do you do to find it ?
6. I = how many whole ones ? | ? -^ ? | ?
7. 4=? | = ? Jj^ = ? -V^=?
8. A fraction is an expression of what operation?
9. How may we find the value of a fraction ?
10. Define the value of a fraction.
Mnd the values of:
11 I 15. -\Q- 19. -2g5- 23. \3. 27. 2L^Sl 31. 1|
12. f 16. I 20. Aj^ 24. If 28. ^| 32. f|
13. f 17. J^ 21. i^ 25. ^9^ • 29. ^f 33. ff
14. I 18. ^^ 22. -^ 26. If^ 30. -J| 34. -L\^
95. Written
1. 1|1 3. ^^ 5. i^^ 7. If^il 9. %4
2- W 4- -Vf 6. ^ 8. ^ 10. i||il
11. -fj*^ 14. ^f^ 17. S/ji 20. A|^A
12. il4|i 15. -V^a 18. i'jl^ 21. i|-fA
13. -^ 16. ^ 19. 2J^^ 22. -4111
REDUCTION OF INTEGERS AND MIXED NUMBERS 55
REDUCTION OF INTEGERS AND MIXED NUMBERS TO
IMPROPER FRACTIONS
96. Oral
1. How many fourths in 1 circle? In 2 circles? In 3
circles ? In 4 circles ?
2. How many fourths in 4| circles ? In 2| circles ? In 3|
circles ?
3. How many eighths in 1 circle ? In 3 circles ? In 2
circles ? In 4| circles ? In 2| circles ?
4. How do you reduce an integer or a mixed number to a
fraction ?
Reduce to improper fractions
5. IJ
9.
3|
13. 8f
17.
8J
6. 41
10.
2|
14. 4|
18.
^A
7. 3f
IX.
4f
15. 5f
19.
8t%
8. 5i
12.
21
16. 64
20.
9i
97. TTn^fgw
1. Reduce 38J to a fraction.
38 = 38 X 9 ninths = 342 ninths.
342 ninths plus 7 ninths = 349 ninths.
The work may be expressed thus: 38| = ^^ Ans.
_9
342
7
349
56 LEAST COMMON DENOMINATOR
Reduce to fractions :
2.
9A
9.
49i^
16.
19t^
23.
35^
3.
^^
10.
25^
17.
29A
24.
191^
4.
25f
11.
59A
18.
149f
25.
203 j8,-
5.
16^
12.
67t^
19.
1281
26.
98ii
6.
23^
13.
89M
20.
137^
27.
8711
7.
40i
14.
131|
21.
238iJ
28.
138j2^
8. 3T^ 15. 270|| 22. 491^ 29. 351if
LEAST COMMON DENOMINATOR
98. Fractions whose denominators are alike have a common
denominator ; e.g. 60 is a common denominator of g^^, l|, and |J.
99. Fractions having the smallest possible common denomi-
nator have their least common denominator; e.g. ^-^, ^^, ^.
100. Oral
1. We have found that when we add fractions having dif-
ferent denominators, we must first change them to fractions
having the same denominator. What shall we call that
denominator ?
2. Since the common denominator must contain all the
given denominators, it must be what of those denominators ?
(A number that exactly contains two or more other numbers
is what ?)
3. The least common denominator, then, must be which
multiple of the given denominators ?
4. Reduce |, |, and J to fractions having the least common
denominator.
LEAST COMMON DENOMINATOR
67
How many 12ths in 1 ? (12 --- 4 = 3.)
How many 12ths in | ? (^^^ = — . )
How many 12ths in 1 ? (12 ^ 6 = 2.)
How many 12ths in f ? ('^^^ = —^
^ ^ V6 X 2 12 ;
How many 12ths in J ? (12 -^3 = 4.)
How many 12ths in | ? f^^ = -^.^
^ ^ V3x4 12;
Change the following to fractions having the least common
denominator :
5- hi
6- hi
12. f, I, f
13
6 1
6' 2
14.
J.f'l
23.
h h h i
15.
f f ' f . T^
24.
h I \h i
16.
J' !. f ^
25.
h tV' h J
17.
A' ¥' i
26.
f T^I- |. Jl
18.
f . I'f ' f
27.
f i^ h i
19.
f.*'*
28.
h f . i' ■h
20.
f fA
29.
h h h h tV
21.
,f ' h H
30.
h h h 1^' tV
22.
i'l^' A
31.
i' ¥' 16' 32' 2
58 REVIEW AND PRACTICE
101. Written
Change •:j^, j^^, Jf and ^^ to fractions having the least com-
mon denominator.
7
8
16
17
2
10
15
33
30
3
5
15
33
15
5
5
5
11
5
1 1 11 1
2 X 3 X 5 X 11 = 330 L. C. M.
330
•^ 10 = 33
7 x33 231
10 X 33 330
330
-f-15=22
8 x22 176
15 X 22 330
330
-i- 33 = 10
16x10 160
33 X 10 330
330-
- 30 = 11
17 X 11 187
==
30 X 11 330
131 116. 110 18X A7)fi
330' 330' 330' 330 ^^^'
Change to fractions having the least common denominator:
^- h f' f 6. 1, 1, 1, J 11. i|, ^5_ II
3. I' 1% i 8. 1 |, I, ^ 13. 1|, ^5_, ^
^' fit 9- I' 14' A 14. M'l^^'A
5. f , ^2 , if 10. f , ^5^, f , I 15. ^, ^^, J3, 38_
102. Ora?
REVIEW AND PRACTICE
1. What change should I receive out of $2 for a purchase
of 1.50? 8.75? $.85? i.45? $1.25? 11.79? $.69?
2. Henry bought a top for 3 cents, some candy for 11 cents,
and a pencil for 7 cents. What change should he receive from
a quarter?
3. $240 will buy how many typewriters at $60 apiece?
At $ 80 apiece ?
4. 8 cows at $ 40 a head cost how much ?
5. What is the cost of 2 bushels of potatoes at 20^ a peck ?
REVIEW AND PRACTICE 59
6. 800 + 500 + 700 + 1500 = ?
7. Name the prime factors of 90.
8. Tell the value of -J/; 1^^; 2i; 3^^; J_2j)_(i.
9. Change to improper fractions 8| ; 2|; 17| ; 5^J; 241;
10. What is a fraction ? If you change | to eighths, how
will its value be affected ? How will the number of parts be
changed ? How will the size of the parts be changed ?
11. How does ^ compare with ^ ? Show this by a drawing.
12. Which is larger, 1 of an apple or ^ of an apple ? -^- or J?
fori?
13. Which is greater, | or | ?
14. 29 pounds are how many times 5 pounds? Compare
$250 with $50; 1 qt. with 1 pt.; 80^ with 20)?^.
15. Compare 2 cents with 50 cents ; 2 gal. with 3 gal. ; 8 lb.
with 64 lb. ; $.25 with $1.50.
16. Find the cost of 24 souvenir cards at the rate of 3 for
5 cents.
17. A windmill turned 20 times a
minute with a certain wind. The
owner oiled the bearings of the mill
and then it turned 24 times a minute
with the same wind.
a. How many turns per hour were
gained by oiling the bearings ?
h. How many times as much work
did the mill do after oiling as before
oiling ?
c. What part as much work did the
mill perform before oiling as after oiling ?
60
REVIEW AND PRACTICE
103. Written
Land Surfaces in Square Miles
New York . . .
47620
Rhode Island . . ,
1080
Texas ....
262290
Pennsylvania . .
. 44980
Nebraska . . . .
76840
Connecticut . .
4850
Delaware . . .
1960
Illinois ....
56000
California . . .
155980
Montana . . .
145310
Kentucky . . .
40000
Massachusetts
8040
New Jersey . . .
7450
New Hampshire .
9000
District of Columbia
. . 60
Alaska . . (nearly) 570390
Note 1. — Water surfaces are not included in the above figures.
Note 2. — While answering questions 1-5, keep your geography before
you, open at the map of the United States. By referring to the map,
estimate each answer before computing it, and then compare your estimate
with the result obtained by computation.
1. a. Texas contains how many times as much land as New
York ? h. It contains how many more square miles of land
than New York ?
2. Alaska would make how many states the size of New
Hampshire ?
3. Compare, by division, the land areas of:
a. Alaska and Illinois.
b. Illinois and New Hampshire.
c. New Jersey and Pennsylvania.
d. Rhode Island and Texas.
e. Massachusetts and New York.
/. Connecticut and California.
ff, Montana and Delaware.
h. Rhode Island and District of Columbia.
4. Compare, by subtraction, the land areas of:
a. Nebraska and Pennsylvania.
ADDITION OF FRACTIONS AND MIXED NUMBERS 61
h. Delaware and New Hampshire.
c. Kentucky and Rhode Island.
d. Illinois and Massachusetts.
e. Texas and Alaska.
5. a. Find which of the columns of land surfaces (top of
page 60) indicates the greater number of square miles.
b. What is the difference ?
Make other problems from the above table.
6. Find the prime factors of 1232.
7. Reduce 365J and 66| to improper fractions.
8. Reduce -^^^ to lowest terms.
9. How many 15ths are there in 39 ?
10. Find the value of ^OgO-; %2_4; J^_t
11. How many 40ths are there in 7| ?
12. 7 is equal to what fraction having 7 for a denominator ?
13. Reduce to lowest terms:
«• }lf *• iff 0. Ill d. ii, e. Uf /• ^-h 9- HI
14. Reduce to fractions having the least common denominator :
"• 9' IB' 10 ^- 1?' 21' 35 ^' 11' 3' 8
ADDITION OF FRACTIONS AND MIXED NUMBERS
104. A number is in its simplest form when it is in the form of
an integer, or a proper fraction in its lowest terms, or a mixed
number whose fractional part is in its lowest terms ; e .g. 18, | and
5^ are in their simplest forms ; ■^, |^, -*^ and 8| are not in their
simplest forms. Why ?
Answers should always be expressed in simplest form, unless the
question requires a different form.
105. Oral
Add:
1- hi
5.
ii
2- hi
6.
hi
3- h\
7.
hi
4- f.l
8.
hi
62 ADDITION OF FRACTIONS AND MIXED NUMBERS
9. f,J,3 13. 1|,2^,J
10. i,|, 11 14. 3i,lf,5
11. 21 41 1 15. f, 8J, 4
12. f j.jj 16. ^,^,^
17. A man paid I J for a book, f | for an inkstand, and f J for
writing paper. How much did he spend ?
18. Mary had 8f ; her mother gave her $8 J. How much
had she then?
19. The addends are 7f , 16 J, IQlf. What is the sum ?
20. Mary walked 5| miles on Monday, 4 miles on Tuesday,
5| miles on Wednesday, and as far during the next three days
as during these days. How far did she walk in all ?
106. Written
9 8 6
3 4 1 f If = 2^ S^^-
2x2x3x3x4 = 144, L. C. D.
Add lOf , 7f , and 6|.
o ■" ? 0^ We add the whole numbers and frac-
• t = • ^1^ tions separately, and then unite the
6f =
sums.
2^^ = 2^^Sum,
17.
6f , 8|, f , i
18.
3, f , h 1
19.
f . 4, f, li
20.
1' h iV t\
21.
6f , 81 5|, 7|
22.
9i,5f9,J|
23.
i f . % i
24.
6i, 7f , 91, 45
SUBTRACTION OF FRACTIONS AND MIXED NUMBERS 63
Add:
4. f, i, f 12. f, 1^, I, i
5. JL,-|,| 13. J,2|,|,6
6- h i I' ^ "• i' I' 2i, tV
^' f' A' 2 0' 2 ^^* 4 9' y 3' 2T
25. What is the sum of 14f 9|, IQi and 12lf ?
26. A man travels 25| miles on Monday, 37| miles on Tues-
day, and on Wednesday as many miles as on Monday and Tues-
day. How many miles does he travel in three days ?
27. A farmer has 27 J bushels of potatoes in one bin, 133|
bushels in another, 47^^^ bushels in- another. How many
bushels has he ?
28. How many yards of cloth will I have, if I buy 123| yards,
76| yards, and 58| yards ?
29. 2J yards of cloth are required for a coat, IJ yards for
trousers, and | of a yard for a vest. How many yards are
required for the whole suit ?
SUBTRACTION OF FRACTIONS AND MIXED NUMBERS
107.
7.
Oral
1.
f-i
6.
f-f
11.
12 -H
2.
«-J
7.
T^.-|
12.
22 -l|
3.
l-i
8.
il-f
13.
4-lJ
4.
i-i
9.
7-J
14.
^-^
5.
T\-f
10.
8-1
15.
11 -8f
64 SUBTRACTION OF FRACTIONS AND MIXED NUMBERS
16. 19|-9f
20.
6-i
24. 121-51
"• i-i
21.
3i-i
25. 81 -4J
18. i-i
22.
15-21
26. il-l
19- i-i
23.
10-f
27. 1-1
108. Written
From \^ take |.
1 = 1^
How is
45 obtained?
■Jl Difference
From 29J take IS-,;^.
How do we obtain |4?
15J|^ = 15^ Difference
1. Take 1 from f.
2. From Jf take ^p
3. Find the difference between l|^ and ^.
4. Take 91|- from 1781
5.
H-i
13.
U-i
21.
13,^-3^
6.
H-H
14.
81-51
22.
481^^-232,^
7.
42f-33f-
15.
2101 -109f
23.
862|-46f
8.
198f-49f
16.
12i-5i
24.
2301-140^
9.
H-H
17.
n-H
25.
891-431
10.
16f-8f
18.
461-271
26.
807-jV-298}
11.
l-i
19.
1867f-976|
27.
190| - 28f
12.
l-A
20.
321 _ 26f
28.
281^-371
ADDITION AND SUBTRACTION OF FRACTIONS 65
29. A piece of silk contains 18J yd. How many yards will
be left after 13 J yd. are used ?
30. Mrs. Brown bought 4| yd. of broadcloth and used all but
1| yd. How much did she use ?
31. Find the difference between 256J and 149J.
32. A man bought a lot at auction for $92^ and sold it the
next day for |105|. What did he gain ?
ADDITION AND SUBTRACTION OF FRACTIONS
109. Written
1. From a piece of cloth containing 47|^ yd., 22| yd. were
sold to one lady and 5| yd. to another. How many yards
remained unsold ?
2. A farmer sold a load of hay for f 13^"^ and another for
$16|. He was paid $25. How much was still due ?
3. A lady paid $1-^-^ for a pair of gloves, $S^ for an umbrella,
and 1292'^Q- for dress materials. How much should she have left
from four ten-dollar bills ?
4. What must be added to the sum of |- and 10| to
make 20?
5. From the sum of 109| and 87| take their difference.
6. A grocer drew at one time 9|- gallons and at another
time 15f gallons from a tank containing 44^g gallons of oil.
How many gallons were left ?
7. Mary and Alice live on Bryant Avenue, and their school
is on the same street, between their homes. Mary walks 40^2_.
rods to school, and Alice 25J| rods. a. How much farther
does Mary walk than Alice ? b. How far apart are their
homes?
66 ADDITIOI^ AND SUBTRACTION OF FRACTIONS
8. It took Mr. Farmer S^^ hours to plow a field, and 13|
hours to plant it. a. How much more time was required for
planting than for plowing ? b. How much time was required
for both ?
9. 132\-2^3_ + 7|._4i|^ ? Can ^ou find the result in
two ways ?
10. a. i| + (li-J) = ? 5. 8i|-(li + J) = ?
11. Roscoe gave ^ of his new writing pad to his sister and
^ to his brother. What part did he keep ? •
12. From 16f + 12| + 51 take 18f + 6|. .;
13. Add |, |, ^^2' '^^^ A ' ^^^®^ subtract IJ from the sum.
14. After reading g^, f, and 1 of a book, what part have you
yet to read ?
15. A owns 79^5g acres of land, B 9^^ acres less than A,
and C 25^ J acres less than B. a. How many acres has B ?
b. How many acres has C ? e. How many acres have all three
together ?
16. From 4223^0- take 826|.
17. Take 8^2 ^^^m 17^^.
18. A has I641-. B has $37| less than A. How much
money have both ?
19. 19^g yards of twine were cut from a ball containing
69^ yards. The piece that was left was how much longer
than the piece cut off ?
20. Add |-, |, and ^J, and subtract the sum from 5.
21. Find the sum of |, ^, -f^^ and ■^\.
22. Find the difference between | and -^^y.
23. A boy walked to his grandfather's in three hours, walk-
ing -^^ of the distance the first hour and ^ the second hour.
What part of the distance did he walk the third hour ?
MULTIPLICATION AND DIVISION COMBINED 67
QUICK TEST
110. 1. 39 cents is how much less than one dollar ?
2. 25x20 = 1000-^?
3. Which is greater, |- or J ?
"^ 4. 3400 is the product of 100 and what other number ?
5. Express ||- in simplest form.
6. Express J as 28ths.
7. How far will a motor car run in 12 hours if it runs
at the rate of 50 miles in 4 hours ?
8. What is the L. C. M. of 2, 3, 4, and 5 ?
9. What is the G. C. D. of 21, 35, and 49 ?
10. 16 is the sum of 10| and what other number?
111.
MULTIPLICATION AND DIVISION COMBINED
Written
/
(20^
20-1-
4) X (21 - 7) = ?
4 = 5 21-^7 = 3 5x
3 = 15
Ans.
or.
fxf
20 x21
4x7
= 15 Ans.
20 and 21 are dividends and 4 and 7 are divisors. The result is the same
whether we make each division separately and then multiply the quotients,
or divide the product of the dividends by the product of the divisoi-s. In
many cases the latter way is easier, because we may use cancellation ; e.g.
5 8
I a.(20 + 4)xC21-i-7) = (|x|)=?|^ = 15^«s.;
^7 42
b. (18 + 7) X (28 + 24) X (210 ^ l.S) = ^^-^|i2lM= 42 Ans.
^
68 MULTIPLICATION OF FRACTIONS
Find results :
1. (12-5-ll)x(22--5)x(35-*-6)x(15-f-2)
2. (20 -5- 6) X (55 -^ 10) X (42 -11)
3. (39 -^ 13) X (35 -5- 21) X (12^ 7) X (21 -J- 3)
, 42 36 63
*• T^T^li
5. (27 -^ 18) X (35 -5- 75) x (25 -^ 12) x (12 - 7)
6. (68 -I- 7) X (14 - 8) X (35 ^17)
7. (52 -^ 10) X (34 H- 13) X (125 -h 10)
8. (26 -^ 20) X (68 -f- 13) X (125-^35)
9. (70 -f- 17) X (68 -5- 24) X (35 -i- 7)
7510898 49241720
42 ^ 26 ^ 15 ^^* 56 ^ 34 ^ 5 ^ 3
12. Multiply the quotient of 29 divided by 12 by the quo-
tient of 84 divided by 29.
MULTIPLICATION OF FRACTIONS
112. Any integer may be expressed as a fraction by writing
it as a numerator with 1 for a denominator^ e.g. ; 5 is the
same as -^ ; 19 is the same as -Ij^ ; |- x 7 x ||- is the same as
fx^xM-
113. The word of, between fractions^ means the same as the sign
of multiplication ; e.g. |of| = |x|; |of4x^ = fx^X^^^.
114. An indicated multiplication of two or more fractions is
called a compound fraction ; e.g. f X | ; ^ X Jf X |f ; f of |.
MULTIPLICATION OF FRACTIONS 69
115. Written
1. Find the product of f , f , and -^q.
Each of these fractions indicates what operation ?
Since all the numerators are dividends and all the denominators are divi-
sors, we may find the result by dividing the product of the numerators by
the product of the denominators, as in Article 111, using cancellation :
3
8
15
56
■ Ans.
Find the products .
:
2. |X|
8.
fof^off
14.
fxTxif
3- Axf
9.
ixi\xl-
15.
1 of 1 x 14
4. ioiJ^
10.
Axfxi
16.
Axifx22
5. fofjf
11.
AxJ^<ixf
17.
2xfof iV
^- i^x^
12.
i|x34xf
18.
fT0f34x^
7. .|0f|0f|
13.
f ofl5
Find the value :
19. 1 of 40
22. f of 328
25. ^J of 342
20. fof42
23. f of 721
26. If of 800
21. |ofl6
24. 1 of 90
27. -5^ of 2222
28. Find the areas of rectangles having these dimensions :
a. I in. by | in. e. f|- mi. by ff mi.
h. f yd. by f yd. /. ^\ mi. by |f mi.
c, -^ rd. by |- rd. ff. |- mi. by |^ mi.
d. f ft. by I ft. A. \i mi. by |f mi.
29. What is the cost of J of 24 quarts of milk at 5|- cents a
quart ?
30. A grocer bought 27 barrels of apples and sold ^ of them
at $2^ a barrel. How much money did he receive ?
70
MULTIPLICATTOISr OF FRACTIONS
MULTIPLICATION OF FRACTIONS ILLUSTRATED
GRAPHICALLY
Wi
fo/i
i i
12 I 12
bcl.
io/i
ion
iofi
Jon
ion
iofi
Wi :*•
's*
Fig. 1
h
How many 6ths ?
116. Look at Fig. 1 and answer :
1. How many 12ths are there in |-?
2. How many 12ths are there in |^ of |- ?
3. How many 12ths are there in |?
4. How many 12ths are there in ^ of J?
How many 12ths are there in J of f?
How many halves are there in | of | ? f o^ f = ^ = 2 •
What else is shown in this figure ?
Look at Fig. 2 and answer :
1. A, B, 0, and D together are what part
of Fig. 2?
2. A is what part of | ?
3. A is what part of Fig. 2 ?
4. A and E together are what part of
Fig. 2?
5. J.is what part of ^? J of ^ = ?
6. A-\- B + C= what part of J? A + B+ (7= what part
of Fig. 2 ? I of 1 = ?
Note. — Feet and inches are sometimes indicated by marks, thus: 7'
stands for 7 feet j 6" stands for 6 inches. What does i" stand for ? |' ?
A
1 1
B 1 C ! D
1 1
1 1
E
1 1
1 1
i 1
! 1
Fig. 2
MULTIPLICATION OF FRACTIONS
71
From Fig. 3 answer the following questions
1. How many parts like K are there in
the square inch ? i
2. What part of a square inch is ^?
3. What is the length of JT? The
breadth? The area?
4. The top row of oblongs is what part
of the square inch? ^is what part of that
row ? -^ of I = ?
5. The left-hand column of oblongs is
what part of the square inch ? ^ is what
part of the column ? ^ of ^^ = ?
In Figure 4 :
1. M (the unshaded part) is how long ?
How wide ? What is its area ?
2. jK" is what part of the square inch ?
M contains how many parts like Kl M
is what part of the square inch ?
What does Fig. 5 show ?
Draw on the blackboard a square foot.
, Divide two opposite sides into fourths
and the other two sides into thirds. Con-
nect the opposite division points. Show
that: a. ixi = yV ^' i>^i = ^=h
r
K
Fig. 3
K
^
M
^fe
W!^/<
pJ
WMy///M
'■Wl/,
i-
m
¥.:....
„2^
Fig. 4
Fig. 5
Show as many other facts as you can by that figure.
To THE Teacher. — Many exercises similar ixD the preceding may be
given to interest children and make the topic real to them. We must re-
member, however, that these are mere graphic verifications of the rule for
multiplication of fractions. They neither prove nor derive the principle.
The authority for every operation in fractions is found in the principles of
division and the relation of dividend, divisor, and quotientc
72 MULTIPLICATION OF FRACTIONS
117. Oral
1. How much is 1^ of I of an inch ?
2. Illustrate that \oi ^ oi an apple is \ of an apple.
3. Multiply I by i; i by 1; 1 by 1; 1 by 1
4. Howmuch is^of f? foff? ^off? iof^=?
5. A man owned f of a farm and sold ^ of his share. What
part of the farm did he sell ?
6. James had i|, and John | as much. How much had
both?
7. If a pound of tea costs If, what will \ pound cost?
8. -J of |- of a square yard is what part of a square yard ?
Show it by a drawing.
9. Frank gave Harry |- of his apple and Harry gave away J
of his piece. What part of the apple did Harry give away ?
10. Mr. Greeley, having an acre of ground, took J of it for a
garden. He planted J of the garden to potatoes and |- as much
to corn. What part of an acre of corn did he have ?
118. Mixed numbers may he reduced to improper fractions
and then multiplied ; tJiua^
I|x8jx^x4 =
2
6 ;I7 5 ^ 50 ,., ,
gX^x^xf=g- = 16t^«».
Written
1. 4fx5J 4. 35|x27|
2. ll,\x7J 5. 2^xl^
3. 177fx3 ^ 6. Find 3^ of f of 11 of 8^
MULTIPLICATION OF FRACTIONS 73
7. 4ix7| 9. -3-3^x25fx|
8. 5fx8fx^ 10. 3ix9fx6|
11. Multiply 101 by I by f by 6f.
12. Multiply : a. ISJ by 14f . h. 16 by ■^.
13. 5|x2fx20 15. 9|x^x2| 17. ^g x 4 x 51
14. 7ix5|xf 16. 6|x|iXi\ 18. -f^xS0x5^
19. If a man earns f 2| a day, how much does he earn in 35
days?
20. Multiply f by I by J| by |l by ^\.
21. Find the cost of 16 bushels of oats at 37|^^ a bushel.
22. Mrs. A buys S^ qt. of milk a day. What does she pay
for it at 5 J/ a quart?
23. Show by a diagram that 1 of ^ = J.
24. How far can Joe ride in 3| hours if he rides 9J miles an
hour ?
25. How many square feet of floor are there in a room 12^
ft. by 71 ft?
26. P^ind the cost of 86 cords of wood at f 4| a cord.
27. Find the value of | of a chest of tea weighing 57 J lb. at'
$f per pound.
28. «. I of f of f of f of 1= ? h. -j9^ of f of IJ^ of i\= ?
29. a. T\offfof.3-9^oflfoff = ? h. ioi^^oiiofil^?
30. Mr. Brown earns f 60| a month, and his son | as much.
How much does the son earn ?
31. At 1121 a ton, how much will 9^\ tons of hay cost?
32. What will be the cost of 48| yards of cloth at $f a yard?
74
REVIEW AND PRACTICE
33. A man gave 124^5^. acres of land to his two sons, giving
I of it to the elder and | to the younger. How many acres
did each receive ?
34. If it requires 21| days for a man to dig a ditch, in what
time can he dig | of it ?
REVIEW AND PRACTICE
119. Oral
1. Read 305,027,503,060.
2. Read XLVI ; CXII ; CCIV ; XCIII.
3. If 2 sheep are worth $7, what are 8 such sheep worth?
4. If 12 books, worth f 8 apiece, will pay for a typewriter,
how many books at f 6 apiece would pay for it ?
5. If I make a purchase for $9.15, what change should I
receive for a 1 10 bill ?
6. Express in simplest form
_6_. lA
10 ' 3
¥; If; 6^; Y;
4.021
34'
¥
r^
7. What is the area of the top of this
table ?
8. 280^70 = ? 640-^40 = ?
9. Find the product of 32 and 20.
10. ?^|=| i+^ = ? |^? = f
11. Name two numbers that are prime
to each other.
120. Written
1. Express in figures one hundred twenty-five million, ten
thousand, seven.
2. Find the G. C. D. of 126, 210, and 294.
3. Find the L. C. M. of 720 and 216.
DIVISION OF FRACTIONS 75
4. Divide the product of 144, 25, and 56 by the product of
48, 120, and 105, using cancellation.
5. When potatoes are worth 55/ a bushel, how many bushels
must be given in exchange for 3 jars of butter, each containing
33 lb. at 25 / a pound ? Indicate the work, and solve by can-
cellation.
6. When $150 will buy 189 bushels of wheat, how many
bushels will i^50 buy? (|50 is what part of $150?)
7. Express in simplest form; a. |f| b. ^-^ c. |^^ d. 17^|f.
8. Change 18 to ninths.
9. Reduce |J^ to a fraction whose terms are prime to each
other.
10. How many 40ths are there in 7|?
11. How many 99ths are there in 89|^?
12. A certain block in our city is \^ of a mile long and J|
of a mile wide. What part of a square mile of land does it
contain ?
13. Find the area of both sides of a square piece of cardboard
whose edge is 15| inches.
DIVISION OF FRACTIONS
121. Divide If by f.
Since || is a product and | is one of its factors, we may state
the question thus:
§5^5 ^^^35^^x_?
72 8 ' 72 8 X?
In order to find the required factor we must divide the
numerator 35 by 5, and the denominator 72 by 8, thus:
35-|-5^7
72-8 9*
76 DIVISION OF FRACTIONS
That is exactly what we should do if the question were:
7
72^5 ' 7;2 ^ 9*
9
The latter method is the more convenient, especially when the
numerator of the divisor is not exactly contained in the numer-
ator of the dividend or the denominator of the divisor in the
denominator of the dividend.
Therefore, to divide hy a fraction we interchange the terms qf
the divisor and multiply.
122. Written
1. Divide 4| by 5|.
2
(How do we treat mixed numbers?)
2. Divide 47 by ^.
Solution : 47 -^ GJ = ^ -^ -V- = ^f x ^^ = f | = 7^3 Ans.
(How do we treat integers?)
3.
fi-sV
9.
HHi
IS.
8^A
21.
^Hi
4.
1 ^i
10.
H^A
16.
10-i-f
22.
H^H
S.
iHf
11.
^+5|
17.
1^14
23.
8J^9f
6.
A^l
12.
■^■^H
18.
11^8
24.
n-^u
7.
if^f
13.
il^Si
19.
2i^6J
25.
il^^
8.
il^T^
14.
2+i
20.
^i^H
26.
10|-*-4if
27. By what must f| be multiplied to make f^-?
28. One factor of |^ is J J. What is the other?
DIVISION OF FRACTIONS 77
29. How many pieces | of an inch long can be cut from a
wire that is 10^ inches long?
30. When 3| lb. of beef steak are worth 57| cents, what is
the value of one poUnd?
423. Division of fractions is sometimes indicated hy writing the
dividend above and the divisor below a line. Such an expression
is called a complex fraction; e.g,
A, !_, M M and i^
8|' 16' 25' 7f ^"^^ If -I
are complex fractions.* Read each fraction.
A fraction whose terms are integers is a simple fraction ; e.g,
W is a simple fraction.
7
1. Reduce r-r to a simple fraction.
81" 3-1- 3~1^26~26* ^^*'
_5
2. Reduce 45 to a simple fraction.
6=A^40 = Ax-l = J- Ans
40 17 n^'^jZ) 136* '^^''
74
3. * Reduce ^^ to its simplest form.
5
2
4. li
if
«■'?
5. 18|
6
-8
78 DIVISION OF FRACTIONS
In examples 4—13 change the given complex fractions to simple
fractions by performing the indicated divisions :
8. i^ 10. M 12. i^
9. i 11. 1! 13. i^L^
14. If 1^ of an acre of land is worth f 72^, what is the value
of an acre at the same rate ?
15. There are 5|^ yards in a rod. How many rods in 70|-
yards ?
16. At I 6 J a ton, how many tons of coal can be bought for
$73J?
EXAMPLES FOR PRACTICE
124. 1. 2| X I -f- IJ = ?
2. Multiply l| by i| and divide the product by 1^.
3. a. 14f -*- 71 = ? 5. 71 -I- 14| = ?
4. Change to a simple fraction - — \c)\'
6. What is one third of one hundred seventy-five and one
half?
7. The multiplicand is 1^^ and the product is 2^j. Find
the multiplier.
8. Simplify |L±|.
AVERAGES * 79
9. How many pounds of sugar at 6^ cents a pound will pay
for 12 J dozen eggs at 16 cents a dozen ?
10. When 15 yards of silk cost $ 16J, what is the price per
yard ?
11. Divide 75f by 14f .
12. Find the value of ^-li-
-^2 16
13. In one month Mr. Finlay earned $ 46|, his wages being
$ 2^ a day. How many days did he work ?
14. Divide f of | of 2| by | of f of 7.
15. $ 75 will pay for how much corn at f f a bushel ?
16. Divide the sum of 4| and 5| by their difference.
17. If J of a mile of telephone wire was divided into 14
equal pieces, how long was each piece ?
18. By what must 2^ be multiplied to obtain 2^ ?
19. By what must 3| be divided to obtain l^j ?
20. How many aprons can be made from 10|^ yards of cloth,
if 1 J yards are enough for one apron ?
21. Divide 85f by 14f .
AVERAGES
125. 1. Jacob weighed six of his
chickens, and found their weights to
be 68 oz., 40 oz., 63 oz., 47 oz., 55 oz.,
and 70 oz. What was the average
weight of the chickens ?
Solution
68
oz.
40
oz.
63
oz.
47
oz.
55
oz.
70
oz.
6)343
oz.
Total weight.
57i
oz.
Average weighty Arts.
80 AVERAGES
2. Harry's marks in spelling for a month were as follows:
1st Week 2d Week 8d Week 4th Week
Monday
80
75
78
80
Tuesday
86
90
84
88
Wednesday
83
82
90
92
Thursday
88
80
86
96
Friday
92
76
90
100
a^d. Find Harry's average for each week.
e-i. Find his average for all the Mondays, all the Tues-
days, etc.
j. How much higher was his fourth week's average than his
average for all the Mondays ?
k. On which day of the week did he spell best ?
I. What was Harry's general average for the month ?
3. Here are the standings of seven girls in three examina-
tions :
Marion Frances Dorothy Helen Jessie Hazel Ruth
Arithmetic, 70 93 98 92 70 83 95
Geography, 93 90 76 98 90 84 80
Language, 95 88 95 79 96 85 76
a. Which girl has the highest average ?
5. What is the average of the class in language ? c. In
arithmetic ? d. In geography ?
e. Find the difference between Marion's average and Hazel's.
/. Between Ruth's and Jessie's.
g. Find the general average of the class.
h. Find the difference between Helen's average and the av-
erage of the class.
4. Our outdoor thermometer indicated the following tem-
peratures for the mornings of last week : 76°, 82°, 80°, 67°, 60°,
IDEAS OF PROPORTION 81
70°, 81°. The week before the record was 60°, 63°, 58°, 57°,
70°, 68°, 72°. For which week was the average temperature
higher, and how much higher was it ?
IDEAS OF PROPORTION
126. Oral
1. 2 is what part of 6 ? If 6 quarts of beans cost 45 cents,
what will 2 quarts cost ?
2. 14 is how many times 2 ? What will 14 pears cost at the
rate of 2 for 5 cents ?
3. 18 is how many times 3 ? If a boy is paid 20 cents for 3
hours' work, what should he receive for 18 hours' work ?
4. If a boy works 6 days to earn §4, how long should he
worl^ to earn flO ?
5. What should I receive for 5 weeks' work when I earn $16
in 5 days ?
6. One gallon is how many times 1 quart? 10 gallons are
how many times 10 quarts ? When 10 quarts of milk cost 60
cents, what should be paid for 10 gallons of milk ?
7. If 14 five-pound jars of butter will last a family a certain
time, how many ten-pound jars would last the same time ?
127. Written
1. 3 is what part of 4 ? What should a man pay for three
acres of land when 4 acres are worth i 189 ?
2. How many bushels of wheat can be raised on 42 acres of
land when 159 bushels are raised on 7 acres ?
3. How many tons of hay can be bought for §3600 when 17
tons cost 1300?
82 REVIEW AND PRACTICE
4. How many 10-gallon cans may be filled from a tank of
oil that will fill 155 two-gallon cans ?
5. Five is how many times three ? How far can a man walk
in 5 days if he walks at the rate of 67 miles in 3 days ?
6. Find the amount of cloth needed for 36 suits when 17|
yd. will make 3 suits.
7. How many 8-quart baskets of peaches would it take to
equal in value 5346 bushel baskets of peaches ?
REVIEW AND PRACTICE
128. Oral
1. What is the L. C. M. of 8 and 12 ?
2. Name two other common multiples of 8 and 12.
3. Which of these numbers are composite : 15, 13, 29, 36,
71, 83, 87, 91, 97, 99 ?
4. Name two numbers that are prime to each other.
5. Name a number that will exactly contain 13.
6. What is the smallest number that will exactly contain 3,
4, and 6 ?
7. 1 of J = ? 1 and 1 = ? 1 less J = ? ^ times \ = ?
8. From I take |-.
9. A farmer having 60^eep sold \ of them at one time
and \ at another. How many had he left ?
10. A field of 3|- acres was planted to corn and potatoes.
There were 1| acres of potatoes. How many acres of corn
were there ?
REVIEW AND PRACTICE 83
11. A lady went shopping with f 10. After spending i3| in
one store and $5^ in another, how much money had she ?
12. Change ^| to a fraction whose terms are prime to each
other.
13. Change 4J to an improper fraction.
14. Find the value of -2^-.
15. My mother paid 8 cents for one melon, 7 cents for
another, and 10 cents for another. What was the average cost?
16. Show by means of a circle that l of ^ = 1.
129. Written
1. Draw a clock face, using Roman numerals. Let the hands
indicate a quarter past 9.
2. Find the wages of 8 men for 5| days at f 3J a day.
3. 15 sheep at |2| apiece will pay for how many yards of
cloth at 1 1 per yard?
4. A watch gains 1 J seconds every day. How many min-
utes does it gain in the months of June and July?
5. There are 609 pupils in a school, j- of whom are girls.
How many boys are there ?
6. Divide ^ by M.
7. A man spent | of his money and had $60 left. How
much had he at first?
8. a. 15fxl6J = ? h. 16f-v-33i=? c. 5f-^li = ?
9. (2j4-f + 3)^| = ?
10. If 6^ bushels of rye cost $5|, what is the cost of 1 bushel?
84
REVIEW AiNTD PRACTlClii
11. Change to a simple fraction ^ ^ ,
6|
12. |78| will buy how many barrels of flour at f 4| a barrel?
13. Find the value of 2l.
14. The area of a wall map is 976J square inches. Its length
is 42|^ inches. Find its width,
15. An alley between two houses is | of a rod wide and 7|
rods long. How many square rods of land does it contain ?
16. a. The bunches of bananas hanging in this fruit stand
contain respectively 98, 124, 62, and 140 bananas. What is the
average number of bananas per bunch ?
b. The two smaller bunches are red, and sell at the rate of 2
for 5 cents. What are they all worth ?
e. What will the others bring at 15 cents a dozen ?
d. If the four bunches were bought at 11.09 per bunch, what
will be the entire profit on the sales ?
REVIEW AND PRACTICE 85
17. Mr. Scotese, the fruit dealer, bought 5 bushels of apples
for $3.25. They average 136 apples per bushel. He sells them
all at the rate of 4 for 5 cents.
a. What is the cost per bushel ?
h. What is received for a bushel ?
c. What is the profit on 5 bushels ?
18. a. He sold a dollar's worth of pears at the rate of 3 for 5
cents. How many pears did he sell ?
h. If he should put the pears in baskets, 12 in each basket,
and offer them to you at 14 cents a basket, or at the rate of 4
for 5 cents, which offer would you take ?
19. a. He bought peaches at $1 a basket and sold them at
15 cents a quart. There were 10 quarts in a basket. What
was his profit on 5 baskets of peaches ?
h. If 9 of these peaches would make a quart, and they were sold
at 2 cents apiece, what would be the profit on a basket of peaches ?
20. What was his profit on 50 baskets of Delaware grapes
bought at 12|^ a basket and sold at 18^ a basket?
21. He bought chestnuts at $4.00 a bushel and sold them at
20 cents a quart. What did he gain on a quart ?
22. Spanish shelled peanuts cost him $9.80 a sack, each sack
containing 140 pounds. He sold them at 12 cents a pound.
What was the profit on a sack ?
23. He bought 50 watermelons at 23 cents apiece and sold
them at 40 cents each. What was his profit ?
24. He paid f 8 a hundred for cantaloupes and sold them for
12 cents apiece. How much did he gain on a hundred?
25. John earned $ 6| one week, 1 8f the second, and $ 7^ the
third. What were his average earnings per week?
26. How many pounds of sugar, at b^ cents a pound, are equal
in value to 6J dozen eggs, at 15 cents a dozen ?
86 ALIQUOT PARTS
27. Divide If of -^^ by 3f times l^j.
28. Divide f of f of ^ x 3^ by f of if of f
29. Solve in the easiest way :
a. If 6 acres of land cost $ 438, what will 42 acres cost ?
h. How many loads of earth can be bought for $80 when
$400 will buy 1135 loads?
30. -^ is how many times ^^2
ALIQUOT PARTS
130. One of the equal parts of a number is an aliquot part of
that number; e.g. 8 oz. is an aliquot part of 16 oz. because 8 oz.
is ^ of 16 oz. ; 16 J cents is an aliquot part of 100 cents because
16| cents = J of 100 cents.
Find the number of cents in If; I J; I J; ||; $1; ||;
The answers you have given are all what kind of parts of a
dollar?
Prove the correctness of the following table :
PARTS OF A DOLLAR
5 cents = 1 2V ^H cents = ||-
61 cents = $ Jg . 37^ cents = $|
8J cents = lyV ^^ cents = |l
10 cents = $^^ 62J cents = if
12 J^ cents = -I J 66| cents = If
16f cents = i^ 75 cents = $f
25 cents = $ J 87^ cents = $f
Which column in the table gives aliquot parts ? This table should be
committed to memory like the multiplication table, because its use will
shorten many problems, e.g. 33 books, at $ .16| each, will cost 33 x $^ = $ 5^.
i
ALIQUOT PARTS 87
131. Oral
Multiply :
1. 121- cents by 16 , 7. 37J cents by 8
2. 16f cents by 12 8. 50 cents by 15
3. 25 cents by 20 9. 62^ cents by 8
4. 33^ cents by 27 10. 66f cents by 9
5. 6i cents by 16 11. 75 cents by 4
6. 81 cents by 24 12. 87J cents by 8
13. What is the cost of :
16 pounds of bacon at 12| ^ a pound ?
16 balls at 50^ each?
36 yards of ribbon at 331^ a yard ?
36 pounds of candy at 25 j^ a pound?
8 pounds of tea at 62| ^ a pound ?
14. When 4 geographies cost $ 3, what is the cost of one ?
Of 9? Of 11? Of 15? (There is an easier way to find the
cost of 20 geographies. What is that way?)
15. At $.75 apiece, what must be paid for 3 chairs ? 4 chairs?
6 chairs ? 16 chairs ? 40 chairs ?
132. Written
What is the cost of :
166 pounds of pork at 121 cents a pound ?
348 pounds of veal at 16| cents a pound ?
265 boxes of strawberries at 25 cents a box ?
1215 yards of flannel at 33|- cents a yard ?
3580 pounds of honey at 20 cents a pound ?
748 pounds of tea at 50 cents a pound ?
88 DECIMALS
Oral
1. At 25^ a pound, how many pounds of butter can be
bought for ^8 ? (How many pounds can be bought forfl?
For $8?)
Divide :
2. $3 by 331/ 5. $9 by 121^ 8. $6 by 331/
3. 15 by 25/ 6. II by 6^ 9. |4 by 121/
4. 12 by 81/ 7. 110 by 50/ lo. $2 by 25/
11. $3 divided by 81/ =?
12. At 25 cents apiece, how many hats can be bought forf 6?
13. At 25 cents a pound, how many pounds of cheese can be
bought for 15?
14. At 16| cents a dozen, how many dozen eggs can be
bought for 14?
15. How many pounds of beef can be bought for $4 at 16|/
a pound ?
16. At 33 J / a yard, how many yards of linen can be bought
for 1 10?
17. How many penknives can be bought for 16 at 33|^ cents
iece ?
18. 24 X 121/ = ? 19. f 24 ^ 121/ = ?
apiece ?
DECIMALS
133. Each removal of a figure one place to the right affects
its value how ?
We have used this principle thus far in dealing with in-
tegers only ; but it holds true also for numbers smaller than
one. Thus,
DECIMALS
5000.
.■
500. = 5000.
J- 10
50. = 500. -^10
Moving 5 to the right .
5. = 50.
.5 = 5,
hlO
-10 = ^
\:
.05 = .5 -
-10=ifo
.005 = .05 H
-l^ = ToV
.0005= .005 H
-10 = Tol(
734. = 7340. -- 10
Moving all the figures
73.4 = 734. -^10 = 73^^
to the right
7.34 = 73.4 -f- 10 =7^3^
.734= 7.34-:
L0 = 3^,V
89
Notice that we place the decimal point (.) at the right of
units' place. This shows where the integer ends and the frac-
tion begins. The places at the right of the decimal point are
called decimal places and are named much like those at the
left, thus :
•♦-•
w
(A
4>'
Vt
JZ
T3
C
c«
w
3
o
3
o
M
C
"O
■o
C
■o
f
rt
?
(A
a>
rt
a>
■!-•
vt
I-
JZ
1.
(O
■C
1-
3
■a
(A
(O
■D
3
+<
T5
o
O
c
C
+i
c
C
O
c
C
:=
J=
3
<o
c
0)
3
£
a>
3
—
H
X
f-
3
h
I
h
\-
X
S
4638-754966
.7 is read seven tenths.
.75 is read seventh-five hundredths.
.754 is read seven hundred fifty -four thousandths.
. 7549 is read seven thousand five hundred forty-nine ten-thou-
sandths.
90 DECIMALS
Oral
1. Read .8; .49; .786; .4923; .56249; .38T654.
2. Read 500; 50; 5; .5; .05; .005; .0005; .00005.
3. Read .25; .36; .39; .47; .365; .3; .7; .403; .07; .009.
4. How many cents make one dollar ? What part of a
dollar is one cent ? 7 cents ? 19 cents ? 41 cents ? 97 cents ?
5. We write 25 cents, $.25, because it is 25 hundredths of
a dollar. One dime is 1 tenth of a dollar. How would you
write it ? Write on the blackboard ; 5 dimes ; 7 dimes ; 8
dimes ; 3 dimes.
6. 4321.648 is read 4321 and 648 thousandths.
Notice that and is used between the integer and the fraction.
Read: 6.42; 17.5; 4.23; 583.97; .640; 7.640; 439.018;
9341.215; 68.43; 70.9; 893.047; 903.03; 642.008; 95.249.
Read: 3.4; .0034; .987; 2048.017; .00315; 200.02.
Write in words on the blackboard : 35.08 ; 6.5 ; .084 ;
.6082; 235.235; 64.105; 308.02; 56.081; 30.130.
134. The product of equal factors is a power :
e.g. 4 is a power of 2 because 2x2 =4
8 is a power of 2 because 2x2x2 =8
81 is a power of 3 because 3x3x3x3 = 81
100 is a power of 10 because 10 x 10 = 100
Name 3 other powers of 10.
135. A fraction whose denominator is 10 or a power of 10 is a
decimal fraction ; e.g. -f^, -^^-q, .25, .3421. Only decimal
fractions can be expressed by use of the decimal point as in the
DECIMAL FRACTIONS 91
last exercise. When a decimal fraction is thus expressed, how
may we tell what the denominator is ? Name some decimal
fractions not given here.
136. A number that is composed of an integer and a decimal
fraction is called a mixed decimal ; e.g. 2.5 ; 130.35 ; 21.007.
137. A fraction that is expressed hy writing the numerator
above ayid the denominator below a line is a common fraction; e.g.
h I' 9' if' I'V-
138. Change
to the decimal form:
1- i¥(y
6. fi^^ 11. ^^
16. A
2- t¥o
'• m 12. ^JjV
"• iV
3- A%
8- h 13. ,^
18. 5^^
*• A
9- lb "• T^%
19. 8J^
5- 'Uo '
10- 1^7 15- 1%
20. 64^Vo^
Change to common fractions and read:
21. .36
26. .485
31. 5.6
22. .7
27. .016
32. 5.06
23. .125
28. .16
33. 5.006
24. 12.2
29. .06
34. 5.600
25. 6.25
30. .6
35. 5.060
Write^ first as common fractions^ or mixed numbers, then as
decimals :
36. Four tenths.
37. Seventy-five hundredths.
38. One hundred twenty-five thousandths.
92 . WRITING DECIMALS
39. Sixteen, and forty-eight hundredths.
40. Twelve, and four tenths.
41. Six tenths.
42. Six hundredths.
43. Six thousandths.
44. How many decimal figures are required to express
thousandths ? Hundredths ? Tenths ?
45. Read the numerators only in examples 36 to 43.
Write the following as decimals^ and read the numerator and
denominator of each :
46. Two hundred eighty-two thousandths.
47. Fifty-six hundredths.
48. Seven tenths.
49. Six hundred thousandths.
139. Oral
1. What part of 10 units is 1 unit ?
2. What part of 1 ten is 1 unit ?
3. What part of 2 hundreds is 2 tens ?
4 In the number 555, what is the value of the first 5 at the
right ? The second 5 ? The third 5 ?
5. Upon what does the value of any figure depend ?
6. In the number 555, the value of tho first 5 is what part
of the value of the second 5 ?
7. -f^ is what part of 2 units ?
8. In the number 5.5, the value of the right-hand 5 is what
part of the value of the left-hand 5 ?
READING AND WRITING DECIMALS 98
9. In the decimal .555, what is the value of the first 5 at
the right ? The second 5 ? The third 5 ?
10. Name
.225; .3478;
the denominator of .6 ; .17
.06; .049; .207; .3007.
; .105; .006; .05;
11. Read the numbers in question 10.
12. Read:
a. .368
h. .894
j. 37.005
Tc. 25.2036
s, .421
^^. .691
c. .5328
I. 38.000006
u. .81
d. .2053
m. .498869
i;. .637f
e. 25.623
n, 4.9836
«^. .4378^
/. 7.0063
0, 49.836
X. .809Jj-
g. 28.3005
h. .28962
i. 15.605
j9. 498.36
g. .000400
r. .0004
y. .430^
z. .6842J
TTn'^e deciinally:
1. Eight tenths.
2. 29 hundredths.
3. Sixteen, and 284 thousandths.
4. 4584 ten-thousandths.
5. Twenty -five hundredths.
6. Twenty-five thousandths.
7. Twenty-five ten-thousandths.
8. Twenty-five hundred-thousandths.
9. Twenty-five million ths.
10. 1650, and 464 thousandths.
94 WRITING DECIMALS
11. One thousand one, and 36 hundred- thousandths.
12. Sixteen, and six thousandths.
13. Seven hundred eighty-four millionths.
14. Twelve hundred-thousandths.
15. Seventy-five ten-thousandths.
16. Seven hundred five thousandths.
17. Seven hundred, and five thousandths.
18. Four thousand three ten-thousandths.
19. Four thousand, and three ten-thousandths.
20. Twenty-four, and five hundred-thousandths.
21. Seventy-one, and seven hundred-thousandths.
22. Four hundred thirty-five, and four thousandths.
23. Eight thousand three hundred forty-one ten-thousandths.
24. Ninety-nine, and eighty-six thousandths.
25. Seventy-eight, and four thousandths.
26. Nine thousand seven, and two hundred seven" ten-thou-
sandths.
27. One, and one hundred thousandths.
28. One thousand one, and one hundred one thousandths.
29. Ten, and ten ten-thousandths.
30. One hundred, and one hundred ten thousandths.
31. One hundred one, and one hundred ten-thousandths.
32. One thousand one ten-thousandths.
33. One thousand, and one ten-thousandth.
34. Two hundred seven thousand, and two hundred seven
thousandths.
35. Six, and six thousand ten-thousandths.
ADDITION AND SUBTRACTION OF DECIMALS 95
ADDITION AND SUBTRACTION OF DECIMALS
140. Since decimal figures increase in value from right to
left, like the figures in whole numbers, we may add and subtract
decimals as we add and subtract whole numbers, taking care to
write them so that the decimal points are all in a column, thus :
4.375 391.42
.35 165.70316
28.3065 225.71684 Difference
351.294
384.3255 Sum
The vacant places in the addends and in the minuend are treated as if
they were occupied by ciphers.
Add:
i 1. 2.2 2. 3.25 3. 4.5 4. .004
34.5 7.163 .168 4.1
79.89 15.0032 2.12 16.1563
5. .175+1.75 + 17.5 + 175.
6. 145 + 14.5 + 1.45 + .145 + .0145.
7. 3.2 + 14.0063 + .006 + 25.384 + .1.
8. .8 + .446 + 59.3 + 2.575 + 1.0056 + .3.
9. 1.45 + 2.365 + 96 + .96 + 15.863 + 4.3 + .0004.
10. 446 + 44. 6 + 37562 + 9 + .8 + . 321 + .16.
11. 21.0005 + . 3842 + . 1 + .005 + 3.6 + .158.
12. 1.0006 + 2001.1 + .003 + ^.b + 11.1111.
13. 205.07 + 301.2 + 687.9124 + 83.045 + 200.
14. .308 + 308. + 8.09 + 9.0786 + .859.
15. 2378. + 23.50 + .890 + .089 + 1.0886.
96 ADDITION AND SUBTRACTION OF DECIMALS
.41
. 1. Subtract :
a.
24.3 h.
2.86
C. 4.
d.
2.46
4.5
1.325
1.15
.005
2.
7 -.15
7.
29.325-15.14
3.
1-.004
8.
3.852 -.125
4.
13-2.1
9.
1.1111 -.0011
5.
3.256-1.05
10.
500- .05
6.
256.1-1.256
11.
25.3894-15.005
- •
12. From twenty-eight, and twenty-five thousandths take
fourteen, and twenty-five hundredths.
13. From one tenth take one thousandth.
14. a. Which is the greater, fiJty thousandths or five
hundredths ? h. Three tenths or three hundred thousandths ?
15. Take one thousandth from one thousand.
16. From 5 hundred take 5 hundredths.
142. Find results :
1. 175 - 30.23.
2. .015 + 1.05 + .57 + 5.7 + 1.04 + .0045 + 75.36.
3. 50.4 - .504.
4. 25.006 + 200.00008 + 6.00005 -f 49.005 + 300.059.
5. 2.005 4- 5.5 4- 25.010 - 3.2045.
6. Find the sum of two hundred forty, and four hundred
fifty thousandths ; thirty-four, and three hundredths ; six hun-
dred four, and six hundred four ten-thousandths ; fifty, and
five tenths.
7. A boy had two balls of kite string. One contained
145.3025 yards and the other 84.3502 yards. He made a kite
string 200.02 yards long. How much string had he left ?
DIVIDING BY MtlLTIl^LES Oi" t:EN 97
8. Subtract:
a. 32.854 c. 86.2 e. 21.101
9.378 43.948 7.
b. 19.042 d, 36.015 /. 28.78
16.854 . 24.008 21.987
9. There are four villages on the same road. From the
first to the second is 8.46 miles; from the second to the third,
10.5 miles; from the first to the fourth, 25 miles. Make a
picture of the road and find the distance from the third to the
fourth village.
MULTIPLYING AND DIVIDING BY MULTIPLES OF TEN
143. Oral
1. How can we multiply decimals by 10 ? By 100 ? By
1000 ?
2. Multiple/ by 10 :
5.25; .06; 3.7; 593.207; 6.800; 9.16; 82; 420; .035; .0061.
3. Multiply by 100:
61.843; 3.215; 75.16; 3.18; .65; 2.3; 5; 520.
4. Multiply .0612 by 10 ; by 100 ; by 1000 ; by 10,000 ; by
100,000 ; by 1,000,000.
5. By what must .0503 be multiplied to obtain .503? 503. ?
50.3? 5030.?
6. Moving the decimal point one place to the left is the
same as moving all the figures of the number one place to the
right. For example, moving the decimal point one place to
the left in the number 42.3, it becomes 4.23. How does this
affect the value of the number ?
98 MULTIPLYING BY MULTIPLES OF TEN"
7. What, then, is the easiest way to divide a decimal by 10 ?
8. Divide by 10 :
35.; 247.; 385.; 16.; 24.3; 2.59; 347.69; 8.137; 42.69;
394.68; .725; .042.
9. How may we divide decimals by 100 ? By 1000 ? By
10,000 ?
10. Divide hy 100 :
3567.8; 937.; 635.25; 42304.; 687.96; 485.03.
11. Divide hy 1000 :
986; 5321; 63,485; 983.7; 4284.25.
12. Divide hy 10,000 :
389,076; 42,831; 68,379.5; 425.
13. By what must 8193 be divided to obtain 819.3 ? 8.193 ?
81.93? .8193? .08193? .008193?
14. How must the decimal point be moved to change tenths
to hundredths ? Thousandths to tenths ? Thousandths to
millionths ?
15: Divide as follows :
64.2 by 10; 83.75 by 10; 63.59 by 100; 4251 by 10,000;
33 by 1000; 5 by 100.
16. What is the effect of moving the decimal point three places
to the left ? One place ? Two places ? Four places ? Six places ?
17. Find results :
3.5x10; 6.5-^10; 83-^1000; .987-^10,000; .8432 x
100,000.
18. What is the effect of moving the decimal point to the
left one place ? To the right two places ? To the left four
places ? To the right six places ?
19. How may we multiply a decimal by 10,000 ? Divide a
decimal by 1000 ? Multiply a decimal by 100,000?
MULTIPLICATIO^r OF DECIMALS
MULTIPLICATION OF DECIMALS
144. Multiply 6.41 by 3.2.
6.41 = 641 -- 100
3.2 = 32-5-10
6.41 X 3.2 = 641 X 32 + 100 -*- 10
641 6.41
32 3.2
1282 1282 — -_^
1923 1923 \4
20512 = 641 X 32 20.512 = 641_x 32 -^ 100 -$- 10
How did we divide by 100 and by 10 ?
From this we see that to multiply decimals we multiply the
factors as whole numbers and point off in the product as many
decimal places as there are in both factors ;
e,g, 2.8 1.25 .005 25
8 .6 m jM
22.4 .750 .00015 1.60
Find the products :
1. .18 X .15 8. 13.3 X 1.3
2. 1.0005 X .2 9. 100 X .01
3. 2.5 X .06 10: 100.56 x .0005
4. m X .005 11. 25.32 X 1.05
5. .005 X 1.6 12. 2.84 X .25
6. 25.05 X 1.15 13. 3.28 x 1.125
7. 2.863 X 100 14. 1.111 X 1000
100 DIVISION OF DECIMALS
145. Oral
1. One of the factors has two decimal places and the other
has five. How many decimal places has the product ?
2. When there are five decimal places in one factor and one
in the other, how many are there in the product ?
3. When there are eight decimal places in the product and
five in one factor, how many are there in the other factor ?
When there are six in the product and four in one factor?
Five in the product and three in one factor ?
DIVISION OF DECIMALS
146. Divide 10.96516 by 4.67.
2.348 Quotient
4.67)10.96-516
934 We first divide as in whole numbers. Since the
-— — • dividend is a product and the divisor one of its fac-
tors, the other factor, or quotient, contains as many
1401 decimal places as the number of decimal places in
2241 ^l*® dividend, less the number of decimal places in
the divisor.
Mistakes may be avoided by observing the fol-
lowing
1868
3736
3736
RULE FOK PLACING THE DECIMAL POINT
When the divisor is an integer^ place the decimal point in the
quotient directly over the decimal point in the dividend (or under
in short division^.
When the divisor contains decimal places^ make a dot on a line
with the tops of the figures as many places at the right of the deci-
mal point in the dividend as there are decimal places in the
divisor. Place the decimal point in the quotient directly above
this dot (or below in short division).
MULTIPLICATION AND DIVISION OF DECIMALS 101
Note 1. If there is a remainder after all the figures of the dividend
have been used, we annex ciphers to the dividend, and continue the division
until there is no remainder, or until a sufficient number of decimal places
have been obtained in the quotient.
Note 2. When the dividend contains fewer decimal places than the
divisor, we annex ciphers to the dividend until it has as many decimal
places as the divisor.
Written
Find the quotients and test
1. 60.8-^-1.6
2. .00075^.05
3. 25. 50 -f-. 34
4. 15.2 -3.04
5. .27560-^265
6. 90.978-4-3.54
7. 38.4444-4-177
8. 14.4 -4-. 0018
9. 1.127-4-4.9
10. .9156-4-12
11. 315.432 -.48
12. 1.5906 -4- .6
13. 375 -4- .125 .
14. 125 -4- .125
15. 1000^.001
16. .001-4-1000
17. 25-^.25
18. .25-4-25
hi/ multiplication :
19. 18.65-4-100
20. 266.4^.036
21. 2.107 -4-. 35
22. .100854-4-3.879
23. 125874.^.486
24. 9801. -4-. 99
25. 2976.^4.96
26. 164. 32 --.208
27. 347.76^.368
28. .0006478 -4-. 079
29. 2.5826-5-69.8
30. 98.07^.210
31. 20.852 -4- .52
32. .0023322 - .0026
33. 676.8 -4- .08
34. 1273.998 -4-. 199
35. 357.6-4-2.98
36. .75897-4-810
102 MULTIPLICATION AND DIVISION OF DECIMALS
147. Find the products and test hy division :
1. 3.2x3.6 16. 7.0001 X. 0603
2. 86 X .09 17. 43.55 x .06
3. 9.8 X. 005 18. 2354 X. 008
4. .039x57 19. 39.04x2.08
5. .00356x6.8 20. 6.80x86732
6. 6394. X .029 21. 2400. x .387
7. 864. X .278 22. .0406 x 3080
8. .00967x240 23. 920x63.7
9. 839.42 X .0015 24. .992 x 3001
10. 208.7x30.9 25. 2460 X. 0039
11. 930x6.80 26. 1234. x .56
12. 4203 X .0076 27. 9.924 x .0106
13. 69 X. 00035 28. .0204x20.40
14. 406 X. 000039 29. 78.08 x .025
15. 7.92x1.002 30. .8060x300
148. Oral
.7 = -j"^, or 7 divided by 10.
.305 = %%, or 305 divided by 1000.
.581 =^ , or 58 J divided by 100.
In like manner tell the meanings of the following decimals :
1. .8 6. .89 J 11. .029^ 16. .05 J
2. .416 7. .6f 12. .007^ 17. .034
3. .21 8. .39f 13. .103f 18. .0165
4. .3879 9. .48^ 14. .2134f 19. .00017J
5. .200 10. .873^ 15. .4070^ ao. .OOOf
CHANGING DECIMALS TO FRACTIONS 103
CHANGING DECIMALS TO COMMON FRACTIONS OR MIXED
NUMBERS
149.
.072 =
10 00 —
ilr
-
18.25 = 18^^0 =
= 18i
Reduce to
common fractions or
mixed
numbers in simplest
form :
.1.
.8
8.
.125
15. 16.75
2.
.25
9.
.875
16. .00125
3.
.35
10.
.375
17. .054
4.
.75
11.
.455
18. .0250
5.
.64
12.
.025
19. .01375
6.
.52
13.
.561
20. .342
7.
.38
14.
.368
12
Solution : .342 = 34f ^ 100 = 2^ X yi- =i? Ans.
21.
.12J
31.
2.331
5
41.
.166f
22.
.621
32.
'H
42.
.19^
23.
.06}
33.
42.621
43.
.5621
24.
.18i
34.
97.087J
44.
400.40f-
25.
.03|
35.
56.131
45.
361.41-1
26.
.25f
36.
158.06}
46.
2042. If
27.
.871
37.
409. 6|
47.
79.00f
28.
.66f
38.
•OTA
48.
308.00||
29.
.361
39.
.261
49.
2890.901
30.
16.25
40.
.012f
50.
98.000|f
104 KEDUCTION OF FRACTIONS TO DECIMALS .
150. Oral
Using aliquot parts^ reduce the following to mixed numbers:
1. 158.50 6. 16.05 11. 15.081
2. 139.331 7. I32.12J 12. 422.331
3. |5.16| 8. $19.10 13. 100.14|-
4. 17.25 9. II8.I42 14. 603.16|
5. 119.20 10. 16.02 15. 99.02
151. At sights reduce the following to common fractions or
mixed numbers :
1.
.621
4. 12.875
7. 29.871
10. 43.4
2.
.66|
5. 13.02
8. 3.6
11. 12.371
3.
.75
6. 430.625
9. 7.8
12. 54.375
REDUCTION OF COMMON FRACTIONS AND MIXED NUMBERS
TO DECIMALS
152. Written
Change -{^ to a decimal.
.3125
16)5.0000
48
"20
16 ^g= 5 -5-16 = . 3125 Ans, .
40 Write 9j-^g as a mixed decimal.
32
"so
80
REDUCTION OF FRACTIONS TO DECIMALS 105
Ret
iuce
ifo decimals
.*
1.
t
11.
H
21.
*l
31.
19,^
2.
1
12.
A
22.
If
32.
72T
3.
A
13.
i
23.
M
33.
llr
4.
i
14.
li
24.
H
34.
12^fT
5.
A
15.
3iV
25.
i¥ff
35.
14f|
6.
1
16.
f
26.
i¥ff
36.
StIt
7.
1
17.
^
27.
12A
37.
ISiV
8.
3\
18.
^r .
28.
H
38.
2AV
9.
iV
19.
II
29.
If
39.
5A
10.
If
20.
2#j
30.
^2V
40.
3^
153. ^ fraction in -lowest terms whose denomiyiator contains
other prime factors than 2 arid 5 cannot be reduced to an exact
entire decimal; e.g. |, f, if, j\, If, ^\.
Such a fraction may be reduced to a decimal of nearly the
same value by carrying the division to a certain number of deci-
mal places, thus :
Reduce ^| to a decimal of four places.
,U01^^Ans.
26.)19.0000
182
— 18=_9_ •'^307 is almost equalto ||.
^g 2 6 13 rp^g gxact value of if is .TSOT^V
200
182
"Is
The result may be expressed, .7307-1-
106 REDUCTION* OF FRACTIONS TO DECIMALS
154. Written
Reduce to decimals of three places:
1- 1
7.
if
13.
8A
19.
42ii
2. ^
8.
A
14.
231
20.
til
3. ^\
9.
if
15.
681t\
21.
2¥^
*• f
10.
W
16.
Hy
22.
43A
5. f
11.
2^^
17.
II
23.
16A
6- A
12.
151
18.
M
24.
in
A COMMON
FRACTION
AT THE
END OF
A DECIMAL
155. .21 =
.2+ a
ofi^,
or 2^. or ,
.05).
.2 + .05 =
.25
In a similar manner, we may show that,
.271 = .275, .3841 = .3845, etc.
Also, that .21 = .225, .341 = .3425, etc.
Also, that .8| = .875, .06f = .0675, etc.
Also, that ,^ = .9125, .07| = .07375, etc.
Oral
Express as entire decimals:
1. a.
'^
J. $.17J
c. .3601
d. 71
e.
.0041
2. a.
'H
5. 3.7^
c, 19. 20 J
(^. i.39}
6.
.145^
3. a.
.05f
h. l.lf
c. $21.46f
d, .033f
e.
.090f
4. a.
•7i
6. 6.81
c. 80. 3 J
(^. 12f
e.
1.9i
5. a.
^2^
h, 3. 97 J
c. 150f
(?. 24.01
e.
29.0|
6. a.
1.30f
J. 2.45|
<?. lOOJ
d. 20f
e.
40.001
REVIEW AND PRACTICE lOT
REVIEW AND PRACTICE
156. Oral
1. Read :
.0305; 42.0042; 4.0004; 6.395; .0600; 639.500; 639.00005;
10.10; 36f; 50.00|.
2. Express as common fractions in lowest terms:
.5; .25; .75; .121; .37I; .871; .16|; .33^; .621; .66|;
.081;. .02.
3. Divide hi/ 10 :
62; 93.5; 3.44; .25; .0081; .0384; 1.027.
4. Multiple/ ly 1000 :
8.934; .0245; .612356; 48; 63.05.
5. Grive results:
436x100; 436 -i- 1000; 436x^10; 5-^100; 8314-^10,000;
6.183 X 10,000.
6. Express as common fractions in lowest terms:
.4; .60; .8; .500; .70; .200; .600.
7. How may decimals be divided by 10 ? by 100 ? by 10,000 ?
by 1,000,000? by 1000?
8. A stamp clerk received $4085 for special delivery stamps
at 10 cents apiece. How many such stamps did he sell?
9. Compare $1 with |i ; If with If.
10. Compare 1 bushel with ^ of a bushel ; 5 bushels with |
of a bushel.
11. What will 3 months' rent cost at $ 240 a year ?
12. What will a dozen cabbages cost at the rate of 3 for 25^?
108 REVIEW AND PRACTICE
13. a. Count the eggs in the top
layer of this crate. How many dozen
are there ?
h. How many such layers are there if
the crate contains 30 dozen ?
c. ^ of them are how many dozen ?
How many eggs ?
d. ^ of one layer is what fraction of the entire number ?
e. How many pounds will a dozen eggs weigh if their average
weight is 2 oz. apiece ?
/. What is the entire weight of the eggs in the crate ?
14. If I of a yard of cloth costs 50 cents, what will 3 yards
cost ? (3 yd. are how many times | yd ?)
15. If I borrow one dollar, and agree to pay it back in one
year, with 6 cents more to pay for the use of it, how much must
I pay in all, at the end of the year ?
16. If you lend me f 8 for a year, and I agree to pay five
cents for the use of each dollar, how much will I owe you at
the end of the year?
How much will I owe if I agree to pay 5J cents for the use
of each dollar? 7 cents? 8 cents? 4|^ cents? 6 J cents?
9 cents ?
17. What must I pay for the use of 1 20 for one year, if I pay
5 cents for the use of each dollar ?
If I keep the money three years, paying each year for the use
of it, and then pay back the $20 which I borrowed, how much
will I have paid in all ?
18. Mr. A borrowed 9 dollars from Mr. B. After four years
he paid it back and also paid 5 cents a year for the use of each
dollar. How much did Mr. A pay to Mr. B ?
KEVIEW AND PRACTICE 109
19. A man borrowed 810 from me and agreed to pay me for
the use of it at the rate of 6 cents a year for each dollar. He
kept the money only half a year. How much should he pay me
for the use of the money ?
How much should he pay me in all ?
20. How much should be i)aid for the use of twenty dollars
for one and one half years at the rate of 5 cents a year for every
dollar ?
21. A man paid $5 a year for the use of $100. How much
a year would he pay for the use of §500 at the same rate ?
22. If $7 is paid for the use of a sum of money for 2 years,
what should be paid for the use of the same sum for 6 years ?
23. A woman who lives near the grocery takes her lamp to
the grocery to be filled and pays 5 cents every time. The lamp
holds a pint of oil.
a. How much per gallon does the oil cost her ? ,
h. If the lamp burns one pint of oil in two days, in how many
days will it consume one gallon of oil ?
c. In how many days will it consume four gallons ?
d. How much would she save by buying four gallons of oil
at 121^ a gallon?
24. Margaret's mother buys ^ lb. of sliced bacon twice a
week at 23^ per pound.
a. How much does she pay each time ?
h. How much does she pay for bacon in six weeks ?
c. How much would a 6-pound piece of bacon cost at 20^ a
pound ?
d. How much would Margaret's mother save by buying such
a piece at that price ?
25. What will 20 quarts of berries cost at the rate of 2 quarts
for 25)^?
110 REVIEW AND PKACTlCfi
157. Written
1. Express as common fractions in simplest form: a. .125;
b. .625; c. .025; d, .5625; e, .087500; /. Five hundred five
thousandths.
2. Express as exact decimals:
a. ^; h. lif; c. ^g; d. gf o*' ^- ^q^; /• ^V
3. Reduce to decimals of four places :
1- _3_7_. IIA. 24_. £5.. 14.
3» 111' 120' 1Y5' 91' 29-
4. Express in figures and add: Six and forty-five thou-
sandths; forty-nine and seven hundred seventy-nine thousandths;
twenty -four thousand nine hundred, and five hundredths; six
hundred five thousandths; three hundred forty-three hundred-
thousandths; seventy-nine and five tenths; eleven and thirty-
nine hundredths; eight hundred twenty-five, and forty-five
hundredths; nine hundred fifteen, and six tenths.
5. From fifteen and twenty-five thousandths take five and
six hundred twenty-five millionths. Express the result in
words.
6. Find the sum of 3.7; 36.05; .085; 35.25.
7. From .8 take eight thousandths.
8. Find the sum of .0375, 241, 43.8, and 7-|.
9. Find the cost of Q,^ barrels of flour at $5.75 a barrel.
10. What number divided by 17| will give 5.05?
11. Add 36.048, 25.13, 13|, and 25|.
12. Find the sum of | and .375.
13. A liveryman bought 10.5 tons of hay at $10,375 a ton.
What did it cost?
14. At $60.75 an acre, what is the cost of 40.25 acres of
land?
REVIEW AND PRACTICE
111
15. A lady purchased 4.5 yd. of silk at $1.25 per yard and
7.25 yd. of broadcloth at $3.50 per yard. What change should
she receive from a $ 20 bill and two $ 10 bills ?
16. A farmer divided his farm of 168.8 acres into 16 equal
fields. How much land was there in each field?
17. Divide 53.66523 by .941.
18. 28.8-^.0072 = ?
19. Divide .305996 by .337.
20. Divide 990 by .11.
21. $87.50 will pay for how many piano lessons at $1.25
a lesson?
22. 207.09
,015 = ?
23. A boy's height increased 1.4 inches in his 12th year, 1.5
inches in his 13th year, and 1.6 inches in his 14th year. What
was his average growth per year for the three years?
24. A farmer received $146^ for pigs at $3J apiece. How
many pigs did he sell?
25. The smaller of two numbers is 9346.05. Their differ-
ence is 412.08. What is the greater?
26. 9.2x5.37 -(1.785 + .1318) = ?
27. The following quantities of stamps were sold during a
year at a city post office. Find what was received for all of
them :
1-cent . . . 4,832,500 6-cent . . . 40,000
2-cent . . . 10,550,000 8-cent . . . 47,100
3-cent . . . 117,500 10-cent . . . 70,200
4-cent . . . 161,000 13-cent . . . 3400
5-cent . . . 198,000 15-cent . . . 8800
112
REVIEW AND PKACTICE
PART OF AN AMERICAN WAR FLEET
Battleships
NAME
LENGTH
TONNAGE
NO. OF GUNS
SPEED IN KNOTS
PER HOUR
MEN
Maine .....
Missouri ....
388 ft.
388 ft.
13500
14278
44
44
18
18.2
648
652
Kentucky . . .
Kearsarge . . .
Louisiana . . .
368 ft.
368 ft.
450 ft.
12905
12905
17666
60
56
74
16.9
16.8
18.8
613
650
803
Rhode Island .
435 ft.
16125
66
19.5
812
New Jersey . .
Virginia ....
Alabama. . . .
435 ft.
435 ft.
368 ft.
16125
16125
12170
66
66
48
19.5
19.5
17
812
812
592
Illinois
368 ft.
12170
46
17.5
660
Indiana ....
348 ft.
11403
45
15.5
491
Iowa
360 ft.
12445
48
17.1
520
Armored Cruisers
West Virginia.
502 ft.
13680
60
22.2
648
Pennsylvania .
502 ft.
13780
60
22.9
750
Colorado. . . .
502 ft.
13780
60
22.2
750
Maryland . . .
502 ft.
13680
60
22.4
.648
28. a. If all these ships were formed in a single unbroken
line, how many feet long would the line be ?
h. How many miles long would it be ?
c. What is the average tonnage ?
d. How many guns do they all carry ?
e. How many men are there on all the ships ?
/. How many knots greater is the average speed of the
cruisers than of the battleships?
g. What is the average number of men on a ship ?
REVIEW AND PRACTICE
113
The Battleship Louisiana
29. A knot, or nautical mile, is used in measuring the speed
of vessels. It is equal to about 1.15 common or statute miles.
a. The Missouri's speed is how many common miles per
hour?
h. The Maryland's speed per hour is how many common
miles greater than the Missouri' s?
30. a. A knot is 6080.27 ft. How many feet farther can
the cruiser Maryland sail in an hour than the battleship
Alabama f
h. If the Alabama is 9.9 knots ahead of the Louisiana^ and
both are moving at highest speed, in how many hours can the
Louisiana overtake the Alabama?
31. How many common miles can the 'Kentucky go in 24
hours, if she goes 16.9 knots per hour all the time ?
32. How many feet can the Rhode Island go in 1 minute ?
Make other problems about these warships, using the num-
bers given in the table.
114 REVIEW AND PRACTICE
33. What is the cost of 8.2 bales of cotton, each bale weigh-
ing 412.6 lb. at I J a pound ?
34. Two motor cars start from the same place at the same
time and go in opposite directions, one at the rate of 12.325
miles an hour, and the other at the rate of 14.875 miles an
hour. How far apart are they at the end of 10| hours ?
35. a. What must be paid for the use of $225 for one year
at the rate of 6 cents a year for the use of one dollar ? h. For
two years ? c. For 3| years ?
36. Mr. Scott borrowed il500 from Mr. Moore and agreed
to pay it back in two years, and also to pay Mr. Moore for the
use of it at the rate of 5 cents a year for every dollar.
a. How much did Mr. Scott have to pay for the use of the
money ?
5. How much did he have to pay in all ?
37. If I pay 42 dollars for the use of $600 for a year, how
much do I pay for the use of one dollar ?
38. Find how much I will pay in 2-|- years for the use of $ 2000
if I pay 4|- cents each year for the use of every dollar.
39. Mr. Marvin borrowed $3500 and paid it five years after-
ward. He also paid 4 cents each year on every dollar he owed.
How much did he pay in all ?
40. If I pay at the rate of T cents a year for the use of one
dollar, how much must I pay for the use of $2800 for six
months? (Six months are what part of a year?)
41. a. If 5 cents will pay for the use of one dollar for a year,
$24 will pay for the use of how many dollars for a year ?
J. At the rate of 6 cents a year for the use of a dollar, $24
will pay for the use of how many dollars for a year ?
REVIEW AND PRACTICE
115
^
|piBS
HiJ^^^^HBflHH^
2* -^
3j^
H|^^^H \
^
^a|
J/i
1 '
^^M
^L
42. These loaves of bread are 11^ inches long. The blade
of the bread cutter revolves so as to cut off a slice of bread at
every turn of the
wheel.
a. If the slices are
■^^ in. thick, how
many inches of the
loaf will be cut off
by 18 turns of the
wheel ?
h. How many
inches will be left ?
e. How many
turns are made in
cutting an entire
loaf ?
£?. How many slices are made from one loaf ?
43. A loaf of this bread is made into sandwiches. The
slices are y^g in. thick. The crusts are not used for this purpose.
Each sandwich is made of two slices of bread and one ounce of
meat that costs 16^ a pound. The bread costs 5/ a loaf.
a. How many sandwiches are made ?
h. What is their entire cost ?
c. If they are sold at 5 ^ apiece, what is the profit on all of
them ?
44. There are 90 loaves of bread in the picture.
a. If they are 11^ in. long and cut into slices |- in. thick,
how many slices are there ?
h. If the bread costs 5j^ a loaf and all except the crusts is
served at a lunch counter at 1 ^ a slice, what is the profit on all
of it?
c. The 90 loaves would make how many slices f in. thick?
116 MULTIPLYING AND DIVIDING BY TWENTY-FIVE
45. A barrel of flour weighs 196 lb. Flour costing i5 a
barrel is made into bread containing llj oz. of flour to a loaf.
Some of the bread is bought by Mrs. X at 5^ a loaf.
a. How many loaves of bread are made from a barrel of
flour ?
h. How much does it bring at 5^ a loaf ?
c. How much would Mrs. X save by buying a barrel of
flour and making her bread, supposing that the yeast costs
50^ and the extra coal 80^?
MULTIPLYING AND DIVIDING BY TWENTY-FIVE
158. 25 = 100 -i- 4. Therefore we may multiply a number hy
25 hy multiplying it hy 100 and dividing the product hy 4, thus :
36 X 25 = 3600 ^ 4 = 900.
How did we multiply 36 by 100 ?
159. Oral
Multiply the following numhers hy 25:
12, 24, 16, 20, 48, 36, 44, 32, 8, 28, 40.
1. Multiply by 25 :
a. 63
e. 76.347
i.
1124
m.
2.463
h. 7428
/. 14.231
J-
9.370
n.
2468 tons
c. 231
g. 2.31
k.
21.3
0.
.0934
d. 155
h. .2835
h
84.8
P-
.00080
2. Find the cost of :
a. 97 stoves at $25 apiece.
h. 25 yd. of silk at $1.39 a yard.
c. 130 acres of land at $25 an acre.
d. A 12-pound turkey at 25 cents a pound.
ACCOUNTS AND BILLS 117
160. To divide a number hy 25 we may point off two decimal
'places in the number and multiply the result by 4, thus :
452 -^ 25 = 4.52 x 4 = 18.08
^ 4.52
Explanation ; 452 -^ 25 = ^ -i- i^ = ^ X ^ = 18.08
^ 1 4 1 100
37.5 -^ 25 = .375 x 4 = 1.500
.375
Explanation: 2:1, b-^2b= 2^1. b-^^ = ^x,^ = lMO
^ 4 1 100
161. Written
1. Divide
by
25:
a. 425
e. 6934
{. 63,940
m. $348.5
b. 37.8
/. 5876
y. 6234
w. 12964
c. 239
g, 93.6
k. $934
0. 832 pk.
d. 87.64
h. 98,301
Z. 150.25
^. .637
2. a. 831
-f-
25 = ? b.
6.934
X 25 = ?
c. 25 X ? = .21
3. a. 428
=
25 X ? 6.
389 -T
- ? = 25.
^. ? X 25 = 12.6
ACCOUNTS AND BILLS
162. When your mother sends you to the store where she is
accustomed to buy groceries, giving you no money to take with
you, and tells you to buy certain articles and have them
charged, what does she mean ?
The merchant has a book in which he keeps the names of
persons to whom he sells things not paid for at the time of the
sale, together with a list of the articles sold, their value, and
the date of sale. This list is called an account.
118 ACCOUNTS AND BILLS
The person who sells the goods is the creditor, and the person
who buys the goods is the debtor. The debtor and creditor are
called parties to the account.
A doctor keeps a record of the calls which he makes or re-
ceives in treating his patients, when the calls are not paid for
at the time. This record is called the patient's account. Who
is the creditor ? Who is the debtor ?
If your father works for some one, he keeps an account of his
time and wages. Which party is your father ?
Whenever something is paid toward a debt of this kind, a
record of the payment is put in the account and is called a
credit. The difference between the amount of the debit (or
owing) items and the amount of the credit items is called the
balance of the account.
At certain times, the creditor copies on a piece of paper a
statement of the debtor's account and sends it to the debtor.
This statement is called a bill. Some merchants always send a
bill with the goods at the time of purchase.
163. There are various ways of writing a bill, but it should
always contain these things:
1. The time and place of making out the bill. This is called
the date of the bill.
2. The debtor's name and address.
3. The creditor's name and address.
4. A list of the items — that is, the goods sold, money paid
or services rendered, with the amount of each item.
5. The date of each transaction, if any of them occur at any
other time than that of making out the bill.
6. The amount, or footing, of the bill.
BILLS
119
164.
FORMS OF BILLS
(Form i)
Syracusb, N. Y., June 20, 1907.
Mr. John P. Smith,
713 McBride St.,
Bought of Andrews Brothers,
cor. James and Warren Sts.
2 bu. Apples
$1.10
10 lb. Granulated Sugar .05|
I lb. Tea
.50
12
20
55
25
$3
00
What is the date of this bill?
Who is the creditor? The debtor?
Read the items.
What is the amount of the bill?
Who ought to pay the bill?
Who should receive the money?
When a bill is paid, the creditor "receipts" the bill by
writing at the bottom, "Received Payment," followed by the
date and his own name. This shows that the bill has been
paid. The debtor keeps the receipted bill to show that the
debt has been paid.
When the above bill is paid, who should receipt it ?
Make a bill similar to the one given above, but using different
items. Find its amount, and receipt it.
120
BILLS
(Form 2)
Mr. W. C. Flint,
1213 Maxwell Ave.,
To John M. Semple, M.D., J)r.
207-208-209 Jamieson Building
Spokane, Wash.,
April 1, 1907.
To professional services rendered, Feb. 25 to March
18,1907, $37 50
Received Payment,
April 16, 1907.
John M. Semple.
Name each of the parties in this bill. Has the bill been
paid? How do you know?
Sometimes a clerk, an agent, or a bookkeeper of the creditor
receives the money for payment of a bill. He should then
write the creditor's name under the words " Received Pay-
ment," and under the creditor's name, his own name or initials,
thus :
Received Payment,
John M. Semple,
Per Kate L. Bunn.
3. Receipt the bill in Form 1 as though you were a clerk for
the creditor.
4. Make out bills of the following items, the teacher being
the debtor in each one, and yourself the creditor. Let the
teacher examine each bill, and mark it O.K. if correctly made.
Receipt the bill and return it to the teacher.
VOLUME MEASURE 121
a. 3 bu. potatoes at 75i^ per bushel.
8 lb. lard at 15^ per pound.
5 gal. kerosene oil at 12/ per gallon.
h, 4 lb. coffee at 25/ per pound.
18 lb. sugar at 5|-/ per pound.
5 gal. molasses at 60/ per gallon.
c. 6 bbl. potatoes at $1.80 per barrel.
2 tons hay at §16 per ton.
3 cords wood at f 4 per cord.
d. 16 yd. silk at 11.50 per yard.
4 pairs hose at 50 / per pair.
9 yd. lace at 60/ per yard.
e. 1 chocolate pot, 75/.
6 salad plates at $1.10 each.
15 Haviland bouillon cups at $12 per doz.
1 bread plate, 98/.
VOLUME MEASURE
165. Anything that has lengthy breadth^ and thickness is a solid ;
as wood, stone, earth.
166. A solid hounded hy six square faces is a cube.
167. A solid hounded hy six rectangles is a rectangular prism.
Name as many objects as you can that are rectangular prisms.
168. The lengthy hreadth^ and thickness of a solid are its dimen^
sions.
169. A cube whose edge is 1 inch is a cubic inch.
Show with your hands how long, wide, and high a cubia
inch is.
122 VOLUME MEASURE
170. A cube whose edge is 1 foot is a cubic foot.
Show with your hands the length, breadth, and height of a
cubic foot.
Show how high a cubic foot would be if it were lying on
your desk.
Can you think of some object about as large as a cubic foot ?
171. A cube whose edge is 1 yard is a cubic yard.
Show with your hands how wide and high a cubic yard is.
Show how high it would reach if it stood on the floor by your
side.
Could a cubic yard be put through the open door or window
of your school-room ? Measure and see.
172. The number of cubic yards^ cubic feet^ or cubic inches that
a solid contains is its contents or volume.
Name some object whose volume is about 1 cubic yard.
The measure by which the contents of a solid are measured is
called volume measure.
To THE Teacher. — Cubes of various sizes should be provided for this
lesson. Inch cubes should be put together to make two-inch cubes, three-
inch cubes, and so on.
The teacher should gain from children that each block has six equal
square faces, and therefore is a cube. Make prisms with the cubes, and
gain from the children that each face of the prism is a rectangle ; also that
the volume of a rectangular prism is the product of its three dimensions.
A piece of board one foot square and one inch thick can be used effect-
ively. Mark it off, to show that it contains 144 cubic inches. Gain from
children that 12 such boards, piled one upon another, would make a cubic
foot. Have children draw patterns of inch cubes, cut, and paste them.
Draw on the blackboard a pattern of a cubic foot.
Work with such exercises as are here outlined until pupils are perfectly
familiar with these fundamental ideas of volume measure. Do not take for
granted that children have these ideas, but test their knowledge by requir-
ing them to show and construct.
VOLUME MEASURE
123
[ *
/ \
\ /
V
This is a pattern of a cubic inch, or an inch cube. When it is
drawn on paper, an inch cube may be made by cutting along
the dark lines, folding along the light lines, and pasting the
flaps over to hold the edges together.
How many faces has an inch cube? What is the size of
each face ?
Draw a pattern of a cubic inch. Cut it out and paste it so
as to make a cubic inch.
A foot cube has how many faces V What is the size of each
face?
On a large sheet of stiff paper, at home, draw a pattern of a
cubic foot. Bring it to school and paste it together.
124
VOLUME MEASURE
VnXV v
X
\, \. \ , \.
\
1. Using inch cubes, make a rectan-
gular prism, 4 in. by 5 in. by 1 in.
How many cubic inches does it contain?
5x4x1=?
2. If three such prisms as figure A
were piled up, they would make a prism
containing how many cubic inches ?
5x4x3=?
3. How many cubic feet are there
in a rectangular prism 5 ft. by 4 ft. by
3 ft. ?
4. How many cubic yards are there
in a rectangular prism 5 yd. by 4 yd. by 3 yd. ?
5. A rectangular prism 6 in. long, 3 in. wide, and 2 in. high
contains how many cubic inches ?
Make this prism from paper and find the area of each face
and all its faces.
6. Find the contents of a 2-inch cube. Find its entire sur-
face.
7. A dry-goods box is 6 ft. long, 8 ft. wide, and 3 ft. deep.
How many cubic feet of space will it occupy in a freight car ?
8. Find the contents of a 4-inch cube. Find its entire sur-
face.
9. A block 8 inches long, 2 inches wide, and 2 inches high
will make how many 2-inch cubes ?
10. From a piece of paper 8 in. square, Mabel cut enough to
cover a box 2 in. by 4 in. by 1 in. How many square inches
of paper were used? How many were left ?
VOLUME MEASURE 125
11. How many cubic inches in a cube of soap 4 in. by 3 in.
by 11 in.?
12. If a pasteboard box is 4 in. square, how high must it be
to contain 32 cu. in. ? 4 x 4 x ? = 32. 16 x ? = 32.
13. A stick of wood is 3 in. wide and 2 in. thick. How long
must it be to contain 60 cu. in. ? 3 X 2 x ? = 60. 6 x ? = 60«
14. 2 X 5 X ? = 70. ? X 3 X 4 = 48. 7 x ? x 2 = 28.
15. If a cubic foot of ice weighs 60 lb., what is the weight of
a cake of ice 2 ft. by 1 ft. by 1 ft. ?
16. Will's lunch box is 8 in. long, 4 in. wide, and 3 in. thick.
What is its volume ?
17. If a common brick is 8 in. by 4 in. by 2 in., how many
cubic inches does it contain ? What is the area of one of its
largest faces? How many such faces has it? What is the
area of one of its smallest faces ? How many such faces ?
What is the area of the other faces ?
Note. — Bring a brick to the class and verify these results.
18. Find the contents of a rectangular prism, 3 in. by 4 in.
by 21 in.
19. A cubic foot is how many inches wide ? High ? Long ?
It contains how many cubic inches ? Can you show this by a
drawing ?
20. A cubic yard is how many inches wide, high, and long ?
It contains how many cubic inches ? Show this by a drawing.
21. How many square inches in one face of a cubic foot ?
22. How many square feet in one face of a cubic yard ? In
all its faces ?
23. Learn the table of Volume Measure, p. 173.
126 VOLUME MEASURE
174. Written
1. How many cubic inches are there in one cubic yard ?
2. a. How many cubic inches are there in 5| cubic feet ?
b. In lOf cubic feet ? c. In 25^ cubic feet ? d. In 307 cubic
feet?
3. a. How many cubic feet are there in 70|^ cubic yards ?
b. In 235|^ cubic yards ? c. In .16| of a cubic yard? d. In
21.33^ cubic yards ? e. In 4.66| cubic yards ?
4. 9234 cu. ft. = how many cubic yards ?
5. How many cubic feet are there in 24,192 cubic inches ?
6. How many cubic feet are equal to 84 cu. yd. 17 cu. ft.?
7. Change 38 cu. ft. 347 cu. in. to cubic inches.
8. A cake of ice containing S^ cu. ft. contains how many
cubic inches ?
9. How many cubic feet are there in a bowlder that contains
7| cu. yd. ?
10. Change 13 cu. yd. ; a. to cubic feet ; b. to cubic inches.
11. Change 513,216 cu. in. to cubic feet.
12. A carload of earth containing 19.8 cu. yd. contains how
many cubic feet ?
13. 7 X 8 X ? = 280. The contents of a rectangular prism
are 280 cu. in. Two of its dimensions are 7 in. and 8 in.
What is the other dimension ?
14. The dimensions of a water tank are 10 ft., 11 ft. and
2 J ft. What is the volume of the tank?
15. A gallon contains 231 cubic inches. Seven gallons are
how many cubic inches less than one cubic foot ?
VOLUME MEASURE 127
16. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high con-
tains how many cubic feet ?
17. A stick of timber 8 in. square and 45 ft. long contains
how many cubic inches ? (First change 45 ft. to inches.)
18. a. A block of granite 16 ft. by 3J ft. by 2| ft. contains
how many cubic feet ? h. How many cubic inches ?
19. A flagstone 8 ft. long, 5 ft. wide, and 6 in. thick contains
how many cubic feet ?
Hint : The stone is what part of a foot in thickness ?
20. A street 100 rods long must be lowered 3 ft. in order
that a pavement 36 ft. wide may be laid. a. How many yards
long is the street? h. The cut is how many yards wide?
c. How many cubic yards must be cut out ?
21. a. A box car 30 ft. long, 9 ft. wide, and 7 ft. high con-
tains how many cubic feet of space ? h. How many cubic
yards ?
22. A school-room is 39 ft. long, 30 ft. wide, and 12 ft. high.
a. How many cubic feet of air space are there in the room?
h. How many cubic yards ? c. If there are 40 pupils in the
room, how many cubic feet of space are there for each pupil ?
d. How many cubic yards are there for each pupil ?
175. Oral
1. What three numbers multiplied together equal 12 ?
2. What dimensions could a cake of maple sugar have to
contain 12 cubic inches ?
3. 4 X 3 X ? = 24. A block of wood 4 in. long and 3 in.
wide must be how thick to contain 24 cubic inches ?
4. What dimensions could a box have to hold 36 cu. ft. of
coal ?
128 VOLUME MEASURE
5. What dimensions could a box have to hold :
a. 40 inch-cubes ? h. 18 inch-cubes ? c, 32 inch-cubes ?
6. A box is 4 in. wide and IJ in. deep. It contains 72 cubic
inches ? How long is it ?
7. How many inch-cubes are equal to a rectangular prism
3 in. by 2 in. by 5 in. ?
176. Written
1. A man engaged to dig a cellar 40 ft. long, 25 ft. wide,
and 6 ft. deep. How much has he to dig after he has taken out
3246 cu. ft. of earth ?
2. How many bricks 4 in. by 8 in. by 2 in. are equal to a
cubic foot ?
3. Henry gathered 2 boxes full of hickory nuts. The boxes
measured 6 in. by 14 in. by 10 in. and 7 in. by 9 in. by 12 in.
He emptied the nuts into a box 20 in. long, 18 in. wide, and 1
ft. deep. How much space was left in the large box ?
4. a. A tank 2 ft. by 3 ft. by 4 ft. contains how many cubic
feet? 6. How many cubic inches ? c. If there are 100 gallons
of water in the tank, how many cubic inches of water are there ?
d. How many more cubic inches of water will fill the tank ?
5. If a box of candy 1|" by 4'' by 6" is sold for 19 cents,
what should be the price of a box 6" by 3'' by 6" filled with
the same kind of candy ?
6. a. How many cubic feet of earth are required to fill an
old cellar 30 ft. by 90 ft. and 7^ ft. deep ?
h. How many loads of earth are required if one load will fill
IJ cu. yd. of space ?
7. Make and solve a problem about the volume of a wagon box.
8. Make and solve a problem about the quantity of air in a
room.
PART TWO
To THE Teacher. '-It will be found worth while to give frequent "quick
tests," specimens of which are given in this and the primary book. Dic-
tate the questions one at a time, and allow a few seconds (more or less time
according to the nature of the question) for the pupils to obtain the answer
mentally. Then at a given signal let all write the result at the same time.
Allow no use of pen or pencil except to write results. This exercise will
cultivate power of attention, concentration, and alertness. Let it be short,
sharp, and rapid enough to keep pupils doing their best.
REVIEW AND PRACTICE
177. Oral
1. $8 will buy how many pounds of butter at 25^ a pound?
2. At 33 J ^ a bushel, 13 will pay for how many bushels of
turnips ?
3. $1 is the cost of how much cheese cloth at 6|/ a yard?
4. Find the cost of 24 yd. of ribbon at 8J^ a yard.
5. When $4 will buy 24 lb. of beefsteak, how many pounds
will il buy? How much does a pound cost?
6. When i9 will pay for 72 cans of peas, what is the price
per can? (How many cans for a dollar?)
7. How long will $6.00 worth of stamps last a man who
uses 1.30 worth every day?
8. Find the cost of 300 yd. of flannel at 33j/ a yard.
129
130 REVIEW AND PRACTICE
9. Grive the products :
|x2;lx3;2xll;|x3;lxl;lx|;^x|;ix|.
10. 2 qt. of beans at TJ^ and 3 lb. of maple sugar at S^^cost
how much?
11. How many one-ounce samples can be made from 2^ lb.
of cereal?
12. How many yards are there in 4 rd. ?
13. What is the area of a floor 12 ft. by ^ ft. ?
14. Express ^f- in simplest form .
15. J of J of a gallon is how many gills?
16. When 3 cents will buy \ lb. of maple sugar, what is the
cost of 2 lb. ?
17. |- of 75 ft. are how many yards?
18. J of ^ of a yard is how many inches?
19. Frank began at Chapter XVII in his book this morning
and has read 12 chapters to-day. Express in Roman notation
the number of the last chapter which he read,
20. Multiply 2.04056 by 1000; by 100 ; by 10,000.
21. Divide 89,345 by 10 ; by 1000 ; by 10,000 ; by 100.
22. 42.86 = 4286^?
23. 8903.4 = 8.9034 X?
24. 3.984 x? = 3984.
246 _13 ^ 246 x 13
' 1000^100 ?
26. 24.63 X. 029 = 2463 X 29 -^?
27. Name and describe the parties to an account.
REVIEW AND PRACTICE 131
178. Written
1. Express in Arabic notation eight hundred million eighty
thousand eight, and seventy thousandths.
2. Write in words 4040.0700.
3. Find the prime factors of 176; 482; 1260; 775; 385;
1920.
4. Divide the product of 36, 45, 20, and 14 by the product
of 80, 27, 35, and 72.
5. Find the quotient of 27 x 28 x 30 divided by 18 x 35
x36.
6. Find the G. C. D. and L. C. M. of 182 and 196.
7. Find the smallest number that will exactly contain 42,
63, and 105.
8. What is the greatest number that will exactly divide
1176 and 1848 ?
9. Change ^-^-^ to simplest form.
10. How many ninths are there in 18 ?
11. Change 2^^ to 33ds.
12. 8f-2lf + 33-5, = ?
13. A man bought 5^ acres, 6^ acres, and 10| acres of
land. He then gave his son 11| acres. How many acres had
he left ?
14. The sum of two fractions is ^^. One of them is A-. What
is the other?
15. The product of two fractions is -^q. One of them is ||.
What is the other ?
16. Yesterday you bought from your grocer 60 lb. of sugar
at 5|^ per pound, 60 clothespins at 5/ per dozen, and 1 of a
barrel of flour at $4.64 per barrel. To-day you paid the bill.
132 REVIEW AND PRACTICE
and the grocer receipted it and gave it to you to take home.
Write a copy of the bill.
17. I of the difference between -^j and -^^ is what ?
18. Find the cost, at $.Q2^ a bushel, of the seed oats for a
fieli of 7^ acr^s, if the farmer sows 2| bu. on an acre.
19. Draw a plan of a rectangular lawn 4 rd. by 2 rd., using
-^Q of an inch for a foot. That is, draw it to the scale of 1' to
re •
20. Find the cost of 325| bu. of wheat at il.l6| per bushel.
(Express i.l6|as a fraction of a dollar.)
21. 8 J sq. rd. = how many square yards ?
22. A merchant bought | of a piece of cloth containing
59^ yd. at 37|^^ a yard. Find the cost.
23. How long will it take a motor car to travel 87| mi., if
it travels 12| mi. in half an hour ?
24. Divide .1024 by the product of .064 and .01.
25. How many cubic yards of earth must be removed to make
an excavation 123 ft. by 24 ft. by 12 ft. ?
26. Make a problem about 7 tons of coal, $43.75, and 28 tons
of coal. Find the answer.
27. A mile is 5280 ft. A rod is 16J ft. What can you find
from these numbers ? Find it.
28. The product of .025, .85, and another number is 1.7.
Find the other number.
29. Find the number of cubic inches in a cake of ice IJ ft.
by 2J ft. by li ft.
30. Find the number of cubic feet in a bin 6J ft. long, 4J ft.
wide, and 3J ft. deep.
31. A room 16 J ft. long and 10| ft. wide must be how high
to contain 1425^ cubic feet of air ?
PRODUCTS AND FACTORS 138
PRODUCTS AND FACTORS * i^^
179. Complete each of the following statements and tell
which numbers are products and which are factors.
1. Since 1 rod contains ft., 4 rods contain ft.
2. At 3J^ each, 8 cucumbers will cost cents.
3. At apiece, f 47.50 will buy 10 baseball suits.
4. 11.25 is yV of •
5. 18 isf of .
6. |of| = .
7. .0Tof$25 = .
8. When I of Paul's earnings for a week are $1.50, he
earns .
9. 16.35-^.05= .
10. .75 of = 225bu.
11. .17|- of the cost of a piano was f 35. The piano cost
12. t of a farm is worth f 1200. The whole farm is worth
"§"
13. A city lot 47 ft. wide cost 11950.50. It cost per
front foot.
14. 54J is times 2^-^.
15. 13|^ yd. of broadcloth at a yard cost $51^.
16. A rectangle 4 ft. 6 in. by 8 ft. 3 in. contains square
inches.
17. 7^V = ^If.
18. 19.25 will pay for the use of I when $.05 will pay
for the use of one dollar.
134 STATEMENTS AND QUESTIONS OF RELATION
180. STATEMENTS AND QUESTIONS OF RELATION
Queation.
4x7J=?
4 X ? =30
?x7| = 30
|off = ?
|of? = A
Which term in division is a product ?
Which terms in division are factors ?
When the factors are given, what must be done to find the
product ?
When the product and one factor are given, what must be
done to find the other factor ?
Every problem that depends on the relation of factors and
product may be solved by one of these operations ; e.g.
1. a. An acre of land costs $80. What is the cost of 12
acres ?
Here we have two factors given, and the product is to be
found.
Statement of Relation: 12 x $80 = cost of 12 acres.
12x$80 = ?
Solution: 12 x $80 = $960 Ans.
Solution.
4x7|
= 30 Ans.
30 H
-4
=n
Ans.
30 -i
3
= 4
Ans.
?x2
_ 9
Ans.
^
7
14
2
3
2
9
.5
9
x| =
.6
Ans.
14 "
4
7
3
3
■7
9
6
_ 9
T
3
Ans.
X - =
14 *
7
H
2
2
^4
STATEMENTS AND QUESTIONS OF RELATION 135
h. When 12 acres of land cost 1 960, what is the cost of one
acre ?
What have we given in this problem ?
What is to be found ?
Statement of Relation : 12 x (cost of 1 acre) = $960.
12x? = !^960.
.^ Solution : ^ 960 -^ 12 = $ 80 A ns.
c. If 1 acre of land cost i80, how many acres can be bought
for f 960?
What is given in this problem, and what is to be found ?
Statement of Relation : (the number of acres) x $80 = $960.
?x ISO = $960.
Solution: $960 -^ $80 = 12, the number of acres Ans.
We may make use of the relation of factors and product in
solving problems containing fractions ; e.g,
2. a. At $80 an acre, what is the cost of f of an acre of land?
Statement of Relation : f of $ 80 = cost of | of an acre.
fof$80 = ?
10
Solution : 5 x ^^ = $50 Ans.
^ 1
h. When | of an acre of land cost $50, what is the cost of
one acre ?
Statement of Relation: f x (cost of 1 acre) =$50.
fof? = $50.
Solution: $50^| = $80 Ans.
c. At $80 per acre, what part of an acre of land will $50
buy?
Statement of Relation: (number of acres) x $80 = $50.
? of $80 = $50.
Solution : $50 -f- $80 = |g = f, number of acres Ans.
136 STATEMENTS AND QUESTIONS OF RELATION
3. a. What is the volume of a box which is | ft. by X ft.
by 2 ft. ?
Statement of Relation : ^ x -^^ x 2' = volume.
|xJjx2 = ?
3 7 2 7
Solution : 4 ^ To ^ 1 ~ 8 ^^^* ^^'^ ^'*^*
2 4
h. What must be the length of a box | ft. wide and ^ ft.
deep to contain | cu. ft. ?
Statement of Relation : (| x -5^) x length = J.
|x^x? = f
4
Make the statements for finding the height and width of the
box.
In each of the following problems let the steps be taken in
this order :
• First. — Read the problem and determine what is given
(two factors, or product and one factor), and what is to be
found (the product, or the missing factor).
Second. — Make the statement and question of relation.
Third. — Give the solution.
181. Written
1. 99 is how many times 12 ?
2. A man sold his farm for f 7200. What did he receive
for .125 of the farm ?
3. 36.48 is how many times .012 ?
PRODUCTS AND FACTORS 137
4. How many feet are there in 15 rd. ?
5. Find the cost of 98| bu. of oats at 40^ a bushel.
6. How much money have I if | of it is ^100 ?
7. f of a man's salary is $1200. What is his salary ?
8. 16 J is the product of 33 J and what other number ?
9. A boy spent | of his money and had 12.80 left. How
much had he at first ?
10. A man owning | of a vessel sold | of his share for 14800.
a. What part of the vessel did he sell? h. What was the
whole vessel worth ?
11. Wilfred spent $2.40, which was | of his money. How
much money had he ?
12. By what must we divide 5| to obtain 3| ?
13. $75 will pay for how much wheat at f | per bushel ?
14. The multiplier is 4^ ; the product is 16^ ; find the
multiplicand.
15. William earns $630 in a year, which is ^ as much as
his father earns. What can you find ? Find it.
16. I of a yard of cloth cost $|. Make the question and
answer it.
17. a. A farmer bought 13|- acres of land at $25| per acre.
Find the cost. h. He paid for it in wheat at $|^ a bushel.
How many bushels of wheat were required ?
18. 2.3 acres of land for $149.50 is how much an acre ?
19. Find the price of a yard of cloth when f of a yard cost
$1.25.
20. a. If I of the price of a piece of timber land is $4200,
what is the price ? h. What is the price of f of it ?
138 PRODUCTS AND FACTORS
21. a. If ^ of a ton of coal 'cost $4, what is the price per
ton ? h. How much coal will $151 J buy ?
22. A man owning |^ of a store sold | of his share for i 1000.
a. What part of the store did he sell ? h. What was the whole
store worth at that rate ?
23. a. 12| lb. of sugar is how many times 9| lb. ? 5. If 9|
lb. cost 46|- cents, what will 12| lb. cost ? c. In the same way,
find the cost of 16^ tons of coal, when 2\ tons cost $12.60.
24. I of the value of Mr. Blank's house is $2400. a. What
is the house worth ? h. Find | of its value.
25. I owned f of a farm and sold ^ of my share, a. What
part of the farm did I sell? h. If I received $1248, what was
the farm worth ?
26. How many oranges at 7J cents apiece will cost as much
as 60 pears at 2|^/ apiece ?
27. .025 of $5600 is how much money ?
28. .35 of my money is $700. How much money have I ?
29. A grocer buys flour at $1.44 a sack. He sells it so as to
gain .121 of the cost. What does he gain on a sack ?
30. A merchant sells cloth so as to gain $.20 on a yard.
This is .40 of the cost. Find the cost of a yard.
31. Joseph earned $17, and used .62| of it to help his
mother pay for the rent of her house. How much did he give
toward the rent ?
32. If Mary earned $1.60 and gave her mother $.40, what
decimal part of her money did she give her mother ?
33. A merchant sold cloth for cloaks at $8.30 per yard.
This was 1.65 times as much as it cost. What did it cost ?
PRODUCTS AND FACTORS 139
34. .22^2_ Qf a rod is how many feet ?
35. Lewis wants a bicycle. In 21 days he can earn .42 of
the money with which to buy it. How many days must Lewis
work in order to earn the bicycle ?
36. Mr. Johnson's store caught fire, and his goods were
damaged by smoke and water, so that he sold them for ,35 of
their cost. a. What did he receive for 38 yd. of lace that cost
$.25 a yard? h. What was the cost of a coat that sold for
$ 7 ? c. How much did he lose on a table that cost |15.00 ?
37. .49 of a number is 19.6. a. Find the number. 5.
Find .62 J of the number.
38. y\ of a certain number is 6^. Find .37 of the number.
(How many statements of relation ?)
39. Mr. Byrne bought a house for 11200 and sold it for
$1800. The gain was what decimal part of the cost ?
40. 3 inches are how many hundredths of 3 feet ?
41. I borrowed $250. When I paid it back, I paid my
creditor $250 and .05 as much for the use of the money. How
much did I pay in all ?
42. The width of my garden is 48 feet. The width is .80 of
the length. Find the length.
43. $|is what part of $10?
44. Eldred sold his bicycle for $10.50, which was .35 of its
cost. What did it cost ?
45. There were 50 words in the spelling lesson and Charlotte
missed two. How many hundredths of the words did she
spell correctly ?
46. Raymond shovels the snow from 80 feet of sidewalk.
After shovelling .25 of the walk, how many feet more must he
shovel ?
47. A xMx? = 4l
140 PERCENTAGE
PERCENTAGE
182. Decimals in hundredths are used very generally in busi-
ness calculations. The merchant calculates his gain or loss as a
certain number of hundredths of the cost of the goods. Banks
compute interest in hundredths. Agents who sell goods some-
times figure their earnings as a certain number of hundredths
of the selling price of the goods. The relations of numbers
are expressed generally in hundredths. •
Per cent is another name for hundredths. Six per cent means
six hundredths ; ten and one half per cent means ten and one
half hundredths.
Instead of writing the words per cent^ the sign % is used ;
thus: 5% of $8 = .05 x|8 =8.40.
12i % of 4 in. = .12-1 x 4 in. = .50 in.
4% of 80 yd. = .04 x 80 yd. = 3.20 yd.
108% of $12 = 1.08x112 =iil2.96.
Oral
As above, tell the meaning of each of the following ex-
pressions and find its value:
1. 8% of 50 9. 1% of 12100
2. 50% of 200 10. 5% of 100 boys
3. 12 % of 100 miles 11. 80 % of 20 horses
4. 10% of 60 sheep 12. 20% of 400
5. 50% of 300 men 13. 33 J % of 900
6. 2% of 30 bushels 14. 10% of 2000
7. 25 % of 64 days 15. 2| % of 30 days
8. 10% of 150 bushels 16. 7J%of$100
PERCENTAGE 141
17. 25% of 200 minutes 20. 90% of 1 60
18. 33 J % of 175 21. 150% of 20 pounds
19. 5% of 32 22. 300% of fl
183. The statement and question of relation aid in solving
problems in percentage ; e.g.
1. Donald gave his mother $3.75, which was 75% of his
week's wages. How much a week did he receive ?
Read the question, using the word hundredths in place of pel
cent.
Statement of Relation: 75% of (Donald's wages) = $3. 75.
.75 of ? = 13.75.
Solution: $3.75 -e- .75 = $5 Am.
The statement of relation shows that §3.75 is a product, .75
one of its factors, and Donald's wages the other factor, which
we are to find.
2. What per cent of 1 38 is $24.70 ?
Reading hundredths instead of per cent, the question is,
"How many hundredths of $38 is $24.70?"
Statement of Relation : of $38 = 24.70.
Solution : $24.70 -r- $38. = .65 or 65% Ans.
Observe that the steps in solving percentage problems are:
a. Read the question, using hundredths in place of per cent.
b. See which terms of relation are given, and which are to
be found.
c. Give the statement and question of relation.
d. Make the solution.
In finding the per cent, or number of hundredths, how many
decimal places must there be in the quotient? Then how
must the number of decimal places in the dividend' compare
with the number of decimal places in the divisor ?
142 PERCENTAGE
Therefore,
Before dividing^ to find per cents, arrange the dividend and
divisor so that the dividend contains two more decimal places than
the divisor. This may he done hy annexing ciphers to one or the
other of these terms, as may he necessary.
If the quotient is not exact when two decimal places have been
reached, express the remainder as a common fraction, in the quo-
tient; thus :
3. 1.65 T. is what per cent of 4.2 T. ?
Statement of Relation : of 4.2 T. = 1.65 T.
Solution : 1.65 T. -^ 4.2 T. = .39f or 39f %.
.8911 = .39f or 39^ % Ans.
4.2)1.6-50
126
390
3 78
12
4. Mr. Moore earns $140 a month and saves $84. His
earnings are what per cent of his savings ?
Statement of Relation: of $84 = $140.
Solution ; fiUO -^ $84 = 1.66f or 166|% Ans.
____L66|f = 1.66| or 166f % Ans,
$84.)$ 140.00
84
560
504
560
504
56
To THE Teacher. — It is well to follow this plan until pupils acquire a
considerable degree of skill in the process. After pupils have become
well grounded in the fundamental idea that per cent and hundredths are iden-
tical as expressions of ratio, it may be desirable to give practice in expressing
fractional parts of 1% as decimal approximations, as 8.33% instead of 8^%.
PERCENTAGE 143
Written
Solve the following, using the steps indicated above :
5. Find:
a. 40 % of 120 h. 35 % of 700 pupils
b. 121% of 160 lb. I 1% of 190
c. 18|% of 365 y. 16| % of 66 miles
c?. 36% of 250 yd. k. 48% of $8.50
e. 81% of 360 L 100% of $24.70
/. 6|%of64bu. m, 200% of 118.79
g. 37% of $16.50
6. a. 6.8 is 17% of what?
b. $3.95 is 5% of what?
c. $289 is 50% of what?
d. 75 doz. is 2^ % of how many dozen ?
e. 5 is 5 % of what ?
7. a. What per cent of $240 is $80 ?
b. 150 is what per cent of 900 ?
c. What per cent of $113 is $39.55 ?
d. Find what per cent 495 years is of 825 years.
€. Find what per cent 12.96 feet is of 96 feet.
8. 45 bushels are 90 % of what ?
9. 3 J miles is 70 % of what distance ?
10. What is 81% of 690 1b.?
11. What per cent of $ 920 is $230 ?
12. 15 minutes are what per cent of 25 minutes^
13. a. 1 quart is what per cent of 4 quarts ?
b. 1 quart is what per cent of 1 gallon ?
14. Find 181% of 362.
15. Of what number is 3.71 twenty-five per cent?
144 PERCENTAGE
16. 15 9J of a number is 10.50. What is the number ?
17. Whatisl|% of 640?
18. 37 J % of what equals 15 lb. ?
19. What per cent of 85 lb. is 17 lb. ?
20. 35 yd. are what per cent of 105 yd. ?
21. Henry raised 80 chickens and sold 75 % of them. How
many chickens did he sell?
22. Mary has an allowance of f 25 a year. If she puts 12%
of it in the bank each year, how much will she put in the
bank in 5 years ?
23. There were 50 words in the spelling test. Dorothy
spelled 98 ^^ of them correctly. How many did she have
right ?
24. In a school of 660 pupils, 55 % were girls. How many
were girls?
25. 45 % of a class of 40 pupils were boys. How many
were girls ?
26. 60 % of a class were boys. a. If there were 21 boys,
how many pupils were there in the class? h. How many were
girls ?
27. In a class of 50 pupils, there were 27 girls. What per
cent of the class were boys ?
28. In one day a grocer sold 14| % of a barrel of sugar,
containing 350 lb. How many pounds did he sell ?
29. Mr. Williams has 40 acres of timber land. This is 25 %
of all his land. How many acres has he ?
30. .375 is what per cent of .875 ?
31. A man's house and furniture are insured for $3600.
40 % of this insurance is on the furniture. What is the insur-
ance on the furniture ?
PERCENTAGE 145
32. Mr. French borrowed $ 725 of Mr. Rich, and paid him
6 % of that sum for the use of it. How much was paid for
the use of the 1 725 ?
33. A man borrowed some money and paid 1 75 for the use
of it. If that was 5 % of the sum borrowed, how much was
borrowed ?
34. A bushel of potatoes weighs 60 lb. If 75 % of this is water,
how many pounds of water are there in a bushel of potatoes ?
35. It costs a man $1200 to support his family for a year. If
this is 80 % of his salary, what is his salary ?
36. 920 lb. or 23 % of a load of grain is wheat. How many
pounds does the load weigh ?
37. a. In a baseball game, Fred's team scored 14 runs.
If Fred made 2 runs, what per cent of the 14 runs did he
make ? 5. If the other team made 6 runs, what per cent of all
the runs did Fred's team make ?
38. A bat and ball cost $1.50. If 33 J % of this sum was
paid for the bat, what did the ball cost ?
39. A huckster bought berries at 8^ a quart and sold them
at 12^. His gain was what per cent of the cost ?
40. 35 gallons of Jersey milk contained 8.05 gallons of
cream. What per cent of the 35 gallons was cream ?
41. A man lost $127.50 by selling a village lot. If this
was 15 % of the cost, what did the lot cost ?
42. A merchant sold goods for $25.63 less than the marked
price. If this reduction was 10 % of the marked prioe, what
was the marked price ?
43. A grocer bought a crate of cherries containing 32 quarts
for $2.88. He sold them at 12 cents a quart. His gain was
what per cent of the cost ?
146
REVIEW AND PRACTICE
REVIEW AND PRACTICE
60'
Popcorn ^
Sweet Com *
Tomatoes "^
Cucumbers
I Squash *^
; i
Early • „ ,. ^ 1 Late
, _ 1 Radishes \ ,
Lettuce j j Lettuce
1 Spinach j Peppers"^
Cabbages «
Cauliflower w
ti i
t\ Parsnips i Salsify
Carrots \ Onions \ Beets «
i 1 !
Flowers «
This is the phin of a real garden which Joseph and his father
cultivated. All the st^iteraents in the exercises are facts.
Oral
1. Take your rule and find out the scale of the plan.
2. Find the number of square feet of land planted to
radishes ; to late lettuce ; to spinach; to onions ; to beets.
REVIEW AND PRACTICE
147
n
184. Written
1. The corn was planted so that each hill occupied one square
yard of ground.
a. How many hills of pop corn were there ?
b. How many hills of sweet corn ?
2. There were six hills of cucumbers, and the same number
of hills of squash. How many square feet of land did each hill
occupy ?
3. There were 15 tomato plants. How much space did each
plant have ?
4. The cabbage and cauli
flower plants each had
square feet of space. How
many plants were there ?
5. a. What per cent of
the entire garden is planted
to flowers ? 5. To cabbage
and cauliflower ? e. To
spinach ?
6. The family table was
supplied with vegetables
from the garden, a correct
account of them being kept. Those not needed at home were
sold by Joseph during his spare time, mornings and Saturdays.
Find the value of each of the following productions :
a. Sweet corn .... 312 ears $.13 per dozen.
b. Tomatoes .... 6 bu. $.04 per quart.
c. Popcorn 54 1b. $.041 per lb.
d. Cucumbers .... 219 3 for 10 cents.
e. Squash 88 1b. 3^^ per lb.
*.:• :;*i^
148 REVIEW AND PRACTICE
/. Cauliflower .... 20 heads $.18 apiece.
g. Lettuce (early) . . 100 heads $.04 apiece.
h. Lettuce (late) ... 70 heads 2 for 5 cents.
i. Spinach 4 bu. 18^ per pk.
j. Radishes QQ bunches 3 for 5 cents.
k. Parsnips 1 bu. 25/ per pk.
I. Salsify 25 bunches 5/.
m. Peppers Ill 16/ per doz.
n. Young onions ... 40 bunches 2 for 5 cents.
0. Ripe onions .... J bu. 4/ per qt.
p. Cabbage ..... 22 heads 9/ apiece.
q. Carrots 22 bunches 2 for 5 cents.
r. Beets 30 bunches 2 for 5 cents.
«. Beets (full grown) . 3 pk. 60/ per bu.
7. What was the total value of these productions?
Make other problems using the numbers given above.
8. The first ripe tomatoes were gathered on the 14th day
of August and the latest on the 17th day of October. How
many days did they last ?
9. A crop of peas was grown before the cabbage and cauli-
flower plants were set out. Peas were raised, also, on either side
of the rows of tomatoes, cucumbers, and squash before those
vines began to spread. The crop of peas was 2|^ bu. and they
were worth 33/ a peck. Find the value of the crop of peas
raised here.
10. Beans were planted between the rows of corn and
matured before the corn was high enough to shut out the sun.
The three bushels of string beans thus raised were worth how
much at 5/ a quart?
II. Some pumpkin seeds were planted with the seed corn, and
yielded 27 pumpkins worth 12 cents apiece. What were they
all worth?
REVIEW AND PRACTICE 149
12. Late sweet corn was planted where the radishes, spinach,
lettuce, and part of the peas had grown, and yielded 148 ears
worth 12 cents a dozen. What was the crop worth?
13. Adjoining this garden is a space of the same size de-
voted to fruit, flowers, and perennial vegetables. The flowers
were used for the adornment of the home and the pleasure of
giving them to others. The extra fruit and vegetables were
sold.
Find the total value of the following:
Asparagus, averaging one bunch per day from May 1 to July 1
at 9 cents a bunch.
Twenty bundles of pie-plant at 5 cents per bundle.
Three pounds of sage at 35 ^ per pound.
Fifty-three baskets of grapes at 20/ per basket.
Six and one half bushels of plums at 20 / per peck.
Four bushels of cherries at 12/ per quart.
Two bushels of pears at 30/ per peck.
Two and one half bushels of peaches at 8/ per quart.
Two bushels of currants at 9/ per quart.
14. The expenditures were :
For fertilizers 16.50
For seeds 1.38
For Bordeaux mixture for spraying trees, etc. ... .40
For Paris green for spraying .30
For trees, vines, and plants 2.38
For implements worn out .65
Joseph received as his share $25.20, which was 25% of the
,value of all that was raised, after the expenses were paid.
a. What was the value of the produce less expenses?
5. What was the total value of the produce?
150 PER CENTS EQUIVALENT TO COMMON FRACTIONS
PER CENTS EQUIVALENT TO COMMON FRACTIONS
185. All percentage problems involving the relation of prod-
uct and factors may Jbe solved in decimals. But in many cases,
as we shall see, the work may be shortened by changing the per
cents to common fractions.
Oral
1. The whole of anything is how many hundredths of it ?
What per cent of it ?
2. J of anything is how many hundredths of it? ^? -|?
tV? I? I? *? V V V f? V V F V V iV?
3. What common fraction is the same as .10? .20? .30?
.40? .50? .60? .70? .80? .90? .25? .33J? .14f?
.62 J? .371? .66|? .121? .87^? .75? .16f? .83^? .08^?
.05?
4. What per cent is the same as ^? ^? J? |? J? |? \? ^V-
A? 2V? ^? 2V? F I? V V i? f? V F
5. Learn this table :
1 = 50% i = BSl% | = 62|%
1 = 25% i = ^H% 1 = 87^%
1=76% J = 16|% ^=10%
i = 20% f = 83i% ^ = 8^%
1 = 40% | = 14|% 5'(r=5%
1 = 60% i = 12i% TV = 6i%
1=80% 1=371% 5>j = 4%
PERCENTAGE
151
Answer the following questions, using common fractions instead
of decimals when it is easier to do so.
6. Find:
k. 6 J % of 16 days
I. 60 % of 50 ft.
m. 371% of 64
n. 87| % of 96
0. 83 J % of 18 trees
p. 121% of 600
q, 331% of 60 years
r. 4% of 150
«. 81 % of 3600 people
t. 25 % of 836 miles
a. 331% of 12
5. 10% of 200
c. 16f% of 30 da.
d. . 81 % of 144 sq. in.
e, 75 % of 28 gal.
/. 66|%of 27cu. ft.
g, 20% off
h. 40% of 1100
z. 90% of $200
y. 62 J % of 24)2^
7. 9 is 25 % of what number ?
8. 32 is 50 % of what number ?
9. 75 is 10 % of what number ?
10. 4 is what per cent of 16 ? (^ = what per cent ?)
11. 6 is what per cent of 9 ? (f = | == what per cent ?)
12. 1 is what per cent of 16 ?
13. 3 months are what per cent of 18 months ?
14. 3 books are what per cent of 15 books ?
15. 17 are what per cent of 1 84 ?
16. 9 men are 75 % of how many men ?
17. 10 min. are 16|% of what?
18. 2 is what per cent of 40 ?
19. 80% of 35^ is what?
20. 20 % of my money is 16 ^. How much money have I ?
21. 8 cents are 66| % of what ?
152 PERCENTAGE
22. Frank is 15 years old. Julia's age is 16| % of Frank's.
How old is Julia?
23. Will is 8 years old and Ethel is 7. Elhel's age is what
per cent of Will's?
24. I gained f 7 in selling my watch. If that was 12 J % of
the cost, what did it cost ?
25. Out of 25 words, Charlie missed one. What per cent
of the words did Charlie miss ?
26. Edith had 9 examples right. If there were 10 in the
lesson, what per cent of them did she have right ?
27. — ■ = how many hundredths ?
100 100 ^
28. 100 % — 90 % = what per cent ?
29. If I spend 90 % of my money, what per cent do I have
left ?
30. Having spent 90 % of my money, I had % 2 left. How
much money had I at first ?
31. A grocer sold 90 % of a barrel of sugar and had 35 lb.
left. How many pounds did the barrel contain at first ?
32. Hubert gave away 75 % of his apples and had 6 left.
How many had he at first ?
33. Harry took a silver dollar to the store and bought 2 lb.
of cheese at 18 ^ a pound. On the way home he lost 12J %
of the change. How much change did he lose ?
34. A boy sold 66f % of his chickens and kept 20. What
per cent of them did he keep ? How many had he at first ?
How many did he sell ?
36. Sarah answered correctly 80 % of the questions that,
came to her and missed one. How many questions came to
her ? How many did she answer correctly ?
PERCENTAGE 153
36. Alfred attended school 98 % of the days of the term.
If he was absent 2 days, how many days were there in the term ?.
37. A boy was sick and stayed out of school 5 days in one
month. If there were 20 days of school in that month what
per cent of the time did he attend school ?
38. 62 1 % of a cask of vinegar leaked out. If there were 15
gallons left, how many gallons did the cask hold ?
39. .20 + .10 = how many hundredths?
40. 20 % + 10 % = how many per cent ?
41. A butcher bought 200 lb. of beef. He sold 20 % of it
to one man and 10 % of it to another. What per cent of the
beef did he sell ? What per cent was left ? How many pounds
were left ?
42. What per cent of this oblong is -4.?
J5? O? m If A is 50 sq. in., what is
Bl (7? i>? El What per cent of the
oblong are A^ jB, and Q together ?
43. In a shipwreck, \ of the crew were
lost. What per cent were saved ? If 80 men were saved, how
many were lost ?
44. Alice, having read 75 % of a book, has 50 pages yet to
read. How many pages does the book contain ?
45. A man has traveled 60 % of the distance from New York
to Chicago and has 400 miles yet to travel. What is the whole
distance ?
46. When 25 days of the month of November are past, what
per cent of the month is yet to come ?
47. If your schoolroom is 40 feet long and 30 feet wide, its
width is what per cent of its length ? Its length is what per
cent of its width ?
A
B
C
D
E
154 PERCEKTAGE
186. Written
1. An army of 19,000 men went to the front. 12| % of
them were killed in battle, and 25 % of them died of wounds
and sickness. How many were left ?
2. A collector for a newspaper started out with bills
amounting to ^840. He collected $156. a. What per cent
of the bills did he collect ? b. What per cent did he fail to
collect ?
3. 35 % of the apples in an orchard were unfit for market
and could not be sold. If 1300 bushels were sold, what was
the entire yield ?
4. 39 % of the 4700 blossoms on a cherry tree were blasted
and the rest became fruit. How many cherries did the tree
bear ?
5. 98 % of the men in a certain city can read and write.
If there are 1398 men who cannot read and write, how many
men are there in the city ?
6. 23 % of the men in a certain city work in factories. If
there are 9200 men who work in factories, how many men are
there in the city ?
7. 45 % of a jeweler's goods were stolen, a. If he had
i 16,500 worth of goods left, what was the entire stock of goods
worth ? h. How many dollars' worth were stolen ?
8. In an orchard of 3600 trees, 25 % were pear trees, 15 %
peach trees, 10 % plum trees, and the rest apple trees. How
many apple trees were there ?
9. My gas bill for one month was f 1.80. Five per cent of it
was deducted for prompt payment. What was saved by pay-
ing promptly ?
10. A merchant bought a piece of cloth for f 65 and sold it
for 130 % of its cost. What did he receive for it ?
PERCENTAGE
155
11. This load of hay weighs 2200 lb., the wagon 1200 lb.,
and the team 2600 lb. a. The weight of the hay is what per
cent of the entire weight ? h. The weight of the wagon is
what per cent of the entire weight? c. The weight of the
team is what per cent of the entire weight ? d. How many
tons must the bridge support ?
12. a. In 1905 the Chicago baseball team won 92 games
and lost 60. What per cent of all the games played did they
win ? h. The Boston team won 78 and lost 74. What per
cent of all the games did they lose ? c. The New York team
played 149 games and won 47^9^"^ % of them. How many games
did they lose ?
13. By selling paper at 150 % of its cost, a stationer receives
90 cents a package for it. What is the cost of a package of
this paper ?
14. Twelve pounds of seed for a lawn contained 21 lb. of
white clover seed. What per cent of the mixture was white
clover seed ?
156 A FRACTION IN THE MULTIPLICAND
15. a. If a pine plank weighs 45 lb. and an oak plank of the
same size weighs 72 lb., the weight of the pine is what per cent
of the weight of the oak? h. The weight of the oak wood is
what per cent of the weight of the pine ?
16. A grocer bought 100 lb. of soda for $3.50 and sold it
for 142|^ % of its cost. What did he receive a pound for it ?
A FRACTION IN THE MULTIPLICAND
187. In multiplying a large mixed number by an integer,
time may often be saved by multiplying the whole number and
the fraction separately, then adding the products, thus :
314| X 5 = ?
3141
5
3| = I X 5
1570 = 314 X 5
1573f = 3141 X 5
Multiply:
1. 248f by 5 6. m^-^ by 8 li. 224J^ by 9
2. 39jl- by 3 7. 35f by 7 12. 42^^ by 12
3. 42^ by 6 8. 49^^ by 13 13. 65|-| by 16
4. 8501 by 15 9. 207fby3 14. 201ff by 16
5. 292 by 11 10. 38^gbyl6 15. 431|J by 72
188. In division, if either dividend or divisor contains a
common fraction that cannot be easily reduced to a decimal, it
is sometimes helpful to multiply both dividend and divisor by
the denominator of the fraction, thus making both dividend
and divisor integers, or simple decimals ; e.g. :
A FRACTION IN THE DIVISOR 167
.05f 148.74
Multiplying both dividend and divisor by 7,
2814.
Quotient
.37 1041.18*
(Multiplying the dividend and
74
divisor by the same number
301
affects the quotient how ?)
296
51
37
148
148
>.
1. 2.295 -^ .05|
6.
.3125 -*- .02^
2. 10.44 -J- .04|
7.
787.2 ^ .931-
3. 36 -^ .13J
8.
1.024 -^ .005 J
4. 96.9 ^ .15|
9.
218.24 -^ 1.13|
5. .0256-1- .071
10.
385.35 -h 5.2f
11. Find the number, of which :
a. 72 is 5{ %
/•
701.4 is 4| %
h. 10.5 is 11^%
9-
284.4 is 1051 %
c. 24.64 is 391%
h.
5.775 is 116f %
d, 12.834 is 14f%
i.
.3155 is 901 cj^
e. 1263 is 171%
h
.833 is 108y\ %
12. Mr. Fitch rained 14.60 in se
illinof a waffon. This was
6| % of its cost. What was the cost of the wagon ?
13. The average attendance in a certain school was 640
pupils. If this was 91 f % of the number registered in the
school, how many were registered ?
158 REVIEW AND PRACTICE
REVIEW AND PRACTICE
189. Oral
1. What number is composed of 5 units, 7 tens, and 3
thousands ?
2. Read XLIV; CCLXII; DCXCI; MC IVIII; CDLIV.
3. G-ive results rapidly^ adding or subtracting the tens* figures
first: 36 + 45; 29 + 32; 57 + 76; 93+28; 93 -27; 84-45;
72 + 39.
4. Grive quickly the number of:
a. Quarts in 98 pt. j. Feet in 2 rd.
b. Pecks in 28 bu. k. Dollars in 36,000 cents.
c. Hours in a week. I. Gills in a gallon.
d. Seconds in 1 hour. m. Days in two common years.
e. Inches in 2 yd. n. Tons in 1600 lb.
/. Square inches in 2 sq. ft. o. Square rods in 10 A.
g. Square yards in 450 sq. ft. p. Yards in 10 rd.
h. Cubic feet in 2 cu. yd. q. Days in 14 wk.
i. Dimes in |15. r. Days in a summer.
«. Cubic inches in a box 5 in. by 2 in. by 1 in.
5. A half dollar, a quarter, 2 dimes, and a nickel are how
many cents ?
6. i + i + A = ? 10. 15-J = ?
7. 3x8x? = 48 * 11. 18- If = ?
8. 8x9 = 6 X? 12. 5| + 13f=?
9. 88h-? = 8 13. 7|-f=:?
14. When 36 men can earn a sum of money in 15 da., how
long will it take 12 men at the same wages to earn the same
amount ? 9 men ? 6 men ? 72 men ?
REVIEW AND PRACTICE 159
15. If 6 men earn 8 dollars in a certain time, how many men
can earn $ 16 in the same time at the same wages ? $ 32 ?
164? 14?
16. If 10 men earn $ 200 in 8 days, how many dollars will
twice as many men earn in that time at the same rate ? '
17. fof8bu.= ? 21. 36 is y9^ of what?
18. I of? = 14. 22. iof^f = ?
19. 27 is -fj of what ? 23. f of what= 15 qt. ?
20. What part of 18 is 15? 24. | of what = ^y?
25. What are the prime factors of 84?
26. Name two numbers that are prime to 12.
27. How many 42ds are there in |?
28. What is the least number that exactly contains 6, 15, and
20?
29. Name three numbers of which 7 is an exact divisor.
30. Give two composite numbers that are prime to each other.
31. How may we tell whether a number is prime or not ?
32. What is the greatest number that will exactly divide 26
and 39 ?
33. What is the smallest number that 6, 8, 12, and 16 will
divide ?
34. Crive results at sight :
a. 362 -- 10 /. 14 x 200 k. .06 of 500
h. 4900-1000 ^. 99-f-.l I. lof = 7|
e. 29 X .01 h. .224 T = lb. m. 5% of 25
cZ. 834-10,000 z. 23,400-200 n. 121% of = 7
e, 29x1000 y. .12x50 o. ^V = %
35. What per cent of a ton is 400 lb. ?
36. Compare .84 x 25 with 84 x .25.
160 REVIEW AND PRACTICE
37. (16 + 4) X (43 -23) = ?
38. What must we do with |, |, and | to find out which is
greatest ?
39. Numerate 23516.00562.
40. Compare the values of |, |^|, and i^.
41. Find the cost of:
a. 64 lb. of pork at 12 J ^
h, 600 boxes of berries at 8| ^
c. 96 gal. of molasses at 50^
d. 16 doz. oranges at 37 J ^
e. 54 yd. of matting at 33 J ^
/. 48 1b. butter at 25^
g. 16 knives at 62 J ^
h. 2 doz. sleds at 87|^ apiece
^. 80 lb. rice at 6^^
42. How many bushels of beets will §12 buy at 33^^ a
bushel?
43. f 16 will rent a boat for how many hours at 16^ an hour ?
44. What change will be left from a f 20 bill after paying for
12 hours' labor at 37^ ^ an hour ?
45. At 16|^ a dozen, how many ears of corn will $2 buy ?
46. ^ of ^ lb. = oz.
47. What is the area of a rectangle 5^ inches by 4 inches ?
48. What is the cost of a dozen tomato plants at the rate of
4 for 5 cents?
49. At 10^ a dozen, how many sheets of sandpaper will 5^
buy?
REVIEW AND PRACTICE 161
50. A man spent 20 % of his salary for rent, 10 % for cloth-
ing, 5 % for fuel and light, 25 % for food, and 20 % for other
things. What per cent of his money did he spend ? What per
cent did he save ? If he saved |400 a year, what was his
salary ?
51. What is 120% of 5 miles ?
190. Written
1. Write in figures forty-two thousand, and two hundred
five ten-thousandths.
2. 209 X 87,000 0,6%^).
3. 235,404 -^ 468 (fesQ-
4. 23,945 -^ 160 {te%t^,
5. 302,050 - 92,059.
6. 17 hr. 35 min. = how many minutes ?
7. 639,800-- 700 (te%t),
8. 4320 square yards = how many square rods ?
9. When 60 bu. of oats grow on an acre of ground, how
many bushels grow on a square rod ?
10. The highest ten batting averages in the National Base-
ball League in a certain year were .377, .363, .356, .328, .317,
.316, .315, .311, .308, .304. What was the average of all
these ?
11. Using cancellation, divide 48 x 54 x 200 by 18 x 108 x
25.
12. How many pieces of sheeting, 39 yd. in a piece, worth
12^ a yard, would pay for 52 hours' work for 36 men at 32 ^ an
hour ?
162 REVIEW AND PRACTICE
13. Find the L. C. M. of 23, 37, 32, 36, and 56.
14. Find the G. C. D. of 48, 60, and 78.
15. Change to lowest terms:
16. Reduce to simplest form :
a. Iff. h. 82fA. c. ^i-. d. ^^. e. ^. /. ^^.
ff- H^-
17. Change to improper fractions :
a. 399|J. b. 48|J. c. 18^. d. 5&^\.
18. Change : a. 7| to 24ths. b. U|| to a fraction whose
terms are prime to each other, c. |, |, and -^^ to fractions
having the least common denominator.
19. A wagon which cost $52 J was sold for f 46|. a. What
was the loss ? h. What per cent of the cost was lost ?
20. A salesman cut 19| yd. of cloth from a piece containing
38|^ yd. How many yards remained ?
21. The remainder is 632|^ and the minuend 965|. What is
the subtrahend ?
22. a. After John had spent J of his money for a book, ^ of
it for a knife, and ^ of it for oranges, what per cent of it was
left ? b. If he had $.36 left, how much had he at first ?
23. What can 6 men earn in 4J wk. at $12 J a week ?
24. ^x8xifx6|x|f=?
25. A man sold a horse for $160 and a cow for | as much.
What did he receive for both ?
26. Find the area and the perimeter of a rectangle 12^ inches
by 8|^ inches.
27. Find I of f of -^-q.
REVIEW AND PRACTICE 163
28. A man sold | of his farm and had 90 acres left. How
many acres did the farm contain ?
29. Simplify^.
3
30. A man put in the bank f 252, which was f of what he
received for a wood lot. What was the selling price of the
wood lot ? •
31. A merchant lost | of his money and has §123.50 left.
How much had he at first ?
32. Simplify ¥^.
i of 2i
33. The sum of two fractions is -^||. One of them is -Jf .
What is the other ?
34. Add ten and five thousandths, three and seven tenths,
forty-seven millionths, five hundred five thousandths.
35. Find the sum of eight and thirty-five thousandths,
seventeen and fifty-three thousandths, fifty and fifty-four
millionths, five hundred two and nine ten-thousandths.
36. Find the sum of 6.06; 70.50; 6.0765; .00365; 101.09;
28.56741; 50.005.
37. Express in figures and add : 25 thousandths, 12 hun-
dredths, 26 ten-thousandths, 8 hundred-thousandths, 7 mil-
lionths, 2375 hundred-thousandths.
38. Add seventeen thousandths, eighteen ten-thousandths,
sixty-four millionths, fifteen ten-millionths, five hundred two
hundred-thousandths, and from the sum subtract eighty-four
hundred-thousandths.
39. a. .0375 is how much greater than ^^0^?
6. 12.5-9.0025 = ?
164 REVIEW AND PRACTICE
40. Reduce to common fractions or mixed numbers:
a. .00125. h. .0875. c, 3.625. d. 4.032. e. .83J-.
41. Multiply and test your work :
a. .00375 by 400 /. 34.05 by |
5. 5.275 by 5000 g. .000568 by 1.07
c, 5.64 by .006 h, 4.32 by .15
d, 35.005 by .008 i, '.0316 by .58
e, 350.5 by 8.04 j. .375 by 2.05
h. Four and twenty thousandths by twenty-six and nine
tenths.
42. How much a ton do I pay for coal when .375 of a ton
costs me 81.875?
43. At what rate per hour is a launch running when it goes
132.3 miles in 13.5 hours ? .
44. A farmer buys groceries and sells farm produce to the
grocer as follows :
Groceries Farm Produce
10 gal. oil at 2^f 25 lb. cheese at 18 f
50 lb. sugar at 5|^ 40 bu. potatoes at 58/
2 boxes soap at |3,25 J T. hay at $12
3 dozen oranges at 40 >^ 20 doz. eggs at 25/
5 gal. molasses at 40/ 7 bu. pears at $1.50
20 lb. coffee at 35 / 72 lb. smoked ham at f .13
In whose favor is the balance of this account, and how much
is due him ?
45. 10% of a man's income was paid for rent. If his rent
was il5 a month, what was his income per year?
46. After using 70 % of his month's wages, Jerry had 114.40
left. What were his month's wages ?
REVIEW AND PRACTICE 165
47. a. A schoolroom 30 ft. square and 12 ft. high contains
how many cubic feet of air ?
h. If there are 30 pupils in the room, how many cubic feet
of air are there for each pupil ?
48. a. Supposing a cubic foot of ice to weigh 62 J lb., what
is the weight of a pile of ice 12' by 10' by 8' ?
5. How many three-ton loads would it make ?
c. If 17| % of the ice melts in handling, how many pounds
can customers receive from this pile of ice ?
49. Amos sells vegetables for Mr. Robbins, the gardener, and
is allowed to keep 12| % of all the money he takes in. a. If
he earned 13.41 in a week, what was the amount of his sales for
that week? h. How much did Mr. Robbins receive ?
50. I bought 75% of a carload of sugar and sold |^ of my
share. What per cent of the carload did I sell ?
51. 97% of the pupils of a certain school are present. If
21 are absent, how many pupils belong to the school ?
52. Three days are what per cent of a week ?
53. A piano was sold for 1700. This was 140% of what it
cost the dealer, a. How much did the dealer pay? h. How
much did he gain ?
54. H. J. Howe, jewelry merchant, sold to Mrs. James R.
Hazzard \ doz. silver table-spoons at $30 a dozen, one dozen
silver table-forks at $25 a dozen, one tea urn $15.50, one
kitchen clock, $2.00. Who is the debtor? The creditor ?
Make out the bill and receipt it.
55. What is the amount of my bill for 4J lb. of mutton
steak at 16^, 1^ lb. of tea at 48^, 5 lb. coffee at 34^, and two
lb. raisins at 15^?
166 DENOMINATE NUMBERS
DENOMINATE NUMBERS
191. A number that is composed of units of weight or measure
is a denominate number; e,g, 10 doz., 215 cu. in., 2 gal. 3 qt.
1 pt.
192. The name of a unit of weight or measure is a denomina-
tion ; e.g, ounce, square foot, minute.
193. A denominate number that is expressed in two or more
denominations is a compound number; e.g. 1 yd. 2 ft. 7 in.;
2 lb. 14 oz.
194. TABLE OF LIQUID MEASURE
4 gills (gi.) = 1 pint (pt.).
2 pints = 1 quart (qt.).
4 quarts = 1 gallon (gal.).
Oil, vinegar, molasses, and other liquids are shipped in barrels
or casks of various sizes. But for the purpose of indicating the
capacities of vats, tanks, reservoirs, etc., 31 J gallons are called
a, barrel (bbl.) and 63 gallons a hogshead (hhd.).
195. Oral
1. 5 gal. = pt.
2. 1 hhd. = bbl.
3. What will 48 pt. of cream cost at f 1.20 per gallon?
4. 1 bbl. is what per cent of 1 hhd.?
5» How many pints in 10 gal.?
6. At 4 ^ a pint, what is the cost of 6 qt. of milk?
7. 4 gal. 2 qt. 1 pt. = pt.
DRY MEASURE 167
8. A tank contains 10 bbl. of oil. How many gallon cans
will it fill?
9. 63 qt. is what part of a hogshead ?
10. A gallon contains 231 cu. in. How many cubic inches
are there in J of a gallon ? In ^ of a gallon ? In -jl^ of a
gallon? In 1 qt.?
11. How many gallons and quarts are there in 50 quarts ?
12. A cistern that holds 10 hhd. of water holds how many
barrels? How many gallons?
13. One pint is what per cent of one gallon?
14. 10 % of a barrel is how many gallons ?
15. If 1 qt. of sirup can be made from 20 oz. of maple sugar,
how many ounces will make a gallon of sirup ?
16. 33i % of a hogshead is how many gallons ?
196. TABLE OF DRY MEASURE
2 pints (pt.) = 1 quart (qt.).
8 quarts = 1 peck (pk.).
4 pecks = 1 bushel (bu.).
197. Oral
1. 64 qt. = pk.
2. 1 bu. = qt.
3. If 2 qt. of cherries fill a jar, how many jars will 2 bu. fill ?
4. 1 pk. is what per cent of a bushel?
5. 1 qt. is what per cent of a peck?
6. Elsie, Nina, and Robert gathered 4 bu. of chestnuts and
sold them for 10^ a quart. How much did they receive ?
7. 10 bu. = pt.
168 AVOIRDUPOIS WEIGHT
8. ^hu. +^ pk. = qt.
9. 8 qt. = what part of 2 bu. ?
10. What is gained on a bushel of hickory nuts bought for
$2 and sold at 10^ a quart ?
11. A barrel of potatoes containing 2| bushels will sell for
how much at 20^ a peck?
12. 3 bu. and 3 pk. of apples at $1 a bushel cost how much?
13. If a bushel of oats weighs 32 lb., what is the weight of
3} pk. ?
14. How many bushels of apples at 25 ^ a peck can be bought
for 120?
15. A bushel of corn and a peck of wheat are ground to-
gether. What per cent of the mixture is corn? What per
cent is wheat ?
16. 37 J 9^ of a bushel is how many quarts ?
198, TABLE OF AVOIRDUPOIS WEIGHT
16 ounces (oz.) = 1 pound (lb.).
2000 pounds = 1 ton (T.).
2240 pounds = 1 long ton.
100 pounds =1 hundredweight (cwt.).
The term hundredweight is used less than formerly, although
its value (100 lb.) is still taken as a unit in quoting freight
rates and prices of various articles, when the quantity used
makes this a convenient unit of weight.
The long ton is used in wholesaling certain mining products.
The ton of 2000 lb. is sometimes called a short ton.
LINEAR MEASURE 169
199. Oral
1. How many ounces are there in 1 ton ?
2. At 48^ a pound, what must be paid for 4 oz. of tea ?
3. 1 % of a ton is how many pounds ?
4. 1 cwt. is what per cent of a ton ?
5. What is the cost of a ton of corn meal at f 1.25 per hun-
dredweight ?
6. How many short tons equal 1 long ton ?
7. What is the cost of 500 lb. of hay at $12 a ton ?
8. How many pounds of coal at $6 a short ton can be
bought for 11.50 ? y
9. A car was loaded at the mines with 10 long tons of coal.
How many pounds of coal did it carry ?
10. One ounce is what per cent of a pound ?
11. 5 lb. of candy will make how many 4-ounce packages ?
200. TABLE OF LINEAR MEASURE
12 inches (in.) = 1 foot (ft.).
3 feet = 1 yard (yd.).
5J yards
or =1 rod (rd.).
lejfeet
320 rods = 1 mile (mL).
201. Oral
1. How many inches in 10 yd. ?
2. One foot is what per cent of a yard ?
3. One inch is what per cent of one foot ?
170 LINEAR MEASURE
4. How many rods are there in 2 mi.? In 10 mi.? In
100 mi.?
5. 12 yd. 2 ft. = ft.
6. 33|^ % of 2 rd. = how many feet?
7. 10 rods are how many feet ?
8. 12J % of a mile is how many rods ?
9. 8 J % of a foot is how many inches ? 25 % of a foot ?
50%? 16f%? 331%?
10. How many rods are there in the perimeter of a lawn
that is 33 feet square ?
11. Draw on the blackboard a line 1 yd. long, using no
measure. Measure and correct it. On your paper draw a line
3 1 in. long, using no measure. Measure and correct it.
12. Estimate the length and breadth of your schoolroom.
Test your estimates by measuring.
13. How many feet high do yo\i think your schoolroom is ?
Can you find a way to measure it without climbing ? Measure
it, and see how nearly correct your estimate is.
14.
This oblong represents a field. ^ inch stands
for 1 rod. Measure the sides, and tell how many
rods long and wide the field is.
15. Draw on the blackboard a plan of your schoolroom floor,
using ^ inch for 1 foot ; that is, draw the floor to the scale of
1' to J".
SUKFACE MEASURE
171
16. This square represents a square
mile. Can you tell what the scale is ?
Each small square is what part of a
square mile?
How many rods of fence would be
needed to inclose a square field con-
taining ^ of a square mile ?
17.
320 rd.
The width of this floor is 16 ft.
Can you find the scale of this plan ?
What is the length of one long side?
What is the greatest length of the
room?
18. If you go 30 inches at a step, how many steps will you
take in going 30 feet?
19. If a row of corn contains five hills to a rod, how many
hills are there in a row a quarter of a mile long?
202. TABLE OF SURFACE MEASURE
144 square inches (sq.in.) = 1 square foot (sq. ft.).
9 square feet = 1 square yard (sq. yd.).
30 J square yards = 1 square rod (sq. rd.).
160 square rods = 1 acre (A.).
640 acres =1 square mile (sq. mi.).
172 SURFACE MEASURE
203. Oral
1. Without a measure draw a square inch. Measure and
correct it.
2. Without a measure draw on the blackboard a square foot
and a square yard. Measure and correct them. Divide the
square yard into square feet. Divide the square foot into
square inches.
3. How many square inches are there in two square feet ?
4. Estimate the number of square yards in the floor of your
schoolroom. Measure it, and see how nearly right your esti-
mate is. Make an estimate of the area of each wall, and test
it by measuring. Measure your school lot, and find what part
of an acre it contains.
5. How many square inches are there in -|- sq. ft. ?
6. One square yard is ^ of how many square feet ?
7. A 5-inch square contains how many square inches ?
Draw it.
8. A rectangle 6'' by 12" is what part of a square foot ?
9. How many tiles 6'' square will cover a floor 10 ft. by
5 ft.?
10. 8 sq. in. are what part of an 8-inch square ?
11. How many square yards are there in 4 sq. rd. ?
12. 40 sq. rd. are what per cent of an acre ?
13. A room is 15 ft. by 12 ft. and 9 ft. high. How many
square yards are there in the floor ? Draw a plan of it to the
scale of y = 1'.
How many square yards are there in one long wall ? In one
short wall ? Draw a plan of each wall.
VOLUME MEASURE
173
How many square yards of plastering are needed for the
ceiling ?
14. How many acres are there in a farm 160 rd. long and
100 rd. wide ?
15. How wide must a field be to contain 10 A. if it is 40 rd.
long ?
16. A 10-acre field is 20 rd. wide. How long is it ?
B
A
17. The scale of these plans is 16' to 1''. Find the perimeter
and area of the surface represented by each.
204. TABLE OF VOLUME MEASURE
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.).
27 cubic feet = 1 cubic yard (cu. yd.).
205. Oral
1. Define a cube.
2. How many edges has a cube ? How do they compare ?
3. How many cubic inches are there in a 6-inch cube ?
174 TABLE OF TIME
4. Prove that a tight tin box 7 in. by 11 in. by 3 in. will
hold one gallon.
5. A block of wood 4 in. square must be how long to con-
tain 96 cu. in. ?
6. 18 cu. ft. are what part of a cubic yard ?
7. A box 12 ft. long and 3 ft. wide must be how deep to hold
4 cu. yd. of sand ?
8. 8 J % of a cubic foot is how many cubic inches ?
9. A 2-inch cube is equal to how many 1-inch cubes ?
10. How many 2-inch cubes would make a 4-inch cube ?
206. TABLE OF TIME
60 seconds (sec.) = 1 minute (rain.).
60 minutes = 1 hour (hr.).
24 hours = 1 day (da.).
7 days = 1 week (wk.).
365 days = 1 common year (yr.).
366 days = 1 leap year.
Ten years are called a decade^ and one hundred years make
a century^ but these terms are not used in arithmetical cal-
culations.
The four thirty-day months may be remembered easily by
the following old rhyme :
" Thirty days hath September,
April, June, and November."
February has 28 days, with 29 in leap year. The other
months have 31 days.
TABLE OF COUNTING 175
207. Oral
1. How many minutes are there in a working day of 8 hr. ?
2. A man who works for 30 ^ an hour receives how much a
minute ?
3. A train that is running at the rate of 2 miles in 3
minutes goes how many miles in an hour? In 10 hours?
In 24 hours?
4. A boy who is idle 15 minutes in every hour wastes what
per cent of his time ?
5. How many hours are there in a week ? In the month of
June ?
6. How many hours have we for work in a morning session
of school if it begins at 9 o'clock and closes at 11.45, allowing
a quarter of an hour for recess ?
7. How many days are there in the fall months?
8. Close your book. Recite the table of time and the names
of the months, giving the number of days in each month.
208. TABLE OF COUNTING
12 =1 dozen (doz.).
12 doz. = 1 gross.
20 =1 score.
209. Oral
1. How much apiece do oranges cost at 40^ a dozen?
2. One dozen is what per cent of 1 score ? Of 1 gross?
3. How many pens are there in a gross ?
4. If I buy pens at 72 ^ a gross and sell them at 1 f^ apiece,
how much do I make on a gross ? On a dozen ? On a pen ?
I gain what per cent of the cost ?
176 PAPER MEASURE
5. "Fourscore and seven years ago" was how many years
Ago?
6. A merchant bought fiber pails at f 3 a dozen. How
much apiece did he pay ? If he sold them at 35 ^ apiece, what
did he gain on one ? On a gross ? What per cent of the cost
did he gain ?
7. A merchant buys shoe brushes at 11.20 a dozen and sells
them at 15 ^ apiece. How much does he gain on one ? What
per cent of the cost does he gain ?
8. What is the cost of 6 cans of Alaska salmon at 98 ^ a
dozen cans ?
9. What is the cost of a gross of pencils at 40^ a dozen?
10. A man's age is threescore and ten years. How many
years old is he ?
11. Bars of soap at 1 9 a gross are how much a dozen ?
210. TABLE OF PAPER MEASURE
24 sheets = 1 quire.
20 quires = 1 ream.
The terms bundle (2 reams) and hale (5 bundles) are seldom
used. The denomination quire is used mostly in measuring the
finer grades of writing paper. Wrapping paper is sold by the
pound or by the thousand sheets. Many kinds of paper are
sold in packages of five hundred or one thousand sheets. Pack-
ages of five hundred sheets are sometimes called reams,
211. Oral
1. How many sheets of paper are there in 2 quires ? In 4
quires ? In ^ of a ream ? In 3 quires ? In |^ ream ? In
10 reams ?
ARC AND ANGLE MEASURE 177
2. One quire is what per cent of 1 ream ? Of ^ ream ? Of
•| ream ?
3. What is the profit on 10 quires of paper bought at 14
cents a quire and sold at a cent a sheet ?
4. A package of 500 sheets of paper contains how much more
than twenty quires ?
5. A stationer sold 10 quires out of a package of 1000 sheets
of paper. How many sheets were left ? What per cent of
the package was sold ? What per cent was left ?
6. A stationer made a dozen tablets, each containing 72
sheets of paper. How many quires were used for each tablet?
The paper cost 40^ a ream. What was the cost per quire ?
What was the cost of the paper in one tablet ? What was the
cost of the paper for a dozen tablets ?
If the backs and labor cost 28^ for a dozen tablets, what
was the entire cost of a dozen tablets ?
If they were sold for 10^ a piece, what was the gain on a
dozen tablets ? The gain was what per cent of the cost ?
What was the gain on a gross of tablets ?
7. One quire of paper will make how many leaves if each
sheet is folded into 8 leaves ?
8. If 12 sheets of a certain kind of paper weigh one pound,
how many pounds will 5 quires weigh ?
9. 960 pages in a book would require how many leaves ? If
one sheet makes 4 leaves, how many sheets are required ?
How many quires ?
212. TABLE OF ARC AND ANGLE MEASURE
60 seconds (") =1 minute (')•
60 minutes = 1 degree (°).
An arc of 360° = 1 circumference.
178
ARC AND ANGLE MEASURE
213. The difference in direction of two
angle ; e.g.
that meet is an
214. The lines that meet to form an angle
are the sides of the angle.
Lines are read by means of letters placed
at their extremities. Angles are read by
means of letters placed at the extremities of their sides.
In the angle ABQ the lines AB and BO are the sides.
215. The sum of all the angles that can he formed around a
point in a plane is 360°,
120
120
In figure 1 there are three angles about a
point. Add the numbers of degrees.
Fig. 1
In figure 2 there are five angles about a
point. Add the numbers of degrees.
90°
90'
90'
90'
Fia. 3
Fig. 2
In figure 3 there are four angles about a
point. Add the numbers of degrees.
Draw eight equal angles about a point.
How many degrees are there in each
angle ?
Make other questions about these angles.
ARC AND ANGLE MEASURE 179
216. Oral
1. How many angles are there in Fig. 3, page 178 ? How do
they compare ? Each of these angles is a right angle. How
many degrees are there in a right angle ?
2. When the hour hand of the clock is at 12 and the minute
hand is at 9, they form what kind of an angle ? At what other
number could the minute hand point to make a right angle
with the hour hand at 12 ?
An angle of 90° is a right angle.
An angle that is greater than a right angle is an ,
obtuse angle.
An angle that is less than a right angle is an
acute angle.
3. Draw a right angle. How many degrees are
there in it? Divide it in the middle by a line. How many
degrees are there in each of the angles thus formed ?
4. Make a drawing of a wagon wheel with 6 spokes. The
spokes form angles of how many degrees ? Put in twice as
many spokes. How many degrees are there in the angles ?
Double the number again, and tell the size of the angles.
What kind of angles are these ?
5. Draw an angle that you think is about an angle of one
degree.
6. One minute is what part of a degree ? Can you think of
something that is like an angle of one minute ?
7. The minute hand of a clock passes through how many
degrees in 12 hours ? In 1 hour ?
8. What kind of angle (right, obtuse, or acute) is formed
by the hour and minute hands of a clock at two o'clock ? At
five o'clock ? At eleven o'clock ?
180
ARC AND ANGLE MEASURE
9. 10° = how many minutes ? 20° ? 40° ? i° ? J/ ?
10. 1° = how many seconds ?
11. Stand facing north. Turn 90° to the left. In what di-
rection are you facing ? Turn 90° farther. In what direction
are you facing? Turn 180° farther. In what direction are
you facing ? How many degrees have you turned in all ?
217. A plane figure hounded hy a curved line^ every point of
which is equally distant from a point within, called the center, is
a circle.
218. The boundary line of a circle is its circumference.
219. Any part of a circumference is an arc.
The number of degrees in an arc is always the same as the
number of degrees in the angle at the center whose sides meet the
extremities of the arc, thus :
The angle AOB is -J the sum of all the
angles at the center, or 90°. The arc AB
is \ of the circumference, or 90°. Can you
tell the number of degrees in the arc BQl
In the angle BOQl
Note. — The number of degrees in any angle may be measured by
means of a protractor, an instrument with the degrees marked and num-
bered.
A Protractor
UNITED STATES MONEY 181
220. TABLE OF UNITED STATES MONEY
10 mills = 1 cent.
10 cents = 1 dime.
10 dimes = 1 dollar.
The gold coins of the United States are the 85, f 10, and
$20 pieces, once called the half eagle, eagle, and double eagle.
Gold dollars are not in general circulation, although a few of
them have been coined.
The silver coins are the dollar, half dollar, quarter dollar,
and dime. Silver half-dimes are no longer coined. Most five-
cent pieces are made of nickel. Most 1-cent pieces are made
of bronze, though some nickel and copper cents are in
circulation.
The mill is not coined.
221. Oral
1. One dime is what part of a dollar ? What per cent ?
2. One cent is what per cent of a dollar ?
3. One mill is what part of a dollar ? What per cent ?
4. 79 cents is what decimal of a dollar? 7 mills is what
decimal of a dollar ? 19 cents and 7 mills ?
5. Express as decimals of a dollar :
85 cents 6 mills; 10 cents 8 mills ; 4 cents 7 mills; 8 cents ;
29 cents 1 mill ; 3 mills.
6. 5 mills are what part of a cent ? 4 mills ? 8 mills ?
7. The value of a $5 gold piece is what per cent of the
value of a 110 gold piece ? Of a §20 gold piece ?
8. Make other problems about dollars, mills, and cents.
182 TROY AND APOTHECARIES' WEIGHTS
222. TABLE OF TROY WEIGHT
24 grains (gr.) = 1 pennyweight (pwt.).
20 pennyweights = 1 ounce (oz.).
12 ounces = 1 pound (lb.).
These weights are used in weighing gold, silver, and some
jewels. To get an idea of the weight of a grain, think of the
weight of a grain of wheat or rice.
223. Oral
1. How many grains are there in 1 Troy ounce ?
2. A silver dollar weighs about 412| grains. This is how
much less than a Troy ounce ?
3. A gold dollar contains 23.2 grains of pure gold, but
enough harder metal is put with the gold to make it weigh 25.8
grains. This weight is how much more than 1 pwt. ?
4. Calling the weight of a gold dollar 1 pwt., what does a $20
gold piece weigh? How many dollars in gold would weigh
a pound ?
5. How many Troy ounces would $1000 in gold weigh?
How many Troy pounds ?
6. What must I pay for a watch chain weighing 240 grains
at $1 a pennyweight ?
224. TABLE OF APOTHECARIES' WEIGHT
20 grains (gr.) = 1 scruple (sc. or 3).
3 scruples = 1 dram (dr. or 3).
8 drams = 1 ounce (oz. or 5).
This table is used by druggists and physicians in compound-
ing medicines ; but medicines are bought and sold by avoir-
dupois weight, except in quantities smaller than one ounce.
REDUCTION OF DENOMINATE NUMBERS 183
Druggists also use a term fluid ounce^ which is not a measure
of weight, but of capacity, and is equal to -^^ of a pint. Thus,
a 2-ounce bottle is a bottle that holds ^ of a pint of any liquid
regardless of its weight.
225. Oral
1. A druggist buys a pound (avoirdupois) of quinine con-
taining 7000 grains. How many 2-grain tablets can be made
from it ?
2. How many 2-grain tablets can be made from 1 3 ?
3. How many 3-grain tablets can be made from 1 3 ?
4. A patient takes 5 gr. of a certain medicine every day.
How long will 1 3 of it last him ?
5. A druggist made 1000 powders, each containing 2 gr. of
ipecac, 2 gr. of muriate of ammonia, and 10 gr. of extract of
licorice. There are 7000 grains in 1 lb. avoirdupois. How
many avoirdupois pounds did all the powders weigh ?
REDUCTION OF DENOMINATE NUMBERS
226. Changing numbers to larger denominations is reduction
ascending.
227. Changing numbers to smaller denominations is reduction
descending.
228. Oral
1. How many gallons are there in 72 pints ? What kind of
reduction is this ?
2. How many minutes are there in 10 ° ? What kind of re-
duction is this ?
184
REDUCTION OF DENOMINATE NUMBERS
3.
Reduce :
a.
2 bu. to quarts.
I.
h.
64 pt. to pecks.
m.
c.
17 T. to pounds.
n.
d.
96 oz. to pounds.
0,
e.
11 yd. to rods.
P-
/.
33 ft. to rods.
?•
9-
5 A. to square rods.
r.
h.
288 sq. in. to square feet.
s.
i.
1728 cu. in. to cubic feet.
t.
h
10 cu. yd. to cubic feet.
u.
k.
20 wk. to days.
V.
^ da. to hours.
^ yr. to days.
J niin. to seconds.
20 da. to hours.
240 sec. to minutes.
96 doz. to gross.
12 score to units.
100 quires to reams.
50 reams to quires.
7' to seconds.
720" to minutes.
4. Reduce :
a, 51 qt. to gallons and quarts.
h. 7 gal. 2 qt. to quarts ; to pints.
c, 35 qt. to bushels and quarts.
d, 1 bu. 3 pk. to pecks ; to quarts ; to pints.
e, 1 T. 370 lb. to pounds.
/. 40 oz. to pounds and ounces.
g. 15 cwt. 50 lb. to pounds.
h, 1 A. 40 sq. rd. to square rods.
i, 4 sq. yd. to square feet.
j, 100 sq. ft. to square yards and square feet,
k. 64 fluid oz. to pints.
I, 130 min. to hours and minutes.
REDUCTION DESCENDING
229. Compound numbers are seldom expressed in more than
two denominations. In measures of time and arcs three denom-
inations are sometimes used.
Long and difficult reductions are seldom necessary.
KEDUCTION OF DENOMINATE NUMBERS 185
Reduce 17 da. 10 hr. 40 min. to minutes.
17 da.
24 number of hours in 1 da.
68
34
408 number of hours in 17 da.
10
418 number of hours in 17 da. 10 hr.
60 number of minutes in 1 hr.
25080 number of minutes in 418 hr.
40
25120 number of minutes in 418 hr. 40 min.,
or 17 da. 10 hr. 40 min.
Reduce 41 A. 20 sq. rd. to square feet.
41 A.
160 number of square rods in 1 A.
2460
41
6560 number of square rods in 41 A.
20
6580 number of square rods in 41 A. 20 sq. rd.
30J number of square yards in 1 sq. rd.
1645 (6580x1)
197400 (6580 x 30)
199045 number of square yards in 6580 sq. rd.
9 number of square feet in 1 sq. yd.
1791405 number of square feet in 41 A. 20 sq. rd.
186 REDUCTION OF DENOMINATE NUMBERS
230. Written
Reduce :
1. 49 da. 7 hr. to hours.
2. 79 A. 50 sq. rd. to square rods.
3. 16 yr. 7 mo. 20 da. to days (allow 30 da. for 1 mo.).
4. 9 sq. yd. 6 sq. ft. to square inches.
5. 78 T. 16 cwt. to hundredweight.
6. 59 cwt. 23 lb. to pounds.
7. 48 lb. 15 oz. Troy to ounces.
8. 12 cu. ft. 384 cu. in. to cubic inches.
9. 78 cu. yd. 19 cu. ft. to cubic feet.
10. 48 gal. 2 qt. to pints.
11. 15 bu. 3 pk. to pints.
12. 25° 30' 15^' to seconds.
13. The degrees in 1 right angle to seconds.
14. 158 gross 5 doz. to dozen.
15. 3 mi. to feet.
16. 1 mi. to inches.
17. 1 sq. mi. to square rods.
18. 21 T. 362 lb. to pounds.
19. 4 cu. yd. to cubic inches.
20. 121 yr. 11 mo. to months.
21. 8 mo. 28 da. to days.
22. 42 wk. 3 da. to hours.
23. 17 cwt. to ounces.
24. 12 yd. 19 in. to inches.
25. 2 yr. 4 mo. 3 da. to days.
REDUCTION OF DENOMINATE NUMBERS 187
26. 3 yr. 6 mo. 15 da. to days.
27. 17° 3' to minutes.
28. 7 yr. 14 da. to days.
29. 15 mo. 29 da. to days.
30. 42° 12' to seconds.
REDUCTION ASCENDING
231. Reduce 3876 sec. to hours, minutes, and seconds.
How many seconds = 1 min. ?
387p sec. 3876 sec. = how many minutes and
6^ min. + 36 sec. seconds ?
1 hr. + 4 min. How many minutes = 1 hr.?
1 hr. 4 min. 36 sec. Ans. ^^ "^^^- = ^^^ ^^^^ ^^^^ ^"^
minutes?
6JZ)
Written
Reduce :
1. 42,876 sec. to hours, minutes, and seconds.
2. 16,307^' to degrees, minutes, and seconds.
3. 8370 da. to years and days.
4. 983 pk. to bushels and pecks.
5. 5834 lb. to tons and pounds.
6. 4376 oz. Troy to pounds and ounces.
7. 892 pt. to gallons and quarts.
8. 508 pt. to gallons and quarts.
9. 8376' to degrees and minutes.
10. 45,360" to degrees and minutes.
11. 4416 sheets to quires.
12. 685 sq. ft. to square yards and square feet.
13. 28,347 cu. ft. to cubic yards and cubic feet.
188 REDUCTION OF DENOMINATE NUMBERS
14. 38,627 sec. to hours, minutes, and seconds.
15. 497' to degrees and minutes.
16. 89,764 lb. to tons and pounds.
17. 49,763 ft. to miles and feet.
18. 42,374 da. to years and days.
19. 94,276 min. to days, hours, and minutes.
20. 13,794 ft. to rods.
VARIOUS FORMS OF REDUCTION
232. Written
1. How many inches are there in |- rd. ? (Indicate the
work thus : f x-^^x-^^-; then cancel.)
2. Find the number of:
a.
Inches in f mi.
/.
Coat-hooks in ||^ gross.
6,
Pints in ^g bu.
9-
Sheets in l| ream.
c.
Pints in f bbl.
h.
Cubic inches in g|^g cu. yd,
d.
Seconds in l\'' .
. i.
Minutes in -^^^ wk.
e.
Ounces in -^^ T.
J-
Square feet in -^^ A.
3. What part of 5 gal. is 1 gal. 1 pt. ?
Note. — Find the number of pints in 5 gal. ; then in 1 gal. 1 pt.
Statement of Relation : of 40 pt. = 9 pt.
4.
What part:
a.
Of 1 T. is 324 lb.?
/.
Of 7 cu. yd. is 5 cu. yd. 9
h.
Of 2 T. is 7 cwt. 40 lb. ?
cu. ft. ?
c.
Of 3 gal. is 2 qt. 1 pt. ?
9-
Of 2 yr. is 1 yr. 3 mo. ?
d.
Of 5 da. is 12 hr. 30
h.
Of a square mile is 200
min. ?
A. 40 sq. rd. ?
e.
Of a circumference is
i.
Of 20 gal. is 5 gal. 2 qt. ?
36° 45'?
J-
Of a week is 18 hr. 8 min . ?
REDUCTION OF DENOMINATE NUMBERS 189
5. 3 bu. 2 pk. of potatoes are sold out of a load of 28 bu.
a. What part of the load is left? 5. What per cent of the
load is sold ?
6. 2 bbl. of cranberries, each containing 3 bu., are worth
how much at 12 J ^ per quart ?
7. The speed limit for automobiles in a certain town is 8
miles per hour. That is how many rods per minute ?
8. What is the cost of 5 T. 500 lb. of coal at 15.40 per ton?
9. \ T. is equal to how many ounces ?
10. How long will 16 bu. of corn last Fred's chickens if he
feeds them 1 qt. a day ?
11. a. What is the profit on a bushel of chestnuts bought
for 82^ and sold at 10 cents a pint ? h. The gain is what per
cent of the cost ?
12. How many minutes are there in April, May, and June ?
13. What is the cost of the milk supply for September of
a housekeeper who buys \ gal. per day and pays 6|^^ per
quart ?
14. What are the yearly wages of a man who earns a cent
in 2 minutes and works 8 hours a day and 26 days in a month
throughout the year ?
15. If a horse eats 12 lb. of hay per day, how many tons and
pounds will he eat in a year ?
16. A box 6'' X 4'' X 2'' contains what fraction of a cubic
foot ?
17. I of an acre of land contains how many square feet ?
18. 16 sq. rd. 11 sq. yd. are what part of an acre ?
Lb.
Oz.
7
8
15
14
23
15
47
5
Add:
190 ADDITION OF COMPOUND NUMBERS
ADDITION AND SUBTRACTION OF COMPOUND NUMBERS
233. Written
Add 7 lb. 8 oz., 15 lb. 14 oz., 23 lb. 15 oz.
15 oz. + 14 oz. + 8 oz. = 37 oz. = 2 lb. 5 oz.
2 lb. + 23 lb. + 15 lb. + 7 lb. = 47 lb.
47 lb. 5 oz. Ans.
1. 19 ft. 6 in., 17 ft. 10 in., 9 ft. 6 in.
2. 13 A. 17 sq. rd., 19 A. 153 sq. rd.
3. 2 hr. 5 min. 30 sec, 8 hr. 53 min. 47 sec.
4. 8 gal. 3 qt., 15 gal. 1 qt., 16 gal. 2 qt.
5. 81° 19' 35'', 2° 50' 29", 3° 4' 50".
6. 6 T. 480 lb., 7 T. 730 lb., 19 T. 900 lb.
7. 5 yd. 2 ft., 16 yd. 1 ft., 18 yd. 2 ft.
8. 6 pk. 7 qt., 3 pk. 5 qt., 2 pk. 6 qt.
9. 7 cu. yd. 18 cu. ft., 12 cu. yd. 19 cu. ft.
10. 6 yr. 7 mo. 3 da., 7 yr. 8 mo. 29 da.
11. 8 lb. 7 oz., 16 lb. 14 oz., 19 lb. 10 oz.
12. 21 bu. 3 pk., 9 bu. 2 pk., 35 bu. 1 pk.
13. 2 wk. 3 da., 19 wk. 1 da., 20 wk. 6 da.
14. 7 hr. 38 min. 21 sec, 5 hr. 47 min. 29 sec
15. 5 yr- 200 da., 7 yr. 321 da., 8 yr. 179 da.
SUBTRACTION OF COMPOUND NUMBERS 191
234. Written
From 18 yr. 7 mo. 14 da.
take 6 yr. 8 mo. 26 da.
11 yr. 10 mo. 18 da. Difference
7 mo. 14 da. = 6 mo. 44 da. „^- , , , , . «
1Q a 1H. 1Q (Why do we make these reductions?)
18 yr. 6 mo. = 17 yr. 18 mo. \ j j
18 yr. 7 mo. 14 da. = 17 yr. 18 mo. 44 da.
17 yr. 18 mo. 44 da. - 6 yr. 8 mo. 26 da. = 11 yr. 10 mo. 18 da.
Suhtraet:
1. 12 yr. 3 mo. 15 da. 5. 4 yr. 2 mo. 18 da.
10 yr. 1 mo. 19 da. 1 yr. 7 mo. 12 da.
2.
9yr.
3yr.
2 mo.
5 mo.
Ida.
29 da.
3.
9 yr.
4 yr.
2 mo.
8 mo.
5 da.
5 da.
4.
12 yr.
8 yr.
2 mo.
16 da.
12 da.
6. 7yr.
3 yr.
5 mo.
6 mo.
18 da.
7 da.
7. 9 yr.
8yr.
6 mo.
5 mo.
13 da.
15 da.
8. 1 yr.
3 mo.
7 mo.
15 da.
• 9. How many years, months, and days are there from
May 30, 1907, to Dec. 5, 1909 ?
1909 yr. 12 mo. 5 da. Note. — December is the twelfth month
1907 vr. 5 mo. 30 da. ^^^ ^^^ *^® fifth. Count 30 da. for a
' month.
Find the time from :
10. July 19, 1827, to Mar. 26, 1878,
11. Sept. 20, 1831, to Nov, 15, 1909.
192 SUBTRACTION OF COMPOUND NUMBERS
12. Jan. 7, 1840, to Feb. 8, 1896.
13. May 4, 1850, to Jan. 12, 1861.
14. Oct. 19, 1760, to Aug. 20, 1860,
15. Dec. 12, 1880, to June 5, 1903.
16. July 10, 1809, to Oct. 2, 1893.
17. May 8, 1899, to Feb. 12, 1908.
18. Feb. 12, 1901, to Jan. 30, 1906.
Subtract :
19. 5 hr. 54 min. 30 sec.
1 hr. 50 min. 50 sec.
20. 122° 31' 15''
60° 20' 45"
21. 23 hr. 54 min. 36 sec.
20 hr. 24 min. 48 sec.
22. 7 ft. 6 in.
3 ft. 11 in.
23.
27 gal.
18 gal.
2 qt.
3 qt.
24.
19° 31'
6° 41'
25.
18 bu.
Ipk.
17 bu.
3pk.
26.
10 A.
56 sq.
rd.
4 A.
106 sq.
rd.
27.
16 1b.
4 oz.
51b.
12 oz.
28.
48 ft.
3 in.
27 ft.
9 in.
29.
42' 13"
35' 5^
l^
30.
5 min.
47 sec.
2 min.
. 48 sec.
31. Find the time between Dec. 21, 1620, and July 4, 1776.
32. How much time elapsed from the beginning of the Civil
War, April 14, 1861, to the close of the war, April 9, 1865 ?
33. Washington was born Feb. 22, 1732, and died Dec. 14,
1799. How long did he live ?
34. How much time has elapsed from Oct. 12, 1492, to the
present time ?
SUBTRACTION OF COMPOUND NUMBERS 193
EXACT DIFFERENCES BETWEEN DATES
235. Written
I. What is the exact number of days between Dec. 16, 1895,
and March 12, 1896 ?
Dec. 15
Jan 31 There are 15 days in December after
■J-, ■■ oq the 16th. January has 31 days, February
29 (leap year), and March 12, making 87
March L£ days. Always count the last day.
87 days. Ans.
Note. — Every year whose number is divisible by 4 is a leap year, except a
centennial year, which is a leap year only when its number is divisible by 400 ;
e.g. the year 1896 was a leap year but the year 1900 was not.
2. Mr. Griffith bought a house, Feb. 25, 1896, and paid for
it, July 12, 1896. Find the exact number of days between the
buying and paying for the house.
3. Find the exact number of days between June 25, 1900,
and Aug. 24, 1900.
Mnd the exact time between :
4. Sept. 6, 1896, and April 7, 1897.
5. Nov. 11, 1898, and Dec. 4, 1898,
6. Aug. 16, 1907, and Dec. 21, 1907.
7. July 4, 1896, and Aug. 10, 1896.
8. Feb. 23, 1897, and June 4, 1897.
9. Oct. 9, 1899, and Feb. 6, 1900.
10. Nov. 8, 1905, and Oct. 6, 1906.
II. A gardener planted an acre of sweet corn on the tenth
day of May. It was ready for market on the third day of
August. How many days were required for the corn to grow?
194
COMPOUND NUMBERS
MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS
To THE Teacher. — Little time should be spent upon multiplication
and division of compound numbers. In solving problems other than those
in longitude and time, it is generally better to reduce the compound num-
ber to one denomination before multiplying or dividing.
236. Written
1. Multiply 8 lb. 3 oz. by 9.
8 lb. 3 oz 9x3 oz. = 27 oz. = 1 lb. 11 oz.
9 9 X 8 lb. = 72 lb.
73 lb. 11 oz. Product 72 lb. + lib. = 73 lb.
Multiply :
2. 7 lb. 9 oz. by 5. 7.
3. 15 gal. 1 qt. by 10. 8.
4. 14 ft. 11 in. by 3. 9.
5. 17 A. 40 sq." rd. by 5. 10.
6. 19 ft. 7 in. by 4. ii.
12. Divide 32° 15' 30'' by 15.
1 hr. 20 min. 20 sec. by 15.
2 hr. 40 min. 30 sec. by 15.
3 hr. min. 30 sec. by 15.
4 hr. 19 min. 30 sec. by 15.
hr. 58 min. 47 sec. by 15.
15 )32° 16' 30'^
2° 9' 6" Quotient
82° 4-15 = 2° and 2° Remainder.
2° = 120'. 120' + 16' = 136'.
136' -T- 15 = 9' and 1' Remainder.
r=60". 60" + 30" = 90".
90" -4-15=6".
Divide by 15 and test your work
13. 30° 16' 15"
14. 61° 1'45"
15. 17° 6' 0"
16. 2° 2' 15"
17. 46° 10' 0"
18. 20° 0' 30"
19. 20° 5' 45"
20. 60° 35' 15"
21. 70° 30' 45"
22. 100° 1' 30"
REVIEW AND PRACTICE 195
REVIEW AND PRACTICE
237. Oral
1. At 8)^ a quart, what will a bushel of berries cost?
2. How many oranges will 50 cents buy at the rate of 3 foi
10 cents?
3. f of 18 is what part of 72? What per cent of 72?
4. From 4|- bu. take 3 pk.
5. What will 15 bu. of potatoes cost if 7|- bu. cost 1 6 J?
6. At 1 15 a dozen, what will 84 chairs cost?
7. A cube 5 inches long contains what part of a cubic foot ?
8. 9 mo. is what part of a year?
9. I sq. ft. = sq. in.
10. Bought a peck of chestnuts for 80 cents and sold them
at 5 cents a half pint. How much did I gain? What per cent
of the cost did I gain ?
11. 25 is what part of 75? What per cent of 75?
12. What part of f is |?
13. 15 gal. 3 qt. = pt.
14. Multiply 7.635 by 10; by 1000; by 100.
15. Divide 5.8 by 10; by 100; by 1000.
16. Divide 16.54 by 1654.
17. 36 lb. of coffee for i9 is at the rate of 12 lb. for how
much?
18. A wagon was sold for $60, which was -J of the cost.
What was the cost ? How much was lost ? What per cent of
the cost was lost?
19. How many square rods are there in -^^ of an acre?
20. A ten-acre lot is 20 rods wide. How long is it?
196 REVIEW AND PRACTICE
21. f of 16 is I of what?
22. Andrew spent -| of his money and had 10 cents left.
How much had he at first?
23. What is the entire weight of three chickens if their aver-
age weight is 2 lb. 11 oz.?
24. What part of |^ is J?
25. ^ is what part of ^?
26. Find 20% of 300; 331% of 72; 16f % of 60; 62J% of
16; 90% of 10.
27. Sugar at $100 a ton is how much a pound?
28. A house was damaged fSOO by fire. This was 10% of
the cost. What was the cost ?
29. 87| % of the cost of Glen's bicycle was 1 28. What was
the whole cost?
30. If 9 eggs cost 27 cents, what will 6 doz. cost ?
238. Written
1. The month of January is how many minutes long ?
2. If school closes for the long vacation on June 24 and
opens on Sept. 6, how many vacation days are there ?
3. If it is now 8.30 a.m., what time will it be in 20 J hours ?
4. Find the exact time from March 15, 1906, to Aug. 2, 1906.
5. How many weeks, days, and hours are there in 9785 hours ?
6. How many rods of fence are needed to inclose a square
field 396 ft. long ?
7. What will it cost to paint a ceiling 24 ft. long and 16J ft.
wide, at 25 cents a square yard ?
8. What is the length of one side of a square garden whose
perimeter is 632 feet ?
REVIEW AND PRACTICE 197
9. How many square feet of land are there in a field contain-
ing 21 A. ?
10. What will it cost at 65 cents a square yard to cover with
matting a floor 21 ft. long and 15 ft. wide ?
11. How many feet of fence will inclose a field 46 rd. by
31 rd. ?
12. Find the perimeter of a room that is 15' 6" by 13' 8".
13. How many rods are there in 2448 inches ?
14. a. If steel rails are 30 ft. long, how many are needed for
8 miles of street railroad track ? b. How many tons do they
weigh if each foot weighs 90 lb. ?
15. At the rate of 660 feet a minute, how many miles will a
street car run in two hours ? (Indicate and cancel.)
16. What is the cost, at $1.80 per rod, of a fence inclosing a
rectangular field 50 rd. by 561 ft. ?
17. Draw a line to represent 1| miles, letting the scale be
80 rd. to an inch.
18. Draw a rectangular building lot 4 rd. by 8 rd., and put
on the lot a house 44 ft. by 33 ft., and a barn 22 ft. square,
using as a scale 22 ft. to an inch.
19. Three barrels of extract of witch hazel are put up in
pint bottles and sold for 20^ a bottle. What is received for
all of it ?
20. A grocer bought 40 bbl. of cider at S 2.50 a barrel, made
it into vinegar, and sold it at 5 ^ a quart, a. How much did he
gain ? b. What per cent of the cost did he gain ?
21. Divide in the shortest way the product of 36, 40, 144, 8,
and 160 by the product of 18, 272, and 6.
22. Find the least number that will exactly contain 27, 60,
and 24.
198 REVIEW AND PRACTICE
23. Find the greatest number that will exactly divide 231
and 385.
24. In the corner stone of a church were cut these two dates,
MDCCCXXI — MCMVII. How many years apart were they?
25. Mr. C. J. Hogan has laid for Mr. Henry Sumner a
piece of Portland cement walk 50 ft. long and 5 ft. wide at 12J ^
a square foot. Make out the bill and receipt it.
26. Divide: a. 8.8 by .0008; h. 20.005 by .005.
27. Divide the sum of two thousand and two thousandths by
two hundredths.
28. How many times will a bicycle wheel rotate in going a
mile if the wheel is 7.2 ft. in circumference ?
29. The product of two numbers is 12.46, and one of them is
24.92. Find the other number.
30. The product of three numbers is .1456. Two of them
are 2.6 and .14. What is the third ?
31. Reduce to common fractions in lowest terms :
a. .025 d. .06 J g. .625 /. .21
h. .0125 e, .871 h. .0875 k. .0495
c. .371 /. .375 i, .012i h 4.25
32. Express as decimals :
d. i^ A. ff Z. 14|^ p. 15|
33. Find the value of:
a. 2i + 32-f.ioff + f of J 5. 314^x5
KEVlEW AKD PRACTiC:e l99
e. 8^x21x^x8 /. t^xl8|
34. A hardware merchant bought a bill of goods for $ 560,
and marked them to be sold at a price which was 140 % of the
cost. a. At what price did he mark the goods ? b. He sold
the goods for 90 (fo of the marked price. For what did he sell
them ?
35. A jobber bought 2000 wheelbarrows at f 1.25 apiece,
marked them at 115 % of their cost, and sold them for 95 % of
the marked price. What did he receive for them ?
36. During a storm 1000 bbl. of sugar, or S^ % of the cargo
of a ship, were thrown overboard. How many barrels formed
the cargo ?
37. Make and solve a problem about $ 420 and 35 % .
38. Mr. Fish paid city taxes to the amount of f 280, which
was 1^ % of the value of his property. How much was his
property worth ?
39. A load of boards weighing 3500 lb. were put into a kiln
and dried. When taken out they weighed 1^ tons. What per
cent of the weight of the boards at first was water ?
40. a. Write a problem in percentage in which factors are
given to find a product, h. Write another in which one of the
factors is to be found.
41. If a pint of cream is used in making two gallons of ice
cream, what per cent of the ice cream consists of other things
than cream ?
42. I of my money is $ 1260. What is ^ of my money ?
200
REVIEW AN^D PRACTICE
43. This walk is made of sawed Ohio sandstone,
cost at 21^^ per square foot.
Find its
7Falk\
Walk
44. A man owned
f of a mine and
sold I of his in-
terest for 11710.
What was the whole
mine worth at the
same rate?
Walk
Scale 5 to ]/i
45. A teamster
mixed 600 lb. of
wheat bran with a
ton of corn meal
with which to
feed his horses.
a. What per cent
of the mixture was bran ? b. What per cent was corn meal ?
46. The population of a city to-day is 130,000, which is 125 %
of what it was 10 years ago. What can you find ? Find it.
47. Mr. Hunter's ranch in Wyoming contains 3200 acres of
land, worth $25 an acre.
a. How many square miles of land has he?
b. What is its value ?
e. One fourth of the land can be irrigated and made worth
$75 an acre. Would it pay to irrigate the land at a cost of
120,000? Why?
48. a. Mr. Hunter has 6000 sheep. They are worth $5.25
apiece in the fall and $5.75 apiece in the spring. What is the
increase in the value of the entire herd during the winter?
h. The increase is what per cent of the value in the fall?
Can you tell why they are worth more in the spring ?
REVIEW AND PRACTICE
201
The Ranch
49. a. The sheep are separated into two equal flocks and are
cared for by two herders, two dogs, and a camp tender. Each
herder receives $40 a month and the camp tender 850. The
supplies cost 5^ 52 per month. What is the cost of tending the
sheep for a year ? !
h. On the twenty-ninth day of June the men started with
their flocks for the "summer range" in the mountains 117
miles from the ranch. They traveled
an average distance of 6J miles per
day. On what day of the year did
they reach the summer range ?
e. In the spring the sheep were
fed 250 tons of alfalfa, worth |5 a
ton. A ton of salt lasts them 6
months and costs I 60. Add these
items to the cost of caring for the
sheep, and find the entire cost of
keeping these two flocks for a year.
Do you know how much a ton of coarse salt costs where
you live ?
Why does it cost more at this ranch ?
On the Move
202
REVIEW AND PRACTICE
50. a. In early summer 5 men sheared the 6000 sheep in 15
days, receiving 8 cents per fleece for their work. What were
the average daily wages of the shearers?
h. The wool was bought by a commission man for a jobber in
the East at 22| cents per pound. The fleeces weighed 8 lb.
apiece, on the average. What did Mr. Hunter receive for the
wool ?
c. The commission man charged the jobber 10J% of this
sum for buying the wool. What did the wool cost the jobber ?
A Wool Freighter
d. The wool was shipped in sacks 8 ft. long and 4 ft. wide,
holding 400 lb. apiece. It was taken to the railroad station
in loads of 22 sacks each, drawn by ten-horse teams. How
many sacks were left after 3 loads had been drawn ?
51. It may be found from the foregoing problems that Mr.
Hunter received §7426 more than he expended on account of
his sheep business this year. At the same rate, what would
be a man's yearly profit from a herd of 48,000 sheep ?
52. If a herder kills ten coyotes in a year and receives from
the government a bounty of $3 for each one, how much will he
add thereby to his monthly income?
ARTICLES SOLD BY THOUSAND AND HUNDRED 203
ARTICLES SOLD BY THE THOUSAND, HUNDRED, OR
HUNDREDWEIGHT
239. Written
1. What is the cost of 8975 bricks at 1 7 per M. ? (M. stands
for 1000.)
8975 = 8.975 M.
Since 1 M. costs $7, 8.975 M. cost 8.975 x $ 7, or $ .. Ans.
2. What must be paid for 980 soapstone pencils at f .30 per
C. ? (C. stands for 100.)
980 = 9.80 C.
Since 1 C. costs $.30, 9.80 C. will cost 9.80 x $.30, or | . Ans.
3. Find the cost of 1550 lb. of new buckwheat flour at 12.50
per cwt.
1550 lb. = 15.50 cwt.
Since 1 cwt. costs $ 2.50, 15.50 cwt. cost 15.50 x $2.50, or $ . Ans.
Note. — In final results, a fraction of a cent, equal to or greater than | cent,
is counted a whole cent. A fraction which is less than ^ cent is dropped.
4. Find the cost of each of the following items and the total cost
of all of them :
a. 27,325 bricks at f 5.15 per M.
b. 4900 cu. ft. of gas at $1.20 per M.
c. 583 lb. sugar at 14.75 per C.
d. 4900 tomato plants at $1.50 per M.
e. 1000 laths at 40 ^ per C.
/. 3125 cu. ft. of city water at $.14 per C.
g. 1 T. fiber paper at $2.50 per C.
h. 5600 paper bags at $2.90 per M.
i. 16 boxes of envelopes, 250 in a box, at $1 per M.
204
MEASUREMENTS
y. 850 Japanese napkins at i.l8 per C.
k. 10 bbl. American lump salsoda, 375 lb. in a barrel, at
1.80 per cwt.
I. 6500 No. 3 butter trays at 11.60 per M.
m. 8950 7-inch picnic plates at $1.75 per M.
n. 13,500 cedar shingles at |4.10 per M.
0, 675 lb. light manila bread paper at 13.75 per cwt.
MEASUREMENTS »
AREAS OF PARALLELOGRAMS
'240. A plane figure hounded hy four straight lines is a quadri-
lateral; e.g.
241. Lines that are the same distance apart throughout their
whole length are parallel lines ; e.g.
242. A quadrilateral whose opposite sides are parallel is a
parallelogram. Which of the above figures are parallelograms ?
243. A parallelogram that has four right angles is a rectangle.
Which of the above figures are rectangles ?
244. Two lines that meet to form a right angle are
perpendicular to each other.
245. The side on which a figure is supposed to rest is its base.
AREAS OF PARALLELOGRAMS
205
246. The perpendicular distance from the highest point of a
figure to the base, or to the base extended, is its altitude ; e.g.
'^
247. Figures are read by means of letters placed at their angles.
Thus, Fig. 1 is read, ''Oblong ABOD.'' Fig. 2 is read, "Tri-
angle ABO.'' Read Figs. 3 and 4. The base of the triangle in
Fig. 2 is AC. The altitude of Fig. 1 is DQ ov AB,
248. Oral
1. In Fig. 5 the part ^compares how with the part Jf ?
2. The area of the par-
allelogram ABCB com-
pares how with the area of
the parallelogram EFOD ?
3. What is the base of
each of these parallelo-
grams ?
4. What is the altitude ? What is the area ?
5. Cut from paper different shaped parallelograms.
6. Change them to rectangular parallelograms by cutting off
a part similar to ^and placing it like M.
7. How is the area of a rectangle found ?
8. If the base of a rectangle is its length, the altitude is what ?
A E
B
F
\k 1,
\
V M i
\ !<
\ 1'
\ I'*
N'
Fig. 5
206
AREAS OF PARALLELOGRAMS
9. If we know the base and altitude of a rectangle, how
may we find the area ?
10. Since any parallelogram may be made into a rectangle of
the same base and altitude, how may we find the area of a par-
allelogram ?
249. The area of a parallelogram is equal to the product of its
base and altitude.
250. Oral
The following examples relate to parallelograms.
Fill in the missing number :
Base Alt. Area Base Alt. Area
1. 8 ft. 6 ft. 5. 3J in. 7 sq. in.
2. 71 in. 8 in. 6. 2 ft.
3. 5 mi. 4 sq. mi. 7. 3| ft.
4. 6 yd. 96 sq. yd. 8. 1 yd. 15 sq. ft.
Alt.
3J in.
6 in.
8 ft.
1yd.
251. The following parallelograms are drawn to the scale of
1' to ^". Find their areas in square feet.
AREAS OF TRIANGLES
207
AREAS OF TRIANGLES
252. A 'plane figure hounded hy three straight lines is a
triangle ; e.g.
253. Oral
Fig. 1
Fig. 2
Fig. 3
Fig. 4
1. Figures 1, 2, 3, and 4 are what kind of figures ? What
-kind of figures are A and B 2
2. In each of the above figures, how does A compare with B?
3. In each of the above figures, how do the base of the
triangle and the base of the parallelogram compare ?
How do the altitude of the parallelogram and of the triangle
compare ?
4. Fold a paper once. By cutting through both leaves of
the paper at the same time, you may cut out two triangles.
How do they compare ? Put them together so as to make a
parallelogram.
5. Repeat this process, cutting triangles, each pair of which
differ in shape from the other pairs.
Each triangle is what part of the parallelogram having the
same base and altitude?
6. How is the area of the parallelogram found? Of the
triangle? ^
208
AREAS OF TRIANGLES
The area of a triangle is equal to one-half the product of its
base and altitude.
7. The following triangles are drawn to a scale. The bases
and altitudes of the triangles which they represent are indicated
in the drawings. Find their areas.
17
nrd.
254. Written
1. Find the areas of triangles having the following dimen-
sions, using cancellation where
possible. When the base and
altitude are of different denominations, make
them similar
before multiplying.
Base
Alt.
Basb
Alt.
a. 51 ft.
42 ft.
y. 15 yd.
15 in.
b, 27 in.
15 in.
h. 2 mi.
160 rd.
c. 6fft.
7Jft.
L 11 mi.
1201 rd.
d. 3 ft. 7 in.
2 ft. 9 in.
m. 4 rd.
3J rd.
e. 9fin.
16 in.
71. 2 yd.
50 in.
/. 18 yd.
27 in.
0. 3^rd.
12 ft.
g, 12 ft.
Hyd.
p, 101 yd.
5 ft. 4 in.
h, 14 rd.
7 yd.
q. 7 yd.
7 ft.
i. 1 mi.
80 rd.
r. 17fin.
A^d.
MEASUREMENTS: THE CORD
209
2. The above figures are drawn to the scale of 10 ft. to J in.
Find their bases, altitudes, and areas.
THE CORD
255. The pile of wood in the center of this picture is 8 ft.
long, 4 ft. wide, and 4 ft. high. How many cubic feet does it
contain ?
128 cubic feet = 1 cord.
In the forest, fuel wood for market is generally cut in 4-foot
lengths like that in the picture, so that a pile 4 ft. high and
210 MEASUREMENTS: THE CORD
8 ft. long contains a cord of 128 cu. ft. The term cord^ how-
ever, is often used to mean any pile of wood that is 8 ft. long
and 4 ft. high, whatever may be the length of the sticks.
256. Oral
1. Hold your hand above the floor high enough to show the
height of the pile of wood in the picture. Stand as many feet
from the side of the room as the pile is long. Show with your
hands the length of the sticks.
2. How many cubic feet are there in a pile of wood 8 ft.
long and 4 ft. high, if the sticks are 1 ft. long ? Show with
your hands the height of this pile. Show its width. Walk far
enough to show its length.
3. How many cubic feet would there be in the above pile if
the sticks were 2 ft. long ? Show the width of the pile with
your hands.
4. How many cubic feet would there be in the pile if the
sticks were 3 ft. long ? 1^ ft. long ? Show these lengths with
your hands.
257. Written
1. Using cancellation, find the number of cords in a pile ot
4-foot wood :
a. 20 ft. long and 8 ft. high.
h. 64 ft. long and 4 ft. high.
c. 12 ft. long and 6 ft. high.
d. 100 ft. long and 7 ft. high.
e. 26 ft. long and 5 ft. high.
/. 40 ft. long and 4 ft. high.
g. 72 ft. long and 7 ft. high.
h. 18 ft. long and 2 ft. high.
MEASUREMENTS: BUILDING WALLS
211
2. If a pile of 4-foot wood is 48 ft. long, how high must it be
to contain 9 cd. ?
3. What must be paid for enough 4-foot wood to fill a shed
26 ft. long, 16 ft. wide, and 12 ft. high at 14.50 a cord ?
4. In the yard of a certain tannery there is a pile of bark
100 ft. long, 24 ft. wide, and 10 ft. high. How many cords of
bark are there in the pile ?
BUILDING WALLS
258. There are no universal rules for the measurement of
masonry. Some masons measure around the outside of a cellar
wall to determine its dimensions, while others make allowance
for the corners. The method of measurement should be specified
in the contract in every case.
Quantities of uncut stone are bought by the cord, and usually
99 cu. ft. are taken for a cord.
From 21 to 23 bricks 8^' X 4'^ x 2'' are estimated to make a
cubic foot of brick wall.
259. Written
7' 6"
\_1
\\ \ \ \ \X vX'xx ^\< v<:\<;\^-^^ff'«'
\\\
1
1 { 1 1 1 1 1 I 1 1
\\X\
\V
x\
s
1. a. How many cubic feet does this wall contain ?
h. If 21 bricks make a cubic foot of wall, how many bricks
are used in this wall ?
212 MEASUREMENTS: FLOOR COVERING
c. What do they cost at 1 6.30 per M.?
d. At the same rate, what would be the cost of brick for a
partition wall 50 ft. long, 11 ft. high, and 12 in. thick ?
e. When 22 bricks 2'^ x 4'' x 8'' make a cubic foot of wall,
how many cubic inches of mortar are used ?
2. A house built in the form of a rectangle 36 ft. by 21 ft.
has a cellar 8|- ft. deep. The cellar wall is 1 ft. 6 in. thick.
a. Draw a plan of the cellar wall.
h. Find the number of cubic feet to be paid for in the cellar
wall, measuring around the outside, and making no allowance
for the corners.
e. If the cellar wall extends 3 ft. above the grolind, how
many cubic yards of earth were removed to make the cellar ?
d. What was the cost of the brick at $6.50 per M. for a
partition wall 12 in. thick, from end to end through the middle
of the cellar, allowing 22 bricks for a cubic foot ?
e. What was the cost of a Portland cement floor in one half
of this cellar at 10 ^ per square foot ?
3. Find the cost of the stone for a wall 297 ft. long, 8 ft.
high, and IJ ft. thick at 16.50 a cord (99 cu. ft.).
FLOOR COVERING
260. The exact number of yards of carpet, matting, or other
covering to be purchased for any given floor is diflicult to deter-
mine because of the waste in fitting and in matching figures.
We may obtain a fairly correct estimate by dividing the
number of square yards of surface to be covered, by the
width of the strips, in yards, and then adding a certain
amount for waste.
A yard of carpet, matting, etc., is a yard of the length of
the roll, regardless of its width.
MEASUREMENTS: FLOOR COVERING
213
261. Oral
1. A piece of carpet 1 yd. long and | yd. wide will cover how
Qyd.
much surface? Draw it full size on
the blackboard.
2. What is the area of this floor ?
3. If a yard of carpet | yd. wide
will cover | of a square yard of floor,
how many yards of such carpet will cover 18 sq. yd. of
floor ?
4. How many yards of matting 1 yd. wide will cover the
same floor ?
262. Written
1. If this floor is covered with carpet | yd. wide, how many
strips, running lengthwise, must be
purchased ?
Note. — When a part of the width of a
strip is needed, a whple strip must generally
be purchased.
2. How many yards of carpet must
be purchased for this floor ?
3. Draw a diagram of each of the floors whose dimensions are
given below, and compute the number of yards of material to be
i purchased to cover it, running the strips the longer way of the floor :
Dimensions
Width of Material
a.
81 yd. X 5 yd.
1yd.
b.
15 ft. X 3 yd.
2 ft. 3 in.
c.
101 ft. X 18 ft.
fyd.
d.
18 ft. X 24 ft. 6 in.
1yd.
e.
17 ft. X 27 ft. .
36 in.
214 MEASUREMENTS: FLOOR COVERING
/.
9 ft. X 28 ft.
|yd.
9'
13 ft. 3 in. X 15 ft.
fyd.
h.
12 ft. X 16 ft.
1yd.
i.
19^ ft. X 29 ft.
fyd.
h
15 ft. 9 in. X 19 ft.
2 ft. 3 in.
k.
11 ft. 3 in. X 14 ft.
27 in.
I.
20 ft. X 10 yd.
1yd.
m.
16 yd. X 6 yd.
54 in.
n.
20 ft. X 38 ft.
fyd.
0.
16 ft. X 22 ft.
1yd.
V'
15 ft. X 18 ft. 3 in.
27 in.
<!'
29 ft. X 16 ft. 6 in.
fyd.
r.
14 ft. X 20 ft.
iyd.
s.
13 ft. 8 in. X 19 ft. 6 in.
1yd.
t.
6 yd. X 23 ft.
ijyd.
u.
100 ft. X 75 ft.
1 ft. 8 in.
V.
31 yd. 2 in. x 13 yd. 1 ft. 6 in.
fyd.
4. What is the expense of covering a kitchen floor 12 ft. x
13| ft. with inlaid linoleum at $1.40 per square yard, allowing
11 sq. yd. for waste in matching the pattern ?
5. Find the cost of covering a porch floor 9 ft. by 30 ft. with
plain cocoa matting 54 in. wide at 50^ a square yard.
6. The living room of a summer cottage is 17 ft. x 24 ft.,
and the floor is covered with plain grass matting 1 yd. wide,
laid so as to make no waste. (Which way must the strips run?)
Find the cost at 40/ a yard.
7. Harold's bedroom is lOJ ft. x 13|^ ft. It is covered with
matting 1 yd. wide, costing 45 / a yard, pieced so as to make
no waste except in turning under at the ends. Carpet paper,
MEASUREMENTS; FLOOR COVERING
215
costing 5/ a square yard, is laid under the matting. Two
5-yard pieces of braid, costing 2^ a yard, are used. Harold
does the work, wasting \ yd. of
matting and -^ sq. yd. of paper.
a. What is the entire cost ?
b. Draw a diagram of the room
showing how the matting is laid
and pieced.
8. This rug is made of Royal
Wilton carpet costing $2.60 a
yard. The making and sizing cost
121 f^ a yard. 22 yd. of carpet
braid were used at a cost of 5^ a yard, sewed on. Find the
entire cost of the rug, allowing | yd. for waste in matching
the pattern.
19'9"
i 16'
r
l_ _
This is Lucy's room drawn to the
scale of I" = V.
9. a. What would a
hard-wood floor in this
room cost at 16J^ per
square foot?
b. Find the cost of
carpeting this room with
three-ply ingrain carpet
1yd. wide at 90 (^ per yard,
paying 5^ per yard for
making and laying, and
5^ per square yard for
carpet paper.
e. Find the cost of
cleaning the carpet at
8^ per square yard.
216
MEASUREMENTS: PLASTERING
PLASTERING
263. There is among builders no universal rule for comput.
ing the amount of plastering in the walls and ceilings of a
building. Some contractors deduct the entire surface of open-
ings, such as doors, windows, etc., and some only one half of
such surface.
264>. Written
1. The scale of this drawing is -^q '^ =
1'. Make a drawing like this, using ^"
for 1'. Cut it out, and fold it on the
dotted lines, making the model of a room.
12'
End
Wall
16'
Side
Walt
12'
End
Walt
16'
Ceiling
16'
Side
Wall
a. What is the entire length of the end and side walls ?
What is the hei^ ht ?
b. How many square feet are there in all the walls ?
c. How many square feet are there in the ceiling ?
d. How many square feet are there in the walls and ceiling
together ?
e. How many square yards are there in all ?
/. What will it cost to lath and plaster this room at 85 cents
a square yard, taking out 5 J square yards for openings ?
2. a. What will it cost to lath and plaster a room 15' by 18'
and 9' high at f .30 a square yard, allowing 10 sq. yd. for doors
and windows? h. What will it cost at |.28 a square yard,
making no allowances ?
MEASUREMENTS: WALL COVERINGS 217
3. Measure your schoolroom to the nearest half, third, or
fourth of a foot, and estimate the cost of lathing and plastering
it at 40)^ a square yard. Allow 3 sq. yd. for each door and
window.
4. Estimate the cost of plastering other rooms in your school
building in the same way.
5-6. Measure two rooms in your house. Find the cost of
plastering the walls at 30 cents a square yard, allowing 2 sq. yd.
for each door and each window. Bring your work to school.
7. Find the cost of lathing and plastering the walls and ceil-
ings of a room 27 ft. by 18 ft. and 9| ft. high, allowing for
4 windows, each 3 ft. by 6 ft, and 3 doors, each 3 ft. by 8 ft.,
at 32 cents per square yard.
WALL COVERINGS
265. A roll of figured wall paper is usually 8 yards long and
J yard wide. How many square yards of paper does it contain ?
Ingrain paper is 30 inches wide.
Paper hangers generally estimate that a roll of paper will
cover from 30 to 34 square feet, after allowing for waste.
Woven wall coverings are sold by the square yard.
266. Written
1. Fanny's mother wished to decorate a room in her house,
and Fanny estimated the cost. The dimensions were 16 ft. by
21 ft., and 9 ft. 9 in. high. There were 2 doors and 3 windows,
each estimated at 2 sq. yd.
a. The paper was estimated to cover 30 sq. ft. per roll.
How many rolls were needed for the walls ? (You cannot buy
a part of a roll.)
b. What would the paper cost at 25^ a roU ?
218 MEASUREMENTS: WALL COVERINGS
c. The molding to extend all around the room at the top of
the walls was sold only in 12-foot lengths, and cost 4J^ a
foot. What would it cost ?
c?. In preparation for tinting, the ceiling was to be lined
with paper at 10^ a roll of 30 sq. ft. What would this paper
cost?
e. They expected to use two packages of tinting material
costing 30^ a package; the putty, glue, flour, etc., were esti-
mated at Q^^\ and the labor at two days' work for two men
at $3.50 per day for each man. What should have been the
total of Fanny's estimate ?
/. If they decide to add to the ceiling some relief work
which costs f 3.81, and the men can put it on in ^ of a day, how
much must Fanny add to her estimate ?
2. Before being tinted, a ceiling 12' x 17' was covered with
sheeting 2 yd. wide. What did the sheeting cost at 25 ^ a
lineal yard ?
3. a. How much money is needed to buy, at 40^ a square
3^ard, enough crash to cover the four walls of a room 18' x 32'
and 10 ft. high, allowing 190 sq. ft. for baseboard and open-
ings, and not purchasing a fraction of a square yard ?
h. What would it cost to paint the walls and ceiling of this
room at f .23 a square yard ?
4. Estimate the cost of painting the walls and ceiling of
your schoolroom at f .25 a square yard.
5. Select a room in your own home; measure it. Find the
cost of decorating it as your mother would like to have it done.
Ask her to tell you what she would like to have put on the
walls ; then you make the measurements, compute the amount
of material and labor, and the cost.
Make and solve other problems in papering.
LUMBER MEASURE 219
LUMBER MEASURE
267. A piece of wood 1 ft. long, 1 ft. wide, and 1 in. thick
is a board foot (bd. ft.).
To THE Teacher. — As material for this lesson, a real board foot — a
piece of board exactly 1 ft. long, 1 ft. wide, and 1 in. thick — should be pro-
vided. Refer to it in obtaining answers to the oral questions below and
whenever pupils seem to answer wide of the mark in this subject. This is
very important.
268. Oral
1. How many inches long is a board foot ? How many-
inches wide ? How many cubic inches does a board foot
contain ?
2. How many board feet piled one upon another would make
a cubic foot ? Show with your hands how wide, long, and high
this pile would be.
3. A board 1 in. thick, 1 ft. wide, and 6 ft. long contains
how many board feet ? Draw it full size on the blackboard,
and mark off the board feet.
4. If the board in question 3 were twice as thick, how many
board feet would it contain ? How many inches thick would
it be ?
5o If it were five times as thick, how many board feet would
it contain?
6. How many board feet are there in a piece of board 1 ftc
wide, 16 ft. long, and 1 in. thick?
7. If this piece of lumber were 2 in. thick, how many board
feet would it contain ? 3 in. ? 4 in. ? 5 in. ? 6 in. ?
8. A piece of inch board 3 ft. long must be how wide to con-
tain 1 bd. ftt
220
LUMBER MEASURE
9. A cubic foot of wood could be sawed into how many
board feet if there were no waste in sawing? The number of
board feet in any piece of lumber is how many times as great
as the number of cubic feet ?
269. We may find the number of board feet in a piece of
lumber by multiplying the number of cubic feet by 12. The
rule commonly used by dealers and mechanics gives the same
result, and is stated as follows :
To find the number of hoard feet in any piece of lumber^ mul-
tiply together its three dimensions, two of them expressed in feet
and the other in inches.
Lumber that is less than 1 in. thick is counted as 1 in. thick
in measuring.
270. Oral
1. About how many feet of lumber (board feet) are there in
the top of your desk ? The teacher's desk ? One end of the
bookcase ? The cupboard door ? All
the shelves in the bookcase ?
2. How much lumber is used in
making this box with cover ? (Take
outside measurements.)
3. Estimate the amount of lum-
ber in a cubical box, including the
cover, made of |^-inch lumber, the
length of the box being three feet.
4. The door in this hayloft is
4|'x4'. Two battens across the
inside are 4' x 6'^ How many feet
of lumber are used in the door, the
lumber being 1'' thick ?
LUMBER MEASURE
221
271. Written
1. How many board feet are there in a piece of timber
18' X le^'x 3^'? (18'xi|'x 3'^= bd. ft.)
2. The floor of a tent 12 ft. by 16 ft. is made of boards
1 in. thick laid close together, a. How many feet of lumber
are used ? h. How much is it worth at $ 30 per M. (thousand
feet) ?
a. Find the number of feet of lumber in this stick. Walk
as far on the floor as the length of this piece of timber.
Show with your hands how high it is. Show how wide it is.
How many feet of lumber does it contain ?
h. What are twenty such sticks worth at f 26 per M. ?
a. Find the amount of lumber in this plank. 5. Find the
cost of ten such planks at | 30 per M.
5. The floor of your schoolroom is |^ in. thick, a. If no
allowance is made for sawing and matching, how many feet of
lumber are there in the floor? h. If \ of the lumber was
wasted in sawing and matching, the floor contains only ^ of the
lumber that was bought. How much lumber was bought ?
c. What did it cost at ^ 42 per
M.?
6. A fence like this, 6 ft. high,
extends around two sides and one
end of a rectangular garden 40
ft. by t)^ ft. Draw a diagram of the garden, a. How many feet
of boards were used ? h. How much did they cost at $ 27 per M. ?
222
LUMBER MEASURE
7. The floor of this bridge is 14 ft. by 8 ft., and made of oak
planks 3 in. thick. What
did they cost at 1 45 per M. ?
8. Is there a board fence
at the rear or side of your
school ground ? If so, find
the number of feet of boards
in it, as a part of to-morrow's lesson.
9. Find the cost of each of the following quantities of lumber :
Number of Pieces
Dimensions
Price per M
a.
21
3^^x
12'' X 18'
124
h.
10
^" X
6" X 20' .
$26
c.
75
2"x
4" X 20'
$26
d.
6
10^' X
14" X 30'
$ 32"
e.
3
11'' X
11" X 10'
$35
/.
49
l''x
5" X 16'
$30
9-
60
2''x
10" X 14'
$24
h.
72
2''x
8" X 14'
$24
i.
m
2''x
6" X 14'
$25
h
121
rx
3" X 12'
$28
h.
2
6''x
10" X 22'
$25
L .
4
6"x
10" X 16'
$25
m.
4
2''x
8" X 18'
$26
n.
8
S^x
8" X 14'
$27
0.
16
y'x
10" X 12'
$40
P-
7
2J''x
8" X 16'
$45
?•
600
rx
51" X 10'
$42
REVIEW AND PRACTICE 223
REVIEW AND PRACTICE
272. Oral
1. Read: 1,003,050 ; XCIX; MCM ; CDIV; CLI ; 2,050,205.
2. What is the greatest common divisor of 12, 28, and 56?
3. What is the smallest number that exactly contains 4, 5,
and 8 ?
4. Give five multiples of 11.
5. Name all the composite numbers smaller than 30.
6. Name the prime factors of 42.
7. What common fraction in lowest terms equals .33J?
.871? .625? .37-1? .20? .80? .60?
8. What decimal is equal to f ? J ? | ? f ?
9. Give results: .8x7; .12-J-6; 1.5x.6; 1.03x5;
I 7.01 X. 6.
10. Give results : 2000 ^ 20 ; 3.528 x 100 ; 53 -r- 1000 ;
950-50; 2.3x1000.
11. 10 is how many times ^ ?
12. What is the cost of 24 oranges, when 8 oranges cost 25
cents ?
13. What is the difference between^ of 10 and ^ oi 10?
14. If 21 yd. of cloth cost 1 7 J, what will 5 yd. cost ?
15. f of 20 are how many times 4 ?
16. J is contained in 7 how many times?
17. Give results: J-i; ^-^1 1-i; -i + J; l + l; | + 1;
A X i'
18. What is the cost of three bananas at 20 cents a dozen ?
224 REVIEW AND PRACTICE
19. What will 4 doz. steel screws cost at 15 i a gross ?
20. If 3 boys can shovel a walk in 15 minutes, how long
should it take 1 boy ?
21. When 3 eggs cost 5 cents, what is the cost per dozen ?
22. A rectangle 18 in. by 2 in. contains what fraction of a
square foot of surface?
23. \ of 25 is what part of 30?
24. Give results : 41 - f ; | of f ; 4 - If
25. A man had a sum of money. He earned \ as much and
then had |9. How much had he at first ?
26. \ inch is what part of a foot ?
27. 40 rd. are what part of a mile ?
28. 40 sq. rd. are what part of an acre ?
29. How many boys are there in a class of 45 pupils if \ of
the pupils are girls ?
30. How many cubic feet of stone are there in a stone wall
8' X 4' X 2' ?
31. Ethel cut 20 roses one morning, of which 40 % were red.
How many red roses did she cut?
32. One dozen is what per cent of one gross ?
33. A peddler sold 36 pencils at the rate of 2 for 5 cents.
What did he receive ?
34. 48^3 + 36-50 = ?
35. 28 days are called a lunar month. One week is what
per cent of a lunar month ?
36. How many days were there in February, 1906 ? In Feb-
ruary, 1493 ?
REVIEW AND PRACTICE 225
273. Written
1. See how quickly you can obtain correct answers and test
them:
a. The addends are 4,798, 6,430, 895, 49,785, 231, 8,942,
700, 98,346, 209, 98,020, 49, 816. Find the sum.
h. The product is 910,386, one factor is 4,398. Find the
other factor.
e. 7,695 and 493 are the factors that make what number ?
d, 2,983, 4,978, 9,399, and 16,897 are the parts that make
what number ?
e. 83,469 added to what number will make 103,497 ?
2. Find the least number that will exactly contain each of
these numbers: 28, 35, 20,42.
3. What is the number whose prime factors are 2, 3, 5, 7,
11 and 17?
4. How many times is the G. C. D. of 48, 36, and 72 con-
tained in their L. C. M. ?
5. Reduce to common fractions in simplest form: a. .025 ;
h. .375; c. .3125; d. .8375; e. .1275; /. .30; g. .625.
6. Reduce -^^ to a decimal, and write the answer in words.
7. The product of three numbers is l^^. Two of the num-
bers are 2^ and 1^^. Find the other number.
8. Which is larger, and how much, | of 64 or |^ of 45 ?
7-^
9. Change -^ to its simplest form.
8
10. I of 72 is if of what number ?
11. Mrs. Hill's new curtains cost $85.40. This was f as
much as her carpets cost. What was the cost of both curtains
and carpets ?
226
REVIEW AND PRACTICE
12. In a certain year 200,000 typewriting machines costing
$12,500,000 were made in the United States by 10,000 men.
a. What was the average cost of the machines ?
h. At the average rate, how many men were needed to
conduct a factory that turned out 20,000 machines in a
year ?
<?. If the labor was all but 40 % of the cost of the machines,
what was the expense for labor in making 327 machines ?
d. 1440 of these machines were shipped to Mexico. If
they were sold in Mexico at an average price of $95 apiece,
how much more did they bring than the cost of manufacture ?
e. One of the machines in the picture has 76 keys, and
the other two have each 42 keys. If each of the three factories
where these machines are made can turn out 50 complete ma-
chines in a day, how many key tops are needed to supply the
three factories for one week ?
13. Find the areas of triangles having the following dimen-
sions :
a. Base 28 rd., alt. 46 rd. c. Base 16 in., alt. 42 in.
b. Base 19 yd., alt. 23 yd. d. Base 64 ft., alt. 47 J ft.
14. Find the altitude of a triangle whose base is 18 ft. and
whose area is 72 sq. ft.
15. Find the base of a triangle whose area is 600 sq. rd. and
whose altitude is 30 rd.
REVIEW AND PRACTICE
227
16. This gable is covered with shingles
that cost $5.50 per M. If 8 shingles cover
a square foot, what did the shingles cost ?
17. What are the weekly wages of a girl
who finishes 4800 buttonholes a day, if she
receives 1^ cents per dozen garments, and
each garment contains three buttonholes ?
18. John Milton was born Dec. 9, 1608,
and died Nov. 8, 1675. What was his age at his death ?
19. The battle of Bull Run was fought July 21, 1861. How
long ago was that ?
Wheat Blockade in the Northwest
20. In two years the United States exported 154 million
bu. of wheat.
a. In how many days would 1000 threshing machines thresh
this wheat if each machine threshes 1375 bu. per day ?
h. How many grain cars would carry this wheat if a car
can carry 700 bushels ?
c. If a bushel of wheat weighs 60 lb., how many tons would
each carload weigh ?
d. Allowing 33 ft. for the length of each car, a train must
be how long to carry the entire quantity of wheat ?
228
REVIEW AND PRACTICE
Harvesting Wheat
21. 48 % of the yield of wheat in a western township in one
year amounted to 73,728 bu.
a. What was the entire yield of this township ?
h. If the average yield was 20 bu. per acre, and 50% of
the land was sown to wheat, how many acres of land were there
in the township ?
c. If the township was a rectangle 6 mi. long, how wide was it ?
22. How many days elapsed from President McKinley's sec-
ond inauguration, March 4, 1901, to his death, Sept. 14, 1901 ?
23. The distance from Covington, Ky., to New York is 996
miles. If you leave Covington at 1.30 p.m. (New York time),
May 1, and travel toward New York at an average rate of
30 mi. per hour, at what time will you arrive at New York ?
24. How many j^ards of carpet 32 in. wide must be bought
for a room 18' x 16' ?
25. Find the cost of the paper at 22 ^ a roll for the four walls
of a room that is 9 ft. 6 in. high, 21 ft. long, and 15 ft. wide,
making allowance for a baseboard 9 in. wide all around the
room, and for two doors and three windows, each 8 ft. by
7J ft. Estimate a roll of paper to cover 30 sq. ft.
REVIEW AND PRACTICE
229
26. Henry was invited to his uncle's camp in the Adirondack
Mountains for a two weeks' vacation. He had saved |20 from
his earnings during the year.
a. With a part of this money he purchased the following
articles :
1 Paragon bait rod, 12.50.
1 multiplying reel, 11.40.
1 enameled silk line, $.90.
3 doz. snelled hooks at $.35 per dozen.
1 canvas hook and tackle book, |.50.
1 willow trout basket, f 1.10.
How much did these articles cost ?
I. He bought a railroad mileage book of 500 miles for |10.
From this he paid his railroad fare for 98 miles each way.
How many mile slips were left in the book ?
c. What did Henry's fare cost ?
d. He paid 1.75 for his ride in a buckboard wagon from the
station where he left the cars to his uncle's camp, a distance of
6 miles. What was the rate per mile ?
230
REVIEW AND PRACTICE
e. He purchased of g. St. Regis Indian woman for 60 cents
a sweet-grass workbasket for his mother, and for 40 cents a
handkerchief box for his sister. His expenses, other than those
mentioned, amounted to ^t) cents. How much money did
Jlenry have when he returned home ?
/. What was the remainder of his mileage book worth, at the
same rate per mile at which he bought it ?
g. What was the entire cost of his vacation ?
A. Henry and his uncle went trout fishing eight times.
Their catches for the different trips weighed as follows ; 3 lb.
5 oz., 2 lb. 5 oz., 7 lb. 4 oz., 1 lb. 12 oz., 2 lb. 2 oz., 5 lb. 4 oz.,
10 oz., 4 lb. 8 oz. What was the total weight ?
i. If the weight of the trout averaged 7 oz. apiece, how
many did they catch ?
27. I of a field is planted with corn and \ is sown with wheat.
The corn occupies how many times as much land as the wheat?
28. What is the value of 6 acres of land when 4 acres of the
same land are worth $420?
29. If a ship has water enough to last 25 men 8 months,
how long will the water last 8 men?
INTEREST 231
INTEREST
274. When we have the use of property belonging to another,
we pay the owner for the privilege of using it. For instance,
if Mr. A lives in a house that belongs to Mr. B, he pays Mr.
B a certain sum for the use of the house. That sum is
called what ? The amount that Mr. A pays depends upon
the value of the house and the length of time for which the
rent is paid.
When we rent a horse and carriage from a liveryman, we pay
for their use. The amount which we pay depends upon the
kind of horse and carriage, the length of time we use them, the
distance we drive, etc.
When you rent a boat at the boat livery, you pay for
it according to the time you use it. Can you give other
illustrations of paying for the use of property belonging to
others ?
Sometimes a man finds it necessary to borrow money from
another. Can he return the exact pieces of money which he
borrowed ? Should he return just as much as he borrowed ?
Should he return more than he borrowed ? Why ?
The sum paid for the use of a large quantity of money should
compare how with the sum paid for the use of a smaller quan-
tity of money ?
The sum paid for the use of some money for a long time
should compare how with the sum paid for the use of the same
money for a short time ?
The price paid for the use of the same quantity of money
for the same time varies in different places.
275. Money paid for the use of money is interest.
276. Money for the use of which interest is paid is the prin-
cipal.
232 INTEREST
277. The sum of the principal and interest is the amount.
278. The sum to he paid for the use of money is always de-
termined hy taking a certain per cent of the principal.
279. The number of hundredths of the principal taken as the
interest for one year is the rate of interest. For instance, if *„
sum of money is borrowed and 6% of that sum is the interest
for one year, the rate of interest is 6 % .
280. ^-Phe rate of interest which is fixed hy law is called
the legal rate. In a majority of the states the legal rate is
6%. In some states it is greater than 6 %, and in some states
less.
A lower rate than the legal rate is always allowed by law if
the debtor and creditor so agree. In a few states a higher rate
than the legal rate is allowed if the debtor g,nd creditor so
agree ; but in most states a higher rate than the legal rate is
forbidden by law. What is the legal rate where you live ?
281. Interest at a higher rate than that permitted hy law is
usury.
282. Oral
1. Mr. Smith borrowed from Mr. Arnold $100 for 1 yr.
*At the end of the year Mr. Smith repaid the money which he
had borrowed and also paid Mr. Arnold 6 % interest. How
much was the interest ? What was the principal ? How much
did Mr. Arnold receive in all ? What is this sum called ?
Who was the debtor ? Who was the creditor ?
2. What is the interest on 1500 at 6% fori yr.? On $ 800 ?
On I 900? On $300? On $1000? On $250?
3. What is the interest on $500 for 1 yr. at 5% 7 At 10% ?
At7%? At4%? At3%? At8%?
INTEREST 233
4. What is the interest on ilOOO at 5% for 1 yr. ? For
2 yr. ? For 3 yr. ? For 8 yr. ? For 10 yr. ?
5. What is the interest on 1 100 for 2 yr. at 6fo ? At 4 % ?
At3%? At9%? At8%?
6. Six months are what part of a year ? 3 mo. ? 4 mo. ?
8 mo. ? 9 mo. ? 10 mo. ? 1 mo. ?
7. What is the interest on $ 600 for 1 yr. at 6 % ? For 6
months? For 3 mo. ? For 4 mo. ? For 8 mo.? For 9 mo. ?
For 1 mo. ? For 11 mo. ?
283. Written
1. What is the interest on $2000 for 3 yr. at 7 % ?
x-tL-x|-=|420 Ans,
1 m 1
How do we find the interest for 1 yr. ? For 3 yr. ?
2 What must be paid for the use of $700 for 4 mo. at 6%
per year ?
2
3. Find the interest on $350 for 3 yr. 6 mo. at 4%. 3 yr.
6 mo. = how many years ?
7 ?
iMx-i-xI-$ Ans
4. JPmi ^^e interest on $800 a^ 6 % :
a. For 7 yr. ' ^. For 21 yr. <?. For 9 mo.
234 INTEREST
d. For 2 yr. 6 mo. /. For 5 mo. h. For 2 yr. 10 mo.
e. For 1 yr. 8 mo. g. For 1 yr. 5 mo. ^. For 3 yr. 7 mo.
Note. — In final results, a part of a cent smaller than | cent is dropped ; a
fraction equal to, or greater than, \ cent is called one cent.
5. Find the interest on il600 at b% :
a. For 5 yr. c. For 2| yr. e. For 1 yr. 3 mo.
h. For 4 J yr. d. For 3 yr. 8 mo. /. For 2 yr. 3 mo.
6. Find the interest on |400 for 3 yr. at 31 %.
7. TF^a^ ^s fAe interest:
a. On $200 for 4 yr. at 5 % ?
5. On $1400 for 1 yr. at ^ % ?
c. On 11800 for 6 mo. at 6f % ?
c?. On 11500 for 3 yr. 4 mo. at 4 % ?
e. On 11200 for 2 yr. 8 mo. at 4| % ?
284. It is the custom of business men in computing interest
to consider one month as 30 days, and one year as 360 days.
Allowing 360 days for a year, 100 days are what part of a
y^ar ? 200 da.? 150 da.? 10 da.? 75 da.? 90 da.?
1. What is the interest on $720 at 6 % for 100 da.?
2
e.
For 90 da.
/.
For 45 da.
{.
For 33 da.
J'.
For 93 da.
k.
For m da.
I.
For 345 da
INTEREST 285
2. Find the interest on $720 at Q%:
a. For 200 da. c. For 10 da.
h. For 150 da. d. For 75 da.
3. Compute the interest on $1800 at i%:
a. For 200 da. e. For 45 da.
b. For 150 da. /. For 75 da.
c. For 90 da. g. For 110 da.
d. For 30 da. h. For 240 da.
4. Compute the interest on $75 at 4% for 2 mo. 20 da.
2 mo. 20 da. = how many days ? How many years ?
^ 2
3
5. Find the interest on:
a. $280 for 3 mo. 10 da. at 3 %.
h. $375 for 7 mo. 9 da. at 8%.
c. $500 for 11 mo. 6 da. at 9 %.
d. $450 for 5|mo. at 8%.
e. $300 for 4 mo. 12 da. at 7|%.
6. What is the interest on $ 450 at 6 % for 1 yr. 1 mo. 10
da.?
1 yr. 1 mo. 10 da. = 360 da. + 30 da. + 10 da., or 400 da.
5 ^
236 INTEREST
7. Compute the interest on:
a, 1300 for 1 yr. 5 mo. 12 da. at 5%.
h. 1900 for 1 yr. 7 mo. 11 da. at 4 %.
c. $360 for 1 yr. 2 mo. 7 da. at 7 %.
d, $840 for 2 yr. 15 da. at 6 %.
8. Compute the interest on $660 for 2 yr. 8 mo. at 7|^5^,
71 15
'^2^' "100 "200
3.30 5 8
xlixll = $132 AnB,
m n
m i
Note. — We cancel 100, then divide 330 by 100 by pointing ofE two deci-
mal places.
9. Find the interest on 175.25 at 8% for 20 da.
3.01 ^
9
10. Compute the interest on :
a. $485.50 for 1 yr. 3 mo. at 4 %.
h. $125.50 for 1 yr. 4 mo. at 41 %.
c, $ 240 for 8 mo. 15 da. at 5^ %.
d, $540 for 1 yr. 4 mo. 10 da. at 5 %.
From the foregoing examples we observe that the interest on
any sum of money, at any rate, for any time, is always the prod-
uct of three factors. What are they ?
In what denomination is the time expressed before multiply-
ing to find the interest?
INTEREST 237
285. Written
In examples 1-24 compute the interest.
Principal
Rate
Time
1.
14320
Hfo
2 yr. 6 mo.
2.
1720
5%
2 yr. 8 mo.
11 da.
3.
$1081.08
7%
6 mo.
20 da.
4.
$5000
Hfo
2 yr. 8 mo.
5.
1901.80
8%
3yr.
24 da.
6.
$1236.48
H%
2 yr. 2 mo.
2 da.
7.
1620.40
5|%
2 yr. 3 mo.
10 da.
8.
I12T5.30
9%
5 yr. 6 mo.
15 da.^
9.
11500
61%
2 yr. 9 mo.
9 da.
10.
$270.27
6%
2yr.
27 da.
11.
$396
10%
1 yr. 1 mo.
9 da.
12.
$444
4%
5 yr. 6 mo.
13.
$84.50
7%
2 yr. 5 mo.
12 da.
14.
$16.75
7%
7 mo.
17 da.
15.
$336
5%
15 da.
16.
$300.50
3%
1 yr. 2 mo.
15 da.
17.
$42.20
4i%
lyr.
16 da.
18.
$51.17
4%
9 mo.
29 da.
19.
$35.50
7%
3 yr. 5 mo.
20 da.
20.
$691.04
5%
1 mo.
3 da.
21.
$640.50
10%
10 mo.
26 da.
22.
$105.10
12%
48 da.
23.
$92.96
7%
4 mo.
3 da.
24.
$31.40
7%
273 da.
238 INTEREST
25. Mr. Ward borrowed 1 10,000 of Mr. Beach at 5 % and
lent it to Mr. Waite at 6 %. Mr. Waite kept the money 2 yr.
6 mo. 18 da. He then paid Mr. Ward, and Mr. Ward paid
Mr. Beach. What was Mr. Ward's gain ?
26. What must be paid for the use of $ 160 for 11 yr. 11
mo. 11 da. at 7i
70
27. Required, the interest on $ 2000 for 3 yr. 7 mo. 12 da.
at 3 %.
28. How much will f 358.50 earn in 1 yr. 8 mo. 6 da. when
put on interest at 7 % ?
29. How much interest will $475.50 earn in 5 yr. 9 mo. 24
da. at 4 % ?
30. How much interest will $840.50 yield in 10 mo. 15 da.
at 41 % ?
31. At 5 % interest, what will $ 75 gain in 11 mo. 10 da.?
286. Oral
1. The sum of the principal and interest is called what ?
2. Mr. Williams borrowed $ 100 of Mrs. Johnson, paying 6 %
interest. What was the principal ? The interest ? The amount ?
3. Name the principal, interest, and amount when $ 100 is
borrowed for 3 yr. at 6% interest.
4. Find the amount of $ 50 for IJ yr. at 6%.
5. Find the amount of 1 1 for 7 yr. at 5%.
6. What is the amount of $ 100 for 6 mo. at 5% ?
7. If I borrow $ 200 at 4% interest, how much do I pay in
three years ?
8. Frank's uncle gave him a New Year's present of $ 100,
which he put in a bank that paid 4% interest. How much
money did Frank have in the bank at the end of three months ?
INTEREST 239
9. What is the amount of $ 300 for four years at 5% ?
10. A man bought a span of horses for $ 150 apiece and a
wagon for $ 50, agreeing to pay for them in one year with inter-
est at 10%. How much did he have to pay ?
11. Mr. B owes $ 1000 on his house and pays the interest
every six months at the rate of 5% per year. How much inter-
est does he pay each time ?
12. I owe a debt of $ 40, due in one year and six months. If
I pay interest at the rate of 10% per year, how much will the
debt amount to when it is due ?
13. What is the interest on $ 200 for five years when the rate
of interest is 3J% ?
14. What is the amount of 1 400 for two years at 3^% ?
15. The amount of $400 for a certain time at 7 % interest
is 1428. Can you find the time ?
16. The amount of 1 100 for 2 years is f 110. Can you find
the rate of interest ?
17. The amount of a sum of money for one year at 7% inter-
est is 1214. Can you find the principal ?
18. The interest of 1 50 for 2 years is 1 6. Can you find the
rate of interest ?
19. The amount of $80 for 1^ years at 5% per year is how
much?
20. When the principal, rate, and time are given, how may
the amount be found ?
21. The principal added to the interest gives what ?
22. The difference between the principal and amount is what ?
The difference between the interest and amount ?
240 INTEREST
287. Written
In examples 1~1S find the amount:
Principal
Eatb
Time
1.
1450
6%
1 JT,
2 mo.
20 da.
2.
$150.50
8%
20 da.
3.
1330
71%
2yr.
8 mo.
4.
17500
4%
2 mo.
20 da
5.
11250
4|%
2yr.
8 mo.
6.
1901.80
8%
lyr.
6 mo.
12 da.
7.
1620.40
61%
lyr.
1 mo.
20 da.
8.
1444.60
3%
lyr.
6 mo.
9.
$42.25
7%
lyr.
5 mo.
12 da.
10.
I960
' 6%
11 mo.
20 da.
U.
1173
6%
8 mo.
16 da.
12.
$1500
8%
2yr.
5 mo.
13 da.
13.
$90
6f%
lyr.
27 da.
14.
What will $1000 amount to in
1 yr. 7
mo. if
put at
intere^
3t at 4%?
15. What will $8450 amount to in 90 da. at 10 % ?
16. A man borrowed $416 at 5%. If nothing was paid on
the debt for 1 yr. 16 da., how much did the man then owe ?
17. If you should borrow $150.25 at the legal rate of inter-
est where you live, how much would you owe 11 mo. 15 dao
after borrowing the money?
18. If a man borrows $146.75 to-day at the legal rate of
interest where you live, how much will he owe 9 mo. 15 da.
from to-day?
19. P'ind the amount of $750.25 for 1 yr. 27 da. at 9%.
INTEREST . 241
PROBLEMS IN WHICH THE TIME MUST BE COMPUTED
BEFORE INTEREST CAN BE FOUND
288. Written
1. What is the interest on |144 from June 12, 1901, to
Jan. 2, 1903, at4%?
1903 yr. 1 mo. 2 da. ^^
1901 6 1 2 ^"^m^m^^^'^^ ^'''*
1 6 20 Difference in Thne ^ ^
2. Find the interest on :
a. 1500 at 6 % from May 7, 1902, to Sept. 7, 1904
h. 172 at 41 % from Apr. 1, 1904, to Apr. 16, 1907.
c. $60 at 51% from Sept. 30, 1899, to June 15, 1901.
d. 1240.60 at 10% from Oct. 25, 1904, to Dec. 10, 1907.
e. $360 at 9 % from Nov. 30, 1903, to May 25, 1905.
/. $1000 at 3|% from Jan. 21, 1904, to June 11, 1906.
g. $48.48 at 8 % from Feb. 25, 1906, to Jan. 5, 1908.
h. $99 at 6 % from Sept. 21, 1904, to Jan. 1, 1906.
i. $36.36 at 10 % from Feb. 2, 1903, to Oct. 22,1905.
y. $900 at 31 % from Dec. 30, 1904, to Jan. 15,1905.
h. $45.90 at 6 % from Jan. 9, 1900, to Aug. 5, 1903.
I. $576 at 4 % from July 4, 1902, to Feb. 3, 1905.
m. $960.84 at 3 % from Apr. 3, 1904, to Sept. 6, 1907.
n. $162.72 at 5 % from May 12, 1900, to May 4, 1905.
242 INTEREST.
INTEREST FOR SHORT PERIODS
289. When money is on interest for less than a year, it is
customary to compute the time in days.
What is the interest on $1575.25 from Jan. 9, 1904, to March
15, 1904, at 3% ?
r 22 da. left in Jan.
The money is on interest for \ 29 da. in Feb.
[ 15 da. in March
66 da. Term of Interest
787.625 11
Wl^'M xjgx ^ = 18.663875, or $8.66 Am,
10
Observe that by keeping (not cancelling) the factors 100 and 10 below the
line we may multiply together the factors above the line and then divide
by 100 and 10 by pointing off three more decimal places in the product.
290. Written
1. Compute the interest on 1721.44 at 4% from April 3 to
July 6.
2. Find the interest on $9000 at 5% from March 4, 1898, to
April 3, 1898.
3. A man borrowed $576.72, May 12, 1896. How much
did he owe Aug. 10, 1896, computing the interest at 5 % ?
4. A man borrowed $3500, Jan. 5, 1902, and repaid the
money with interest at 6% on April 1, 1902. How much did
he pay ? "
5. Find the amount of $250 borrowed June 1, 1907, and
paid Aug. 21, 1907, with interest at 6%.
INTEREST 248
6. Find the interest on :
a. 1600 from Aug. 1 to Aug. 21, 1907, at 51%.
h. 1120 from May 6 to May 31, 1905, at 5%.
e. 1219 at 7 % from June 12 to July 30, 1904.
d. 1638 at 6% from Dec. 15, 1906, to Feb. 8, 1907.
e. $1000 at 7| % from Nov. 18, 1907, to Feb. 17, 1908.
/. 1248.50 at 9% from Aug. 15 to Aug. 31, 1901.
^. 1631.78 at 10 % from Jan. 14 to Sept. 8, 1896.
h. 148.70 at 10% from May 3 to Oct. 7, 1899.
i, $246.42 at 8% from Sept. 30, 1907, to Jan. 1, 1908.
y. 1401.28 at 4| % from July 8, 1906, to Jan. 1, 1907.
k. $283.49 at 6% from Dec. 31, 1902, to March 30, 1903.
I $800 at 5% from Dec. 1, 1903, to April 1, 1904.
m. $12,000 at 61% from Jan. 30 to June 16, 1896.
n. $200,000 at 3f % from Aug. 5 to Aug. 23, 1905.
0. $150,000 at 41 % from March 10 fo July 18, 1907.
p. $76.47 at 9% from April 1 to April 29, 1901.
q. $84.13 at 7 % from Feb. 20 to Aug. 1, 1907.
r. $43,475 at 41% from May 7, 1899, to Jan. 14, 1900.
8. $4376.40 at 6% from May 6, 1900, to March 17, 1901.
7. Find the amount of:
a. $400 at 7 % from Aug. 31 to Dec. 1, 1904.
h. $308.12 at 6 % from Feb. 1 to March 13, 1906.
c. $242.14 at 7 % from Jan. 31 to April 3, 1904.
d. $800,000 from June 16, 1900, to Jan. 1, 1901, at 31%.
e. $140,000 from April 14 to May 19, 1904, at 4| %.
/. $131.13 at 8% from Oct. 31, 1905, to Feb. 27, 1906.
g. $434.25 at 5i% from Feb. 21 to July 3, 1907.
244 REVIEW AND PRACTICE
291. Oral REVIEW AND PRACTICE
1. Expressing numbers by means of figures is called what ?
2. Name the first six places in integers.
3. Define an integer.
4. ReadMCMIX.
5. Numerate 20756.3010.
6. The numbers added are called what?
7. Find the value of 8 + 3 x 2 - 21 -5- 7.
8. Giive the sums rapidly :
18 + 8; 27 + 12; 26 + 19; 49 + 48; 53 + 47.
9. How may we test our work in subtraction?
10. How may we test our work in multiplication ?
11. How may we test our work in division?
12. Define division.
13. Which terms in division are factors?
14. Which terms in multiplication are factors?
15. Grive results rapidly :
43-12; 56-17; 93-56; 85-59; 132-94.
16. Multiply hy 100 :
48; 4264; 408; 37.9; 84.729; .0079. ^
17. Multiply 48 by 25.
18. Divide hy 1000 :
2645.3; 793; 4835; 3.9; 9638.2; 7; 82.
19. When 20 pickles cost 13 cents, how much should be paid
for 80 pickles?
20. ^At the rate of 7 for 3 cents, how many screw hooks can
be bought for 15 cents ?
REVIEW AND PRACTICE 245
21. How much a dozen is paid for bananas when 36 bananas
cost 60 cents ?
22. Name the odd numbers between 40 and 50.
23. Name the prime numbers from 1 to 47.
24. Name the prime factors of 60.
25. What number will divide every even number ?
26. How may we tell whether a number is prime or not ?
27. What is cancellation ?
28. The smallest number that exactly contains each of two
or more numbers is called what ?
29. What is the greatest number that will exactly divide 45
and 60 ?
30. Find the L. C. M. and G. C. D. of 36 and 8.
31. A fraction is always an expression of what operation ?
32. Which term of. a fraction is the dividend?
33. Which term of a fraction is the divisor ?
34. Which is greater, J or J? | or | ? f or f ? ^3_ or ^ ?
35. The " Lincoln Stars " won 5 games of baseball, with the
following scores : 7, 8, 2, 9, 4. What was their average score ?
36. Name five aliquot parts of one dollar.
37. How many packages of cereal at 1.12^ each can be
bought for 13?
38. What per cent of anything is J of it ? ^ of it ? J ? | ? -J ?
f?|?|?f?
39. If the interest on $25 for a certain time is $3, what is
the interest on $200 for the same time, at the same rate ?
40. 62 J % of f 16 is how much money ?
41. Draw a line one yard long without a measure. Measure
and correct it.
246 REVIEW AND PRACTICE
42. Draw a square foot. Measure and correct it.
43. Draw a square 8 inches on a side. Measure and cor-
rect it.
44. How many days are there in the summer months ?
45. Each spoke in a certain wheel makes an angle of 60°
with the spoke next to it. How many spokes are there in the
wheel ?
46. When a stationer buys tablets at the rate of $6 a gross,
how much does he pay for each dozen ? When he buys them
at ^ 9 a gross, what does he pay for each dozen ?
47. I bought a ream of note paper and used 12 quires of it.
How many sheets were left ?
48. One circumference is equal to how many arcs of ten
degrees each ?
49. From the Fourth of July to Christmas is how many
days ?
50. What is the cost of 2500 shingles at $6 per M.?
51. What is the area of a triangle whose base is 20 inches
and altitude is 1 foot ?
52. What is the altitude of a triangle whose base is 12 feet
and whose area is 48 square feet ?
53. How many feet of lumber are there in a board which is
14 ft. long, 6 in. wide, and | in. thick ?
54. What is the amount of 1300 for 1^ yr. at 3 % ?
55. The interest on a sum of money for ten months is $ 35.
What is the interest on the same sum, at the same rate, for two
months ?
56. 1^ of a flag pole 126 ft. high was broken off by the wind.
How many feet high was the piece left standing ?
REVIEW AND PRACTICE
247
292. Written
1. Express in Roman numerals the number of the year in
which you were born.
2. Write in words 500200.00202.
I. Add, and test
your work :
a
b
c
d
358
26
98034
12843
47
544
576
798
968
37
934
6347
7684
829
86
999
9235
5444
715
3857
1386
308
8397
92124
428
9176
869
28315
•79
283
476
76543
9830
706
59
89412
49
2493
987
6347
15730
819
43
48009
2132
6478
3754
90384
4. The sum of two numbers is 80,305. One of the addends
is 79,496. Find the other.
5. One rectangular field is 35 rods by 54 rods ; another is
24 rods by 51 rods. How many more feet of fence are required
to inclose one of them than to inclose the other ?
6. Mr. Walch sold two horses for $275 each, and a carriage
for i295. He then bought an automobile costing four and
three fourths times as much as the horses and carriage brought.
He received how much less than he paid out? Indicate the
work, and then find the answer.
7. Make and solve a problem that requires the following
operations: 16 x 8.45+ 84 x $.70-132.18.
248 REVIEW AND PRACTICE
8. 5.375-^(1.55 -.3) + 142.34x7 = ?
9. Find the prime factors of 589.
10. Find the smallest number that will exactly contain each
of the numbers 105, 56, 84, 220.
11. Find the largest number that will exactly divide 260,
490, 1078, 364.
12. One factor of f f is ||. Find the other.
13. Divide 3^ of f by If of If .
14. Divide I of f by l| of if.
15. Find the value of f X 4| divided by ^ of 8f
16. Change i^rYto to a fraction whose terms are prime to
each other.
17. How much greater is | of 41 than 51^j - 42i|?
18. What decimal is exactly equal to ||^?
19. Mr. Tripp kept a record of the temperature indicated by
his thermometer at noon every day for a week as follows : 76°,
80°, 78°, 83°, 87°, 90°, 89°. What was the average noon
temperature for the week?
20. Make out a bill of four items, using prices found in the
newspaper. Receipt the bill as if you were the creditor's
clerk.
21. Multiply 43.761 by 5.8|.
22. Divide 74.96f by .6^.
23. Multiply 3867 by 25, in the shortest way.
24. Divide 3976 by 25, in the shortest way.
25. A contractor agreed to excavate a cellar 180 ft. by 45 ft.
and 11 ft. deep. What fraction of the work was left undone
when he had removed 175 cu. yd. of earth ?
26. What common fraction is the same as 11| % ?
REVIEW AND PRACTICE
249
27. 15 % of the weight of a certain piece of cloth, was wool.
If the piece contained 25J lb. of cotton, how many pounds did
the piece weigh ?
28. In excavating the side of a hill .for a railroad, it was nec-
essary to remove 8500 cubic yards of clay and rock. If 19| %
of the material removed was rock, how many cubic feet of clay
were removed ?
29. a. A brick wall 41 ft. 6 in. long, 1 ft. 6 in. wide, and
8 ft. high contains how many bricks, if 22 bricks will make
1 cu. ft. of the wall ?
h. They cost how much at 19.50 per M. ?
30. 16,200 bricks were required in building a wall 2 ft. thick
and 9 ft. high. If it required 24 bricks to make a cubic foot
of wall, how long was the wall ?
31. a. A 20-acre vineyard contains 540 vines to the acre.
All the vines in the vineyard are set in 90 equal rows. How
many vines are there in each row ?
Vineyard
250
REVIEW AND PRACTICE
h. The average yield is 1000 baskets of grapes per acre. An
empty basket weighs IJ lb, A filled basket weighs 8 lb. How
many pounds of grapes are raised on an acre ?
c. How many tons are raised in the whole vineyard ?
d. What are they worth at 824 per ton ?
e. If it costs 1 cent per basket to pick and pack the
grapes and y*^ of a cent per basket for cartage, what must
be paid for picking, packing, and carting an acre's yield of
grapes ?
32. a. Three cents a tray are paid for picking wine grapes.
What are the weekly wages of a picker who picks 50 trays of
grapes a day ?
5. If 64 filled trays weigh a ton, and the yield of an acre is
3J tons, including the weight of the trays, what is the cost of
picking three acres of wine grapes ?
c. The weight of an empty tray is 5 lb. What is the weight
of the grapes that fill one tray ?
d. How many pounds of grapes
will fill 64 trays ?
e. At 5 cents apiece, what is
the cost of the trays for an acre
of grapes? (See 5.)
/. At one cent a pound, what
is the value of the grapes from an
acre of ground ?
33. a. A grape grower in Cali-
fornia has 40 acres of wine grapes.
It costs $12 an acre to train and
cultivate his vines and 11.35 per
ton for picking. If the yield is
6 tons to the acre, and he sells the entire crop for $15 a ton,
what is bis net profit per acre ?
REVIEW AND PRACTICE 251
h. Each ton of these grapes will make 150 gallons of wine.
Allowing 32 gallons for a barrel, how many barrels of wine can
be made from the entire 40-acre vineyard ?
c. How many pounds of grapes are used in making one gal-
lon of wine ?
34. The grapevines are supported by wires fastened to posts.
If it requires 609 lb. of wire per acre, costing 842 a ton, what
is the cost of the wire for a 15-acre vineyard ?
35. What will it cost to carpet a room 20 ft. by 23 ft. with
carpet 27 in. wide, costing 11.75 a yard, with 8^ per yard added
for making and laying, running the strips the longer way of the
room, and making no allowance for waste in matchipg the figure ?
36. a. What will it cost to lath and plaster the walls of a
room 80 ft. by 30 ft. and 14 ft. high at 40 cents a square yard,
making full allowance for 20 windows, each 3|^ ft. by 7 ft., and
4 doors, each 3 ft. 3 in. by 8 ft.?
h. The floor of this room is supported by 120 joists, each
16 ft. long, 12 in. wide, and 3 in. thick. What did they cost
at $28 per M. board feet.?
37. Find the sum of 35° 46' 52'' and 72° 13' 38".
38. A game of baseball began at 25 minutes 38 seconds past
2 P.M., and closed at 7 minutes 15 seconds past 4 p.m. How
long did the game last ?
39. The widths of six city lots, lying side by side, are as follows :
bb ft., 40 ft. 6 in., 72 ft. 9 in., 38 ft. 10 in., 80 ft., and ^ ft.
8 in. Find in feet and inches the entire width of all the lots.
40. What is the amount of 1 350 at interest from March 15,
to July 11, 1907, the rate of interest being 5J % ?
41. Find in the shortest way the interest on a sum of money
for 30 days, when the interest on the same sum, at the same
rate, for 120 days, is 137.16.
TABLES
LIQUID MEASURE
4 gills (gi.) = I pint (pt.).
2 pints = 1 quart (qt.).
4 quarts = 1 gallon (gal.).
DRY MEASURE
2 pints (pt.) = 1 quart (qt.) .
8 quarts = 1 peck (pk.).
4 pecks = 1 bushel (bu.).
AVOIRDUPOIS WEIGHT
16 ounces (oz.) = 1 pound (lb.).
2000 pounds = 1 ton (T.).
2240 pounds = 1 long ton.
100 pounds = 1 hundredweight (cwt.),
LINEAR MEASURE
12 inches (in.) = 1 foot (ft.).
3 feet = 1 yard (yd.).
5i yards or 16^ feet = 1 rod (rd.).
320 rods = 1 mile (mi.).
SURFACE MEASURE
144 square inches (sq. in.) = 1 square foot (sq. ft.).
9 square feet = 1 square yard (sq. yd.).
30| square yards = 1 square rod (sq. rd.).
160 square rods = 1 acre (A.).
640 acres = 1 square mile (sq. mi.).
262
TABLES 263
VOLUME MEASURE
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.).
27 cubic feet = 1 cubic yard (cu. yd.).
TIME
60 seconds (sec.) = 1 minute (nain.).
60 minutes = 1 hour (hr.).
24 hours = 1 day (da.).
7 days = 1 week (wk.).
365 days = 1 common year (yr.),
366 days = 1 leap year.
COUNTING
12 =1 dozen (doz.).
12 dozen = 1 gross.
20 =1 score.
PAPER MEASURE
24 sheets = 1 quire.
20 quires = 1 ream.
ARC AND ANGLE MEASURE
60 seconds (") = 1 minute (').
60 minutes = 1 degree (°).
An arc of 360° = 1 circumference.
UNITED STATES MONEY
10 mills = 1 cent.
10 cents = 1 dime.
10 dimes = 1 dollar.
TROY WEIGHT
24 grains (gr.) =1 pennyweight (pwt.).
20 pennyweights = 1 ounce (oz.).
12 ounces = 1 pound (lb.).
254 TABLES
APOTHECARIES » WEIGHT
20 grains (gr.) = 1 scruple (sc. or 3).
3 scruples = 1 dram (dr. or 3).
8 drams = 1 ounce (oz. or 3 ).
EQUIVALENTS
1 gallon = 231 cubic inches.
1 bushel = 2150.42 cubic inches.
1 quart, dry measure = 1| quarts, liquid measure (nearly),
1 lb. Avoirdupois = 7000 gr.
1 oz. Avoirdupois = 437.5 gr.
1 lb. Troy or Apoth. = 5760 gr.
1 oz. Troy or Apoth. = 480 gr.
1 lb. Avoirdupois = lyV? lb. Troy or Apoth.
1 oz. Avoirdupois = ^|f oz. Troy or Apoth.
THE MULTIPLICATION TABLE
255
2x 1= 2
3x 1= 3
4x 1= 4
5x 1= 5
2x 2= 4
3x 2= 6
4x 2= 8
5x 2 = 10
2x 3= 6
3x3=9
4x 3 = 12
5x 3 = 15
2x 4= 8
3x 4 = 12
4 X 4 = 16
5x 4 = 20
2x 5 = 10
3x 5 = 15
4 X 5 = 20
5x 5 = 25
2x 6 = 12
3x 6 = 18
4x 6 = 24
5 X 6 = 30
2x 7 = 14
3 x 7 = 21
4x 7 = 28
5x 7 = 35
2x 8 = 16
3x 8 = 24
4 X 8 = 32
5x 8 = 40
2 x 9 = 18
3 X 9 = 27
4x 9 = 36
5x 9 = 45
2 X 10 = 20
3 X 10 = 30
4 X 10 = 40
5 X 10 = 50
2 X 11 = 22
3 X 11 = 33
4 x 11 = 44
5 X 11 = 55
2 X 12 = 24
3 X 12 = 36
4 X 12 = 48
5 X 12 = 60
6X 1= 6
7x 1= 7
8x 1= 8
9x 1= 9
6x 2 = 12
7X 2 = 14
8x 2 = 16
9x 2= 18
6x 3 = 18
7x 3 = 21
8x 3 = 24
9x 3= 27
6x 4 = 24
7x 4 = 28
8x 4 = 32
9x 4= 36
6 X 5 = 30
7x 5 = 35
8x 5 = 40
9x 5= 45
6 X 6 = 36
7x 6 = 42
8x 6 = 48
9x 6= 54
6x 7 = 42
7 X 7 = 49
8x 7 = 56
9x 7= 63
6x 8 = 48
7x 8 = 56
8x 8 = 64
9x 8= 72
6 X 9 = 54
7x 9 = 63
8x 9 = 72
9x 9= 81
6 X 10 = 60
7 X 10 = 70
8 X 10 = 80
9 X 10 = 90
6 X 11 = 66
7 X 11 = 77
8 X 11 = 88
9x11= 99
6 X 12 = 72
7 X 12 = 84
8 X 12 = 96
9 X 12 = 108
10 X 1 = 10
11 X 1= 11
12 X 1= 12
KOMAN
10 X 2= 20
11 X 2 = 22
12 X 2= 24
Numerals
10 X 3= 30
11 X 3= 33
12 X 3 = 36
I =1
10 X 4= 40
11 X 4 = 44
12 X 4= 48
10 X 5= 50
11 X 5 = 55 ■
12 X 5 = 60
V=5
10 X 6 = 60
11 X 6 = 66
12 X 6 = 72
X=10
10 X 7= 70
11 X 7 = 77
12 X 7= 84
L =50
10 X 8= 80
11 X 8= 88
12 X 8 = 96
C =100
10 X 9= 90
11 X 9 = 99
12 X 9 = 108
D=500
10 X 10 = 100
11 X 10 = 110
12 X 10 = 120
10 X 11 = 110
11 X 11 = 121
12 V 11 = 132
M = 1000
10 X 12 = 120
11 X 12 = 132
12 X 12 = 144
M = 1,000,000
INDEX
Accounts and bills, 117-121.
Addend, 7.
Addition, 7.
of compound numbers, 190.
of decimals, 95.
of fractions and mixed numbers, 61-62.
of integers, 7-10.
terms of, 7.
Aliquot parts, 86-87.
Altitude, 205.
Angles, 177-180.
acute, 179.
around a point, 178.
obtuse, 179.
right, 179.
sides of, 178.
Apothecaries' weight, 182.
Arc and angle measure, 177-180.
Areas of parallelograms, 205-206.
Areas of triangles, 207-208,
Articles sold by the 100, 1000, and cwt.,
203.
Averages, 79, 80.
Avoirdupois weight, 168.
Balance of an account, 118.
Bale, 176.
Barrel, 166.
Base of a figure, 204.
Bills, 117-121.
Board foot, 219.
Building walls, 211.
Bundle, 176.
Cancellation, 39.
Century, 174.
Circle, 180.
Circumference, 180.
Coins, 181.
Compound number, 166.
Contents or volume, 122.
Cord, 209-210.
Counting, 175.
Credit, 118.
Creditor, 118.
Cube, 121.
Cubic foot, 122.
Cubic inch, 121.
pattern of, 123.
Cubic inches in a gallon, 126, 167.
Cubic yard, 122.
Days in a month, week, and year, 174.
Debtor, 118.
Decade, 174.
Decimal point, 89.
Decimals, 88-106,
Degrees, 177-180.
Denominate numbers, 166-193.
Denomination, 166.
Denominator, 51.
common, 56.
least common, 56.
Dimensions of a solid, 121.
Dividend, 21, 51.
Division, 21.
by multiples of ten, 97.
by twenty-five, 116.
fraction an expression of, 51.
of decimals, 100.
of fractions, 75.
of integers, 21-24.
terms of, 21.
Exact differences between dates, 193.
Factors, 17, 21, 36.
integral, 36.
prime, 36.
Figures, 3.
266
INDEX
257
Figures, significant, 3.
Floor covering, 212-215.
Fluid ounce, 183.
Fractions, 51.
at the end of a decimal, 106.
clearing divisor of, 157.
common, 91.
complex, 77.
compound, 68.
improper, 53.
in the multiplicand, 156.
proper, 53.
simple, 77.
terms of, 51.
value of, 51.
which can be reduced to exact deci-
mals, 105.
Greatest common divisor, 44-45.
Hogshead, 166.
Ideas of proportion, 34, 35, 81.
Indicated work, 30, 31.
Integer, 36.
Integral factor, 36.
Interest, 231-243.
Least common multiple, 42-44.
Legal rate of interest, 232.
Linear measure, 169-170.
Lumber measure, 219-222.
Measurements, 204-222.
Minuend, 11.
Mixed decimal, 91.
Multiple, 36.
common, 42.
least common, 42.
Multiplicand, 17
Multiplication, 17.
by multiples of ten, 97.
by twenty-five, 116.
of compound numbers, 194.
of decimals, 99.
of fractions, 68-73.
of fractions illustrated graphically,
70-71.
of integers, 17-20.
terms of, 17.
Multiplication and division combined, 67.
Multiplier, 17.
Naught, 3.
Notation, 3.
Arabic, 3, 4.
Roman, 3, 6.
Number, 3.
Numbers prime to each other, 44.
Numeration, 5.
Numerator, 51.
Of between fractions, 68.
Paper measure, 176.
Parallel lines, 204.
Paral'elogram, 204.
Parenthesis, -30.
Parties to an account, 118.
Per cent, 140.
Percentage, 140-145, 151-156.
Percents equivalent to common fractions,
150.
Perimeter, 10.
Periods, 4.
Perpendicular lines, 204.
Places, 4.
Plas ring, 216.
Power, 90.
Principal, 231.
Principles
of Arabic notation, 32.
of Roman notation, 6.
Product, 17.
Products and factors, 133, 137-139.
Quadrilateral, 204.
Quick test, 67, 129.
Quotient, 21.
Rate of interest, 232.
Receipt of bill, 119.
Rectangle, 204.
Rectangular prism, 121.
Reduction, 51.
ascending, 183, 187.
descending, 183, 184.
of common fractions and mixed num-
bers to decimals, 104.
of decimals to common fractions or
mixed numbers, 103.
258
INDEX
Reduction
of denominate numbers, 18^189.
of fractions to integers or mixed num-
bers, 53.
of fractions to least common denomi-
nator, 57.
of fractions to lowest terms, 52.
Remainder, 11, 21, 23.
Review and practice, 13-16, 25-29, 40-41,
58-60, 74, 82-85, 107-115, 129-132,
146-149, 158-165, 195-202, 223-230,
244-251.
Review of fractions studied in primary
arithmetic, 49-50,
Review of integers, 46-48.
Review of primary arithmetic, 17-18.
Rule
for finding the number of board feet in
a piece of lumber, 220.
for finding whether a number is prime
or composite, 37.
for placing the decimal point, 100.
Separatrix, 4.
Simplest form, 61.
Solid, 121.
Special cases in multiplication and divi-
sion, 32-33.
Statements and questions of relation, 134-
186.
Subtraction, 11.
of compound numbers, 191.
of decimals, 95.
of fractions and mixed numbers, 63-65,
of integers, 11-13.
terms of, 11.
Subtrahend, 11.
Sum, 7.
Surface measure, 171.
Terms of a fraction, 51.
Test
of addition, 9.
of division, 24.
of multiplication, 33.
of subtraction, 12.
Time, 174.
Ton, 168.
Triangle, 207.
Troy weight, 182.
Unit, 3.
United States money, 181.
Usury, 232.
Volume measure, 121, 128, 173.
Wall coverings, 217, 218.
Zero. 3.
H6
^305^83
U/i 163
vy
THE UNIVERSITY OF CALIFORNIA LIBRARY