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MATHEMATICS Liprapy
THE UNIVERSITY
OF ILLINOIS
LIBRARY
The
Frank Hall collection
of arithmetics,
presented by Professor
H. L. Rietz of the
University of Iowa.
atO-FL $13
So ée
joe
Digitized by the Internet Archive
In 2022 with funding from
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ees BS
ESSENTIALS OF ARITHMETIC
ORAL AND- WRITTEN
Book II
FOR UPPER GRADES
GORDON A.-SOUTHWORTH
SUPERINTENDENT OF SCHOOLS, SOMERVILLE, MASSACHUSETTS
LEACH, SHEWELL, AND SANBORN
BOSTON NEW YORK CHICAGO
THE ESSENTIALS OF ARITHMETIC. Book I. For
Lower Grades.
THE ESSENTIALS OF ARITHMETIC. Book II. For
Higher Grades. -
Both books are published with and without answers.
—— Ol oes
¢
Key to Book //. for use of Teachers only.
Copyrienst, 1895, By GORDON A. SOUTHWORTH.
Norwood JBress ;
J. S. Cushing & Co.— Berwick & Smith.
Norwood, Mass., U.S.A.
— aw 5138
a MATHEMATICS LIBRARY
PREFACE.
Boox II. of the present series follows its predecessor after a
considerable interval of time. It is for upper grammar grades, or
for all grades above the primary when but one text-book in arith-
metic is required. The manner of treating elementary subjects pro-
vides for this double adaptation.
Much that has been deemed not to belong among the “ essentials,”
though commonly found in arithmetics, has been omitted, or left
accessible in a subordinate form in the appendix. The order of
presentation is in the main the usual one, though previous acquain-
tance with the rudiments of a subject has often been assumed, and
some subjects have been introduced in a preparatory way a few steps
in advance of the full and formal treatment, which thus becomes
far easier to comprehend.
It is of high importance to be quick with figures, and long practice
is needed: exercises specified as oral, written, for dictation, ete.,
are accordingly given in abundance, alternately upon the subject in
hand, or as constantly recurring reviews.
But the methods suggested call for effort and study, and look to
the mathematical training of older children in something more than
mechanical figuring by imitation. What ought to be perceived or
discovered by thinking and reasoning is not first stated outright in
print, though often led up to by stimulating questions. Such teach-
ing should develop habits of correct and ready expression, with
intelligent and permanent grasp of simple principles and processes.
As to division of time between solving problems and analyzing
them, the teachers must decide; but it has been shown that princi-
iii
404022
Hh PREFACE.
ples and methods cannot be securely fixed by mere repetition, or the
working of many examples. Fewer problems, if solved indepen-
dently and logically analyzed, will do most toward attaining the
highest purpose of arithmetical work.
Many a principle is made conspicuous upon the page; definitions
are collected in five groups, and arranged alphabetically for refer-
ence; set rules are given only as a summary, also for convenient
reference, in the appendix.
An introduction to the study of algebra is included in the
appendix; and throughout the book letters are conveniently used to
represent unknown quantities.
The contents of each section are given in side headings, to which
the following index furnishes a complete guide.
JUNE, 1895.
INDEX.
FULL-FACED FIGURES REFER TO THE APPENDIX.
Cylinder,
Decimals,
Decimal system,
Definitions,
PAGE
188, 224
3-5, 90-103, 2
2
7, 46, 110, 147, 234
Denominate numbers, 8, 43, 70, 81, 85,
86, 100, 105, 111-115
PAGE
Addition, of integers, 10-12
Of fractions, 59
Of decimals, 95
Accounts, 18, 84
Angles and arcs, 115
Annual interest, 14
Algebra, Appendix ii.
Average of accounts, 19
Bank check, 206
Bank discount, 193-199
Of interest-bearing notes, 198
Rule, 6
Bills, 40
Bills of exchange, 207, 208
Bonds, 201
Business forms, 18, 40, 180, 190, 198,
199, 201, 206, 207, 208, 16
Cancellation, 52, 66, 68
Cash account, 18
Check, 206
Circles, 127-133
Comparison of numbers, 47, 76, 78
Complex fractions, 73
Decimals, 91
Commission, 175-178
Compound interest, 191, 6, 14
Cones, 224-226
Frustum, 4
Cube root, 7,19
Customs, 188
Difference between dates, 165
Discount, bank, 193-198
Trade, 170
Successive, 171
True, 204
Division, of integers, 24-29
Of fractions, 70-74
Of decimals, 98
Duties, 188
Equations, 37
Equation of payments, 17
Exact interest, 169, 6
Exchange, domestic, 205
Foreign, 209
Factoring, 19, 58
Fractions, common, 3, 4, 54-90, 2
Complex, 73
Decimal, 90-103
Changes in form of, 54-60
Added and subtracted, 68, 64
Multiplied, 65-70
Divided, 70-74
Practice Table, 70, 74
Greatest common divisor, bed
“i INDEX.
PAGE
Insurance, 173, 174
Interest, general method, 106-109
Bankers’ method, 160-1638
One dollar method, 163-165
Choice of methods, 166, 168
Compound, 191
Exact, 169
Problems in, 204
Rules, 5
Legal rates, 15
Leap years, 8
Least common multiple, 60, 1, 8
Literal quantities, 36, Ap. ii.
Measurements of —
Arcs and Angles, 115; Circles, 127-
133; Cones, 224; Cylinders, 138,
224; Floors, 119; Frustums, 4;
Hypotenuse, 222; Land, 121, 9;
Lines, 111, 222; Lumber, 135;
Pyramids, 225; Prisms, 135-136 ;
Rectangles, 117 ; Rhomboids, 125;
Roofs, 120; Rules for, 3, 4;
Spheres, 228; Trapeziums, 127;
Trapezoids, 124; Triangles, 126;
Wood, 154.
Metric system, 10-13
Mensuration. (See Measurements. )
Mixed numbers, 32, 55, 63, 66, 69, 73
Multiples, 60
Multiplication, of integers, 19-24
Of fractions, 65-69
Of decimals, 96
Notation, integers, 3
Decimal, 91
Roman, tf
Notes, promissory, 180, 16
Discounted, 195-199 |
Partial payments of, by U.S.
rule, 182-187
By merchants’ rule, 16
PAGE
Numeration, 2, 91
Numbers, kinds of, L, 2,0 Os
7, 20, 46
Partnership, Pale iW
Partial payments, 182-187, 6, 16
Percentage, 149-215
To find percentage, base, or rate
per cent, 149-156
Bonds, 201 ; Commission, 175-179 ;
Duties, 188; Exchange, 205-209 ;
Insurance, 173; Profit and Loss,
154-159 ; Stocks, 199-202 ; 'l'axes,
189; Trade Discount, 170-172;
Rules, 4, 5.
Powers, ad
Present worth, 203, 6
Principles of —
Addition, -10, 59, 60; Cancellation,
52; Commission, 175, 176; Dec-
imal System, 2; Division, 27, 98;
Greatest Common Divisor, 58;
Interest, 160, 165, 169; Multipli-
cation, 20, 21, 65, 97 ; Partial Pay-
ments, 184; Profit and Loss, 155;
Proportion, 214; Reduction of
Fractions, 56; Right Triangle,
222; Square Root, 218; Trade
Discount, 171.
Profit and loss,
Promissory notes.
155-159
(See Notes. )
Proportion, 215-215, 6
Pyramids, 225, 226
Quadrilaterals, 116
Ratio, 47, 77, 78
Rectangles, Lae
Review Exercises in —
Fundamental rules, 17-19, 29-34,
38-45, 50-53
Fractions, . 62, 75, 78-90
Decimals, 100-106
INDEX. vr
PAGE
Measurements, 118-123, 181, 159-
142, 232, 233
Percentage, 159, 172, 179, 211
Interest and bank discount, 167, 168,
212
Miscellaneous, 1438-146, 210, 237-262
Rhomboids, 123
Right triangles, 222
Roots, 217
Rules, 1-7
Rule of Three, 33, 2138-215
Savings-bank deposits, 192
Signs, 7, 35, 47
Similar surfaces, 229
Similar Solids, 231
Spheres, 221, 228
Square root, 217-228, 7
Stocks, 199-202
Statement of problems, 24, 67
PAGE
Subtraction of integers, 12-16
Of fractions, 59
Of decimals, 95, 96
Successive discounts, 171
Surveyors’ measure, 9
Tables for practice, 12, 14, 16, 29, 70,
74, 154
Tables of weights and measures, 8, 9
Metric system, 10, 12
Taxes, 189
Time between dates, 165
Trade discount, 170
Trapeziums, 127
Trapezoids, 124, 125
Triangles, 126, 222
True discount, 203
United States money, 5, etc.
Wood measures, 134
Weights and measures, 8, 9, 10-12
THE
ESSENTIALS OF ARITHMETIC.
Book ook IL.
1.— Use of Numbers. What need of numbers has: 1. A mer-
chant? 2 A carpenter? 3 837.64 642.85
1, From c¢ to / inclusive. - 946.89 937.63
2ptoy 1% dtog 14 ftov . 987.63 | 846.75
3. dtom &, ctor 13. e tow Airo 21.14 324.93
4, eton OF 182088 14, dtoa » 608.55 698.79
. 327.83 128.93
5, ig to o 10,7 tOrG 15. c to y » 469.75 648.72
6. etonp, V ll giow . 984.96 562.37
678.94 689.85
627.34 283.97
Subtraction. 234.56 135.42
25.— Taking a 1, One part of | ». 789.12 | 698.57
Part away. 17 eggs is 8 eggs. . 846.89 | 569.38
How would you | » 764.83 | 783.92
find the other part if you had forgotten 758.75 964.83
that 8 from 17 leaves 9? 839.65 385.75
2. Take out 10 stormy days in Janu- . 087.93 978.59
ary: x remain. 931 less 10, or — 10, or = OOABA 628.32
diminished by 10 = a. . 849.64 759.67
38. Make a problem in subtraction, 376.86 314.11
using concrete numbers. Which is sub- | ¥% 978.35 629.55
trahend, which minuend? 4, The other
partis .. The largest part is
5. ‘In subtraction, which terms must be like numbers ?
26.— Finding the Subtract and say whether the result is re-
Difference. mainder or difference : —
1, You have $12 and spend $7. 2. You have
$12, and I have $7. 3. You have $9, and will earn enough to make
it $ 15.
EXERCISES. 13
4, How do you find the third term when you have the difference
and the subtrahend? 5, The minuend and the difference ?
27.—The Terms 1, Which is the larger number, 3 ft. or
in Subtraction must 24 in.? 2 Which is the larger quantity ?
be Like Numbers. 3. How can one be subtracted from the
At sight. other ?
4, A boulder weighs 7000 1b., a stone
block 4aton. The difference in weight is x Explain the process.
Where you can, give two values to x, first like the minuend, then like
the subtrahend : —
5. 4 lb. — 32 0z.=2 9, 10 h. — 240 min. = @
6. 60 mo. —2yr.=2 10. {-—i=2
7. 0.7 —0.03 = a ll, 2T.—21b.=2
8. $250 + a= $525 - 12. «— $166 = $ 34
28. — Rapid Give in one minute or less the difference
Subtraction. between each number and the one below it;
Oral. _ between each number and the one at the
right of it.
A, | 11 9 | 13 7 | 10 Looe ely oo) 18, 1d
| 2 ‘Ti eed ad 8 | 6 bao 7 On ae 5
B. 9 O05 216.) 11 ae ar: Nata a Bees Cs Fe eb 8 16
3 iro S46 341° 9 (AGS S44 9
0; | ib ices OAM ey ela Lb | foe eile tte 8 119. °° 10
7 S14 5| 6 9| 8 7 Pep Bei. 7
Pema 0 1AT 18d 1316 918118 12) 8~ 12
a Et Rane: al i
|
|
}
/
|
14 NUMBERS IN REVIEW.
29. — Rapid - |. Give the difference between 100 and
Subtraction. each of the following numbers. 2 Between
Sight or dictation. each number and the one at its right. 8. Be-
tween each number and the one below it.
a) 11 (88) 44 74.152. 709 SO" DS 60) OF eal es ela
b| 35. 61.) 82 14) 91> 33 | 22°65) 42 163410, 59 Oo eae
ce | 88 30] 23 57 | 89 48] 95 29) 68 32) 84 26) 79 16
O.) 05 81) T3° F221 94 O65 OOS. 03 ee OO neon me
é. | 92 19.1 63: 45>) 96. “1857 86) 46097 Oe 49 eT Ola es
. | 64 77 | 24 28) 34 39] 78 38) 80 25) 97 54] 98 40
4, From 1000 take 120 175 225 350 760 807 901
5, What remains when each of the numbers in the table is taken
Gut Olvl29 7 dole li3r
30. — At Sight.
1, 2, 3, 4, 5, 6.
700 3000 60503 25000 oo111 36459
325 800 40402 371892 46221 47560
7. 34+ 2=48 x + 27 = 80 « — 79 = 23 x + 24 = 150.
8. Replace x with the proper number. Think quickly.
Minuend 48. 62) ~@ (840° 27e a 80. abe = eeu
Subtrahend 16 x LOR PALO x 46 x Lio 4 dee
Remainder x A aes x ab! Bee aks 19 x Lied i
——
31.—OralProblems. 1. If you sleep 8 h. and spend 54 at school,
how many of the 24 remain ?
2. Out of $2 I spend 37¥, a half-dollar,
and adime. What have I left ?
3. What is the hundredth day in 1897 ?
4, What is the difference in latitude between a city 35° north
of the equator and one 34° south of it?
For dictation.
QUESTIONS AND PROBLEMS. 15
5. If a person is 69 yr. old to-day; when was he born? 6, A man
who died in 1879 would have been 100 yr. old if he had lived 18 yr.
longer. When was he born? 7 In what year was a house built
that now lacks 12 yr. of being 150 yr. old?
8, What number is 16 less than 100 — 59 ?
9, At 72 min. after half-past three, what time is it ?
10. Find what remains in counting backward by 13’s from 100.
32. — Oral 1, From 97 count backward rapidly by
Exercise. Gea e- os. bas,
At sight. 2. Count up to 200 by 17’s.
3 1384+2=120 250—x=120 2424=17
What change from a $5.00 bill that pays for —
4, Oysters, $0.75 5. Gloves, $1.25 6. Pens, $0.35
Crackers, 0.38 Scarf, 0.75 ipke Mth
Cheese, 0.62 Pin, 2.50 Paper, 0.874
7. Add the difference between 38 and 67 to the subtrahend.
Find what remains after receiving and paying as shown below : —
Received. Paid. Received. Paid. Received. _—_ Paid.
8. $1.16 $0.93 9, $45.00 $28.00 10. $2.25 $1.75
0.24 O17 95.00 19.00 3.75 2.30
0.60 0.25 70.00 23.00 De201 bac
33. — Written 1, Try subtracting one order at a time, in
Subtraction: the work at the left, giving each figure its
the Process. real value. What is the first difficulty ?
457 from 683. 2. If you had 83 sticks in bundles of 10 each,
Minuend 683 With 3 sticks over, how would you subtract 7
Subtrahend 457 sticks? How many bundles remain ?
38. At the left, 5 tens are to be taken from 2
tens. 4 What was added to the 3? 65 What
may be added to test the work ?
6. Give directions for each separate step in the process.
Remainder 226
16 NUMBERS IN REVIEW.
34. — Written Without copying find quickly the sum of the
Exercises, four differences between —
1. eand f 3 gandh 6 tandj 7. k and J 9, manda
%. fand g 4, h andi 6. 7 and k 8, 7 and m 10. » and e
Find the sum of the ten differences between —
11s Arand?B 13. Cand D 15. A and C
12, Band C 14, Dand A 16. Band D
A, is) C. D.
e. $3764.82 $ 4769.31 $ 5000.37 $ 9000.15
ap 927.35 3468.97 689.82 794.38
q. 860.83 385.68 1348.75 1866.75
h. 1527.96 2487.32 946. 2889.43
1. 3184,98 694.39 37.89 648.95
j. 2876.45 1748.64. 9586.54 1864.37
k. 825.35 4839.87 829.85 624.94
l. 96.47 658.34 1472.98 1739.41
m. 849.383 1987.62 468.52 866.
nm. 3276.41 594.83 5500.31 49.75
35. — Oral 1, Four parts of 75 are 18, 9, 13, and 22,
Problems. The fifth part is a 2. 387 gallons are in a
At sight. tank. Add 17 while 23 run out. What re-
mains ?
8, An engine goes forward 25 rd., back 88 rd.; forward 60 rd.
How far is it from the starting-point ?
4. How much farther is it round a 17-foot square than round a
square 13 ft. wide ?
5. By annexing to 57 the figure 6 how much is added ?
6. Taking the 5 from 275 leaves how much ?
7. Bought a pony and phaeton for $500. Sold the pony for $175,
losing $50, What did the phaeton cost ?
QUESTIONS AND PROBLEMS. 17
8. Having $400 in bank a person draws $25, deposits $150,
draws $75 and $50. How much remains ?
9, One horse is worth $50 more than a second and $150 more
than a third. If the highest priced one is worth $200, what are
they all worth ?
10. If you find 5 eggs one day and 6 the next, how many dozens
will you get at that rate in a week ?
36. — Problems. 1, How much remained in bank to Mr.
Rich’s credit Saturday night, what he put in
and took out being as follows for the week : —
Deposits: $26.95, $793.82, $427.96, $ 839.64, $500, $ 387.28.
Withdrawals: $18.56, $ 689.37, $ 419.28, $ 649.39, $ 600, $125.82.
2. A merchant’s assets are as follows: —
Merchandise in store, $ 24876.39; cash on deposit, $ 1489.38 ; due
from customers, $4897.64; real estate, $28649.27. He owes for
merchandise, $ 16483.56; for real estate, $6498.27; on promissory
notes, $ 6483.75. How much will his estate be worth if he closes out
his business and pays his debts ?
Written work.
3. I have on hand at the opening of business cash to the amount
of $846.95. I pay out $84.92, $64.87, and have on hand at night
$ 837.69. What have I received ?
4, I received during the day $ 249.85, and I paid out $521.75. 1
had on hand at night $37.62; in the morning «.
5, Thomas Bond begins business January 1 with cash $ 478.37 and
merchandise $1875.28. At the close of the year he has $1487.63
worth of merchandise and $738.29 in cash. How much has he
gained, or lost, during the year ?
6. The sum of two numbers is 346301. The smaller is 89795, the
larger a.
7. What number must be subtracted from one million to leave the
difference between 347689 and 486931 ?
18 NUMBERS IN REVIEW.
8. The distance from A to B is 628 feet, from A to C 1426 feet,
and from B to D 1648 feet, all in a straight line. How far is it from
C to D? Draw a line and mark off the distances.
9. How many days of 1897 have passed before Aug. 15 ?
10. What is the difference between the sum of column A, page 12,
and that of column B?
37.— Cash Nortr. — An account with ‘‘Cash”’ is, as it were, an account
with one’s cash-box or pocket-book. Cash is debtor for all that
Accounts. is put in, and cash is credited with all that is taken out.
Dr. CASH. Or.
1896 1896
Nay /\@n hand /00\00 Nay o by Ndaé. bought L50\00
O|\So Rent ree’d|| 50\00 Ale Kano “ \|250\00
7\ “ Mdae. wold\| 25\00 ea) Clothing “7 25|00
VO CO ™ Lane glivoed //\ batanee || 75|\00
ie GOO\00 GOO|00
Nay 22 bn hand 5|00
1. Cash is charged with having received four amounts which it
owes me and for which it is my debtor. How much was there at
the beginning ?
2. What was added from sales of merchandise? 3. Cash is debtor
for the price received for land. Why is the income from rents charged
to Cash? 4 What is the total amount my cash has received if I -
wish to draw upon it? 5. How much does Cash pay back to me
for the piano purchase ?
6. Why do I credit Cash with my clothing expenses? 7. What
are the total outgoes for the month ?
8. What is the footing of the debit side? 9. What more might I
have spent so as to balance the footings? 10. How is the balance
found ?
QUESTIONS AND PROBLEMS. 19
38. — Written 1. Balance the cash account of Charles
Exercise. Watson. He has on hand $4.21. He re-
ceives at various times $6.24, $7.36, $8.49,
$7.34, $6.75. He pays out $8.75, $9.81, $3.26, $8.39.
2. Monday morning a merchant begins business with $247.84 on
hand. He receives $24.75, $86.91, $ 84.28, $97.25, $164.29. He
pays out $18.99, $ 37.49, $64.91, $83.15. Find the balance on hand.
Find the balance of each of the following accounts : —
3. 4. 5.
Dr. Cr. Dy. Or. Dr. Cr.
$ 987.65 $629.55 $4768.82 $468.34 $649.81 $82.46
1839.76 83.74 947.61 984.59 8439.87 981.32
6482.91 968.71 847.77 1483.22 648.38 641.25
478.85 28.46 3998.64 91.76". 239.86
698.47 318.93 8372.91 728.41
Multiplication.
39. — Numbers A: Unequal numbers. 8B: Equal numbers.
Combined. 94+847+4=28. 7474747, or4x7, =28.
At sight. Il. In A, the combining process is \..
2. Can more than two numbers be added at a time?
3. Under B, the first process is ..; the shortened process is W.
4. Do you know the product of 4 7’s by counting or from memory ?
5. Which number is to be multiphed ? Which is the multiplier ?
6. Why not get the result in A by multiplying? 7 Compare
addition and multiplication.
40.— Rapid Factoring. What numbers multipled together, Le.
Oral. what factors (smaller than 14), produce —
1. 28, 32, 33, 35 4. 65, 66, 72, 77 TeUB ALLO MIT
2. 36, 39, 42, 45 5. 78, 81, 84, 88 8. 121, 130, 132
8. 48, 49, 52, 54 6. 91, 96, 99, 104 9. 143, 156, 169
20 NUMBERS IN REVIEW.
10. The sign means: separate into two equal factors, or find
the square root. W25; V81; V36; V49. 11. Vi44=axy;
V121 = 2; V64 = x; V4 x 25 = a.
41. — Principles in xx $8 = § 24.
Multiplying. ASX ee Were eis I. Only one factor
At sight.—1. Say | can beconcrete. Both
which is multiplier and which multiplicand, | may be abstract.
giving values. 2. What are the factors Il. The product
(makers) of $24? Of 8 sq. ft. ? and the concrete fac-
For dictation. —8. What is anaddend? | tor will be ‘like’
A subtrahend ? A multiplicand? | numbers.
4. Make two examples: the multiplicand Ill. The order in
concrete in one, abstract in the other. | which the factors are
5. Try multiplying by 5 stones or any | used will not affect
concrete number. the product.
6. Show with objects that 3x4 of a
kind are 12 of the same kind. 7. Give the factors of $21; 49 m.;
18 cases.
8. Compare 4 x 5 bu. and 5 x 4 bu.
42.— Rapid For dictation. —1. 9 and 12 are factors of
Multiplying. what? 13 and 6? 19 and-3?. d1land 122
Oral. $8 multiplied by 4? 41? 41°
Give the product : — Multiply by 9 and add 9 :—
8. 3,6%; 10%, 2; 12, 13. Glee S 14> G4GLD oie eles
4913 X POND x Pate ol. Give two factors making —
0. 374, 2; 5,124; $14, 4. tT. 63772748 Ol thy fe Sood,
8. Multiply the following by 8; by 9; by 12: —
7 bales 70 bales 80 rods 50 fathoms 90 feet
At sight.—9. Give each product quickly, stating which factor
is multiplicand : —
$800 600 700 rd. 900 800 m. 400 6000
) I 2ivd. 76s 9.30). Pe ula 16m. 13
————sieg ) oemenanmnaioniceeniiot
QUESTIONS AND PRINCIPLES.
10. Give two factors of 182 sec.; 125 in.; 144%; 108 h.
IL; Take 4 x 7 from 9 x 7. 13. Add 18 x 13 and. 2'« 13.
12. Take 6 x 8 from 8 x 9. 144, 3x9in.+6x9in=®@.
15. 8'=3 x 3=9; 4?; 67; 88; 72; 9; 122; 207; 502
21
43.— 1st: Written Arrange as shown at the right, then complete
Work; 2d: Oral Ex- the equations. 1. Find 8 x 858.
planation. ei fe aiye & 112x543 = 8x 8=
Think what is required and write equations from the oe 0
following datu: 3 x 9¢ = 27¢. Pee
4, At 2 for a quarter what will 14 baskets cost ? 8 x 858 =
5. 387 poles at $4 each. 8. 1031 lb. of 6¢ sugar.
6. 6 doz. barrels, $2.50 each. 9, 8.h. 20 min. a day, 6 days.
7. 12men,10d.,$2aday. 10. $1 a week for 2 yr.
44.— Written By an integer of one figure. —
Multiplication. 1. Under C what are added to get
For oral analysis. the result? 2, Explain their position
and real value. 8. Show how the re-
A. B. sult may be got without setting down
183 183 the partial products.
By 10, 100, 1000, ete. —4. Annex
a cipher to 28, and give the values
of the 8 and 2 before and after the
2954
change. 6. How would you multiply by
Every cipher an- 10,000? +6, Compare the work under A
nexed to an integer and B, and make a rule for multiplying by
multiplies by 10. any number of 10’s, 100’s, ete.
10, by 100, by 1000:—
37 64 8357 946 3472 8476 29,475
8, Compare 90 x 100 and 100 x 90. 9. 120 x 300 =.
7. Read these numbers multiphed by
rays NUMBERS IN REVIEW.
45.—For Rapid Fig- Find the product :—
uring. 1, 67x 4763 =a 2, SO eo Leesa
and 6789. 4. 5987 by 7. 5, 84,965 by 80.
6. 8 x $12,039. 7. 848,794 and 300. 8, 1,203,900 x 50. 9 1 ft.
Sin. x 3100. 10. 34,000 and 90,000.
46. — Multiplying 1. In the work at the right, read the multi-
by any Integer. plier. 2 What three partial multipliers are
For oral analysis. used? 38 Read the 2d
partial product. 4 4 x 578 578
would be what? 5, Explain how the 3d partial - 346
product is got. 6. Would it affect the result if we 6 x%578= 3,468
should multiply first by 300? 7% In ordinary 40x 578= 23,120
work how much of this may be omitted? 8 Where 300 x 578 = 173,400
is the lowest figure of a partial product tobewritten? 346 x 578 = 199,988
47.— The Process Give directions for six steps in multiplying :
ESE le 1, Arranging the factors.
Choosing a Multiplier. ae :
2. Beginning to multiply.
Gee 3. Setting down and carrying.
$7 multiplied by 4, Arranging partial products.
2378 must equal 5. Finding entire product.
16646 .
% 6. Testing the work.
For 2378 ;
paulerplied ir 7 a. In finding the cost of 2378 bbl. flour, at
equals 16646 7, what is the true multiplicand ?
b. To shorten the process, why must we use
abstract numbers as shown at the left ?
48.—For Rapid Figuring. 3, 37¢ x 432 7. 64 oz. x 976
4, 427 x 83 Geet oe
1. 5386 x 846 5. 329 x 847 9. 846 x 9372
2. 3976 x 597 6. $687 x 4395 10. 387 T. @ $8
:
|
ANALYSIS. 23
49. — Examples Oral.—1. Perform the work at $4.37
with Decimals. the right aloud, giving each figure 19
its value in dollars or cents. 2, Ex- $39.33
plain the position of the decimal point in the product. $ 43.7
$ 83.03
Written. — 3. 479 ed. @ $12.00 ? Write the product in cents ;
4, 4868 T. hay @ $27 ? then in dollars :—
5, 25,789 bbl. flour @ $5.00 ? 9. 239 men get $5.75 each.
6. 787 M. brick @ $16? 10. 2958 lb. tea @ §$ 0.67.
1. 193 x T cents ? 11, 234 million half-dimes.
8, $0.07 x 795? 12. $5.623 x 8 x 3.
50. — Oral Explain the process that you use :—
wi te ake 1. Take 16 in. from. yd. 2 Add 1 yd.
and 4+ ft. To what may both be reduced ?
8. 7x 4.0f 120 ='2.
4, =, of $57 =a ond ae cate} bee 6. x= 200 x 84
qin Of GST =y Ne OUU ane yY = 71, of 2375
10 x $0.57 =z 13 x 800=-2 BLEU
ie 10 gh a7 gal. 8. 2 yd. @ 18¢ 9, 31 lb. @ 20¥
LOS i=: “yd. 1 doz. @ 16¢ 3 lb. @ 16¢
4. Ib. = & 07, 1 yd. @ 32¢ 2 qt. @ 121¢
< lb. = @& 02.
Te a O45 OV OF er,
51.— Analysis of 1, Make a problem in which the equation
Problems. 4x 162¢=~2 will indicate the work to be
Oral. done. What are the two equal quantities ?
2. Which is easier to do: reason about a problem so as to show
how it may be solved; or figure out the result after being told how
to do it? From which do you learn more? 38 Uxplain the maxim:
“ Well understood is half done.” 4 Define an equation.
24 NUMBERS IN REVIEW.
od
52.— Problems to 1, $2.50 was the expressage on 19 tables
be Stated. (@ $12.74, and 28 chairs @ $2.58; a is the
For written work. total cost.
Statement. — $ 2.50 +19 x $12.74 + 28 x $2.58 = a.
Make an equation showing all that must be done to find the value
of x; then find tt.
2. 130 men @ $2 a day, 47 @ $1, and 8 @ $ 3.50 receive x dol-
lars in one day. If paid weekly, they receive y.
3. w is what is paid for — 4, The amount received = a.
24 and 64 lb. tea @ $1.19 217 A. wheat ;
59 bu. potatoes (@ T5¢ 27 bu. to an acre;
30 Ib. coffee @ 34¢ sold for 83¢ a bushel.
5. A nursery contains 1000 trees; 75 are dead; the rest are to be
sold @ $2 each. They will bring $a.
6. 3 house lots cost $1260.80 each and sell for $1500 each. The
total gain is a.
Use a short method in finding —
7. The sum of 827 x $9.28 and 183 x $ 9.28.
8, The difference between 649 x $12.84 and 149 x $12.84.
Division.
53.— Finding an For dictation. —1. The product of two
Unknown Factor. factors is 48. One is 6; the other, —.
2. How many 9’s in 54? What goes 9 times
in 63? 3 12 is the multiplicand; how many times is it taken to
make the product 84? 4, What multiplicand, repeated 12 times,
makes the product 108 ?
5, Suppose one factor and the product are known; how is the
other factor found ? Tlustrate, using $6 x a= $42; and $a x 5=
$45. 6, Why is the process called Division?
PRINCIPLES AND PROCESSES. 95
At sight.—7. Show by the examples in 5 that —
(a) The product becomes the dividend (something to be divided) ;
(6) The known factor becomes the divisor ;
(c) The unknown factor, when found, becomes the quotient (show-
ing how many times, or the size of each part).
8. Give the quotients: «x15=30d.; $9 x2x=108;
=
oO, 96+-24=a@; T2:2=4; alae eG
18. Pe he
9. Describe the four different ways of indicating division shown
in the preceding line.
10. Find two factors of 96 leagues; 91 d.; 168 h.; 182¢.
54.— The Process 1, Division is the reverse of ... 2 How
of Division. may the multiplication table help to find a
) quotient? 38 Without that table how might
one find the number of 12’s in 60? 4 Find by subtraction the
number of 24’s in 96.
5. How many 12’s in 1740 ?
A. B. C.
145
12)1740 12)1740 12)1740
1200 = 100 12’s 12 , 145
pesca f 6. Are there 200 12’s
sath e nei in 1740? 7. Are there
60 60 100? 8, Subtract them:
60= 65 12's 60
what remains? 9, How
many 12’s in 540?
10. Subtract 40 12’s: what remains? 11. 60 =@ 12’s; subtract
them: what remains ?
12. How many 12’s in all have been taken from 1740 by the three
subtractions? 18, Explain the changes under B. 14 Perform
aloud the work of C.
Total = 145 12’s
26 NUMBERS IN REVIEW.
15. What is the difference between long and short division ?
A. be 16. Explain process
207935 A of finding how
235)48674 235) 48674 many 230’s in 48674.
47000 = 200 235’s 470 | 17, Explain process B.
1674 1674 18. Why are there no
1645 = 7 2835's 1645 tens in the quotient ?
29 207 235’s 29 19. What part of an-
other 235 is found in the remainder, 29? 20. By how much
should the dividend be increased to give 208 for a quotient ?
55. — Examples. 1, How many 9’s in 4752? 2 8’s in
Written. 9896? 38 12’s in 3300? 4 15’s in 4650?
5, 25)16325. 6. 17784+312. 7% 19998
8. 27 x «= 40527. 9, Product= 9672; quotient = 372; divisor =a.
10. 96 and 75 are the factors of what dividend? Give proof.
11. Show that 33810 + 245=138. 12. Multiply 245 by 138 and
find the partial products in the work of Example 11.
56.— For Dictation. 1. How many 99’s in 500? 1000? 10,000?
Oral. 2, Count by 99’s to 500. By 98’s to 500.
8. Define dividend; divisor; remainder.
4, 4 of 12 ist of what? 5. $12+24=$2. 6, What kind of
number is the quotient when the divisor exceeds the dividend?
7. If 6 shillings make $1, one shilling is worth a cents.
8. Knowing one factor of 48, how can you be sure of the other?
9, What are the three factors of 18? Give 4 divisors of 18. 10. The
largest divisor of 72 is a Of 48? The greatest divisor of both is
what ?
57. — The Given Fac- 1, How many $10 bills make $300?
tor Like the Dividend. 2. One factor of 60 yd. is 6 yd.; the other
At sight. is .. 8 In multiplying two factors to
) make a product, which factor is always
abstract? Which may be concrete ?
PRINCIPLES AND PROCESSES. ve
4, If 7 ft.x 6=42 ft., 42 ft.
+7 ft. =a, and 1 of 42 ft. =y. A divisor that is like the
5. Show by the last example dividend is one of its equal
that when the dividend (or prod- | Parts. The quotient tells their
uct) and the given factor are alike number, showing how many
the factor to be found must be times the divisor can be sub-
abstract. 6. Show that the quo- tracted from the dividend.
tient might be found by repeated
subtractions.
7. 90 in. +6 in.=@.
8. 28 qt. in 84 qt. # times. 10. 45% +15%=~2.
58. — The Given Fac- 1, 8 hats cost $ 40, 1 costsa. x«x8= $40;
tor Abstract; Dividend 1 of $40=a. 2 When a product or divi-
Concrete. dend is concrete, are the factors like or
For dictation. unlike ?
3. Which factor shows —
(a) the number of equal parts
united ?
(b) the fractional part of the many equal numbers make
dividend to be found? the dividend.
(c) the size of the parts? The quotient is one of
these numbers.
9, 400 ft.)8000 ft. (a
An abstract divisor of a
concrete dividend shows how
4, Kach woman gets ;, of $60.
The number of women is @.
5. When 20 books cost $200, what part of it will one cost ?
59. — Examples. 1. 6482 ft. + 90 ft. = m.
Written. 2. One factor of $475,000 is $250, the
other is 2.
. 84 equal numbers make 6300 yd. Find one.
360 m. = multiplicand; 25,520 m. = product; # = multiplier.
Multiplier 125 ; product 100,000 bales ; multiplicand a.
. Divisor x quotient = $16,750; divisor = $ 670, quotient = 2,
. Dividend = 197 x $461; quotient = a,
AO - ©
98 NUMBERS IN REVIEW.
8. After subtracting 220 atimes from 44022, what is the least
that must remain ?
Find cost of one acre, when —
9, 64 cost $ 367 x 28. 10. 128 cost $ 1000.
60. — Division: the
Process Described. ah ag fee Sit
144 pens)10,000 pens 25)%176.95
8640 pens $175
1, A box of pens 1360 pens $1.95
contains a gross. Make 1296 pens $1.75
a problem for the work 64 pens $0.20
under A. 2 Under B.
38. Explain why in each case all the numbers but one are like
numbers. 4 What numbers when added make the dividend ?
5. Why not divide the $0.20?
6. Give directions for six steps in division.
J. Arranging the numbers.
II. Choosing 1st partial dividend.
Ill. Writing quotient figure.
IV. Finding product to subtract.
V. Completing 2d partial dividend.
VI. Finishing the process.
61. — Rapid First column (p. 29). 1. Divide quickly by -
Division. 2; 3; 4; 5; 6. Give integral ea and
Oral. remainder. 2 Find 4+; +; 4; 4; 7. Give
the exact size of the equal parts.
Second column. 8 In each number how many times will 10 go?
100? 200? 300? 40? 4 Use as divisor 50; 60; 80; 90; 70.
Give remainders.
Third column. 6. Give quotient and remainder in cents after
dividing by $0.50; $0.25; $0.80; $1.10; $1.20.
Fourth column. 6. Give results in dollars and cents of each
dividend + 1,000; + 2,000; + 200; + 4,000; + 3,000,
EXERCISES. 29
1. 2. 3. 4. 5. 6.
a. 21 50 $1.50 $ 4261 $ 567.82 347,694
b. 32 520 1.25 8937 739.75 932,976
c. 43 636 1.75 6425 947.50 843,207
d. 54 724 1.38 8034 842.90 600,898
e. 65 837 2.75 6481 838.38 347,291
ooo 964 3.25 8972 496.81 468,394
qg. 87 523 4.50 4729 149.85 729,831
h. 98 649 5.40 6834 328.74 476,984
i. 89 732 9.60 9287 692.48 294,765
a (8 807 7.23 3199 728.47 300,041
62. — Practice in First column. 1, Divisors: 18, 14, 15, 16, 17.
Division. Use short division.
Written. 2. Divide by 18, 19, 20, 21, 22. -
Second column. 3. By short division find the other factor when
mmeris 00,93, 97, 96,-95.
Third and Fourth columns. 4 Change both numbers to cents,
then use the larger as dividend.
Second and Sixth columns. 6, Divide numbers in 6 by numbers
in 2 and give remainders.
6. Col. 4+ (col. 3+ col. 5). 7% Col. 6+ col. 1 x col. 2,
63.— Oral Problems. 1. 1, of 10 min.=@ sec. 2 3 sec. in 10
At sight. min. #times. 38 @ repeated 125 times = 750
min. 4. Three persons share $12,630 un-
equally. How much may each receive? If they share equally,
what must each have ?
5. $62,500 is separated into # packages containing $125 each.
6. One of the equal parts of 16,250 is 250. How many more such
equal parts are there ?
7, If 208 tickets are distributed one at a time to each of eight
persons, how many will each have when the tickets are half dis-
tributed ?
30 NUMBERS IN REVIEW.
8, How many bags will hold a million dollars if there are 100
twenty-dollar gold pieces in each bag ?
9, Take = of $2.10 from +; of it.
10. Make two examples: one with quotient abstract; one concrete.
64. — The Funda- 1, Which of them have to do with combin-
mental Processes. ing several numbers into one? 2 When is
the shorter process used? 8 Contrast sub-
traction and division. How are they alike?
4, Is the number which equals 10 8’s a sum or a product ?
5. If you pay $2.38 with a $10 bill, what is left? 6, Four num-
bers make 87; 12, 16, and 5 are three; the fourth is what?
7. 87 —15=2 x what? 8 91s what part of 12? 9, 2 wk. = how
many hours? 10. Why pay 13¢ for 4 yd. at 25¢? For 3 yd. I
Pye
For dictation.
65. — Short Without copying, find the difference : —
Examples. Ik 2. 3
Weiner $ 478.36 $ 548.79 379.64
$ 1399.78 $ 693.78 1633.99
4. $847.21 — $368.27 5, $2000 — $367.41 6, 932.61 — 878.95
7. Take the sum of the last three subtrahends from the sum of
the last three minuends.
8. If 6 bbl. oil cost $47.70, 29 bbl. cost a.
9, Multiply 648 by 81. 10, $24.84 x 97, =a.
66. — Oral Exercise. Supply values of x and y.
At sight.
1, yt 3. 4,
27+43=2 16 SC9e=7 * of 630 =9 x= 7 — 3h
26--a¢ = 44 Txe=91 Si0 =o xX 2 4.0L (=
58 —19 =@ 4e¢=—144 15=t of 2x3i=2@
GO Sep 2 % =, of T2 6 =~ of 144 e= 7 + 3h
e+y =100 4 of «= 90 (20 + «= 180 oy Xe
QUESTIONS AND PROBLEMS. 3)
5. The divisor is 7, the quotient is 346. How many 7’s are sub-
tracted from the dividend in finding the quotient ?
6. In dividing 9000 into 4’s, how many 4’s do we at first subtract
from the dividend? 7, What is 1 of what remains ?
8. Compare 6 lb. and 1 lb.; the cost of 6 lb. and the cost of 1 lb.
6 lb. cost 84, 1 Ib. costs W. of 84¢, or a.
9, Dis tofex. 27 lb. cost $1.80, 9 lb. cost y.
10. If 14 lb. cost 84¢, what will 10 lb. cost? 10 x zy of 84 = a.
a
67. — Statement of 1. If 16 cd. wood cost $120, 24 ed. cost
Problems. what? In solving such a problem which of
Oraland written. these suggestions seem most important ? —
I. What is to be found out ? (Cost of 24 ed.)
II. Facts that help to find it. (16 cd. cost $ 120.)
III. Process, by steps, briefly set down. (24 x p, of $120 = cost
of 24 ed.)
IV. Indicated work performed. (24 x +5 of $120 = $180.)
V. Whether the result is reasonable. (24 ed. should cost 14 times
as much as 16 cd.)
Apply the preceding suggestions, and explain orally : —
2. Bought 12 lb. tea @ 75¢, and 20 lb. coffee @ 40¢. How much
butter at 30% would cost the same ?
12 x $0.75 + 20 x $0.40 __
$ 0.30
3, Exchanged a 60-acre farm worth $ 2400 for 200 acres of wood-
land valued at $13.75 an acre. Gain?
4, Gave 3000 sq. ft. of 20-cent land for a span of horses and $ 75.
What were the horses valued at ?
5, Sixty-four men are employed 25 days in digging a sewer. The
contract price was $1200. Nothing was gained or lost. What were
the men paid each per day ?
Statement. — ~
ay NUMBERS IN REVIEW.
6, A train runs 280 miles in 11 hours. Seven 3-minute stops are
made, and a hot axle makes a detention of 39 minutes. The rate per
hour was @ miles.
7. Six men buy 640 A. @ $125, and sell for $95,000. Each
man gains w [} of ($95,000 — 640 x $125) =each man’s gain. |
In the statement what represents the cost of the land? The pro-
ceeds of the sale? The whole gain ?
8. Bought 59 bbl. flour (@ $4.75; sold 15 bbl. @ $5, and the
remainder @ $5.25. Required, my gain.
9, Three 1-1b. packages will go by mail each for 1 an oz. plus
registration; by express for 25¢ each. Which way is cheaper ?
10. A peck, 2 bushels, and 5 quarts are to be divided equally
among seven persons. Any two receive @ quarts.
68.— Product of Oral. —1, Explain the process used in each
Mixed Numbers. example: 3x12=a; 1 of 12=y; 34x12=z.
2. Give results: 31x12; 81x10; 51x15;
7 xX 82.
of 9; $ of 16; 2 of 20; 4 of 28.
4, Give results: 2 of 45; 7 of 56; $ of 72; #4 of 63.
5. Give results: 0.6 of 20; 0.8 of 60; 0.9 of 70; 0.08 of 400.
576 X
Ino
3, Give results: <
No
8] 6. Supply omissions in the work at the left.
72 — } of 576 Show what might be omitted in ordinary work.
504 = § of 576 7. Give directions for each step in multiplying
4608 =~ x ~
by a mixed number. ..
5112 = 87 x
Written. — Carefully arrange partial products and results.
8 98 x 280 1l, 780 x 19,9, 14, 13.5, x 280
9. 18% x 942 12, 603,32, x 2000 15, 143 ft. x 784
10. 110%, x 144 13, 18% x 1728 16. 914 Ib. x 1080
17. Compare 32 with 10. Show quick ways of multiplying by 34,
334, and 3334.
QUESTIONS AND PROBLEMS. 33
69. — Oral Review. 1. Count by ea roe 100 to 0. 2. Count
en Letitia to 300 by 3873's. 8. To 500 by 622’s.
4. What two numbers pee than 1 give 7
as product? 6. Give the sum of 64, 34, 6, 4, 54, 43, 2, 3, 83, 14, 5,
and 7. 6. Compare the time 6 men need to Sir a road with the
time required by 2 men. By 15 men.
7. If I spend 2 of my money and give away $8, I shall have noth-
ing left. What have I now ?
8. What is 5 mo. rent of a house hired for $300 a year? 9. One
year’s interest is $40; 21 years’ will be what ?
10. Count from 180 to 0 by 18’s.
70. — Review. 1. tof 4864 = 3. 300 is 1 of a
At sight. 2. «= of 54180 4, .. minutes = 21 h.
6. 164 ft.=1rd.; 10rd.=aft. 6. 14 reams = 2x_quires.
7. Give rapidly the following fractional parts of 100: —
NUS OS aged SAS Mpeg A ieee Ob ale Se Race
89 49 8? 22 89 4) 8) 6) 3% 2) 37 6°
8. If I divide an integer by 356, the largest possible remainder is
what ?
9. 2 of 24 is a more than $ of it. (4 of 24) + a= 3 of 24.
10-210 «0.034, 11. § 2000 is contained a times in $80,000.
@=100 x 0.034. 12. wis sj of 100 thousand.
71.— A Rule of Three 1. If 5 lb. cost 42 ¢,10 lb. cost what? Why
Applied. is it needless to find the cost of 1 lb.? What
Oral: at sight. | would 24 lb. cost ?
2. When 21 lb. cost $3.21, 7 lb. cost aw, and 3 lb. cost y.
8. 5 for $1.70 makes 35 cost x 4. 9 for $1, 6 for a.
5. 18%, or 18, of $600 is profit. $36 is what part of the profit ?
6. $7070 is the value of the crop of a 56-acre market garden
16 A. at that rate yield $a. 40 A.?
34 NUMBERS IN REVIEW.
7, What 42 men can do in a week 7 men could do in WV, and 28
men in .W..
8. Supplies for a regiment of 1000 men would maintain 100 men
oo or, O00 meno
9, 10 papers a week, or 2a year. 10, 1000 ft.in 12 sec.; # an hour.
72.— Problems for 1, Compare 8000 and 2000. Find a short
Study. way of multiplying 599 by 8000 and divid-
Written. ing the product by 2000. What is 5955 of
ia B. CQ. 8000 times a number ?
257.36 — 129.28 = x 2. Copy the equations at the left, giving
385.91 — 236.99 = y values to a, y,z Add A; add B. Compare
536.84 — 327.45 = z the difference of their sums with the sum
of C, and explain.
3, In its circuit round the sun the earth traverses about 567 mil-
lions of miles in a year of 365 days. How many miles a day?
4, Compare 52 and 364. If 52 bbl. apples cost $175, what will
364 bbl. cost ?
5. Find the average weight of 6 men weighing respectively 135,
176, 180, 138, 207, and 156 lbs.
6. What would be the duty at 20%, or 4, on 325 bu. beans at
$1.69 a bushel ?
7. Mr. Fisk leased an office for 3 yr. @ $374 a month. What
had the use of it cost him at the end of 21 yr. ?
8. A barrel of flour fills 8 bags, and costs $4.50. What is the
gain on 3 bbl. sold at $ 0.624 a bag?
9, Hans can haul as much sand in 15 d. as Knut can haul in 20 d.
Which should receive higher day wages? 10. Knut, working with
a cart and horse, got $60 for 20 days’ work. If Hans had taken
the job for that amount, what could he have earned a day ?
QUESTIONS AND PROBLEMS. 85
73. — The Use of Express in words, giving values to x :—
Signs. lox b+ T= 2 27+3—-VY/g=2
Oral. 2 (9+6)x(8—5)=" w=—18—3+./95
[See pp. 7 and 47.] 3, 57—-5x2=3?+~2
4, Give name and meaning of each sign just used.
Why are the results wilike in the upper and lower lines : —
5. 38+4x5=34200r23 6 6x12—8+2=72 —4 or 68
(3+ 4) x 5=12 x 5 or 60 6x12—8+2=6x4+2Zo0rl2
7. When a number has x or + on one side and + or — on the
other, which process must be performed first?
8, Compare in value 15 — 5 x 2+ 8, and (18 — 5) x (2+ 8).
Prove that —
9, 36+4—45+9=12x3+48-+11.
10. 2+5x6= twice (2 x 546).
ll. 4 of (7+14)—1 of 67-9) =8 x 9—3) —48 = (5+ 4)’
— (36 + 4)”.
74.—To Write from 1, 96+12 2. Sq. root of 36
Dictation. mult. by mult. by 9
200 + 25 less (7 — 2)
=4x# = (fc
8. V36 4, 3 mult. by 5. The sq. root
x (9 — 5) (6 + 5) of 3 of 48
= sq. root of less 4 of added to
(36 — 11) (37 — 10) 2 of 35
+19 = 30 =a OL
75. — At Sight. 1, What expressions are here marked to be
treated as one number ?
8 x16+8+48 x (146—9)=V3x9x3x108+12—10+1
2, What signs are used to show that two or more numbers are
to be treated as one number ?
36 PRINCIPLES AND PROCESSES.
3. What number is to be divided by 11 ?—
6x 544x9)+11 =[6 45) x (0 —4)]+ 11
Show why it was better to use brackets [ ] than curves ().
4, w= (12'+24)x V54—5 6 12x44+6x12)+ V100=2
6. (4800 + 100)+ 0.01 of 6€00=a 7% x=[(7+3) x2—4 of 39}
8 w=[6 x8—4 x (14+4) + 60] +100
7 2 ‘ ee ed
3 le ie: =) + Vor =x 10. [(V25)'— Vex t] + fof 72 =2
co
ry
(
76. — Substitution The first letters of the alphabet are often
of Numerical for Lit- used to represent quantities whose value is
eral Quantities. known. 6a=6 xa; abe=a~xbD xe.
Oral. SUPPOSe eel a Dee
Che | CL a eemet
Find the value of —
red, One aie ae hl eecaoor 16. db +e VAs
Die (ce Gf =e 12a ile. sf DO eee
i at+d $
ah af Ue a De LOL 18. 23. d
C
4-19 Gao ono aie ee 19, Z 24, de?
C
5. 3d 10. 2f—38e 15. abe 20. abce 25, be?
iE wate Substitute the following values for the letters
in the problems, and solve them :—
ai (44 6 39sec =a On Sd == Lose a afi lo:
1, If } pounds cost a dollars, one pound will cost 2.
2, What will e yards cost if ¢ yards cost $d?
3. dyards =ainches. 4 cmiles=afeet. 5. be-+a+ef=2.
6. Add a hours and e days.
ff cde 8. fbe 9, a+f+d
w d c+e
EQUATIONS. 87
78. — Equations. Use addition or subtraction in finding the
Oral. unknown number or quantity represented by
x, y, or Zz. Explain how you find it.
l «= 12425 6 42— #2=19 ll, 19+11+4+ 2#=50
2. 38+ .12= @ Y Fe elf ON 9 12, 724+ 2+14=96
8 44— 19= @& & 28+ e2=50 13, 404+ 20— a=50
4, «x=100—72 9 ¢— w«=3 14, $2.75 +4= $ 4.50
5 a2— 24=—48 10. J lb.—aw2=120z. 15. «—$7.30= $2.84
16. [ am & years old; in 8 years my age will be 356 years.
(x + 8 yrs. = 36 yrs.)
17. After taking $14, $16, and $12 out of a sum of money $3.75
remained. There were $2 at first. 2—$14— $16 — $12 = $38.75.
18. A prize cup contains 23 oz. of gold, 10 oz. of silver, and a oz.
of alloy. The cup weighs 42 ounces. (Make an equation.)
19, 25 gallons run into a tank, and 46 run out. When the faucets
were closed, 80 gallons remained. There were 2 gallons in the tank
when the faucets were opened. (Equation.)
20. Make a problem about the weather in March to suit this
equation: 31 d.=12d.+ad.+10d.
79. — Equations. Use multiplication or division in finding the
Oral. value of X, y, or z. Hapluin your method.
Note. —3@ is the same as 3timesa@ dy=5x y.
meter FO 6.82 =400 11 2 = 8 16s 4e45— 21
Bey b= or PW Ty = 91. 12:
I
Aer iy es, 30
8. Lofe=16 8142 =700 18, 18, 18y x 10 = 180
|
Bese e= 4 9250 =—625 ~-14., 19, 42y+21= 70
1
5.144+ec=16 10 44a2= 45 16, 20. tof 16% =120
|
-~
~
Sle 8/E Ele slBaw
|
=
Or
838 PRINCIPLES AND PROCESSES.
21, At $3 a day a mason earns $a in a week. (@=6 x $3.)
22: At 2 cents a mile I can ride a miles for $ 4. ( 2= oe |
23. My brother is 8 years my senior. This is + of my age. How
oldam I? My brother is #yearsold. (#w=4x8+8.)
24, ae =S8m. The distance to the city is required.
0;
25, 3 times a number added to7 times it = 280. (8a+ 7% = 280.)
26. 4 of my money taken from 6 times it leaves $55. (Make an
equation. )
27, 30a = $150; fr =15. (Make problems for these equations.)
80. — Oral Explain the process; or give the reason.
Exercises. 1, How many 5’s in half a million?
For analysis. 2. Divide 80,000 by 4 of itself, and the
quotient is a.
3. Compare 2 men’s work with 6 men’s in quantity; in cost.
4, A board bill for 8 days @ $10.50 a week is $a.
5. $111 = _.$ Vs. 6. ~ bu. = @ qt.
7. Compare 62 and 20. If 20 gal. cost $15, 62 gal. costs 1 as
much or 2.
8. 900 miles of railroad cost $18,000,000 at the rate of $a a
mile. $900 is a % of $3600.
9, Compare the interest of $480 and that of $160. [§ 211.]
10, Compare 8 mo. interest with 11 yr. interest.
11. If you know the cost of 7 rape how will sou find the cost
of 9 of the same kind ?
12, Find the cost of housing 14,000 Ib. coal @ 25¢ a ton.
13, At $40 per M., what do I pay for 25,000 ft. of lumber ?
EXAMPLES. 39
81.— For Dictation.
1, Give 6 multiples of 25
2. 373 + %= 1000
3. How much fencing is re-
quired for a 30 ft. square ?
4, A man leaving his office at
8 a.m. is absent 20 h. At what
o'clock does he return ?
5. Square 2; 3; 4; 5; 6
Kemduater so 309s 11; 12
7. What number multiplied
by itself equals 900 ?
8. What is ;8, or 6% of
$ 500 ?
Umit pe 2230, 1a or
what ?
Tato = (0.4 = What?
1l. Give two exact divisors
of 52; 51; 57; 58
12. Divide the cube of 17 by
its square.
1S
18, Divide a million by ten
thousand.
14, V36; V49; 121; V81
15. Give three multiples of —
fd SG Ds apace oa bys
16. Give 12 divisors of 144
17. Make an example in which
the result is an amount; a re-
mainder; a product.
18. The factors of 10a are.
82.— At Sight.
1, 2 yd. of 75¢ satin cost
2. At $0.25, 31
cost x
8, 33 yd. cloth @ 162¢
4, Beef @ $0.25: 72 Ih.
cost x
5, $0.121 raisins: 48 lbs.
6. Coffee, 95 Ibs., @ $ 0.832
7. If I strike out from 39
its largest factor, I divide it by @
and get y for quotient.
8, 22%=a2 V625=y
9, 123—1728. V1728=-a
What should be paid for —
10. 61 yd. of 25¢ ribbon ?
11. 21 doz. buttons @ 15¢ ?
12. At$1.00ayard, 2 yd. silk ?
13. $2 less i of it=2
14, Explain fis difference be-
tween 8%, V8, 8 +8
15. Which represents the cube
of 3:3.x3, or 8+3+3, or
Diane Moet o
16. Find a common factor in
2oOG mano Le
17. Which is larger, subtra-
hend, or remainder in 2148 m.
— 1075 m.?
18. Express more simply,
$0.75, $3
0.25’ $3 3
doz. eggs
40 PRINCIPLES AND PROCESSES.
83. — Business 1, Go over the computations in the follow-
Forms, Invoices, ete. ing bill or invoice to find the errors it contains.
Cuicaco, Aug. 1, 1895.
Ne. Ifemry dS. Warner
Bought of FOHN V. FARWELL & CO.
Nay /8\ 23 yd. Brussels Carpeting, B/,50 \|\84\50
fune 22| 164 yd. Black Silk, 1.75 \|\/8|98
Yuly 6| 8& yc. Wamaeutla Cotton, 0 S2; re (5
08 |\2oO
Leaw ///0 5|62
D2 | #/
Cr.
fune 2 vy Caan, $ 25,00
“BB i Whee 20.00 45\00
heewved payment, | ogee
JoHN V. FaRweLL & Co., |
By Smith.
Make out bills in proper form. Supply dates and names.
2, 13 tons Franklin Coal (@ $ 7.25
6i tons Lackawanna (@ 5.50
1 Cord Hard Wood @ 11.00
34 bbl. Cement (QQ 3.25
3, 2500 ft. Spruce Flooring @ $13.75 uve M.]
2500 ft. Western Pine (@ 46.50
1700 ft. Whitewood @ 30.00 “
Freight, 8.48
4, A grocer’s receipts for one week were $365.18; $193.75;
$96.48; $89.24; $198.65; $479.83. The average of his daily
expenses was $128.00. What was the cash increase ?
INVOICES. A
5. Monday, Jan. 1, 1894, Sam’l Chase had $32.76 to his credit
ina bank. If he deposited $25 every week-day during the month,
and $100 extra every Saturday, what amount could he draw against
Feb. 1? ‘
6. In buying 785 music books (@ 85 ¢ a discount of 1 or 20% is
allowed on cash payments. The net cost is 2.
7. Bill 62 lb. Formosa Oolong Tea @ 60 ¢
30 lb. Maracaibo Coffee @ 241 ¢
2 bbl. “ Bridal Veil” Flour @ $ 5.25
Discount 2% [for cash].
8. Bill 371 yd. Dwight Cotton @ 11¢
ou 1 yd. Scotch Gingham (@ 25 ¢
113 yd. India Silk (Y B1.75
Credit mdse. returned, $8.75
9. A firm buys goods billed or invoiced at $1837, less three dis-
counts. 30% is allowed “to the trade.” After deducting these ;',,
5% is allowed on “large lots.” The amount due is then lessened
by 2% “for cash.” The net charge is 2.
10. Invoice 3% gro. No. 314 Eagle Pencils @ $4.20
gro. No. 404 Gillott’s Pens @ $ 0.574
gro. 4to Blank Books (@ $3.66
rs
*.
ee ~~ (
Nl
84.— Oral Review. 1. 2 x 16, or —, is } of 2, 4 of y, + of z.
At sight. 2. A gross of pens at 4% each costs the
same as } doz. pins at . ¢ each.
8. If12 yd. silk cost $36, 72 yd. cost . 72 x qt, of $ 386= 4 of a.
If > =6, ¢ = what ?
4. 9x, or 9 x w, = the sum of the 9 digits.
5. P=; V144=_. 6. The cube of 3 is . W125 =
1. Add 64, 36, 47, 53, 39, 61, 54, 46, 17, 83.
8. (64+ 8)+2= V2. 9. Va=648+2.
10. + of the sum of 5 numbers is their average. Find the average
of 2, 3, 4, 5, 6, 7, 8.
492 PRINCIPLES AND PROCESSES.
85. — Problems. 1. Make an example to show whether it is
For written work. necessary to begin multiplying with the ones’
figure of the multiplier.
2. What is the value of 795 books at $1.38 each? 8. Write in
words the second partial product.
4. If a young man earns $36 amonth, in how many years would
he earn $ 5670 ?
5. An importer is charged $45 on a hundred at the custom-house
on a shipment of goods worth $18,000. What duty does he pay ?
6. In one minute divide 200,000 by 1728. 7. Write the second
partial dividend in words. 8. How many such divisors may be
subtracted from the dividend, leaving a remainder of 9920 ?
9. A cubic foot of hickory cord-wood weighs 494 lb.; what will
a cord weigh ?
10. How many feet inamile? If 10 feet of wire weigh 9 lb.,
what will 2 miles weigh ?
11. At 15%, how much more than 5000 Ib. of cotton can be
bought for $800.10?
1
86. — Oral Review. 1. + of 15 is z Glee
Al sight. 2. 40f @=124=—2 of.
3. The ee on $2 for a year is 12 ¢ or
6%. The interest for a year on $1 at 6% is what? On $5? $380?
$100 ? $1000 ?
4. A year’s interest on $1000 is $60 at 6% ; at 7% it is a.
5. Give the factors of 28, of 70. Their common factors are
6. V64 — V64 = Va. 7. #& =64; Ve=2.
8. 25% of $800 =a 1=what %? 80% of aton=-2 lb.
9. What fractional part of a dollar is $0.25? $0.331? 12} ¢?
75% of a dollar ? 162% of a dollar ?
10. Find the largest common factor of 24, 60, and 100.
11. 9+5)x3=6e 12 6Gr=945x8 18, 2=V/6i%< 16.
DENOMINATE NUMBERS. 43
87. — Denominate 1. Among the tradesmen in your town a
Numbers. “shilling” is what part of a dollar? What
Oral. other value is given to a shilling ?
2. The last decade of the XI Xth century includes the years . to
... The first decade of the XXth century includes the years 1901
to .. To which century do the years 1899 and 1900 belong ?
How many fortnights in a year? Days in a quarter ?
How many cents = a double eagle ?
At 4 for 7 cents, 2 doz. cost a, and 4 a gross cost y.
4 a dozen, 4 a score, and } a gross make how many ?
IOXAB wf
If there are 25 envelopes in a package, how many packages are
needed for a ream of note paper? For 5 quires ?
8. Ten pencils to a box, how many boxes will a gross fill? st 40 24 50. Balt 160
SS 16 “be 30" 160° 200: -20° 240
Par e.. Goo es see 9 Peer ied @ ia, lt 10
-
— —» ——9 ——=» Lp, toe . Sy ahr ees eae, 01 trem le erer el ere
21 49 28 42 84 91 CUneesUeriew ta oe) b
10 fete “Oooo LU ae 0h ae 80. 100
th laps Eiewer | mrs ? ’ eas
few oo) 40) te. 9 ee ae 00g re or
11. Without changing the value, make the denominators of these
et) 4 BR. 4 1 8 Et 102 3 1
fractions 156: 4, 24, 38, 44, 42%, ade 4 7iv
12. In changing a fraction to lower terms, how do you affect the
size of the fractional units? Their number ?
o
=
‘
“
|
Ne
5
8
13. Why does multiplying both terms of
unchanged ?
@ by 8 leave the value
109. — Rapid Give an equivalent fraction in larger or
Changes. smallest terms : —
At sight.
| ae g 21 64 27 36 73
"4 56 y . Tahewie = 06. 108-505
Plat 568 94 Of :
2. on 7 = 7. 75%, 0.50, 80%, 0.90
4_w 5_y qty ee aye (Oped ye
5 45° 9 45 ~~ 105" 7 500 yy «20
alae 2 v 9
4. = we 12 to 48ths 9, Lies x — 105 _ : ao :
16 24 96 ae AO ae ear) are
5 48 3 96 16 12 2 »v 288
.—~ -——», — to 60ths 10. ——= = = ___ =
12 120 4 144. es-y~. 24 288)" u
110.— Greatest = = 1. Why do we call 72 a composite number
Common Divisor of and 75 a prime number? 2 What com-
Two or More Num- mon divisors have 36 and 108? 60 and 90?
bers. 3. What is their greatest common divisor
(g.c.d.) ?
58 FRACTIONS.
4. Numbers without a common divisor are prime to each other.
Explain which of these are prime to each other: —27 and 35;
16 and 45; 27 and 45.
5. What advantage is there in using the g.c.d. in changing fractions
to smallest terms? 6. Change to smallest terms, using the g.c.d.:
2500108 69 Ph 20 6
Tes) TH, $2, 80%, 0.125, 4%.
111. — Finding For small divisors. 1-8. Which of these
G.C.D. when not numbers are multiples of 2? of 3? 4? 5? 6?
readily seen: Sno aoe OLE |
360 Dige 8397 2160
1728 6984. 6624 3240
Notre. — Any number is a multiple
Of 2, if it ends in 2, 4, 6, 8, or 0; i.e. when it is even ;
Of 3, if 3 divides the sum of its digits ;
Of 4, if the last two figures are zeros or express a multiple of 4;
Of 5, ifit ends in 0 or 5;
Of 6, ifit is even and divisible by 3 ;
Of 8, if the last three figures are zeros or express a multiple of 8 ;
Of 9, if 9 divides the sum of its digits ;
Of 10, if it ends in 0.
For large divisors. 9. Using 8 and 12, or any other two numbers
having a common divisor, show that —
Any divisor of two numbers is a divisor —
I. Of their sum. Il. Of their difference.
Ill. Of any multiple of either.
To change 244 to smallest terms, applying the principles just
stated.
10. Dividing the greater number by the less and
247)533(2 the last divisor by the remainder, we find 13 to be
494 the g.c.d. Show —
39)247 (6 a. That any divisor of 247 and 533 must
234 divide every 247 in 538 (III.), and also the
g.c.d. = 13)39(3 remainder 39 Clays
39 b. That, therefore, the largest common divisor
cannot be more than 39;
ADDED. 59
c. That any divisor of 247 and 39 must divide every 39 in 247
(IIL.), and the remainder 13 (IL.);
d. That 18 divides both, but no larger number would. Hence 13
9 53: 18 | 247 — 19
is the g.c.d. of 247 and 533. 13 | 247 = 18.
ry
i“
~~
>
fr)
=)
bo
—
112. — Fractions a. 10.
ols
saalco
205 6 9_ 1
2 171 465 1325
to Smallest Terms. 9 136 261 8. 333 1], 1281
fT 290 ee aoie | Se SOG
ritten. 278 209 123 1656
Written 38. 353 6. 333 9. 333 12. 3335
113.— Like Frac- 1. What are hke numbers ? Give examples.
tions to Add and 2. What is an integer? A fraction? Isa
Subtract. fraction a number? 3. Show the difference
Oral. between integral and fractional units.
4. With regard to each of the follow-
ing numbers mention.(a@) the integral unit,
(6) the size and kind of the fractional
unit, (¢) the number of fractional units : —
$ pk.; 7% yr.; 48; $3; 7 m.; $0.15; 6%
of a day.
5. Which of the following fractions have units (a) of the same
size? (b) Of the same kind? (c) Of the same size and kind?
Which are like fractions? #yr.;4yr.;¢yd.; 7 day; $4; 3 yd.
6. Why not add 7 1lb. and $3}? #2 wk. and i wk.?
Like fractions
have units of the
same size and kind.
7 pee AN e 147 +21 7 % .
7 —+—+—=— 49. ——=— ll. 72% — = 0.48
271272 1D 36-36-36 2h — 599
fue eee a. =e 050) 19) 16212188 = 2
64 64 64 64
13. How are like fractions added? How subtracted ?
114. — Unlike 1. 14d.4+3 wk. =2 wk. or y days.
Fractions to Add 2. Mention several unlike fractions. Why do
and Subtract. you call them unlike ?
Oral. 3. 4,3,4,3. Which of these fractions have a
common numerator? A common denominator ?
Which are like fractions ? Which unlike ?
60
son Suede Siu sige gaan
the like fractions first.
Why change 18ths to 6ths
Why change 8ths to 24ths ?
7. Which term is common to like frac-
tions? Change 2 and ;§& to a common
denominator. To the least common denom-
inator. To 20ths.
FRACTIONS.
=a ed
6£§ t+i1=2 7 a
40 Add Ly 03 Before finding
their sum or differ-
ence, fractions must
be changed to like
fractions.
8. In adding 3 and 3} shall we use 4ths or 60ths as the common
fractional unit? 4 or 60 as the common denominator? 9. Why
is it easier to use the largest common units — or the smallest common
denominator —in adding? 10. Give three steps in adding 7 and 5%.
115. — Oral or Written.
. Change to a common un
i
2 3
. Add or subtract numerators. ye 5 aes
, 4
4
1
2
53. Simplify the result.
4. Find short methods.
9.
10. $+ 4+de+at
ll. 20% +4+4+0.20 +
1. F+E4+ E4404
13, 4 haere
14. 18% +0124 3+4
20. + wk. + 325 yr. +
116. — Multiples:
2o+s Oa
ae tet 3 6. $—-@
100 le er
#+3 8. 25% + $F
fo aecte aes
75 lo. ¢typete—ats
$ 16. § +43 + 36 + 23
25% 1% 34-2484
18. 36% —4+4+0.14—4
: _ Wee +e-Ht a
id. — 5k mo. + 180 min. = «
1. Show that 36, 60, 72, and 120 are multi-
Common Multiples; ples of 12.
Least Common Mul-
2. Show that 50, 60, 80, and 120 are multi-
tiples. ples of 10.
ae
Norr. — The term “‘ divi-
dend’’ may be substituted for
‘*multiple.””
3. Show that 60 and 120 are common multi-
ples of 10 and 12. 4. Show that 60 is the
least common multiple (1.c.m.) of 10 and 12.
LEAST COMMON MULTIPLE. 61
5. 2, 6, 3, 10, 15, 5 are factors of 30. Which of them are the
prime factors of 30? . What is the product of these prime factors ? |
6. 2, 6, 3, 14, 7, 21 are factors of 42. Select from them the prime
factors of 42 and give their product.
7. The product of the prime factors of a number is always
8. Show that any multiple of 30 contains all its prime factors.
9. Show that any multiple of 42 contains all its prime factors.
10. Show that any common multiple of 30 and 42 contains all the
prime factors of each.
117. — Finding the To find the least com- 80=2x3x5
L.C.M. mon multiple of 80 and 42=2 x3 x7
Aero et Hemerimilgiplartic Ne tate ee ee
containing only such prime factors as are
needed to produce each number separately.
1. Will a number whose factors are 2, 3, 5, 7 be a multiple of
2x3x5 or 30? Of 2x3x7T7 or 42? 2. Why need not the 2
and the 3 be used twice in this multiple ?
3. What factor not found in 42 is needed in the common multiple ?
4. Find the l.c.m. of 60 and 84.
5. Find the least number exactly divisible by 60, 72, and 108.
Oral and written.
6h Fo & eG 6. What prime factor of 60 is not found
72=2x2x8x8x2 among the prime factors of 108? Of 72?
108 =2x2x3x3x3 7. What is meant by the least common
5x 2x 108=« multiple of several numbers ?
8. What is the l.c.m. of 60, 84, and 132 ?
9. Find the l.c.m. of 45, 90, 100, and 200.
10. Is a multiple of 90 a multiple of 45?
45 . :
ay ee eae Compare 200 and 100 in this respect. How
100 then may the process be shortened ?
200=2x5x2x2x5 Nore 1.— The prime factors of large numbers may often be got
200 XX B= 2, by finding composite factors first, and then the prime factors of
these. Thus:—
120 100 12 eX O) Ke Mie OCS) LOU LO xe LO = (SO) 0 (YIX'S %.B),
Norte 2.— For other methods of finding the ].e.m. see the Appendix, p. 8,
62 FRACTIONS.
Find the least common multiple of —
al 15 2145
12.16, 18, 27, 72
13. 16, 25, 80, 100
14. 12, 18, 96, 144
15. 34, 85, 51, 68
16. 480, 600, 1000
Practice in Changing Fractional Forms.
118; — Oral.
l. Read as mixed numbers:
39 47 289 365° 300
29°
CE) ye) Sey BR
2 Putinte Sa he form:
6 F&F 6 4
45, 58, 102, 20%, 62, 134.
3. Change to ecatteat terms:
2'8 V4 6:85 325 64596 09 1.0'S
44). 51? 72? 72? GE 84) 120°
4. Simplify the form:
JA AMO EE op Ete i i hen Ss
Ci mle t oe LAUT 0.160.
5. Read as 144ths: 1, 2, 54,
Si TE ay A SOD)
167 167 7.27 2:38:87? 14402 Teale
6. Change to smallest terms,
} - 25 2 52 48
using g.¢.d.; ee eb 36 30
32 Olea? 72
480? 108 288? TET
7. Find l.c.m. of 5, 7, 35, 70;
8, 16, 128.
8. 64 = how many 8ths ?
3 2 2 @
Ope ean pee
79-66 2 eee 63
Di tn0 Ate
0 0 6 12 8° 9
11. What parts of 144 are 1, 2,
Beess, OMe. 729
12. Compare in value:
Pee Wey ee Myer ary eee
695) 175) 98? 99) 56) 6S
18. Read in order of size:
By Sb wie Be eke, leh. 3! 12
TiO? oAmige iso) wlelvedeo? sel eisai
119. — Written.
1. Write two large fractions,
using four figures.
2. Express the same values
with 12 figures.
3. Change .to mixed num-
bers: 8248; 1990; 700, $251
4. Simplify the ae of $139
486 doz.; 352% ed.
5. Change 75%
terms.
209
to smallest
as 625ths.
1. 52%, in smallest terms =?
8. Change 27 and 45, to like
fractions.
18
6. Express 48
9. Find the least common
multiple of 16, 48, 96, 108.
10. Change to smallest terms,
Sa CETE
3080
1l. Express 24% with four
figures.
12. Condense this expression :
13. Shier ae 19613 and 2234.
14. What part of i7 28 is 576?
63 x 2000 x 31
te, ete
125 x 1260x70
120. — Fractions
Added or Subtracted. adding or subtracting
Process.
15=3~x 5
18=2x3sx38
sO = Py 1, CO:
or l.c.d.
6
LL 90 _ 66
13 °° 90° 90
5
7 _ 36
13 °° 90 90
Sum = 49) = 143
121.— Written.
ADDED OR SUBTRACTED.
63
1. What is the first step to be taken in
it and 54? 2. Is the
common fractional unit readily seen ?
38. What is the lem. of 15 and 18?
4. What then will be the smallest common
fractional unit, or the least common denom-
inator ?
5. How many 90ths =
how many 90ths? 54?
7. How are the size and the number of
fractional units changed by multiplying both
terms of 14 by 6?
?
Sag eS wl. a pe Dee
Lioiee Lidwe ~ Lice
8. What remains to be done after the
fractions have been changed to like fractions ?
l ~+A=2 11. Add 7%.+ 44 to the sum of $4 and 41.
2. $f+ii=a 12. What shall be added to 5% to make 49?
8. Fe t+e=H 13. Which is greater, °, or; ? How much?
4, 294 298=¢% 14. Add 23 to 4%. What should be the
5. 38; + 38255 = a first step here; and why ?
6. 0.8—0.625 =a 16. 2444 2314 723+ 16541818 Add
7. 80% —3=2 the fractions mentally.
8 $4 5428 = 16. Add 138, 16%, 0.027, and ;48,.
9. 518 + 2149 = 17. 323 4+1714=¢2
10. «7+ 13 = 729
122. — Subtracting
Mixed Numbers.
Oral.
Process
W575 = 1535 = T44P
5711 = 5733 = 5733
Difference = 17$4
1. In subtraction of integers, what 1s done
when a digit in the subtrahend is larger than
the one above it in the minuend ?
2. How is the 4%,
at the left? The 74?
3. Tell how to take 73 from 154.
109 obtained in the process
64 FRACTIONS.
4.123—-Ti=a 6. 10—3§— 25 8. 98 — 514
5. 100 —aw= 83 1. 52 — 33 — 3 9. 334 — 121
10. 20 —{—{-— {4-4 12. 15 — 28 — 28 — 26
ll. 93 — ¢ — 5,3, 18. 203 — 3 —75% — 0.75
123. — Written. 1. 6313 — 417 3. 725 — 1514
2. 944 — 18,3, 4, 1913 —517
. 14,9, — 6.51 6. 0.875 — 0.625 +, By + hts
. What is the differ ence between 874 % and 662 %?
. What added to 17,9 gives 29%,
10. From 184 take 9.935
5
7. 11411 — 1632 + 8419 — 1623
8
9
=
124. — Problems 1. 4, 2, and + of a number=110. Is the
with Fractions. unknown Aine larger or smaller ?
2. A barrelis $full. Draw off 4 of a barrel
and 2 of a barrel. What part remains ?
3. A stone wall cost $1 arod. What costs 3 days’ work, or 63 rds.,
53 rds., and 72 rds. ?
4. What % of an income is collected when 0.125 of it, 25%, 0.35,
and 0.025 are paid in ?
5. A chimney contains 182 courses of brick. =; are under ground,
24 roofed in; how many courses are ara ?
6. How een cords of wood in 2% ed. sawed by hand, {24 ed. by
machine, and 7% cd. chopped ?
1. Two dane contribute + and 1 toward filling a reservoir,
springs contribute 4+ and 4, surface ma the rest. How much more
do the pumps yield than other sources ?
8. Ina10 acre marsh lot three men cut 4, =3,, and 2 of the whole.
What part remains oF a fourth man to cut ? f
9. If you Bn eh , of what you have in one way and 3 in another,
what remains ?
10. At $3 a day what is due a man for working half a day, 2 d.,
$d., and 213 d.?
Oral or written.
MULTIPLIED. 65
125.— To Multiply 1. 9x7 units =a units. Does it matter
a Fraction by in- whether these units are integral or fractional ?
creasing the Number 2. Then it follows that 8x3 fifths =@
ee fifths, and 9 x gigs,
SOLS
A : = - or 3 and 7. The product of 22° by T=a
4. 6x35 =2, ory andt+ 8. 24x $2=a
6. §4+8+4+8+4848=% 9, 100x $4— 4
6. ;4 multiplied by 11 = a 10. 50x $8 =a
1 23 a 12. Multiplicand 54, multiplier 9, product ?
13. In the preceding exercises, have we changed the number or
the size of the fractional units?
126.— To Multiply l. James has $3, Henry has twice as
a Fraction by Increas- much. Has Henry 6 quarters or 3 halves?
ing the Size of Parts. 2 x }=—3or $? .
2. If I have ten }’s of a dollar and change
them for as many coins of double the size, what fractional parts do
I get, and what increase in value ?
3. Compare ;, andi. 4x i =- lb. What change is made
t
4
‘ |
in value when instead of 16ths, we take as many 4ths ?
ind the product by increasing the size of the parts : —
4,.6xq 7. 10 x 34 10. 15 x 4
5. 8 x 13 8. 12 x 48 yd. ll. 25 x 348;
6. 83xi 9. 18 x 4 m. 12. 36 x 7;
127.— Multiplier, 1. Which is larger, product or multiplicand ?
a Fraction; Multipli- 2. Is it proper to say either 4 times $6, or
cand, an Integer. $6 multiplied by 4? 3. What about the
product here? Is anything really multiplied ?
66 FRACTIONS.
4. Infinding } of $8 to be $2, do we multiply or divide? 5. 2 of
6 yr. (or 2 x dof 6 yr.) is 4 years. What part of the multiplicand
has been taken? What does multiplying mean ?
6. Show that multiplying by a fraction is finding one or more of
the equal parts of anumber. 7. Is this more like multiplication or
division ?
8. 2of 24h. 13. $ of 830=5 x 42 = 20139 = 218. Explain.
9 tof 36in, 14 2 of 3) oe ae = 21f. Explain.
(
10. 14 of 84mo. 15. % of 20 18. =3, of 100
ll. fof 60d. 16. 3, of 50 19. 0.23 of 200
12. 42 0f $100 17. Sof 100 20. 24% of 500
128.— The Productof 1. Show that 7, of $20 =20 x $54
Fractions and Integers. 2. What is the a of series
Last 3. Compare 4 not 106 = = x with 105 ae
pp 5
ee au
4. — of 144=—-—=404. Explain.
60)
5
5. 13 of 105. 7. 18% of 250. 9. 17 of 225 tons.
6. 65 x +5. 8. 8 of 85. 10. 840 x 44 yd.
129.-— One Factor 1. At $3.50 a yard what will 2 yard cost ?
a Mixed Number. 2. If one revolution of a an requires
Written. & of a second, how long will it take it to
revolve 1000 times ?
38. 20f 400 =a; 16 x 400 = y; 162 x 400 =2+ ¥.
4.17 x8=2; 17 x 300=y; 17 x 3008 = a+ y.
5. At $0.95 a pound what will 742 pounds of tea cost ?
PROBLEMS. 67
Process.
$0.99 Copy the accompanying process, supplying
_ 145 values for the letters.
9) $6.65 or a x $0.95
0.738 = § of b Nore. — When the cost includes a fraction of § cent or more, it is
3.80 =c x $0.95 customary to count the fraction as another cent. Why is this done?
Answers given in this book conform to this custom.
66.50 =d x $0.95
$71.03§ or 6. Price $8.75; quantity 183 cords; total
$71.04 RE
7. Weight of one bag of coffee, 352 lbs.; 19 bags weigh what ?
In the process at the left give the values of
Process. a, b, ¢, and d.
355 Ib. 8. 15 boxes; one weighs 183 lb.; all?
Doe 9. $6.25 each; 273 yards; cost?
8)133 or 19 x a 10. 15,4 miles an hour; 25 hours ?
163 Ib. 19 x b 11. Rent, $28; time, 7; months.
alte, Coe : rat :
12. Time, 2911 d.; sailing rate, 185 miles.
30 Od Ko) : :
13. 2 of $785 was lost in speculation;
681g + ataey
what remained ?
130.— Problems. 1. What remains of a 49-yd. piece of cloth
For written work. after selling $ of it, 3 of the rest, and 4 yd.?
What is the remnant worth at 83¢ per yard ?
2. Another piece of 47 yards is damaged. One half sold at 7¢.
Of the other half 23 yards were unsalable, but the rest went at 5¢.
Give the total receipts.
3. At the rate of $12 a day, figure a board bill in dollars and
cents for 3 months from August 1.
4. If a glacier moves uniformly a hundred feet a year, how far
does it go in 181 days ?
5. A sawyer works up wood at the rate of {9 cd. a day. What
can he do in 26 weeks if he takes a half-holiday each Saturday ?
6. A wind storm passes over ,), m. in 3,3, sec. In what time
would it go a mile? 7. If it travels +3, m. each second, how far
would it go in an hour?
68 FRACTIONS.
8. Find the cost of seven 50-gal. barrels of oil at three for $16.71.
9. Supposing an empty barrel to be worth $1.25, what is the oil
worth per gallon ?
10. An unfailing spring flows 374 barrels daily. How much
would it yield in October ?
131.—To Find One 1. 2 of 6 things (apples, dollars, fourths)
or More Equal Parts =v.
y) y)
oe er or 2. = of 10 twelfths =a; = of e a = ses
D Og Lele
ans yp IP as
SUS ore a 5. of 22 =a 1. ps of 38 =a
4. tofis§=—-2x 6. 75 of 42 =a 8. tt of 72 =a
To separate * into 5 equal parts and give the value of 3 such parts ;
that is, to find 3 of #, or to multiply 2 by 2.
Process. 9. How does increasing the denominator affect the
lof3=,3, size of the fractional units? 10. If we make the denom-
of }=,% inator 5 times as large, how is the value of the fraction
: : ; = s changed? tot#—. ll? If of a-traction=— 3. oF
2)
it will be how many times 53, ?
12. Which part of the process gives the product of the numera-
tors? Of the denominators? 138. Make a rule for finding the
product of two fractions (i.e. for finding one or more of the equal
parts of a fraction).
132. — Fractions 1. Find 19 of 24, or multiply 24 by 19.
Multiplied. 2. Of what use is cancel- Process.
Cancellation. lation? 8. On what prin- 12% 3$=220=7
ciple is it based? 4. Which a i h
is easier in the process at the right, to change the 12° 338.7
product to lowest terms or to cancel first ? ee
5. ay of #9 THX Hs t
6. +4, of 32 8. 28 x 24 10. 96% of 23
MULTIPLYING FRACTIONAL NUMBERS. 69
133.— Multiplying 1. Find the product of 14 and 24
i
Fractional 2. What is 2 of 74 (2 of 43)?
eos 8. 88x 63—-a (First step ?)
Written. 4. How may the product of small mixed
numbers be found ?
6, 22 x 73% 1. 4 of 8 of 34 9. 16% of 54
6. 44 x 158 8. 22 x 44 x Th 10. 94 x 1219
11. 168 x 246 = what? [§ 68.] 12. 68 x 1373 = what?
i aie, 13. 84 x 126 Explain the process.
xplain the process. “
¥ 14. 149 x 72 187%
246 ; $ 63
65 15. 83 x 2094 :
6 x 137 = 822
16. 68 x 1943
ee \=(5 x 246) +8 Ne Sar 8 x 137 = 1028
163} 17. 264 x 7952 6xi = 33
P10 OX DAG Fe 15
‘ 18. 84% x 641,5, x' = #
3 ;
ice 19. 962 x 109.8, 92834
134.— Business Make out bills in full for —
Problems. 1. 9.
For written work. 17 doz..@ $ 1.624 178 yd. .@10¢
152 doz. ..@ $ 1.00 13 yd. —.@ 621
3. Find the cost of 92 tons of coal at $ 7.41.
4. ‘Twenty pounds of sugar bought @4,% are sold for $1.25.
At this rate what is gained on a barrel of 200 lbs. ?
5. Oil is bought at $3.50 for a 42-gal. barrel and retailed at 121¥.
The gain is what part of the cost?
6. Oranges bought at 3 for 5¢ are sold at 4 for 9¢. What is
gained on a box of 9 doz., 1 in 12 of which are worthless ?
7. I can buy blank books of one dealer at the rate of $1.25 a
hundred; of another at $1.60 a gross. How much is one offer
better than the other ?
70 FRACTIONS.
135. — Products at Multiply each fraction in the table by the
Sight. number at the end of its line or column.
Oral. Change any fraction in the product to a
lower eter et when possible. Thus :—
xo yd. = 20 yd. = 22 yd. = 2 yd. 8 in.
vu
10
& sq. yd.
8 :
15 M1.
Division of Fractions.
136.—Fractions to %+5. To divide % into 5 equal parts.
Bea ake 1. Each part will be t of 3 or — 8
[See §§ 107, 131. ] 2. What is meant Dads § 42
4. Divide 2 ft. into 6 equal igen 5 x “+ @ = ah.
6. 16+4= a 7. 43+4= a 8. Using the two preceding
examples as illustrations, give two ways of dividing a fraction into
equal parts. 9. When is the second method used? 10. Which is
shorter ?
ll. 4 of 24, 14. 51, of 4 17. 0.15 +6
(Dy yet: Woe LE aby 18. 48% + 16
1gaaeees 16. 4, + 20 Vid eee erin
DIVIDED. 71
137. — The Divisor 5+3. To find how many times 2 is con-
a Fraction; the Divi- tained in 5.
dend an Integer. 1. How many times is } contained in 1, or
how many 4ths in 1?
OP od
a eS 5 x
2. How many 4thsin 5? 5= 7k 8. 3’sin20? 249+2=2.
4. 2ft.+3in.=2; or, since dividend and divisor must represent
y] ?
like units, 24 in. + 3 in. = @.
a eect: Geay ie ee 8 j » t]
5. 72 1. T+2=a. 9. 58 in 8, x times.
6. 8+4=2. 8. 2 in 7, # times. 10. Divide 17 by 3.
—J
138.— The Dividend #+2. To find how many times % is con-
and Divisor both Frac- (tained in 3.
tional. 1. Which is the larger, 2 or 2? Are the
: ay OA se ;
units of the same size? 2. ~=—; -=>~3 w+y=11.
Oe Be
3. 4 or = is contained how many times in ¢ or 12? 12+5=2a.
4, +2 6. §+32 8. 55, in $ 10. Divide 22 by 55
5. $m 7. +e 9. $+F 11. Divide 0.9 by 30%
12. What is the first step in dividing days by hours? In dividing
4ths by 5ths? One fraction by another ?
139. — Denominate 1. 2 of a case of slates at $43 a case cost
Numbers. $a. 2. A ten-pound box of marbles contains
Drat: 4 yellow, 2 blue, and 48 red. The box con-
tains how many? 8. The box cost 30 ¢, and
the marbles sell at ten for a cent. The profit is a.
4. A girl who earned 2 of a dollar gave 3 of it to one who had
nothing. With the rest she bought three things that may have cost
x, y, and z cents.
5. A dealer bought $10 worth of oranges. After selling a fifth of
them for $3, he sells the rest for what the whole had cost. What
was the profit ?
ke FRACTIONS.
6. 1 of a bushel of berries are picked; 4} of them are sold to one
man, 1 of the remainder to another. How many quarts are unsold ?
7. A certain sand-glass runs ten minutes. It runs out twice during
© of an hour are spent in the morning, and
practice time at noon;
In 6 days it would be a.
jz at night. Give the sum in hours.
8. At 51¢ each 64¢ buys a, and 80 ¢ buys y.
9. $10,000 in postage stamps are divided among 4 offices. If 3
of them are twos and the rest ones, how many ones does each office
get? 10. How many twos are distributed ?
140. — The Divisor (a) +3 To find how many times 3 is
Inverted. contained in +
Oral and Written. First Process.
1. To what common unit are the fractions changed ?
2. How many times are 24 units contained in 35 units ?
Second Process. Analysis of Process. + is contained
in 1 five Pens 3 is contained in 1,
3 : ;
4 of 5 times or 3 times. Since 3 is con-
at 1 it will be aon < of 3 times or
ala
tained in 1 3 times, in
3 times = 141 times.
But 3 is the divisor 3 inverted. Hence to divide one fraction by
another, we may
Multiply the dividend by the divisor inverted.
3. What advantage has the second process over the first ?
4. What disadvantage may it possibly have ?
Apply the shorter process to the following and explain it : —
5. #+2 7 $ini 9. What part of 3 in 3?
6. ~§+¢4 8. £in 54 10. What part of 5% in 3?
Notre. — Cancellation will often shorten the process in division as in multiplication of fractions.
25. 2) pei a PR, es eB es 6 340 +
11. Tt eatee BR) 12. 44° 71 18. 250 ° %5 14. 625 > as
DIVIDED. 73
141.— Division of 1. Divide 123 by 32. (Change mixed num-
Mixed Numbers, bers to improper fractions.)
Complex Fractions. 2. 158 + 93 5. 822 + 322
Written. 3. 733 + 574 6. 1000 + 662
799 - A498 321
24. 3
8, ~2 1s a way of indicating the division.of 2} by 73. 2=~.
i> =
3
Such expressions are called complex fractions. To ahenne them
to simple fractions, multiply each term by a common denominator
of the fractions. Thus:—
7k x6 44
2 3 ue 3
_ 165 10, 18% Taos 12, #
100 100 304 aor
142. — Mixed 1. What are the two steps in dividing
Numbers Divided. 16453 by 9?
Written.
2. 34768 +8 =a ST OLCSS.
5
8. Divide 73294 by 12 ee ,
7 ¥é
a UA1Sp I = & a. }of 16453 = 182,
5. How many times is 11 contained and 73 remaining.
in 8764? b. Zot TZ = 4% of A= fe
6. Divide 2893 by 26. To what common fractional unit are both
dividend and divisor changed ?
Process.
26) 280% 7. 3672 + 24 9. 47232 + 105
m4) 4 8. 8462 + 39 10. 69481 + 216
104) 1159(11y4%; ll. If 75 boxes weigh 847,55 lb., what
= will one weigh ?
104 12. 89 rd. = 14683 ft.; 1 rd. = x ft.
rbzg0f 15 = 104
74
143. — Dividing by
Mixed Numbers.
Process.
$ 248) $ 1428
oe 8
195 ) 11424(58
975
FRACTIONS.
Oral. —1. If one chair costs $248, how
many can be bought for $1428, and how
much will remain ?
2. Why do we change dividend and divi-
sor to 8ths? 3. The remainder is always
a part of what? 4 What is the unit of
the first dividend? Of the second? Of
the remainder ?
Written. —5. How many times is $428
contained in $2500 ?
6. 1 sq. rd. = 2721 sq. ft. ; 1728 sq. ft. =a sq. rds.
7. Find the quotient and remainder; 2000 cu. ft. + 182 cu. ft.
8. At $ 0.873 each, how many spoons can be bought for $75, and
how much remains ?
9. Of $525 I spend as much as I can for bicycles at $1257 each.
With the remaining money, how far can I travel at 2 cents a mile?
144. — Fundamental Processes applied to Fractions.
PRACTICE TABLE.
. b. c d. é. ag q. | h. ds
1. 2 3 i} 23 63 152 516} $ 100
2. 3 2 3 12 181 284 4932 250
3. 2 3 a O35 975 213 6412 500
4. i Ts a5 62 103, | 642 8274 576
5. 4 ops Fire 98 152 853 9364 600
6.| 3 35 BL 84 212 | 905% | 14641 640
1.) 45 ee 13 dit If 164 | 2525; 720
8} 44 pe Te 93% | 124 | 721 47693 800
9. 2 3 zis | 104, | 183 362 84613 960
10.| i4 + es 348 | 2014 | 253 | 755055 L000
FOR PRACTICE. 75
To tHe TEAcnER. — Each of the following 45 combinations may be applied to each number in
the designated column so as to furnish ten examples, which may be assigned consecutively from
I to 10, making 450 in all. How much to use depends on the degree of accuracy and _ facility
attained or desired.
Addition. Subtraction. Multiplication. Division.
l. 6+2 ll, 20 —e Sher Se.0 ol. ba
2. ¢+3 12. f— 54 Boe LG 32. a+c
38. d+ 13. b—e 23. axg 83. b+¢
4, b+c¢ 14. d—corc—d 24 bofd 34. e+d
5. c+d 16. f—e 20.16: % oh 35. f+e
hep ¢-d 16. g—f 26. bx exd2 80. oF
1 e+f 17. 947 —g ahs OO Gee ST 9
8. f+g 18. h—g 28. fxg 88. h +75
9 e+ftyg 19. g—d 29. 67 xh 39. i+e
10. e+ft+gt+h 20. g—e-f 30. g xi 40. i+g
Mixed. —41. b+c—id 42. g—(e+/f) 43. cof f+e
44. gy+exf 45. (b of e)+(c of f)
145.—To be 1. An heir gets ¢ of an estate, then loses 3
Formulated. of his share. What part of the estate does
Written. he keep ?
2. I buy at 20% discount. What is the total cost to me of goods
sold regularly for $1.42, $3.98, $57, $0.162, and 9 pieces at
$ 0.314?
3. If 84 tons of coal cost $487, what is the cost of 68 tons?
4. Property which cost $5000 is rented for $434 a month; what
is the annual income to the owner after paying a tax of $15 ona
thousand ?
5. Three cheeses, weighing respectively 344, 423, and 474 Ib.,
were sold for $20.60; what was the price per pound ?
6. J. F. Sampson bought 721 bu. potatoes at 621% a bushel, and
sold 3 at 641, the remainder at 75¢; what did he gain ?
76 FRACTIONS.
7. An electric launch was sold for $285, or 32 of the cost. Find
2% of the cost. |
8. 2 of a ton of hay (@ $20 pays for 14 tons of coal at how much
a ton?
9. 161 ft. of 2-in. pipe @ 61¥, and 1020 ft. of 1-n. pipe @ 41f,
are exchanged for 120 lb. tubing at 1149, and 134 ft. @9¢. What
is the difference in value ?
10. Two trains start together in the same direction. How far
apart will they be in an hour if one goes a mile in 1,3, min. and the
other in 85 sec. ?
146.— Fractional Parts Compare with 100:—
of 100. 125055 25, 91.5, 920, ee LO oo 1 0,
At sight. Oi. 1D) Ole GS
Repeat rapidly, until thoroughly learned, the values of the following
parts of 100: —
Quit. bee oi euros ieee y fae ee is tee BAT a Fa OL
Sas 3 Bp! 5 5 y SFU 12 16 20 2D ee O 100
42 4 Hl es Oe oe on Pea a Sake Me ei oe i!
ie Pa 5 «6 6 he eke 8 == ARH) 12 16 DRY Ue ANE yl
5a he AS See 0 oe See ees a) ee
ak} 10 12 16 20 PiDmeDIO ee 16 40 3 16 16
Bris eZine 4207 ea eae 10s cia ey hs Seed Be
ert sO 6 6 8 8 8 LD 16 12 40 16 L 2
11. A nurseryman sells 2500 strawberry plants @ $6 a hundred.
They cost him 3 as much to raise, and he gives an agent 4 of the
profit. How much does he gain ?
147.—To Find “Compare one number with the other in each
what Part or what of the following columns. Thus: —
per cent One Num- 8 is 3 or 834% of 24.
ber is of Another. 24 is 3 times 8 or 300% of 8.
The ratio of 8 to 24 is 4.
The ratio of 24 to 8'is 3.
Saye
1
2
3
4.
5.
6
is
8
9
10.
148. — Finding
NUMBERS COMPARED.
[T. IS:
25, 64 90, 18
314, 10 Oe) ot,
3,4 2 gal., 3 qt.
72, 60 30, 50
37k, 64 4, 0.16
3, & 0.68, 0.51
3 103, 5}
10%, 20% = Foo Yor
$0.75, $1.25 $1, 100¢
374, 874 334, 100
Va
Palen ie
1 wk., 1d.
5 min., 25 sec.
144, 48
16
_80
100° 100
77
177 2 a ir
Ax 3 XS
0.98, 0.31
1, 200
$1, $1.50
1.831, 1.00
What number compares with LOO as —
what %.
4 with 28 15 with 40 6. 85 with 105 = 3d.with 1 wk.
2G) 280 6% 142 7. 4in.withiyd. 2mo.with1 yr.
25 “ 150 24 “ 42 BEE OZ Stee Llores dai. 88s cL ini
LE ois 8: aS eee one ue Looe t feta. LOU 1B." er 1 1”
18 “ 54 PGuee meceeeme Or ite Ca al at fee a coe gl
149.— A Part Given; 1. 12is tof 5. 28 is 2 of &
the Whole Required. 2. 16is tof a 6. 36 is 3 of &
Oral. 8. 24islLof a 7. 72 is ¢ of &
4. 19isiofz 8. 100 is 19 of &
9. What is 7 of 960 ? 11. 450 is 43 of what?
10. What part of 23 is #? 12. 175 is 23 of what ?
13. 47 or 85% of x tons of coal were sold.
14. What part of 1200 is found in 480 ?
15. 33 is + of what ? 18. 7 is $ of what ?
16. 123 is 3 of what? 19. What part of 2 is 7; ?
17. 142 is 2 of what ? 20. 183 is what part 62} ?
78 FRACTIONS. -
150. — To find the 1. 16 is 4 or 50% of x
Whole when a Part or 2. 241s 3 or 75% of x
Per Cent of it is known. 3. 32 18 2 or 662% of x
4. 40 is 2 or 831% of x
5. 56 is % or 874% of a
6. 20 is 25% of w 9. 2118 10% of x 12. V25 is 5% of &
151s 124% ofa 10. 321s 331% ofa 18. 361s 18% of x
eeu is 3710 ofa Ill. fis 20% ofa 14. 9? is 27% of x
a
151. — For Frequent 1. 35 of 10 10 x 10+ 45
Practice. 10 is = of what number ?
Oral. 2. Change to fractions of a dollar in lowest
terms: $ 0.124, $ 0.375, $ =25,, $ 0.662, 831%.
3. Express in cents these parts 6. What is 1000 times —
of a dollar: — $ 2 + + 7
aoe ern Rae t. Give the fewest cents that
will pay for 1 when the price of
4. Of 1000 find — 12 is—
+ 4 32 § fF 1p 16 18 20 25
30 oo 38 40 42
5. Use each of the following 4° 50) 65 70 88
numbers as divisor of the one at 8. Add the following frac-
the right or left of it: — tions : —
ou 1d Liesl 65 26 3 4 +9 4 rs
& 10 621 1000 125 2000 2 4 335 4 oy
Give rapidly the following parts of 100
9. 4, 4, 4, etc, to. 10. 2, 3, 4, 5s $, etc., to 49
2? 39 4
152. — General Give numbers when needed to explain your
Questions. answers.
1. What are the processes of combining
fractions ? Of separating them ?
2. Of what use is it to change the form of a fraction and not its
value? Explain the principle.
PROBLEMS. 79
3. State the method of finding the sum of two fractions if their
units are not alike.
4. What is meant by “higher terms” and “lower terms ” ?
5. Show a connection between % and fractions.
6. Why is the product of two fractions less than either? 7. How
can a quotient be larger than its dividend ?
8. Compare common and decimal fractions.
9. In working with fractions what is the need of finding a greatest
common divisor? of a rod?
c
js; mM. = rods.
$9lin.=azit.+yin. 8 3ft. x15} x 320=1m.=-2 ft.
9. Learn the distance from home to school by measuring and
counting your steps, or in some more exact way.
eae I eS
10. Estimate in rods and then in feet the dimensions of your
schoolroom; schoolhouse; school lot; the width of the road or
street. Test your estimates by measuring.
213.— Length 1. 3 yd.=«@ ft.yin. 2. 621% of a mile
Measures. = @ rods.
At sight. 3. Ata cent a foot, 4 rods of picture cord
will cost # cents.
AAT 12054 yard, picture moulding for a room 265 ft. long and
20 ft. wide will cost a.
5. drd.=aft. 6 j-rd.=axft.oryin. 7. 100 in.=2 yd. y ft.
gin. -8: 1000 trd.=a2m. 9. 2m. 40 rd, =-2 rd.
10. At the rate of 3 m. an hour, how many rods can you walk
in 15 min. ?
214. — Length 1. An ocean steamship 660 feet long is
Measures. what part of a mile in length?
Written. 2. Steaming 22 miles an hour is at the rate
of x feet every second. 3. Crossing the Atlantic, a distance of 3100
miles, in 5 days 74 hours, the average rate is # miles per hour,
a Bs MEASUREMENTS.
4. Mt. Everest is said to-be 29,002 feet or a miles high.
5. The distance from the equator to the north pole is ten million
meters. Calling a meter 39.57 inches, what is the distance in miles ?
6. One wheelman rides 24 miles an hour. Another rides 1 mile
in 34 seconds. Compare the distance per minute each one rides.
7. A race-course was 30 knots. The time of the winning* yacht
was 3 hours 25 minutes. This was x feet per minute.
8. A horse trotted a mile in 2.04, or at the rate of w rods and y
feet per second.
9. In May, 1893, the Empire State Express ran from Syracuse
to East Buffalo, 145.6 miles, in 2 hours 21 minutes. Find the rate
per hour.
10. The driving-wheel of the locomotive was 64 feet in diameter.
Calling this 5%, of the circumference, and making no allowance for
slipping, how many revolutions would it make ?
215. — Surface 1. What is a plane surface ?
Measures. 2. What are the boundaries of surfaces
[Review Tables, p.9.] Called? 38. What kind of lines bound rectilin-
ear surfaces (rect- meaning right or straight) ?
4. What is the shape.of most of the common units of surface
measure? 5. Describe a square; an oblong, or rectangle.
6. Name the five square measures of surface, beginning with the
smallest, and giving the length of each. 7. An acre is not a square
measure. It contains square rods. 8. 12?=a; (51)?=y; (164)’?=z.
9. Give the length of a square yard in yards; in feet; in inches.
10. The length of a square rod in yards; in feet; in inches.
11. What is the length of a square mile in rods? in yards? in
feet ?
216.— Of Surfaces 1. Draw a diagram to show what a square
or Areas. inchis. Isit aninchsquare? Might it con-
tain a square inch of surface and be of some
other shape ?
OF SURFACES. 113
2. Draw a diagram as an example of a square foot. If your paper
is too small, draw it on a scale of 4 or 4 or 4; that is, make it } or
4 or 4 of its actual length. 8. Divide your drawing to show the
number of square inches in a square foot. How many are there ?
4. 5. 6.
= 3q Thi. =%. 89.10 = 84 fre Sole i 36 sq in, = 1 sq ft
2 a
1 1 5
3 84 t= 47 80; in 3 84: tis ence Lh 60 sq. in. or sq. ft.
; Beit. ==4y BQ4 10, Bde 2.50. tt | LOS Sq I =3 sq. ft.
7. 662% of a square foot =x sq.in. 8. 14 sq. ft. = sq. in.
9. Represent a square yard in outline; scale, 4. Separate it into
square feet. 10. How many square inches in it?
217. — Of Land LL roa yeaa ott. 2 i.
Areas. 2. Draw a diagram on a scale of =; (44 of z
Written. inches long) to represent a square rod. From
one corner mark the yards along two sides.
3. Separate it into square yards. You find that you have 25 squares,
a half-squares, and y quarter-squares.
4. If convenient, outline a square rod on the schoolroom floor;
imagine one on the ceiling, and show how far it would extend; or
have one marked off in the school-yard.
5. How many square yards in an acre of land? 6. Draw a fig-
ure to represent an acre, 10 rods wide and 16 rods long; scale, + in.
to arod. Divide your drawing to represent square rods.
7. What part of an acre does the school lot equal? 8 7A.=2
sq. rd.
9. How many acres in a square mile? In 10% of it? In }
of it ?
10. A western township is 6 miles square. It contains w square
miles or sections, and the distance around it is y miles.
11. Find some piece of ground which you can show to contain
about 1 A.
114 MEASUREMENTS.
218. — Surface 1. Change 20,000 sq. in. to square feet.
Measures. 2. Change 12,371 sq. ft. to square rods.
Written. 3. Change 287 sq. rd. to square feet.
4. 1350 sq.m.=a A. 65. Change an acre to square feet.
6. Bought 4 A. for $400, and sold it at a dime per square foot.
How much did I gain or lose ?
7. A farmer owns five fields or lots measuring as follows: 80 A.,
200 sq. rd.; 2 sq. m.; 874% of an acre; and 435,600 sq. ft. What
is the acreage of this farm ?
8. In 20,000 sq. ft. how many square rods ?
Ode: 10. What will be the cost of a
17 sq. ft. 19 sq.in. school lot containing 32,670 sq. ft.
19 sq.ft. 75 sq: in. at $5000 per acre and $12.50 per
42 sq. ft. 108 sq. in. — square rod for filling and levelling ?
96 sq. ft. 121 sq. in.
219. — Lines. 1. Draw horizontal, vertical, and inclined
or oblique lines. 2. Name the three kinds
that you have drawn and describe them according to their direction.
Try to make your description exact and brief before consulting
page 142.
3. With reference to each other, two lines may be parallel or per-
pendicular. How many pairs of parallel lines on pages 116-117 ?
4. What are parallel lines ?
5. Describe straight lines and curved lines. 6. What is a line?
220. — Angles. 1. Draw an angle and show its sides and
vertex, or their point of meeting.
2. Two lines having different directions and
meeting at a point make an angle. To measure
an angle is to measure this difference in the
direction of the lines. Repeat the table for
circular measure (p. 9).
3. If you prolong the sides of an angle, do
you increase its size ?
OF ARCS AND ANGLES. 115
4. Draw two intersecting lines so as to make four equal angles.
5. The lines thus drawn are perpendicular to each other and the
angles are right angles. Define perpendicular lines. 6. What is a
right angle ?
7. Draw two inclined lines perpendicular to each other.
8. Compared with right angles what are acute angles? Obtuse
angles? 9. Show how many of each of the three kinds are on
pages 116 and 117.
10. Adjacent angles have one side in common. With two strokes
of a pencil draw four angles; then draw four figures showing the
four pairs of adjacent angles that you have made.
11. Unequal adjacent angles are oblique angles. “Oblique”
means ~ ?
221.— Divisions of 1. What is a circle? 2. What name is
Circle; Degrees of given to the curve that bounds it? 3. A
Arcs and Angles, diameter of a circle bisects it. How is the
Oral. diameter indicated in the figure? 4. How
many radii are drawn ?
5. The surface enclosed by bca is a quad-
rant; by bcg, a sextant. These same names
are apphed to the arcs ba and bg, as the cir-
cumference is often called a circle. What
part of a circle or circumference is bea or ba?
beg or bg?
Oo
ts)
g f 6. For convenience in measuring arcs and
angles, every circumference, whether large or
small, is divided into 360 equal parts or degrees (360°). How many
degrees is a semicircle? a quadrant? a sextant? a sign or 12th of
a circle ?
7. Each degree is divided into 60 minutes (60'), and each minute
into 60 seconds (60").
1S = wy! . 300! = w® An are of 30° contains y/
Baek Or == oi 600" = a! 8 of a circle contains 2°
116 MEASUREMENTS.
9. Over how many degrees does the long hand of a watch move
inan hour? in 80 min.? nth.? 10. In 20 min.? in 25 min.?
in 85 min. ? in 4 day?
ll. Of the six angles in the figure which is the right angle ? How
many degrees in the arc between its sides? What arc measures a
right angle ?
12. Which three angles in the figure are equal? What is the
size of the arc that measures each? Each is an angle of a.
13. How many degrees in half a right angle?
14. Draw angles of 90°; 45°; 60°; 105°; 120°. Which are obtuse ?
Which acute ?
222.— The Six 1. Notice how many sides these figures
Quadrilaterals. have, and define a quadrilateral.
Oral. 2. Which have their opposite sides paral-
lel? What is a parallelogram.
8. Which of the quadrilaterals has only
l two parallel sides? What is a trapezoid?
4. Which is a quadrilateral without paral-
an : lel sides? What is a trapezium ?
5. Which two parallelograms are equilat-
™ 0 eral? LEquiangular? Rectangular ?
l
6. What name is given to a quadrilateral
with four right angles? 7. To an equilateral
rectangle ?
8. Which of the parallelograms have only
oblique angles ? What is a rhomboid ?
l 9. Which rhomboid is equilateral? What
is a rhombus ?
3
ic)
10. Show the propriety of each of the
following terms as applied to A: quadrilat-
eral, parallelogram, rectangle, square, equi-
angular, equilateral.
3
FS
9
QUADRILATERALS. 117
11..Of each parallelogram which line is
7
the base 2? 12. Which shows the altitude or
height of the parallelogram ?
m7 re 18. The altitude and base must always be
perpendicular to each other. Try to tell why.
l
14. A straight line, like Jo, that joins the
vertices of opposite angles is a diagonal.
Prove by cutting —
‘ (1) that a diagonal bisects a parallelogram.
(2) that the opposite angles of a rhomboid
are equal; and by measuring prove,
(3) that the sum of the angles of a quad-
rilateral equals four right angles.
0
223.— Of Rectangles. 1. An inch-wide rectangle 12 in. long con-
tains wsq.in.; a 12-inch-wide rectangle of the
same length contains 12 x @ sq. in., or y sq.
in. (See the figure.)
2. What is the area of a rectangle 15 in.
square? 8. Of a rectangle 1 ft. 6 in. square?
4. A foot-wide rectangle 24 ft. long contains
# sq. ft.; a 12-ft.-wide rectangle of the same
length contains 12 x x sq. ft. or
y sq. ft.
5. Find the area of rectangles
15 ft. long, ‘7 ft. wide; 13 it. x
20 ft.; 164 ft. long, 164 ft. wide;
5i yd. x 54 yd.
@ = :
ST Ss«*6. A piece of land measures
yg 25 rods one way and 20 rods
the other way. Find its area in square rods; in acres. Which of
118 MEASUREMENTS.
the following statements or equations is right for the second
answer ?
25 x 20 sq. rd. cL 25 x 20 sq. rd.
160 160isq;.50 ee
7. A kindergarten table 4 ft. 3 in. long and 20 in. wide is marked
off into inch squares; how many are there ?
8. How many square yards in a web of cotton 404 yd. long and
3 ft. wide ?
9. In a flag 103 ft. long and 2 as wide, how many square yards
of bunting, not allowing for seams ?
10. A patchwork quilt measuring 3 yd. by 3 yd. is made of 4-inch
silk squares; how many are there ?
224. — Superficial 1. What shall I pay Mr. Bates for concret-
Contents of ing a walk 60 ft. long, 4 ft. wide for half its
Rectangles. length, and 3 ft. wide the rest of the way ?
Written. His price is 75 cents per square yard.
Norr. — Draw diagrams to 2. Mr. Cross fenced his strawberry patch,
illustrate ; make statements,
and cancel when possible.
which was 4 rods wide and 100 feet long,
with three lines of barbed wire at 11% per
foot. The posts cost $7. How many quarts of berries at 5f a
quart must he sell to pay for the fence ?
3. When an acre of land is 40 rods long, what is its width ?
4. Mrs. Fiske bought the equivalent of a square yard of 4-inch
ribbon for $4.50; what was the price per running yard ?
5. How many square tiles 9 inches long will lay a floor 12 ft.
wide and 27 ft. long ?
6. A roll of oilcloth 72 in. wide is 30 ft. long ; what is it worth
at 621¢ per square yard?
7. How many square feet of glass in your schoolroom windows ?
8. Of blackboard surface in the room ?
9. Drawing paper measuring 24 x 36 is cut into 9 x 12 pieces.
How many pieces will a ream furnish ?
OF FLOORS. 119
225.— Of Carpet- 1. Ingrain carpets are generally woven in
ing, Tiling, etc. strips a yard wide; other carpets, three-
quarters of a yard. What two advantages
come from running the strips lengthwise of
the floor rather than across it ?
Oral and written.
2. On floors of the following widths which width of carpet could
be used without either cutting or turning under any strip ?
12ft. 15ft. 22ift 227i. 13ift. 20ft. 18 ft.
3. How many strips of ingrain carpet will be needed for a room
18 ft. square? How many running yards? How many yards must
be bought if a quarter-yard is wasted in matching each two strips ?
4. How many strips of brussels or tapestry carpet will be needed
for a room -15 ft. wide and 21 ft. long? How many yards, if it
matches without waste? Find the cost at $1.25 a yard.
5. Find the cost of covering a floor 14 ft. by 20 ft. with yard-
wide carpet at 75%, no strips to be cut, nor allowance made for
waste. 6. What will it cost using three-quarter carpet at $1.50, on
the same conditions? 7. Using 4-ft. oilcloth at $1?
Find the cost of carpeting floors under the following conditions (strips
that are cut cost as if whole) :—
Length of Width of Width of Allowance for Cost
room. room. carpet. matching. per yd.
els. ff: 14 ft. 1 yard 13 yds. — $0.90
9. 22 ft. 18 ft. 3 yard 21 yds. 4.25
Mose rior tip! 182 ft. 2 dyad t yd. 0.874
Tree th 20 e 8 yard 23 yds. 1.374
12. How many 8-in. marble tiles are required to cover a hearth
2 ft. by 4 ft. 8in.? 18. To cover a floor 20 ft. by 464 ft. ?
14. The areas to be tiled about a fireplace are: one of 5 ft. 3 in.
by L.ft. 9 in.; two of 2 ft. 104 in. by 1 ft. 9 in.; one of 1 ft. 9 in.
by 6 in.; one of 5 ft. 3 in. by 2 ft. 44 in.. Find the total area.
16. How many tiles 1} in. by 3 in. are required ?
120 MEASUREMENTS.
16. Find the area of the surfaces shown
at. the left. 17. How many 2-in. tiles are
required to cover them? [°, ', p. 148.]
18. A room is 134 by 18, and 8 ft. high,
How many rolls of wall paper are required,
each being 8 yd. by 18 in., no allowance
made for doors, windows, or baseboards ?
226.— Of Roofs, 1. 93 squares of slating are required to
Pavements, etc. cover a certain roof. This is equal to how
many square yards? If the slates are 8 x 16,
and each course overlaps 10 inches of the one below it, find the
number of slates used.
2. How many blocks 6 x 4 inches will be used in paving a four-
rod square ?
38. How many tin plates 13 x 19 must be used for 1 square of
roofing, if they are lapped or folded 4 in.
on each side ?
4. Three piazza roofs about a house
measure in feet 308, 24x7, 74123.
How much less than 5 squares do they
contain ?
5. A house lot contains + A. How
many sq. ft.? The house is 274 x 40.
What would it cost to sod the remainder
at $1.50 a square ?
6. Let this figure represent the out-
line of a cellar. Copy, and divide it
into 5 rectangles. From the given
dimensions find those of each rectangle.
7. How many square yards of cement
would be required to cover the bottom of the cellar ? |
OF RECTANGLES. 121
227.— Areas and 1. A chess-board contains 64 squares 14 in.
Perimeters of long. What is its perimeter? If it has an
Rectangles. inch-wide border, what ?
For oral analysis. 2. In a 2-inch square how many 14-inch
squares ? How many +-inch ?
3. Compare the perimeter of a 4-foot square and an equal surface
8 feet long.
4. My sidewalk is 10 ft. wide besides the curb, and 100 ft. long.
How many 4 x 8 bricks in it?
5. Compare a 4inch square and a 12-inch square as to length
and area.
6. A marble slab 4 ft. by 21 ft. was sold for $4.50; price per
square foot ?
7. What is the area of a square that can be set off with 200 feet
of rope ?
8. How many boards 9 inches wide make a close fence 8 feet high
around three sides of a square lot 180 feet long ?
9. A hall measures 12 feet by 36 feet. How many breadths of
yard-wide carpet would be needed? How many yards, allowing
3 yards wasted in matching the pattern ?
10. A room 14 ft. by 18 ft. is to be covered with yard-wide carpet
at $1. Which is the cheaper way to run the strips? Why?
228.— Of CityLand. 1. Mr. Sharp bought land bordering on
Written. Spring Street between Poplar and Maple at
[See next page.} 3 cents per square foot, which he cut up into
building lots. He first laid out a 40-foot street in the rear, which
he called Leland Street. What did he pay for the land?
2. He sold lot C to a civil engineer for his services in surveying
and making plans, plus an additional 2 cents per square foot. What
did the survey, ete., cost ?
122 MEASUREMENTS.
After reserving lot A for his own dwelling-house, he sold the
remaining 10 lots by auction at the following prices.
Be ee ee ee ee
Spring
Poplar
Maple
2 Leland
as TS Se ay ar
Find the proceeds of the sale of each lot.
8. Lot B for12i¢ 6. Lot F for183¢ 10. Lot J for 193¢
4. Lot D for 15¢ 7. Lot G for 21¢ 11. Lot K for 173¢
5. Lot E for 223¢ 8. Lot H for 201 12. Lot L for 251?
9, Lot I for 173¢
13. Before the sale, he opened and laid out a 16-ft. alley from
Maple Street to Leland. What did the alley cost him, $85 being
paid for labor ?
OF RHOMBOIDS. 123
14. The grading of Leland Street cost him $3.75 per square rod.
What did the street cost, including land and labor ?
15. Mr. Sharp laid a sidewalk 8 ft. 8 in. wide on two sides of his
own lot A. The 8-inch edge-stones cost him 80 cents per running
foot. The brick cost $12 per thousand, and the labor $58.25. The
bricks were 8 x 4 x 2 and laid flat. What did his walk cost? (Make
statement. )
16. The owner of lot I paid an average of $3.50 per rod for fenc-
ing. It cost him $2 if he paid for only half of the division fence.
17. The abuttors combined, and concreted the alley at 561 cents
per square yard. What was the total cost? 18. What part of the
whole cost should the owner of J pay? 19. What is the assessment
of the owner of I?
20. Leland Street is accepted by the city and paved at a cost of
$3 per square yard, the abuttors agreeing to pay 25% of the cost of
the part adjoining their property. What is the assessment on lot L?
229. — Of Rhomboids. 1. Cut a rhomboid from paper.
2. Divide it along any altitude line.
3. Adjust its parts so as to form a
rectangle.
4. Compare the base and altitude of
the rectangle with the base and altitude
of the rhomboid.
5. How is the area of the rectangle
found ?
6. How, then, may the area of the
equivalent rhomboid be found? Area =
base x altitude.
Find areas of rhomboids with —
7. Base 122 ft., altitude 74 ft.
8. Base 16 rd., altitude 40 ft.
Ae As Faas Ce ai Re let ve
10. A = 83 ft., B 54 in.
124 MEASUREMENTS.
230. — Of Trapezoids.
Oral.
1. Cut out a trapezoid
having two right angles.
2. Divide it along a middle line parallel to its parallel sides.
3. Adjust the parts so as to make a rectangle. Notice where the
parallel sides of the trapezoid are to be found
in the rectangle.
4. Compare the area of the rectangle with
the area of the trapezoid.
5. Show that one dimension of the rectangle
equals the sum of the parallel sides of the
trapezoid.
6. Show that the other dimension of the
rectangle equals one-half the altitude (length
or width) of the trapezoid.
7. How is the area of the rectangle found ?
8. The base and altitude of the rectangle
correspond to what lines in the trapezoid ?
9. How, then, may the area of the trapezoid be found ?
231.— Of Trapezoids. At Sight. —1. In connection with this
trapezoid explain this statement:
(12 + 16) sq. in. x $=112 sq. in.
2. Show that the average or mean
or middle length of the trapezoid is
(12 + 16) + 2.
8. Is there any difference in value
eres x8, 02416) x5
oF a2) x8»
between
OF TRAPEZOIDS. 125
4. In what three ways may you state the process of finding the
area of a trapezoid ?
5. What is the altitude of trapezoid B? ithe sum
of its parallel sides, or mean width? Explain -
8 + 16
2
(mean length x width, or) _
(mean width x length ;
x 56 =o.
6. In trapezoids
Written. — Find the area of trapezoids measuring —
7. Parallel sides 16 ft. and 24 ft.; altitude 13 ft.
8. Parallel sides 25 in. and 24 in.; altitude 44 in.
9. Parallel ends 13 in. and 16 in.; length 14 ft.
10. A trapezoidal board is 74 in. wide in the middle and 16} ft.
long.
232. — Of Oblique- 1. Cut an oblique-angled trapezoid along
Angled Trapezoids. its middle line and place
its parts end to end to
form a rhomboid.
2. What lines of the trapezoid form base and
altitude of the rhomboid ?
3. How may the area of the rhomboid be
found ?
4. Show how an oblique-angled trapezoid may
be changed to an equivalent rectangle.
5. Find the area of a trapezoid measuring 223
ft. in altitude, 723 ft. and 853 ft. along its paral-
lel sides.
6. A 10-ft. wall is 128 ft. on the ground and 122 ft. along the top.
What will it cost to paint both sides at 9% a square yard ?
7. Draw a rectangle to represent the area painted. (Scale, ;,/55)
126
233. — Of Triangles.
Written
re
MEASUREMENTS.
1. Show by measuring with a protractor, or
by cutting and laying the angles together, that
the sum of the angles of a triangle equals two
right angles (180°).
2, How many right angles may a triangle
contain? How many obtuse? How many of
60° ?
3. Find the size of the third angle when
two angles of a triangle measure 90°, 30°;
120°, 40°; 65°, 35°; 624°, 874°.
4, Triangles are named from their largest
angles, — right, obtuse, and acute. How many
of each kind are represented here ?
5. Named from their sides, triangles are
equilateral (three sides equal), isosceles (two
sides equal), or scalene (three sides unequal).
Give examples of each from the drawings.
6. Cut from paper any one of the four par-
allelograms (pp. 116, 117) and find its area.
7. Bisect it along one of its diagonals.
What is the area of the resulting triangle ?
8. Compare the base and altitude of the
parallelogram with those of the triangle.
9, How is the area of the parallelogram
found? The area of the triangle ?
Find areas of triangles of the following
dimensions : —
10. Base = 40, alt. = 18.
1]. Base = 60, alt. = 25.
VIB Sit Balt. =.9 i, 137 3B 4 rh ali it.
14, 191 ft., 21 yd.
16. Show that B x }
15. 38 rd., 224 ft.
alt., Bx alt., or 4(B x alt.) =
OF CIRCLES. 12a
234.— Of Trapeziums, 1. Draw or cut out a trapezium.
Written. 2. Separate it into two triangles along one
of its diagonals, as lo.
3. Find the dimensions of each tri-
angle and its area.
4, What will the area of the trape-
zium be ?
5. The diagonal of a trapezium is
24 inches; the altitudes perpendicular
2 toit are 18 inches and 15 inches, re-
spectively. What is its area ?
6. The diagonals of a trapezium cross at right angles. The point
of intersection is 50 feet from the upper end of each diagonal. One
diagonal is 100 feet long, and the other 150 feet. Find the area.
235.— Of Circles: 1. Bring to school the results of very care-
Diameters and fully measuring the diameter and circumfer-
Circumferences. ence of several circular objects: plates, rings,
Oral and written. covers, wheels, or coins.
2. In each case divide the circumference by
the diameter, carrying the division to several
decimal places, and compare the quotients.
3. If you have measured and divided accu-
rately, the quotient in each case will be 3.1416—-.
What does this show ?
4, In like manner divide your diameters by
the corresponding circumferences. Your quotient
should always be 0.31831. What does this show ?
5. How many diameters make a circumference ?
6. What part of a circumference equals a diameter ?
7. The diameter of a circle is 10 ft.; the circumference = 3.1416
x 10 ft., or x ft.
128 MEASUREMENTS.
8. The circumference of a circle is 10 ft.; the diameter = 0.31831
GLO ity or eT
9. Compare 10 + 5.1416 and 0.31831 x 10. Which is easier, to
divide by 3.1416 or to multiply by 0.31831 ?
10. 3.1416 = the ratio of the circumference to the diameter. It is
represented by the Greek letter 7 (English p). D= diameter;
C= circumference; A= radius. Interpret the following: —
C=) xr DO; T _ 0.31831; D= Gxe
Tv Tv
Find the diameter or circumference or radius. Forecast the result.
1 Wp #7 1b reed CS 146 C= F402 San. ee
TAP (Ch 20D ith ae) ea 16.2 Da 167 an Cae
ISR == 90 Tee Ce. 167 C= iy dy ets hee
236.— Circles With Objects. —1. Cut a circle from paper.
changed to Hqui- Bisect it and eut each half into fourths.
valent Rectangles. 2. Arrange the eight sectors as in B, forming
a figure somewhat lke a rhomboid.
3. Cut another equal circle, divide
it into sixteen equal sectors, and ad- —
just them to form a figure still more
eae like a rhomboid.
os 4. Of the two rhomboids, which is
What
more nearly a rectangle ?
part of the circle is its base? Its
altitude ?
5. Imagine a circle cut into a
thousand equal sectors, arranged as
B before. Would the figure formed be
a rhomboid or a rectangle? 6. Com-
pare its base with a straight line.
What can be done to make its base
more nearly straight? 7. The circle
C would then be changed to a rectangle
OF CIRCLES. 129
having a base equal to . and
an altitude equal to ... How
would the area of this rec-
tangle be found ?
8. Explain the figure D.
9. Interpret the equation:
D Area of circle = R x 5
ae, es es C= ahout2o «Ai.
Find the area of circles : —
ll. Radius, 6 ft.; Cir. x 13. Cir. 100 ft.; Diameter, x
12. Diameter, 10 ft.; Cir. 14. Cir. 50 ft.; Radius, x
237.— Diameter of Oral. —1. efgh is the square of the di-
Circle Given, to find ameter of the circle. A circle is what part
the Area. of the square of its diameter ?
2. Explain the following (as shown
in §§ 255-6) : —
(a) A=Dx &; but C= D x 3.1416;
hence,
(b) va gaa Bp
2X a22* ; cancelling,
we have
(oye A176) 50.7804; bute Dx
D=D*; hence
(dq) A= D’ x 0.7854; that is, a circle
is 0.7854 of the square of its diameter.
3. Inthe figures at the left the shaded
portion is the area of the circle. What
decimal part of the square is it ?
What decimal part of the square is
not shaded ?
130 MEASUREMENTS.
4. A circle is what part of its circumscribed square ?
5. How is the area of a circle found from its diameter ?
Written. —6. What is the area of a 5-foot circle ?
7. What part of 3.1416 is 0.7854 ?
8. Find the area of a 12-inch circle.
9. What is the cost of a circular piece of aluminum at 30 cents
per square inch, the radius being 3 inches ?
10. What is the area of a circular pond 200 feet in diameter ?
238.— The Area of 1. In the following figure we have three
Circles. rectangles each equivalent to the circle.
Which one is twice as long and half as wide as H? Which is half
as long and twice as wide as H?
2. The length of each is what part of the circumference ?
3. The width of each is what part of the diameter ?
4. The diameter of a circle is 10 feet. Whatis its area? Explain
each of these three methods of finding it.
0.7854
[ Rect. H.|] — a ae # poe) = 78.54. Formula: R x c= A.
1 0.7854 R
[Rect. 1] x (10 x 3.1418) = 78.54. Formula: > x C= A.
0.7854
yl
[Rect. J.] 10 x (Sree) = 78.54.. Formula: D x — A,
PROBLEMS. 131
5. The circumference of a circle is 12 feet. Explain this method
of finding its area: —
C = 12 feet. D = 0.381831 of 12 feet.
3
(0.31831 x 12) x i= 11.45916.
Written. — Find the area of circles when —
6. Diameter = 40 8. Radius = 24 10. C = 200
7. Circumference = 80 9. Diameter = 36 We Des bo
239.— Oral Review. 1. What objects before you are nearest in
length to a yard? to a foot? to a rod?
2. How many degrees measure aL? Can you find as you look
about any except right angles ?
3. After going 1 round the world, «° complete the circuit.
4. If the angles of aA=2L/’s, each angle in an equilateral A
measures @°.
5. How may the area of a triangular park be found ?
6. How much of an 8-in. square is not covered by a 7-in. square ?
by a 6-in. one ?
7. A rhombus containing 3 sq. yd. contains how many sq. ft. ?
8. Which measures more, a rhombus or a square with the same
perimeter ?
9. The top of a round table has what RE of the area of that of
a square table of the same diameter ?
10. A triangle = 4 the area of _.
240.—Drawingand 1. Draw a 2-in. circle; then the largest pos-
Measuring of Figures. sible square inside, and one of its diagonals.
2. Calling the diagonal the base of the tri-
angles, what is the area of each?
3. The area of the largest square drawn in a 3-in. circle = a.
4. Draw a 1-in. square and a 1-in. rhombus. Are their bases the
same? ‘Their altitudes? ‘Their areas ?
abs y: MEASUREMENTS.
ol
5. Using 1-in. lines, make a rhombus with an altitude of + in.
Its area will be —..
tole
6. Using 1-in. lines, make a figure of as small area as you can.
What is its shape ?
7. As a ring is flattened does its capacity change? Does the
length of its perimeter ? |
8. Draw a trapezoid; the horizontal and vertical lines may be
1, 14, and 2 inches. Divide it into a rectangle and a triangle. Find
the areas of each, and add; then find the area of the trapezoid in
the usual way.
9. Draw a rectangle and a second figure with the same length of
line, but noL’s. What is its shape? How do the two differ in area ?
10. Explain what dimensions you need to know in order to find
the area of a trapezium that you have drawn.
241.— Problems in 1. A button is 4.7124 in. round. How long
Measuring Circles. a button-hole is required ?
Written. 2. Find the circumference of the base of a
lamp chimney that is 23 in. across.
3. A circus ring is 414,45 ft. round. Find the distance to the
centre in rods.
4. A hogshead is a little over 121 ft. round the middle. Will
it go through a doorway that is 3 ft. 10 in. ?
5. If a mountain is 10 m. round, what distance might be saved
by tunnelling ?
6. A pie is cut accurately into 6 equal pieces. Which is longer,
the curved edge or the straight one ?
7. The hubs of two wheels are alike, but the spokes of one are
3 in. longer. How much greater is its circumference ?
8. If a barrel is 18 in. over the chine, how much strap iron will
be required to make 100 end-hoops with 3-in. laps? Make a state-
ment.
OF SOLIDS. 133
9. In a lawn 100 ft. square the circular basin of a fountain is
40 ft. from each side. Draw a figure, and find the area of the
greensward.
10. When you know the area of a circle, how can you find the
radius ?
242. — Of Solids. 1. Lines have one dimension; viz. W.
y]
[See p. 9.] 2. Surfaces have two dimensions; viz.
OT wets
3. Solids occupy space and have three dimensions, viz. —, ~,
BuO 2
4. Mention the three common solid measures in the order of their
size. 6. Compare each one with the one next larger or smaller.
6.-10 cu. ft. = 2 cu. in. 8. 10 cords contain x cu. ft.
feat curve 7 CU. Lt 925°720 cunits=—& cu. yd.
10. State the method of finding the number of cubic feet in 20,000
cubic inches.
243. — Of Cubes. Oral. —1. What is arectangle? 2. A solid
bounded by six rectangles is a rectangular solid.
3. A cube is a solid with six square faces. How many corners,
edges, angles, has a cube ?
4. Is acommon brick a cube? Is it a rectangular solid? What
is an equilateral rectangular solid ?
5. Describe an inch-cube, or a cubic inch ;
a cubic foot; a cubic yard; a 2-foot cube.
6. What is a 9-inch cube? How many
cubic inches in a 9-inch cube ?
[See the figure.] Along one edge of a
cube there is a row of x cu. in.; 9 such
rows make a tier of 9X cu. in. or y
cu. in.; 9 such tiers contain 9 x y cu. in.
or 729 cu. in.
DiMemetiaae x9 % 9 CU. in, = 2 Cu, 10,
134 | MEASUREMENTS.
Written. — In a similar or better way show the contents of —
7. A 6-in. cube 9. A 5-foot cube ll. A 10-yard cube
8. An 8-in. cube 10. A 12-foot cube 12. A 20-in. cube
TS MD tied , pO Am Cle O° et Usa oe ames,
14. V/64? 1/216? 1728? W512? ¥/1331? 729? ~/27000?
244. — Rectangular 1. Prisms are named from their bases, —
Prisms. square, rectangular, triangular, hexagonal, ete.
_ Name some familiar objects that are rectangu-
lar prisms; that are square prisms. Which kind includes the other ?
2. Find the cubical contents of a rectangular prism
whose dimensions are 9 in., 4 in., and 5 in.
Notice in the figure the number of cubic inches in
a row, the number of rows in a tier or layer, and
the number of tiers‘in the prism; and then explain
the statement: 9 x 4 x 5 cu. in=~# cu. in.
3. How many inch cubes may be put into a box
10 in. long, 8 in. wide, and 5 in. deep ?
4. A trunk measures 3 ft., 20 in., and 18 in. Find
its cubical contents. Why not multiply 3 by 20 by 18 instead of
36 by 20 by 18 ?
Find the contents of rectangular prisms of these dimensions : —
Length. Width. Height. Length. Breadth. Depth. ©
Deel Ont. cel 4G eo ts 8. 421 ft. 20 ft. 13% ft.
TEs Ras Ra ade 8 ES Pinal 9. 12iyd. 10ft. 16in.
(Fare harem sega 2 cok 10. 204 ft. 172 ft.) Orin
245.— Of Cord- 1. Wood for fuel, sold by the cord, is usually
Wood. in sticks of what length? 2. In what form
Oral and written. are they piled to make a cord? 8. Give the
dimensions of a cord; a half-cord; 3 of a cord
or a cord foot.
OF WOOD. 135
4. What kind of solid does a half-cord resemble? 6. Explain:
8x4x4cu. ft.=2 cu. ft. as applied to cord-wood.
6. A pile of 4-foot wood, 4 ft. high and 8 ft. long, contains a cord.
Tf16 ft. long? 24ft.? 32 ft.? 96 ft.?
7. A pile of 4-ft. wood of the usual height must be how long to
~
contain 10 cords? 12 cords? 25 cords?
8. How many cords in a pile of 4-ft. wood, 4 ft. high and 18 ft.
long? Explain the following statement, and show what may be
cancelled : —
4x4x 18 cu. ft.
128 cu. ft.
9. Bought a pile of 4-foot wood 30 ft. long and 8 ft. high at $6
per cord.
4x8 x 30
128
In the statement what represents the number of cubic feet? The
number of cords? The cost of all ?
x $6 =2.
Find the value of piles of wood, as follows : —
Length. Width. Height. Price. Length. Width. Height. Price.
10. 24 ft. 4f. 6ft $4. 13. 24 ft. 4ft. Tift. $3.50
Pisa te 5G: TLS Te. oD. 14. 20 ft. 3 ft. 12 ft. 5.00
Bee RO Ste 10 fer TE: 6. 16. 163 ft. 44in. 22ft. 4.25
246. — Of Lumber. 1. Timber sawed for building purposes is
Written. lumber. What forms can you mention besides
boards, planks, joists, and beams ?
2. In measuring lumber no attention is paid to the thickness
unless it exceeds an inch. A board 12 ft. long and 12 in. wide
and 1 inch or less in thickness contains 12 sq. ft.; if 10 in. wide,
it contains 19 or 2 of 12 sq. ft. or = a sq. ft.
3. A board 15 ft. long, 8 in. wide, and # in. thick contains how
many square feet ?
Nore. —If 15 ft. long and ove foot wide, it would contain @ sq. ft.; being % of a foot wide, it will
contain, etc.
136 MEASUREMENTS.
4. 10 16-ft. boards averaging 9 in. in width contain a square feet.
Explain: 10 x 2 of 16 sq. ft.
5. A board 1 inch thick and a foot square is
a board foot. « of them piled together would
make a cubic foot.
6. A board 10 ft. long, 12 in. wide, and 1
inch thick contains 10 bd. ft. If 2 in. thick, it
would contain 2 x as many bd. ft. ora. If 14
in. thick? If 14 in. thick? If 24 in. thick?
If a ft. thick ?
7. Find the contents of a 3-in. plank 15 ft. long and 10 in. wide.
Explain: 3 x 15 x 2=~@.
8. 12 joists, 16 ft. long and 4 in. square, contain & board feet.
Find the contents in board feet of lumber measuring as follows : —
9. 6 boards, 16 ft. long, 14 thick; width in inches: 8, 10, 12, 13,
14, 9.
10. Fifteen 3 x 5 joists, 18 ft. long.
11. A stick of timber 18 ft. long and 12 in. square.
247.—To find the 1. How many faces has a rectangular
Surface of a Cube. prism? 2. What name is given to a rectan-
gular prism when all its faces are equal ?
Megat 3. Find the entire surface of a 5-inch cube.
Explain the statement: 5? x 6 = 2.
4. The entire surface of a cube is 150 sq. in. How long is the
cube? Explain the statement: (aes a (eh
Find the entire surface of — How long a cube has—
5. A 9-in. cube 8. An entire surface of 384 sq. in. ?
6. A cube 10 in. long 9. An entire surface of 600 sq. ft. ?
7. A 16-in. cube 10. An entire surface of 294 sq. in. ?
OF PRISMS. 137
\
248.— Of Rec- 1. Compare with each other the ends of
tangular Prisms. a square prism. 2. Compare its four sides.
Written. 3. Find the entire surface of a square
prism 8 in. fong and 3 in. wide. Ex-
plain the equation : —
2x(8x3)+4x(8 x 3)=@.
4. Compare the opposite faces of
any rectangular prism. 9. Find the
entire surface of a prism measuring
Gby 4 by 2. Explain the statement : —
z
2x(2x4)+6x 44+442+4+2)=2.
6. Explain the figures at the left.
4 Find the entire surface of prisms —
7. 10 in. long, 6 in. wide, 4 in.
; thick.
§.=12 it, long, 9 ft. wide, 6 ft.
high.
9. 20 ft. long, 14 ft. wide, 10 ft.
high.
10. 16 by 18 by 4; 20 by 1 by 1.
11. 12 by 9 by 8; 23 by + by 16.
12. 12 by 12 by 6; 2 by 3 by 73.
249. — To Find the 1. Mention several common objects that
Contents of a are perfect cylinders; that is, of uniform
Cylinder. diameter, and with ends (or bases) that are
rite equal parallel circles.
9. How might a cylinder be turned from
a square prism of the same diameter ? 3. Recalling the formula
for the area of a circle, 0.7854 of D? (p. 129), what part of the prism
would be shavings, and what part cylinder ?
138 MEASUREMENTS.
4. Give the contents of the largest cylinder that may be turned
out of a square prism 25 in. long, 4 in. wide.
Explain the statement : — 4’ x 25 = « = contents of _.
0.7854 of « = y = contents of
5. A circle is ——. aan of a square of equal diameter.
A cylinder is ——— of a square prism of equal diameter and
length.
6. Find the contents of a cylinder 10 ft. long and 2 ft. in
diameter.
Explain the statement : — 0.7854 of (2? x 10)= @ cu. ft.
7. A cylindrical pail 6 inches in diameter inside and 12 inches
deep contains # cubic inches. Forecast the result, observing that
0.7854 is a little more than 3; thus, 3 of 6? x 12 = 324+.
8. A cylindrical tank is 10 ft. deep and 8 ft. in diameter.
9. A well is 32 ft. deep and 5 ft. in diameter.
10. A gallon contains 231 cu. in. To hold a gallon, a pail measur-
ing 33 sq. in. on the bottom must be w inches deep.
10000
*
250.—To Find 1. In form, the ends of a cylinder are
the Surface equal ..s. The rest of the surface is the
of a Cylinder. convex surface.
2. Suppose the diameter of a cylinder
to be 4 inches; its circumference = 2, or
3.1416 x D (§ 235).
Written.
OF CYLINDERS. 139
3. The circumference of a cylinder is
8 inches; its diameter is a, or 0.31831 x C.
4. A cylinder is 20 inches long and 4
inches in diameter. Find the area of its
ends. Explain the statement :—
(0.7854 of 47) x 2= 0.
5. Roll an oblong paper to form a cyl-
inder. Give the length and circumference
of the cylinder thus made.
20
6. Unroll the paper and give the di-
mensions of a rectangle equivalent to the
convex surface of the cylinder. Explain
the diagram at the left.
7. The convex surface of a cylinder
=Cx wl. Explain. [Z =length.]
8. A cylinder is 25 inches long, 4 inches in diameter. Its con-
vex surface is % Explain: (3.1416 x 4) x 25=2.
9. A cylinder is 20 inches long, 5 inches in diameter. Entire
surface ?
10. If we allow 17 square inches for seams and the flange of the
cover, how many square inches of tin are actually used in making a
coffee can 6 inches in diameter and 8 inches deep? Show why there
must be an allowance for waste.
251. Oral Review. 1. A bookcase has 10 feet of space right
and left and 6 feet up and down. It is 10 inches or “ feet deep, and
y
the cubical.contents = z.
2. The sides that make the right angle of a triangle are each 10
feet. The area is a.
3. Give approximately the area of the surface of a lead pencil 4
inch in diameter and 8 inches long.
4. An old tree is 22 feet round; how far is it through ?
140 MEASUREMENTS.
6. Which takes more room, a cord of wood ora 5-foot cube ?
6. 4a circle = a rectangle having the radius for one side and
for the other.
7. A 20-foot log averages 1 square foot in the cross section. The
cubic contents are a.
8. How many cubic yards of earth will a bin hold that is
patts XPLOUE. xe,
9. About how many cubic yards does your schoolroom contain ?
10. What is the approximate capacity of a well 40 feet in depth
and 7 feet in area of opening ?
252. — Review 1. Give the dimensions of three dissimilar
Problems. rectangles each containing 56 square inches.
Written. Give the perimeter of each.
2. A square rod contains # square feet.
The wall of a rectangular cellar encloses 2 square rods. One of its
dimensions is 20 feet, the other a.
8. The boards of an old floor are 18, 14, 12, 10, and 6 inches
wide. If used in equal proportion, what is the average width? To
cover 2 squares, how many running feet would be required?
4. A panelled ceiling contains 72 squares 14 feet wide. It is 12
feet on one side, # on the other.
5. Divide the area of a square on the diameter of a circle by the
area of the circle. The quotient is a.
6. Explain the formula C=7 x2 R.
7. A board 6 feet by 6 inches contains 324 cubic inches actual
measure. How thick is it ?
8. How many oranges 3 inches in diameter will go into a box
2x1x1 feet if packed in equal rows ?
9. On a scale of 1 inch to 1 mile, represent a tract of land 2
miles by 3 miles. Divide into square miles by dotted lines. Draw
a mile square in the middle, and divide the rest into 4 equal tracts..
10. Each of the four contains @ acres.
MISSING FACTOR FOUND. 141
253. — To find a IP ee umes. Oe se Tex 1 98:
Missing Factor. 2, Multipheand = 25; product = 400.
Written. How is the multiplier found ?
& 186+ a%='31.
4, Dividend and quotient being given, how is the divisor found ?
5, When product and multipher are given, how is the multiph-
cand found ?
6. What is the area of a rectangle 12 feet long and 614 feet wide ?
7. A rectangle containing 108 square inches is 9 inches wide.
How long is it? (9 x @ sq. in. = 108 sq. in.)
8, A lot of land is 200 feet long and contains 24,000 square feet.
How wide is it ?
9. A sidewalk 50 feet long requires 50 square yards of concrete.
How wide is the walk ?
10. One-half an acre of land is taken for a new street 40 feet
wide. How long is the street ?
ll. The area of a triangle is 325 square inches; its base is 25
inches. What is its altitude ?
12. The altitude of an isosceles triangle is 14 feet; its area is 126
square feet. What is its base ?
18, At 30¢ a board foot a mahogany board one inch thick and 12
feet long cost $2.70. How wide was it?
14, The area of a rhomboidal field is 12 acres. Its length being
20 rods, what is its altitude ?
15, A square contains 400 square inches. How long is it?
16. The perimeter of a square is 1000 feet. Its area?
17. The radius of a circle is 5 feet; its area is 300 square feet.
What is the circumference? (}C x R= A.)
18. What is the area of a circle 100 feet in diameter?
oie 0.7854 — A)
19, What is the diameter of a circle containing 7854 square feet ?
(2 x 0.7854 = 7854 sq. ft.)
20. A circle contains 28.2744 square inches. What is its diameter ?
142 MEASUREMENTS. :
254.— Contents and l1138xT7x$a= §$ 910.
Two Dimensions of a 2. I hired 15 men at $2.50 per day each.
Solid given, to find the At the completion of the work I paid them
Third Dimension. in all $150. How many days did they
work ?
3. A box on my table holds 432 cubic inches. It covers 72 square
inches of the surface of the table. How high is the box ?
4. The area of the floor of your schoolroom is 900 square feet.
The room contains 10,800 cubic feet. How far is the ceiling from
the floor ?
5. A packing box is 48 inches long and 30 inches wide. How
deep must it be to hold 10 cubic feet ?
10 x 1728
Statement : SRE ane a depth.
Explain the statement, and show a short solution.
6. A closet 8 feet high and 27 inches deep will contain 72 cubic
feet. How wide is it?
7. A pile of 198 cords of 4-foot wood covers 16 square rods. How
long is it? How high is it ?
Explain the statements : —
— 198 x 128 _
16 x 2724
8. A cylindrical oil-tank holds 10 gallons. Standing on the
floor it covers 77 square inches. How high must it be ?
A eed om B
9. A bookcase holding 32 cubic feet covers a wall space of 24
sq. ft. How far must it project into the room ?
10. I have room in my stable for a grain bin 8 ft. by 4 ft. How
deep shall I make it to have it hold 72 bushels ?
11. A grindstone 4 ft. in diameter contains 6.2832 cu. ft. How
thick is it? Explain the statement: 6.2832 + (4° x 0.7854) = a.
12. In digging a trench 3 ft. wide and 43 ft. deep 330 cu. yds. of
earth were removed. How long was the trench ?
PROBLEMS. 143
255. — Miscellane- 1. I buy a corner lot 120 ft. by 50 ft. and
ous Problems. use the earth obtained by digging a cellar 60
Written. ft. by 30 ft. by 10 ft. to raise the grade how
many feet ?
2. A circular standpipe 75 ft. high is 25 ft. in diameter. When
2 full, how many gallons of water does it contain, reckoning 7}
gallons to a cubic foot ?
3. A speculator buys a field 600 ft. long and 500 ft. wide for
2500. He runs a 40-ft. street through the centre in each direction
at an expense of $425 for labor. He sells the land at 20 cents a
square foot. How much does he make or lose ?
4. At $3.75 per square yard what will be the cost of paving 3 of
a mile of street 81 ft. wide ?
5. A reservoir supphes a town with 4,575,800 gallons of water
daily. If its surface area is 7 acres, how much will the water be
lowered in a week, provided one-half as much runs in as runs out ?
Call 1 cu. ft. equal to 74 gal.
6. In a house of 36 windows a glazier finishes drawing the sash
of 4 windows in 3 h., spending twice as long on the inside as on the
outside. He can do the outside of them all in 1 day. How long is
a day’s work ?
7. A water glass has two bands round it, each containing ten
figures. It takes three seconds to cut each figure. What will it cost
to decorate 2,1, gross at four dollars an 8-hour day ?
8. Find 80% of as many articles as can be bought for $200 at
162¢ each. If sold ata profit of 100%, how many would be sold
for $2?
9. What will settle a debt of $127.50 that has been drawing 9%
interest for 248 days ?
121 x $36 x 162 x 75
10. 183 gg ame a oe: SEAT |
if © 81 x 374 x 374 z
1l. From a lot of land 40 rods square I sold 40 square rods.
What is the remainder worth at $230 an acre ?
144
MEASUREMENTS.
256. — EXAMPLES FOR PRACTICE.
For dictation.
1. Give the perimeter of a 6-
inch square.
2. Of 4
3. What is the ratio of the
diameter of a circle to its circum-
ference ?
a 6-inch square.
4. 8 feet 3 inches is what part
of a rod?
5. How many square feet in
4 a square rod ?
6. T'wo angles of a triangle
measure 30° each. What -~does
the third angle measure? Of
what kind is the triangle ?
7. How many cubic inches in
a cube 2 of a foot long ?
8. Find the entire surface of
a 6-inch cube.
9. Of 3 a 6-nch cube.
10. 4 cord feet cost $ 5.
will 3 cords cost ?
11. Your schoolroom is 12
feet high and contains 10,800
cubic feet. Length of floor ?
12. Contents of a 4-inch cube ?
What
13. Of one twice as long?
14. Rods in 31 miles ?
15. Length of a square con- |
a square mile ?
taining 900 square feet ?
At sight.
1. 0.0001 of 24,765 = a.
2. 2 of 1 rod = @ feet>
3. Days from Oct. 17 to Dee.
25, inclusive.
4. Average temperature for a
week, if the thermometer read:
7°, —4°, 10°, — 6°, —18°, 12°,
20°?
5. Area of circle 100 feet in
diameter ?
6. Cost of 1 pound if # pound
cost $2?
7. What is the exact middle
of February, 1900 ?
8. Area of rhomboid when
base and altitude are 24 inches ?
9. Area of square 163 feet
long?
10. Number of board feet in a
board 12 feet long, 8 inches wide
at one end and 10 inches at the
other ?
ll. V1i44 — V81i =2.
9 25 x 6} x 38 _
19x13 x -V25 |
18. 9 yards @ 137¢ cost = a.
14. Area of triangle 61 by 3.
15. How many acres is 61% of
PROBLEMS, 145
257. — Practical 1. 6}, 8, 44, are the dimensions of my
Exercises in Mensu- coal bin. Reckoning 90 pounds to a cubic
ration, etc. foot, what will a hin full cost @ $5?
Written. 2. Quincy granite weighs 1653 pounds to
the cubje foot. What is the weight of 6 pieces of curbing 8 inches
thick, 2 feet wide, and half a rod long ?
3. Find the cost of carpeting a 9-foot hallway 22 feet long with
three-quarter carpeting at $0.87}. Cut no strip, and allow 11 feet
per strip for matching.
4. How many tons of 15-inch ice may be cut to the acre, a cubic
foot weighing 57} pounds? Apply your knowledge of cancellation.
5. What is the capacity, in 42-gallon barrels, of a cylindrical oil-
tank 31 feet in diameter, 22 feet long? Make a statement and
cancel.
6. What is the area of a sector of 120°, its radius being 24 inches?
7. A ball ground 375 feet long and 280 wide is enclosed by a
tight board fence 8 feet high. What will the boards cost at $24
per M.? Add 10% for waste.
8. Bought 12,000 long tons of coal at $4.00 and sold it at the
same price per short ton. What did I gain?
9. What will it cost to polish the visible portions of a shaft of
red granite 6 feet by 2 feet by 22 inches at 62¢ per square inch ?
10. Draw a6-inch square, a rectangle 9 inches by 4 inches, and one
} inches by 12 inches. Compare areas and perimeters. “What infer-
ence do you draw ?
258.— Examples for 1. What decimal of a square prism becomes
Practice. shavings when the largest possible cylinder is
Written. turned from it ?
2. What number subtracted 88 times from 80,005 will leave 13
as a remainder ?
3. A railroad company fences 13 miles of its road at 732 cents a
rod.
4. How many square feet of zinc will line a cubical cistern 5 ft.
10 in. deep ?
146 MEASUREMENTS.
5. The time of the operatives in a mill was increased from 52 to
58 hours, and their wages increased ;45. Was this a gain or a loss to
them ?
6. Bread sells for 10 cents with flour at $5.00. Flour goes up
to $6.50. What should bread sell for on this basis ?
7. Ina city of 7200 school children there are 2720 cases of tardi-
ness in a year during which there are 400 sessions of the schools.
The average attendance is 6800. How often is each child tardy ?
8. Find the cost of six 8 x 10 sills 18 ft. long at $24.75 per M.
9. In a school containing 567 white children every tenth child is
colored. How many children in the school ?
10. A schoolroom measuring 32 ft. x 284 ft. x 13 ft. seats 49
pupils. Each one needs 1800 cu. ft. of fresh air an hour. The
room full would last the class « minutes.
259.— Problems for 1. How many sheets of paper folded into
Analysis. 16 leaves will make a, 400-page book ?
Oral. 2. At $10.50 a week what is the amount
of your board bill from noon of Aug. 21 to
noon of Sept. 25 ?
38. What is a £100 Bank of England note worth in New York at
its face value ?
4. Cost of 82 yds. at $ 0.374 per yard.
5. Compare a 5-inch square with one half as long.
6. A circle is 10 feet in diameter. How long is an arc of 36°
in its circumference ?
7. A cubic foot of distilled water at a temperature of 38° F.
weighs 1000 ounces. How will you find the weight of a gallon ?
8. # of an acre produces a crop that sells for $360. How much
is this for every 12 sq. rds. ?
9. I pay $1.80 for having my cord-wood sawed into 3 sticks.
What ought I pay when it is sawed into 4 sticks ?
10. A trapezoid is twice as wide at one end as at the other. It
measures 12 in. in the middle, x in. at one end, and y in. at the
other.
DEFINITIONS.
147
260. DEFINITIONS.
[FOR REFERENCE. ]
Acute Angle.
than a right angle.
Altitude. Height. Measured by a
straight line perpendicular to the line
of the base, and extending from it to
the highest point.
Angle. The divergence from a com-
mon point of two lines having different
directions.
An angle sharper
Are. Any portion of a circum-
ference,
Area. The size or total contents of
a surface.
Base. ‘The line or surface on which
a figure is supposed to stand.
Chord. A straight line joining the
ends of an arc.
Circle.
by a curve every point of which is
equally distant from a point within
called the centre.
Circumference.
boundary of a circle.
Convex Surface. The surface of
a solid excluding that of its bases.
Cube. A solid with six square
faces.
Curvilinear surfaces
bounded by curves.
Cylinder. A solid having for its
bases equal parallel circles, and hay-
ing a uniform diameter.
Degree. 0:
For 10 mo.= 0.
For 24d... = 0.
Total = $ 0.
4. *Ehe int, Of 1 15.
Hor (3 Yio = pV:
Hor Moses" 47
HOralp dee =:
Tobabe= 90).
281.— Interest by
the One Dollar
Method.
At Any Rate.
i
Process.
of $1 at 6%.
— $0.18
0.035
0.0031
$ 0.2182
48.96
816
39168
4896
9792
6)10.68144
1.7802 at 1%
$12.46 at 7%
48.96 = Prin.
Interest
For 3 yr.
LST fae va}
re LO Cs
it;
III.
IV. $61.42 = Amount.
INTEREST.
18. The interest of $1.00
For. 4 yr. = $0.
Hor, dm 0:
Hor-10.0 See.
Total = $ 0.
The int. of $1. 16. The int. of $1.
OL) <2 ta 0), For S547. = $0)
For’-9 mo, = 770. Hor. omg. see
For 22d. = 0. Hor desea
Total = 0. Total = $0.
I hire $48.96 at 7% for 3 yr. 7 mo. 19 d.
What shall I pay at settlement ?
Explain these four steps of the process:
I. Finding the interest of $1 at 6%.
II. Finding the interest of the given
principal at 6%.
III. Finding the interest at the given
rate.
IV. Finding the amount.
2. In II. why is the smaller number
used as a multipher ?
Norr.—The work should be carried to four decimal
places, and results given to the nearest cent.
38. What will discharge a debt of
$475 which has been drawing 5% inter-
est for 2 yr. 11 mo. 24 d.?
4. Find the amount of $7000 at 4%
for 3 yr. 3 mo. 18 d.
5. I hold two notes of $731 each,
one at 5% interest, the other at 8%.
They have been running 4 yr. 8 mo.
17d. What shall I receive at settlement ?
TIME BETWEEN DATES. 165
Find the amount under the following conditions :
6. Principal, $84.75; rate of interest, 4%; time, 3 yr. 15 d.
7. Principal, $942; rate of interest, 5%; time, 4 yr. 1 mo. 7 d.
8. Principal, $193; rate, 7%; time, 18 mo, 27 d.
9. Principal, $64.50; rate, 8%; time, 5 yr. 5 mo. 5 d.
10. Principal, $712.10; rate, 9%; time, 7 yr. 4 mo. 29 d.
ll. 4 yr. 6 mo. 21 d., $425.50, 3%.
282.— Time between 1. To tind the time in years, months, and
Dates. days from June 24, 1895, to Aug. 18, 1896.
Explain each process.
Process A. Process B.
From June 24, 95, to June 24,96 = 3 yr., or 6/93 to 6/96 = 3 yr.
From June 24, ’96, to July 24, 96 = 1 mo., or 6/24 to 7/24 = 1 mo.
From July 24, ’96, to Aug. 13, 96 = 20 d., or 7/24 to 8/13 = 20 d.
2. To find the time from Sept. 14, 1891, to Mar. 11, 1895. Ex-
plain each process.
Process A. Process B.
From Sept., 91, to Sept., "94 = 3yr., or 9/91 to9/94= 3yr.
From Sept. 14 to Feb. 14= 5mo.,or 9/14 to2/14= 5 mo.
From Feb. 14 toMar. 11=25d., or 2/14 to 3/11 = 25 d.
3. What advantage has process B over process A? 4. Why is
it well to know the months by their numbers as well as by name ?
5. In process B, what is found at the left of the inclined line? At
the right of it?
6. Napoleon was born Aug. 15, 1769, and
Process. died May 5, 1821. How long hact he lived?
8/1769 to 8/1820 = 51 yr. Explain the process.
8/15 to4/15 = 8mo. ; : ; ;
4/15 to5/5 =204. 7. Find the time from May 22, 1890, to
June 12, 1895.
8. Find the time from Dee. 25, 18938, to Mar. 3, 1896.
9. How long from 4/18, ’96 to 3/11, ’99 ?
10. Find your exact age to-day.
166 INTEREST.
283.— Interest: Choice Meruops or ComputTinG INTEREST.
of Methods. I. A general method, page 106.
If. The bankers’ method, page 160.
III. The one-dollar method, page 163.
Any of these three methods may be used exclusively, but as no
one method is always the best, 1t is well to learn to choose the one
that will give an accurate result most quickly.
Written.
Solve the following problems by each method, compare results, and
tell which method you prefer, and why.
1. Find the interest of $ 360 at 7% for 207 d.
2. What is the amount of $75 at 8% for 3 yr. 4 mo.?
3. What shall be paid for the use of $723.60 for 85 days at 10%
interest ?
What is the interest under the following conditions ?
Principal. Time. Rate. Principal. Time. Rate.
4. $648. llld 4% 9. $432. leyr-3 38 anos es a
5. $324. 167d. 5% 10. $767.80 3yr.11mo.9d. 534%
6. $750. 200d. 9% ll. $50.40 10 mo. 41%
7. $427. 93d. 12% 12° 3.8737) 114 d. 6%
8. $865. 48d. 4% 13. $137.77 Ayr.9mo.25d. 74%
14. Interest is the product of what three factors ?
15. Which method of finding interest is best when principal, rate,
or time is divided by 4? By 44? 6? Q9ori2? Why?
16. Which method uses the aliquot parts of the time ?
17. Which are “6% methods”? Why so called ?
18. Which is the best method when there are years, months, and
days in the time, and when cancellation is impossible ?
19. What is the interest of $ 400 at 10% for 21 years ?
EXERCISES.
167
284. — For FREQUENT PRACTICE.
At sight.
1. Find 200 months’ interest
of $ 87.56 at 6%.
2. Find 20 months’ interest
of $ 300 at 3%.
3. Interest of $500 for 4 yr.
at 10% ?
4. Interest of $1 for 6d. at
6% ?
5. Interest of $569 for 60
days at 6% ?
6. Interest of $100 for 4 yr.
at 8% ?
1. 2 yr. interest of $400 at
9% ?
8. 8 mo. interest of $200 at
9% ?
9. 22 mo. interest of $500 at
6% ?
10. 24 d. interest of $800 at
6% ?
11. 0.003 of the principal is
the interest for « days at 6%.
12. A principal gains as much
as itself at 6% in x months.
18. 11 mo. interest of $ 400 at
9% ?
14. A principal gains 25% of
itself in # months at 6%.
15. ;1, of principal =@ mo.
interest.
For dictation.
1. 4 yr. interest of $500 at
5% ?
2. 34 yr. interest of $100 at
8% ?
3. 6 mo. interest of $120 at
| 9% ?
4. 7% interest of $500 for
2 yr.?
5. 60 d. interest of $ 567 at
6% ?
6. 20 mo. interest of $1200
at 6% ?
7. 10% interest of $1 for
06 d. ?
8. 6% interest of $1 for 17
mo. ?
9. 6% interest of $1 for
112 days?
10. 5 mo. interest of $ 240 at
10% ?
1l. At 6%, what part of the
principal = 50 mo. interest ?
12. 11 days’ interest is what
part of a year’s interest ?
138. 28% of the principal is
how many years’ interest at 5% ?
14. Find the interest of $600
at 8% for 4 mo. For
15mo. For 15d.
15. $372 is the interest of
$ 372 for how long at 6% ?
For 5 mo.
168 INTEREST. _
285. — Interest: Nore. — The method to be employed in the
Choice of Methods. solution of the following problems is shown
Written. by the Roman numerals J, JJ, or IIT (p. 166).
1. What is the interest of $840 for9 mo. 17d. at 4%?
2. Find the amount of $722 for 156 d. at 12%. J.
3. What will settle an account of $425 that has been drawing
interest at 5% for 5 yr.5mo.? III.
4. May 17, 1893, I borrowed $ 284 at 21%. Aug. 15, 1895, how
much interest had accrued? J/I.
5. In 43 years how much will be received on a $5000 railroad
bond paying 2% semi-annually? J.
6. May 27, 1898, I paid a note of $ 475.28 that had been drawing
4% interest since Dec. 31, 1894. JIT.
Find by inspection the best method of solving the following problems,
and use it in finding the interest. Try to forecast the result.
7. $9000 on interest 7 mo. 24 d. at 4%.
8. $728 draws 5% anereat for 20 months.
9. 34% interest of $900 from Jan. 15 to Nov. 2.
10. Principal, $72.59; time, 125 days; rate, 121%.
11. What shall I pay for the use of $500 ror 50 days at 5% ?
12. $320; 74%; July 7, 1845, to August 4, 1859.
13. $720; 8%; October 19, 1890, to May 11, 1893.
14 470 3305 © eligeds 20. 3% $872 4yr.8d.
15. $648 41% 8mo. 21. 5% $ 5000 16 mo.
16. $800 21% 1804. 99. 219, $178.91 104d.
17. $950 9% 2484. 93. 8% $64.87 2954.
18. $2000 7% 191 mo, 24.1% $3294 17% mo.
19. $4000 4% 42 yr. 95. 419, $700 412 yr.
RECKONED EXACTLY. 169
286. — Exact 1. In computing interest for parts of a
Interest : year we commonly consider 30 days a month
+ « 26 OUVQa « T » i 1
365 Days in an Inter- and 360 days a year. In taking 31) of a
est Year. year’s interest to find the interest for 1 day,
do we take too much or too httle, considering the actual length of a
year ?
2. Exact or accurate_interest is reckoned for the actual number of
days in the given time, and counting 365 days to the year. It is
used by the United States government and sometimes in other busi-
ness transactions. It differs from common interest only as applied
to parts of a year. What part of a year is August? February ?
The last three months of 1896 ?
3. ai, is what part of =4,? Explain this process : —
eg ihe ee OO) 80 0 2272
360 — 865 I 36 :
4. If 1 day’s accurate interest is 72 of 1 day’s common interest,
what is the accurate interest when the common interest is $ 146 ?
ed el
365
5. If from the common interest I deduct +>, of itself I shall
have the exact interest. Explain.
6. Find the accurate interest of $ 500
for 90 days at 4%.
7. Find the exact interest of $1000 at
5% from May 9 to Sept. 4.
Common interest de-
as ; 1 . .
creased by =; of it-
self is exact interest.
Nore. — The exact number of days must be found; that
is, 22+30+31+31+4=118.
Find the exact or accurate interest of —
8. $800; 6%; Aug. 11 to Oct. 9.
9. $720; 8%; Jan. 4 to Mar. 15.
10. $1200; 3 mo. 12.d.; 5%.
11. $1500; 72d; 10%.
12. What is the exact interest of $ 1000 for 2 yr. 8 mo. 9d. at 6% ?
(Find common interest for 2 yr. + exact interest for 8 mo. 9 d.)
18. Find the accurate interest of $5000 for 5 yr. 9 mo. at 8%.
170 INTEREST.
287. — Wholesale 1. Show the difference between grower
or Retail. For Cash or producer, importer, wholesaler, retailer.
or on Credit. 2. From which class of dealers do you buy ?
Oral. 3. With whom do wholesalers have to deal ?
4. The regular price of a pear-tree is $ 1.50.
If I get it for $ 1.25, what is the discount or deduction ?
5. If I buy a dozen at one time, I pay only $12. What per cent
of the highest price is this? What is the rate of discount ?
6. A man is trusted for goods billed at $100. He is to pay in 3
months. How long is the term of credit ?
7. The dealer offers to sell the same goods for ¥ 98 cash. Why is
this? What per cent does he discount ?
288.— Trade Discount. 1. My bill is $15, less 10%, as I am
“in the same trade.” What must I pay ?
2. Price per dozen, $2; for 30 dozen I pay $50. Without dis-
count the cost would be $a. The discount was y%. 3. If I had
bought 100 dozen, the net price, or what I actually pay, would have
been only $1.60. Is this a larger or a smaller rate of discount ?
Why ?
4. Discount on a carload of coal is 10%, or $4. What would it
be on 20 carloads at double the
rate of discount ? List Price or).
= Base.
5. It is common tohaveaper- | Amount of Bill)
manent list of prices and to change Discount = Percentage.
the rate of discount as may be Net Price = Difference.
necessary.
List price, $40; net price, $32; rate of discount, x%.
6. When the discount changes to 25%, the difference in price is y.
7. $5 is 20% of list price. The net price is 2.
8. Discount, 10% ; net price, $90; list price ?
CASH DISCOUNT. ATE
289.— Time and Oral.—1. A $4000 house is offered at $3500
Cash Discounts. cash. The discount = $a. The rate=y%.
2. Bought $200 worth of flour. If I need
not pay for 6 months, what do I save? Explain.
3. Which customer receives the larger discount, one who pays in
3 months or one who pays ina year ?
4. January 1 I buy $400 worth Discount is always a part
of wool, and am promised a time or per cent of the price which
discount of 2% if I pay by April 1. it reduces.
By paying when the goods were
bought, I should be allowed 4%. What is a cash discount ?
Written. — Find the missing terms : —
Amt. of bill. % off for Net cost. Discount for List price. %.
cash. cash in 30 days.
5. $2000 5 x 8. $24 $ 600 ny
6. $900 x 810 oe es $ 150 1
if x 2 490 10. $20 x 4
290. — Successive 1. From 100 take 60%, from the remainder
Discounts. 25%, from that remainder 10%, leaving x.
Written. What per cent of 100 have you deducted ?
2. A box of pens is listed at $1, but a retailer buys it for 50¢,
the trade discount being 7%. 38. When he buys 100 boxes, he gets
a second discount of 20% from the
lower price, each box costing $y. Successive discounts are
4. A third cash discount of 1% taken from the price as
makes a box cost $z. 5. Have we already reduced.
deducted
(50% + 20% +1%) of $1 or 50% of $1 + 20% of 50 +1% of 40¢ ?
Find net prices : —
List. % off. List. % off.
6. $15.40 20 then 5 9. $14.85 60, 10, and 2
7. $49.50 50 then 2 10. $320.15 20, 5, and 1
8. $600 45 then 3 ll. $4000 30, 12, and 3
172 DISCOUNT.
291. — Problems. 1. A library buys its books 35-% off. An
invoice of $10,000 calls for how much net ?
2. A bill is made—“ Terms: cash in 60 days.” What discount
may be expected for cash at time of sale? (Money at 6 %.)
3. When money is worth 12%, a dealer gives 4 mo. credit.
What discount for cash may be expected ?
4. Which would be more profitable in the end —to-sell for $ 100
cash, or to charge $ 103, giving 6 mo. credit? Is the cash discount
here more or less than 3 % ?
5. Cash or net price, $760; discount, 40 % of .; list price, a.
6. An invoice of jams is charged at $2500 on 6 mo, time, or
with time discounts of 1-% a month. What amount will pay the
bill in 30 days ?
7. One buyer gets 30% off, another gets 25% andi%. Give
net cost to each on a shipment of $ 2000 gross value.
8. A merchant who gives 90 days on all bills allows 5% for
cash. You infer that money is worth to him w % a year.
9, A furniture maker allows 15 % from the list price. Find the
net cost on an order amounting to $12,458, including $118 for
carting, which is without discount.
10. Tubing listed at $10,000 is billed less 60 % and 2 % for cash.
Net price = a,
11. The trade discount on certain goods is 70%. Large buyers
receive a second discount of 10%, making the total discount x%.
12. List price, $500; net, $425; discount, #; rate, y %.
13. List price, $488.90; net, $591.12; discount, w; rate, y.
14. 1800 ft. of moulding at 20¢, less 12 % to the trade and 1 %p
for cash, cost...
15. A shipment of sugar invoiced at $11,000 is subject to a rebate
or reduction of 5%. Terms: 15 days. 414% off for cash makes
the net cost 2.
16. The discounts on a $ 1000 invoice are not 45 %, but are 30 %,
10%, and 5%. Find the net price.
INSURANCE. 173
292. — Insurance. 1. If the owners of a hundred ships agree
ear to share the loss 1f one ts wrecked, who might
profit by the arrangement ?
2. A man owns a house worth $3000. By spending $30 he
can be sure that loss by fire will be made good. Many others
do the same, and from their money his loss is paid. What % will
he save ?
3. An insurance company promises security in case of loss to
those who have paid a certain per cent or premium on the insurance
of their property. What will it cost to insure goods for $250 at
1g?
11%:
4. 100,000 persons pay 25 ¢ each to an accident insurance com-
pany. If it pays $15,000 in claims for injuries, and $4000 for
expenses, the profit is a Who is the insurer 2 Who are insured ?
5. The agreement to make good a loss on certain conditions is
printed in a policy made by the wnderwriter or insurer, and held
by the insured. The cost of a $ 25,000 policy at 3% a year is $ a.
6. By paying an annual premium a person may be assured that
at his death or at a certain age, his family, or he himself, will
receive a specified sum. How will this money have been obtained ?
7. A ship costing $210,000 is insured for 2 of its value at 2 %.
If lost, the owners receive « The underwriters lose y.
8. $40 pays for five years’ insurance on a brick store which cost
$5000. The insurance valuation is $4000. What is the annual
rate ?
9. A wooden tenement house two miles from a fire-engine is
insured for 3%, but only for one year. If the valuation is $4000,
the cost for five years is w The property insured is called a risk.
Compare the last two risks.
10. Insurance provides for sharing loss due to what causes ?
11. A schoolhouse is insured for 5 years at }% premium, which
is $300. The insurance valuation is 2 of the cost of the house.
What is the underwriters’ loss, if it burns ?
174 INSURANCE.
293. — Examples. 1. A stock of goods is worth $12,000. The
Written. premium for a year is 1%, or $100. What
is the insurance valuation? If destroyed,
what will the underwriters pay, and what will the owner lose besides
the premium ?
Supply values for «:—
Valuation. Rate %. Premium. Rate %. Premium. Insurance.
2. $20,000 x $ 100 40) $ 12.50 z
3. 3,900 x 42 6. & 62.50 $ 5000
4. a $ 30 1. 2 AY 4TAO
8. Why is property usually insured for less than its full value ?
A $7500 house is insured at 11% for $62.50. The insurance valu-
ation is @.
9. A man is insured in a mutual company, sharing all gains and
losses; $60 insured his $8000 house in full for five years. What
was the rate? After five years $20 with interest was returned to
him. This reduced the rate to a.
10. A ship worth $30,000 is insured for 2 of its value at 21%.
The possible loss to the owner, including premium, would be a. |
11. $3.75 was the premium on # the value of some furniture at
1% ayear. What was its insurance valuation ?
12. One company offers to take a $12,000 risk at 14% for five
years; another at 1% a year. Which is cheaper, and why ?
13. $234.69 is the amount of a policy on some window glass.
The premium is 2%; the difference, or $a, equals the value of the
glass, which is ¥% of the amount of the policy.
14. $2000 is 98% of the amount insured. Premium = 2% of a.
15. If a stock of goods is worth $6930, what insurance at 1%
will include that amount and the premium? 6930 = 99% of a.
16. A block is insured for $5000 in each of 15 companies at
an average rate of 4% for five years. Find the annual cost of
insurance, not counting interest. What would each company pay if
the building suffered $50,000 damage ? Why is it safer to divide a
risk among several companies ?
COMMISSION, 17d
294.— Commission. Oral.—1. I send goods to a person in town
Selling through to be sold. I am the principal; he is my
Agents. agent. He sells them for $100, and I pay
him 13% for the service. He keeps $a as
his commission, and sends me the net proceeds, which are $100 — $a,
or $y.
2. What is 2% commission on a sale of 3000 melons at 50¢
each ?
3. My agent, a commission merchant in New York, sends me
95% of what he collects, and keeps $20. He collects $ a.
4. The gross proceeds, or sum col-
lected, is $150. The net proceeds are Commission for sell-
$ 147, less $15 expense of transporta- ing 7s a percentage of
tion. The commission is $a, or y% | the sum collected.
ae
5. A consignment of goods to be sold is sent by the consignor to
the consignee. The sum collected minus the 5% commission is
$285. What are the net proceeds? What per cent does the con-
a0 =
signee return? 1% = , or $3. The gross proceeds are $ a.
Written. —6. Four near! of peaches are consigned to a factor,
or agent, who is to receive 5¢ a basket. The price realized is $1,
but the charges for freight equal three times the commission. The
owner receives what per cent of the gross receipts ?
‘ 7. Net proceeds of a sale of cocoanuts, $100; charge for storage,
$5.60; commission, $4.40; gross receipts, By; rate, «%.
8. Find the commission on a consignment of rubber shoes sold
~at 10% off $300, the consignee retaining 3%.
9. A shipment of strawberries sells for $138.15, from which the
agent pays out $125 for freight. His 5% commission amounts
to $a.
. . 4 ie Le ‘
10. Commission = 100 of gross, or of net receipts ?
Commission + expenses + net proceeds = what ?
176 COMMISSION.
295. — Commission. 1. If I employ a correspondent, or agent,
Buying through to buy goods for me, I must pay him a per-
Agents. centage of the amount which he expends
forme. If I send him $1000 with which
to buy corn, $ 1000 1s my remittance to him.
If he invests it all, how much more must I remit to him to pay his
3% commission ?
Oral.
2. If I had sent his commission when I remitted the amount he
was to expend, how much must I have sent for each dollar he was
to expend ?
3. If I send 100% of the amount he is to expend, do I send him
enough to pay his commission? If I send him $1.05, on what sum
will he receive 3% commission? Why should the remittance be
105% of the amount to be expended ?
4. Should an agent receive a commission on his own pay, or on
only so much as he expends for his
employer ? Commission for buying is
6. An agent buys $ 2000 worth of | @ percentage of the amount
copper, charging 1% commission, or | ¢*Pended.
bw. What must his employers re-
mit to pay both bills? For every dollar he spends they must remit
exactly _..
6. Remitted $102.50 to buy umbrellas; commission 24%. How
much of my remittance will buy $1 worth of umbrellas and also
pay the agent? The agent can buy $a worth.
7. An agent charges 12% for buying chair stock. If he is to
spend $ 1000, how much must I remit?
8. Remittance, $400; goods purchased by agent, $337.50. He
pays $25 for storing and forwarding them. The gross amount ex-
pended is On what sum should he receive his 10% commission ?
9. Remittance $9.45, less 5% commission = amount of purchase.
Explain: $9.45 + $1.05 = a, -
FOR BUYING. LTT
296.— Commission for 1. At 5% commission, how many
Buying. dollars’ worth can be bought for $ 126,
leaving enough to pay for the service ?
2; The principal in a_ transaction
remits $2050 for a purchase of apples, less 2}% commission.
$ 2050 is a % of the amount to be invested. 1% of it = y, 100%
Oru. %, OF ax
3. The agent above mentioned spends only $1500. How much
of the remittance must he retain ?
4. 11% or $ 20 is a charge for buying hides. Find the base and
the entire remittance.
5. When the commission is 13%, the amount expended is what
100 x 4 __
1018 x4
Written exercise.
fraction of the remittance due ?
_—
In Selling :
Amount collected or |
Gross receipts, bah
Commission = Percentage.
In Buying:
Gross amount |
expended }
Commission Percentage.
Base
Remittance +
Commission.
Net Commission
ob
l Expenses.
r
\
|
)
Proceeds = Base — -
297. — Examples. 1. $927 is the amount sent to purchase
granite and pay the agent 3%. By what
Written. must you divide $927 to find 1% of the
base? What is the agent’s commission ?
2. Remitted $ for the purchase of glassware. The commission
was $ 42, or 31% of $y.
3. Sent my agent $1200 for purchasing wheat. His commission
is 5% of the purchase, or $ a.
4. Forwarded $ 287.50, and received in return goods worth $ 250.
The commission at 1% would have been $2. Actual rate, 7%.
5. A person bought from my agent $547 worth of straw. When
a 2% commission and $47 expenses have been deducted, I receive $a.
178 COMMISSION,
Supply values for x and y (dealings with selling agents): —
Gross Proceeds. | Expenses. — Rate of Commission.
6. $437 0 2%
7. a $ 47.50 1%
8. $ 250 x y
9. x 0) 2%
10. $1200 $ 100 a
1l. $1680 x 4%
12. a 0 y
298. — Examples.
Written.
Commission.
Net Proceeds.
wv Yy
$ 27.50 y
$ 25 $ 200
y $ 980
$ 18 y
y $ 60
$ 13.52 $ 437.18
1. $4380 is received from a sale of linen.
After retaining 14 % commission and paying
$ 2.50 for advertising the sale, what is the
balance to be remitted ?
2. What value of goods can be bought on 5% commission from
a remittance of $577.50, allowing $ 24.50 for advance charges of
forwarding the purchase ?
3. A correspondent retains 44% on the receipts of a certain
sale, and after paying $4.37 for carting, etc., remits $200. The
gross receipts include $2 + y+ $z.
4. A dealer sends to his agent $ 20,500, including a commission
of 2% on what the agent will spend, which is $100 for insurance
and sundries, and the balance $ a for wool.
Supply values for « and y (dealings with purchasing agents) : —
Remittance. cebe gamed
5. $595.80 yy
6. $179.76 $148.30 + $ 27.94
if x $ 755
8 a y
9. $1293.75 .
10. % $ 684.10 + $ 88
Am/’t of Purchase + Freight, R
ate of Commission. | Commission.
y $ 6.60
v y
y $ 22.65
5 % $ 9.90
PROBLEMS. 179
299. — Problems: 1. On a bill for hardware amounting to
Discount, Insurance, $480, I received four successive discounts
Commission. of 10% each. What is the amount to be
paid ?
2
2. My residence is insured for 2 its value in the Provident
Insurance Co. at }%. The premium is $40. What is the value of
my property ?
3. My agent in Mobile bought 40,000 lb. of cotton at 97%. His
commission is }% and his expenses are $143.75. What shall I
remit him ?
4. A real estate broker sells a farm for $8000 at a 5% com-
mission. What are the net proceeds of the sale, and what is his
commission ?
5. I can buy 1000 bbl. of oil at $1.12} with 3 % off in 30 days,
5 % for cash. What shall I save by oe the better offer,
money being worth 6 % ?
6. The estimated loss of property at a large fire was $ 275,000.
The insurance received was $180,000. How much must be taken
in new risks at an average of 2 % to cover this loss to the under-
writers together with $ 5000 expenses ?
7. I receive from my agent in London a draft for $3860, the
net proceeds of a sale of flour at 351% commission. What were
the gross proceeds ?
8. A drummer earns $2500 annually. $1000 is a guaranteed
salary; the remainder is his commission of 5%. What are his
annual sales ?
9. A broker negotiates a loan of $6500 on a real estate mort-
gage. His commission of 2 % and the expenses of examining title,
etc., are $72.37. What does the mortgager receive ?
10. Bought 1000 gross of screws at 27 cents, with a discount
of 15,10, and 5. I sold the lot at cost plus 30%. What was my
gain ?
180 INTEREST.
300. — Promissory May 17, 1895, Edward Rich lends to Thomas
Notes. Poor $180 to be repaid when the lender asks —
it, together with interest at 5%; as evidence
of the loan and security for its payment the lender receives from the
borrower a promissory note like the following:
/Haneneter, Nay ff ad Oe
Qn demand, after date, %. promise to pay to the
_Gdward Reh
One /fundred Gighty
order of
with interest, at five per cent.
Value received. A
Shomae Loor.
1. Who is the maker of this note; i.e. the promissor ?
2. Who is the payee, or the one to whom promise of payment is
made ?
3. What is the face of the note, or the sum namedin it? 4. When
is the note payable ?
5. In what way does the maker acknowledge the receipt of the
note, or its equivalent ?
6. Why is such a note as this called a demand note ?
7. Why is it called an interest-bearing note?
Promissory notes are a kind of property, and may be bought and
sold like other property.
8. Show that on May 17, 1896, the above note is worth $189 to
the owner or holder. What will the owner gain or lose by selling it
for $ 200, 2 years from date ?
PROMISSORY NOTES. 181
9. Whenever the payee of a note
transfers it to the ownership of another
person he first indorses it; that is, he
places his signature on the back of it.
6dward Reh. What is an indorser 2? An indorsement?
10. The payee of a note may indorse
it in blank as in A, or he may make a
special indorsement, as in B.
A. blank indorsement makes the note
payable to the holder. A special indorse-
ment makes it payable to the person
named by the indorser as payee. Copy
and indorse the note of Thomas Poor.
Say to the order of Every indorser of a note is responsible
for its payment unless the words “ with-
out recourse” precede his signature.
6dward Reh. 11. The holder of the note on the
preceding page demands payment of
Mr. Poor Aug. 27, 1897. What is the
amount then due ?
/remry K¥all
12. If the words “one year,” “four months,” “sixty days,” ete.,
were substituted for “on demand,” when would the note be payable ?
Nore. — In some states three extra aye after the expiration of the time named in the note are
allowed the maker for its payment. They are called days of grace. Interest is exacted, how-
ever, for the days of grace, [See § 310.]
13. If the note were a four months’ note, at what date would it
be payable without grace? With grace? When, if it were a 2
months’ note? A 6 months’ ?
Notes mature, or are legally payable, on the day when the time
named in them expires, or on the third day thereafter, when grace is
allowed.
All notes that contain the words “with interest” draw interest
from date unless otherwise specified. All other notes draw interest
from maturity. When no rate of interest is specified, the legal rate
is understood. [See Appendix, p. 15. ]
182 INTEREST.
301. — Promissory Notes. Make interest-bearing notes answer- —
ing to the following conditions, and
compute the amount due at settlement.
In finding the day of maturity, allow three days of grace. |
Written Exercises.
Date. Face. Time to run, Payee. Rate. Settled.
3/17,’95 $240 Ondemand A. P. Rice 6 9/14, ’96
8/12,’96 $800 One year E. F. Foss C1275 oo
4/21,’96 $725 Fourmonths Wm. Ward 5 Maturity
6/15, ’93 $1800 Six months Thos. True 4 9/21, 796
1/19,’95 $610 ‘Two years A.M. Bates’ 9.5 )12/25on
2/24, ’96 $280 Ondemand Rk. E. Nye 4i 7 /16,’98
eo $75 Sixty days E. B. Hale 12 Maturity
Se oe ace ome aaa
8. Write a note without signature, making yourself payee, and
transfer the note, properly indorsed, to your teacher.
9. A note matures Aug. 16, 1897. Its face is $1200, and it has
been running at 4% since May 12,1894. What is paid at settlement ?
10. Draw up a note in which Thos. Talbot hires $1700 of Samuel
Strong, at 4%, agreeing to make payment on demand. Demand for
payment is made May 14, 1895, the note being made March 12, 1893.
What amount pays the debt ?
302. — Partial Payments 1. I pay $80 on a note whose face is
of Notes. $150. What part do I pay ?
2. I pay 30% of a note of $1500.
What is the partial payment ?
8. A note of $300 draws 10% interest. What amount would
discharge the note at the end of the first year? Suppose that
instead of the note being paid in full at that time a partial payment
of $100 were made, what would then be due ?
4. Would the $100 pay all the interest due? How much of the
face or principal would it also pay ?
PARTIAL PAYMENTS. 183
5. How much of the original $500 does the maker of the note
continue to keep? On what sum, therefore, should he pay interest ?
6. Ifthe remaining $ 230 should be used another year, the interest
on it at10% would be $a, and the amount due would be $ 230 + $ a,
or Sy.
7. If another partial payment of $100 should then be made, a
remainder of $ ¥ — $ 100, or $z, would still be left at interest in the
hands of the maker of the note.
8. Give the values of a, y, and z in the solution of the following
problem :
On my note, payable to you for $300, I make a partial payment
of $100 at the end of each year for three years. What is then
due you, 10% interest heing charged ?
SOLuTION.
I. Of yourmoney Ihave foruse ..... . $800
For a year’s use of itat 10% ITowe . .... . x
At the end ofthe yearIoweyou. .... . . $380
I make a partial payment to you of ee. __ 100
II. This leaves a balance for me to use Dt ri ee SoU
For a year’s use of this sum I owe you . __ 23
At the end of the second yearlowe you .... $ y
I make a second partial payment of . . ... . 100
III. Inow have of yourmoney only ..... $ @
For a year’s use of thissum ITowe you... . . 15.30
I owe you at the end of the third year . . . . . $168.30
PE Paes OU se Teen wg ete be per cy.” 100
PVeeUalialh Btu OWe VOU...) es ne eg oh Ba OB.80
9. I borrow $500, giving my note at 6%. At the end of each
year, for two years, I pay $100. How much remains due?
10. A note for $800 draws 5% interest and is dated May 1, 1894.
May 1, 1895, $200 is paid; May 1, 1896, $100 is paid. What is
due May 1, 1897 ?
11. Two partial payments of $ 400 each are made on a $ 1000-note,
dated Aug. 20, 1895. The rate of interest is 5%. The first pay-
ment is made Aug. 20, 1897, and the second, Aug. 20, 1899. What
amount remains due ?
184 INTEREST.
When partial payments of a note are made,
the owner records the amount and date of
each on the back of the note.
To find the amount due on the following
note Dee. 31, 1901:
303. — Partial Pay-
ments of Promissory
Notes. To find
Amount Due.
§ 720 — Shringpretd, Aug. /4¢, 1895.
On demand after date 4% promise to pay to
te Ake LE al terry SYoward Y Eo.
seven /fundied Swenty Dollars
the order of
with interest at a2 per cent.
Value recetved. ¥. Bae
Jhomar SF. Sowell,
INDORSEMENTS ON Back OF NOTE.
The Supreme Court of the
United States has decreed
that, —
Reeetwed on the within note:
Dee. 26, 1896, § 209
Sept. /¢,/899, 175
Partial payments of notes
must first be used to cancel
dee. 3/, 1900, 4.00 the interest due. Any bal-
Settled, Lee. 3/,/90/.
ance remaining may be used
to lessen the principal.
1. Who puts these indorsements on the note? 2. Will any other
receipt be requested for the $200 paid Dec. 26, 1896, and if so,
by whom? 8. What needs to be done before the note can be trans-
ferred to a third person ?
PARTIAL PAYMENTS. 185
SOLUTION.
From date of note to Ist payment.
8/14, ’°95to 8/14,°96 =lyr. . . .* $0.06 $ 720
5/14, 96 to 12/14, "06 =-4'mo. . .)\. «. 0.02 0.082
12/14, 96 to 12/26,°96 = 12d. . . . . Daa 44
Intol Serie .eee S52 8 0;088 57 6
Interest due when lst paymentismade ...... . $59.04
Beeee: Rote, Or lst principal
To find the Principal. Divide the interest by the product of the rate and
the time in years.
To find the Rate. Divide the interest by the product of the principal
and the time in years.
To find the Time. Divide the interest by the product of the principal
and the rate per cent. The quotient is the time in
years.
To find the Amount due on a Note on which Partial Payments
have been made. United States rule: Find the amount of the principal to
the time when a payment or the sum of several payments shall equal or exceed the
interest due at the time. Subtract such payment or sum of payments from the
amount, and with the remainder as a new principal, proceed as before to the time
of settlement.
Compound Interest. Find the amount of the principal for the first period
of time. Treat this amount as a new principal, and find its amount for the
second period, and so on for the entire time. The last amount less the given
principal will be the compound interest.
Present Worth. Divide the given debt by the amount of $1 for the time to
elapse before the debt is due. The debt less the present worth ts the true discount.
10. — DISCOUNT.
Bank Discount. Compute bank discount as tf it were simple interest on the
Jace of the note for the term of discount. The face of the note less the bank
discount will be the proceeds.
True Discount. See Present Worth, § 9.
11.— PROPORTION.
Rule of Three. Make that number in the problem which is of the same kind
as the desired result the third term of a proportion.
If, from the conditions of the question, the result is to be larger than the third
term, use the two like numbers in making the jirst ratio of the proportion less
than 1; but if the result is to be smaller, make the first ratio greater than 1.
Divide the product of the means by the given extreme, and the quotient will be
the fourth term of the proportion, or the result desired.
APPENDIX I. 7
Partnership. (Give each partner such part of the whole gain or loss as his
capital for any time is part of the whole capital for the same time.
12. = ROOTS:
Extracting the Square Root.
I. Beginning at the decimal point, separate the given number into groups of
two figures each.
Il. Find the greatest square in the left group and place its root at the right ;
subtract the square of this root from the left group, and to the remainder annex
the next group for a dividend.
Ill. Divide this dividend, omitting the last figure, by double the root already
Sound, and annex the quotient to the root, and also to the divisor.
IV. Multiply the divisor as it now stands by the last root figure and subtract
the product from the dividend. 2
V. If there are more groups to be brought down, proceed in the same manner
as before.
Extracting the Cube Root.
I. Beginning at the decimal point, separate the given power into groups of
three figures each.
Il. Find the greatest cube in the left group and place tts root at the right.
Subtract the cube of this root from the left group, and to the remainder annex the
next group for a dividend.
Ill. Annex a cipher to the root already found and take three times its square
for a trial divisor. Divide the dividend by this trial divisor and place the quo-
tient as the next root figure.
IV. Multiply the number last squared by the last root figure and add three
times this product and the square of the last root figure to the trial divisor for a
complete divisor.
V. Multiply the complete divisor by the last root figure, subtract the product
From the dividend, and to the remainder annex a new group.
VI. Form a second trial divisor, using two figures of the root with a cipher
annexed, and proceed as before until all the groups have been used.
13. — Roman Seven capital letters were used by the Romans to
Notation. represent numbers. They are of almost no use in
computations.
Vv xX L C D M
5 10 50 100 500 1000
8 APPENDIX I.
To represent other numbers, these letters are combined according to the
following principles :
I. Repeating I, X, C, or M, repeats its value.
II. When I is used before V or X, X before C or L, and C before M, the
difference of the values is to be taken.
III. When any numeral follows one of greater value, a sum of values is to be
taken.
IV. A dash (—) over any numeral but I increases its value 1000 times.
Show how these principles are illustrated in CC; IX; LX; CM; MM; C;
MDCCCXCV. Mention four uses of Roman numerals.
14.— Least Com- The least common multiple of two or more numbers
mon Multiple. contains only such prime factors as are needed to pro-
duce each number.
The following method of finding the l.c. m. is a useful one, though not differ-
ent in principle from that given on page 61.
To find the l.c.m. of 24, 40, 72, 108.
2.124 40'..72> 108 EXxpLaNaTIon. — We discard 24, for
20 ~=.36 54 any multiple of 72 is a multiple of 24.
10 18 27 We then divide the remaining numbers
5s) Cae by any prime factor common to any two
7 ie ar ee of them, until quotients are obtained that
5 1 eS are prime to each other. The product
2x2x2x3x3x5x3=1080, l.c.m. of the divisors and the remaining quo-
tients is the desired least common multi-
ple. Select from the process shown above the prime factors of each number.
15.— Leap Years. A True or Solar Year is the exact time in
which the earth revolves once around the sun. Its
length is 865 d. 5 h. 48 min. 49.7 sec., or about 11} min. less than 365} days.
To avoid the confusion and inaccuracy of the methods of reckoning time
then in use, the Roman dictator, Julius Cesar, 46 B.c., reformed the calendar
by establishing what is now known as the Julian year, of 3651 days. To avoid
the inconvenience of counting the fractional part of a day every year, he
decreed that three successive years should consist of 365 days and the fourth
year of 366 days, the extra day being added to the month of February. The
year containing the extra day is called bissextile or leap year.
But this arrangement of the calendar made the civil year too long by about
114 minutes, an error that amounted to 1 day in about 130 years. To correct
this and other errors, Pope Gregory XIII. struck out ten days from the calen-
APPENDIX I. > 9
dar, calling Oct. 5, 1582, Oct. 15; and ordaining that thereafter only those
centennial years should be leap years whose numbers are divisible by 400.
The Gregorian year is now the civil or legal year in nearly all civilized coun-
tries but Russia and Greece, where the Julian calendar is still in use, and the
dates 12 days behind ours.
The Gregorian calendar was ‘not adopted in Great Britain till 1752. The
error had then amounted to 11 days, and hence the third of September was
called the fourteenth. Old style dates are according to the Julian calendar, new
style dates conform to the Gregorian calendar.
When the number of a year ts divisible by 4, it is a leap year; but centennial
years whose number is not divisible by 400 are exceptions.
16.— Land Measure- Government Lands are divided by parallels
and meridians into townships six miles square,
containing 36 sections or square miles. Each
section is divided into half sections and quarter sections.
A township is designated by its number north or south of a base line running
east and west, and east or west of a principal meridian running north and south.
Thus, C is Township 4 N., Range 3 FE. Whatis A? B?
The 36 sections into which a township is divided are numbered as in the
accompanying figure. Point out section 15.
Half and quarter sections are designated as W. or N. half sections, etc. ; and
S. W. or N. E. quarter sections, etc.
ments.
Township. Section 15.
Surveyors generally use, in measuring land, a steel chain 100 ft. long,
divided into foot links, or a steel tape line of the same length graduated to feet
and tenths. Sometimes a Gunter’s Chain is used. It contains 100 links,
each 7.92 in. long. The chain is 4 rods, or 66 ft., or 792 in., in length. 80
chains, or 320 rods, measure a mile.
10 APPENDIX I.
17.—The Metric The Metric System of weights and measures is
System of Weights named from the Meter, from which all the units of 3
length, surface, volume, capacity, and weight are
derived.
The Meter is approximately one ten-millionth of the distance from equator
to pole on the earth’s surface.
and Measures.
Notr. — The Metric System is in general use by nearly all civilized nations except Great Britain
and the United States. It is used by some departments of the United States government, and in
the sciences.
The Metric System is a decimal system, ten units of one denomination making
one of the next higher.
Decimal parts of the standard units are denoted by Latin prefixes ; multiples
of the standard, by Greek prefixes.
Milli means 1000th Myria means 10000
Centi means 100th Kilo means 1000
Dect means 10th Hekto means 100
Deka means 10
In the tables units in common use are in étalics.
Length Measures. Standard unit, the Meter.
Table. Equivalents.
10 millimeters (™™) =1 centimeter (™) =0.8937079 inch
10 centimeters = 1 decimeter (#™)
10 decimeters = aecer (=) = 9.57079 inches
10 meters = 1 dekameter (P™)
10 dekameters = 1 hektometer (#™)
10 hektometers = 1 kilometer (=) tees
~ (0.621382 miles
10 kilometers = 1 myriameter (™™)
Surface Measures. Principal unit, the Square Meter.
Norr. — As the units of surfaces are squares whose dimensions are the corresponding linear units,
it takes 10? or 100 units of one denomination to make one of the next higher.
Table. Equivalents.
100 sq. millimeters (84™™) = 1 sq. centimeter (9ae™) = 0.155 sq. inch
100 sq. centimeters = 1 sq. decimeter (#1 ¢™)
~ 100 sq. decimeters = 1 sq. meter (°4™) = 10.764 sq. feet
100 sq. meters = 1 sq. dekameter (#1 Dm)
100 sq. dekameters = 1 sq. hektometer (#1 Hm)
100 sq. hektometers = 1 sq. kilometer (25™) = 247.114 acres
Nore. — When used in measuring land the square meter is called a centare (ca), the square deka-
meter an are (a), and the square hektometer a hektare (Ha),
APPENDIX I. heh
Volume Measures. Principal unit, the Cubic Meter.
Norr. — As the units of volume are cubes who edges are the corresponding linear units, it takes
108 or 1000 units of one denomination to make one of the next higher.
Table. Equivalents.
1000 cu. millimeters (2 ™™) = 1 cu. centimeter (C%e™) =,0.06103 cu. inch
1000 cu. centimeters = 1 cu. decimeter (cu am)
1000 cu. decimeters = 1 cu. meter (c%™) = 35.314 cu. feet
Nore. — In measuring wood the cubic meter is called a Stee (18t = 0.2759 ed.) ; a decistere (1%*)
is one tenth of a stere.
Measures of Capacity. Principal unit, the Liter = a cu. decimeter.
Table. Equivalents.
10 milliliters (™) = 1 centiliter (*!) = 0.6102 cu. inch
10 centiliters == 1 deciliter (4!)
oe : 1.0567 liquid quarts
— ] —
10 deciliters = 1 liter {*) = { 0,908 diy quart
10 ljters = 1 dekaliter (DP!)
26.417 gallons
= Hl) — 5
10 dekaliters = 1 hektoliter (™) = 2.8375 bushels
10 hektoliters = 1 kiloliter *)
Notre. —The dizer is used in measuring liquids and small fruits, the hektoliter in measuring
grain, vegetables, and liquids in larger quantities.
Measures of Weight. Principal unit, the Gram.
Table. Equivalents.
10 milligrams (™8) = 1 centigram (°) = 0.15482 grain
10 centigrams = 1 decigram (28)
10 decigrams = 1 gram (8) = 165.432 grains
10 grams =1dekagram (2)
10 dekagrams = 1 hektogram (8)
10 hektograms =1 kilogram (®8) = 2.20462 pounds
10 Kilograms = 1 myriagram (M8)
10 Myriagrams =1 quintal (&)
10 Quintals =1 metric ton(T) = 2204.621 pounds
Nore. — The gram is the weight of a cubie centimeter, the kilogram of a cubic decimeter, and
the metric ton of a cubic meter of distilled water at its greatest density.
The gram is used in mixing medicines, and in weighing jewels, precious metals, letters, ete.
Ordinary articles are weighed by the kilogram (commonly called ilo), and heavy articles by the
metric ton,
12 f APPENDIX I.
18.— TABLE OF EQUIVALENTS.
Common. Metric. Common. Metric.
1 inch = 2.54em 1 cu. foot = 28.31 7cu dm
1 foot = 30.48em leu. yard =0.7645cum
1 yard = 0.9144™ 1 cord = 3.6248
1 rod = 5.029m 1 liquid quart = 0.94631
1 mile = 1.6093Km 1 gallon = 3.785!
1lsq. inch = 6.452sacm Ldry quart: #11013
1 sq. foot = 9.290384 dm 1 bushel = 0.3524H1
1sq. yard =0.8361sam 1 grain = 0.06488
1 sq. rod = 0.25298 1 ton = 0.9072met ton
1sq. mile =2.59saKm 1 troy ounce = 31.10358
1 Acre = 0.4047Ha lav. ounce = 28.358
leu. inch =16,387cucm lav. pound = 0.45386Ks
Approximate Equivalents.
1 decimeter = 4 inches 1 liter = 1.06 liq. qt. or 5%, dry qt.
1 meter = 3 ft. 33 in. 1 dekaliter = 1 pecks
1 dekameter = 2 rods 1 hektoliter = 22 bushels
1 kilometer = $ mile 1 gram = 15} grains
1 acre = 4 sq. rds. 1 kilogram = 21 av. pounds
lhektare = 23 acres 1 metric ton = 2200 pounds
1 stere = } cord
19.— Metric System. ‘The units of the Metric System form a decimal
system. Hence the following principles apply : —
I. Excepting in square and cubic measures, any
metric number may be changed from one denomination to the next smaller. or the
next larger by moving the decimal point one place to the right or left, as the
case may be.
Written Exercises.
II. Jn square or surface measures this reduction is effected by moving the
point two places, and in cubic or volume measures three places, instead of one.
e
Ill. Any denomination may be taken as the unit, the number at the right of
the point being read as its decimal.
Explain the following changes or reductions : —
1, 3247.28™ = 324728 = 32,4728Hm — 3,24728Km — 3247280mm,
9, 67317.9694m = 673.1796s4 dm — 6,73179624 m = 0.0673179684 Dm,
APPENDIX I. 13
8, 8.3724H8 — 837.249 — 83724¢4 = 8372454 ™,
4, 47.2384cum — 47,234st — 47234cu dm — 47234000cu em,
5, 247.831! = 2.47831H! — 24783. 141.
6, 1846.982% = 1.346982K = 134698.2¢s.
7, In 847.2K, how many grams? How many pounds?
8, Change 75 bushels to hektoliters.
9, How many square meters in a rectangle 18 ft. by 10 ft. ?
10, An importer pays duty on 1,200 meters of cloth. How many yards ?
11, How many square rods in a square hektometer ? |
12, How many liters in a cubic meter ?
18, An importer buys 250! of liquor at $0.75 a liter. He sells it for $3a
gallon. What does he gain or lose ?
14, A rectangular stone is 1™ long, 54™ wide, and 24 thick. How many
kilograms does it weigh, being eight times heavier than water ?
15, How many kilograms of flour in a barrel ?
16, Add 18.32K™, 648m, 94.8Hm, 38 ,4dm,
17. What will a stere of wood cost at $12 a cord ?
18, How many hektares in a field 144™ long and 40D™ wide? How many
acres ?
19, How many gallons in a cubic meter of water ?
90, How many times is 164™ contained in 1,.28K™ ?
91, If goods are bought at $2.55 per yard, at what price per meter must they
be sold to gain 25%? (1 meter = 39.37 inches.)
99.. A hektoliter of fruit weighs 63 kilograms, and 32 liters of syrup can be
obtained from it. How many kilograms of fruit will it take to make a hekto-
liter of syrup ?
93, The distance between two places on a map is 12.5 centimeters. What is
the actual distance between the places if the scale of the map is 1 to 60,000 ?
94, If a certain stone is 2.83 times as heavy as water, what is the weight
of a piece of this stone which is 5.59™ long, 17.364" wide, and 52.6e™ thick ?
14 APPENDIX I.
20. — Comrounp InTEREST TABLE.
vay 2 per cent. | 24 per cent. | 3 per cent. |34 per cent. | 4 per cent. | 5 per cent. | 6 per cent.
1 1.020000 1.025000 1.030000 1.035000 1.040000 1.050000 1.060000
2 1.040400 1.050625 1.060900 1.071225 1.081600 1.102500 1.123600
3 1.061208 1.076891 1.092727 1.108718 1.124864 1.157625 1.191016
4 1.082432 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477
5 1.104081 1.181408 1.159274 1.187686 1.216653 1.276282 1.338226
6 1.126162 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519
7 1.148686 1.188686 1.229874 1.272279 1.815932 1.407100 1.503630
8 1.171660 1.218403 1.266770 1.316809 1.868569 1.477455 1.593848
9 1.195093 1.248863 1.304773 1.362897 1.423312 1.551828 1.689479
0 1.218994 1.280085 1.848916 1.410599 1.480244 1.628895 1.790848
11 1.248374 1.312087 1.384234 1.459970 1.539454 1.710339 1.898299
12 1.268242 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197
~ a: 1.293607 1.878511 1.468534 1.563956 1.665074 1.885649 2.182928
14 1.319479 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904
15 1.345868 1.448298 1.557967 1.675349 1.800944 2.078928 2.896558
16 1.372786 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352
aT 1.400241 1.521618 1.652848 1.794676 1.947901 2.292018 2.692773
18 1.428246 1.559659 1.702433 1.857489 2.025817 2.406619 2.854339
19 1.456811 1.598650 1.753506 1.922501 2.106849 2.526950 3.025600
20 1.485947 1.638616 1.806111 1.989789 2.191123 2.653298 3.207136
21.— Annual Interest. 1, I borrow $800, agreeing to pay 6% interest
at the end of each year. I do not, however, pay
any interest until the end of 4 years 8 months, when I pay principal and interest.
What should I then pay ?
In cases where there is an agreement to pay interest annually the custom is
to charge simple interest on the principal and on the overdue interest for the
time each interest payment is overdue. ‘This is called annual interest.
Process.
Each payment of annual interest should have been $48.
The 1st annual interest payment of $48 has been due 3 yr. 4 mo.
The 2d annual interest payment of $48 has been due 2 yr. 4 mo.
The 3d annual interest payment of $48 has been due 1 yr. 4 mo.
In all, interest on $48 has been due for . . . Tyr.
APPENDIX I.
Interest of $800 for 4 yrs. 3 mo. .
7 yr. interest of $48
Total interest due
Principal due
Amount due at arrieaane
Find the annual interest of —
9, 31,200 for 4 yr. 6 mo.
3, 1,800 for 5 yr. 8 mo.
4, 4,200 for 3 yr. 2 mo.
5, 640 for 5 yr. 7 mo.
at 6%
at 5%
nite
at 5%
co co “I OD
15
ae Mere ae ye $ 204
20.16
$ 224.16
800
$ 1024.16
$900 for 9 yr. 6 mo. at 7 ¥,
720 for 3 yr. 8 mo. 12 d. at 44%
. 618 for 4 yr.5 mo. 17d. at 3%
, 427 for 6yr.8 mo. 2d. at 12%
22.— Taste sHowinc RATE oF INTEREST ALLOWED IN THE STATES AND
TERRITORIES.
The legal rate is given in the first column.
in the second column. * Any rate.
upwards on collateral security.
States. ape
per cent.
BAIR DATING ceietais acre Selec
A YIZOUS ed cas ote ota 7 *
TATEKSNIBAR S Aotaticteye noe 6 | 10
Californidentesncec oes vi *
COlOTAAO: © a4 oes carne ae S| *
Connecticut.......... 6
Dela wareiis secrete 6 6
Dist. of Columbia....| 6] 10
OEIC A ccc itatt cee taal ig
EOTTAseldtis «os owen es q 8
TdANO scene < one cites eb ats:
linotsisaece ean os ele a
Tridiana.n.2 ss eee e 6 8
LOWS eine here 6) 8
TCANBBSE. wees at 6 | 10
IONTUCE Yat ass 6 6
}
States eee
per cent.
OUisiaN Sass eo secs 5 8
MERIT OP a ocr tore eyes ere arr 6 *
Maryland. swe ofecacs's 6 6
Massachusetts....... 6 ma
MICHIE ST precepts ee sais 6 8
IVETTAT ESO US teietstete stele, ss ed ady)
Mississippi pew fees 6 | 10
Missoutine- aeons. 6 8
IM OTICATIONS amtsye- cosy: 10 *
INGDTASKie conics «assis 7 | 10
IN GY Ad Saee was xe ies q *
New Hampshire..... 6 6
New Jersey ......... 6 6
New Mexico:........ 6 | 12
IN GW X Of katstetean «tes 6 6
North Carolina ...... 6 8
North Dakota >..
14. 3a=a+a-+a; the factors are3anda. 3a?=%
The coefficient of a literal quantity consists of the factors pre-
ceding it in the same term, usually the numerical factor, which
comes first. Thus, 3 is the coefficient of 3 ayz, or 3 ay is coefficient
of z. When no figure is written, 1 is understood: a=1 xa, or 1a.
Give the numerical coefficient, and factor each term, thus:
3m=mMm+-m+-m, or 3xXm; =m xX mMxXM.
15. n 18. 2 a?m Bie 24. b*(a + b)— c(d — e’)
a
Oe ee eee LOS ry 22. 3ab—bB® =. Ba(b+c)+2y(f—g)
9
fs 300 20. 3 am? 23. 2cd+5z 26. ab(a—y)— 2)
&
Similar terms must contain the same powers of the same literal
quantities. The numerical coefficients and the preceding signs may
differ.
mb and bm? are similar, but in each the 6 should be written
before the m.
27. Select similar terms among the following : —
7 mb —2a*n 5 an — 2 vy Teal
an 3 0'm? — 2 bm? 2 vy” — any
— 4 oy? — 5 bm’ 4 ay? 2 bn —9an
13.— The Value of
the Unknown Quantity
found by Subtraction.
I. If the same quantity be taken
from equal quantities, equal quantities
remain.
APPENDIX IL. 31
Apply this axiom or self-evident truth in finding the value of # in
these equations.
1. «+4=10 — Subtracting 4 from each member, (1) « = 10—4
Performing the process indicated, (2) « =6
9. 4ea=3e24+17
Taking 3a from each member, we have (1) 4%-—82=17
Performing the subtraction, we find (29) Me AT
8. x£+ 24 = 32 5. dl+a= 64 7. «+ $19 = $82
4. B5&a@=—42+4+ 20 6. 27=15-+¢a@ 8. e+ $23 = $20
9. $5 added to my money 10. The price of corn has risen
will give me $28. How much 13¢ and is now 63¢. What was
have I? it at first ?
Let 2 = my money now. 11. The watch and chain cost
Then «+5 = { my money after $5 $495. The chain cost $30.
is added.
Then x + 5 = 28, etc.
12. Show that subtracting a quantity from each member of an
equation is the same as transposing it to the other side with a minus
sign.
14.— The Value of Il. If the same quantity be added
the Unknown Quantity to equal quantities, the sums will be
found by Addition. equal.
Apply this second axiom in finding the value of a.
th e— {= 12
Adding 7 to each member, we have (1)*%—-7+7=12+4+7
Performing the processes indicated, (2)'%= 19
What is the effect of taking away 7 and then adding 7 to any
number ?
2. a—$$12= $19 3. e—#=2 4. x — $24.75 = $ 82.75
39 _ APPENDIX II.
5. The cost was $18; the 6. He was in school 26 half
discount was $43. What was days, and absent 12. How long
the list price ? had school kept?
Let # = the list price. 7. 327 miles had been trav-
(1) « — $41 = the cost. elled. The journey was x miles,
(2) « — $45 = $ 18, ete. and 487 yet remained.
8. Show that in an equation hke «—3=10, adding 3 to each
member is the same as transposing — 38 to the other side, with its
sign changed. :
15.— The Value of Til. If equal quantities be divided
the Unknown Quantity by the same quantity, the quotients will
found by Division. be equal.
Apply this third axiom in finding the value of a.
1 "0 Wee 24
Dividing each member by 3, we have (1) = 5
If 3a = 24, 1a or x = q of 24 or 8.
Anel2ee 00 4. 2ie= 25 OU Sea
8. 19y =57 §. 184 = $52.13 7. 2a=90
8. I sold my house for $6,000. If this was three times the cost,
what did I gain ?
9. Seven times a number less 5 equals 51.
10. Eight times my money and $40 is 12 times my money.
16.— The Value of IV. If equal quantities be multi-
the Unknown Quantity plied by the same quantity, the products
found by Multiplication. will be equal.
Apply this fourth axiom in finding the value of 2. if ao i
Multiplying each member by 8, we have Cl) a= 21
If 1 of x = 7, the whole of ¢ = 38 x 7 or 21.
APPENDIX II. 83
x 3 5a T2
Wee a owe 4’ 90 6 KM = 8 210
12 te 6 i?
6. Lost 2 of my money, but $3805 still remained. How much
had I?
7. 35; of the whole distance and 21 miles cover the journey.
8. 18 of the 30 miles of the yacht’s triangular course were sailed
in 72 minutes. The full time was what ?
9. A third and a fourth of my money made $18.60. How much
had 1?
(1) ; Z — $18.60. Multiply both members by the 1. c. m. of 3 and 4.
10. 2 of what I received was silver, ? was gold, and the remaining
$ 24 was in bank bills.
17.— Reduction The value of # in any simple equation may
of Equations. be found by applying one or more axioms;
that is, by increasing, diminishing, multiply-
ing, or dividing both members of the equation by the same quantity.
The steps of the process should be taken in this order :—
I. Combine similar terms; reduce to lowest terms.
Il. Clear the equation of denominators by multiplying both mem-
bers of it by their l. c. m.
Ill. Transpose unknown quantities to the first member, and known
quantities to the second, by addition or subtraction.
IV. Combine similar terms.
V. Divide both members of the equation by the coefficient of x.
Pa Pere
1. .Give 3
4
eh tae by 12, we have (1) 8a +9x2+ 108 = 10% + 192
Subtracting 108 and 10a, (2) 8a#+9a— 10% = 192 — 108
Combining similar terms, (3) 1@.= 84
Dividing by 7, (4) G12
+ 16, to find the value of a.
2. Given “ = r+ Rather mots 6 = -— ah 17, to find the value of a.
34 APPENDIX II.
To verify an equation is to prove that its members are equal by
substituting numbers for the letters that represent them.
> M
Thus, in the equation in Example pg: 3 mt ei —9= a + 16, we found that
x=12. Using 12 in the equation wherever x occurs, we have 7
5) 3 2 le ‘
eee Kes Rae SN Ty
ss) 4 6
24 36 60
or 3 + — mi +9= gt 16,
or acon iaettettr
or 26 = 26.
Find the value of x in the following equations, and verify each:
3. 132 15 — aoe 8. oat ee a8
PAZ a lost Se ie 9, 204 °2_17=0
5. 108+ 20%4+14=127+4+19e 10. ange aa
6. 18a—7416=52418 —2 1 Te
7 e+et+e=_t 30 12. Pete a 5g
14: 2a 2 451-983
18. — Problems. 1. A horse and wagon cost $280. The
horse cost 3 times as much as the wagon.
2. Together the girls picked 45 quarts of berries. One picked
4 as much as the other.
3. Divide 23 into two parts, one 5 less than the other.
APPENDIX II. 35
4. The sum of two numbers is 87, and their difference is 9. What
are the numbers ?
5. I bought 6 lb. of coffee and 5 lb. of tea for $6.40. The tea
cost twice as much as the coffee. Find the price of each per pound.
6. Of three candidates at an election, A had 3 times as many
votes as B, lacking 220, and C had 2? as many as B. If there were
3300 votes cast, how many did each ‘candidate receive ?
7. Four times a certain number divided by 3 added to 2 the
number equals the difference between the number and 518.
8. A certain number diminished by 40 is the same as 40 dimin-
ished by 4 of the number.
9. Take from a number 2 of itself, 4 of itself, and 40, and nothing
remains.
10. A man bought an equal number of horses, cows, and sheep.
For each horse he paid $125; for each cow, $50; and for each
sheep, $12. How many did he buy for $1,309?
19. — Problems. 1. A’s capital in business is twice B’s. A
loses $5,000, while B gains $3,000. They
then have $15,000 together. What did they have originally ?
2. A farmer sold 4 his wood at $3 a cord, 4 of it at $4, and +
of it at $5. He received $44 for the lot. How many cords dia
he sell ?
3. A certain sum of money was divided among F, G, and H. F
and H received $150; F and G, $216; and G and H, $178. How
much did each receive ?
4. Two tanks contain an equal quantity of water. But after 75
gallons have been taken from one, and 50 gallons added to the other,
one contains twice as much as the other.
9. Divide 30 into two such parts that 4 times the greater shall
equal 6 times the lesser.
36 APPENDIX IL.
6. Take twice a number from 19, divide the remainder by 3, and
add the number. The sum will be the same as if half the sum of the
number and 10 were taken. Find the number.
7. The sum of three consecutive odd numbers is 39. What are
they ?
8. The sum of three consecutive multiples of 7 is 273. What
are they ?
9. A grocer sells 80 pounds of tea at 50 cents a pound. But this
tea is a mixture of poor tea at 45 cents with a better quality worth
65 cents. How many pounds of each kind in the mixture ?
10. A has $250, and B has $75. How much must one give the
other, that they each may have the same sum ?
20.— Addition: the As in arithmetic only like numbers can be
Signs + and —. combined in one sum, so in algebra only simi-
lar terms can be added. To add da, 26, and
leis in a sense like adding 8 acres + 2 bushels + 1 cent; we can
only indicate the addition of the different units, thus: 3a+ 2b+1e.
1. How many n’s are 4n, n, 2n, 13?
2. Arrange similar terms in columns; then add —
Zab, dc, Tax, 4bx, c, tab, bx, 3aa, Te, Lab, ax, bx.
Indicate the sum of the four amounts just found.
Suppose a man begins the week with no money, and in his book
puts a + before each amount that he gains and a — before each
Gain $ 3 amount that he loses. Instead of Lay
Gain $14 entering each transaction as shown aay
Loss $ 2 at the left, he might keep the avy A
Gain $ 3 account as at the right. He would 4 34
Loss $ 4 first add the + sums which in- ae
crease his property, then the —
sums which diminish it, and then + 19d and —
put the two together. If it were CO etaed
+ 19d and +7d that he put together, the sum would be + 26 d, but
adding — 7d to + 19d has an opposite effect. The sum of his gains
and losses is + 12d.
$19 gain less $7
loss = $12 gain
APPENDIX II. 37
Consider that the signs.+ and — belong to the quantities that
follow them, showing their character as gains or losses, and we may
write (+3d)+(+14d)+(—2d)+(+38d)+(—4d)=(+19d)+(—7 d)
=(+12d). Adding —7 is like taking away + 7, and as all quan-
tities are + unless marked —, we may write 19d —-7d=12d.
3. Add a loss of $10 to a gain of $15; what results ?
Add + and — quantities separately.
4. —4a 6, + 227 6. 8 (b +c) —22
+ 3a —Ilzl —2(b+¢) — 32
+a 4+ 4 27] —3(b+.¢) nee
—5a — 227] 5 (b + ¢) — 52
+ 5a + 7 al (b + ¢) en
21. — Problems. 1. A yacht goes 10 m off shore and 3 m back.
Give these distances opposite signs, and add.
2. Again she sails from port + 7m, — 6m, + 6m, +4m, — 10m.
How far from port is she at the finish ?
3. A man has no money, but there are due to him $10, $3, $4.50,
$2.50. He owes $2, $6, $0.75, and $1.25. If all these sums
should be paid, would you mark the balance + or —? In adding
the eight sums given, which four should be marked — ?
4. A bin is kept for corn and oats mixed. One bagful = (¢ + 0).
The changes in a month are 10 (e+0) received, 4(c+0) sold, 25(c+o)
received, 2(¢ + 0) received, 9(¢ +0) sold, 1(e+ 0) sold. Arrange
in one column with the proper signs, and add.
5. Suppose a man’s debts are $100 greater than the money that
he has together with what is due to him. Which will represent the
amount of his property, + $100 or — $100? : ¥
6. If the sum of the — or negative terms is greater than the sum
of the + or positive terms, will the result be + or — ?
7. If you go forward 100 miles (+ 100m) and backward 150 miles
(— 150m), is your final position in reference to the starting point
+ or —?
38 i
Negative Quantities.
When plus means
APPENDIX II.
Any quantity may be taken in two opposite
senses, positive and negative, which we indi-
cate by the signs + and —.
1. What is the difference between 10°
22.— Positive and
sty ‘negative above zero and 5° below zero ?
above = x | below 2. If we take away a man’s gains (+
forward : _ back quantities ), is his property diminished or
} before Eater increased? If we lessen the amount of
: ae, é inca his losses (— quantities), will he have more
excess \ deficiency dollars or fewer? Taking away a posi-
tive quantity makes the minuend smaller.
Taking away nothing leaves the minuend the same.
Taking away a negative quantity makes the minuend larger.
10 —0= 10
10 —(+3)=7
Let us compare these four cases : —
To add
——_sSa—_— eee 7
Like signs.
+ 6 — 6
ee Rae
Pine
Here the first
quantity 1s in-
creased in the
same direction
by adding the
second quan-
tity.
Unlike signs.
30 eb
pee eae
42 —2
Here we com-|
bine opposing
quantities. SOIT Ol eOIOd
ie)
~y
jor a aor ~ajpo
a.
BO bolt BIO bol BIO DO BIO Bie
RrPwwnwm Nee
Loe bd}
on or)
HP A]
Do + LO), bolt boILO bot boItO Bl rol Wl pty,
Nr Dp
Rio |
bol bo}
Ov CO Pe B® PRP OOD DY ee
1 Oe Co
aR
wen clo cjo
° : °
772 ; 294 rem.
b. 1794;
c. 13825; 507 rem.
“et See Foe
Sag Roe) OF Fe 08 eer
_
829 ; 202 rem.
414; 775 rem.
485; 854 rem.
1395 ; 246 rem.
734; 618 rem.
402; 501 rem.
371; 644 rem.
joo to 0000 GIO
A . ° °
OO HOO HG OKO pl
OO HO WIP DM °
CON FO RO RISO LE
ce]
wo
on
o
30 i
rs
SO Br WO oF
eA ie hse geht
Pe FAQMSAS SS
foal
caja fen cee cy,
ey)
oO
cont
="
REN
me to ON
Oo
Waa
He he
—
Hm HH IO OD GO On
_
oo
on
> color PID AY
I crloo Goj~r SO}
inte
ie SO
WI BIO cole “ID oi
“my Ol
orl WO Oo
. .
SAPO cps ej BODO
Be FANS ROR
Cpr C709 Ores HA}
Dinr onleo wo
sale i
Art. €5.
$ 921.42,
$ 144.99,
1254.35.
$ 478.94.
$ 1632.59.
$ 53.66.
$ 2165.19.
$ 230.55.
5346.
$ 225.63.
ANSWERS.
Art. 67.
565 1b.
$350.
$ 525.
$ 0.75.
28 m.
$2500., each
man’s gain;
$80000., cost of
land; $95000.,
proceeds of sale;
$15000., whole
Dap ww
ae
gain.
8. $15.75.
9. 3¢ cheaper to
send by mail.
10. 22 qt.
Art. 68.
8. 2640.
9. 17741.
10. 15924.
11. 15522.
12. 1,206,060.
13. 32616.
14. 3690.
15. 11466 ft.
16. 10710 lb.
Art. 72:
rR 128.08,
y = 148.92,
Zz = 209.39.
Sum of A=
1180.11.
Sum of B =
693.72.
Sum of A— sum
of B = sum of C
or 486.39.
8. 1,553,424249
4. $1225.
See aT e OO ves Or ae
SOs ee Coe ap Nene, bag
="
>
Beano eat Si CA «pms 009
-
- ©
1652 Ib.
. $109.85.
. $1050.
. $1.50.
Hans.
$4.
Art. 77.
$16.
$ 192.
- 4608 in.
84480 ft.
5185,
50 d.
B41 ee:
245.
128,
Art. 83.
$ 1632.
152.38.
210.11.
655.13.
1107.76.
533.80.
21.46.
25.66.
LIVELT:
39.36.
inn te hey
$ 1097.10.
$ 124.20.
13255 Yr.
$8100.
1154289,
27200.
110.
6288 lb.
9504 lb.
504 Ib.
cin hoa eee
163
_
S
Ce eerie
OM AUD
pepe hel ot. ta, ke
Art. 88.
83,8; d.
1750 rm.
3122 bu.
694 54, sq. ft.
390 8% ¢c.
5625 sq. m.
115428° cu. ft.
282169 m
2894 boxes.
$2 T.
Art. 89.
26000 pckg’s.
18944 pt.
5824 pt.
$ 5,366,760.
2500 quires or
60000 sheets.
86400.
443520.
12375 ft.
4840 sq. yd.,
43560 sq. ft.,
6,272,640 sq. in.
. 8,317,760 cu. in.
Art. 90.
$ 38.40.
$ 11424.89,
$6198.43.
$ 5255.57,
$ 604,
$ 51.60.
$15.08 saved
pe buying of
second firm.
. $17280.
$ 19430.65.
. $22076.28.
. 145.8 ft.
13.
14.
. 7 passengers
fewer than
when $110.55
was received
from 33 trips.
$ 0.02.
8335) in. or
6944 ft.
Art. 96.
1. 51,973,650 °
coins.
2. $35,506,987.50
3. $14,989,278.60
4. 2640 ft. ;
o20 ft. ;
660 ft. ;
4620 ft.
§. 4320 sq. rd.;
5200 sq. rd.
6. 23528 cu in.;
315 ft. 2 in.
7%. 253,800.
8. 6696; 58590;
334,800.
9. 3180 lb. iron
10.
11.
for 1 ]b.silver;
716 lb. iron for
1 lb. nickel ;
8 lb. iron for 1
lb. lead.
37 854, cu. ft.
$ 91.25.
30 -
+ bb]. =24¢4 lb.;
$0.35 gain on
1 bbl.
$100 =
£ 205940 ;
10.
—
oO
>
OOO Oe OE Na et
ANSWERS.
$100 =
518,28, fr. ;
420 40, M.
. $0.674.
. $2 13 ;
52 wk. 2d:
. 49 gal.—11319
Cleanse
3 bu.=6451,26
cu. in.
. No profit.
. 332° bu;
$ 19.8331,
100 bbl.
63 bbl.
Art. 101.
24.
420.
ibe
2112 mM,
. 42 periods.
$ 9216.
13m.
$ 270.
. 30 pieces.
. 4.53 8-hour
watches.
Art. 102.
. 388492 deposi-
tors and a re-
mainder of
$ 181.04.
. $1075.57 ;
$ 185.43.
. 888,24, ft.
. $610 loss.
. Sun’s diame-
ter = 1093338
—
i
Soe Rs Oak uene Co. tot
x earth’s di-
ameter.
. 234 min.
. 4164 lb.
. 2,280,960.
. 92,795,826 m.
769.2 -— 1m,
oF
on
no &
oe)
BS
Ne) ~! DS WlR wo
Ss CO Co see
: his
oo
ge
~
—
HA\bo
|co
Rie
00 eo
oly wis +]
so
Oofoo bs]
© el et cs of
° ° ° ;
to
wi?
11.
12.
eee Ol ie, OE
Loe
—
OojUT CYDO BOHN cole *
. ° ° . a
Col Ol *
m .
mS one RR RS PO Oe ae RS eS On
vo °
bon
>
bo}
sepa
1]
°
et |
Ri oO}
o,
2 ah ye
. 288.
. 1020.
. 12000.
Art. 119.
—
COR.
ee ee oe
aS OV
_
_
S00 Ott Oe Se
873 ;
6 ro00:
$235
403 doz ;
27 oy cd.
— et ore it
Hal Ol Co Rl Oco
on °
~I
* ow oo °
»|
|
on
we bd aT bw eH +
08 Ts a J] vie
ne a On Mh ale °
. . on
~I
31
is 455 great-
ie]
than 3’5.
oO
Fr ol
. 123.
-
OMIA TP w
. 84,
SOMATA WD
CHOIR MAP WW
ANSWERS.
Co
ON RR RR
i eis Rl
ni
mor Oo
e °
7 OK}
pits ST et on
co
or .
a
20% %.
. ify TE
173
. 8550:
Art. 124.
the un-
known num-
ber is larger.
er, oo
oe
mi2o.
Art. 129:
$ 1 Ps:
8331 sec.
6560.
. 51063.
$ 71.038.
. $160,781.
, 681% Ib.
. 2814 Ib.
. $173,958.
(10. 38975 m.
11. $2013.
12. 5534,%, m.
13. $2943.
Art. 130.
1. 6 yd. ; $0.513.
2. $2.69.
8. $149.50.
4. 4943 ft.
5. 80,; cd.
6. 102 sec.
7. 45) m.
8. $38.99.
9. $0.09.
10. 1161, bbl.
Art. 133.
1. #43
2200.
3. 56}.
5. 193.
Gre 0.
yiiec e
8. 84.
9. $$.
10. 1222
Art. LSS:
1. $2.914.
9. $2.952.
8. $0.51,%.
2.50.
4. $0.79}3
0.29 Ps.
5. $0.50.
1.83}
6. $933
7% $3.94
8. $45.80
9. $2.85.
10. $71.14
_
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. $3.62}.
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13.
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$ 0.57%.
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gross better.
4
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12. 161 ft.
Art. 143.
. 581 times.
6,47, sq. rd.
OTA N
: EE eoocnS.
$ 0.411 re-
maining.
9. 1075 m.
Art. 144.
Cf le
WIL, RIS OlOe Cojo bo} Col
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ANSWERS.
(9) io: (10) 2433.
(10) 13335: 8. (1) 22 2.
4. (1) 15 (2) 46°;
(2) 1y5 (5) 8037
(3) 134: (4) 7455-
(4) 153: (5) 101;
(5) 34. (6) 112}.
(6) 13% (7) 263.
(7) 135 (8) 843}.
(8) 1 (9) 5523.
(9) 1b ae 4675
(10) $3 . C1) 24255
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(2) 3% (3) 8075
(3) 1?z (4) 8138
(4) iL. (5) 11123.
(5) #6: (6) 121 y5-
(6) }. (7) 317%
(7) 236 (8) 9455.
(8) Ye (9) 6534.
(9) 3. (10) 5055
(10) 322 10. (1) 341,37
6. (1) 225 2) 54TH.
(2) 188 (3) 67744
(3) 152. (4) oe
(Ale. (5) 1048.25
(5) 148. (6) 15853
(6) 15. (7) 25568
(7) te (8) 48635
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(10) 1.9% 11. (1) 172
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(2) 2548 (3) 147%
(3) 1433 (4) 135
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(5) 2511 (6) 114.
(6) 8037. (7) 157.
(7) 14.3%. (8) 10%.
(8) 2275. (9) 954.
(9) 2832 (10) 16,5.
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(2) 4653.
(3) 6208.
(4) 76325.
(5) 8511,
(6) 18734.
(7) 25083.
(8) 46971.
(9) 84245.
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(2) 562.
(3) 641.
(4) 2568.
(5) 4283.
(6) 5452.
(7) 1153.
(8) 578,
(9) 3308.
(10) 258,
24. (1) 42.
(2) 3.
(3) 49.
ANSWERS.
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(2) 5152.
(3) 194.5).
(4) 66122.
(5) 1343,5,.
(6) 198413,
(7) 16215,
(8) 8864.
(9) 6881},
(10) 540,%.
29. (1) 21205}.
(2) 330752.
(3) 429971.
(4) 554628,
(5) 627703.
(6) 980991.
(7) 169,181,%,.
(8) 319,548}.
(9) 566,915%.
(10) 505,896,%,.
80. (1) $1583}.
(2) 7083}.
(3) 106873.
(4) 36992.
(5) 51450.
(6) 58176.
(7) 11880.
(8) 57800.
(9) 352455,
(10) 25800.
LON LEN ON ON FN FS EOS
on oo» Ww &
ee
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(6) 20. (7) 138.
3 1 473
9 856°
(7) 19. (8) 5333 (9) $2633, (10) 136355.
(8) 423. (9) 1333. (10) $3834%. | 45. (1) £25.
(9) 27. (10) 144%. 41, (1) 38. (irlhds
(10) 26,5. 37. (1) 35§. (2) Ys (3) $2.
33. (1) 1}. (2) 5434. (3) 145 (4) 4°58.
(2) 23. (3) 7135 (4) 335 (5) 2t.
(3) 1. (4) 9134. (5) 425 (6) 144s-
(4) 67%. (5) 1047%5. (6) lis CT) 3343.
(5) 43. (6) 16234. (7) 33$- (8) 333o3-
6) 228. (7) 28012, (8) 139. (9) 1424,
1H : i 1433
(7) 144. (8) 52953. (9) 175. (10) 383?
(8) 45. (9) 9405. (10) $63:
(9) 21, (10) 83822. 42. (1) 6193. Art. 145
(10) 14%. 38. (1) 441. (262s li
34. (1) 248. (2) 6431 (3) 61212, 2. $52.30!
(2) 204. (3) 8183 (4) 47%. 3. $391.
(3) 113%, (4) 1134 (5) 6023. 4. $445
(4) 238. (5) 1283, (6) 60235. 5. $0.16498
(5) 46. (6) 1947 (7) 1497. 6. $4491.
(6) 24,9, (7) 33182 (8) 50,4. 7. $493.8.
(7) 1434. (8) 63;%'5 (9) 733. 8. $6.
(8) 351% (9) 112,88. (10) 42, 9. $19.011.
(9) 1528 (10) 100182 43. (1) 1113 10. 3127 m.
(10) 10,%,. 39. (1) $38.5. (2) eS,
Sees, (2) $8234, (8) 188yy. page hae
(2) 27435 (3) $9453 (4) 7%. 1 F
(3) 1335 (4) $84i7 (5) sis 2. 500%; 20%.
(4) 1308 (5) $603. (6) 382. 3. 40%; 250%.
(5) 130% (6) $72.4. (7) 38%6- 4. 662%; 150%.
(6) 2o%%5- (7) $1524. (8) We5- 5. 25%; 400%.
(7) 2335. (8) $803%%5 (9) 33% 6. 50%; 200%.
(8) Lies (9) $9420, (10) 25535 7. 1500% ; 62%,
(9) 1333 (10) $25373. | 44. (1) 38253. 8. 1%; 10000%.
(10) 55/37. 40. (1) $6y%. (2) 6725. 9. 50%; 200%.
36. (1) 27%. (2) $817. (3) 86335. 105 Ose
(2) 1333. (3) $2344 (4) 96544. 100,000 %.
(3) 224%. (4) $8333 (5) 136333:
(4) 653%. (5) $6343 (6) 223335. II.
(5) SPY. (6) B7gy%5- (7) 34430. 1. 400%; 25%.
(6) 4392. (7) $4377. (8) 881493 2. 335%; 300%.
S Comin» PP w
_
SCOMWAH AP wWD
OID oP © WO
. 250%; 40%.
. 120%; 831%,
. 600%, ; 162%.
. 50%; 200%.
. 125%; 80%.
. 60%; 200%.
. 60%; 1662%.
. 426% ; 2331 9,
Il.
. 500%; 20%.
. 61%; 1600.
. 2662% ; 871,
. 60%; 1662 %.
500% ; 20%.
133}% 5 75%.
. 200%; 50%,
- 20%; 500%.
. 100%; 100%.
. 831%; 300%,
Vx
. 2400 % ; 44.4,
. 700%; 142%.
. 1200%; 81%,
. 800%; 331%,
. 500%; 20%,
. 1331%; 75%.
. 300%; 331%,
Lof 1%;
20,000 %,.
. 662%, ; 150%.
10.
210/-FA 6 0
1834 3 048, 0°
ATt. 153:
2 66297.
. 662% Swedish;
31 0
51% other
sources,
. $1000.
i)
, 80% ;
SCOMWWAATH
ANSWERS.
. 8% brass.
6} %.
$ 50.
& or 555%.
2 or 60%.
5h 3
A See Aa
. 333% ; 6% ;
121%; $0.99.
Att. Loi
. TLA. at 10¢a
foot costs
$8963.20 more
than 560 A. at
25¢ a sq. rd.
. 1877 bu; 492¢
rem. ; $99.95.
. $0.095.
. 50h. ; 1680 m.
293.
$ 15.20.
179! cans.
$ 11662.
Peete rem.
. $2227.98.
Art. 159.
. $9364.88.
23
. 73:
378
. 10255 full
steps.
1440
tions.
5
36°
revolu-
3
+ zo M.
iwi L DY al,
Tigaleiis
Mar. 14;
May 1.
13
i a or)
oe be
-
oO
Pee
oe ah
SCOMMHTRPWHNH
CO 095 52 Oe CRE Soar
$ 15.52.
$ 2001.81).
3512 Ib.
263 Ib. ; 7999,
$ 243.52,
. $1621.92.
. $698.76.
. $212.68.
$ 15.44.
. $426.65.
ASC eLOL:
$ 3.983.
$2.85.
$ 13.
$870.50.
$ 1.574.
. $216.
9073 sq. yd;
4 squares ;
36 squares ;
32670 squares.
19.
20.
_
Se oe ee
)
1j Tay ie
89} in. ; 742 in.
12 in.
Art. 162.
$ 58.80.
. 8800 boxes.
17s. ; 204d.
/ ATAU ET Ga!
nile is 73.3 ft.
lessthan 1}! xa
common mile.
“OO: VOs 520 sable
5 2
ee 2 ge eas
189022 Ib.
2
ve
. $2,819.
Art. 163.
. $105.
1
+ bu.
. 59 posts.
44d.
Mary, 3;
Sarah, 4;
both in 17h.
. John, 4; his
brother, +;
both in 22d,
. $8000.
. 80 jars ; 18750
eggs,
4
Ie Aro:
10.
20 bu.
ATLA LOo:
. $135.
. Clifford,
kee
60?
Clifford and
10
_
oO
OW WWD. HM
Sos OP ig 3) Cag in ten og Oo
rol Raila sent tee kL Pil oe
Leonard,
Leonard,
Leonard
.
)
40?
can
dh,
do the whole
in 40 min.
lt sis 10/54
Thursday p.m.
in New York.
. 0.04 Friday
A.M. in Holy-
head.
$ 8400.
. 36 meters.
$ yd.
$ 11.25.
Art. 166.
16 tables.
$ 195.12.
39 bu.
$ 48.
. $5964.
Art. 172.
100
or 4910 ft.
1 meter.
Art. 177.
He ele olen | |
ale ae § al of
. . . e .
en
by
. The father.
. 23d.
yA DOUG as I;
ANSWERS.
DoH onion
; A
ly Hie BIH
i
Poe
BI) RL, BL, BL, ole BL BY
al wt al” a" = of af
Be CR SAT al a
_j-
ts
iH ot 4]
wi~r °
2
i)
i> a
jon = bo
S|
co
w
J
or
Cal iss:
PArtsee so:
, 0.588}.
. 0.8381.
. 0.5624.
. 0.2662.
. 0.4284,
. 0.5558.
. 0.2381.
0ST TR
. 0.4662.
. 1.0623,
. 0.8333.
. 0.9166.
. 0.5555.
. 0.5454.
. 0.1838.
. 0.4666.
. 0.0016.
25.
26.
27.
oR oO OD HAIHM TARP
et BE IS peal N NDE |
0.5888.
0.2307.
0.9280.
Art. 181.
. 206.568.
. 335.722.
21.8725.
. 22,32589.
700.108.
. 680.40235.
234.696.
25.1429,
. 222.585.
123.27114.
ATC toe:
. 226.472.
. 396.2494,
152.807.
3.193.
23.5237.
Art. 184.
1.74.
1.5056.
31.13.
8.774.
3.749,
0.3976.
L767;
1.5306.
27.926.
. 5.87.
. 5.893.
. 2.5416.
. 23.8352,
eeO-00 Ls
Art. 185.
182.62.
. 2.886.
=
f=)
—_
i=)
CA and pid Bode Os ORAS oe ots, an
ENE Pd Mie
BORER ON CoC bn
1.775.
0.162,
0.0162.
7958.2,
0.235. -
6.0.
1.62.
. 5.562.
Art. 186.
2.975.
Aiea 7c
99.75.
0.501.
ree0es
Art. 190.
. 79.88904.
. $8.064.
6.963744.
4.23,
0.46875.
0.675.
46.656.
$ 480.
51.20.
A Ag Sate
Art. 191.
$13.49.
$ 16.92.
$ 1050.
1372.5 T.
$ 8.55.
520.2.
750 Ib.
. 0.5625.
$ 80.
. $6000; $120.
_
Oo
td ir haha gel pg Aine At ee
iP go gO
T0
mn Lt.
Art. 193.
3600.
. 0.289.
78.4.
0.39.
70.5.
113.5.
$ 8.9125.
. 640.0287. ©
. 55.1286,
. 8560.
Art. 196.
14s.
| bole
°
16
00°
is 0.04d
more than
£0.004,
50 %.
2 doz.
. 48 sheets.
5%.
. 61%,
. 15%,
1.12.
Art. 199.
. $89.25.
. $89.25.
10400,
(st ward ;
12800,
2d ward ;
8400,
3d ward ;
8400,
4th ward.
. 9159.2 ft.
en ee ee
Seomont our wonr oO
so
eee
$05 O05 Es Ge ON ee 002 Sos
ANSWERS.
, 258 balls ;
1082 oz.
175.
0.049.
0.025.
. 0.33338.
fs LS tet
0.183; 4%.
Art. 200.
. 2.005.
0.0831.
0.081.
0.10.
0.68.
0.075.
0.0125 ; 4.
0.125 ; 4.
0.0025 ;
a
400°
. 0.2124.
Z4.0.0t
. 0.0625.
. 0.015625.
. 4g).
Pieced
. 2.25 ft.
, 18it.
. 2 sq. it.
a1 Sisq. it.
‘ATE 20.
. 208,876.8.
. $0.75.
. 5500 Ib.
4750 lb.
10334 Ib.
15040 Ib.
35624 lb.
. $7.20.
, 21%,
of the
men receive
10.
© @ =
0.15 more than
$1aday; the
rest 0.388 more.
. 0.081; 0.125;
0.002723.
. 0.638341 yr.
. Sept. 29.
$ 1.
. 13} loaves; 3.
Art. 202.
eee Lie
128’ 128’
52 |
128
. 0.8089,
. 0.5218 ; 0.071;
0.052,
. 0.3771,
. 0.0075 ;
0.0028 ; 0.9990.
0.183.
yO!
Art. 203.
1
B°
Oe ie
. 8453.775 ft., or
3756,225 ft.
4. 1m.
5. $180.60.
6.
7
8
9
124 sheep.
. 90 sq. in.
- yr.
. 25m.
64 ed.
oo 0
11
Art. 204.
. $1840.
yt ee Wt WF
78.0195.
on
>
CO BR aH Dip Oo}
Om of Hl
mM
OQ +
=
—
Or Hy
=~]
(520.977.
Lh P
Otek hoe
ee
1. 252 d.
2. Lata; 607%.
4.
OHrHys
. $38.88.
1 $20-bill ;
1 $10-bill ;
1 $5-bill ;
1 $2-bill;
1 $1-bill;
1 half-dollar ;
1 quarter-dollar;
1 dime ;
3 cents. _
11 pieces.
£15;
795 er.
7 oz. 4 dr.
. $0.212.
$412.
4 OZ.
15 gr.
1 pwt.
Art. 206.
. 3216} m.
463°.
. 691m.; 144m,;
1014 ft.
10.
17,5 m.
720 ft.; 690 m.
37 double
eagles.
20 ecu. ils
bi CU sin, 5
113} gal.
537.604 cu in.;
41.85 + bags.
48 pt.; 80 gi;
2 pk. 4 qt.
20; 17 cwt. 50
IG, eae pk:
SSG peel Dt
4 gi,
Art. 208.
147.
87.
64,
1000.
67.67.
Art. 209.
$4.
1.88.
LEYS
Oe hoe
9.33.
ooreG.
45.
11.68.
Art. 210.
$ 26.55.
134.81.
127.80.
12.51.
90.71.
81.58.
© ©
10.
oI wo
Qa Reve
ANSWERS.
«epi OGs
195.70.
21.08.
8.09.
188.85.
50.25.
16.67.
31.06.
51.79.
526.46.
Att; 214;
1
z m.
62 it,
. 2455 m,
Pat Oat b
6215227 m.
Ist rider, 2 m.
in vi mins
“ 15
2d rider, $3m.
We Rnbah Aa
. 890154 ft.
. Qrd. 918 ft.
. 6145 m
10.
37632 revolu-
tions.
Art. 217.
. 140 sq. rd.
. 640 A.; 64.A.;
560 A.
36 sq. m. ;
24 m.
Art. 218.
1588 sq. ft.
45 479, sq. rd.
781854 sq. ft.
864000 A.
. 43560 sq. ft.
PLUS SSC
Oana P w Ww
. $3411.50.
3821) A,
. 135935 sq. rd.
. 176 sq. ft. 35
eqyins
$ 5250.
Art. 223.
AV PANE (e fae VER
. 324 sq. in.
288 sq. ft.
260 sq. ft. ;
272} sq. ft. ;
504 sq. yd.
fee A.
if
. 403 sq. yd.
. 84 sq. yd.
10.
1020 sq. in.
2916 squares.
Art. 224.
$17.50.
389 qt.
4 rd.
50¢.
. 576 tiles,
. $12.50.
. 3840 pieces.
Art 2225,
. Carpets 1 yd.
wide could be
used on floors
L2Sth el ete
27 it...and: 18
ft. wide; car-
pets $yd. erie
on floors 224
ft. and 154 ft.
wide.
ee a
On Powe
Neither width
could be used
on floors 20
ft. wide.
be strips; oo
yd.; 374 yd.
7 strips; 49
vd.; $61.25.
$ 25.
$ 70.
. $26.67.
$ 28.35.
. $258.90.
. $24.83;
- $90.92.
. 21 tiles:
. 20924 tiles.
3245 sq. ft.
. 1048 tiles.
21,7; sq. ft.
770 tiles.
-- 14-rolls:
' Art. 226.
. 2850 slates.
. 261386 blocks.
6212 plates.
13 at ft. more.
$ 147.
: 48° sq. Lhe
200 sq. ft. ;
64 sq. ft. ;
LO SO sat tas
140 sq. ft.
712 sq. yd.
Art. 228.
. $4098.60.
300.
937.50.
4. $1350.
§. 2559.38. !
S © 1728.
%. 2245.32.
8. 1975.59.
9. 2602.33.
10. 1447.88.
ii; 937.20.
12. 1929.60.
13. 272.68.
14. 604.08.
15. 407.15.
16: - 94.08.
Ling ool
18. 67.50.
19. ~ 124,
20. 320.
Art. 229.
7. 93% sq. ft.
8. 10560 sq. ft.
9. 11} sq. it.
. 3& sq. it.
Art. 231.
- 260 sq. ft. ;
110} sq. in.
. 1614 sq. ft.
10. 10/5 sq. ti,
Art. 232.
5. 180133 sq. ft.
6. $25.
Art. 233.
10. 360.
11. 750.
12. 11 sq. ft.
18. 231 sq. ft.
. 73} sq. ft.
. 69753 sq. ft.
ANSWERS.
Art. 234.
. 896 Sq. in.
. 7500 sq. ft.
Art. 235.
. 62.832 ft.
. 66.25365 ft.
. 665.488 ft.
. 1.485442 ft.
. 61.051 in.
. 1.195664 ft.
Ia, Peel oe
. 60.
113.0976 sq. ft.
. 78.54 sq. ft.
. 795.775 sq. ft.
198.94575 sq.
ft.
Art. 237.
19.635 sq. ft.
1
4°
113.0976 sq. in.
$ 8.48.
. 51416 sq. ft.
Art. 238.
. 1256.64,
509.296.
1809.5616.
. 1017.8784.
ts) Rote
. 1963.5.
Art. 241.
. 16+ in.
RABE
. 4.00009 + rd.
4.
oO
14.
. Curved
OHARA P ww
. 600 cu. ft.
10.
Diameter of |
hhd. = 3.978
+ ft. ;
Diameter of
door = 3.833
+ ft.
1.8169 m.
edge
eon AC)
Straight edge
= §7.2958°.
. 18.8496 in.
. 496.24 ft.
. 9685.84 sq. ft.
Art. 243.
216 cu. in.
J eoL2: Gus 1D;
ip Om it
Sly 2S cu.ft,
1000 cu. yd.
8000 cu. in.
125; 348;
12 © 12S
216 ; 1000;
133 ; 8000.
Aas PADS S
TG oo.
Art. 244.
. 180 cu. in.
400 cu. in.
74 cu. ft.
1280 cu. ft.
4 cu. ft.
4608 cu. ft.
115831 cu. ft.
1772 cu. ft.
10.
Re
12.
13.
14.
15.
C2) Sie Bee
10.
i
13
Art. 245.
$ 18.
100.
112.50.
19.69.
29.30.
5.50.
Art. 246:
10 sq. ft.
120 sq. ft.
12 boards.
2Osbdy dies
15 bd. ft.;
123 bd. ft.;
20° bd. Tt.:
a >< 1LOsbas Te
ove bd. ft.
256 bd. ft.
131 bd. ft. ;
162 bd. ft. ;
20 bd. ft: ;
21% bd. tts
231 bd. ft. ;
15 bd. ft.
110 bd. ft.
537% bd. ft.
216 bd. ft.
Art. 247.
. 486 cu. in.
zo GOOLEn. in,
. 1536 cu. in.
8 in.
10 ft.
eer hall:
Art. 248.
114 sq. ft.
. 88 sq. ft.
14
OOIP A
10.
© © FP wo WO
. 0.2146
. 248 sq. in.
. 468 sq. ft.
1240 sq. ft.
. 8485 82.
. 552; 973.
. 576; 224.
Art. 249.
shav-
ing;
0.7854
der.
cylin-
oh G elie ins
7854 . 7854.
10000 %9 10000°
31.416 cu. ft.
399.2928 cu.in.
. 602:656 cuit,
7 O28.562-ClLeLG.
ian?
Art. 250.
12.5664 in,
. 2.54648 in.
25.1828 sq. in.
. 314.16 sq. in.
. 353.43 sq. in.
_ 224.3456sq.in.
Att ecoe:
Rab aii eC se alas
perimeter 30
ee
S) oli oo dll yes
perimeter 26
ing
porte. Pe tehiee
perimeter 24
in.
. 27.225 ft.
ANSWERS.
3. 12 in. average | 8. 30 in.
width ; 9: 11-ft.
200 running | 10. 2.8 + ft.
feet. 11. 4 ft.
4. 13} ft. 12. 660 ft.
§. 1.2784.
7. 3 in, Art. 255.
8. 128 oranges. 1. 3 ft.
10. 800 A. 2. 1840781 gal.
3. $48595 gain,
Art. 253. | 4. $ 133650.
ey ies
1. 24x6=144;] 6 op
Me Sera Wt Re 7. $2.50,
ae suit 8. 960; 6.
ae 9. $135.41.
6.- 78 sq. ft 10. $9.
pate 11. $2242.50.
8. 120 ft
9. 9 ft. Art. 257.
10. 5443 ft. 1. $50.63.
eA, 2. 109394 Ib.
12. 18 ft. 3. $26.98.
et ee 4. $1505.44.
1420 rd. 5. 37.6992 bbl.
ee INE 6. 603.1872sq.in.
16. 62500 sq. ft. 7. $276.67.
17. 31.416 ft. 8. $5760.
18. 7854 sq. ft. 9. $476.80.
oe ne ft. 10. Perimeter of
20. 6 in. 6-inch square
Art. 254. Ofc oneeleree
1. $10. tangle =26 in.;
2. 4d. of 3 x 12 rec-
3. 6 in. tangle = 30in.
4, 12 ft.
5. 12 in. Art. 258.
6. 4 it. 1. 0.2146.
7. 1089 ft. ; 2. 909.
5°, tt, 3. $6106.88.
Oop
170, sq. fi,
1,5 % loss.
$0.13.
. Once in 1000
times.
. $17.82.
. 630.
. 83% min.
Art. 262.
. $5376.
. 4801 bu. corn;
3542 bu. oats ;
4807 bu.wheat.
. 9354 T,
. 68.8 m.
. 9552 A.
. $2.44,
. $209.25.
. $61.19.
. $400.
. $56000.
. $560.
. $434.
20. 46946.
Art. 265.
ool Au
. 6300 volumes
in library ;
1764 works of
fiction.
(Aes
96%.
30000 T.
. $17634.37, or
453%.
824),
. $106.25.
444 9/,
ee ee
a Pp we
ee
See Pee Oak Ne We COLD” bt
Art. 266.
Aliare equally
profitable.
. 2ofsomething
is 100 x 2% of
it.
25 %.
. 581%,
. 45%,
95 %,
162 %,
. 4; 800; 2%
7 3
. 6; 1600; 1%.
. 112;
83} i
163 yee
13; 140; 3
3
2910). 21.
: 331%; 31;
25 Y,
- 625; 2000;
1000.
. $1241.67.
. $ 12000.
. 864%.
Art. 267.
12,
960.
16%.
40 A.
92%,
$ 1728,
564%.
225 cd.
7800 T.
4%,
. 18%.
. 138%,
- $9600.
5
re
10.
SIAR ewe
ean eee ots an DO ars
oP oO De
ANSWERS.
Art. 268.
. $6500.
$ 24640.
- $60; $72.
331 94,
162 %,
25 9,
33! 9,
LO ]
. 31% more is
gained by
buying for $4
and selling for
$ 4.80.
. $9.
. 81%,
Art. 269.
$ 4.80.
$72.
$ 6000.
(pk
500 A.
$ 50000.
. $2000.
60.
35%,
Art. 273.
80 %,.
$ 60000.
. $50000.
$7.81.
- $5000.
1B ie
co
oO ROOST CAC, teas ro Oates
Art. 274.
$ 96.
$ 48.
$ 32.
$ 30.
$ 30.
$ 35.
25 Yf,.
. $20 or 25%.
. 20%, import-
er’s gain ;
162%, retail-
er’s gain.
142%
$870
Art. 275.
. 183% gain on
meat; 142%
gain on pota-
toes.
. $10.
a 3
. 1g saved by
buying to-day.
. $48000;
$ 50400.
. $140.
. 15E 9%,
. bY,
east he
. 60%,
AS TE0
147% %
. 84,
. $5200.
Art. 277.
. $44.80.
137.60.
28.05.
15
6. $5.88.
if 298.54.
8. 3785.
9. 104.99.
10. 125.16.
Art. 278.
(oe Re Ree
8. 9.92.
9. 1.56.
10S is
lives tea:
192 seis
135 720/80;
14570211653
15. 38.65.
16. 0.05.
Art. 279.
9. $4.49.
Die 10.672
12. 15.53.
133 4-13.27.
14. 14.21.
15. 6.438.
16. 1.90.
17. = 4.09.
18. 9.06.
190 ec:
20. 8.68.
21. 196.67.
pV Wed laste
9300 16716:
Art. 281.
3. $546.86.
4. 7920.11.
5. 1909.96.
6. 95.06.
16
7. $1135.24.
8. 314.28.
Ors .02.02;
10,4 -11S8tt26;
11. 4838.69.
Art. 282.
(a prye'4 ies
8. 2 yr.2 mo. 6 d.
9. 2yr.10mo.13d.
Art. 283.
1. $14.49.
2g. fe ee:
feet ee sO!
4. 7.99.
i a af PavAs
6. 937.50.
Nee calouw es
8. 461.
9. 21.85.
10. 166.45.
1. 1.89.
127 6 1
13. 49.80.
LSPS thee
Art. 285.
1. $26.79.
2. 759.54.
8. /640.11.
ANP51506;
5. 900.
6. 540.02.
Te 9234,
8. 60.67.
OF 202205
10. 3.15.
phar ve
PR By eye
ANSWERS.
18. $147.52.
14. 4.41.
15. 19.44.
1Gseee 0:
17. 58.90.
18. 227.50.
192% 746.67;
20. 105.22.
21. 385.33.
O22. 20:
23, 4.25:
24. 48.95.
Obs viisl2.50:
Art. 286.
4. $144.
6. 4.93.
71o0..0,00.
Soe ies G:
9. 11.05.
1016.7 72
11. 7:29.59:
12. 160.98.
1353 coc t.Gia
Art. 289.
5. $1900.
6. 10%,
7%. $500.
8. 4%.
9 $1.50.
10. $500.
Art. 290.
12s los
Bre OU Uae
3. $0.40.
4. $0.593.
5. 50% of $14 ;
20 9, of 50 ¢+ ;
1% of 40¢.
—
SHMARBAP wD
11:
$ 11.70.
. $24.26.
$520.10.
$ 5.24.
. $240.88.
. $2390.08.
Art. 291.
$ 6500.
19,
40.
. Cash discount
is 3%.
$ 1266.67.
$ 2375.
- $1400; $1425.
20 %,
. $ 10472.
. $8920.
73,
. 15%,
. 20%.
. $313.63,
. $10423.88,
. $598.50.
Art. 293.
. $10000.
$ 10000,
$ 2000.
£%:
119,
$ 4000.
$ 62.50.
144.
$31.60.
$ 5000.
30/7. 17
: o%5 40°
$ 5625.
$ 375.
Do P oo
© oc =-t
12.
Ree Ly
2d risk is 1%
cheaper.
. $230.09 ;
9 8 a % .
. $2040.82.
. $7000.
. $120; $3388},
Art. 294.
80 ¥,
. $110; 4%,
$ 8.10.
. $6.91.
. Of gross re-
ceipts.
Art. 296.
. $120.
- 1023%; $20;
$2000.
ee yee
. $1600; $1620.
400
£07°
Art. 297.
103 ; $27.
. $1200 amt. in-
vested ; $1242
remittance.
. $34.95.
. $2.50; 15%.
. $293.06.
$ 8.74 ;
$ 428.26.
. $2750; $2675.
. $25;
. $1000; $20.
10.
1}:
10%,
14%; $1082.
$ 1552.80 ;
$ 67.20.
$ 450.70 ;
3%.
pl
oO
ae
Fete ee oe
| aioe
y ©
Art. 298.
$421.05.
526.67.
200.00
9,65
4.57
$ 214.00
20000.
589.20 ;
112% +.
3.52 ;
2.37% +.
777.65 ; 3%.
207.90 ;
198.
1250 ;
45.75.
787.49 ;
15.39.
Art. 299.
$314.95.
12000.
4123.38.
7600 ; $400.
16.87}.
9250000.
4006.
30000.
6207.63.
58.87.
Art. 301.
$ 261.52.
987.60.
76.58.
1356.55.
1847.71.
10.
1b
10.
je
Se tie as Se
ooh
ANSWERS.
Art. 302.
$ 355.80.
600.60.
370.
Art. 304.
$780.
3180.
576.31.
Art. 305.
$ 156.10.
865.49.
2280.01.
750.97.
197.74.
806.99.
Art. 306.
$45.
$45.
. $128.65
Specific duty
$40 more.
. $60.75 ; 34%,
, 1221%; $1650.
$ 132.60.
20 %,
15 %.
. $0.32 gain.
Art. 307.
$ 300,000.
$50,000 ;
10,000.
$150n $1000; |
14 ¢ on $1; |
$ 8000.
13%; $0.012; |
14%.
112; $1.48.
iO
0407 SRE Py
. July 7.
10’ >?
$ 120.88.
Art. 308.
. $78.81.
206.46.
449.95.
1030.30.
38.14.
Art. 309.
$366.94 +.
424.48.
270.61.
812.06.
428.85.
Art. 310.
. Mar. 3.
Sept. 8.
May 31.
Mar. 3.
. Nov. 80.
Feb.-22:
Feb. 28.
Apr. 18.
Nov.
Mar. 3.
Apr. 29.
Art. (SOLE:
$ 590.
1187.60.
7 per SLOVO; |
10.
un.
“12.
13.
14.
17
Art, 612,
$ 790.70
(grace) ;
$791.
$7135.70
(grace) ;
$ 714.
. $8.97 (grace);
$8.75.
. $594.75
(grace) ;
$ 595.
$ 523.56
grace) ;
$ 523.69.
$ 906.54
(grace) ;
$ 906.68.
$ 310.82
(grace) ;
$311.04.
$ 784.50
(grace) ;
$ 785.
$ 953.28
(grace) ;
$ 953.60.
$ 715.58
(grace) ;
$715.74.
$ 836.25
(grace) ;
$ 836.59.
$ 870.90
(grace) ;
$ 871.05.
$ 91.32
(grace) ;
$91.54.
$ 5907.75
(grace) ;
$5910.
18
15.
16.
ma IE Pag ine! A 8 ag ae
$ 4241.50
(grace) ;
$ 4243.29.
$ 814.93
(grace) ;
$ 815.43.
Art. 313.
. July 7/10;
27 d. or 80 d.
. Feb. 15/18;
63 d. or 66 d.
= Octibs3-
37 d. or 40 d.
. June 17/20;
47 d. or 50 d.
2 Aug.4 77.
57 d. or 60 d.
. Feb. 24/27 ;
26 d. or 29 d.
. Jan. 19/22;
70 d. or 73 d.;
2 mo. 9 d. or
Foe Le eee
. May 11/14;
66 d. or 69 d.;
2 mo. 5 d. or
2 mo. 8 d.
Art. 314,
. 63 d. or 66 d.
24 d.
$ 438.
$ 994.86.
$ 795.33.
. $2980.50.
$ 447.65
(grace) ;
$ 447.80.
$ 705.06.
10.
14.
15.
16.
it:
ANSWERS.
. $794.22
(grace) ;
$ 794.55.
$ 1195.50
(grace) ;
$ 1196.45.
$1195.97
(grace) ;
$ 1196.69.
. $247.60
(grace) ;
$ 247.69.
$ 1716.07.
. $188.74
grace) ;
- $188.80.
$ 273.24
(grace) ;
$ 275.48.
$ 275.18
(grace) ;
$ 273.87.
$ 345.44
(grace) ;
$ 343.70.
$ 752.65
grace) ;
$ 752.85.
$ 752.78
erace) ;
$ 752.97.
$ 4988.89
(grace) ;
$ 4989.93.
Art. 315.
. $1223.64.
. $502.24
(grace) ;
$ 502.15.
| 9.
dined rant iD AAR et La
. $44,166,666.67 |
p25 |
$ 726.92
(grace) ;
$726.81 (time
inexact days).
Art. 316.
1
. $60; $8.
Art. 317.
. $5; 5%,
85.
$ 1125.
$ 20.
60 shares.
$ 101.50 ;
$ 1.50.
$ 370.
$ 100.
Go,
6%.
. $3450; $138;
4%.
. $5; $200.
Art. 318.
$ 500.
$700.
5,
$ 4000.
150.
1102,
- $10100.
10.
11;
$ 4906.25.
$15 nearly on
a 100-dollar
bond.
Art. 319. |
90 9,
130 %.
. $2838.38.
. $9712.50.
. $4015.
. $4280.
. $3940.
. 8.84%,
. 549%.
Art. 320.
. $476.19.
694.44.
1045.48.
697.12.
83.50.
753.33.
Art. 321.
2 yr.
Bday
. $200.
. 4yr. 6 mo.
. $840.
. Be,
. $960.
Rare A'§ veces tele)
. 43%.
. $179.28.
Art. 322.
63 ¢.
. 80%.
. $20.76; 30%.
Art. 325.
oy AA AS
. $6.25;
$ 1494.25.
. $1485.50.
OO:
after
date.
11.
12.
——s
Pa Sag eed ee a Ok os eee Berar
_
12.
13.
14.
15.
16.
17.
18.
. $3920.
. M. 3640.
. $387.60.
Oop o wo
. $73.46.
. $1266.25.
- $87.70;
$ 493.50. 19.
. $490; $489.75 | 20
grace).
$3958 (grace);
$3940.
$ 10.25.
£ 2010.
£ 860 10s.
Art. 327.
. 2872.50.
$ 21.56.
217.01}.
2085.71.
$ 6185.
$ 12.09.
$5.19.
$ 40.41.
_
oO
$ 89.24
(grace).
$ 49.61 ;
$51.14
(grace).
—
wo
$ 195.95 13.
14.
(grace) ;
$ 194.74.
$ 289.59.
$ 58.80.
$ 256.10.
$ 845.56.
$ 941.11. 15.
OOH oP
OHMIRH HP ww
ANSWERS.
$ 1886.70.
$ 1213.21,
Art. 328.
$ 250.
. $4761.90.
Art. 326. 3.
Eig AB aT)
200 /o OF
0.605 %.
$ 156.25.
. $33.39 loss.
. 6535.95— bu.
. $4189.47.
. $1590.
. Gain $ 14.25.
10.
$ 1960.78.
Art. 329.
. $402.38.
- $751.82,
$ 9.93.
$ 563.20.
$3.35,
$ 820.
24 yr.
8%.
. $905.66.
- $690.08.
. $497.78;
$ 497.57
grace).
$ 212.24.
$48; $47.36;
$ 48.72 ;
$ 48.00
(grace);
$ 48.30 ;
$44.44.
$ 20.
5.
. 59
re)
, ae
ob et Ph
Art. 330.
ok es
. 300.
Se HE
pee
1
#10 Vd.
. $20.
75.
240.
1536,
65.
43.5.
Art. 331.
. $15.
. $378.
. $126.
. O¢h.
. 223d,
Beebe lt,
. $133.35.
ee TAS
. 62 02.
teat UeoU,
Art. 332.
ye |
$ 1200 ;
B. $300.
. S. $560;
B. $140.
$2000 ;
$3000 ;
$ 4000.
$300 ; $450.
. $10,000.
| 10.
rhe
12.
13.
wwnwnnnnndw wnnrerere
VKH SCHoHKBNTATHEONMOORA
il
or ©
13.
14.
15.
Se eS | Sli Ot. Ns OB. BO be
$ 0.844, ;
$ 507.70.
$7411.76.
$ 6000 :
$ 4000 ;
$ 3000.
$ 20,000.
Art. 335.
EE
Fee:
, sok
. 42,
. 46.
. 54.
68:
. 68.
75.
. 84,
= U4;
. 73.
4 On.
» 92:
. 58,
. 99.
Art. 336.
532.
547.
636.
746.
869.
1462.
0.75.
0.96.
6.5,
. 0.88484.
. 0.9485+.
. 4.41214.
28.7210+4+.
4.12734.
0.8.
19
20
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
-
rOoOMBADTP WD
Fo oe
SODARAH WD
12.
13.
14.
0.2529-+.
43.9590-+4.
27.9991+-.
15.03.
1430.
8.74.
0.89444.
0.559+-.
23.
2.380+.
9.099-+.
(),5422+.
Lo o2oor
1.4790-+.
Art. 337.
307.
0.85384.
3.7249+.
1004.
3136.
7921.
45.09.
. 0,.9682+.
11.
. 999.
Os.
64.
4a
i 44 jee.
, 0.193642:
11.2745+4.
. 28.0178+.
. 1856.
530.25244+.
. 0.860815+4.
Art. 338.
44,
21.9094.
29,9334.
ANSWERS.
15. 61.8464. Art. 342.
i sels bene a 1. 36 sq. ft.
Ti 180.00. 2. 2937 cu. ft., or
a 29.629+ cu. ft.
19. 186.051+. 4. 471.4 cu. in.
cata SUL EEE 5. 500 cu. in.
917 47745
Art. 343.
Art. 339. 3. 384 sq. in.
1. 2 ft. 4. 412.300 cu. in.
2. 40rd.; 160rd.| 5. 4 sq. ft.
3. 24.1664 ft. 10. 105 sq. in.
4. 21.9814 ft. lis 375 sq-sin:
5. 18,32 esq: tt. 12. 255.62 sq. in.
6. 4ft.2.9+4 in. | 18. 301.593+ sq.
4. 20.784. ft. in.
8. $141.42: 14. 463.624 sq.in.
9. 15.8664 ft. 15. Contents.
10. 108.1662+ sq. | 16. 0.2146.
ye Art. 344.
Aaa: %. 452.3904.
8. 12,566,400.
1 G69282 24-39. 9. “Bd Gee
‘ : Qs Jit.
Ce 10. 200,000,000.
3..5:4.
4. 21.089 rd. Art. 345.
5. 208.71+ ft. 7. 113 cusin:
6. 97.616+ ft. 10. 4.1888 cu. ft.
7. 63.639 ft. 11. 4,188,800,000.
8. 176.568+ rd. | 49 0.4764:
OF 1427 arden 0.5236.
235.5+ ft.
10. 101.98-+ ft. A THEaaG:
9. $2700.
Art. 341. 10. 4=9:
by, 1178-10; TVs 1:50.
6. 670.208. 12. 194.
4. 942.48. 13: oh
8. 11.3184 ; 14:03 7876:
189.554. 15. $21.09.
OHARA
Dorp owe
Art. 347.
64; 8.
20 lb:
1171.874 bu.
96 h.
Art. 348.
$ 9.25,
. 681.7875+4 sq.
it:
167d.
. 423 sq. ft.
so LOSdr ite
$ 32.26.
124.686.
875.
. 1.273824 ft.
AST 8
768 sq. ft.
. 4.1888 cu, Tt.
. 14.499 sq. ft.
Art. 349.
1080.
. §.2832 cu. in.
12.65+ in.
530.145 sq. in.
. 24902.18142+.
180 rd. ;
169.68+ rd. :
150.40— rd.
. 94.814.
. 26.5294.
a 4:37.
1607.8125 Ib.
14.9354+ ft.
. 28.2744.
18,000 Ib.
14.
15.
. 2108.
Ses
=
ae ee ee)
706.86 sq. ft.
70.71+ ft.
Art. S65,
11352,
$ 63.28 ;
$ 2763.88.
$ 59.42.
1349 cu. in.
. 274 yd.
$2.24.
. 499842,
24
605°
Art. 366.
$ 1504.29.
$ 2745.76.
Neither.
6750.
$ 320.
16.96 % ;
$16.11.
Loss $13,351.
Cash.
113.
38377.
Art. 367.
120° E. of
starting point.
913
b5}3.
88.
821
1372
&
ANSWERS.
5. 2 min. 3932) 4. 20°.
sec. 5. $527.15 gain. |
G.lnteal 6. $3.24. |
7. 10722 + q. 209.
8. 300 Ib. 8. $12000.
9. 16. 9. $6.75.
10. 8.24. 10. The latter by
Art. 368. li. 10,
1. $552.49, 12. $1045.45+.
2. $2687.44, 13. 18.8%.
3. $20.
4. $27.61, Art. 371.
5. $4378.79. gS a ae
6. 1413.72 sq. ft. | 92 Int. for 4} yr.
7. 460.195 + lb. to:
8. 1583.34 +. $F L380 2m.
9. $92.36. 4. 389.70+' sq. ft.
10. $337.27. 5. $100.
6. 2.652.
oe SALE 7. 6.6338-+ Ib.
1. $4372.60. 8. $450.
2. 173%. 9. $ 107.623.
3. 10.63-+ in. 10. $789.78.
4. 35%.
5. $141.20. Art. 372.
6. $421.85. 1. 92¢.
Ue he hs Q. 2714,
8. $108,800,000. | 3 & 1980.
9. 63.8 : 36.2. 4. 4129,
3 /0
10. $0.01375+. 5. $2924.10,
11. 752.52— ft.; | § 9 409,
twice as much 7. $ 145.90 +.
appa 4’ 8. 17.72 + ft.
12. fae eae | 9. 1232 cu. in.
tsar OAS: 10. A $ 10800;
| B $7200. |
Art. 370. 11. $851.27 + or |
1. $217.50. | $851.59 +.
2. $11.55+. 112. $83}.
3. $5.91. 13. 9.36 + %.
14.
15.
25 %.
Lan:
Art. 373.
$ 26.25.
» $256000.
36 shares ;
$50 rem,
20
62 %.
$ 3668.75.
5 0
5%
$ 178.76.
$ 20937.50.
Increase $ 15.
. The former by
5
$ 10250.
. $6300.
75° East,
10.89 + in.
. 93.55 — sq. in.
84.82 + sq. in.
Art. 375.
1. $9.5.
2. 874%.
3. $2400.
4. 160 bbl.
§. $ 58500.
6. $1980.43.
%. A$12;B$?o:
C $24.
8. 623 A.
9. 444 ft.
10. $9.21.
Lt. 31-66 36:
12. $1440.46.
Art. 376.
1. 50%.
SB. .12.i6.
99 ANSWERS.
3. 614. 9. 40¢. 5. $538.77 ;
4. 7360 bu. 10. $105.80. $ 538.45
5. $160; $208. | 11. 2332.64. (grace).
6. 1, 12. $371.20. 6.
Tan0e Vy, 13. 222%. 7 STD:
8. 81 yd. 14. 32638 m.- 8. 140.064 sq.in.
9, OA. 9. Nothing.
10. Bonds .5, %. Art. 379. 10. $285.
1205-7. ; 11. Rich $800;
12. 162 sq. ft. cues Foster $1200.
13. 116 ft. 8} in. - 8 10.50. 12. 336.13+.
14. 413013: T. 4. $8663. 13. 622, sq. ft.
Art. 377. ee tbe Art. 382.
6. 331%.
oa es 7. $48 gain. 1. 5h.
2. UTE Ye 8. $ 9774.72 ; 2. $ 18,000.
3. 22%. $ 64162. 3. $625.
4. $13.82. 9. $29.75. 4. $115.63.
5. 173 yd. 10. $212.61. 5. $3010.
6. 98.81— rd. 11. $6.25. 6. 3125.
Toate ¥, $817.16.
8. 19683 sq. in. 8. 42.1+ yd.
9. 9in. x 9in. Art. 380. 9. $12.10.
10. $ 140.623. the 164 ite 10. e176.
11. $ 1871.09. 2. $6150. 11. $26,000.
12. 162%. 3. $8240.
18. A $3240; 4. $3761.25. Art. 383.
B $ 1260; 5. 616.11 + sq. re
Soh x, 7 ee $38;
14. 399.999975. 6. 35.99— gal. C $ 26. ;
15. $48. 752564 bd. ft A aeane
en Gion é ate
1. 125%. 10. -75 DO} 6. $152.
SOt meq tte lt ames 7. $72.
Sere 12. 8. 1765.17 gal.
4. $11,880. Art. 381. 9. 9929.4.
5. $20,000. 1. 13d. more. 10. 12.44 ft.
G12 4. 2. 64 ft. 1131-710:
7. $3696. 3. $103.25. 12. $16.80.
8. 500 men. 4. 53h. 118. 564 sq. ft.
aa
= ©
—
© eo
10.
rat
. 44.74 — 9,
CO) 00) Sh OF. OE eee
So Soa
Art. 384.
. $33,3331.
74%.
$ 266.48.
50 m.
12 d.
$115.20.
12 rd.
2 ft. 4 in.
. 32.725 Ib.
. $3.3894+.
. The former by
$70.30.
. 64 men.
<9.0z:
Art. 385.
14 bu. 2
4 qt.
$ 37.70.
$ 620.123.
$ 4504.50.
pk.
. $249.64;
$ 249.52
(grace).
2.6— cu. ft.
as
$ 55.
. A $91.30;
B $114.13;
C $ 144.57.
$ 55,000.
$ 1030.64.
Art. 386.
. $72,000.
. $76.
5
C #53
. $577.40.
Ls
SSS So
$386 ; 600 fr.
$ 82,187.50.
116 ft.
$ 1059.87 + ;
$ 1059.67
(grace).
$ 2125.33!.
Of B; $1.39.
159.155 cu. ft.
Art. 19.
8472009 ;
1867.75+ Ib.
26.4347.
16.7284 ™,
1312.359 yd.
. 395.38 sq. rd.
1000,
. $10.63 gain.
96049,
. 88.9056,
. 2,845184400,
. $3.3108.
. 5GHa. 138,384,
264.17 gal.
800.
. $3.21.
196.87549,
. 7500",
13.9286947,
oP @ to
O DAD MTP ww
ANSWERS.
Art. 387. 7. 20.014 m.
$1.54. 8. 21,3.
$7.20. 9. $16.80.
$ 5700. 10. $80,960.
6. 11. $10,552.
504 bu. ; 12. $147.
$ 290.64.
. 0.02078125. Art. 388.
1. 38.
APPENDIX I.
Art. 21.
$ 356.40.
567.
554.40.
198.39.
774.90.
126.30.
86.87.
454.78.
Art. 23.
2. $605.02.
iv)
OW ADP w
. $1175.77.
Art. 25.
3 mo. 10d.
1 mo. 24 d.
2) ADT. 20,
July 26.
June 11.
Dec. 21.
10.
11;
12.
CS race
OID Tw ow
$ 843.36 ;
Ta do
Aug. 20.
Jan. 4.
Aug. 23.
Art. 26.
May 31.
Oct. 24.
May 25.
Nov. 13.
Jan. 5.
Art. 28.
23.
32.
36.
42,
47,
aha
G0 Se St
23
$ 2007.50.
$ 2.912 ;
$ 912.912.
$ 9956.86—.
$ 1337.
Makes $6.50.
$ 3.54,
61%
i AD ing
a Lb. 1-4 in;
156.
Ceo bole colar wip *
. . ° . (nd
—
on
te
.
STO lt HI
. 662 sq. ft.
24 ft:
—
°
(or
CO Oe Can le
Mf
apt er
25.
50.
74 bbl.
$100.
8.
$ 1000.
20.
2 —— 0.
6.
Li Ep ee ka
9 x 10,
Art. 9.
a — (30 > 3d).
_ 20 x8
2
150
~~ 100 = 25:
_ _ 160
Ui ——=
3
r= i — 2.
a= x t of 189 +2,
e200 ae
400”
or 7 = 400 2:
==) DLS SA
or eee
144
Art. 10.
m(a +b).
ANSWERS.
APPENDIX IL
mt
ie}
SY &
~
~
caotanPr ww
[on
wee
ry
oO
—
=
Art. 16.
12.
12:
24,
$ 300.
. $56.
36m.
120.
$ 31.8852.
$ 84.
OOD AMP ww
_
od
Axt. “17.
SHOHMPABDARwD
se
Loe
wo
(<2)
144.
12:
4,
Art. 18.
$210; $70.
20 qt. ; 25 qt.
9714
48; 39.
$0.40; $080.
A 2180; B 800;
C 320.
168. 9. 150.
60. 10a
Art. 19.
$ 10,000 ; $5000.
12.
3. F $94; G $122;
Oe
H $56.
200 gal. each.
132
8.
Lis entlo:
84; 91; 98.
60 ; 20.
$ 871.
Art. 20.
0.
10 x7.
7(b+ 6).
— 12%.
Arteais
SET ViDy
1m.
24 rece’d.
Lb.
. @bex?.
be Gert 1) + 2.
: a+(b-—c)-e.
x—(y—z2)+a.-
Art. 25.
10 abe.
18 abe.
36 axy.
48 bey.
228 abxy.
63 bedz.
2(abcx)?.
4 atc?mat.
8 ab.
6 abcrny.
a.
(m + n)?.
6(a — Db).
6(a — b)?.
(m+ n)?.
12 (“+ y)?.
343+ 3ay? + 3 az.
. yz 4+ ay?z + yz.
. Gam.
10 axty? + Ls ar ey? + |
20 ax? y?z. )
14a+21b+7e. |
(Combine _ before
multiplying. )
10.
1
2.
3.
aYZ + Ak2Z + ALY
ANSWERS.
Art. 29,
Art. 30.
lbay, be.
10 a2’ 10 ab
3%, 2am.
6a2y’ 6 ay
a Dae ae
48 ab2m2’
3 b?m?z
48 ab?m?
Samy
48 ab?m? ’
Art. 31.
4a+2b?
xy
or
LYZ
aye + £2 + LY) |
LYZ
22m — 3am
nin
|
25
2 wyz? +12 22+ 16 oy?
8 ey?Z2
4¢c— 2ad*
cdy
3y(a—b)+3a(a+d)
rY
adn + ben — bdm.
bdn
a(emy + dn) |
a(b? + x)—y,
ad
ite
4 ay
2@mnxy ,
3 azz
" 2bced
(a+ b)\@+Y)-
Art. 33.
bo
a
xn &
bo
5 m?
<~ ar |
2
—
arb
oO Xe
2(2a + b*)
K(2a + b)ay?
1¢ca
" Dbz
26
> OO 3
10.
3.ab
aly + da)
3 cd3y
Ded
ab?
Art. 34.
Beit:
S53.
15; 4.
$75; $100. | 18.
—_ :
FOO OA ar & Ww
ANSWERS.
bs enee
2
. 30; 24,° 3
18; 12. 4.
= Alea es 5
LS:
27; 74.
Art. 36.
A $50; 10.
B $40.
. $550; $300.
. 64; 36.
40; 24,
. Tea ib¢:
Coffee 40 ¢.
70; 30.
63 ; 27.
20; 12.
A $270;
B $252.
4
9°
at
me
&
oh
aye
ik ;
ee
bag
Ai de eat hi
fh iy :
aoe
7
=
Tele
a
eh.
er a
~ a er
eet te — — “
<3
eacstat Seater} 3;
Serbeteittstmes
errs etree
Sette
Sorse-sesesses
Setpetriieciese
ors
tissseseciss
So BS ye p+
wey eeegtse pasesaee
af
stectess
ia
Baas
Esser
UNIVERSITY OF ILLINOIS-URBANA
513S08E C001 V002
THE ESSENTIALS OF ARITHMETIC ORAL AND WR
TN
3.0112 0171
streSete Syed e7
SOI