9FAL16~LY~<
ECLECTIC EDUCATIONAL SERIES.
R A Y' s
A RlITHMETICAL KEY:
CONTAINING
SOLUTIONS TO THE QUESTIONS
AND TO SOME OF THE MOST DIFFICULT QUESTIONS IN
THE SECOND BOOK:
ALSO
AN A P P E N DIX
OF TEST EXAMPLES FOR THE SLATE OR BLACKBOARI).
BY JOSEPH!AY, MI. D.,
PROFESSOR OF M3ATHEMATICS IN WVOODWARD COLLEGM.
NEW AND IMrPROVED EDITION.
PUBLISHERS:
CLARK, AUSTIN & SMITH, NEW YORK;
WVINTHROP B. SM3IT-I & CO., CINCINNATI.
M JATHEI IATICAL WORKS.
By PROFESSOR JOSEPH RAY.
TYPE ENLARGCED-N>EW AND IMPROVED EDITIONS.
EachL Boost of Ray's Aritfhmetical Course, also of the Algebraic,
is a covmplete work ins itself, and is sold separately.
FIRST BOOK,
PRJIfARY LESSONS AND TABLES, simple and progressive,
for little learlners.
SECOlID BOOK.
INTELLECTUAL ARITHIMETIP'IC, by Induction and Analysis;
a thorough course of meontal exercises.
T I D BOOK, i
PRACTICAL ARITHMETI[C by Induction and Analysis; a simple and thoroulgh work for schools antd privazte students.
KEY TO PibY'S A1RITHMIETIC, THIRD BOOx.
IIGHER ARITHMBETIC. Principles of Arithmetic analyzed and
practically applied. For advanced students.
KEY TO RAY'S HIGHER ARITHM-IIETIC.
ELEIENT ARY ALGEBRA.
RAY'S ALGEBRA, FIRST Boor, for Common Schools and Academies; a simple, progressive, and thorough treatise.
H-IGt E1 ALGEBRA
i AY'. S ALGEBRA, SECOND BOOIc, br advanced students in
a academies, ancl for colleges; a, lucid and comprehensive work.
KEY TO RAY' S ALGEB3RA, FQ.IRST AND SECOND BOOKS; complete. ji ione v:1liasm, l2meo.
Entered accordinog to. Act of Congress, in the year Eighteen Iliundred and
FortS-firve, 1byev Ws5erifo, B. SrlITr,: in the Clerk's Office of the I)istrict Court
of the ULnitedtrates, for the District of Ohio.
C. F. O'DI9ISCOLL & CO., STErEOTYPERS, CINCINNATI, O.
PREFACE.
The leading objects in the preparation of the following work,
have been:Fiirst, to aid the teacher, who may not at all times have leisure,
in the midst of his labors in the school-room, to furnish the pupil at
once with the aid which he requires. For this purpose, the work
contains full solutions to all the questions in Third Book, as far as
Compound Numbers; after that, the method of working each example is indicated by arithmetical signs, so that any one acquainted
with the elementary rules, can readily perform the required operations.
Second, to assist young teachers. For this purpose, besides
what has been already named, the work contains solutions to all
the difficult questions in the author's Mental Arithmetic, (Second
Book); remarks designed to illustrate more fully the nature and
principles of several parts of arithmetic; with suggestions on the
best methods of instruction.
The examples for exercise at the blackboard have been inserted
at the request of several of the best teachers, who have found it
both interesting and:useful to the learner, to propose questions
not found in the text-book.
It has been a complaint with some respectable teachers, that
answers are given to the examples in the Third Book. When the
pupil is required to explain, at the blackboard, the method of
working out an example, it is desirable that he should rely on the
correctness of his own reasoning, and, if he understands the method of
obtaining the true result, it is of little moment whether he has
seen the answer or not.
The experience of many eminent teachers is, that the use of
a book without answers has an immoral effect on the pupils.@
A superior pupil will obtain the answers correctly, while other
members of the class will procure them privately, either from
him or others, who have previously worked them.
It is the author's firm conviction, founded on an experience of
many years, that the only proper test of a learner's knowledge
is, to place him at the blackboard, to solve and explain the questions; and, with this test, it appears to him that the objection to
furnishing answers to the questions in the book is removed.
A celebrated Boston teacher once said, that an Arithmetic without answers,
tended to make the pupils both rogues and liars.
o~~~~P _____ __ I -- -C-r. -s r -
N OW PUBLIStHED.
IAY'S THIGH ER ARITTHMETIC, embracing the
principles of the science, analyzed and applied;
designed for advanced students and business men.
An admirable and highly comimended treatise, peculiar for its
freshness, philosophical accuracy and adaptation to the purposes
of instruction.
Its rapid introduction into schools of the highest order, and
the continued preference given to it, over other admired and
popular Arithmetics, may be regarded as evidence of decidedly
superior excellence.
OBSERVE.-That it is especially adapted to those desiring
enlarged intelligence with reference to Mercantile, Mechanical
and Agricultural pursuits.
Just Published.-A KEY to Ray's Higher Arithmetic,
embracing full and lucid solutions to the examples
contained in that work.
(4)
KEY
TO
RAY'S ARITHMETIC,
THIRD BOOK.
SUGGESTIONS ON TEACHING WRITTEN ARITHMETIC.
IN teaching written, or, as it is frequently termed, practical
arithmetic, the first step is to make the learner well acquainted
with numeration and notation. After this, pupils should be frequently exercised in reading and writing numbers.
Similar remarks apply to the numeration and notation of decimals; when these have been properly studied, the subject of decimal fractions presents few difficulties to the student.
It is scarcely necessary to mention that every pupil should be
able to perform Addition, Subtraction, Multiplication, and Division, with ease and correctnzess, before proceeding further. In order
to do this, it is frequently necessary to review these subjects more
than once, and to require the pupil to solve examples additional
to those contained in the text-book. The instructor and pupil
should both feel that no time-is lost in dwelling upon and reviewing the elementary rules, until the pupil is thoroughly master of
them; it is the most certain means of insuring his future rapid
and successful progress.
In explaining the nature and operation of many of the rules,
the instructor will find it of great advantage to illustrate the subject by simple mental exercises. Every competent teacher can
generally frame such as are suitable, and present them orally to
the class. In this way, they mnake a more decided impression on
the pupils, than when presented in the text-book.
Mental exercises, suitable for the illustration of those subjects,
when they can be lised with advantage, will be found in a
succeeding part of thlis volume, under their' appropriate heads,
and precedinig the examriples for exercise at the black-board.
6 IKEY TO RAY'S THIRD BOOK.
tWhen practicable, the pupils should be arranged in classes, clue
regard being had to their ages, acquirements, etc. After this, the
proceeding in the best schools, is somewhat as follows:
A certain number of examples is arrangefl for a lesson; it will,
also, frequently be necessary that a. part, or even the whole, of the
lesson shall consist of the illustration of principles, or the memorl.
izing of definitions or rules. When the class meets for recitation,
each pupil passes his slate into the hands of the pupil next above
him, except the pupil at the head, who passes his to the foot scholar.
The teacher then reads the answer to the first question, while each
pupil examines the slate lie holds, to see if the answer is correct
and properly obtained.
In addition to reading the answer, the teacher, in many cases,
such, for example, as proportion, should state the general method
of working the question. The pupils mark the answers that are
wrong, or obtained improperly. In the same manner, each question is examined and marked. Instead of the teacher reading the
ansswers, the pupils in succession may read them.
When there is a blackl-board (and there should be at least one
in every school-room, 4 or 5 feet wide, and as long as the room
will permit), each pupil should be required to work out one or more
of the examples, and give the reasons for th[e method of performing the operation. The time required to examine the questions is
generally short, while the lhabit of closely scrutinizing each other's
work, improves the perceptive faculties of the pupils.
The instructor will find the suggestion contained in the Note,
Article 138, with reference to the examples having a star, -, orthy of particular attention.
in: order to teac]h either intellectual or practical aritlhzetic well, the
instructor nmust be Jfaniliar with the subject, so as to be able to explain,
pr'omptly and clearly, the solution of every question, as well as the priniciples on uwhich each rule is based. IIe should have, also, a strongy desire
to render his pupils fanciliar with the subject: for, it zmay be stated as
an accknowledlecl truth in education, that, a teacher who is indifferent to
the cadvancemenz t of his pupils, will renzder them indifferent likewise;
(land that, on the coati rn'yar, onTe that is deeply interested in the student's
I. p'c/e.dsu, wil n c/el/! fareli, to e/nkiznd7le the salice spirit.
OF SYMBOLS. 7
O9f SYMBOLS.
1. The sign- is called the sign of equaZity; it denotes that
quantities between which it is. placed are equal. Thus, $1 —
100 cents; read, one dollar equals one hundred cents.
2. The sign + is called pluts; it denotes that numbers
between which it is placed are to be added together. Thus,
4+3 —7; read, four pzus three equals seven.
3. The sign - is called nminus; it denotes that the number
placed at the right of it is to be taken from the number on the
left. Thus, 5 —3=2; read, five 7minus1t three equals two.
4. The sign X is called the sign of mzutii2lication; it denotes that the two quantities between which it is placed are
to be multiplied together. Thus, 4X3=12; read, four times
three equals twelve,
5. The sign --- is called the sign of division; it denotes
that the number on the left of it is to be divided by that on
the right. Thus, 12-_ 3-4; read, twelve divided by three is
equal to four.
Division is also denoted thus, Gl, or 3)12; either of which
indicates that twelve is to be divided by three.
6, The parenthesis, ( ), when it incloses several numbers,
indicates that these are to be regarded as one single numler.
Thus, (5 +3)X4-32; read, five plus three multiplied by four
is equal to thirty-two,
If the parenthesis were wanting, the expression would stand
thus, 5 +3X4-17; this indicates that five is to be added to
the product of three multiplied by four, and that the sumn is
seventeen.
7. A number placed over another number, a little to tile
right, is called an expoznent. Thus, 2 and 3 are called the exponents, in the expressions, 492, 5', etc., 43, 53, etc.
The exponent of a quantity denotes that the quantity is to
be used as a factor as many times as there are units in the
exponent. Thus, 42 —=4X4, 2-=2X2X2, etc.
The exponent 2 depotes the seconId )pow-i, or squafre.of the
number over7 whtihel it is placed. The exponent 2 denotes the
1 8 KEY TO RAY'S THIRD BOOK.
thlird power, or cube. Thus, 3'= 3 X 3 -9-the square of
three; and 33=2X2X2=8 —the cube of two.
8. The sign -/ denotes that the square or second root of
the number before which it is placed, is to be extracted.
Thus, 1/16=4; read, the square root of sixteen is equal to
four.
In a similar i-manner the sign g/ denotes that the cube or
third root of the quantity before which it is placed is to be
extracted. Thus, g/27-3; this is read, the cube root of
twenty-seven is equal to three.
1KU I DRATION.
SUGGESTIONS TO TEACIES. —Be careful to impress the mind
of the pupil with a clear idea of a unit. Show him that numbers
are formed by the successive addition of units. Explain the distinction between units of different orders; this may be done conveniently by a reference to the coins of the United States, stiff
pieces of paper being used to represent units of the first order;
those of the second order will be represented by cents; those of
the third, by dimes; and those of the fourth, by dollars. MIake
the pupils well acquainted -with readbing numbers under one thousand before proceeding higher. Until the pupil is familiar with
the subject, require him to point the numbers into periods of
three places each, before attempting to read them. Let the pupil
be made well acquainted with the method of writing numbers
under 1000; then, by the use of points, he may soon be taught
to write any number with ease and accuracy.
TO TEACHERS.-Where the word "ARTICLE" occurs in this
volume, it has reference to the corresponding "A' T." in "Ray's
Third Book.'"
The examples in the Key are numbered, and refer to the corresponding number of example, under the same artiole, in the
Arithmetic.
ARTICLE 13.
1, Five., Sixty-three. I 6, One hundred anId forty3, Ninety. 4, One hundred,. seven.
5, One hundred and f;ur. 7, Two hundred and eight.
&, _e —------------ ~ —-- -----
NUMERATION. 9
8, Two hunldred and eighty. 37, Seven hundred and four
9, Four hundred and three. thousand two hundred and
10, Seven hundred andtwen- eight.
ty-nine. 38, Eight hundred thousand
11, Seven hundred and ten. one hundred and forty-one.
12, Nine hundred and one. 39, Nine hundred thousand
13, One thousand. and sixteen.
14, One thousand and five. 40, Six hundred and one
15, One thousand and fifty. thousand and twenty.
16, One thousand and eigh- 41, Seven hundred thousand
ty-five. four hundred.
17, One thousand one hun- 42, Eight hundred thousand
dred. and two.
18, One thousand one hun- 43, Nine hundred and ten
dred and eight. thousand one hundred and
19, Three thousand and three. three.
20, Four thousand and fifty. 44, Seven hundred thousand
21, Three thousand and for- one hundred.
ty-five. 45, Twenty thousand and
22, Nine thousand six hun- twenty.
dred and ninety-nine. 46, One hundred and six
23, Ten thousand. thousand three hundred and
24, Twelve thousand. seven.
25, Thirteen thousand two 47, Four hundred thousand
hundred. and one.
26, Fifty thousand and four. 48, Three hundred and two
27, Sixty-two thousand and thousand four hundred and
one. four.
28, Seventy thousand four 49, Eight hundred thousand
hundred. and ten.
29, Eighty thousand and 50, Seven hundred thousand
ninety. and ten.
30, Ninety-seven thousand 51, One million, thirty thouand ten. sand seven hundred and twen31, Forty thousand three ty-five.
hundred and five. 52, Four millions, fifty thou32, Seventy-six thousand sand six hundred and seven.
and fifty-two. 53, Six millions, six hun33, Eighty-three thousand dred and one thousand.
nine hundred and ninety-one. 54, Seven millions, four hun34, Two hundred thousand. dred and six thousand and
35, Four hundred and fifty- thirty-five.
six thousand. 55, Nine millions, seven hun36, Six hundred and eighty- dred and twenty-five thousanld
two thousand three hundred. and fourteen.
10 KEY TO RAY'S THIRD BOOK.
56, Three hundred thousand hundred and eighty-nine thouthree hundred. sand six hundred and two.
57, Nine hundred and nine 66, Ninety-nine millions,
thousand and ninety. ninety-nine thousand and nine58, Eight millions, six hun- ty-nine.
dred thousand and fifty. 67, One hundred and thirty
59, Two millions, one hun- millions, six hundred and sevdred and two thousand one enty thousand nine hundred
hundred and two. and twenty-one.
60, Four millions, eighty 68, Six billions, nine hunthousand four hundred. dred millions, seven hundred
61, One million and eleven. and two thousand and nine.
62, Three millions, thirty- 69, Twenty-three billions,
six thousand. four millions, ninety thousand
63, Nine millions, one thou- nine hundred and one.
sand and three.' 70, Nine hunldred and forty64, Thirty-three millions. two billions, twenty-nine mnilseven thousand eight hundred lions, three hundred and seven
and twenty. thousand and twenty-nine.
65, Sixty-one millions, one I
ARTI CLE 14.
1. 100. One hundred. 9. 5 in the first order is five,
2. 203. Two hundred and or five units. 4 in the secthree. ond order is forty. 3 in the
3. 3020. Three thousand and sixth order is three hundred thousand. 8 in the
4. 50204. Fifty thousadl two third order is eight hun-.hun50red Fiand ftousra tv cdred. 7 in the fourth orhunre a o. cler is seven thousandcl. 9
5. 9000500. Nine million, five in the seventh order is nin
hun7dred. millions. 4 in the fifth or6. 70050. Seventy thousand tder is forty thousand, 2 in
and fifty. the eighth order is twenty
7. 600004000. Six hundred millions. 8 in the 6th order
million, and four thousand. is eight hundred thousand.
S. 20050200. Twenty mil- 4 in the 9thi order is f),u
lions. fifty thousand two b hundre'd millio,,s. 5 in the
| undtr~u:i'edn., l 10e11h,'(ler1 i: five billion,.l
NOTATION-ADDITION. 11
NOT AT ION.
A RTICLE 15.
1. 4 13. 65 25. 222 37. 503 49. 45083026
2. 16 14. 97 26. 999 38. 827 50. 870743579
3. 118 15. 87 27. 807 39. 1000 51. 2804
4. 20 16. 104 28. 870 40. 1020 52. 4029
5. 24 17. 120 29. 901 41. 25006 53. 6006
6. 28 18. 130 30. 606 42. 345 54. 22765
7. 30 19. 175 31. 309 43. 760 55. 80201
8. 32 20. 203 32. 190 44. 3406 56. 90001
9. 37 21. 340 33. 202 45. 42030 57. 30030
10. 41 22. 435 34. 101 46. 30000 58. 41.0205
11. 46 23. 502 35. 21 47. 163000 59. 800669
12. 59 24. 625 36. 300 48. 341563 60. 900001
61. 500050 64. 100010001 67. 700010001
62. 100010 65. 91007060 68. 1001000040
63. 909090000 66. 70000004 69. 40000200005
70. 726050001243.
ADDITIONI).
SUGGESTIONS TO TEAcnERS. —Simple Addition is the method of
adding numbers of the same name or kind into one sum. We
can not add numbers of diffierent kinds together. If a reason is
asked for this, the reply is, that it is self-evident; thus, 2 apples
and 3 apples added, make five apples; but, we can not add 2
apples and 3 pears together; the sum would be neither 5 apples
nor 5 pears. If it is asked why we write unitst under units,
tens under tens, etc., the answer is, since we can only add numbers of the same name together, we write those of the same denomination under each other, merely for the sake of convenience
in performing the work. We carry one for every ten, because
ten units in any place make a unit of the next higher place or
order. If the base of our scale of Notation had been twelve,
instead of ten, we should have carried one for every twelve.
After the pupil has become acquainted with Compound Addition,
he will better untderstand the reason for carrying in Simple Addition. Celerks, wh-o add long columns of figures with great
rapid(ity, never call the names of the figures in adding; thev
"Yv Ad) Ace fto
12 KEY TO RAY'S THIRD BOOK.
look7 at each figure as they proceed, and either mentally or orally
name the sum which it makes in connection with those that precede it. In this manner the addition is performed in about half
the time that it could have been, if each figure had been named
al.ou(d.
ARTICLE 20.
(Es. 3) sheep. (Ex. 4) dollars. (Ex. 5) 4321 (Ex. 6) 50230
43 210 1254 3105
21 142 3120 423
14 35
- Ans. 8695 Ans. 53758
Ans. 78 Ans. 387
ARTICLE. 22.
(Ex. 11) Ans. 21023. (Ex. 12) Ans. 27910.
(Ex. 13) Ans. 89569. (Es. 14) Ans. 2499593.
(Ex. 15) Ans. 24194086.
(Ex. 1.6) 23 (Ex. 17) 45 (Ex. 18) 51 (Ex. 19) 263
41 19 48 104
74 32 76 321
83 74 85 155
16 55 4
— A2S. 843
Ans. 237 Ans. 225 Ans. 264
(Ex. 20) 94753 (Ex. 21) Jan. 31 (Ex. 22) June 30
2847 Feb. 28 July 31
93688 TMarch 31 Aug. 31
9386 April 30 Sept. 30
258 May 31 Oct. 31
3456
Ans. 151 Ans. 153
Ans. 204388
(23) (24) yds. (25) dollars.
First five, 1,51 50 245
Next five, 153 65 325
November, 30 42 187
December, 31 89 96
Anis. 365 Ans. 246 Ans. 853
ew —-------— L~m-P —
SUBTRACTION OF SIMPLE NUMBERS. 13
(Ex. 26) 1732 (27) 1656
67 11.64
180
Ans. 1799 1004
Ans. 4004
(28) (29)
sheep. lambs. barrels. dollars.
65 43 30 285
187 105 18 144
370 243 23 235
416 95 34 408
Ans. 1038 486 Ans. 105 1072
(30) 287 (31) 337
596 49
841 431
607
fourth, 1724 146
Ans. 3448 Ans. 1570
(32) 3005 (33) 275432 (34) 880000889
42627 402030 2002002
105 300005 77436000
307004 872026 206005207
800791 4002347 49003
320600 990019919
Ans. 5851840
Ans. 1474132 Ans. 2155513020
SUBTRACTION.
SUGGESTIONS TO TEACRERS. —For the same reasons as those already given with reference to simple addition, we can only find
the difference between numbers of the same kind. The order of
writing units under units, is adopted for convenience. It brings
the individual figures, between which the subtractions are to take
place, in the most convenient position with reference to each
other. It is a matter of indifference which of the numbers is
placed uppermost. The method of placing the number to be subtracted under that from which it is to be taken, is generally
supposed to be the more convenient.
| 14 IKEY TO RAY'S THIRD BOOK.
ARTICLE 25,
(2) sheep. (3) dolls. (4) 734
245 875 531
123 751
-Ans. 203
Ans. 122 Ans. 124
(5) 8752 (6) 529 (7) 79484
3421 8 25163
Ans. 5331 Ans. 521 Ans. 54321
APRTICTLE 27.
(12) 4444444 (13) 91516171 (14) 63046571
1234567 15161718 34992884
Anls. 3209877 Ans. 76354453 Ans. 28053687
(15) 153425178 (16) 100000000
53845248 10001001
Ans. 99579930 Ans. 89998999
(17) cents. (18) dollars. (19) feet. (20) years.
63 137 75 1837
17 65 37 1492
Ans. 46 Ans. 72 Ans. 38 Ans. 345
(21) dollars. (22) dollars.
1840 10104
475 7426
Ans. 1365 Ans. 2678
(23) dollars. (24) 20010
100000 19019
11
Ans. 991
Ans. 99989
(25) 91201.0 (26) 4000000 (27) 2020930
50082 4004 1009006
Ans. 861928 Ans. 3995996 Ans. 1011924
g M —- M-nmr -m-un ~-Aaar~l-p —-----
MULTIPLICATION OF SIMPLE NUMBERS. 15
ii ULTIPLICATION.
SUGGESTIONS TO TEACnaiRs.-It is important that the pupil
should understand the nature of multiplication, namely, that it
is a short method of performing addition, when the numbers to
be added together are all equal to each other.
The multiplier shows howv many times the multiplicand is to be
repeated, hence it is always an abstract number. The multiplicand, however, may be either an abstract or a concrete number;
thus, the abstract number 5, repeated three times, makes 15;
and the concrete number, 5 apples, repeated three times, makes
15 apples. It is obvious from this, that the product is always
of the same name as the multiplicand.
Some, who have only examined the subject superficially, suppose that two concrete numbers may be multiplied together, as
in the following question: What will 3 yards of cloth cost, at 5
dollars a yard? The answer, it is obvious, is 15 dollars. Now,
it is asserted, that in this case we multiply 5 dollars by 8 yards,
and that the product is 15 dollars. This view arises from an
ignorance of analysis and the elementary principles of numbers.
The correct solution is as follows: The price of 1 yard is 5 dollars, acnd 3 yard.s being three timzes as much as 1 yard, the price qwill
be three times Cs nmuch, that is, 15 dollars. (See Ray's Arithmetic,
Second Book, page 37.)
ARTICLE 31.
(12) 49 (13) 57 (14) 128 (15) 367
3 4 5 6
Ans. 147 Ans. 228 Ains. 640 Anzs. 2202
(1s) 1427 (17) 19645 (18) 44386
7 8 9
AlAs. 9989 Ans. 157160 Ans. 399474
(19) 708324 (20) 964578 (21) 96432
7 9 10
Ans. 4958268 Ans. 8681202.Ans. 964320
o —. - -- --
16 KEY TO RAY'S THIRD BOOK.
(22) 46782 (23) 86458 (24) 680323
11 12 11
Ans. 514602 Ans. 1037496 Ans. 7483553
ARTICLE 32.
(25) 1236839 (29) 235 (30) 346
12 13 19!
Anes. 14842068 705 3114
235 346
Ans. 3055 Ans. 6574
(31) 425 (32) 518 (33) 279
29 34 37
3825 2072 1953
850 1554 837
Ans. 12325 Ans. 17612 Ans. 10323
(34) 869 (35) 294 (36) 429
49 57 62
7821 2058 858
3476 1470 2574
Ans. 42581 Ans. 16758 Ans. 26598
(37) 485 (38) 624 (39) 976
76 85 97
2910 3120 6832
3395 4992 8784
Ans. 36860 Ans. 53040 Anls. 94672
(40) 342 (41) 376 (42) 476
364 526 536
1363 2256 2856
2052 752 1428
1026 1880 2380
Ans. 124488 Ans. 197776 Ans. 255136
g~~~ mcRE~~~~ i~~ - -------- --— ~ —---------------------- ~~~~~~~~~~~~~~~~~~~~~~~I-I
MULTIPLICATION OF SIMPLE NUMBERS. 17
(43) 2187 (44) 3489 (45) 1646
215 276 3'65
10935 20934 8230
2187 24423 9876
4374 6978 4938
Ans. 470205 Ans. 962964 Ans. 600790
(46) 8432 (47) 6874 (48) 2873
635 829 1823
42160 61866 8619
25296 13748 5746
50592 54992 22984
2873
Ans. 5354320 Ans. 5698546
Ans. 5237479
(49) 4786 (50) 87603 (51) 83457
3497 9865 6835
33502 438015 417285
43074 525618 250371
19144 700824 667656
14358 788427 500742
Anls. 16736642 Ans. 864203595 Ans. 570428595
(52) 31624 (53) 126 (54) 823
7138 6 12
252992 Ans. 756 dollars. Ans. 9876 dollars.
94872
31624
221368
Ans. 225732112
(55) 675 (56) 496 (57) 152
13 24 28
2025 1984 1216
675 992 304
Ans. 877a cents. Ans. 11904 cents. Ans. 4256 miles.
Key.-2
18 S:KEY TO RAY'S THIRD BOOK.
(58) 1760 (59) 365 (60) 1130
209 24 109
15840 1460 10170
3520 730 1130
Ans. 367840 yards. 8760 hours. Ans. 123170 feet.
8
-Ans. 70080 miles.
(61) 2029 (62) 80401 (63) 101032
1007 60007 20001
14203 562807 101032
2029 482406 202064
Ans. 2043203 Ans. 4824622807 Ans. 2020741032
ARTICLZE 33.
(24 () 124 (3) 2395 (4) 2873
124 124 8
6 8
---- 19160 25857
744 or 992 7 6
4 3
- -- Ans. 134120 Ans. 155142
Ans. 2976 2976
(5) 1673 (6) 2874 (7) 8074
9 9 12
15057 25866 96888
7 8 9
Ans. 105399 Ans. 206928 Ans. 871992
ARTICLE, 34.
(l) 245 (2) 138 (3) 428 (4) 872
100 1000 10000 100000
Ans. 24500 Ans. 138000 Ans. 4280000 Ans. 87200000
DIVISION OF SIMPLE NUMBERS. 19
ARTICLE 35.
(1) 2350 (2) 80300 (3) 10240 (4) 9600
60 450 3200 2400
Ans. 141000 4015 2048 384
3212 3072 192
Ans. 36135000 Ans. 32768000 Ans. 23040000
(5) 18001 (6) 8602 (7) 3007
26000 1030 9100
108006 25806 3007
36002 8602 27063
Ans. 468026000 Ans. 8860060 Ans. 27363700
(8) 80600 (9) 70302 (10) 904000
7002 80300 10200
1612 210906 1808
5642 562416 904
Ans. 564361200 Ans. 5645250600 Ans. 9220800000
DIVISIOiN.
SUGGESTIONS TO TEACHERs. —Since division is only a short
method of performing many subtractions, when the numbers to
be subtracted are all equal to each other; and since we calln only
find the difference between numbers of the same kind, it is obvious that the divisor and dividend are always of the same name.
Thus, if the dividend is dollars, the divisor must be dollars; if
the dividend is an abstract number, the divisor must be the same.
Since the quotient shows how many times the same number has
been subtracted, or, which is the same thing, how many times the
divisor is contained in the dividend, it is always an abstract
number. In the question, I-ow many yards of ribbon at 2 cents
a yard, can be bought for 6 cents? the answer is evidently 3
yards. It is not, however, to be inferred from this, that the quotient of 6 cents, divided by 2 cents, is 3 yards. The analysis
of the question is, Since 2 cents buys 1 yard, and 6 cents is 3
i~ --- _.. —-.. _.......... __ -.. w
20 KEY TO RAY'S THIRD BOOK.
times 2 cents, therefore 6 cents will buy three times as many
yards as 2 cents, that is, 3 yards. (See Division, in Ray's Arithmetic, Second Book, page 43.)
Since we commence on the right in Addition, Subtraction, and
Multiplication, it may be asked, why do we commence on the left
in Division? The reason is, that remainders are often left in the
higher orders, which must necessarily be carried to the lower
orders.
ARTICLE 42.
(4) 3)60 (5) 2)400 (6) 3)6000 (7) 4)80000
Ans. 20 Ans. 200 Ans. 2000 Ans. 20000
ARTICLE 43.
(2) 3)693 (3) 4)848 (4) 2)4682
Ans. 231 Ans. 212 Ans. 2341
(5) 4)8408 (6) 3)36936 (7) 2)88468
Ans. 2102 Anis. 12312 Ans. 44234
ARTICLE 44.
(4) 3)462 (5) 5)1170 (6) 4)948
Ans. 154 Ains. 234 Ans. 237
ARTICLE 45.
(12) 2)8652 (13) 3)406235 (14) 4)675043
Ans. 4326 Ans. 1354112 Ans. 1687603
(15) 5)984275 (16) 6)258703 (17) 6)523408
Ans. 196855 Anis. 43117- Anis. 872344
(18) 7)8643275 (19) 8)6032520 (20) 9)9032706
Ans. 12347534 Ans. 754065 Ans. 1003634
(21) 10)1830024 (22) 11)603251 (23) 12)41674008
Ans. 183002-A4 Ans. 54841 Ans. 3472834
and~~~~ --- ---- -M ——`.~~ 4 —s
DIVISION OF SIMPLE NUMBERS. 21
(24) 3)894 (25) 4)140 (26) 4)812
Ans. 298 oranges. Ans. 35 cents. Ans. 203 barl.
(27) 9)423 (28) 12)540 (29) 4)321276
A1zs. $47 Ans. 45 years. Ans. 80319 gals.
(30) 8)1736 (31) 7)734566 (32) 11)495
Ans. 217 barl. Ans. 104938 weeks. Ans. 45
(33) 9)3582 (34) 2)56 (35) 2)3725
Ans. 398 Ans. 28 Ans. 1862',
(36) 3)147 (37) 4)500 (38) 5)1945
Ans. 49 Ans. 125 Ans. 389
(39) 6)4476 (40) 7)2513 (41) 8)5992
Ans. 746 Ans. 359 Ans. 749
(42) 9)8793 (43) 10)1090 (44) 11)4125
Ans. 977 Ans. 109 Ans. 375
(45) 12)5556 (46) 4)144 (47) 5)195
Ans. 463 3)36 each son's share. 3)39
Ans. 12 Ans. 13
ARTICLE 46.
(3) (5) (6)
13)4056(312-Ans. 14)11577(826-i Ans. 15)48690(3246 Azs.
39 112 45
15 37 36
13 28 30
26 97 69
26 84 60
13 90
90
22 KEY TO RAY'S THIRD BOOK.
(7) (8)
23)1110960(48302: 4- Quotient. 67)122878(1834 Quotient.
92 67
190 558
184 536
69 227
69 201
60 268
46 268.
14
(9) (10o)
53)12412(234'S Quotient. 72)146304(2032 Quotient.
106 144
181 230
159 216
222 144
212 144
10
(11) (12)
54)47100(87214' Quotient. 88)71104(808 Quotient.
432 704
390 704
378 704
120
108
12
(13) (14)
66)43956(666 Quotient. 99)121900(1231i Quotient.
396 99
435 229'
396 198
396 310
396 297
130
99
31
DIVISION OF SIMPLE NUMIBERS. 23
(15s) (16)
112)25312(226 Quotient. 123)381600(3102156A4 Quotient.
224 369
291 126
224 123
672 300
672 246
54
(17) (18)
204)105672)518 Quotient. 1234)6o0000(486.. Quotient.
1020 4936
367 10640
204 9872
1632 7680
1632 7404
276
(19) (20)
4674)47263488(10112 Quot. 2034)26497935(13027,1 Quot.
4674 2034
5234 6157
4674 6102
5608 5593
4674 4068
9348 15255
9348 14238
1017
(21) (22)
9876)48905952(4952 Quotient. 12345)4049160(328 Quotient.
39504 37035
94019 34566
88884 24690
51355 98760
49380 98760
19752
19752
24 KEY TO RAY'S THIRD BOOK.
(23) (24)
973)5521G60000(567482s4g Quotient. 15)3465(231 Ads.
4865 30
6566 46
5838 45
7280 15
6811 15
4690
3892
(25)
7980 26)364(14 days.
7784 26
1960 104
1946 104
14
(27)
(26) 19)1083(57 dollars. 107)9523(89 bushels.
95 856
133 963
133 963
(29)
(28) 63)14868(236 hhd. 365)25000(68 dollars per day.
126 2190
226 3100
189 2920
378 $180 rem.
378
(31)
(30) 365)37960($104 1235)6571435(5321 Ans.
365 6175
1460 3964
1.460 3705
2593
2470
1235
1235
~~-~ —------------— __ —
DIVISION OF SIMPLE NUMBERS. 25
(32) (33)
405)1247400(3080 Ans. 1006)10401000(10338ts7 Quotient.
1215 1006
3240 3410
3240 3018
0 3920
3018
9020
8048
972
A RTICLE 47.
(3) 9)2583 (4) 4)6976 (5) 4)2744 (6) 6)6145
7)287 8)1744 7)686 7)1024-1 rem.
Ans. 41 Ans. 218 Ans. 98 Ans. 146-2
6X2+-1=13 rem.
(7) 11)19008 (8) 8)7840
12)1728 8)980
Ans. 144 Ans. 122-4
8X4-=32 rem.
(9) 8)14771 (10) 9)10206
9)1846-3 9)1134
Ans. 205-1 Ans. 126
8X1+3 —11 rem.
ARTICLE 48.
(2) 110)26812 (3) 1100)47100 (4) 1100)372101
Ans. 268 -0 Ans. 47 Ans. 372~-f-o
(5) (6) (7) (8)
1100)462150 1100)620134 11000)181003 11000)3751000
Ans. 462-i-ou Ans. 62018 4 Ans. 18 T Ans. 375,~~~~8 K D'C
26 KEY TO RAY'S THIRD BOOK.
ARTICLE 49,
(3) 41000)731005 (4) 91000)361001
Ans. i.8-0 -- Ans. 4-g 1
(5) 111o000)10781000 (6) 1810)401617(22327 Quotiezt.
36
Ans. 98
41
36
56
54
2
(7) ()
21100)9072137(432!TgiG Quot. 64100)3640106(565@S Quot.
84 320
67 440
63 384
42 56
42
(9) 2510000)7654]6037(306~%%Vo7 Quotient.
75
154
150
4
(10) 634100)435637154(687 f 95j1 Qu-otient.
3804
5523
5072
4517
4438
79r
DIVISION OF SIMPLE NUMBERS. 27
ARTICLE 50.
(1) (2)
7895)434225(55 Ans. 327)106929(327 Ans.
39475 981
39475 882
39475 654
2289
(3) 2289
375)10741125(28643 Ans.
750
(4)
3241 60352)63550656(1053 Ans.
3000 60352
2411 319865
2250 301760
1612 181056
1500 181056
1125
1125
ARTICLE 56.
(1) (2)
96 500 243 1St. 243 1265
120 271 61 304 772
55 - - 225
- Ans. 229 304 2d. - Ans. 493
271 79 772
225 3d.
(3) (4)
157 428 9503 57068
264 186 586 16967
305 - 4794
97 614 1234 Ans. 40101
123 850
946 16967
614.Ans. 332
28 KEY TO RAY'S THIRD BOOK.
(5) $123078 $237 spent in Jan. 538
8706 301 spent in Feb. 10 months left.
131784 538 5380
5918 5380
Ans, $125866 $5918 spent.
(6) 72 (7)
4 yards. yards.
73 88
288 w. 9 12
24
657 1056
1152 657
576
Ans. 1713
Ans. 6912 lights.
(8) days. (9) 39 vols. (10)
365 774 86)31173(362 Ans.
49 258
156 -
3285 273 537
1460 273 516
17885 30186 pages. 213
99 2 172
160965 60372 columns. 41
160965 67
Ans. 1770615 422604
362232
4044924 lines. (12) 63 gallons.
28 (11) 10 5
- $ 40449240 words. 15)315(21 Ans.
25)1400(56 Ans. 30
125 4044924 lines.
47 15
150 15
150 28314468 15
16179696
190111428 letters......................
EXERCISES IN PRECEDING RULES. 29
(13) 73900 (14) 148 148
70 56 56
214)73830(345 Ans. 204 sum. 92
642 92 diff.
963 408
856 1836
1070 23)18768(816 Ans.
1070 184
36
23
138
138
$ (15) $ (16) 15)360(24=gallons filled
60 15 30 in 1 hour.
8 14
- _- 60
480 60 60
210 15
- ~~- ~ 210)3610
6)690 210
18=gallons emptied in
115 Ans 1 hour.
24
(17) 18
70 6=gallons left in cistern
per hour.
2100
$6000
2100 6)360
25)3900(156 acres. 60 hours. Ans.
25 70 acres.
140 226 acres Ans. (18) 222
125 125
150 Ans. 347
150
30 IKEY TO RAY'S THIRD BOOK.
(19) 500 (20) 45 (21) 340 (22) 27
135 15 11 98
Ans. 365 225 9)351 125 diff.
45 246
39
675 15 Ans. 371
65
- Ans. 24
Ans. 740
ARTICLE 61.a-CANCELLATION.
4X1$ 41x11
(2) -— =4 Ans. (3) -- 11 Ans.
3
15X 32;4x4
(6) - =30 Ans. (7) — =12 Ans.
37 X 1 3 36X
(8) - -111 Anzs. (10) — =12 Ans.
14X9 7X>X$Axx3
(11.) --- =21 Ans.
ARTICCLE 61.b
(1)
42X25X18 _XX2X$X5X$X6
2 -1 --— =2X5X6=-60 Ans.
21X 15 9,xx O$x
2 1X 23
(2) =2 X 23=46 Ans.
14X15 2XTX3X0
(3) - -=6 Ans.
35 - xo
cg d b —------------— ~~ —--------------
COMPOUND NUMBERS.-U. S. MONEY. 31
(4)
21X11X6X26 _XX11XlX ~X3X$X 113..11X3=33
13X3X14X2 OX$X.X XA Ans.
21X 15X 33X 8X14X17
(5)
20X34X 22 X27
_7x ~x x x xx 1x x x x x TXtX x~
7X$X=X — X=7XX$XAXAX7XfXT 7 =49
AX XXX iXX 1X X$X$X$ XX Ans.
(6)
21X95X6 _ x$x$X19X2X$.....= 19 X 2=X 38 Ans.
35X9 tX0X$x $
FEDERAL, OR U. S. $ MONEY.
ARTICLE 67.-NUmIERATION.
(1) Eighteen dollars, sixty-two cents, five mills. (2) Twenty
dollars, thirty-two cents, four mills. (3) Seventy-nine dollars,
five cents. (4) Forty-six dollars, three mills. (5) Seventy dollars, one cent, five mills. (6) One hundred dollars, twenty-eight
cents, three mills. (7) One hundred and fifty dollars and two
mills. (8) One hundred dollars and three mills. (9) Six dollars
and twelve cents. (10) Three dollars and six cents. (11) Four
dollars and thirty-one cents. (12) Five dollars and forty-three
cents. (13) Twenty-nine dollars. (14) One hundred dollars
and three cents. (15) Twenty dollars and five cents. (16) Forty
dollars and seven mills.
ARTICLE 63.-NOTATION.
The examples in the Arithmetic are all answered there. It is
recommended to the teacher to exercise the pupils with additional examples, should it be found that they are not sufficiently
acquainted with the subject, after going through with those in
the book.
ls -- -- --------— ~ —-------
32 KEY TO RAY'S THIRD BOOK.
ARTICLE 71.-RED UCTION.
Since the operations in this section consist simply in adding
ciphers or removing them, or erasing points or inserting them
between the different denominations, it is deemed unnecessary to
occupy space, as the whole solution, when presented to the eye,
would consist in notling more than writing down the question
to be solved, and then placing the answer under it.
ARTICLE 72.-ADDITIONo
(2) $ cts. (3) $ cts. m. (4) $ cts.
17.15 18.04 1 43.75
23.43 16.31 7 29.18
7.19 100.50 3 17.63
8.37 87.33 8 268.95
12.31 718.07
- Ans. $222.19 9
Ans. $68.45 Ans. $1077.58
(5) $ cts. m. (6) $ cts. m. (7) $ cts. m.
200.00 0 504.06 3 5.07 0
43.87 5 420.19 7 30.20 3
56.93 7 105.50 0 100.00 5
8.50 0 304.00 5 60.02 0
2.31 3 888.45 5 700.01 1
1000.10 0
Ans. $311.62 5 Ans. $2222.22 0 40.00 4
64.58 7
Ans. $2000.00 0
ARTICLE 73e-SUJBT RACTIO N.
(2) $ cts. m. (3) $ cts. (4) $ cts, (5) $ cts.
29.34 2 46.28 20.05 3.00
17.26 5 17.75 5.50.03
Ans. $12.07 7 Ans. $28.53 Ans. $14.55 Ans. $2.97
(6) $ cts. m. (7) $ cts. m. (8) $ cts. m. (9) S cts. m.
10.00 0 50.00 0 1000.00 0 1000.43 5
1 50 5 1.01 1 900.68 7
Ans. $9.99 9 Ans. 49.49 5 Ans. $998.98 9 Ans.n 99.74 8,, —-----------------
COMPOUND NUMBERS.-U. S. MONEY. 33
ARTICLE 74, —MULTIPLICATION.
(2) $ cts. (3) $ cts.m. (4) $ cts. m. (5) $ cts.
7.83 12.09 3 23.01 8 35.14
8 9 16 53
Ans. $62.64 Ans. $108.83 7 138108 10542
23018 17570
Ans. $368.28 8.Ans. $1862.42
(6) $ cts. (7) $ cts. (8) cts. m. (9) $ cts.
125.02 40.04 12 5 3.28
62 102 17 38
25004 8008 875 2624
75012 4004 125 984
Ans. $7751.24 Ans. $4084.08 Ans. $2.12 5 Ans. $124.64
(10) $ cts. (11) $ cts. (12) 465 Here, instead of
1.06 5.75 7 multiplying 7
338 38 - cents by 465,
Ans. 32.55 it amounts to
848 4600 the same thing
318 1725 to multiply 465
318 by 7, and is more
Ans. $218.50 convenient.
Ans. $358.28
(13) cts. (14) $ cts. (5) $ cts.
34 5.67 2.69
89 24 169
306 2268 2421
272 1134 1614
269
Anns. $30.26 iAns. $136.08
Ans. $454.61
Key. — 3
3-4 KEY TO RAY'S THIRD BOOK.
(I6) $ cts (17) 73 (18) 281
1.25 63 4
691 --
219 1124 lb.
125 438 65
1125- -
750 4599 gal. 5620
55 6744
Ans. $863.75
22995 AnSo. $73.050
22995
Ans. $2529.45
(19) 35 (20) 312 (21)
10 11 18is $ cts.
3 1.25
350 yd. 3432 hours. - 54
1 13 Bushels 54
500
Ans. $3.50 10296 625
3432
-Ans. $67.50
Ans. $446.16
(22) cts.rm. (23) 17 cts. n.
10.00 1 51 14 7
150 - 867
1 17
500).050 85 1029
10001 -- 882
867 lb. 1176
Ans. 1500.15 0
Ans. $127.44 9
AP TICLE, 75. —DIV IsION.
(4) (5) ets. yd. (6) Bhl.
8)280 25)300(12 Ans. 805)16100(20 Ans.
25 1610
Ans. 35 yd. -
50 0
50
COMPOUND NUMBERS -U. S. MONEY. 35
(7) 75)1200(16 Ans. (8) 1125)234000(208 bu. Ans.
75 2250
450 9000
450 9000
(9) 4)92.250 (10) 8)57.500 (11) $ cis.
16)25.76(1.61 Ans.
Ans. $23.062+ Ans. $7.187+ 16
97
96
16
16
(12) cts. m. (13) $ cts. m.
755)328.425(.43 5 Ans. 313)800.000(2.55 5+ Ans.
3020 626
2642 1740
2265 1565
3775 1750
3775 1565
1850
1565
(14) $ cts. m. cts.m. (15) $ cts. m. cts. nm.
133)10000.00 0(75.18 7+-Ans. 154)2705.01 0(17.56 5 Anps.
931 154
690 1.165
665 1078
250 870
133 770
1170 1001
1064 924
1060 770
931 770
I,:-~ r~ ~~ ~ ---
36:KEY TO RAY'S THIRD BOOK.
(16) 25 (17) 235
15 8
m n. cts.m.
125 lb. 1880)122200(65=6 5 Ans.
25 11280
~ ~ cts. cts.
lb. 375)6000('16 Ans. 9400
375 9400
2250
2250
ARTICLE 76.
(1) $ cts. (2) $ cts. (3) $ Ct8. (4)
47.50 35.25 18.38 $ cts. $ cts.
38.45 23.75 81.62.75 5.00
15.47 -.35 3.10
19.43 59.00 100.00.50
59.00 100.00 1.50 Ans. $1.90
Ans. $120.85 1.00 100.00 -
$3.10
Ans. $119.00 Ans $300.00
(5) (6) $ cts.
cts. cts. $ cts. 50.00
8.10.75 20.00 30.50
1.65 8'1975
6.00 19.50
4.00 $6.00 Ans..25 cts. 6
$19.75 Ans. $117.00
(7) $cts. (8) $ cts. (9) 143 (10) 435
3.85 37.06.13 45
1.25 200.85 -
2.50 400.00 429 2175
1.50 236.75 143 1740
124.34
9.10 $18.59 $195.75
999.00 12.60
$ cts. 889.25 - 400.00
21.75 - Ans. $5.99 195.75
9.10 Ans. $109.75 $204-25
AnsAns. $12.654.25
Ans. -12.65
COMPOUND NUMBERS.-U. S. MONEY. 37
(11) 365 (12) 21 (13) 19 (14) $ cts..65 3 4 2000.00
163.75
1825 63 76
2190 35 23 5)1836.25
$237.25 315 228 5)367.25
189 152
$400.00 -- -Ans. $73.45
237.25 Ans. $22.05 Ans. $17.48
Ans.$162.75
(15) $ cts. (16) $ cts. (17) $ cts.
4)516.00 410)910.00 22= —2Xl 1000.00
_ls)s ~~.500.00
4)129.00 10)2.25 500.00
cts. 2)1500.00
43)32.25(75 Ans. Ans. cts. 22 5 m.
301 11)750.00
215 Ans. $68.18+
215
(18) $ cts. (19) $cts.
9 lbs. at 16 cts.........$1.44 8 yards at $1.10........8.80
4 lbs. at $1.25...........5.00 18 yards at 25 cts.......4.50
45 lbs. at 9 cts.......... 405 25 yards at 15 cts.......3.75
17 lbs. at 13 cts.......2.21 12 yards at 35 cts....... 4.20
6 yards at 65 cts.......3.90
Ans. $12.70
Ans. $25.15
(20) $ cts. (21) $ cts.
5 copies at 35 cts.......1.75 27 copies at 19 cts......... 5.13
12 dozen at $10.50....126.00 25 " "27 "......... 6.75
6 M. at $1.60............9.60 8 " " $4.50........ 36.00
5 quires at 25 cts.......1.25 27 " " $2.25.........60.75
3 copies at $4.50......13.50 18 " " $1.50.........27.00
75 " 31cts........23.25
Ans. $152.10
Ans. $158.88
38 KEY TO RAY'S THIRD BOOK.
COM iPOUND ITUM iBEIRS.
In the remaining part of this work, instead of writing out the
solution in full, we have merely indicated the operations to be
performed. When a person understands the elementary rules,
and the meaning of the signs, the method of solution is just as
easily understood, as if written out at length. Pupils, however,
should be taught to work the questions, after the manner given
in the book.
In Reduction, pupils should be frequently required to give the
reason for multiplying or dividing by the numbers used: also to
point out the name of the remainder, and give the reason for it.
ARTICLE 81.-DRiY MEASURE.
bu. pt. pk. pt.
(11) 2X4XSX2-128 Ans. (12) 12X8X0 —192 Ans.
bu. qt.
(13) 8X4XS=256 Ans.
bt. plc. pl. pk. pt.
(14) 1X4+1=-l5: 5><8X2=80 Ans.
bu. pk. pk. qt. qt. qt. pt.
(15) 2X4-8: 8XS+2-66: 6GX2-132 Ans.
bu. pk. pk. pk. qt. qt. qt. pt.
(16) 4X4+-2=18: 18X8+1=145: 145X2=290 Ans.
bu. pk. pk. pPk. qt. ct. qt. pt. pt.
(17) 7X4+3=31: 31X8+7-=255: 255X2 —1=511 Ans.
bu. pk. pk. qt. qt. pt. pt.
(18) 3X4 —12: 12XS=96: 96X2+=1 93 Ans.
pt. qt. qt. pk. pk. bu.
(19) 384-. -2192: 1928 —8=24: 24 *. 46 Ans.
pt. qt. pt. qt. pk. qt. pk. qt. pt.
(a0) 47.e2=-23 1:23 — 8=2 7. Ans. 2 7 1
pt. qt. pt. qt. pk. qt. pk. bu. pk.
(21) 95 — 2=47 1: 47 —8=5 7: 5. 4 — 1 1: collecting the
different remainders, the Ans. is, 1 bu. 1 pk. 7 qt. 1 pt.
~~ —-- ----------- -- - - - -— ~~~~~~~~~~~~~~~~~~~~~~~~ef
REDUCTION. 39
pt qt. pk. qtt. t1. u,. pk. bu.pk.cit.
(22) 5-8 * -,2. 234-.-8=-31 (): 31-:-4=7 3. As. 7 3 6
ARTICLE 82. —TrtoY EIGHT.
lb. gr. lb. oz. oz.
(2) 4X12X20X24=2 — 3040 Anms. (3) 5X12+4=64 Ans.
lb. oz. oz, oz. pwt. pwt.
(4) 9X1'2+3111: 111X20+5=22 25 Ans.
lb. oz. oz. oz. pwt. pwt.
(5) 14X12+11-179: 179X204-19 —3599:
pwt. gr. gr.
3;599X24- 923-86399 _'ts.
lb. oz, Oz, OZ. pAwt. pwt.
(6) 8X1-1t't-9=l105: l5X20 15 -L3 — 213
pwt. g. gr.
2113X'24 —+17-=50729 Ans.
fgr. pwt, gr.
(7) 171 24z-7 3 Anls
gr. pwt. gr. pwt. Oz. pwIt. 0o. pwt. gr.
(8) 505. -24'21 1: 21 20=1 1. Ans. 1 1 1.
gr. pAwt. gr. pwt. oz. pt. oz. lb1) Z.
(9) 12530-9-24-522 2: 5'22 20,26 2: 26 12-2 2.
lb. oz. pwt. gr.
Alns. 2 2 2.
pwt, oz. pwt, oz. lb. oz. lb. oz. pwt.
(10 805 — 20-4() 5 400 1 —-3 4. Ans. 3 4 5.
gr. pwvt. gr. p\wt. oz. pwt. oz. b. oz.
(11) 25591-+4 —1066 7: 1066 20 = 53 6: 53 —12=4 5.
lb). oz. pwt. gl-.
Ans. 4 5 6 7.
A.RTICLE 83.-APOTHECARIES, -NVT EIIGHI-'T.
lb>~~ ~ gr.
(1) 3Xl2X8X3X20= —:'172SO Ans.
40 KEY TO RAY'S THIRD BOOK.
lb 3 3 3 3 D
(2) 4X12=48: 48X8+5=389: 389X3=1167:
3 gr. gr.
1167X20+2 —23342 Ans.
lb 8 8 3 3 D.
(3) 7X12=84: 84X8=672: 672X3-+2=-2018 Ans.
b i b 3 8 3 3 D 3
(4) 7X12+2-86: 86X8=688: 688X3+1=2065:
gr.
2065X20-41300 Ans.
(5) 67 —; 12=5 7 Ans.
3 33 3 Tb t1l 3
(6) 431 —-8=53 7: 53l12_4 5. Ans. 4 5 7.
D 3 3 8 3 8 ltb 3 3
(7) 975- -3=325: 325 8=40 5: 40.-12-3 4. Ans. 3 4 5.
gr. 3 gr. 3 3 3
(8) 6321+e 20=316 1: 316 -3=105 1: 105-8=-13 1:
b I is I 3 D gr.
13 -.12-1 1. Ans. 1 1 1 1 1.
gr. D gr. D 3 3 3
(9) 30941-.-20=1547 1: 1547 33=515 2: 515-.-8-64 3:
3 lbf i lb 3 3 D3gr.
64__12=5 4. Ans. 5 4 3 2 1.
gr. 3 gr. D 3 3 8 3
(10) 29239 -20=1461 19: 1461_. 3=487: 487- 8=60 7:
3 Blb lb3 gr.
60. 12=5. Ans. 5 7 19.
ARTICLE 84. —AVO TRDUPOIS WEIGHT.
cwt. lb. cwt. qr. qr. qr. lb.
(1) 2X4X25=200 Ans. (2) 3X4+3=15: 15X25=375
[Ans.
REDUCTION. 41
T. cwt. cwt. cwt. lb.
(3) 1X20+2222: 22X4X25=2200 Ans.
1T. cwt. owt. qr. qr. qr. lb.
(4) 3X20 —60: 60X4+3=243: 243X25 —6075 Ans.
cwt. qr. qr. qr. lb. lb.
(5) 4X4+1= —17: 17X25+19 —444 Ans.
T. cwtt.. qt. qr. qr. qr. lb. lb.
(6) 5X20=-100: 100X4+3-403: 403X25+15=10090
[Ans.
cwt. qr. qr. qr. lb. lb. lb. oz. oz.
(7) 2X4+3=11: 11X25+2277: 277X16+-12=4444
[Ans.
cwt. qr. qr. lb. lb lb. oz.
(8) 2X4-=8: SX25+ —17=217: 217X16=3472:
oz. dr. dr.
3472X16+3=55555 Ans.
T. cwt.cwt. cwt. qr. qr. lb. lb.
(9) 1X20-+6-26: 26X4=104: 104X25+4=2604:
lb. oz. oz. oz. dr. dr.
2604X16+2=41666: 41666X16+10-666666 Ans.
lb. qr. lb. qr. cwt. cwt. lb.
(10) 4803- 25=192 3 192 -4=48. Ans. 48 3.
lb. qr. qr. cwt. cwt. T. cwt.
(11) 22400~'. 25896896:96-4 —224: 224-*-20=11 4 Ans.
dr. oz. oz. lb.
(12) 2048000-.-16=128000: 128000-. 16=8000:
lb. qr. qr. cwt. cwt. T.
8000 —. 25-320: 320 -4-4=80: 80-. 20-4 Ans.
dr. oz. dr. oz. lb. oz.
(13) 64546 16 —4034 2 4034-. 16=252 2:
lb. qr. lb. qr. cwt. qr. cwt.qr. lb. oz. dr.
252 25 —10 2: 10. 4-2 2. Ans. 2 2 2 2 2.
oz. lb. oz. lb. qr. qr. cwt. qr.
(14) 97203 — 16-6075 3: 6075-. 25-243: 243. 4-60 3:
cwt. T. T. qr. oz.
60 -. 20-3. Ans. 3 3 3.
42 KEY TO RAY'S THIRD BOOK.
drl. oz. OZ. lb. oz.
(15) 544272-. 16 —34017: 34017-:16=-126 1:
lb. qr. lb. qr. cwt. qr. cwt. T. cwt.
2126 — 25=85 1:85- 4-=21 1:21 — 2031 1.
T. cwt. qr. lb. oz.
Ans. 1 1 -1 1 1.
lb. lb. lb. qr: lb. qr. cwt.qr.
(16) 18X52=936: 936 —. 25-37 11: 37- 4=9 1.
cwt. qr. lb.
Ans. 9 1 11.
A TICLE 85.-LONG ]MEASUREo
yd. ft.. ft.. ft. in. i nI
(1) 2X3+-2=8: SX12+-7 —103 Ans,
yd. ft. ft. in. in.
(2) 7X3=21: 21X12l2+11-263 Ans
lIii. fur. fur. rd.
(3) 12XS=96: 96X40,-3840 Aas.
mi. fur. fur. fur. rd.
(4) 7X8+6=62': 62X40=2480 Ans.
mi. fur. fur. rd. rd.
(5) 9X8 —72: 72X40+31=2911 Ans.
in. ft. in. ft. yd. ft. yd. ft. in.
(6) 133-.-12=11 1: 11 —3=3 2 A-ts. 3 2 1
in. ft. in. ft. yd. yd. in.
(7) 181 —12=15 1: 15 —3=5. Ans. 5 1.
rd. fur. fur. mi.
(8) 2240-:-40=56 56-.-8=7 Ans.
rd. fur. fur. mi. fur.
(9) 2200 —40=55: 55 -8-=6 7 Ans.
~~ —-- ------ ---- ---- ----— ~~~~~~~~~~~~~~~~~,
REDUCTION. 43
ARTICLE 88.-LAND IMEASURE.
sq. yd. sq. in. A. P.
(1) 8X9X144=10368 Ans. (2) 4X4X40=640 Anzs.
sq. mi. P.
(3) 1X640X4X40-102400 Ans.
sq.yd. sq.ft. sq.ft. sq.ft. sq. in.
(4) 2X9+3-21: 21X144=3024 Ans.
A R.R. R. P. P.
(5) 5X4+-2-2: 22X40+20-900 Ans.
P. R. R. A.
(6) 960 —40-_24 24-.-4=6 Ams.
sq. in. sq. ft. sq. ft. sq. yd.
(7) 3888 —144-27: 27 — 9=3 Ans.
P. R. P. R. A.R. A.R.P.
(8) 243 —40=6 3: 6 — 4=1 2. Ans. 1 2 3.
P. R. P. P. A.R. A. R. P.
(9) 603-4-40=15 3: 15 —4=-3 3. Ans. 3 -3 3.
sq. in. sq. ft. sq. ft. sq. yd. sq. ft.
(10) 4176 —144=29: 29-.-9-3 2 Ans.
ARTICLE 89.
ft.. ft. sq. ft. yd. yd. sq. yd.
(2) 16X12-192 Ans. (3) 5X4-=20 Ans.
ft. yd. ft. yd. ft. yd. ft. yd.
(4) 18 — 3=6: 12 —3 —4: 21 —3=7: 15-.-3=5:
sq. yd. sq. yd. sq. yd.
6X4-24: 7X5=35: 24-1-35-59 Ans.
ft. ft. sq. ft. sq. ft. sq. yd.
(5) 18X14-252: 252 -9=28 Ans.
rd. rd. P. P.. R. R. A.
(6) 35X32=1120: 1120, —40-28: 28 4 —4-7 An?.
44 KEY TO RAY'S THIRD BOOK.
ARTICLE 90.
sq. ft. ft. ft. sq.yd. sq. ft. sq. ft. ft. ft.
(1) 132~. 11=12 Ans. (2) 30X9=270: 270 —18=15 Ans,
A. P. P. rd.
(3) 9X4X40=1440: 1440-. 45-32 Ans.
A. P. P. rd.
(4) 21X4X40-3360: 3360-.-35-96 Anzs.
ARTICLE 93.-CUBIC MEASURE.
cu. yd. cu. in. C. cu. ft.
(1) 2X27X1728=93312 Ans. (2) 28X128=3584 Ans.
C. C. ft. C. cu. in.
(3) 34X8=272 Ans. (4) 1X128X1728=-221184 Ans.
cu. in. cu. ft. cu.t. cu.yd. cu.ft.
(5) 63936 1728-37: 37 —27-1 10 Ans.
cu. in. cu. ft. cu. ft. tn. cu.ft.
(6) 492480 —1728=285: 285 ~.40-7 5 Ans.
ft. ft. ft. cu.ft. yd. yd.yd. cu.yd.
(7) 8X5X4=160-Ans. (8) 8X5X2=80 Ans.
ft. ft. ft. cu.ft. cu.ft. cu.yd.
(9) 18X15X7-1890: 1890-. 27=70 Ans.
ft. ft. ft. cu.ft. cu. 0f C.
(10) 40X12X8-3840: 3840 128=30 Ans.
ARTICLE 94.-CLOTHt MEASURE.
yd. na. E. F1. na.
(1) 19X4X4=304 Ans. (2) 14X3X4-168 Ans.
yd. qr. qr. qr. na. na.
(3) 5X4+2=22: 22X4+3-=91 Ans.
E. En. qr. qr. qr. na.
(4) 13X5+1=66: 66X4-264 Ans.
REDUCTION. 45
E. Fr. qr. na. na.
(5) 23X6+3=141: 141X4+2=566 Ans.
la. qr. na. qr. yd. qr. yd. qr. na.
(6) 159 —4=39 3: 39 4=9 3. Ans. 9 3 3.
na. qr. na. qr. E.Fr. na. EFr. qr. na.
(7) 287 4=71 3: 71 6=11 5. Ans. 11 5 3.
yd. qr. qr. E.F1.
(8) 6X4=24: 24-.~3=8 Ans.
yd. qr. qr. qr. E. F1.
(9) 9X4+3=39: 39-. 3=13 Ans.
yd. qr. qr. qr. E. Fl. qr.
(10) 12X4+1=49: 49. 3=16 1 Ans.
E.Fl. qr. ar. yd. qr.
(11) 37X3=111: 111 —. 4=27 3 Ans.
E.En. qr. qr. E.Fr.
(12) 36X5=180: 180-6 —=30 Ans.
E.En. q q. qr. E.Fr.
(13) 22X5+4=-114: 114 -6=19 Ans.
E.En. qr. qr. yd. qr.
(14) 47X5-235::235. 4 —58 3 Ans.
ARTICLE 95,-LIQUID MEASURE.
gal. pt. gal. gi.
(1) 17X4X2=136 Ans. (2) 13X4X2X4=416 Ans.
hhd. pt.
(3) 2X63X4X2=1008 Ans.
T. gi.
(4) 5X4X63X4X2X4=40320 Ans.
T. hhd. hhd. hhd. gal.
(5) 3X4+3=15: 15X63=945 Ans.
46 KEY TO RAY'S THIRD BOOK.
hhd. gal. gal. gal. qt. qt. pt. pt.
(6) 1X63+60-123: 123X4=492: 492X2+1 —985 Ans.
hhd. gal. gal. gal. qt. qt. qt. gi.
(7) 2X63+17=143: 143X4+3-3=575: 575X2X4-4600
[Ans.
T. hhd. hhd. gal. gal. gal. pt. pt.
(8) 2X4=8: 8X63+62=566: 566X4X2+1=4529:
pt. gi.
4529X4=18116 Ans.
gi. pt. pt. qt. qt. gal.
(9) 96 —4-24: 24. 2=12: 12 — 4=3 Ans,
gi. pt. pt. qt. qt. gal.
(10) 6048 —4=1512: 1512 —-2_756: 756 —4=189:
gal. hlhd.
189 —-63=3 Ans.
gi. pt. pt. qt. qt. gal.
(11) 32256-4=8064: 8064 —2=4032: 4032-.-41008:
gal. hhd. hhd. T.
100 —63 —16: 16 —4=4 Ans.
gi. pt. gi. pt. qt. qt. gal. qt.
(12) 4050 —4-1012 2: 1012-2=506: 506- -4-126 2:
gal. hhd. hhd. qt. gi.
126 * 63-2. Ans. 2 2 2.
gi. pt. gi. pt. qt. qt. gal.
(13) 30339 —4 —7584 3: 7584 —-2=3792: 3792 -4 —948:
gal. hhd. gal. hhd. T. hhd. T. hhd. gal. gi.
948-1-63=15 3: 15 4 —=3 3. Ans. 3 3 3 3.
gi. Pt. gi. pt. gi. pt. qt. pt. qt. gal. qt.
(14) 10125+ 4-2531 1: 2531-.-21265 1: 1265 4=316 1:
gal. lhh. gal. hhd. T. hMd. T. hhd. gal. qt. pt. gi.
316-)'63: —z5 I: 5-i-4=1 1.:InsS. 1 1 1 1 1 1..~~ — I —6
REDUCTION. 47
pn. gi.
(15) 3X84X4X2X4-=8064 Ans.
p. qt.
(1G) 5X126X4=2520 Ans.
pt. qt. qt. gal. gal. tr.
(17) 5712 —— 2856: 2856 4=714 714-42=17 Ans.
A R T I L 96.-BEER MEASURE.
hhcld. pt.
(1) 4X54X 4X2'-1728 Ans.
hhd. gal. gal. qt. q't. qt. pt.
(2) 7X54=378: 37SX44-3=1515: 1515X2 —3030 Ans.
pt. qt. qt gal. gal. bl. gal.
(3) 1000 —-=500: 500 —4 —125: 125 —-36=3 17. An1s.
p-t. qt. pt. qt. gal. qt. gal. hhcl. gal.
(4) 443. —2-221 1: 221-*4=55 1: 55-.54=1 1.
lhhd. gal. qt. pt.
Alns. 1 1 1 1.
ARTICLE 97.-TIMIE.
hr. SCG. da. min.
(1) 2X6OX60=7200 Ans. (2) 7X24X60-10080 Ans.
da. hr. hr. hr. min. m. min. sec. sec.
(3) 1X24+3=27: 27X60+44=1664: 1664X60+-3=99843
[A.zs.
wk. da. da. da. hr. hr.
(4) 9X7+6-69: 69X24+10=-1666:
hr. min. min.
1666X60-+40=100000 Ans.
mon. wk. wk. da,. da. da. hr.
(5) 1X4=4: 4X7+-3=31: 31X,-4=744:
hr. mlin. n in.
744X60+4=446)44 Ars.I.
-.-. M
48 KEY TO RAY'S THIRD BOOK.
yr. da. da. da. hr. hr. hr. min.
(6) lX365+20=385: 385X24+19=9259: 9259X60+15
min. min. sec. sec.
-=555555: 555555X60+33=33333333 Ans.
da. hr. hr. hr. min. min.
(7) 365X24 —5=8765: 8765X60-148=-525948:
min. sec. sec.
525948X60+48=-31556928 Ans.
sec. min. min. hr.
(8) 10800-.:60=180: 180 60=3 Ans.
sec. min. min. hr. hr. da.
(9) 432000-. 60=-7200: 7200 —60=120: 120 — 24-5 Ans.
sec. nin. sec. min. hr. min. hr. min. sec.
(10) 73227. 60=122 2: 122-. 60=2 2. Ans. 2 2 2.
rlin. hr. min. hr. da. da. min.
(11) 4323-60=72 3:72. 24= —3. Ans. 3 3.
mnin. hr. hr. da. hr. da. -wk. wk. hr.
(12) 20280 -60=338: 338 — 24=14 2:14-7=2. Ans. 2 2.
min. hr. min. hr. da. da. wk. da.
(13) 41761 —60-696 1: 696 —24=29: 29-~ 7=4 1;
wk. mon. mon. da. min.
4 -4=1. Ans. 1 1 1.
ARTICLE 98.-CIRCULAR MEASURE.
(1) 50X60+3'=303' Ans.
(2) 8oX60+41l=-521': 521'X60+45"-31305"/ Ans.
(3) 3sX30=900: 90~X60 —25=-5425' Ans.
(4) 1CX360X60X60=1296000/ Ans.
(5) 244// —. 60=4' 4i" Ans. (6) 915/' 60-15~ 15' Ans.
(7) 1861-. 60-310 1/: 31~-.-30-1s 1~. Ans. 1s 1~ 1/.
REDUCTION. 49
ARTICLE 100.
bu. pt. cts. cts. $
(1) 2X4XSX2 —128: 5X128=640=6.40 Ans.
bu. pl. plc. cts. ets. $
(2) 3X4 —2 —14: 50X14=700=7.00 Anzs.
pk. qt. qt. qt. pt. cts. $
(3) 3X8+-3=-27 27X2-54: 3X54=-1.62 As.
$' cts. ts. cts. Ptc. pkl. bLi.
(4) 3=-300: 300 —15=20: 20 —4. 5 A'zis.
cts. cts, cts. qt.
(5) 1.66166: 166-1-4=41 tl d 2 cts. over, which will l,uy
qt. plk. qt. pk.' bu. pk.
1 pt. at 4 cts. a qt. 41-.-8=5 I: 5-.-4=1 1.
bu. pl. qt. pt.
A2,i. 1 1 1. Or' thus: 4 cts. a quart is 2 cts.
cts. cts. pt. bu. pkl. qt. pt.
a pt.; and 166 -2=83 —1 1 1 1. lAns.
bu. plc. lk. p1c. pk. plk. bags.
(6) 3 2 —14: 91=364:36 4 —14_=26 Ans.
lb. pwt. gr. cts. cts. S
(7) 1 3=5832: 3X5832=17496=174.96 Ans.
lb. oz. pwt. pwt. oz. pwt. pwt. pwt. pwt.
(8) 2 5 5=585: 2 5=45: 58-. 45=13 Ais.
lb. oz. pwt. gr. gr. pwt. gr. gr. gr. gr.
(9) 1 8 1S 1=10033: 7 —127: 10033 -127=79 Ans.
lb 3 gr. gr. gr. dose. cts. cts. nm.
(10) 1 4=6000: 6000 — 15=400: 12 5=125
m..111
125X400 —. 50000=50 Ans.
lb 3 3 9 cts. cts. $
(11) 1 1 1-315: 4X315 —160=12.60 Anis.
cwvt. fctr. lb. cts. cts. $
(12) 6 1=625: 625X3 5=575=18.75 Aiis.
Key. — 4
50 KEY TO RAY'S THIRD BOOK.
T. cwt. cwt. qr. $cts. $
(13) 1 l121=84: 2.25)(84=189,00 Ans.
lb. oz. oz. cts. cts. $
(14) 7 8=120: 5X120=600=-6 Ans.
gr. gr. lb 3 3 D
(15) 20X316=6320-1 1 1 1 Ans.
lb. oz. dr. dr. dr. dr.
(lV) 15 9 12-3996: 3996 —. 4=999 Ans.
cwt. qr. lb. oz. lb. oz. oz. oz. oz.
(L7) 44 2 2=71232 9 15=159: 71232 159=448 Anss.
cwt. qr. lb. lb. lb. lb.
(18) 14 1 3=1428:1428-84=17 Ans.
cw;t.qr. lb. lb. lb. lb.
(19) 7 2 6=756 756- 12=63 Arts.
cwt. lb. oz. lb. oz. oz. oz; oz.
(20) 6 10=9760: 3 13-61: 9760-. 61=160 An.
A. R. P. P. cts. cts. $
(21) 2 3 5=445: 20X445=8900()89 Ans.
sq.yd. sq.ft. sq.ft. sq.in, cts. cts. $
(22) 2 2 = 20=2880: 2880>)5=14400=144 Ans.
sq.yd. sq.in. in. in. sqin. sq.in. sq.in.
(23) 1=1296: 3X2=6: 1296 —. 6=216 Ans.
A. R. P. P. A. R. P. P. P. P.
(24) 16 1 13=2613: 1 1 1=201: 2613 — 201=13 Ans.
cu. f. cu.ft. cu.in.
(25) 2X2X2=8 8X1728=13824 Ans.
oz. oz. lb. oz.
(26) 1000X5=5000=312 8 Ans.
oz. oz. T.
(27) 1000X128=128000=4 Ans.
C. cu.ft. o oz. T. cwt.
(23) 2X128=256: 950X256=243200=7 12 Ans.
--- - -- ------ ~~~~~~~~~~~~- — ~~~u
REDUCTION. 51
yd. qr. na. na. yd na. na. na. pieces.
(29) 13 3 2=222: 666=10656: 10656 —222=48 Ans.
yd. qr. qr. yd. qr. qr. qr. suits.
(30) 5 1=21: 147=588: 588-.21=28,Ans.
yd. qr. na. na. yd. na. na. na.
(31) 2 1 1=37: 37 —592: 592 — 37-16 Ans.
yd. qr. qr. E.FI. qr. qr. qr. suits.
(32) 3 2=14: 70=210: 210. 14=15 Ans.
blid. gi. cts. cts. $
(33) 1=2016: 5X2016=10080=100.80 Ans.
gal. qt.. qt. qt. qt. cts. cts. $
(34) 31 2=126: 126X5 —630: 10)X630 —6300=63 Ans.
$ cts. cts cts. cts. pt. gal.
(35) 2=200: 200 —5-40 40=5 Ans.
$. cts.m. m. m. gi. gal.
(36) 12=12000: 1 5=15: 12000 —15-800=25 Ans.
hhd. pt. qt. pt. pt. pt. pt.
(37) 1=504: 3 1=7: 7X12=84 in 1 doz. bottles:
pt. pt. doz.
504 — 84=6 Ans.
gal. qt. pt. pt.:hhd. pt. pt. pt. pt.
(38) 4 3 1=39: 1-=504: 504 -39=12 and 36 left:
pt. gal. pt. gal. qt.
36=4 2. Ans. 12; and 4 2 left.
gal. pt. gal. qt. pt. pt. pt. pt.
(39) 165=1320 1 1 1=11: 1320 —. 11=120 Ans.
hhd. qt. cts. cts. $
(40) 1=216: 3X216=648=6.48 Ans.
da. min. beats. beats.
(41) 1=1440: 70X1440=100800 Ans.
Qa —---, is - ---
52 -- KEY TO RAY'S THIRD BOOK.
(42) 1840 is a leap year, because it is exactly divisible by 4;
hence, February has 29 days:
29 days=2505600 seconds. Ans.
(43) 1839 is not a leap year, not being exactly divisible by
4; hence, February contains 28 days. January has 31
days, which is 3 days more: 3 days=72 hours. Ans.
(44) By dividing by 4, it will be found that 1,838 and 1839
are not leap years, and that 1840 is one; by adding 4
to this, we find that 1844 and 1848 are both leap years.
da. hr. min. sec. sec. sec. sec.
(45) 365 5 48 48-31556928: 1609403328-. 31556928
yr.
— 51 Ans.
yr. da. da. min. min. da. hr.
(46) 60X365=21900: 30X21900-657000=456 6 Ans.
wk. da. hr. hr. mi. 1mi.
(47) 3 2 3=555: 8X555=4440 Ans.
da.
(48) 2s 40=64~: 64~ —. 2~=32 Ans.
min.
(49) 90~-324000'// 43/ 12"=2592": 324000" /-2592"'/125
hr. min.
=2 5 Ans.
cts. Cts. $
(50) 2X20=40 quires: 20X40=800=8 Ans.
cts. cts. $
(51) 3X24-72 sheets: 2X72=144=1.44 Ans.
doz. cts. cts.
(52) 3-36 apples: 2X36=72 Ans.
(53) 1 dozen dozen-12X1 2=144, and 144X4=576 oranges:
cts. cts. $
3X576=1728=17.28 Anzs.
(54) 1 gross =12 dozen, and 5 gross =60 dozen:
cts. cts. $
5X60=300=3 Aiis.
fa ------------- -- -----
REDUCTION. 53
cts. cts. $
(55) 4X196 —784=-7.84 Ans.
m1. mi. $
(56) 125X200=-25000-25 Ans.
(57) 1 dozen dozen =144: 144X6=864 eggs in all: half a
dozen dozen -72: 864-72-792 eggs left:
et. ets. $
lX792 —792-7.92 Ans.
NoTE.-In the solutions to several of the preceding examrples,
it may appear that we have not adhered to the principle of always regarding the multiplier as an abstract number; for instance, in solution to question 22, it might be supposed that we
multiply 2880 by 5 cents. In this case, however, the 2880 is to
be regarded as the multiplier, and the 5 cents as the multiplicand. It is an established principle, that where two abstract
numbers are to be multiplied together, either of them may be
made the multiplier. The same principle applies where one of
the numbers is concrete: thus, in example 22, the 5 cents may
be taken 2880 times, or 2880 cents may be taken 5 times; the
result, in either case, being 14400 cents. For convenience, the
smaller number is generally made the multiplier.
For the sake of occupying the least possible space with the
solutions, we have frequently represented the quotient of one
concrete number divided by another, as a concrete number;
though, in all cases, it is really abstract, as stated in the remarks on Division, page 19. Thus, in the solution to question
6, preceding, pecks divided by pecks are represented as giving
bags for a quotient; the analysis of the question, however,
shows, that 14 pecks is contained in 364 pecks 26 times, and as
it requires 1 bag to hold 14 pecks, it will require 26 bags to
hold 364 pecks; hence, the answer to the question is 26 bags.
A similar method of analysis applies to several of the other
examples. The quotient in all cases of division is an abstract number, and the analysis of the question under consideration always
determines what name or denomination is to be given to it.
54 KEY TO RAY'S THIRD BOOK.
ADDITIONi OF COMPOUND NUMBERS.
SUGGESTIONS TO TEACHEiRs.-The attention of the pupil should
be particularly called to the nature of compound numbers, and
the analogy that exists between them and simple numbers.
In the latter, the number of each order of units required to
form a unit of the next higher order is uniform, ten units in
any order always making one unit of the next higher order.
In compound numbers the rate of increase is not tenfold, nor is
it the same for the same table. Owing to this peculiarity, in
compound addition, instead of carrying one for every ten, as in
simple addition, we sometimes carry one for every two; again,
we carry one for every three, etc. The reason for carrying in
compound addition is easily understood by every pupil, and advantage may be taken of this, to explain more clearly the reason
for carrying in simple addition.
ANSWERS TO THE QUESTIONS IN COMPOUND ADDITION, TO
WHICH THERE ARE NO ANSWERS IN THE BOOK.
ARTICLE 101.
(4) 871b. 93oz. 17pwt. 22 gr. (5) 191I 11 3 3 23 1 gr.
(6) 79 T. 15 cwt. 3 qr. 20 lb. (7) 94 cwt. 3 qr. 5 lb. 5 oz. 5 dr.
(8) 38 mi. 5 fur. 19 rd. (9) 12 yd. 4 in.
(10) 299 A. 3 R. 30 P. (11) 51 sq. yd. 4 sq. ft. 73 sq. in.
(12) 49 C. 58 cu. ft. 519 cu. in.
(13) 143 cu. yd. 2 cu. ft. 990 cu. in.
(14) 26 yd. 3 qr. 2 na. (15) 37 E. Fl. 2 qr. 3 na.
(16) 46 E. En. 2 na.
(17) 26 T. 34 gal. 2 qt. (18) 128 gal. 3 qt. 1 pt. 3 gi.
(19) 8 mon. 4 da. 8 hr. 49 min. 35 sec. (20) 3290 5' 52"
g~~Dt~ I~PISQ. god~P~L-R~ —~~-~C
COMPOUND NUMBERS. 55
(21) bu. pk. qt. pt. (22) bu. pkt. Pt. (23)ib. oz. pwt. gr.
21 3 2 1 2003 2 10 15 21
14 0 5 0 143 0 2 1 ] 1 16 14
23 2 0 1 400 3 0 0 2 8 5 12
18 I 0 1 255 1 5 0 Ans.6 8173
As 71. 1000 0 0 0
Ans. 100 0 0 0
(24) lb 3 3 9 gr. (25) cwt. qr. lb. (26) lb. oz. dr.
3 4 1 0 8 1 11 13 11 15
4 3 2 0 4 2 14 17 13 11
4 18 5 0 19 14 14 0
6 5 2 18 7 3 O 1G 0 10
Ans. 1 3 2 O 16 7 3 9 19 7 12
Angs. 33 3 3
A-ns. 99 9 9
(27) mi. fur. rd. (28) A. R. P. (29) sq. yd. sq. ft. scin.
104 1 10 186 3 14 17 3 119
95 6 30 286 0 17 18 0 141
jAls. 200 0 0 113 2 9 23 7 00
nAns. 586200 2 0 29 5 116
Ans. 586 2 0
Ans. 88 8 88
(31) yd. qr. na.
(30) C. cu.ft. 17 3 2 (32) hhd. gal. qt. pt.
7 78 13 2 1 73 43 3 1
16 24 23 0 2 27 3 O 0
35 127 27 1 2 15 0 3 1
29 10 29 1 2 162 0 1 0
Ans. 88 111 Ans. 111 1 1 Ans. 277 48 0 0
(33) bl. gal. qt. pt. (34) Jan. 31 (35) Jan. 31
1 28 0 0 Feb. 28 Feb. 29
1 17 0 0 March 31.March 31
5 2 0 0 April 30 April 30
1 2 1 May 1 layy 31.
7 2 1 June 30!8 3 0 Ans. 121. July 4
33 0 0
Ans. 186
An's. 10 0 0 0
(36) Cape HIorn, 55~ 58' 30/" S.
New York, 40 42 40 NAns. 96 41 10
~ —--- - ~ ~ ---- ~ —-- ---
56 KE(~Y TO RAY'S THIRD BOOK.
I SUBI3TR AC T IO N.
SUGGESTIONS TO TEACHERIS.-After the pupil clearly understands the method of borrowing when the lower denomination
is larger than the upper, his attention should be directed to the
method of borrowing in simple subtraction, as he will then be
better prepared to understand the rule there given.
ANSWERS TO QLUESTIONS IN COMPOUND SUBTIRACTION, TO WHICI
THIERE ARE NO ANSWEIRS IN THME BOOK
ARTI C LE 02.
(4) 8lb. 1 oz. ls13pwt. 7gr. (5) 4T. 2cwt 3 r 5 b. 15. S oz. 3 dr.
(6) 7 mii. 38 rd. (7) 1 ydcl. 2 ft. 11 in.
(8) 250 A. 38 P. (9) 2 sq. yd. 8 sq. ft, 104 sq. in.
(10) 55 da. a5 hr. 55 min. 55 sec. (11) 90 59' 56"
1bu. pk. qt. bu. plk. qt. Pt. lb. oz. pwt. gr.
(12) 4 0 (13) 100 0 0 (1s) 19 9 0 0
2 1 1 24 0 0 1 9 9 10 10
Ans. 1 2 7 Ans. 75 3 7 1.Ans. 9 11 9 14
lb. oz. pwt,. gr. b g 9 3 gr. lb. oz.dr.
(15) 8 2 11 0 (16) 3 3 1 1 12 (17) 46 9 0
2 4 7 16 1 7 0 2 18 19 0 4
Ans. a 10 3 8 Ans. I 8 1V 14 Ans?. 27 8 12
cwt. qr. lb. mi. fur. rd. A. R. P.
(18) 32 2 16 (19) 25000 0 0 (20) 146 2 0
8 2 17 100 1 1 86 2 14
AnIs. 23 3 24 Ans. 24899 6 39 Ans. 59 3 26
COMPOUND NUMBERS. 57
C. cu. ft. E.Fi. qr. na. yd. qr.na.
(21) 8 50 (22) 25 0 0 (23) 11 0 0
3 75 14 1 3 3 2 2
Ans. 4 103 Ans. 10 1 1 Ans. 7 1 2
gal. qt. pt. gi. da. hr. min. see. yr. mon.da.
(24) 63 0 0 0 (25)5 10 27 15 (27) 1857 3 5
51 1 0 2 2 4 13 29 1776 7 4
Ans. 11 2 1 2 Ans. 3 6 13 46 Ans. 80 8 1
yr. mon. cda. yr. mon. da.
(28) 1848 8 1 (29) 1832 6 28 (30) 55~ 58' 30"
1]822 6 24 1807 11 25 33 55 15
Ans. 26 1 7 Ans. 24 7 3 Ans. 22 3 15
quires. sheets.
(31) 71~ 10' (Y' (32) 500 0' (Y' (33) 20 0
36 6 30 10 25 48 5 11
Ans. 35 3 30 Ans. 39 34 12 Avns. 14 13
(34) 1 dozen dozen - 12X12_144
6 dozen dozen =144X 6 —864. a dozen dozen -144 — 2- 72
Ans. 792
AR,TICLE 104.- IJULTIPLICAT ION.
bu. pk. qt. pt. lb.oz.pwt.gr. lb. oz. pwt. gr.
(4) 4 3 3 1 (5) 2 11 6 (6) 10 10 10
12 6 10
Ans. 58 1 2 0 1 3 7 12 8 9 4 4
58 KEY TO RAY'S THIRD BOOK.
Bt 33 D gr. T.cwt. qr. lb. T. cwt. qr. lb.
(7) 4 o 2 15 (8) 8 2 (9) 10 3 14
8 9 7
Ans. 2 8 7 1 0 Ans. 3 17 2 8 Ans. 3 16 0 23
mni. fur. rd. mi. fur. rd. A. R. P.
(10) 208 4 16 (11) 30 4 10 (12) 130 3 30
15 12 4
Ans. 3128 2 0 Ans. 366 3 0 Ans. 523 3 0
cu.yd.cu.ft.cu.in. yd. qr. na. gal. qt.pt.
(13) 23 9 228 (14) 26 2 2 (15) 62 1 1
12 6 8
Ans. 280 1 1008 Ans. 159 3 0 Ans. 499 0 0
gal,. qt. pt. da. hr.min.sec.
(16) 123 2 1 (317) 365 5 48 48 (18)s 40 11/ 15//
5 12 8
Ans. 618 0:1 Ans. 4382 21 45 36 Ans. 1 3 30 0
doz. bu.pk.qt. m;.fr. r A. A. P.
(19) 144 (2a1)2 3 5 (22)3 5 16 (23)6 3 30
12 4 6 8
1728 11 2 4 22 0 16 55 2 0
3 6 10 7
Ans. 5184 Ans. 69 3 0 Ans. 220 4 0 Ans. 388 2 0
T. cwt.qr. lb. oz. dr. hhd.gal. qt. pt. gi.
(24) 8 2 14 12 13 (25) 3 2 1 1
11 8
95 0 12 12 15 29 1 0 0
2 2
Ans.9 10 I 0 9 10 1 0 9 14 58 2 0 0.6 3 07
Ans. 6 31 2 0 0
COMPOUND NUMIBERS. 59
T. cwt. qr. lb. yd. qr. na. hhd.gal. qt.pt.
(26) 16 2 24 (27) 37 3 2 (28) 47 3 1
119 89 59
Ans. 99 12 0 6 Ans. 3370 3 2 Ans. 44 52 2 1
(29) mi. fLr. rd. (30) C. cu. ft.
27 3 35 7 98
31 17
Ans. 852 0 5 Azs. 132 2
ARTICL 106.-] D IVISIO N.
(4) (5) (6)
bu. pk. qt. pt. lb. oz. pwt. gr. oz. pwt. gr.
5)67 3 4 1 8)14 8 16 16 10)8 12 12
Ans. 13 2 2 1 Ans. I 10 2 2 Ans. 17 6
(7) (s)
c-wt. qr. lb. oz. dr. Imni. fur. rd.
11)35 1 17 3 7 7)39 7 8
Ans. 3 0 22 0 5 Ans. a 5 24
(9) (11) (12)
yd. qr. na. bu. pk. qt. lb. oz. dr.
5)62 0 3 10)490 2 4 5)266 9 10
Ans. 12 1 3 10)49 0 2 10)53 5 2
Ans. 4 3 5 Ans. a5 5
(13) (i)
lb. oz. pwt. gr. da. hr. min. see.
6)339'7 9 18 6)114 22 45 35
7)56 7 4 23 9)19 3 47 35 5 rem.
Ans. 8 1 0 17 Ans. 2 3 5 17 2 rem.
2)X6+5=17 sec. rem.
~~ ~ —pl~~~opl.~~~- c~~i~,~~-~~ c~i-~r~~~l~~-~ —spm*~lu ~~~~~P~~-* —'Ic
60 KEY TO RAY'S THIRD BOOK.
(16) (17)
lb. oz. dr. lb. oz. dr. lb. oz. pwt. gr. lb.oz.pwt.gr.
23)1027 1 8(44 10 8 Ans. 245)17 7 6 6 (0 0 17 6
92 12 [Ans.
107 245)211(0 oz.
92 20
15 245)4226(17 pwt
16 245
91 1776
15 1715
23)241(10 oz. 61
23 24
11 250
16 122
74 245)1470 (6 gr.
11 1470
23)184(8 dr.
184
(18) (19)
bu. pk. qt. hu. pk. qt. imi. fur. 1rd. mi. fur. rd.
78)309 2 2(3 3 7 Ans. 319)788 4 9(2 3 31lAns.
234 633
75 150
4 8
78)302(3 pk. 319)1204(3 fur.
234 957
68 247
8 40
78)546(7 qt. 319)9889(31 rd.
546 957
319
319
RE-MARK.-For cases involving the division of a compound
number by a fraction; also, one compound number by another,
see " Ray's Ihigher Arithmetic."
Ob —-. —-.- ------ --------
COMPOUND NUMBERS. 61
ARItTICLE 107.
(1) (2) (3)
lb 3 D gr. oz. p-t. gr. A. RP. P.
1 0 0 0 0 8 10 12 104 2 37
4 2 0 13 9 3 5 87 1 38
8 9 7
7 5 2 7 192 0 35
3 1 2 14 26 3 0 40 1 0
5 12 >
Ans. 4 3 2 13 12 3)151 3 35
Or, add the two q 3an1i- 1 As. 50 2 25
ties together, and take 5)8 12 12
their sum from 1 lb. As. 1 14 12
(4) (5) (6)
mi. fur. rd. yr. mon. da. 350 30/ 0//
3 2 5 1845 2 3 Sub. 20 35 0
4 1843 6 27 14 55 0....___ i ______- 14 55 0
1 6 2u 1 7 6 Add 14 20 0
30 5
29 15 0
Ans. 54 3 0 Ans. 8 0 0 Sub. 25 4 30
4 10 30
Add 6 19 20
Ans. 10 29 50
ARTICLE 108.
(1) 15)18~ 25' 30// (2) 30~. 15=2. Ans. 2 hr.
Ans. 1 hr. 13 min. 42, see.
(3) 15)71~ 4' 0"// (4) 15)10~ 35' 0//
Anls. 4 hr. 44 min. 16 sec. Ans. 0 hr. 42 min. 20 sec.
min. sec. hr. min. sec. hr. min. sec.
(5) 37 20 (6) 1 4 56 (7) 5 8 4
15 15 15
Al.s. 9 20' 0// Ans..16 114' 0'" Ans. 770 1/ 0//
62 KEY TO RAY'S THIRD BOOK.
A.RTICLE 109.
(8) hr.min. sec. (9) hr.min.sec.
Time at C. 12 0 0 Time at N. Y. 11 0 0 A. M.
Add diff. 37 20 30~- 2 0 0 to be added.
Ans. 12 37 20 Azs. I 0 0 P. M.
(10) hr.min.sec. (11) hr.min.sec.
Time at Ph. 12 0 0 Time at N. Y. 11 0 0
Subtr. diff. 37 20 Subtract diff 1 4 56
Ans. 11 22 40 A. M. Ans. 9 55 4 A. M.
(12) 124~-80~ 42'-43 18': 43~ 18' — 15:=
2 hr. 53 mJin. 12 sec.
lir.min.see. N o T E. — In performing the subTime at W. 1 0 0 traction we can not take 3 hr. from
Subtr. diff. 2 53 12 1 hr., but 1 P. M. is the 13th hour
Ans. 10 6 48 A. 5M. from midnight, from which, after
taking 3 hr., the remainder is the
10th hour from midnight, or 10 A. iI.
ARTICLE 113.-FACTORING.
N o TE.-Most pupils in a short time may be taught to give the
prime factors of all the composite numbers, from 4 to 100, mentally, or by inspection. The great simplicity of the first 21 examples, renders it unnecessary to present their solution here.
(22) 105. -3-35: 35-+-5.7. Anis, 3, 5, 7.
(23) 168-. 284: 84.2-42: 42 — 221: 21-+ 37.
Atns. 2, 2, 2, 3, 7.
(24) 216- -2-108: 108- -2 —54: 54 -22=27: 27 — 3-=9:
9 —3=3. Ans. 2, 2, 2, 3, 3, 3.
(25) 330 — 2=165: 165-. — =55: 55 —5511. An.s. 2, 3, 5, 11.
(26) 110=2X5Xll: 210z-2X3X5X7. Ans. 2, 5.
(27) 105-3X5X7: 231=-3X7Xl1. Ans. 3, 7.
(28) 330-2X3X5X11: 390=2X3X5X13. Ans. 2, 3, 5.
(29) 231-3X7X11: 330-2X3X5X11. Ans. 3, 11.
~~4 Ib~~~ —- ----------------- ------— ~ —--------------- ~~~~~~~~~~~~~~~~~~~~~~~~~1.1
GREATEST COMMON DIVISOR. 63 1
AraTIcLE 114.
(1) 42=2X3X7: hence the divisors are, 2, 3, 7; and
2X3=6; 2X7 —14; and 3X7=21. Ans.
(2) 105=3X5X7: hence the divisors are, 3, 5, 7; and
3X5 —15; 3X7=21; and 5X7 —35. Ats.
(3) 20O2X2X5: hence the divisors are, 2, 5; and 2X2-4;
and 2X5l10. Ans.
(4) 24-2X2X2X3: hence the divisors are, 2, 3; and 2X2
=4; 2X3=6; 2X2X2=8; and 2X2X3=12. Ars.
A RTICLE 116.-RULE I.
(4) 42-2X3X7: 54-2X3X3X3: hence 2X3=6 gr. c. d.
(5) 70-2X5X7: lO= —2XX:l hence 2X5-10 gr. c. d.
(6) 105=3X5X7: 165=3X5Xll: hence 3X5=15 g. c. d.
(7) 60=2X2X3'X5: 90-2X3X3X5: hence 2X3X5
=30 g. c. d.
(8) 140=2X2X5X7: 210-2X3X5X7: hence 2X5X7
_70 g. c. d.
(9) 66-2X3Xl: 154= —-2X7X11: hence 2Xll-=22 g. c. d.
(10) 154-2X7XlI: 2980=2X2X2X5X7: hence 2X7
-14 g. c d.
(11) 231-3X7Xll: 273=-3X7X13: hence 3X7=21 g. c. d.
(12) 30=2X3X5: 42 —2X3X7: 66=2X3Xll: hence
2X3=6 g. c. d.
(13) 60=2X2X3X5: 90=2X3X3X5:150-=2X3X5X5:
hence 2X3X5=30 g. c. d.
ARTICLE 117.-RULE II.
(4) 495 — 42=11+33 r.: 42 —33=1+9 r.: 33 — 9=3+6 r.:
9-6=13 r.: 36 —3=2. Atzs. 3.
(5) 323 —247=1+-76 r.: 247-+-76-3+19 r.: 76 -19=4.
Ans. 19.
(6) 465. 285=1+180 r.: 285 —. 1801+105 r.: 180 —105
-=1+75 r.: 105_75=1+30 r.: 75 o. 30-2+15 r:
30. — 15=2. Ans. 15.
64 KEY TO RAY'S THIRD BOOK.
(7) 1274-. 532=-2-210 r.: 532 —210=2q-112 r.: 210~-112
=1+98 r.: 112 —98-1+14 1r.: 98 —. 14=7. Anls. 14.
(8) 759,337=2+85 r.: 337 — 85-3+82 r.: 85o 82
-1+3 r.: 82 —-3 7-1 r.: 3 — 1==3. Ans. 1.
(9) 69087 — 9873=6+-9849 r.: 9873 —. 9849 —=1+24 r.
9849 9 — 24=410+9 r. 24 — 9=2+6 r.: 9 —6= —1+3 r.
6 3-3=2. Ans. 3.
(10) 1814 —. 1776138 1 r.: 1776 —38-46+28 r.: 38 —. 28
-d1-10r.: 28 — 10=2+8 r.: 10 -8-1+2 r.: 8 — 24.
AIns. 2.
(11) 1815 —693 —24'29 r.: 693 —429=1-1-2641r. 429+264
-1+-165 r.: 264- 165=1*-99 1r.: 165 —99- 1+66 r.:
99 -66=11-33 r. 66 _ —33-2. Ans. 33.
(12) 3471 — 2145 —1-1326 r.: 2145 -1326=1+819 r.:
1326 — 819- 1-+507 r.: 819 — -507_-1+312 r.: 5073-.312
z=1-.195 r.: 312 —1951=1+117 r.: 195 — 117=1d-78 r.:
117 -78=1 —39 r.: 784-39-22. 1A1.. 39.
(13) 1701l7 —840-20217 r.: 840-.-217=3+189 r.: 217 —189
— 1+28 r.: 189 —28-=6+21 r.: 28 —21=1+7 r.: 21 —7
— 3. Ands. 7.
(14) 153452.66284-2+20884 r.: 66284 —20884
=3+3632 r.: 20884 —3632=5+2724 r.: 3632 —2724
1-f-90S r.: 2724 —908 —3. Ans. 908.
(15) 55-4-40=1+15 r.: 40 —-15-2+10 r.: 15i —10=1+5 r.
10-.-5=2. Again, 105 —-5=21. Ans 5. Or, by resolving
the numbers into their prime factors, and takingl, the pr(oduct of those factors that are common. 40X —2X2X2X5.
55=5X11. 105-3X7X5. Since 5 is the only factor
common to each of the numbers, it is their only common divisor.
(16) 154 —70-2+14 r.: 70- 14-5. Again, 819 —14
-58+-7 r.: 14 —7-2..Ans. 7.
(17) 168 —120=1q+48 r.: 120 —48=24-24 r.: 48 —-242.
Aain, 1768 —24-73+16 r.: 24 —16-1+48 r.: 16- 8-.
A s.. OS. 0, i20(-2X2X2X3X5. 168-=22X2X3X7.
1768-2X>2X2Xl3X17, hence, 2X2X2=-z, is the greatest common divisor.
d. — - ------------------— g
LEAST COMMON MULTIPLE. 65
ARTICLE 119o- RULE I.
(3) 12=2X2X3 (4) 6=2X3
30= X $5 10=-x5
2X2X3X5=60 Ane. 18=-s=x X3
2 X 3 X X3=90 Ans.
(5) 6=2X3 () 6-=2 A 3
8=gX2X2 15=-$x5
9 —=X3 35= x7
2X3X2X2X3 —72 Anv. 2X,3X5X7=-210 Ans.
(7) 10=2X5 (8) 9=3X3
12= X2X3 15=-X5
t15=,x 18=2XtX$
2X5X2X33=60 Ans. 24 — X2vX2X$
3X3X5X2X2X 2=360 Ans.
(9) 8= —2X2X (1o) 14-='X
15 3X5 21=-3X7! 12-xx 320=' X$XS
30o=xox 35=0xI
22X2X 3X5=120 Ans. 2X7X3X5=210 AnEs.
REMIARI.-When the humbers are small, or when they are
fewt Rule 1st is preferable; but in other cases Rule 2d is the
several of the same exsamples by both rules.
ARTICLE 120.-Ru-E, II.
(2) 2)6 10 (3) 3)15 21 35 (4) 3)9 12
3 5 5)5 7 35 3 4
2X3X5=30 Ans. 7)1 /> 4=3.6 A1.s
7)1 7 7
1.
32X5><7=1-05 An..
(5) 7)14 21 (6) 3)6 9 15 (7) 2)4 14 3)5
2 3 2 3 5 7)2 7 35
7X2X3 —42 Ans. 3X2X3X5=-90 Ans.
2X7X2X5 —140 Ans.
Ke.. - -5 _
1 66 KEY -O RAY'S THIRD BOOK.
(8) 2)6 - ) 1O 1' (9) These numbers contain
no common factor, there3 5 1 9 -ore their least common
g 5 )1 t) tj 3 llmultiple is their product.
1 1 1 3 7X1X13X3:3003 Ans.
X 2X 3X3 - 90 Ans.
(10) 2)G312 SQ 7 (11) 3)54 81 63
2)63 6 42 7 3)18 27 21
3)63 3 21 7 3)6 9 7
7)21 1 7 7 3 7
3 1 1 1 s3X3X3X2X3X7=-1134 Ans.
2X2X3X7X3z:252; Ais.
(12) 2)8 12 20 24 25 (13) 2)9 10 24 25 32 43
2)4 6 10 12 25 2)9 5 12 25 16 45
2)2 3 5 6 25 2)9 5 6 25 S 45
3)1 3 5 3 25 3)9 5 3 25 4 45
5)1 1 5 1 25 3)3 5 1 25 4 15
1 i I 1 5 5)1 5 1 25 4 5
2><2<X2X3X5Xs 5=60 flos. 1 1 1 5 4 ]
2X2/2X3X3X5X5X4 X
=7200 Ans.
(1)2)9 72 64 21 18 (15) 2)2 34 5 6 7 8 9
2 )49 36 32 21 9 2)1 3 2 5 3 7 4 9
)49 S 18 6 21 9 3)1 3 1 5 3 7 2 9
3)14 3 9 1 1 1 5 1 7 2 3
i-9 8 7 1811
7 1 8 1 1 A
2X2X2x3X3X7X7X$ —28224 Ans.
COMMON FRACTIONS. 67
COM1MOQN OFRACTION~S.
SUGGESTIONS TO TACHIERS.-A thorough knowledge of fractions
is indispensable to a proper understanding of all the most important and useful parts of Arithmetic, and is also of great
utility in almost every branch of science; therefore, no pains
should be spared to render the pupil perfectly acquainted with
the subject.
Previous to commencing fractions in written arithmetic, it is
desirable that the pupil should have studied Mental Fractions.
(See Ray's Arithmetic, 2d Book, page 86, following.) When this
has not been attended to, the instructor will find it advantageous,
at the commencement of each case, to prepare questions to be
solved mentally, similar to those which follow.
As far as it may be practicable, let the pupil be made to understand clearly the nature of every operation he is required to perform. In doing this, the examples and illustrations derived from
mental arithmetic will be found of great value.
ARTICLE 138.-CASE I.
NOTE.-For mental exercises, and the method of illustrating the
subject, see Ray's Arithmetic, 2d Book, page 95.
The only difficulty in solving the examples in this section, consists in finding the greatest common divisor of both terms of the
fraction. The following are the greatest common divisors.
(2) 6. (3) 30. (4) 6. (5) 15. (6) 30. (7) 14.
(8) 16. (9) 5. (10) 18. (11) 14. (12) 15. (13) 97.
(14) 111. (15) 101. (16) 23. (17) 117. (18) 199.
ARTICLE 139. —CASE II.
From the 1st and 2d. questions in this Article, it will be readily
seen that the reduction of an improper fraction to a whole or
mixed number, involves the same principle as reduction from a
lower to a higher denomination. Thus, to reduce 17 pecks to
bushels, and to reduce 17 fourths to units, or 17 to a mixed num
68 G~s KEY TO RAY'S THIRD BOOK.
ber, all involve the stie sme principle, and are perfbormed in the same
mainnCr.
NOTE. —For mental exercises embracing the principles of this
case, sce Ray's Arithmetic, 2d Book, page 90.
The examples in this Article being all performed by division,
the exhibition of the manner of doing the work would consisl.
simply in writing dcown the question and then the answer; thus,
Q. (6), 3- =8
ART I L40.-CASE I1I.
From the first six examples it will be seen that the rednuction
of a whole or mixed number to an improper fraction, involves
the same principle as the reduction friom a higher denomination
to a lower.
NOTE.-For mental exercises embracing the principles of this
case, see Ray's Arithmetic, 2d book, page 89.
(7) 8X4+3-35. Ans. 3. (8) 12X5+~3=63. AIs. a
(9) 5X10d-3=53. Ans. -. (10) 15X6+3=93. Ans. 93
(11) 26X24+13=637. Ans. 2ms
(12) 8X2+1r=17. Ans. 27. (13) 5X4+1-=21. Ans. 2.
(14) 3X55+17=182. Ang. 1 82
(15) 46X8+5=373. Ants. 383
(16) 21X583+117-12360. Ans. 1 260 O
(17) 1X1000+999=1999. Ans. T1000
(18) 14X71+6=1000. Ans. 1 ~o~
(19) 10Xlll+l=111.1. Ans. 11 t1111
ARTICLE 141.
4X7 28 8X9 72
(2) 4=- — Ans. (3) 8=- -.Ans.
1X7 7 1X9 9
19X19 361 37X23 851
(4) 19 —- Ans. (5) 37=- Ans.
I X19 19 1 X23 23
COMMON FR,tCTIONS. 69
ARTICLE 142.-C-A- SE I V.
(7)..4- 13; then, 9-.7 13_ i3 9
ARTICLE 143,
(z2) x - x -=- Ans.. (13) -X - x Ans.,499 9 4 t 9
7 3 8 t 8
/, 4 4 1~ 4 4
6 13 / 1$ 1
1 7 X$ I4'$ 2
21 $ 42
1 ~ I
(18) -X-X-X-X-Xx —=$ T$CLE 444.- - E 0 16
1X4><5 20 1X3X4 12 1X5X7 35
(5) > (6) (7)
2>X4X5 40 2X3X~4 244 2X5X7 70
3X2X5 30 1X2X4 8 3X2>(7 42
4X2X5 40 3X2X4 24 5X2X7 70
4X2X4 32 1X2>(3 6 6X2X5 60
5X2X4 40 4X2X3 24 7X2X5 70
70 KEY TO RAY'S TIIIRD BOOK.
1X3 X4X5 60 2X5X7X8 560
( 2s 3X4X5 12a 3X5X7XS 840
2X2X4X5 80 2X3X7X8 336
3X2X4X5 120 5X3X7X8S 840
3X2X3X5 90 3X3X5X8 360
4X2X3X5 120 7X3X5X8 840
4X2X3X4 96 5X3X5X7 525
5X2X3X4 120 8X3X5X7 840
6X7X13 546 (11) - of s-. (12) i- of -=5.
1 iX 7 X13 1001 7,, 3v 74 XllX13 572 3X3X1_ 9 5X7 35
7 X11X13 1001 SX3X1 24 SX7 56
8 X11X 7 616 7X8X1 56 IXS s
13X. 1X 7 1001 3XSX1 24 7X8 56
3X8X3 72
I.XS><3 24
ARTICLE 145,
In solution of questions in this Article, the multiplier of both
terms of each fraction is supposed to be obtained mentally, by dividing the largest denominator by the denominator of the fraction
to be reduced.
1X2 1X4 l X2 1X5,
(2) (3) (3) () 3X2 (5) 2X5
4 —a 1X<2 1X3 2X2
4X2 2X3 5X2
1_d 553 5-
c~, —--— ~ —---------------— ~ —6
COMMON FRACTIONS. 71
( X4 (- X = (3) 3X'4 6X4 o
5/2;~. 5 >\2 X3
6X2- 7/2 32- 42.2 —-4'
- J'7 91 o 11 I II I
T-2 —T7~ J 4 — 14 1 r I 6 -' 6
ARTICLE 146. —CASE VI.
(4) Least common multiple of 2, 3, and 4=!2. (Art. 119, 1920.)?6, iS =4,
4 4
1 6- \46,1X- 3X1-3 As. _4 3.
(5) Least common multiple of 3, 6, and 9 —18. (Art. 119, 1.'20.)
18, 11-, U 182
1\/6-69 /13=3, 1\/2=2, AnTs..5
(6) Least common multiple of 2, 4, and 5:20.
20:0o, 4~=5,?50_4.
1X><10=10, 3X5=5, 4X=16G. Als.,,.
(7) Least comm1ou multiple of 8, 5, and 10= -40
4 0-5 4o=', 4o__.;
3X5=15, 4X8=32, 9X4-36. Ans. -i 4.3.
(8) Least common multiple of 3, 4, and 8 —24.'2 4=g Z-'-6, =r __3;
2X8 —46, 3X1 =8, 7X=21 Ans., li
(9) Least commnon multiple of 4, 8, and 9-72.
74z 18,s9,: $;S
3X18=54, 5X9-45, 5X8-40. Ans. 4, -, 4-0j
(10). Least common multiple of 7, 14, 7, and 2S —28.
v -4 28_, ~ t8=4, Is-=;
2X4 —8, 5X/=10, 3X4-12, 11><1 —11..Z!1. 8 I 0 1 2 1 I
Ins..-,,.
(11-) Least common multiple of 2, 3, 4, 5, 6-60.` 390, 6;~- 20, 5O0-1 5, 1 -120 60-2 10:
1X30=30, 2/20: 40, 3X15-45, 4X12=4s,
5/Xo=50o. rins 0 40 45 4, 580
~5Xl~O-50. An"s. ~fGap, Ug, 40 0 9o, 0U-U.
72 KEY TO:RAY'S THIRD BOOK.
(2 ) 1 20 5.f2 2 ]2
Thle fractions then are, 9, 9, -, and (I'
Least common multiple of 4, 8, 6, and 5=120.
12 4 12 ___0
120=30, 120=15, 2=20, 1~:o241;
3X30=90, 7)\15=-105, 5)(20-=100, 12)>24-288.
9a0s. 90, 1 o 100 288
SUGGESTION, —In working the questions, until the pupils are
well acquainted with the operation, they should be required to
putdclown the work after the manner of Ex. 1, Art. 146, Arith.,
as this presents the reason why the values of the fractions are
not altered.
A.:RTICLE 147, -ADDITION.
(2) Ar,. =(3) A ns. 902 (4) lls. 7 2.
(s) As. 16-2. (6) Ans. t-1. (7) s. 37
T IcLE 148.
(5) 4+17+29+-48+76= 174; hence, sun-l s71=2 A2j AS.
(6) 11 + 9+29+101=160; hence, sumn- 1 4 - ~- 9 A.
(7) 999-1-888+777=2664; hnce, su,- -8-4 o=a
— 2853 1.,.
(8) ~+'-, 1,, o _9,= 24 1 _ 1 I
(9) I A+fS.
(3zO) 2 1 + 2'~2 4a
(10) 8+1 _ 12. -, 88! 1424 24 1S.
(12) 8 1 +4 3 3 1 n Ans.
a aI 1.4
('V i +-1Y - + jo= 1o4 +5 + 4n 4+1 G 37 i
1= 629 -5 4&' An1s
( 15 ) 1 +S.=3+1 5 9 4
COMMON FRACTIONS. 73
(:L8) I+- 4 off'; _-o8=15;_
5315+2 lo~ -- Ans.
( )i-4 1 + _ _ 1304i = 3 4-_;
16 —12+8+2-38; 38+2-1=4 0 2'LD Ants.
I I 3 __ 3 0 1 5
(l9) -of 3 1- 6 - 4 O1f 5 1I+- 3 -4
-3 Ans.
(l9) -D of 1'= of 3- 3 o f2-of 5-= of of
~1, 3 5 + 1 8 10+6 _ +3 6
=2 3 1 -0 Ans.
q 1_ rl__ 37 3+ -3 0= 1 8O ~0 + gA 1 3 - I 6 8
T 332o o -- &B0 0l Ans.
ARTICLE 149.-SUBTRACTION.: - 1 o I
RT TICLE 150.
(5) * —_-5 - 1 ALns. (6) i A -- -.- t
(3) 3I__= 19 4 7 AnlS.
(9) 2- 9 8 - Ans.
(10) -1 1 = _ _4 Ans.
(11) 8 3 -8- 4 -4' — 49 A 3.
(13) 45`10 -—'- A s — s.
(18) 13 - =4n 2 ns.
(19) 4 T__ 1= 9 - 9 =1_ A s.
4) o8 3 I — 9 - 7 1 -~:Ans.
(12) 57 -1-2-54 - 91-'1 Ans.
2139) 5' 141 54-4=12 A2s.
(21) 1 -__1 ____q 2 3 ns.
74 KEY TO RA.Y'S TIHIRD BOOK.
(22) s8 -- — 7 q 74t Anls.
(23) li -4 99 41_ 58=6 Ans.
(24) 8-33=713 3 t4-n0 Als.
(25) of ~-O-6;'x of Q =4; 6-4=2 Ans.
of 3_ 3 f*.3 5 2 O 7 O -7 4Aui7s.
sz 6 6i GT9i
(27) -2of fI 9 —8 41 4i7 57 38-171 152-19
=-1 Ans.
-- T'-2 l ~ 74-f
(28), ~ 1; 5 --
(28) of 21e3 of 72o7-l1h3; 1 35 63=7
=143 Ans.
1U L T I PLI CATI 0 N.
SUGGESTIONS TO TEACIdEnS.-When the pupil comes to the multiplication of fractions, he will probably be surprisedl, at first, to
find that the product is firequently less than the multiplicand,
while in the multiplication of whole numbers, it is always
greater. The pupil should be reminded that the multiplier shows
how zmany timzes the multiplicand is to be taken. Thus, if the multiplier is 2, the multiplicand is to be taken two times; if 1, it is
to be taken one time; if, it is to be taken half a time, etc.
Hence, if the multiplier be less than 1, the product will be less
than the multiplicandc also, the less the value of the fraction by
which we multiply, the less the product: when the multiplier
becomes infinitely small, the product becomes infinitely small,
and wh!n the multiplier is 0, the product is also 0.
It should also be noticed, that to multiply by 2,'-, etc., is the
same as to divide by 2, 3, etc.; also, that to multiply by 2, is the
same as to multiply by 2, and divide by 3, etc. Indeed, it is a
general rule, that when an2y result is obtained by dividcig one nLmber
by another, the same'esutlt may be obtained by mzultiplying the given
number b7y the divisoa svrittenz in a fractional form, aCnd inzverted.
ARTICLE 151. —CASE I.
(14) X11-12=- 132 7 37 Axns. (15) \9,<7- 3 Ails.
T i 3~ 17` 172
rps ------------------------------ 9I
COMMON FRACTIONS. 75
o 25
(16) WX9 -=1Z Ans. (17) -5=32 Ans.
36_
9X21 9X3 27 An,(18 ) -__X21.......
14 2
(19) { X10 X 1 2= — 23-71 Ans.
15 3
(20) 5X9=5X95 Ans.
9
(22) 18-X8= — X8+18X8=6+144=150 Ans.
(23) 16'2X3=- X3+16X3 —2+48=50 Ans.
(2a4) 10-X7-X71X7 7:4,9 -70-75-, Anzs.
(25) 10-X9 —-X99-10X9=.-1490 =97'- Ans,
ARTICLE 152,-CASE II.
(12) 2SX= 4 =4X4=16 Ans.
7
(13) 36X7=36X7=4X7=28 Ans.
9
(14) 50X —0 XO-5X9-45 Ans.
10
(15) 31X X3 2 lX==202 Ans.
3 3
(16) 29Xl 29X_3 7_ =2 n As.
437
(17) 37X4 -- 37X4'=209 Ans.
(19) 25X8 o=X 453=5X43=215 Ans.
(20) 45X64-45x='- 12 " 981 AsS.
(21) 55X93= —X6-6=30 ~5184 Ans.
(22) 64X8=-'6-X7~'l=78X71=568 Ans.
76 IKEY TO RAY'S TItIRD BOOK.
ARTICLE 153.a-CA-sE III.
13X2
(9) 8- 35x 1- 47-= 4 s.
(:O 10-) - L- T%-1X 9 Xl9 171 =15 As
~11 I 11
(JJ) 25 2X-/ t6~4 Ans.
(12) 161 33 33Xv3 —_1089 —2721.Ans.
2X5
(13) -1 of 4-; &- of 5 —5;x A — s.
3X3
(14) 7of? o 3 of13A, o; 3 o 7=7
3 T3 z —2 3 —;
6X-7=13 -4 Ans.
(15) g of 9 -- of 37= 1; i of 17
1l\/h34 o3f17 6 29=785 Als.
16 ~ __3X17
__ 3Xll a186 —1 a_1of?
(136) - of 1 1 0 of -; I, of -r 9
5
a 3 5 10
1 5 -O 17 Ans.
(l1) 7X <0 X X 9 AQ.
(19) X X l X - X<X1= il Ans.
~~i~94 3 1
COMIMION FRACTIONS. 77
(20) X-X-X- -X - XX — = Ans.
8
(21) X-X X X X-=24.Ans.
ARTIC L E 153.b
2
$ 7
(l) 1)1 -;2 2 -; -Xi-4= 4 ct2. At s.
(2) 3 X 3=2; 2X55= o= —3; 2 X 7 —4=4];
$ 13 8
3X m 1P 4 2 3 2 X 23-2?, 4 6
S 2
(3) 44- 4: ct,. 0,os,.
^ ~-= 16 or-s. Ans.
15 4
(4) 18-75=; 3a=1-6; AX = $60 JS.
(5) 3-3=.; x0=~1 54.
4
4 4
(6) 2,1 =( >~2 14Ans.
10 2
(7) 5 -- 10 "-i - X=,- o $6'2 Ans.
( — ~ 0 Xt 3= —; 3X( 1) 5 Ta7-= 1 X 3=4 -3 —=425 mi. Ans.
(8 5',E 74 4
78 KEY TO RAY'S THIRD BOOK.
2
(9) W Of 3 X $=x Ans.
11 5
$ 5 $$: 2' 7 ~ 5x x5 97X4
ARTICz LE 154_.-CAsOf I.
tient will always be less than the dividend, as in the division of
whole numbers; thus, to divide ~- by 4, we really find what part
of a time 4 is contained in 3; we also find the -- of ~, since to divide by 4, is the same thing as to multiply by I.
(10) P, -5it s (i 51 -1 1 >'I;lS
In the solution of the following question s we have not marked
the extra space occupied by this kind of type: in op rations performed on the slate, or at the blackboard, the canceled factors
should always be maiked in owder to avoid mistakes.
(s~) 1.49.34 4
(7) y. A4 is. (8) -tt a =l-7 Aby.I
(9) -4 -t4 Ans. (10) - -T0
(12) 3 A ns, 5 5 A. 5 —~S(13).As. 1.: 7-=15 As
(14') caneld 9= Ars. (15) Le-2o4 24 8eo Ac S.
(16) 5. A=3oss.. (17) 12 - 8q; - -3-'.i? AAs.
o 4o
(19) 475- - t-; a_152 7=; _-_ _-3, 4. Ars.
392 C' 4- Ans. (1) 5195
(20) A3O ns. (n) 44Xyi8 4= Ai3s.
(2C) 313 9. l;3 99'. T-33 9 -' 5 -i I A X
K~~~~1,_ ~
COMMON FRACTIONS. 79
ARTICLE, 155.-C0ASE IIL
(4) 4 2 10 Ans. (5) 16 -16X4 4=211 As.
24 3
(6) 8 8Xl54X15-60 A8
2
(7) 6 -=6X=2X77=14 Ans.
(8) 13 - 3 -13X 5-212 Ains
(9) 21..11.X =3X1=33 Ans.
(3-) 2 12; 6, X= 5 X= = 2 yd. Ans.
12
(12) 34 30 — -2X 4-S lb. -Ans.
15
(13.) 4_ 31; 50 — 31__350-X7-30~ —i I A _7s.
31
49 -
-PupRTICE ofte t.-CA losSE. y i
RE1IAA. RTupiCs are often at -a loss to undlerstandl why it is
that the quotient of one proper fraction, dividel by another, is
sometimes a whole number, or greater than unity. The teacher
should be careful to explain this subject, by means of familiar
examples, such as may be found in Ray's Arithmetic, 2d Book.,
Section xxi. It should also be shown, that if we take any dividend, and divide it by different numbers, that as the divisor becomes less, the quotient becomes greater; so that, by making the
divisor sufficiently small, the quotient may be made as large as
80 KEY TO RAY'S THIRD BOOK.
we please. Thus, the quotient of I divided by 1, is 2; by,, is 4;
by T1Oj is 8; by ~0-oUUU1 is 2000000, etc. It is on this principle, that mathnematicians, say, that the quotient of any number,
divided by 0, is infinitely large.
~(5)-4- 43 *l. 1(6) 2 X /= —X 2 AtS
(, ) ~ ~=-' X2= A.ls. (8) _.,2,,=5X3. A4_.
(O) 7 27, L5. ) X1+ —40 Ais,
(3La4) 9.- 4j 958 8; a93 3 2 - 519X2 i
41
4:!.3 ) 2-4 —9; 7 -12 4.19. a =4 1 5-= _ ___-__o___- 41
(3.4) e * \/o= U=1=25 Ans..
(s:.6) 51.f 4_1 v1 4 3_ 3 1- T 4 01 5
ARTICLE 167.
(2) i 2 4 1. I__v Ans._'_
( ) -- 2;1 5'V - I -1 4 A.
(3) 8-.7v 457X1 =X34 _4 An.
(i) 8X. o Ans.
(6) 2* 4. 3 fv 3r- v 3 pa=- Ants.
(7) 12,=; 4j-" 7X-J X4='9!3, A??,,.
Tv — ~XT G 3 3 - 3
(8) 6 S'O 5 In solving this example e we fincl 12-,l3 i
I - f ~, 1 andc 8- o45. When the latter is;jjx:$ S ine ted, 41 becomes a divisor, and, therefore, is placed on the left.
6 =5.Ans..
6
COMMON FRACT.)I..
ARTICLE 158.a
2 -. —- A s Ans.
1
I ($) "' 97.S, 7747 t
2(9) 7 =; 97=;
9 = o; 1- 113; ) 2 5Io~ g 3 3
19
47 I 33
(p (3 2 -3 33=].1 Ans.
47 19X2 — A'.s
A R. T ICLE 15g.
(2) 2, 93 37 l. An.
(3) 2- 4
Je/3 X. 6 _ 1 1 i-\/ - ~
(4) 4-2 - 1 1 33 >< 1-3_ _1/- yd. As. 1
45 3 lf,.-,x
82 KEY TO RAY'S THIRD BOOK.
(5) 10. — - 11v8_8-o 26' Ans.
(6) 3 of 1 Of =.4; 3= o- 24=;
b24' 924-,, 4- 61~ -n7 1 4/\3
2 1 3 3 3-
(7)'A4' of 27= - 5,of5=10; 1 of 21-1=3G Of 8
\1j=_51; 10. 11X- 8_=8 129 Ans.
(8)'. — X3 2 2_= 12 X-I2X =72 2!~2.
RTICLE 59.
(1) (2)
$ cts. cts.
16.06-4 9.12-1
9.122 4.4335*434i 1)O97.l4
5.435~ ~ 0.370
Ans. 833.43- Avf s. 615.56-j
(3) (4) (5), cts. $ cts. $ cts.
50.25 32.31J 5.81'
8 4
(10) 1.8 11. 129 1 3.i9
Als.`$209.061- As. AL7.18 Ans. $4.621
(6) 121X93-10 \8 +4 t 112 cts.=4$1.1254 Ans.
(7) 61X21=26+511=1-1- cts.=-$1.31 4 zs.
(8) $3.18:3X15:47.70-. 114-$47.81-~ An.s.
1(9) 6,X5 1='.. 1.1 o375- 343-3 cts.-=$3.433? Ans.
(13) 17>? ---:1., -:1)X 6 -- cs$2.342,.
(3.0) IS 3-5 ~ 13 2 — T7 X >..-5 25X9- -225 cts_=$2.425 Ans.
(:12) 8$3.37v'',s4 S 7 5-I=3459- ts.=$34.59(13) 17 3g =3: 375X =125X53=6625 cts.=-$66.25 A4ns.
3,> 3 3
COMMON FRACTIONS. 83
(14) 225 -18: X — — 3-X4 12 yd. lns.
(15) 5S 4l-, 137 1_2 3 25 5 3X1-X —-=154 bu. Ans.
(16) $11.56}-~-5-$2.31k- Ans.
(17) $31.0614 — 7=4.4s3 Ans.
mi. fur. fur. rd. rd. ft.
(18) 5X8=40: 40X40=1600: 1600X16-=26400
ft. in.
26400X12=316800 Ans.
mi. fur. fur. rd. rd. rd. ft. ft.
(19) 2X8=16: 16X40+2=642: 642X161+2=10595 Ans.
ft. rd. ft. rd. fur. rd. fur. mi.
(21) 15875 —.16=-962 2: 962 —. 40-24 2: 24 8=3.
Ans. 3 mi. 2 rd. 2 ft.
in. ft. in. ft. yd.
(22) 142634 —12=11886 2: 11886 —. 3=3962:
yd. rd. yd. rd. fur. fur. mi. fur.
3962 —51 —720 2: 720 —. 4018: 18 — 8-2 2.
Anjs. 2 mi. 2 fur. 2 yd. 2 in.
mi. in. f. in. in. in. in.
(23) 2=126720: 2 8=32: 126720 — 32=3960 Ans.
mi. in. ft. in. in. in. in.
(24) 65=4118400: 9 2=110: 4118400-. 110=37440 Ans.
A. R. R.R. P. P. P. sq.yd. sq.yd.
(25) 1X4+3=7: 7X40+16=296: 296X301+25=8979 Ans.
sq.yd. P. sq.yd. P. R. P.
(26) 7506-.301=248 4: 248-.+40=6 8:
R. A. R A. R. P.sq.yd.
6 —4=1 2. Ans. 1 2 8 4
ch. 1. 1. in. in. in. in. in.
(2 7)5 15=515 7-9 oo =T7f9: 19X515=47OCso8 0=40784 Ans.
84 KEY TO RAY'S THIRD BOOK.
P. P.
(28) 40z8=%1 ~: 1X32=81X16=1296: 1296-. 40
R.... A. A.P.
-32 16: 32-.-4=8. Ans. 8 16.
da. da. da. hr.
(29) 3651-X4=1461: 1461X24=35064 Ans.
hr. da. hr. da. yr. da.
(30) 914092 —.24-38087 4: 38087. 365 —104 101:
yr. cen. yr. cen. yr. da. hr.
104- I100=1 4. Ans. 1 4 101 4.
da.
(31) 238545 31 —7695: 7695 —3651-30780 —1461=21 yr.,
and 99 quarter days remaining, which reduced to days,
by div-iding by 4, makes 24- days. Ans. 21 yr. 243 cda.
ARTICLE 160.-CASE I.
lb. OZ. lb. oz.
(3) 1* Xl6=-=vg AAns. ( 1) Xi12-=-= Ans.
yd. qr. qr. na.
(5) 1-AX4i=: -X4=4 Ans.
A. R. R. P.
(6) T -6IX4d —= 32UX40-= Ans.
ct.
(7) 3-5XloO=-.~-3 Ans.
da. hr. hr. min.
(8) Tk4 X245='1 _: o-X60=6=l Ans.
bu. pk. pk. qt. qt. pt.
(9) y3UX4 —: oX -3 X2 Ans.
ARTICLE 161.-CASE II.
na. qr. qr. yd.
(3) 4 4-4: - 4-.i -' Ans.
gr. pwt. pwt. oz. oz. lb.
( -) ~24+=Q: * o=20z: ~ 1+12=-gI Anls.
--------— 6-. -gou ----- — g ---— U 9 ----
COTMMION FRACTIONS. 85
D 3 3 lb
(5), 3 — 4 4 8- - 12=B:~-g- Als.
9 -~, — 2 I 1;,1' - -~ 4 Ans.
pt. qt. qt. pk. pk. bu.
(6) 3- 2=- 8= —' 4 — 3 Ans.
oz. lb. lb. qr. qr.. ct.
(7) 4 16=1 25- 4-1- As.
in. in. na. na. qr. qr. E.En.
(8) 5. 2 X3 4 __ 4 4 a 1 I 5 AS.
min. h'. hr. da.
(9) I 60-= 2 -2 a. 24= ~ -- Ans.
dr. oz. oz. lb. lb. qr.
(10) -7 T 16=41 7 I T 1= T-25- F -o- Aus.
ARTICLE 162.-CASE III.
cts. cts.
(3) s of 100 3~~=60 Ans.
fur. fur. fur. rd. rd. fur. rd.
(4) - of 8='1 —3 =5 of 40=8. Ans. 3 8.
oz. oz. oz. pwt. pwt. oz. pwt.
(5)4 of 123-=:-9: ~ 4 of 20=-~ —=12. Ans. 9 12.
oz. oz. oz. oz. dr. dr. oz. dr,
(6) 4 of 16=?74=9, of-' 16=1-6=2. Ans. 9 42
R. R. R. P. P. R. P.
(7) 4 of 4=4=21: 4= of 40=-20. Ans. 2 20.
hhd. hhd. hhd. gal. gal. gal. qt. qt.
(8) of 4-=3: of 63=31: = — of 4=2.
Ans. 3 hhd. 31 gal. 2 qt.
ARTICLE 163. —CASE IV.
hr. min. min. da. min.
(5) 13 30=810: 1=1440:'s=-40J Ans.
fur. rd. rd. m. rd.
(6) 3 25=145: 1=320: - -45 —49 As.
-- --- - I- I -3 - --
86 KEY TO RAY'S THIRD BOOK.
ft. in. in. yd. in.
(7) 2 8 —32: 1-36: 3-= Ans.
(8) 49G_- Ans., that is, 96 pages are - of 432 pages.
mni. fur. rd. rd. mi. fur. rd. rd.
(9) 15 3 3-4923: 35 7 7=11487: -4t9 2g993 AZs.
A. P. P. A. R.P. P.
(10) 168 28==26908 37 2 14=6014: Ans.
oz. dr. dr. lb. dr.
(11) 7 17- 10124: 1=4A,3O04 -024__ Ans.
(ll-9 79 2 3: 4 —qt. pt. pt. bu. qt. pt. pt.
(12) 2 1 3_136 1 1 I 3203 Ans. 12-.
yd. ft. in. in. yd. ft. in. in.
(13) 1 1. 54 838G 3 2 I8 -9sG6 10846
f1 —- f - 1 -3I-a
3836 191 8 Ans.
AR TICLE 164.
(2) (3) (4.)
hr. min. da. hr. mlin. da. hr. mlin. sec.
da.=16 0 Q wk.=l 18 0 wk. - 4 16 0 0
- hr.= 45 da. = 6 0,da. = 13 20 0
Ans. 1645 41 hr. = — 15 - hr. - 40 0
Ans. 2 0 15 3 min.= 40
Ans. 5 6 00 40
(5) gal. qt. pt. gi. (6) A. R. P.
gal. = 3 1 1 2 P. 9 2 34q
3 hhd.- 5 1 0 0 - A.- 2 262
Ans. 6 0 1 3 28q
Ans. 2 1 91(7) (8) (9)
hr. min. sec. cts. pwt. gr., da.=18 40 0 $ —6 oz. -12 0
Ts hr.== 3 20 4 d..= 7- -- pwt.= 21
Anis. 18 36 40 Ails. 55 Ans. 11 3
__ ~~~~~
CO-IMMON FRACTIONS. 87 1
(lo) (11) (12)
fur.rcl. ft. in. hr. min.sec. qr. na.
mni.-l 31 1 10 - da.=20 34 17- - E.E.=4 121
7Tfur.= 25 7 6 - hr.- 40 0;I yd. =1 0q
Ans. 1 5 104 4 Ans. 19 54 174 Alns. 3 01 -
Or, 1 5 10 10
ARTICLE 165. —PROMISCUOUS EXAMPLES.
299 8 92999X9/ 1 Ans.
29999X19
(2) _ 4+ 8 =27 +43
- 36 9 3 6;. — a 3 -: + =3 — 5, 1 3 +
9 24 14
(3) 43- 2 1 2 1 ~ -1 1 8 3- 3 4
2 T.5_{)_ 2Ma 6 - 3 —=9 of V-=- of
23 6 6 127 3 9 8 3 o 3
_7 29 7 87 28__ 5 9 21 AnS.
(4) of 7 - of - 7 ~ -+7 _. - o_. %
( s 6. Ans.
(5) 52 %1-. 31a 1 Xt 8 4 5;
4 + 3 5- + 8 - 4 T _2 - 83- A s.
( W) 1{(_ -_, —Go ~- - - - (r71 —0o0 16 Ails.
(8) If 4 of any thing are taken from it, there will be 4 left;
if 16 is 4-sevenths, 1-fourth of 16, w-hich is 4, is 1-seventh,
and if 4 is 1-seventh, 7X4-28, is the number.
(9) If to any thing there be added 3 of it, the sum will be
+, — 10; if 20 is 10-sevenths, 1-tenth of 20, which is 2,
is 1-seventh; and if 2 is 1-seventh, 7 times 2, whicllh re
14, is the number.
(10) - of = 45, and 4 -_4_ part' left..
Or, the part left may be found thus. If he sell -4 of his
share, he has - of it left, and 2 of 8 ~-4.1-.t
i52 of $900-4, —2=$375 Ans.
7,h.... - = -...............h -- D
88 KEY TO RAY'S THIRD BOOK.
(:L) j of = 19441 7 o
3 T —-~ _iU _ 9, 7
Having 7-thirty-sixths, if we divide by 7, we get 1-thirtysixth; then, multiplying this by 36, we get the value of
the whole. Hence the answer will be found by multiplying the g6* of the value by 3-60 1 —00-X3a 6
= $10000 Anis.
(2) - of 2=-4; 3-=: Ans.
(13)..= ASS..|
16X23
(14') I 1 L9~t'. L 333 148 312 -793
2 7 9 -3 I 8 00 793 _ 1007 A23.
(15) 4j% -;E 1 - of __ of 4 — =13; 1 31
T4 T2 TT-1T140 i T I 0
2X4X2 [Ans.
5 4_ x7 1 4 5. I=056-9\ 1 1 - 1. I
4 1 1_i24 1 C 59
(1'7) If a man walk 2044 rods in 7-fifteenths of an hour, lhe
will walk 1-seventh of 2044 rods in 1-fifteenth of an hour,
and 15 times this distance in an hour; this, multiplied
by I14, will give the distance walked in 1 1 4 hours. Thus,
- of 2044=292, alnd 292X!5 — 4380-rods walked in an
hour; and 4380X1- 4-1 80X i — 8468 rods Ans.
O0 0-44y1X > 2044X29 292X29-8468 Ans.
Or0, ~.o44x] _? _8468 Ans.
7 1
(18) 11 feet- 15 inche 3- 0 0 2 Ans.
(19)1 3~'~:: — ~~c~I ~ 3 3 4 5 _o
-9%) 3 1 _L 11 + 5 Ans. — 3 4 and
$ $ $
(20) 8 of 2400=5>300 —1500: 1500+500=2000:
If { of B's money is $2000, ~ is; of $2000, which is
$400, and the whole will be 4 times $400, which is
$1600 Ans.
(21) Since 2200 is five-twelfths of what the elder son received,
one-fifth of this will be one-twelfth of what he received;
and twelve times this last sum will be the whole of the
elder son's share:
DECIMAL FRACTIONS. 89
of 2-2 —440, and 440Xi125280-elder son's share;
for a similar reason, dividing this last sum by 16, and
multiplying by 35, will give the whole estate:
y U of 5280=330, and 330X35=11550, whole estate:
Youngest son's share 2200 11550
Eldest son's share.. 5280 7480
7480 3)4070
Eaach daughter's share-$13562
DECIMAL FRACTIONS.
SUGGESTIONS TO TEACHERS.-AS it is impossible for pupils to
understand the-primary rules without first becoming well acquainted with Numeration and Notation, so in studying decimals it is equally important to become familiarl with their
numeration and notation; pupils should, therefore, be drilled
at the black-board until they can read and write decimal fractions with ease and accuracy.
A R T ICE 1 75.-N UM1E RIATION.
To uE REAnxD.-lst Colulmn. (1) Five tenths. (2) Six huncdredths.
(3) Three thouZsandths. (4) Twenty-eight thlousanZdths. (5) Thre[
hundred ancd forty-one thousandths.
2d Column. (1) Three ten-thousandths. (2) Six hundred and
twenty-five ten-thousac.ndths. (3) Two thousand three hundred
and seventy-four ten-thousandclths. (4) Two thousand and six
ten-thouszndclths. (5) One hundred and four ten-thousandths.
3d Column. (1) Four hundred-thousandths. (2) One hundred
and thirty'-seven hundred-thousandths. (3) Two thousand three
hundred and seventy-six hundred-thousandths. (4) Thirty-one
thousand four hundred and fifty-six hu7ndred-thousandths. (5)
One thousand and seven hundred-thousandths.
4th Column. (1) Seven smilliolzths. (2) One hundred and
thirty-three millionths. (3) One thousand seven hundred and
sixty-eight millionths. (4) Forty thousand and thirty-five szillionths. (5) Three hundred and sixty thousand and four soillionths.
- b. -- do
90 KEY TO RAY'S THIRD BOOK.
5th Column. (1) Eight ten-millionths. (2) Nine hundred-nillionths. (3) One million ten thousand one hundred and one
ten-mnillionths. (4) One hundred thousand three hundred and
four hundred-nmillionths. (5) Forty thousand and five hundredmillionths.
6th Column. (1) Six units and five tenths. (2) Sixty units
and four hundredths. (3) One hundred and eighty-four units
and one hundred and seventy-three thousandths.
7th Column. (1) Six tenths, or Six million ten-mzillionths. (2)
Eight thousandths, or Eighty thousand ten-millionths. (3) One
hundred and two thousandths, or One million and twenty thousand ten-nzillionths.
8th Column. (1) Forty tlousand five hundred and four units,
and one thousand and thirty-seven hun.sclred-thousandths. (2)
Fifty-four million units, and fifty-four millionths. (3) Thirty
million seven hundred and one thousand units, and one million
thirty-seven thousand and twenty-five ten-suillionzths.
ARTICLE 176.
-() 1T0jj5; 0ThO=D; -~o oo-6-.013n5; - 30 0o=.002036
T aoa --—.0004 006
(2) 137,-6-13.7; 2410o=24.OS; 30T o-o-30.025;
16~00004'6.0004; 8oW 0UU —S.000105
(3).4 (9).4125 (15).300004 (21).00000907
(4).26 (10).020'5 (16).000203 (22).20020003
(5).35 (11).0008 (17).000007 (23).01010001
(6).08 (12).20304 (18).0000024 (24) 20.25
(7).005 (13).00605 (19).0080006 (25) 106.037
($).304 (14).00009 (20).00000002 (26) 1000.001
(27) 200.025 (30) 200.0000000002
(28) 29.000029 (31) 65.006005
(29).001000005
ARTICLE 177.
(32) 14.3 (33) 32.04 (34) 1.0502
(o! -.
DECIMAL FRACTIONS. 91
ARTICLE 178.-ADDITION.
(2) 3.25 (3) 21.611 (4) 6.61 (5) 4.8
6.4 6888.32 636.1 43.31.35 3.4167 6516.14 74.019
67.1234 11.204
Ans. 10.00 Ans. 6913.3477 1 ul3. 133.333
5 3Ans. 133.333
Ans. 7231.0967
(6) 45.019 (7) 432.432 (8) 16.041
7.00071 61.0793 9.000094
93.4327 100.07794 33.27
6.0401 6.009 8.969
Ans~. 151.49251 1000.1001 32.719906
Anls. 1599.69834 Ans. 100.000000
(9) 204.0009 (10).0035
103.00000009.00035
42.009099.000035
430.99.0000035
220.0000009
A -ls..0038885
Ans. 999.9999999
ARTICLE 179.- SIBTRACTION.
(2) 5.03 (3) 24.0042 (4) 170.0035 (5).0142
2.115 13.7013 68.00181.005
Ans. 2.915 Ans. 10.3029 Ans. 102.00169 Ans..0092
(6).05 (7) 13.5 (8) 3.00000 (9) 29.0029.0024 8.037.00003 19.003
Ans..0476 Ans. 5.463 Ans. 2.99997 Ans. 9.9999
(10) 5.000 (11) 10000.0000 (12) 1.000000 (13).025.125.0001.000001.000025
Ans. 4.875 Ans. 9999.9999 Ans..999999 Ans..024975
ARTICLE 181.-MULTIPLICATION.
(2.).a (5).15X.7= _ o —~ -.105 Ans.
7
Ans. 2.45 (8) 1.035X17- -17.595 Ans.
Ans. 2.45
92 IKEY TO RAY'S THIRD BOOK.
(9) 19 (10) 4.5 (11).625 (12) 61.76.125 4 64.0071
95 Ans. 18.0 2500 6176
38 3750 43232
19
Ans. 40.000 ens..438496
Ans. 2.375
(13) 1.325 (14) 79000 (15).1 (16) 1..0716.079.01.0001
7950 711000 Ans..001 Ans..0001
1325 553000
9275.....9275 Ans. 6241.000
Ans..0948700
(17).043 (18) 40000 (19).09375.0021.000001.1000064
43 Ans..040000 37500
86 56250
93'5
Ans..0000903
Ans..09375600000
DIVISION.
SUGGESTIONS TO TEAcHERS. —The division of decimals is generally a troublesome subject to pupils; this arises from a want of
attention to the rule. Should the pupil be at a loss to understand
why, in some cases, when the divisor and dividend are both decimals, the quotient should be a whole number, let him read the
remarks on the division of fractions, page 79. When the divisor
contains more decimal places than the dividend, it is best, before
commencing the division, to reduce them both to the same denomination, that is, to make the number of decimal places the same in
both; the quotient will then be a whole number.
ARTICLE 183.
1.125 2 2.000
(5) =-37.5 Ans. (6) -— _ — =250 Anls..03.008.008
DECIMAL FRACTIONS. 93
37.2 37.20 86.075
(7) - =7.44 Ans. (8) -- 3.13 A1S.
5. 5. 27.5
24.73704 206.166492
(9) 24'77 7.191 Ans. (130) -50.004 Ans.
3.44 4.123
100.8788.000343
(11) 100 2222 Ans. (32) 0003 0001 Ans.
454 3~43
9811.0047 9811.004700
(13) -..90300 Ans..108649.108649.21318
(14) = 1,.122 Ans..19
102048 102048.0000
(15) 320000 Ans..3189.3189.102048 9.9 9.9000
(16) 124 000032 Ans. (17) *- 440 Ans.
3189.0225.0225
(18) 10).10 (19).1)1.0 (20).01)10.00
Ans..01 Ans. 10. Anis. 1000
1.7 1.7000000
(21) - =.-0265625 Ans.
64 64
(22) 80).080 (23) 1000).001000 (24).001).001
Ans..001 Ans..000001 Ans. 1.
(25).0000001).ooo00ooo1000 (26).000004).010000
Ans. 1000. Ans. 2500.
(27) 1-5000000=2.142857+.7.7
AL RTICLEr 185.-CASsE I.
( 1.000 A9.0 Ans.
(2) ~='~~~=.125 -Ans. (3)'~
8 5
-----— ~ —--------------------- 0
94 KEY TO RAY'S THIRD BOOK.
4 4.00 3.000
(4) 24' =.16 Ans. (5) -43 300. 075I.
(6)> Z 6 —.9375s16 (n)s.- 07=5 An0As.
25 40
1 15.0000 1.0000
(6) 59375Ans. (7)T-.1 0008 Ans.
16 1250
9.0000
(8) 6-j= —.0225 Ans.
400
1.00000000
(9).~.00390625 Ans.
256
5.0000
(10) 8- -..8333+ Ans.
4.000000
(12) - 1.000000 090909+ Ans.
11
ARTICLE 186.-CASE II.
(2).= As. (3) 5.2-= 4T=q Ans.
(4) As.s. (7).033=5 4 bu.s AnAns.
(8).5625=6 Ans. (9).34375=Ans.
(10).1484375j,4s4 s A19n.
ARTI0CLE 187. —CASE III.
(3).125 bu. —.125X4-.5 pk. Ans.
(4).7 pk. -.7 —-4=.175 bu. Ans.
(5).0625 lb.X12-.75 oz. Ans.
(6).05 yd.X4X4=.8 na.
(7).00546875 A.X4X40=.875 P. Ans.
Q - _ _ _- - - --— d
DECIMAL FRACTIONS. 95
(8).0004375 mi.X8X40-.14 rd. Ans.
(9).25 pt. — 2-.125 qt.:.125 qt. —— 4=.03125 gal. Ans.
(10).6 pt.- -2-.3 qt.:.3 qt. —.8=.0375 pk.:.0375 pk. — 4 —.009375 bu. Ans.
(11).3 min. —*60-.005 hr.:.005 hr. —24-.0002083 da. — Ans.
(12).7 rd.-~-40 —.0175 fur.:.0175 fur. —8.-.0021875 mli. Ans.
ARTICLE 188.-CASE IV.
(2).875 ycd.X4-3.5 qr.:.5 qr.X4=-2. na. Ans. 3 qr. 2 na.
(3).7 lb.X12-8.4 oz.:.4 oz.X20=8. pwt. Ans. 8 oz. 8 pwt.
(4).8125 bu.X4=3.25 pk.:.25 pk.X8=2. qt.
Ans. 3 ph. 2 qt.
(5).3375 A.X4-1.35 R.:.35 R. X40-14. P. Ans. 1 R. 14 P.
(6).04318 mi.XS=.34544 fur.:.34544 fur. X40
=13.8176 rd.:.8176 rd.X5-=4.4968 yd.:.4968 yd.X3
-1.4904 Rft..4904 ft.X12-5.8848 in.
Ans. 13 rd. 4 yd. 1 ft. 5.8848 in.
(7).33625 cwt.X4 —=1.345 qr.:.345 qr.X25=8.625 lb.:.G25 lb.X16=10 oz. Ans. 1 qr. 8 lb. 10 oz.
ARTIcLE 189. —CASE V.
(2) 2 pk. 4 qt.=20 qt.: 1 bu.=32 qt.: o0-5 025 AIs.
(3) 13 hr. 30 min.=810 min.: 1 da.=1440 min.
-1 - L -9 -.5625 Anzs.
(4) I lb.-=256 dr.: -,=&.03515625 Ans..028
(5) 1 A.-160 P. 02 —.000175 Ans.
160
(6) 1 da. —-1440 min. y 7a-U=.0048611+ Ans.
(7) 4 gal. 1 qt. 1.28 pt.=35.28 pt.: 1 hhd.=504 pt.
35.28
5 =. 07 Ans.
504
_- -- -- --- --- - -
96 KEY TO RAY'S THIRD BOOK.
(8) 3 pk. 7 qt. 1 pt.=63 pt.: 2 hbu. 2 pk. 4 qt.-i68 pt.
-?-=-=. 375 A>7s.
(9) 99 pages= —9, of 512 pages: -99=.193359375 A, s.
(10) 55 A. 2 R. 17 P. —-8897 P. ss 97 175
229 A. 2 R. 16 P. —36736 P. 3673GARTICLE 190.
(1) $0o.4X9=$3.6: $0.1875X12=$2.25
$3.6+$2.25=$5.85 Ans.
(2) $0.45X2.3=$1.035: $0.375X1.5=4.5625:
$1.035+$.5625=$1.5975 Ans.
(3) $2.6875X16=$43: $2.6875X16 —=$43.671875 Ans.
(4) $35.25 — $0.75=47 bu. Ans.
(5) $98.4 —$2.5625 —38.4 yd. Ans.
(6) 6 cwt. 2 qr.=6.5 cwt.: $3.25X6.5=$21.125 Ans.
(7) 7 hhd. 23 gal.=7.365079-+
$49X7.365079=$360.8888+- Ans.
(8) 343 yd. 3 qr.=343.75 yd.: $0.16X343.75-$55 Ans.
(9) 14 bu. 3 pk. 4 qt.=14.875 bu.
$.625X14.857=-9.296875 Ans.
(10) 13 A. 2 R. 35 P.=13.71875 A.
$17.28X13.71875-$237.06 Ans.
(11) $19.065, $1.24-15.375 yd.=15 yd. 1 qr. 2 na. Ans.
(12) $9.296875 —.3125=29.75 bu.=29 bu. 3 pk. Ans.
(13) $59.265 —. $4.32=13.71875 A. —13 A. 2 R. 35 P. Ans.
(14).34 yd.X4-1.36 qr.: 1.36 qr.+.325 cjr.=1.685 qr.:.685 qr.X4 —2.74 na.: 2.74 na.+.4 na.-3.14 na.
Ans. I qr. 3,14 na
yd. qr. na.
(15) 1.53 yd.-l 2 0.4S
1.32 qr. — 1 1.28
Ans 1 0 3.20
DECIMAL FRACTIONS. 97
(16) 365.25X.05=18.2625 da..2625 da.X24=6.3 hr.
6.3 hr. —.5 hr.=5.8 hr..8 hr.X60=48. min.
Ans. 18 da. 5 hr. 48 min.
(17).41 da.-9.84 hr.: 9.84 hr. —.16 hr.=9.68 hr.:.68 hr.X60=40.8 min.:.8 min.X60-48. sec.
Ans. 9 hr. 40 min. 48 sec.
(18).4 T.X4=1.6 hhd.: 1.6 hhd.+.3 hhd.=1.9 hhd.:
1.9X63=119.7 gal.: 119.7 gal.-.8 gal.=120.5 gal.:
120.5 gal.X4=482 qt.=964 pt Ans.
(19) 365.25 da.X.3=109.575 da.:.575 da.X24-13.8 hr..8 hr.X60=48 min. Ans. 109 da. 13 hr. 48 min.
(20) 365.25 da.X.005X24X6OX60-=157788 sec. Ans.
(21) 1 cord=128X1728=221184 cu. in.
Adi z=.000004+ Ans.
(22) 17 mi. 3 fur. 15 rd.=17.421875 mi.:
$690.35X17.421875-$12027. 19140625 Ans.
REfMARnS.-Pupils must have a thorough knowledge of common
fractions, before they can understand fully the reason of the rules
in decimals.
When a pupil is in doubt with regard to the accuracy of the
result in any operation involving decimals, let him convert the
decimals into common fractions, and then perform the work; the
results, in both cases, ought to be the same.
It is a useful exercise to perform the same operations in
equivalent common and decimal fractions. Thus, they may be
required to perform the operations indicated in the following
examples, by the rules for common fractions; then to convert the
common fractions into decimals, and work by the rules for decimals.
EXAMPLES.
By common fractions. By decimals.
(1) 4 + 1. +-=+ 3= A-ns. 4=.25
-— =.375
-z=.625 Ans.
Key.-7
98 KEY TO RAY'S THIRD BOOK.
(2) + —. Ans. -5-.3125 (3) 1 AMS a375
(4G)!X.Ig AnES.,3L-.09375 ( -- 7. 4. Ans. 1:-=1.4
(6) 9 — 3 Ans. T 9 —.072
The great advantage of decimals is, that the operations are
performed with the same facility as in whole numbers; and, that
the most complicated arithmetical expressions may be reduced to
equivalent decimals that are comparatively simple.
PROPORTION.
SUGGEasTIONS TO TEACHERS.-Simple proportion is justly regarded
as one of the most important subjects in Arithmetic; particular
care should be taken, therefore, to render the pupil well acquainted with its principles. First, let him thoroughly understand the meaning of ratio; if the questions and exercises in the
maly deem necessary. It should be kept before the pupil that
there is no such thing as ratio between quantities of different
kinds, anid hence, that the first and second terms in every proportion must be of the same kind.
In performing the work necessary to obtain the fourth term, it
may be asked, do we not use a concrete number as a multiplier,
and, also, as a divisor? The answer is, we do not; we multiply
and divide by abstract numbers that represent the ratio of the
first. term to the second. Thus, the question, if 2 yards of
cloth cost 5 dollars, how many dollars will 6 yards cost? being
stated, is,
yd. yd. $
As 2: 6:: 5 to the answer.
In performing the operation, we multiply the 5 dollars by 6,
and divide by 2; because, the ratio of 2 to 6, is the same as that
of 2 yards to 6 yards. The same method of reasoning will apply
to every case, both in simple and compound proportion.
ARTICLE 203.
(8) As 4 hats: 10 hats:: $14: $35 Ans.
PROPORTION. 99
(9) As 3 caps: 11 caps:: 69 cts.: $2.53 Ans.
(10) As 4 yd.: 9 yd. $7: $15.75 Ans.
(11) As 8 yd.: 12 yd.:: $32: $48 Ans.
(12) As 12 yd.: 8 yd.:: $48: $32 Ans.
(13) As $32: $48:: 8 yd.: 12 yd. Ans.
(14) As $48: $32:: 12 yd.: 8 yd. Ans.
(16) As 12 men: 8 men:: 24 da..: 16 da. Ans.
(17) As 2 da.: 8 da.:: 60 men: 240 men.Ans.
(19) As 6 gal.: 2 hhd.:: 65 cts. Ans.
Or, 6 gal.: 2X63 gal.:: 65 cts.
I gal.: 21 gal.:: 65 cts.. $13.65 Ans.
(20) As 5 cwt. 3 qr. 10 lb.: 35 cwt. 1 q.:: $21.06:
$126.90 Ans.
(21) As 1 yd. 2. qr.: 1 qr. 2 na.:: $2.50: $0.625 Ans.
(22) As 90 bu. 450 bu.:: 6 da: 30 da. Ans.
(23) As 5 da.: 15 da.:: 6 men: 18 men Ans.
(24) As 60 bu. p.: 140 bu. p.:: 15 bu. c.: 35 bu. Ans.
(25) As 3 cwt. 1 qr.: 16 cwt. 1 qr.:: $22.60: $113.00 Ans.
(26) As 4 ft. 6 in. in shadow: 180 ft. shadow:: 3 ft.:
120 ft. Ans.
(27) As 12 hr.: 9 hr.:: 60 da.: 45 da. Ans.
(28) As $1: $2200:: $.60: $1320 A's, Ans.
As $1: $1800:: $.60: $1080 B's, Ans.
(29) $800.30+$250+375.10+$500+$115=$2040.40
As $2040.40: $612.12:: $1: $.30 Ans. 30 cts.
(30) As $6: $8:: 9 oz.: 12 oz. Ans.
(31) As $300: $250:: 6 mon.: 5 mon. Ans.
(32) 27X7= —189, 36-27=9.
As 9 mi.: 189 mi.:: I da: 21 da. Ans.
(33) 9 hr.: 12 hr.:: $152: $20.881, one month's services
when he works 12 hr. a day. $20.88'X4_=$91.91' Ans.
2 t
100 KEY TO RAY'S THIRD BOOK.
(34) As 5 lb.: lb.:::' j Ans.
(35) As 6 yd. 74 yd.: $53~ $6' Ans.
(36) As bu. bu.: e$8 * - A. YXX=9
(37) As 14 yd.: 2 yd.. 74. A ~ 4X — ~.X
(38) As $294: 314:: 59. yd.: Ans.
TX4'X1 19=1 25-=62' yd. Ans.
(39) As.85 gal.:.25 gal.:: $1.36: $0.40 Ans.
(40) As 61.3 lb.: 1.07 lb.:: $44.9942: $0.78538 Ans.
(41) As yd. E. En. $ Ans.
q qr.: 45 qr.':: $4: Ans. oX, 9X84 Ans.
(42) As y yd.: 17! yd.:: $44: Ans.
-7X' 9Xq=$ —8178.38~ Ans.
cogs. cogs. rev. rev.
(43) As 26: 35:: 1: 1.e. Hence, the smaller wheel gains
09 of a revolution in each revolution of the larger wheel.
rev. rev. rev. of larger.
Then,.:!: 10: 1: 28 =revolution of larger, Ans.
(44) 1 gal. =32 gills, 32-1=31.
As 32: 31:: 100 gal.: 968 gal. Ans.
(45) As 70 p.: 20 p.:: 60 sec.: 174 sec. Ans.
1142X174=19577' ft.=3 mi. 5 fur. 145 yd. 2- ft. Ans.
(46) As 25 ft.: 25 ft. 5.25 in.:: 643 ft. 8 in.
654 ft. 11.17 in. Ans.
(47) As 60 lb.: 2136 lb.:: 75 cts.: $26.70 Ans.
(48) As 60 lb.: 1225 lb.:: 81 cts.: $16.53+ Ans.
(49) As 56 lb.: 1000 lb.:: 63 cts.: $11.25 Ans.
(50) As 32 lb.: 3000 lb.:: 24 ets.: 822.50 Ans.
ARTICLE 205.
(2) 2 da.: 10 da.: 24 mi.: 240 mi. Ans.
4 hr,. 8 hr.
SUGorESTIoN To TEACHERS.-In stating questions in Compound
Proportion, it will be found both interesting and instructive for
PROPORTION. 101
the pupil to give the reason for the arrangement of each pair of
terms, as in the following statements.
NOTE. —For convenience, the third term is placed below the
second.
(4) As 18 rd.: 72 rd. The more rods, the more men.
8 da.: 12 da. The less days, the more men.:: 16 men. Ans. 96 men.
This statement is read thus: As 18 rods are to' 72 rods,
and as 8 days are to 12 days; so are 16 men to the
required number of men.
(5) As 6 p.: 15 p. The more persons, the more dollars.
8 mon.: 20 mon. The more months, the more dollars.:: $150. Ans. $937.50.
(6) As 7 da.: 9 da. The more days, the more miles.
6 hr.: 11 hr. The more hours, the more miles.:: 217 mi. Ans. 5111 mi.
(7) As $100: $75. The less dollars, the less interest.
12 mon.: 9 mon. The less months, the less interest.::$6. Ans. $3.375
(8) As 10100 lb.; 100 lb. The more lb., the less miles.
20 cts.: $60.60 The more money, the more mi.: 20 mi. Ans. 60 mi.
(9) As 12 cwt. 3 qr.: 10 T. The more weight, the more money.
400 mi.: 75 mi. The less miles, the less money.:: $57.12 Ans. $168.
(10) As 20 men: 18 men. The more men, the less days.
40 rd.1.: 87 rd. 1. The more length, the more days.
5 ft. h.: 8 ft. h. The more hight, the more days.
4 ft. t.: 5 ft. t. The more thickness, the more da.:: 15 days. Ais. 58'" days.
(11) As 100 men: 180 men. The less men, the more days.
200 yd. 1.: 180 yd. 1. The less length, the less days.
3 yd.w.: 4 yd.w. The more width, the more days.
2 yd.d.: 3 yd.d. The more depth, the more days.
8 hr.: 10 hr. The less hours, the more days.::6 days. Ans. 24.3 days.
CPI_
102 KEY TO RAY'S THIRD BOOK.
ALIQQUOTS OR PlRACTiCE.
ArTICLE 207.-(CASE I.
(3) $24-cost of 24 yd. at,$1 (5) $173
50 =- l2i=cost of 24 yd. at 50 cts. 25-1i,$ 43.25 lAns.
12 —-t 3=: " " at l121 ctls.
$15 Ans.
(6) $45 (7) $54
25 _' 11.25 25 -' 13.50
61=l 2.81k 12,=A 6.75
$14.06~ Ans. 0 = 3.37nk
$23.62- Anls.
(8) $32 (10) 80 (11) 3B
(,21 s
so=1 162 8
25 =1 8 160 288
12=1 4 25= 20 50o = 18
61=k s_=2 121 10 25 = 9
i29I-1 4.50
Ans. e30 Ans. $190 2
Ans. $319.50
(12) 77 (13) 175 (14) 224
1 14 3
77 700 672
50 O = 38.50 175 25=- 56
12_= a 9.620 1 2 2 8
Ans. $125.12-. 25- 43.75 64-= 14
12, —z 21.87,1 Ans. $770
Ans. $2515.62r
(15) 462 (16) 185
1 1
462 185
61 —' | 28.87~ 33 1 —{ 61.66Ans. $490.87~ Ans. $246.66~ —------- ---- -------------------— ~ —""=- ~:
PRACTICE. 103
(17) 150 (18) 24
3 1
450 24
9339 1- 50 62= 4
3
33=3 50 Ans. $28
Ans. $550
ARTICLE 208.-CAsE II,
(2) $10 (3) $12.50 (4) $12
14 28 5
140 10000 60
2.=1 5 2500 1.= 3.00
Ans. $145 350.00 10 P 4 5
2 R. — 6.25 Ans. $63.75
I R.g 3.121
Ans. $359.37-y
(5) $18 (6) $12.50 (7) $1.75
12 14 3
216 5000 5.25
1 R.=4 4.50 1250 2qr —'.871
1o P.~= 1.121l 175.00 Ans. $6.12,
Ans. $221.621 2 R.; 6.25
1 R.= 3.12'
20 P.=. 1.5614
5 P.-4 39T1
Ans. 186.32o-{(8) $1.50 (9) $17.25 (10) $6
4 56 83
6.00 10350 498
2 qr. —.75 8625 2 pk.-= 3.00
1 qr.-.37 966.00 1 pk.= 1.50
Ans. $7.12' 2 qr.=- 8.622 2 2 t.4 377
1 qr.-= 4.311 Ans. $502.87~
Ans. $978.93,b...................... _.... 6
104 rKEY TO RAY'S THIRD BOOK.
(l1) $4 (12) $3.20 (13) $2.36
24 40 17
96 128.00 1792
2 pk.- 2.00 2 pk.=~ 1.60 256
1 pk.='- 1.00 1 pk.=4.80 43.52
4 qt.=-.50 4 qt;.=.40 1pk -i.64
2 qt.=..25 2 qt..20 08
I qt. —.12' 1 qct.-.10 I pt.=.04
Ans. $99.87 1 --— 05 Ans. 844.28
Ans. $131.15
(14) $.24 (15) $2.56 (16) $3.20
5 3 17
1.20 7.68 2240
8 oz.-=1.12 8 oz.=g 1.28 320
2 oz.= 1.03 4 oz.==.64 54.40
Zo=.-.011 1 oz.==.o16 2Ro.=1 1.60
Ins. $1.36~ 8 dr.=z-.08 1R.i. 80
4 dr.==.04 20 R.-I.40
2 dr.=-.02 10 P.-.20
1 d.=4.01 s p.==.10
Ans.;9.91 4 P.-'.08
Ans. $57.58
ARTICLE 209.-PERCENTAGE.
7,%o.07: 4, I-o=.045: 5- %-=.0525: 10 %=.105:
37-;%=.375: 120 %=1.20
ARTICLE 210.-CASE I.
(2) 7 o=-.07 $175X.07 -$12.25 Ans.
(4) 6;o=.06 $250X.06 — $15.00 Ans.
(5) 7 %o=.07: $162X.07 =$11.34 Ans.
(6) 5 % —.05: $11SX.05 =$ 5.90 Ans.
(7) 8 %o=.08: $ 11X.08 =$ 0.88 Ans.
(8) 1 =.01: 278X.01 -$ 2.78 Ans
(9) 2-{ I-=.02 —: $ 68X.02 —$ 1.53 Ans.: —-----— ~I4 4 4
PERCENTAGE. 105
(10) 4, 2-=.045: $220.50X.045-$9.922+ Ans.
(11) 71 7-.075: $115.42X.075=$8.656+ Ans.
(12) 54 yo=.o05: $243.16X.05V=$13.981+ Ans.
(13) 34.034 $1250X.03=-$40.625 Ans.
(14) 25 =.25: $25X.25=$6.25 Ans.
(15) 101 % =1.015: $2002X1.015=$2032.03 Ans.
(16) 208 %o=2.08: $650X2.08-$1352. Ans.
(17) 1000 %= —10: $24.75X10=$247.50 Ans.
(18)' %o=.001: $400X.001-$0.40 Ans.
(19) %-o.008: $464X.00=$81.74 Ans
(20) Yo fr=.00.: 81950X.00-1-=$1.624 Ans.
(21) 8 %=.081': s72X.08=$ 6.12 ~ $72 —6.12
-=65.88 Ans.
(22) 36 Yo=-.36: $800X.36-$288: $800-$288-$512 Ans.
(23) $56X.133=8$7.35: $51X.14 —-$7.48: $7.48-7.35
-$0.13 Ans.
(24) $1764X.23g=$405.72: $1764X.31X$546.84:
$1764 —.09-8158.76: $1764X.034 —57.33. Byadding
together the amounts expended, we find their sum is
41168.65, which, being deducted from 81764, the remainder is $595.35, the amount paid for muslin.
Or, by adding the several per cents together, and deductingtheir sum from 100 %o (the whole), the remainder is 33
2~ to be expended in muslin: and $1764X.33=$-595.35
(25) 75 lb.X4=300 lb.: 300X.12 —37 lb.
300 lb.-37 lb.-263 lb., and 14 cts.X263-$36.82 Ans.
(26) 160X.35-56: 160 sheep+56 sheep-216 sheep Ans.
(27) 320X.05 —16: 320-16=304: 304X.25= —76:
304-76-228 Ans.
(28) 18+15+23+12 —68: 100-68=32'o left:
$800X.32= —256 Ans.
106 KEY TO RAY'S THIRD BO3OK.
ARTICLE 211.-CASE iI.
(2),15X.01 —.15: $3.00 — $.15=20 Ans.
(3) $50X.01=$.5: 6. $.5-12 Ans.
(4) $75X.01 —.75: $4.50-:-$.75=6 Ans.
(5) -1.331 —33 3-1 70 Ais.
(6) $25X.01-8.25: $0.25 —.25-1 Ans.
(7) 8142.60X.01=1.426: 67.13 —1.426=5 Ans.
(8) 89X.01=-.09: $9 —*$.09=100 Ans.
(9) $9X.01=$.09: $13.50 —. $.09=150 Ans.
(10) $243X.01=$2.43: $8.505 — $2.43 —3.5=-3 Ans.
(11) $2X.01=.02:.002-.02=.1= —jL Ans.
(12) 63532X. 01=835.32: $13.245 — $35.32-.375= Alns.
(13) $300X.01-83 25-o-.83=8 A7ns.
(14) $500X.01=-5: $75 — 8515 Ans.
(15) z=.60=60 % Ans.
(16) 5 bu.-160 qt. To-g-m-(o=.03 —— o3 % ins.
ARTICLE 213.-COMMISSION.
(1) $240X.05-=12.00 Ans. (2) $460X.021= $11.50 Ans.
(3) $180X.04=$7.20: $119X.03= — $3.57:
$7.20+$3.578$10.77 Ans.
(4) $240X.03=$7.20: $225X.05=$11.25:
$7.20+$11.25=$818.45 Ans.
(5) $48OX.02=-$10.80: $275X.03=-$8.80:
$216X.02N=$ 5.76 $10.80-+-$8.80+$5.76
— 25.36 Ans.
(6) $275X.03=$8.25: $341X.15=$51.15:
$964X.25-$241.00: $217X.02-=4.8825:
68.25+$51.15+$241.00+$4.8825=$305.2825 Ans., —----- -- ----- ----- -- -
PERCENTAGE. 107
(7) $13X25=$325: ~$325X.02-=-8.12. Agent receives:
$325-$8.12!$316.87- Owner receives.
(8) 1114 lb. X15=16710 lb.: 8 cts. X16710-$1336.80:
$1336.80X.034 —$43.446 commission:
$1336.80-$43.446=$$1293.354 Ans.
(9) $15X250=$3750: $7X175=-$1225: 0.25X1456
=$364.00: $3750+$1225+$364=$5339: $5339X.03
=$160.17 com.: $5339-$160.17=$5178.83 Ans.
ARTICLE 214.
(11) $1.00+.04=$1.04 $312 —. $1.04=$300 Ans.
(12) $1.00Q-.08=$1.08: $1323.54 — $1.08=$1225.50:
$1323.54-$1225.50-$98.04 Ans.
ARTICLE 215.-INSURANCE.
(1) $2250X.01 —$33.75: $33.75+81.00=$34.75 Ans.
(2) $12600+$14400-$27000: 2 of $27000-_$18000:
$18000><.042-$810.00: $810+$1 — $811 Ans.
(3) 3 of $5000-$,3750: $3750X.00-=$18.75:
$18.75+$1.50= —20.25 Ans.
(4) 4 of 21000-$12000: $12000X.121-$1500:
$1200X.Ol-l$18.00: $1500+$18+81+$1-=$1520 Ans.
ARTICLE 216.
(6) 1-.01.99: $2475 -.99-$2500 Ans.
(7) 1 —.125=.875: $13125 —.875=$15000 A7ns.
(8) 1-.01 =1-.0175=.9825: $2358 -.9825=$2400 Ans.
(9) 1-.00-=:1-.00375-.99625: $2287.39-.99625
-$2296:.2296 —$2287.39=$8.61 Ans.
Or, 1 —.00-.99-= 7'" 7: $ 2287.39 —. 7' 7=$2296, etc.
As —--— Or, a 8 -9.
108 IKEY TO RAY'S THIRD BOOK.
ARTICLE: 217. —STOCKs.
(1) $1400X.04=$56 $1400+$56=S1456 Ans.
Or, 104 -o'1.04: $1400X1.04=$1456 Ans.
(2) $1400X.96=$1344.00 Ans.
(3) $50X11=$550 $550X.05=$27.50: $550+$27.50
-=$577.50 Ans.
(4) $75X15=$1125: $1125X.10=$112.50:
$1125- 112.50-=1012.50 An~s.
(5) 108 — 104-=-3: $1500X.03$=$57.50 Ans.
Or, $1500X1.04 =81567.50: $1500X1. 08=r81625.00:
$1625.00-$1567.50=857.50 Ans.
(6) 3+5-S=8: $1300X.08=-$110.50 Ans.
Or, $1300X.97 —$1261.00 cost: 81300X1.05'
— 1371.50 sold for: $1371.50 —1261.00
=$110.50 Ans.
(7) 4+21=6,: $860X.061=$55.90 Ans.
Or, 8860X1.04=$894.40 cost: $860X.97- =$838.50
sold for: $894.40-$838.50=855.90 Ans.
ARTICL E 218.-BRO KERAGE.
(l) $600X.01-$9.00 Ans. (2) $289X.01O -83.61- Ans.
(3) $360X.003 -$1.35: 8360 —$1.35=8358.65 Ans.
(4) $134X.02=y-3.35: $134-$3.35=$130.65 Ans.
(5) $200X.00-$-1.25: $200+$1.25z=$201.25 Ans.
(6) 1.+-=-. $500X.00O-$3.75 Ans.
Or, 8500X.00-=$2.50 gain in buying. $500X.00'
=81.25 gain in selling. $2.50+$1.25==$3.75 Ans.
ARTICLE 220.-CASE I.
(3) $65 X.05==$3.25: 83.25X4-$13.00 Ans.
(4) $200X.08=$16.00 Ans. (5) $150X.05=$7.50 Ans.
PERCENTAGE. 109
(6) $300X.06=$18.00: $18.00X2=$36.00 Ans.
(7) $275X.06-$16.50: $16.50X3=849.50 Ans.
(8) $187.50X.05=-9.375: $9.375X4=$37.50 Ans.
(9) $233.80X.06=$14.028: $14.028XO1=$140.28 Ans.
(l1L) $215.00X.06=$12.90: $215+$12.90=-$,227.90 Ans.
(12) $45.00X.08=$3.60: $3.60X2-$7.20:
$45+$7.20=$52.20 Ans.
(13) $80.OOX.07=$5.60: $5.60X4=$22.40:
$80+$22.40-$102.40 Ans.
(14) $420.00X.05~ =$22.40: $420.00+$22.40=$442.40 Ans.
(15) $237.16X.033=$8.8935: $8.8935X2=$17.787:
$237.16+$17.787=$254.947 Ans.
(16) $74.75X.04-$2.99: $2.99X5 —14.95:
$74.75+-14.95=$89.70 Ans.
(:17) $85.45X.06=$5.127: $5.127X4 —$20.508:
$85.45+$20.508=-$105.958 Arns.
(18) $325.00X.059=-$17.55: 17.55X3=852.65
$325-+52.65 —$377.65 Ans.
(19) $129.36X.04 =-$5.6595: $5.6595X4422.638:
$129.36+$22.638=:$151.998 Anzs.
ARTICLE 221.-CASE II.
(2) $240X.08-=$19.20~ 2 mon.-x —1 yr.:
6 of $19.20=$3.20 Ars.
(4) 8360X.06=$21.60: -Z of $21.60o=$1.80:
2 da =,.r=- r mon. of $1.80=$.12 Ans.
(6) $86X.06=$5.16: 3 mon.-I~ yr.: { of $5.16=$1.29 Anrs.
(7) $50X.08=$4.00: 4mon.=- yr.:' of $4.00=$1.33- Ans.
(8) $150.25X.08-$12.02: 6 mon.= yr.:
{ of $12.02=$6.01 Ans.
(9) $360X.05=818.00: 7 mon.=-7 yr.:
> of $18.00=$10.50 Ans.
~~a- - - -- -- - --- ---- - - -- ---- ---— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- ---''P
110 KEY TO RAY'S THIRD BOOK.
(10) $204X.07=$14.28 11 mon.= -- yr.:
} of $14.28=$13.09 Ans.
(11) $726X.06=-$43.56: ~1, of $43.56=$3.63:
10 da.=~ mon.: 3 of $3.63=$1.21 Ans.
(12) $1200X.06=$72.00: Ti' of $72.00=$6.00:
15 da.=- mon.: ~ of $6.00 —$3.00 Ans.
(13) $180X.08=$14.40: T of $14.40-$1.20:
19 da.=-m-o mon.: -s-3 of $1.20=$0.76 Ans.
(14) $240X.07-$16.80: -1l of $16.80=$1.40:
27 da, -— 1 — mon. TU9 of $1.40-=$1.26 Ans.
(15) $100.80X.05-$5.04: ~-' of $5.04=$.42:
28 dla. -1 4 mon.: o of $0.42 —$0.392 Ans.
(16) $228X.06= 13.68: 9 mon.=-9= yr.:
-, of $13.68-$10.26: $2284-+$10.26=-$238.26 Ans.
(17) $137.50X.06-$8.25: 8 mon.= -1-= yr.:
a of $8.25-$5.50: $137.50-+$5 50 —$143.00 Ans.
(18) $150X.05=-7.50: -' of $7.50=$.625:
18 ~dx.=- =5 mon.:, of $.625=$.375:
$150+$.375=$150.375 Ans.
(19) $360X.06=-$21s.6: ~' of $21.60=$1.80: 11 da.=i4o
mon.: 3i 0 81.80 —8.66 8360-q -$,360.66 Ans.
(20) $264X.06-$15.84: -- of J15.84 —1.32:
9 da. =, -,3o mon.: y-i0- of $1.32 —$.396:
$264+$.396_=264.396 Ans.
ARTICLE 222.
(2) $350 (3).150 (4) 375.40.04.06.06
14.00 9.00 22.5240
7 4 6 mon.=: 11.262
98.00 6.00 2 mon~ - 3'754
3 mon.=-I 3.50 2 mon.- - 1.50 Ans. $37.540
Ans. 4101.50 Ans. $37.50
PERCENTAGE. 111
(5) $92.75 (6) $500 (7) $560.06.06.08
5.5650 30.00 44.80
3 1 mon.=,l 2.50 2
16.695 15 da. = 1125 89.60
4 mon.=~ 1.855 3 da. = 25 4mon.-= 14.93:
1 mon.={.463- Ans. $34.00 15 da.= { 1.86:_
Ans. $19.013+ Ans. $106.40
(8) $750 (9) $456 (10) $216.06.05.10
$45.00 $22.80 21.60
4 3 5
180.00 68.40 108.00
3 mon.= 11.25 4 ron. —- 7.60 6 mon.=jl- 10.80
6 dca. 15 0.75 1 mon.= z 1.90 1 mon. — 1.80
Ans. $192.00 15 da. =$.95 15 da. =.90
3 da. =-.19 10 da. =.60
Ans. $79.04 2 da. 5.12
Ans. $122.22
(11) $380 (12) $300 (13) $250.15.06.06
1900 $18.00 15.00
380 3 6 mon.=- 7.50
57.00oo 5400 1 mon.= 1.25
3 6 jmon.= — 9.00 Int., $23.75
171.00 2 mon. 3.00 250.00
6 mon.= - 28.50 Int., $66.00 Ans. $273.75
3 mon.= l 14.25 300.00
9 da. = — 1.425 Azs. $366.00
Ans. $215.175,a. —--— ~ -------------
112 KEYr TO PAY'S THIRD BOOK.
(14) (15) (16)
$205.25 $150.62 $210.25.06.05.07
$12.3150 $7.5310 814.7175
2 3 2
24.630 22.593 29.4350
6mon.=-1 6.1575 4 mon.-=l 2.510+6 mon.=- 7.3587+
2mon.=-/ 2.0525 1 mon.=.627 1 mon.=zl 1.2264
15 da. = —.5131 12 da. --.251 20 da. =.8176
Int., $33.3531+ Int., $25.981+ Int., $38.8377+
205.25 150.62 210.25
Ans. $238.6031+- Ans. 8176.601+ Ans. $249.0877+
(17) (18) (19)
$57.85 yr. mon. da. yr. mon.da..05 1849 4 19 1849 4 27
$2.8925 _1847 1 9 1848 2 15
2 2 3 10 1 2 12
5.7850 $150 $240
3 mon. -,7231+.06.08
15 da. =f1 -1205
15 do 1205G 9.00 19.20
d.0482 2 2 mon.= 320
2 d. 0.0160 6.2
18.00 12 da, --.64
Int., $6.6928+ 3 ou.on.-l 2.25 Ans. 823.04
57.85 42 20
10 da..25
Ais. 64542 Ans. $20.50
PERCENTAGE. 113
(20) (21 (22)
yr. mon. da. mon. da. mon. da.
1845 8 28 11 27 8 28
1.843 5 14 7 3 3 1
2 3 14 4 24 5 27
$180 $137.50 $125.40.07.09.o___
12.60 12.3750 100320
2 6270
2 mon.-. 4.125
125.20 20 da. =.6875 $1 0.6590
3 mion.-= 3.15 4 dc.- -.1375 4 monI. —' 3.553
10 da. — Aim 84.9500 1P on - o88*
2 cda. =.07
2 dal. -.07 Od 0.la..296,Ai.s,,s28.84 ^ 050
Int. $5.240-.-.
125.40
(23) As. 130.64+ —
(23)
yr. mon. da.. (24.)
1848 3 9 yr. mon. da.
1847 8 2 1847 7 24
1846 10 25
7 7
8 29
$234.60 8 29
3.05o0 $153.80.05
11.7300
5865 7.6900
12.3165 6 mon. — 3.845.2 -' 1,2816+6 mon.= — 6.15825
6 Cnl 6.1.5825 20 da. 42 72
1 mon.= 1.02637+ -
5 da.' 1068
6 da. =,.20527 4
1 da.:[.03421
int., $5.7460
TInt., $7.42410 153.80
234.60
|Ans. S159,546+Ans. $242.024 A
114 H1KEY TO tAY'S THIRD BOOK.
AR TICLE 223.
(3) 16 mon. 13 mon. 4 mon. 18 da. 7 mon. 12 da.
Ai7.S.,C Ais. $.065 Ans. $.023 $.035
Ans. $.037
10 moen. 13 do. 5 mon. 17 da. 11 mon.
iAns..052- $.025 iAns. $.055
Aas. 0, 0277'2 mon. i do. 9 Imn. 3 da. 14 mon. 4 da.
i.ts. (.0101 As...0070-!I7 mon. 27 do. 33 mon. 20 da..os085 $165
44 33
Ans. 8.0891- Ans..168
ARITICLE 224.
(2),.04><65 —$2.60 Ans. (3) $.o055X36=$1.98 Ans.
(4) $.0828=-$2,24 Ans. (5) $.125X500=862.50 Ans.
(6) $.022X75T=$1.65 Ans.
(7) $.048.X186=$9.083 Ans.
(8) M.072X125-$9.00 Ans.
(9) 1.204X210.25-$402.891 Ans.
(10) $.087-X134.45-$,11.764+ Aits.
(11) $,138~X144-.24$119.929+ Ans.
A:RT'IC E 226.
(2) Principal............. $200.00
Interest to st payment, (12'mon.,)... 12.00
Amount duo, Jan. 1st, 1846 212.00
1st payment, (greater than int.,) to be deducted 70.00
Balance due Jan. 1st, 1846. 142.00
interest for 1 year..... 8.52
Ans. Amount due Jan. 1st, 1847. $150.52
x ~"U"~~"~~.~~""m`w
PERCENTAGE. 115
(3) Principal.... 300.00
Interest to 1st payment, (6 mon.,) 9.00
Amount due Jan. 1st, 1844. 309.00
1st payment, (greater than int.,) to be deducted 109.00
Balance due Jan. 1st, 1844,. 200.00
Interest to 2d payment, July 1st, 1844 (6 mon.,) 6.00
$206.00
2d payment, (greater than int.,) to be deducted 100.00
Balance due July 1st, 1844....... 106.00
Interest to Jan. 1st, 1845, (6 mon.,) 3.18
Ans. Amount due Jan. 1st, 1845. $109.18
(4) Principal $150.00
Interest to 1st padyment (i yr. 4 mon.,).. 12.00
Amount due Sept. 10th, 1851.162.00
1st payment (greater than int.,) to be deducted 32.00
Balance due Sept. 10th, 1851.130.00
Sept. 10th, 1852, paid a sum less than the
interest then due $6.80
Interest from Sept. 10th, 1851, to Nov. 10th, 1852,
(1 yr. 2 mon.,)........... 9.10
139.10
Deduct 2d payment....... 6.80
Ans. Amount due Nov. 10th, 1852. $132.30
(5) Principal.... $200.00
June 5th, 1842, paid a sum less than the
interest then due........ $20.00
Dec. 5th, 1842, paid, (making in all a sum
greater than the interest).. 50.50
Interest from Mar. 5th, 1841, to Dec. 5th, 1842,
(1 yr. 9 mon.,)....... 35.00
$23.o00
Deduct the sun of the 1st and 2d payments. 70.50
Balance due Dec. 5th, 1842,.164.50
Interest on balance to settlement (1 yr. 6 mon.,) 24.675
Ans. Amount due June 5th, 1844. $189.175
116 KEY TO RAY'S THIRD BOOK.
(6) Principal.. $250.00
June 1st, 1845, paid a sum less than int.,
then due.$....6.00
Jan. 1st, 1846, paid (making in all a sum
greater than int. then due)... 21.50
Interest to 2d payment, (12 mon.,).... 17.50
267.50
Deduct the sum of the 1st and 2d payments. 27.50
Balance due Jan. 1st, 1846... 240.00
Interest to July 1st, 1846, (6 mon.,)... 40
Ans. Amount due July 1st, 1846. $248.40
(7) Principal.......... $180.00
Interest to ist payment, (6 mon.).... 5.40
Amount due Feb. 1st, 1845.... 185.40
1st payment, (greater than interest,).... 25.40
Balance due Feb. 1st, 1845 160.00
2d payment, less than interest then due $4.30
3d payment.30.00
Int. on bal. to Jan. 1st, 1846, (11 mon.,) 8.80
168.80
Deduct sum of 2d and 3d payments... 34.30
Balance due Jan. 1st, 1846... $134.50:Interest to July 1st, 1846, (6 mon.,).. 4.035
Ans. Amount due July 1st, 1846. $138.535
PERCENTAGE. 117
(8) Principal.. $400.00
Sept. 1st, 1845. Paid, a sum less than
interest then due.$10.00
Jan. 1st, 1846. Paid in all, greater
than interest then due. 30.00
Interest to Jan. 1st, 1846, (10 mon.,).. 20.00
$420.00
Deduct sum of 1st and 2d payments... 40.00
Balance due Jan. 1st, 1846... $380.00
July 1st, 1846. Paid a sum less than
interest then due. $11.00
Sept. 1st, 1846. Paid in all, greater
than interest then due.80.00
Interest to Sept. 1st, 1846, (8 mon.,)... 15.20
$395.20
Deduct sum of 3cd and 4th payments. 91.00
304.20
Interest to March Ist, 1847, (6 mon.,)... 9.126
Ans. Amount due March 1st, 1847. $313.326
(9) Principal. $450.00
ist payment, Jan. 1st, 1847, less than
interest then due... $20.00
2d payment, April ist, 1847,'less than
interest then due.14.00
3d payment, July 16th, 1847, in all,
more than interest then due... 31.00
Interest to July 16th, 1847, (15 mon.,).. 45.00
495.00
Deduct sum of 1st three payments.... 65.00
Balance due July 16th, 1847..$430.00
4th payment, Dec. 25th, 1847, less than
interest then due. $10.00
5th payment, July 4th, 1848, less than
interest then due. 18.00
Interest to June 1st, 1849, (22 mon. 15 da.,). 64.50
494.50
Deduct sum of 4th and 5th payments... 28.00
Ans. Balance due June 1st, 1849. $466.50
_~ —----------------- _s
118 1KEY TO RAY'S THIRD BOOK.
(10) Principal.......... $1000.00
1st payment, May 1st, 1840, less than
interest then due. $18.00
2d payment, Sept. 4th, 1840, less in all
than interest then due...... 20.00
3d payment, Dec. 16th, 1840, less in all
than interest then due...... 15.00
4th payment, Ap. 10th, 1841, less in all
than interest then due...... 21.00
5th payment, July 13th, 1841, more
in all than interest then due... 118.00
Interest to July 13th, 1841, (18 mon. 12 da.,). 92.00
$1092.00
Deduct sum of 1st five payments..... 192.00
Balance due July 13th, 1841,..900.00
Interest to Dec. 23, 1841, (5 mon. 10 da.,). 24.00
Amount due Dec. 23, 1841......... 924.00
6th payment, (greater than int.,)..... 324.00
Balance due Dec. 23, 1841,.600.00
Interest to Oct. 1st, 1843, (1 yr. 9 mon. 8 da.,) 63.80
i4ns. Amount due Oct. 1st, 1843. $663.80
REMARK. —In accordance with suggestion preceding Example
2d, in Arith., in all the preceding solutions the computation of
interest has been deferred, until the period when a payment, or
the sum of two or more payments, exceeds the interest due.
ARTICLE 227.
(1) Principal...........$...... $320.00
Interest to Jan. 1st, 1847, (1 yr.)...... 19.20
339.20
Ist payment, May Ist, 1.846..... $50.00
Interest to Jan. 1st, 1847, (8 mon.,). 2.00
2d payment, Nov. 16th, 1846... 100.00
Int. to Jan. 1st, 1847, (1 mon. 15 da.,).75
$152.75 152.75
Ans. Balance due Jan. 1st, 1847. $186.45'I,-"~~~l~" —----— ~cl
PERCE NTAGE. 119
(2) Principal........... 540.00
Interest to Jan. 1st, 1848, (10 mon.,)... 36.00
$576.00
1st payment, May 1st, 1847..... $90.00
Int. to Jan. 1st, 1848, (8 mon.,). 4.80
2d payment, July 1st, 1847... 100.00
Int. to Jan. 1st, 1848, (6 mon.,)... 4.00
3d payment, Aug. 1st, 1847.. 150.00
Int. to Jan. 1st, 1848, (5 mon.,).. 5.00
4th payment, Oct. 11th, 1847... 180.00
Tnt. to Jan. 1st, 1848, (2 mon. 20 da.,) 3.20
$537.00 537.00
Ans. Balance due Jan. 1st, 1848 $39.00
A TICLE 228.
(l) Principal of the note.......... 875.000
Interest to Aug. 10, 1834, (43 mon.)... 188.1125
Amount............... $1063.125
1st payment deduct...... 260.000
Due Aug. 10, 1834, forming a new principal, $803.125
Interest to Dec. 16, 1835, (16 mon. 6 da.,). 65.053
$868.178
2d payment deduct........... 300.000
Due Dec. 16, 1835, forming a new principal, $568.178
Interest to Dec. 16, 1836, the time between
the payments being less than 1 year.. 34.090
8602.268
Amount of $50 fromn March 1, 1836, to Dec. 16,
1836, (9 mon. 15 da.,) to be deducted. 52.375
Due Dec. 16, 1836, forming a new principal, $549.893
Interest from Dec. 16, 1836, to Dec. 16, 1837 32.993
$582.886
Amount of $150 from July 1, 1837, to Dec. 16,
1837, (5 mon. 15 da.,).154.125
Due Dec. 16, 1837, forming a new principal, $428.761
Interest to Sept. 1, 1838, (8 mon. 15 da.,). 18.222
Ans. Due on settlement. A446.983+
- 0- — ~ —-- ---
120 KEY TO RAYN'S THIRD BOOK.
ARTICLE 230.
(2) $ 6OX.05-$ 3.00: $12 —$ 3=4 yr. Ans.
(3) 8375X.08=$3.000: $90. — 30=3 yr. Ans.
(4) 8225X.04=-$ 9.00: 66 —$ 9=71 yr.
7-~ yr.=7 yr. 4 mon. AJIS.
(5) 2500X.06=$15.00: $34.50-. $15
=2.3 yr.=2 yr. 3 mon. 18 da. Ans.
(6) 860X.06-$3.60: 813.77. $3.6
-3.825 yr.=3 yr. 9 mon. 27 da. Ans.
(7) $800-$500=8300: $500X.10 — $50:
$300-. $50=-6 yr. Ants.
(8) $7938 — 600-$198: $600X.09=$154:
$198_ $54=3- yr.=3 yr. 8 mon. Anrs.
(9) $200X.06-$12: $200-.-$12=162 yr.=16 yr. 8 mon. Ans.
(10) $100-.$4=25 yr. Arts. (11) $100-.o$5=20 yr. Ants.
(12) $100- - 7=14- yr. Aits. (13) 100 *.$8S=121 yr. An-s.
(14) $100 r$ 10-10 yr. Ains. (15) $100.$12-=8 yr. Ans.
(16) 4110* —$16=64 yr. Ans.
(17) $100X-3=300: $300-('100=$200: $200. +815
-131 yr. Ans.
$100X4=$ 400: $400 — 100= 300: 8300-s-$15
=20 yr. Anls.
ARTICLE 231.
(2) $80\,.01=$.80: $.80X5=$4.00: $28 $4=7 % Ants.
(3) $300X.01=83: 3X1~=$5: $3085-56 % Ans.
(4) $1000X.01=10: $10X2~-=$25: $200-.-$25=S c ArZs.
(5) 8260X.01:$2.60: $2.60X2=5.20:
$23.40 — 85.2=4.5=4- % Ans.
(6) $640X.01=$6.40: 86.40X6=$38.40:
$110.40-.-$38.40-2.875-=2 7 Ans.
_a —- _- ---- -- -- - -~~~l~r""~
PERCENTAGE. 121
ARTICLE 232.
(2) $.05X3=$.15: $8.25 — $.15=$55 Ans.
(3) 8.05X3=$.15: $341.25 -$.15=-2275 Ans.
(4) Int. of $1.00 for 16 mon., at 6 % (Art. 223)=-$.08:
92.26 —. $.08=$28.25 Ans.
(5) $1023.75-.-$.05=$20475 Ans.
(6) Int. of $1 for 12 yr. 3 mon. 20 da. at 9 %o (Art. 222)
-$1.1075: $525.398 —. 1.1075=$474.40 Ans.
ARTICLE 233.
(2) $200X.08=16: $200+$16-$216, 1st yr.
$216X.08=$17.28: $216+$17.28=-$233.28, 2d yr.
$233.28-$200-$33.28 Ans.
(3) $500X.06 —$30: $500$-830=$530, 1st yr.
$530X.06=$31.80: $530+$31.80=$561.80, 2d yr.
$561.80X.06-8$33.70:
$561.80+-833.708 —$595.508 Ans.
(4) $800X.06 —:$48.00: $48+$800=8848, 1st yr.
$84SX.06-= 50.88: $50.88+$848=$898.88, 2d yr.
$898.88X.06=$53.9328: $53.9328+-$898.88
-=952.8128, 3d yr.
$952.8128X.06=$57.168d-: $57.168+952.8128
=1009.98+ Ans.
(5) $250X.06=$15: $15+$250=$265, 1st yr.
$265X.0615.90: $15.90$: $15 9265-=280.90, 2d yr.
$280.90X.06-$16.854: $16.854+$280.90
=$297.754, 3d yr.
$297.754-$250 —$47.754 Ans.
(6) $300X.05 —$15: $15+$300 —$915, 1st yr.
$315X.05=$15.75: $15.75+$315=-$330.75, 2d yr.
$330.75X.05-,$16.5375: $16.5375+$330.75
=$347.2875, 3d yr.
$347.2875X.05= —$l7.364+-: $17.364+$347.287
=-364.65+, 4th yr.
$364.65-$300=64.65- Ans.
~ —---- ----- - ------- - - ---
122 KEY TO RAY'S THIRD BOOK.
(7) Int. of $1.00 for 6 mon. at 6 % (Art. 233) =.a03
$.03X200=$6: $6+$200 —-206, 1st 6 mon.
$.03X206=-$6.18: $6.18+$206=$212.18, 2d 6 mon.
$.03X212.18:$6.3654: $6.3654+$212.18=$218.5454,
3d 6 mon.
$.03X218.5454- =$6.5563+: $6.5563+$218.5454
— $225.101+, 4th 6 mon.
$225.101-$200-$25.101 Ans.
(8) 20 ro per annum is equal to 5 %o quarterly, the question
then is the same as to find the amount of $500 for 8
years, at 5 % per annum.
$500X.05 —$25: $25-$500=$525, 1st quarter.
$525X.05=$26.25: $26.25+-$525:$551.25, 2d quar.
$551.25X.05:$27.5625: $27.5625+$551.25
=$578.8125, 3d qr.
$578.8125X.05-$28.9406+: $28.9406+$578.8125
=-$607.7531, 4th qr.
$607.7531X.05-$30.3876+: $30.3876+$607.7531
-$638.1407, 5th qr.
$638.1407X.O5 —31.907 -: $31.907+$638.1407
=$670.0477, 6th qr.
$670.0477X.05-$33. 5023+ $33.5023+$670.0477
=8703.55, 7th qr.
$703.55X.05=35.177+: $35.177+8703.55
=8$738.727+ Ans.
(9) $300X.06=$18.00: 818.00+$300=318, Ist yr.
$318X.06=$19.08: $19.08+$318=$337.08, 2d yr.
$337.08X.06=$20.2248: $20.2248- -2=$10.1124:
$10.1124+- 337.08 —347.1924: $347.1924 —$300
=$47.1924 Ans.
(10) $100OX.06-= 60: 860+$10O00-$1060, Ist yr.
$1060X.06=$63.60: $63.60+$1060 —$1123.60, 2d yr.
Int. of $1123.60 for 8 mon. 15 da. (Art. 232) -=$47.753:
$47.753+$1123.60=$1171o 353: $1171.353-$1000
=$171.353 Ans.
PERCENTAGE. 123
ARTICLE 234.
(12) Tab. num. 1.790848X345-$617.84+ Ans.
(14) 30 yr.=20 yr.+10 yr. Tab. num. for 20 yr.-=3.207135:
Tab. num. for 10 yr.==1.790848. 3.207135X890
=2854.35015: 2854.35015X1.790848-$5111.70d$5111.70-$890-$4221.70+ Ans.
(15) 70 yr.-25 yr.+25 yr.+20 yr. Tab. num. for 25 yr.
-3.386355: Tab. num. for 20 yr.=2.653298.
3.386355X200-677.271: 3.386355X677.271
-2293.48: 2.653298X2293.48
-$6085.28+: $6085.28-$200.=$5885.28-'- Ans.
ARTICLE 236.-DISCOUNT.
(2) $.04X2=8.08: $1+-$.08=$1.08: $81 — $1.08
-$75 Ans.
(3) $.06X5 — $.30: $11q-$.30-=$1.30: 1300 —.1.30
=$1000 Ans.
(4) $.06X3=$.18: $1$.18 —$1.18: $4720-.- $1.18
— $4000 Ans.
(5) $1+$.06=$1.06: $257.50-$1.06=$242.924+ Ans.
(6) Int. of $1 for 1 yr. 10 mon. (Art. 223)-%.11:
$1-l4-.11=$1.11: $199.80 —$1.11-$180.00 Ans.
(7) Int. of $1 for 5 yr. 10 mon. (Art. 223) -$.35:
$1-+$.35=8$1.35 $6.75 —-1.35= — 00 Ans.
(8) Int. of $1 for 5 mon. (Art. 223) =$.025
$1-.$.025 —$1.025: $307.50 — $1.025 —$300.00 Anzs.
(9) Int. of $1 for 7 yr. 9 mon. 20 da. (Art. 223) =$.468:
$1+$46 1.4 —$1 68~: $493.20 —$1.4683
-$335.89+ Ans.
(10) $.06X4-$.24: $1+.24=1.24 24: $496 —$1.24
— $400, present worth. $496-$400-$96 Ans.
(11) $.06X2=$.12: $1+$.12=$1.12: $276.64 —$1.12-$247,
present worth. $276.64-$247.00- -$29.64 Ans.
124 KEY TO RAY'S THIRD BOOK.
(12) Int. of $I for 3 yr. 4 mon. (Art. 223)=$.20:
$1+$.20=$1.20: $330 —.$1.20-$275 present worth.
$330 —$275=$55 Ans.
(13) Int. of $1 for 8 mon. (Art. 223)=$.04:
$1+$.04=$1.04: $260 —$1.04=$250 Ans,
(14) $1+$.12=-1.12: $235.20 —. $1.12=$210 Ans.
(15) $1+$.07 —=1.07: $401.25-$1.07=$375 present worth.
$401.25-$375=$26.25 Ans.
(16) $1080X.06-$64.80: $64.80X10=$648=interest.
$.06X10-=$.60: $1+$.60=$1.60: $1080 —$1.60
=$675. $1080-$675= —$405 discount. $648-$405
= —$243 Ans.
(17) 1o of $.12-$.10: $1+$.10=$1.10: $1221+. $1.10
=$1110. $1122 —$1110=$12 Ans.
ARTICLE 237.
(18) $956.34 — 3 —$318.78, each payment.
$.05+$1 —=$1.05: $318.78 —. $1.05$303.60, present
worth 1st payment.
$.05X2-.10: $1-K.10=$1.10: $318.78-i-$1.10
=$289.80, present worth 2d payment.
$.05X3=$.15: $+1+.15=~1.15: $318.75+, $1.15
=$277.20, present worth 3d payment.
$303.60+$289.80+$277.20:$870.60 Ans.
(19) $1440.2 —=$720, 1st payment: $1440 — 3=$480, 2d payment: 81440-($720+$480)=$240, last payment.
Int. of $1 for 3, 6, and 9 mon. (Art. 223) is
$.015, $.03, $.045
$1+$.015=$1.015: $720 — $1.015=$709.3596+,-,
present worth 1st payment.
$1+$.03=$1.03: $480. — $1.03=$466.0194+,
present worth 2d payment.
$1+$.045=$1.045: $240 —. $1.045=$229.665+,
present worth 3d payment.
Adding the present worth of the three payments together,
their sum is $1405.044+ Anzs.
~~9- -- -- -- -- --— ~~~~~~~~~~A
PERCENTAGE. 125
(20) $700-($60+$180+$-260)-=$200, last payment.
Int. of $1 for 6, 12, 18, and 24 mon. (Art. 223) is
$.03, $.06, $.09, $.12
$1+.03=$1.03: $60 — $1.03-=58.2524+, present worth 1st
payment.
$1+$.06=$1.06: $180 —. $1.06=$169.8113+, present worth
2d payment.
$1+$.09=$1.09: $260. $1.09=$238.5321+, present worth
3d payment.
$1+$1.12=$1.12 $200.$1.12=$178.5714+, present worth
4th payment.
Sum of payments $645.167+ Ans.
ARTICLE 238.
(1) $.06X5=$.30: $1+$.30= $1.30: $650.-$1.30
=$500 Ans.
(2) $.05X9=$.45: $1+$.45=$1.45: $725 —$1.45
-$500 Ans.
(3) $.05X4=$.20: $1+$.20=$1.20: $571.20- $1.20
=$476, principal.
$571.20-$476=$95.20, interest. Ans.
(4) $.06X2==$.15: $1+$.15=$1.15: $690 — $1.15
=$600, principal.
$690 —$600=$90, interest. Ans.
ARTICLE 240. —BANIK DISCOUNT.
(2) $137X 93 — 60=212+. Int.-$2.12+Ans.
(3) $1780X93 -60=89X31=2759. Int.=$27.59 A8ns.
(4) $375X 33 -60=75Xl —. 4=2061. Int.==$2.06~ Ans.
(5) $165X 63- 60=33X21-.-4=1731. Int.=$1.731 Ans,
(6) $140X123 -60= 7X41=287. Int.=$2.87 Ans.
(7) $ 80X183 —60- 4X61=244. Int.=$2.44 Ans.
(8) $180X 33 60= 3X33=99. Int.=-99 cts.
8180 —$.99=$179.01 lAns.
~o~................. —------ - --- - ---— g
126 KEY TO RAY'S THIRD BOOK.
(9) $960X33 * 60=16X33-528. Int.= —5.28
$960-$5.28=-954.72 Ans.
(10) $875X93 —. 60-175X31-.4-1356-. Int,=$13.561
$875-$13.561z= $861.443 Ans.
(11) $3900X63 —. 60=-65X63=4095. Int.=8$40.95
$3900-$40.95-=3859.05 Ans.
(12) $2580X103 —. 60=43X103=4429. Int. at 6 %.
=$44.29 Int. at 5 %. (Art. 225, Rem. 3.,)
=- of $44.29-$36.908-. $2580 —$36.908
=$2543.09+ Ans.
(13) $3.50X225-$787.50, purchase price. $4X225-=900:
$900X183-60 —15X 1832745. Int.=$27.45
$900-$27.45-$872.55, proceeds of note.
$872.55-$787.50=$85.05 gain Ans.
(14) $1+$.07=$1.07: $535$. $1.07=-$500: $535-$500
=-$35, true discount. $535X.07-$37.45, bank discount.
$37.45-$35-$2.45 Ans.
(15) $.06X4=$.24: $1+$.24=$1.24: $1209+- $1.24
=8975: $1209- 975==$234, true discount.
$1209X.06-$72.54: $72.54X4=$290.16, bank discount.
$290.16-$234=-$56.16 Ans.
ARTICLE 241.
(2) Int. of $1 for 3 mon. 3 da. (Art. 223) =-.0155
$1-$.0155=-$.9845: $393.80 —-$.9845=$400 Ans.
(3) Int. of $1 for 63 d,. (Art. 223) )-$.0105:
$1-$-.0105 —.9895: $500 —$.9895=$505.305+ Ans.
REMARK.-For a full discussion of the difference between Bank
and true discount, see Ray's Higher Arithmnetic.
ARTICLE 243.-CASE I. PROFIT AND Loss.
(3) $150X.10=$15 Ans. (4) $874X.25-$218.50 Ans.
(5) $500X.12=860: $500+$60=$560 Ans.
(6) 8382.50X.04-$15.30: $382.50-$15.3.0-,367.20 Ans.
PERCENTAGE. 127
ARTICLE 244.-CASE II.
(2).x90X./ 25=22 90 cts.-+22'-=81.12' Ans.
$.90X.10=.09: 90 cts. —9 cts.=81 cts. $.81 Ans.
(3) $4.37>X(.33=1$1.458+: $4.375+$1.458=$5.83+Ans.
$4.37!X.20=.877: $4. 371,-$.871 —$3.50 Ans.
(4) $8.50X.20=$1.70: $8.50+$1.70=$10.20:
and $8.50-41.70=$6.80 Ans.
(5) $300 -40=$7.50 $7.50X.20 —$1.50: $7.50+-1.50
-$9.00 Ans.': $7.50-$1.50=$6.00 Ans.
ARTICLn 245.
(6) 10cts.+ y' of lOcts.-=llets. 15cts.+-i of 15cts.=16-cts.
20cts.+- 1- of 20cts.=22cts. 30cts+-l of 30cts.=33 cts.
40cts.+-T1 of 40cts.=44cts. 50cts.-+-l of 50cts.=55 cts.
60cts. -i —, of 60cts. —66cts.
(7) Scts.-+ of ScS.-t= ts. 12cts.+ ~ of 12cts.=13ects.
16cts.+ of Icts.=18Sts. 20cts.+ 8 of 20cts.= —22cts.
(8) lOcts -4- of l0cts.-12cts. 15cts.+- of 15cts.=18cts.
25cts.+- of 25cts.=30ct-.
(9) *$11+- of $1-$1.25 8$1.20+-1 of $1.20=$1.50
$1.50+- of $1.50=.$1.87- $2+f of $2=$2.50
$3+-! of $3=$3.75 $4+- of $4-$5.00
$6+-4 of $6=$7.50
(10) 25cts.+-1- of 25cts.=33~cts. 50cts.+-~ of 50cts.-66hcts.
(11) $2+- of $2=$3.00. $3 — of $3=$4.50
$4++ of $4=$6. $5+- of $5=$7.50 $7+1 of $7.
=$10.50
ARTICLE 246.-CASE III.
(2) $105 —75=$30 ~: -=.40-40 %o Ans.
(3) $ 40-$30=$10 ~='=.33!=33-1 2 Ans.
(4) 30 cts.-25 cts.=5 cts.:,5 =.20=20 o A s.
Q~~~~~~~~~~~~~~~~, 5 —
128 KEY TO RAY'S THIRD BOOK.
(5) 21 cts.-20 cts.=l ct.: — =.05=5 %o Ans.
(6) $9-$8$1: =-.12-=-12 o Ans.
(7) 30 cts.-24 cts.=6 cts.: --— =.20-=20 % Ans.
(8) $40X40=$1600:.-f664= -=.44-44. Ans.
ARTICLE 247 —CASE IV.
(3) $1+$.20 =-$1.20: $6- -$1.20-$5 Ans.
(4) $1-$.25 =$8.75:83-..75-$4 Ans.
(5) $1+$-.25 =$1.25:75 —. $1.25=$60 Ans.
(6) $1 —$.25 =8.75: 875 —8.75=$100 Ans.
(7) 81 —tI.10 =$1.10:.22-$1.10-=$.20 Ans.
(83) 81+-8.1281.1. 125 S X 8.10 1.125=$7.20 Ans.
(9) $1 —8.15 $=.85: 1.19 —$.85=$1.40 Ans.
ARTICLE 248. -PR.OBISCUOUS EXA IPLES.
(1) 6 cts.X30-$1.80: $1.80X.25-$.45 Ans.
(2) $250 __40=-6.25, purchase price per yard.,$6.25X.20
-81.25: $6.25+$1.25 —7.50 Ans.
(3) 1 hhd.=252 qt.: $.40X252-$100.80: 8100.80 —$75
=$25.80: 2580o=.342=342 a Ans.
(4) 2428921-1918604-510317 ~ 1. 3o,-7c-.265+
=26.5+ % Ans.
(5) 1519467-937903 581564: 9 o -A=. 62+
=6G2+ % Ans.
(6) 81+$8.10=$.10: $.42~. 81.10=$8.38-ll, purchase price.
$.51-$.3_8_ —.1= 2 -- 1T -=- 4O- T14=334 %O Ans.
o.S =~ —=T7 s
(7) 81+$.25-=$1.25: $1- 1.25=$.80, purchase price of rye.
$1X.25=,.25: 81-$.25 —$.75, selling price of wheat.
$.80 —.75=$.05 Ans.
b —--------- - - Q g
PERCENTAGE. 1 2'9
(8) $9.60X14 —$134.40, purchase price: $134.40X.20$26.88: $134.40+$26&.88,$161.28, what the whole must
be sold for.
814.40X5=$72.00: $12X4-$48.00: 14-(5+4)
— 5 pieces left.
$72.00+-848.00=$120.00: $161.28-$120-841.28,
what the remaining 5 pieces must sell for.
=$41.28-.-5-$8.256 A1S.
(9) $1-$.14=$.86: $1.29-~-..86 —1.50, purchase price,
$1.50X.14=-$.21: $1.50+$.21= —1.71 Ans.
(10) $1-$.15=$.85: $1.36 —.$.85-$1.60: $1.856 —81.60
=$.256:.~256=.16=16 O Ans.
(11) $1+$-.12-$1.12: $1.96. $1.12 —$1.75, purchase price.
$1.75X.16=$8.28: $1.75-$.28-$1.47 Anls.
(12) 8$1+-.16= $1.16: 8,1.682_ _$1.16=$1.45, purchase price.
$1.$45-$1.247=.203:?i. ~ -—.14=-14 %o,Ans.
(13) X.2,5-81.25: $100 -$1.25 —80 yd. A1s.
(14) $2.50X.20$.50: $2.50-$.50=83.00, selling price of
each cask. $3.00-.$.05-60 lb. Ams. The number of
casks has nothing to do with the solution.
(15) 812-8 $15=1.253: $1 —$1.25=$. 80-purchase price per
bu. $.80X.35=$.28: $.80+$.28=$1.08=selling price
per bu. $59.40 — $1.08=55 bu. Ans.
A R T I C LE 250.-A SS ESSMENT o T Xs.
(1) $500 — $125000=.004, or 4 mills on the dollar.,1650X.004=$6.60 Aizs.
(2) $1.25X1258-$-1572.50, aimount raised on polls.
$9057.60-$-1572.50=$7485.10, amount to be raisecl on
property. 87485.10 —$534650=.014
=i et. 4 m1. on $1?Ans.
130 KEY TO PAY'S THIRD BOOK.
(4) $ 00,) is taxed,14.00 (5) $3000 is taxed $42.00
5().) 2.8. 0 500 7.00
40. 50 80. so 1.12
3..'6.042 0..126
3 polls.. 3.75 4 polls 5.00
Aas. $21.152 Ans. $55.246
ARTICLJE 251.-AMIERICAN DUTIES.
(1) 100+182+264-546: Draft=1-2-3=6 11).
546 —-=540 1b.: 540X.15=81 lb.: 540-81=459 1b.
2 cts.X459-$9.18 Ans.
(2) 400 lb.-4 lb. draft-396 lb.: 396X.05 —19.8 lb. tare:
396-19.=8-376.2, net weight of each bale.
376.2X40=150S 1b1). net. $.25Xl15048-$3762.
$376209X.20= — 752.40 Ans.
(3) 196 lb.-2 lb. draft= —194 lb. 194X.15=29.1 lb. tare:
194 —29.=164.9 net: 164.9X75=-12367.5 lb. $.05X
12367.5 —$618.375 Ans.
A R,l T II GL E 252. -— P ARTNEtSl I P.
(2) i -['-8-'- of M232 —$87, A's share,
1,% 0, g of -;2322 —-- 145, B's share.
(3) $0i0-1t,0l0 —$0 -$300, whole stock,
o~ —o'-3 7 of $120-$28L, A's share.'<t00gQ —;: G of $120-$60, B's share......'. of 1~0,
I'o8O-0- t 1-A — ('f $120 $32, C's share.
(4) $200+400+$600 —$1200, whole s iock.
0 0 — I -1 4 — 00 1-. 0o — I
" of,427.26= 871021, A's share: - of $427.26
-— $142.42, B's share: and I of b427.26
l$213.63, C's share.
$427.26 —:-$1200-=35605:35.605 % gain. Ats.
(5) 1+-3+5 —9. - of $90=,-10: ~ —- of $90-$30:' of $90=850 Ans.
PARTNERSHIP. 131
(6) 2-1-3-+5-+7=17. v of $735.93=-$86.58:
7 of $735.93 —$129.87: -' of $735.93=-216.45:
-T7 of $735.93=$303.03 Ans.
(7) 3-*-6+9+11+13+1 l 7=59.
g9 of $22361 —1137: l6 of $22361=$2274:
9 of $22361=$3411 I: of $22361 —$4169:
1: of $22361=$4927 ~ of $22361-$6443 Ans.
(8), 4o 7` } o Since the denominators are the
same, the firactions are to each other as their numerators.
40+72+105=217. 4~V of 8692.23=$127.60:
72 91 -o 0
7,, of $692.23=$229.68: I of $692.23
=$334.95 Ans.
ARTICLE 253.-BANKRUPTCY.
(9) $175+$500+$600+$210-F$42.50+$20+$10=$1557.50.
As $1557.50 $175 $934.50 $105.00, A's share.
As $1557.50: $500 $934.50: $300.00, B's share.
As $1557.50: $600:$934.50 $360.00, C's share.
As $1557.50: $210 $: $934.50: $126.00, D's share.
As $1557.50: $42.50 $934.50 $25.50, E's share.
As $1557.50: $20:: $934.50: $12.00, F's share.
As $1557.50: $10 $: $934.50: $6.00, G's share.
(10) $234+$175+$326-$735: $492.45- 8735=$.67
- sum paid on each dollar of indebtedness.
$234X.67=$156.78: $175X.67=$117.25:
$326X.67=$218.42 Ans.
ARTICLE 254.-GENERAL AVERAGE.
(11,) 4 o -4 2; % o; 2 4 of 45=20, A's loss.! of 45=a15, B's loss: ~ of 45=10, C's loss.
(12) $100001500500 $25000. $11 25 $25000=.04.=
4a %~, general average. S2150X.04-=$96.75, A's loss.. of $10000=$3750: 93750X.04-=-$168.75 Ans.
g~~~~~~~ S~~~~~~~ g~~~~$70
132 KEY TO RAY'S THIRD BOOK.
ARTICLE 255.-PARTNERSHIP WITH TIMIE.
(3) 23X27=621: 21X39=819: 621+-819=1440.
621 6;9. 519 91.
T44 —Y-T6 9 19 -: -9-1 69iof $54-$23.283, A pays.
1-%u of $54=$30.711, B pays.
(4) $300X(5=$1500: $400X8=$3200: $500X3 —$1500.
$1500+$3200+$i1500=$6200.
1500o15 15.,5
6o2 ~ oo 3-22 e of $100-$24.191 I, A's aud
0200-02' 6200-3 1 0J 31
C's loss -f) of $100l $51.61i-9. B's loss.
(5) 6X30180: 5X40-=200: 8X28224.
180+200+224-604: 180 - - 200=__- 224 56jS~ of $18.12=$5.40, A: ~ ~~r of $18.12=-6, B
-1,'f- of $18.12=$6.72, C.
(6) A $300X 8-$2400, $300+100=$400,
$400X8=$3200, $2400+3200=$5600.
B $600\X10=$6000, $600-300-$300,
$300X6=$1800, $6000+1800=$7800.
C $500X12=$6000, $500+400=$900,
$900X 4=$3600, $6000+3600=$9600.
$5600+7800-i9600=$23000.
As 23000: 5600 $759: $184.80, A's.
23000: 7800:: $759 $257.40, B's.
23000: 9600:: $759: $316.80, C's.
(7) $800X12=$9600: $500X12=-$6000: 12 mon.-7 mon.
=5 mon.: $9600-$6000=3600: $3600 — 5=$720 Ans.
EQIUATIONI OlF PAYIMENTS.
ARTICLE 257.
(2) $2X4-=$ (3) $8X5=$40
6X8=48 4X8= 32
$8) 56(7 mon. Ans. $12) 72(6 mon. Ans.
ALLIGATION M.EDIAL. 133
(4) $250>2-=$500 (5) $100X 6-=600
500X5 —2500 75X 8 — 600
750(X8=6000 125X12=1500
$1500) 9000(6 mon. Ans. $300) 2700(9 mon. Ans.
(6) - of,200=$40.00
- of 200= 80.00:40X 0= 0
O8X 5-400
80X 10-800
-4200) 1200(6 mron. Ans.
ARTI CLE 258.
(2) Counting from April 2d, it is 90 days to the first payment, and 150 days to the second.
$200XA 90-=$18000
$300X150- 45000
$500) 63000(126 da. from April 2d, i-which will give
Aug. 6th Ans.
ARtTICLE 259.
(8) $103-.-$1.03 —100
$106 —$1.06 —$100
$103+ 1006-$209
$209 —,$200$ 9
$200X.06=,9$ 12
$9- -$12=.75 yr. —9 mon. Ans.
ALLIGATIO1N MEDIAL.
ARTICLE 260.
(2) 6 lb. at 3 cts.=18 cts. (3) 25 lb. at 12 cts.-$3.00
4,1b. at 8 cts.=32 cts. 25 lb. at 18 cts.= 4.50
I0 lb. cost 40 lb. at 25 cts.= —10.00
10 lb. cost 50 cts.
50 cts. — 10=5 cts. Ans. 90 lb. cost $17.50
$17.50 —*90-$.19 4 Ans.
- ------ ----- -- - - --- I-I-9
134 Y T'O RAY'S THIRD BOOK.
(4) gal. co"st $0.00 (5) 10 at $3-$30.00
12 gal. at 50- 6.00 12 at 4 — 48.00
- at 90 7:2.00
15 gal. cost $6.00
$6.00 —. 15=$.40 Ans. 30 worth $150.00
$150- -30 —$5 An..,
(6) 6 to 10= 4 hr.: 63~X 4=252~
10 to 1- 3' 70~X 3 —-210~
1 to 3= 2 75 X 2 —150~
3 to 7 — 4 73~X 4=292~
7 to 6=11 hr. 550X11-605~
24)1509~(62"~ Als.
ANALYS IS.
REMAaRxs. —The subject of analysis is one of the most important in Arithmetic. With a thorowuh knowledge of the elementary rules, fractions, and analysis, nearly all the operations
that are purely arithmetical may be solved without the aid of
special rules; because these rules are mostly derived from analytical solutions of particular questions. The pupil, however,
who desires to understand analysis fully, and apply it readily,
must first make himself well acquainted with Mental Arithmetic.
See Ray's Arithmretic, 2d Book,
ARTI3CLE, 261.
(2) $36 (3) $24 (5) 7bl). (6) 10 lb.
(7) lO yd. (9) 7' y. (10) 4. (i4L) 43.
(13) 2: 20. (14) 28: 27. (16) 0. (17) s.
(19) 18. (20) 28. (23) 16 cts.
(24) If i-fifth of a bushel cost 14 cents, 1 bushel will cost 5
times 14 cents, which are 70 cents; and 9T of a bushel
will cost y% of 70 cents, which are 63 cents. Ans.
(25) i-eighth will cost i-third of 3-eighths; if 8 cost 21 cents,
-} will cost - of 21 cents, which is 7 cents, and if S cost
7 cents, 1 yard, or 8-eighths, will cost 8 times 7 cents,
which are 56 cents; then q of a yard will cost -. of 56
cents, which are 48 cents. Ans.
ANALYSIS. 135
ARTICLE 263.
(2) $2.16 —24 —9 cts.: 9 cts.X>23-$2. 07 Anl
(3) $32.50-: 13=-2.50: 02.50X1i48-35 Ans.
(4) $3-:.4=~$2: 24 — $2 — 12 yd. Ans.
(5) $20.67-0 159 —$.13: 34.71~-L.13-267 yd. Ain s.
(6) $147. 0341=S —---: $41X9- 838 —" Ans.|
(8) 25 da.X17=425 da.: 425 da. — 10 42'- da. Ans.
(9) If 6 men can build a wall in 10 days, it will require O1
times- 6-60 men to build it in 1 day, and 1-fifteenth of
60 men-u4 men, to build it in 15 days. tAns.
(10) If 5 men eat a barrel of flour in 12 days, 1 man will eat
a barrel in 5 times 12 days, which are 60 days, and 4 men
Awill eat a bahrel in i-fourth of the time of 1 man, that is,
in - of 60 days —15 da. Ans.
(11) 10 hir.X7=70 hr. to do the work: 70 hr. — 12 hr.=5o
da. of 12 hr. each. Ains.
(12) If 18 men reap 72 acres in 7 day}s, 1 man would reap it
in 7X1S —126 days, and 8 men in - of 126 —15:3 da. Ans.
(13) Each 9 sheep are worth 5 hogs: in 54 sheep there are
6 times 9 sheep; then 54 sheep are iworth 6 timies 5 hogs,
which are 30 hogs. Als.
(14) In 42 gal. of wine there are 14 times 3 gal. of wine;
hence 42 gal. of -Wine are worth 14 times 7 gal. of cider
-98 gal. Atns.
(15) If a 3 cent loaf weigh 8 oz., when flour is $4 a barrel,
if the flour were $1 a barrel it should weigh 8 oz.X4
=-32 oz.; and when flour is $5 a barrel, it should weigh
1-fifth of 3. oz. —64 oz. Ans.
(16) If 3 stacks of hay keep 12 horses 5 mon., they would
keep 1 horse 5 monr.X12=60 mon., and 20 horses.l-tentieth of 60 inon. —3 non. Ans.
(17) If o of a yard cost 40 cts., - of a yard will cost - of 40
cts.=20 cts., and w will cost 3 times 20 cts. —-60 cts. Asis.
136 i:KEY TO RAY'S THIRD BOOK.
(18) If 3 yd. cost $2, one yard will cost 3 times -2= $6, and
3 yd. will cost 4 of $6-$4.50. Ans.
44 1 of
(19) If A of a tun cost $4.;25; one-seventh will cost of
$4.25 —$.85, and a tun will cost 7 times $.85 —= 5.95:
then —' of a tun will cost - of $5.95 —3.85 Ans.
(20) If T9 of a cord cost $2.52, one-eleventh will cost -a of
S2.52$ —.28, and a cord will cost 11 times $.28 =3.08
then -1 of a cord will cost'- of $3.08 —-2.86 Acs.
14'j' 14
(21) If a lb. cost e$, one-seventh will cost - of $> —-, and
a lb. will cost 7 times $'Y —4; then 83 Ib. will cost, of: —,44 ~A's.
ARTICLE 264.
(23) If 2 men can earn $16 in 4 days, 1 man will earn ) of
$16=88 in 4 days, and in 1 day he will earn l of $8
=$2: then if 1 tman earn $2 in 1 day, 7 men will earn 7
times $2=$14, in 1 day; and in 3 days they will earn
3 times $14=$42 Ans.
(24) If 2 men can build 12 rd. in 9 days, 1. man can build 4
of 12 rd.=6 rd. in 9 days, and in 1 day he can build W -
of 6 rd. —= rld.: then if 1 man can build 3 rd. in 1 day,
7 men can build 7 times' rd. —4 rd. in I day, and in 6
days, they can build 6 times 42 rd. —28 rd. Ants.
(25) If 11 A. can be plowed in 5 da. with 15 oxen, it would
require 15X5=75 oxen to plow 11 A. in one day; and,
since 33 A.-3 times 11 A., it would require 75 oxen/3
— 225 oxen to plow 33 A. in 1 day; and to plow 33 A. in
9 days, it would take 4 of 225 oxen —25 oxen Arms.
ARTICLE 265.
(26) Since the first pipe fJills 41 gal. in 1 hr., and the second
emp)ties 6 gal. in the same time, therefore there are left
in, 41 —6-35 gal. per hour; it will therefore require
75o _05 0 hr.=:7 hr. 8 min. 342 sec. to fill the cistern.
(27) 600 —. 8-75 gal. filled by A in I hr.: 600 —. 12 —50 gal.
filled by B in 1 hr.: then 75+50-125 gal. filled by both
in 1 hr.; and fi00_.24_444 hr. Ans.
c~- ------------ ~ —4~ hr..Ans. —----
ANALYSIS. 137
(28) 900 —: 1090 gal1. filled by A in i hr. 900 — 12-75 gal.
eMlnlied by B in 1. hr.: then 90-75=15 gal. left in per
hr., and 900 — 15 —=0 hr. to fill the cistern.
(30) A does.,10 B 1 and C -1 in 1 day, hence in 1 day they
all do I —'r a- - )+6-d- 0 in one day, and
G 3 d -6-ay
therefore all do it in 1 -- das. Ans.
(3)1) Reasoning as in the preceding example, A does y, B 1',
C l-, and D1) in 1 day, and in 1 day they all do I- 1 J
_L 1l] + 40 Ef3Bo >2 0o o- 1 5%3; hence it will
require 1-s. — 8 —4t4 days to do the whole.
(32).9- - -- s_7, and 1 —I 1-4- 4 da. Ans.
Or, suppose the meadow to consist of 72 parts, then A
mows I of 72=9 parts'in one day, and B I of 72=8
parts in one day 9' —8=17 parts both do in one day;
72 —: 17=4f7 da. Ans.
(33) Since A and B together, mow it in 12 days, they mow TAin one day, and since A can mow it in 20 days, he can
mow ~6 in one day, hence B can mow -3
8- -3-0 in one day; hence he dan mow it in 1 —. 31
— 30 da. Ans.
(34) Since both eat it in 6 days, they eat ~ in 1 day, and since
the woman eats it in 15 days, she eats jl in 1 day; hence
the man eats ~ —a - in 1 day, therefore he will eat in 1 —T-= —10 da. Ans.
(35) The first will empty 1, the second 4j and the third 1 in
1 minute, hence they will all empty a-l + 1 - oe 3 A
-boo-__3 in 1. minute; and 1 ~-o o3fo1 mi n.
154- sec. Ans.
(36) Since A can mow 5 in 6 days, he can mow -- in - of
6 days, which is 2 days, and since he can mow I in 2
days, he can mow the whole in 5 times 2 days=10 days;
therefore he can mow 1 in 1 day. Again, since B can
mow - in 10 days, he can mow q in a of 10 days, which
is 2 days, and since he can mow -} in 2 days, he can
rev —------------ ---- - -- - --- - - ~
138:K'EY' TO RAY'S THIRD BOOK.
mow the whole in 7 times 2 days=14 days; therefore
he can mow ~- in 1 dty.
da,-L 7 L h — =-e -, the part both mow in one
ece 5 days, the time in which both can
10o it.
(38) $3000-$5500 —S2500: $2500-_ 2= —1250, less share,
and 4(1250-+$500-=$1750, greater share.
(39) The wife received %3$00 more than the son, and the son
$300 more than the daughter; hence, the wife received
$800+-f$300=81100 moloe than the daughter.
$300+$1100 —81400, whlat the wife and son received
more than twice the daughlter's.
$3500 —$1400-$2100:.$2100 —3-$700- daughter's
share: $;700I-$300-$1000, son's share: $1000+-$800
-=S1800, wife's share.
(41) In this case the minute hand mlust gain 25 min. on the
hour hand. NowT, according to the solution of Example
40, in the Arithmetic, }- of this 25 mil. is the distance
over whicih the minute hand must move to overtake the
hour hand. of mi. Ans.
(42) When the hour hand is at VIII, the minute hand is at
XII. If the hour hand did not move, the minute hand
would have to pass over 10 minutes to be opposite to the
hour hand, but since the hour hand does move, the
minute hand must gaia 10 minutes before it is opposite to
the 1,iour hand. It is then required to find what distance
the minute hand must move to gain 10 minutes on the
hour hand. Reasoning as in the Solution to Example 40,
we find T? of 10-10 0 min. Ans.
(43),s of $3276=$1365 Since -L of what A put in, was
equal to 6 of what B put in,. of 81365=$273, is -' of
what B put in; hence $273X9:2457, iwh-at B put in.
] of $45i7$1001; $3276 — $1001 —:$2275, w hich is.5 of what C put in; hence of 9975 —$45 1 of what
C put ill; hence C put in $4oo55X'X 4- 80. Since A
received a sum the o' cif w-ich was equal to the amount
--------------- ------- - - J
ANALYSIS. 139
he put in,. of the amount he put in, multiplied by 5,
loust equal his slhare; of $3276=,$546, and $546X5
=$2730, A's share. 6 of 1820 —$2184, C's gain.
$7000- ($2730-F$21 84) — $2086, Bs gain.
ARTICLE 266.
(45) By taking 1 lb. at 4 cts. I gain 1 ct.; and by taking 1 lb.
at 8 cts., I lose 3 cts.; hence, in order to equalize the
gains and losses, as often as 11). at 8 cts. is taken, there
must be taken 3 lb. at 4 cts.
(46) By taking 1 lb. at 3 cts. there is a gain of 3 cts., and
by taking 1 lb. at 8 cts. there is a loss of 2 ets.; hence,
to equalize the gains and losses, there must be taken
2 lb. at 3 cts., which will give a gain of 6 ets., and 3 lb.
at 8 cts., which will give a loss of 6 cts.
(48) On each lb. at 11 cts. there is a loss of I ct., hence on
10 lb. there will be a loss of 1 ct.><10-10 cts. On each
lb. at 8 cts. there is a gain of 2 cts.; hence, if 1 lb.
gains 2 cts., it is merely required to find how many lb.
miust be taken to gain 10 cts., which is eviden tly as many
as 2 cts. is contained times in 10 cts., that is, 5 lb. Alns.
ARTICE, 26{7.
(51) 1 hence -; of the n1umber is 8, and the number
is 12 times 8-96 Ans.
(52) 7 ——' I'; hence -r' of the number is 7, and the number
is 10 times 7-70 Anis.
(53) o:o —-, and 1 —8=, the part of his money left.
Then -7- of his money is $35, and - is - of 35, which is,,5n, and 15-fifteentihs, or the whole, is 15 timeos $5, which
is 87 Ans.
(54) 1I.-o +I 3o__...; hen ce 3 —73, and.o is v3 of 73, which is 1, and 3 o or the whole, is 30
times 1 —30 As.
(55) 1 4 -, part remaining; lihe then had left 20+15=35
cts.; - if is 35 cts., wh of 35 cts., hich is ts is, and
the whole is 9 times 7 cts. —63 cts. 4A1s.
I140 ICKEY TO RAY'S THIRD BOOK.
(5r6) i —; 43a of -1-:. 1_ l2 part left; then 10
cts.; and a of 10 cts., which is 5 cts, is 1'1; hence 5is
or the whole, is 15 times 5 cts.-75 cts. Ans.
(57) Of each 4 days he worked 3, earning $6, and idled 1 day,
forfeiting $1; hence, for each 3 days he worked he received 85, and since he received $25, which are 5 times
$5, he worked 5 times 3 da.=15 da. A7ns.
(58) -1 I~ ]__ 0l _ 44 o 1 9 I = part bearing
pears; hence o is 4, and 4O-, or the whole, are 20 tinmes
4=80 Anls.
(59) As many, (1,) as many more, (1,) half as many more,
({-,) and one-fourth as many more, (1-,) are equal to 1+
1i1+ -- 4 - --; then since 1 of the number are 110,
one-fourth will be ~w of 110, which is 10, and the whole,
or 4-fourths, will be 4 times 10=40 Ans.
(60) 1-1 —,-:=part left at close of 1st day.
I of 4-13; 4- 9 1- — part left at close of 2d clay.
4 196o6-94; 9 6 7part lef at close of 3d day.
4 of 674'2 7 27_ _ -Part leftatcloseof 4th day.
Then since 5 —$1.62, I-eighty-first part of $1.62, which
is 2 cts., is part of the whole; hence 256, o1 the
whole, are 256 times 2 cts.=$5.12 Ans.
(61) i +-3:4part of capital at close of 1st yr.
-- Of ( —; f+- 1 2 — partof capital at close of 2d yr.
{ of 12 3; 15'- 3a=9part of capital at close of 3d yr.
Then since 4=-$3375, one-ninth of $3375, which is $375,
will be one-fifth of the capital, and s, or the whole, will
be 5 times $375-81875 Ans.
EXCHANGE OF CURiRENCIES.
ARTI C L E 269.
(1)',r5X20+,3 os.=103 s.: 103 s.X12+2 d.-1238 d.:
1238 d.X4+1 far.=4953 far..Ans.
(2) 8675 far.-;-4=21683 d.: 2168s d. -12-2180 s. 8. d.
180 s. — 20 —9 Ans. ~9 8- d.
X -I —----— c —--- — o
EXCHANGE OF CURRENCIES. 141
(3) z s. d. (4) ~ s. d.., 0 64 17 6 5
5 10 1 15 5 1
2S II-) 4 Ants. 9 0 5'.
Arzs. 1.1 G 0
(5) z S. d. (6) z d.
3 12 2 6)2s o o1.0
Ans. 4 3 54
Ans. 28 17 8
(7) ~.625X\20-12.5 s. ~.5 s.X12-6 d. Ans. 12 s. 6 d.
(8) ~.796875X20-15.9375 s.:.9375 s.X12 —11.25 d.:.25 d.X4=1 far. — d. Ans. 15 s. 11'- d.
(9) 7s. 6d.=-90d,: ~~1-240d.: 90o= -~.375 Ans.
(10) 8 s. 9d.=105 d.: ~1=240 d. ~: I0=1~..43 75 Anls.
ARTICLE 270.
(11).~75 10 s.=~75.5: ~75.5X.04=~3.02: ~3.02X2=~47.55: ~.55X20-1 s. Ans. ~7 11 s.
(12) 12 s. 6d.=~.625: ~85.625X.065=~: ~5.1375X
1- (1 yr. 9 mon.) —~8.990625: ~.990625X20
=19.8125 s.:.8125 s.X2 —=9.75 d.:.75 d.X4=3 far., or
4 d. Ans. ~8 19 s. 93 d.
ARTICLE 271.
(4) ~25X4.84-$121 Ans.
(5) 8 s.=~.4. ~15.4X4.84=$74.536 Ans.
(6) 15 s. 9 d.=~.7875. ~36.7875X4.84=$178.05+ Anzs.
(7) $179.08 —.$4.84=~37 Ans.
(8) $124.388- -$4.84=.~25.7=~25 14s. Ans.
(9) $1000 *. 4.84 —~206.61157+=-~206 12 s. 2{ cl. + A~ns.
e~ ~~ —-— ~ —--- -- t) —----— ~
142 KEY TO RAY'S THIRD BOOK.
ARTICLE 272.
(2) ~150OX4-=$ 6664: $GC66X.08S —$3.3 — $53- prem.
$6i66-+ 853'3 — 87o An7s.
(3) 10s.=~.5 S~0.5X44 —$357.777+: $357.777X.09.=$33.988+ prem.: 8357.777+$33.988 $391.76 — Aizs.
ARTICLE 273.
(1) $120.50X-o-3=~36.15_ —~36 3s. Ans.
(2) $ 75.25X =-~30.1 =-~30 2 s. Ams.
(3),l9SX —~36.75=~36 15 s. Anzs.
(4) ~30 115 s.=9~30.75: ~.30.75 5 —. 1 —-102.50 Ans.
(5) ~25 17s.-~25.85 5- 64.625 A~-s.
(6) ~29 8 s.-=~20.4 ~29.4 -7-4-812G.00 Ans.:LrnTICLE 274.
Ex. 1.166 fr.- -5.30 fr.=$220 LIEns.
ART ICL E 276.-DU D ECI ALS.
(2) ft. in. (3) ft. in. (4) ft. in.
53 5 7 8 6
1 5 1 10 7 3
5 3 //7 5 59 11 3
2 23 4 7 10 2 1 8 3
Ans. 7 5 3 Ans. 10 2 10 Ans. 62 0 1.1 3
(5) ft. in. " (6) ft. in. "/
8 4 6 4 5 6
2 7 4 2 3 5
16 9 0 "' 8 11 0
4 10 7 1 1 4 ////
2 9 6 1 10 3 6
Ans. 21 10 5 0 A'ns. 10 2 2 9 6
_2 ~ ~ ~ ---.,,~ — M ------- -
INVOLUTION. 143
ARTICLE 2979 -INVOLUTION.
(1) 1-4-9-16-25-36-49-64 —81-1100 121.
(2) 1-8-27 —64. (3) 4 4 1
(4) -1-8 27 (5) 16 —32-81. -
(6) 65X665=4225 Ans. (7) 25X25X25z-15625 Alw.
8) 2 = 34 -2.231 X,3-4-2 72 4 1S.
(9) 12.. 2 5 X 25 X 1, -L5S,_a-19O38 Alw.
(9) l,2 — ~ 41953~ Ans.
(10) 13X13X13X1313-28561 Ans.
(12) 9X9X9X9X9X99-531441 Ans.
(13).025X.025X.025X.025-.0000003906250 Ans.
(14) 14 3 — 14X 4X14=2744 Ans.
(15) 194=19X19X19X19=130321 Ans.
(16) 2 1 X- 7 X 7X 7 X7 -80-69 4j Ans.
(17).09 6Q).09X.09X.09X.09X.09X.09=.000000531441 Ans.
ARTICLE 286.
(2) 529(20+3-23 Ans.
400
20 129
2 J29
40
3
43J
ARTICLE 287.
(6) 625(25 Anls. (7) 656i(81 Ans. (8) 30625(625 Ans.
4 64 36
45)225 161)161 122)306
225 161 244
1245)6225
6225
144 KEY TO RAY'S TilIRD BOOK.
(9) 1679616(1296 Ans. (10) 5764801(2401 Ans.
1 4
22)67 44)176
44 176
249)2396 4801)4801
2241 4801
2586)15516
15516
(12) 987606329(31427 lAns.
9
(11) 4;3046'721(6561 Ans. 61)87
36 61
1125)704 6)24)2665
625 2496
1 306)7967 628.2)16963
7836 12564
13121)13121. 62847)439929
13121 439929
(13) 289442169(17013 Ans. (14) 234.09(15.3 Ans.
1 1
27)189 25)134
189 125
3401)4421 303)909
3401 909
34023)102069
102069
~~~ -----— ~~~~~~~~~~~ —---------- --------
EVOLUTION. 145
(15) 145.2025(12.05 Adns. (16).0196(.14 Ans.
22)45 24)96
~~~~44 96
2405)12025
12025
(1.7) 1.008016(1.004 Ans. (18).00822649(.0907 Ans.
i 81
2004)008016 1807)12649
8016 12649
(19) 1/2535I, 1/729=.27, V., s A nes.
(2 0) — 3111 169-13, As I.
(21) 3O... —-", Ans.
(22) 10(3.162277+ Ans. (23) 2(1.41421+ A.ns.
9 1
61)100 24)100
61 96
626)3900 281)400
3756 281
6322)14400 2824)11900
12644 11296
63242)175600 28282)60400
126484 56564
632447)4911600 282841)383600
4427129 282841
6324547)48447100
44271829
Key. —10
1 146 KEY TO RAY'S THIRD BOOK.
(24)) ~2-.666666+(.S1649+Ans. (25) 6=6.4(2.5298+- Ans.
64 4
61i)266 -45)240
161 225
1626)10566 502)1500
9756 1004
16324)81066 5049)49600
65296 45441
163289)1577066 50588)415900
1469601 404704
(26) 1/6241=79 Ana.
ARTICLE 288.
(l) 16=2X2X2X2: V16=2X2= 4 Ans.
(2) 36=2X2X3X3: V/36=2X3= 6 Ans.
(3) 100=2X2X5X5: /100=2X5=10 Ans.
(4) 225=3X3X5X5: /22-5=3X5-=!15 Ans.
(5) V1/(16X25)= 4X5 —20 Ans.
(6) V(36X49)= 6X7-=42 Ans.
(7) /(64XS1)= SX9=72 Ans.
($) /(121X25)=1]lX5=55 AnS.
ARTIC LE 290.
(1) 30-= 900 (2) 812= 6561
402=1600 1082=11664
I/2500-50.Ans. 1/18225 —135 Alns.
(3) 100-=_10000 (4) 452=2025
60-2= 3600 602=3600
1/6400= —80 Ans. 1/5625=75 Ans.
(5) 36 -1296
242= 576
1/720=26.83+ Ans.
EVOLUTION. 147
(6) 60;=3600
37'2=1369
1/2231 —47.2334+=wiidth of street from foot of ladder on one side.
60,2=3600
232= 529
V3071-55.4166+=- width of street firom foot of ladlAs. 1.02.65 der on the other side.
(7) 6002%360000
140'.= 19600
V/340400=583.43+
100 —2= 50.
Ans. 533.43+
(8) 202-400 The square root of 656, will give the
162=256 length of the diagonal line joining
656 opposite corners of the floor of the
room: this is the base of the triSquare of base=656 angle, of which the hypotenuse is
122 —per.2=144 required.
800
1/800=28.28+ Ans.
ARTICLE 291.
(1) 1/4096=64 Ans.
(2) 4X4=16: 16X9-=144: i/144=12 Ans.
Or, /(16X9)=4X3 —12 Ans.
(3) 1/43560=208.71+ ft., side of acre.
43560 —. 2=21780: 1/21780-147.58+ ft., side of half A.
43560. 4=10890: 1/10890-104.35+ ft., side of qr. A.
(4) 10 A.=1600 P.
2 A.-=2000
1/3600=60 Ans.
148 KEY TO RAY'S THIRD BOOK.
ARTICLE 295.
1728)10+2=12 root.
1000
10X0OX3=300 728
10X 2X3= 60
2X 2 = 4
364 728
ARTICLE 296.
(5) 91125(45 Ans. (6) 195112(58
64 125 [Ans.
27125 70112
4X4X300=4800 5X5X300=7500
4X5X 30= 600 5X8X 30-1200
5X5 = 25 sX8 = 64
5425 27125 8764 70112
(7) 912673(97 Ans. (8) 1225043(107
729 1 LAns.
183673 1X 1X300= 300 225
9X9X300-24300 10X1OX300=30000 225043
9X7X 30= 1890 10X 7X 30= 2100
7X7 = 49 7X 7 = 49
26239183673 32149 225043
EVOLUTION. 149
(9) 13312053(237 Ans.
8
2X2X300=1200 5312
2X3X 30- 180
3X3 = 9
138914167
23X23X300- 158700 1145053
23X 7X 30- 48301
7X7 = 49
1635791145053
(10) 102503232(468 Ans.
64
4X4X300=4800 38503
4X6X 30= 720
6X6 = 36
1 33336
46X46X300-634800 5167232
46X 8X 30- 11040
8X 8 = 64
645904 5167232
(11) 529475129(809 A-is.
512
SX 8X300= 19200 17475
80X80X300=1920000 17475129
80X 9X 30= 21600
9X- 9 -- 81
1941681 17475129
-~~~~~~~~~~~~~~~~...
150 KEY TO RAY'S THIRD BOOK.
(12) 958585256(986 Ans.
729
99XX300=24300 229585
9X 8X30= 2160
8X8 - 64
26524 212192
98X98X300-2881200 17393256
98X 6X 30= 17640
6X 6 - 36
2898876 17393256
(13) 14760213677(2453 Ans.
8
2X2X300o=1200 6760
2X4X 30- 240
4X4 = 16
1456 5824
24X24X300-172800 936213
24X 5X 30= 3600
5X 5 - 25
t76425 882125
215X245X300 —18007500 54088677
215X 3X 30- 22050
3X 3 - 9
18029559 54088677
EVOLUTION. 151
(14) 128100283921(5041 Ans.
125
5X 5X300- 7500 3100
50X50X300-750000 3100283
50X 4X 30= 6000
4X 4 = 16
756016 3024064
504X504X300=76204800 76219921
504X 1X 30= 15120
1X 1 = 1
76219921 76219921
(15) 53.157376)3.76 Ans.
27
3X3X300=2700 26157
3X7X 30= 630
7X7 -- 49
3379123653
37X37X300=410700 2504376
37X 6X 30= 6660
6X 6 = 36
417396- 2504376
(16).i99i76704(.584 Ans.
125
5X5X300=7500 741]76
5X8X 30=1200
8X8 = 64
8764 70112
58X58X300=1009200 4064704
58X 4X 30= 6960
4>( 4 = 16
1016176 4064704
152 KEY TO RAY'S THIRD BOOK.
(17) V3216- 6. A (18) U/2744-14. it.,4
Ans3.-7, AAs. T
V]343=7. Is /6859-19 s.
48778 24389 V/24389-29,
118638 59319' V59319-39.
(20) Alo4n. /72 99 Ans -
5-T-T'2, &/125=5
(21) 2(1.25992+ Ans.
1
1XiX300=300 1000
IX2X 30:= 60
2X2 4
364 728
12X12X300=43200 272000
12X 5X 30= 1800
5X 5 -- 25
45025 225125
125X125X300=4687500 46875000
125X 9X 30= 33750
qX 9 = 81
4721331 42491979
1259X1259X300=475524300 4383021000
1259X 9X 30- 339930
9X 9 = 81
475864311 4282778799
12599X12599X300-47620440300 100242201000
12599X 2X 30-= 755940
2X 2 4=
47621196244 95242392488
EVOLUTION. 1 53
(22) 9(2.08008+ Ans.
8
2X 2X\300- 12001000
20X20X300=120000 1000000
20X 8X 30= 4800
8X8 - 64
124864 998912
208X208X300=-1 2979200 1088000
2080X2080X300=1297920000 1088000000
20800X20800X300=129792000000 1088000000000
20800X 8X 30= 4992000
8X 8 = 64
129796992064 1038375936512
(23) 200(5.84803+ Ans.
125
5X5X300=7500 75000
5X8X 30-1200
8X 8= 64
8764 70112
58X58X300=1o009200 4888000
58X 4X 30= 6960
4X 4- 16
1016176 4064704
584X584X 300-1 02316800 823296000
584X 8X 30= 140160
8X 8 — 64
102457024 819656192
5848X 5848X300- 10259731200 3639808000
58480X58480X300=1025973120000 3639808000000
58480X 3X 30- 5263200
3X 3- 9
1025978383209 3077935149627
154 KEY TO RAY'S THIRD BOOK.
(24) 9 =9.166666+(2.092+ Ans.
8
2X 2X300- 12001166
20X20X300-120000 1166666
20X 9X 30=- 5400
9X 9- 81
125481 1129329
209X209X300=13104300 37337666
209X 2X 30- 12540
2X 2-= 4
13116844 26233688
(25) U1953.125 —12.5 ft. Ans.
(26) 3X3X3=27=cu. in. in a cube, of which each side is
3 in. 1728 —27=64 Ans.
(27) 2X2X2=8= cubic inches in the cubical box.
XA4X=g14-=- solidity of a cubical block, each side of
which is ~ of an inch. 8 — - — 512 Ans.
(28) A solid half foot is a cube, each side of which is 6 in.;
it contains 6X6X6=216 cu. in.; half a cubic foot=- of
1728=864 cu. in.; 864-216=648 cu. in. Ans.
cu. ft. cu. ft. ft.
(29) 288X216X48:2985984, Vj2985984=144 Ans.
(30) 1728X3=5184,.J5184-17.306+ inches Ans.
ARTICLE 297.
(31) 63-216; 3 —27. As 216: 27:: 32 lb.: 4 lb. Ans.
(32) 11Xl1X11=1331 Ans.
ARTICLE 298.
(1) 216=2X2X2X3X3X3: V216=2X3=6 Ans.
(2) VJ (27X 64)=3X4=12 Ans.
(3) V (125X343)=5X7=35 Ans.
ARITHMETICAL PROGRESSION. 155
ARITPIHME TICAL PRO GRE SSION.
ARTICLE 31. —CASE I.
(3) 50-1-49; 49X 3+2-2149 Ans.
(4) 100-1 —99; 99X5+9-504 cts.=-$5.04 Ans.
(5) 54 —1=53; 53X2=106; 140-106-34 Ans.
(6) 20 —1-19; common difference=$216-$200=-$16;
$16X);19+$216-$520 Ans.
(7) 99 —1=98; 98X~=854; 329-85-3=243~ Ans.
A R TICLE 302. —CASE II.
(2) 300-3=297; 10 —1=9; 297-9 —-33 Az2s.
(3) 50-5= 45; 10-1 —9; 45-*-9- 5 Ans.
ARTICLE 303.
(1) 49 —544; 44-.-4=-11; 11-1=12 Ans.
ARTICLE 304.-CASE III.
(2) 50+ 2-52; 52X24-1248; 1248-.2=624 Ans.
(3) 1+12=13; 13X12= 156; 156 —-2= 78 Ans.
(4) 10000 —19999; 9999X1+1=10000, last term;
l10000=1000l; 100001X0000=10001000100010000;
100010000 - 2 —50005000 Ans.
(5) The number of terms is evidently 100. The boy travels
6 yards to put the first apple in the basket, 12 the
second, and so on; hence, the first term is 6, and the
common difference 6.
100-1=99; 99X6+6=600, last term.
6+600=606; 606X100-60600; 60600- 2=30300 yd.
30300 yd. —17 mi. 380 yd. Ans.
(6) 30 —1=29; 29X-1-7Z-1; 30 —7-224 mi. distance traveled last day. 30+22z=-,52!-; 521X30 —15821 mi.;
1582_.2 —=7917- mi. Ans.
>4 _ _ __ - - -- --- - -
156 KEY TO RAY'S THIRD BOOK.
(7) Common difference=193X2=386 in.
60-1=59; 386X59+193=22967 inches, distance fallen
in the last second.
193+22967-23160; 23160X60=1389600; 1389600- -2
=694800 in. —57900 ft. Ans.
GEOM(ETRICAL PROGRESSION.
ARTICLE 307.-CASE I.
(3) 212= 4096; 4096X2=8192 Ans.
(4) 48=65536; 262144 —.65536=4 Ans.
(5) Ratio=3; 39-19683; 19683X 10=196830 Ans.
(6) 234-17179869184; 17179869184X1=17179869184 Ans.
(7) 34 —16677181699666569, which, multiplied by 1, gives
the same number for the 35th term.
ARTICLE 308.-CASE II.
(2) 36=729; 729X10=7290, last term. 7290X3=21870;
21870 —10=21860; 21860-+-2-10930 Ans.
(3) 211=2048, 2048X1=2048=last term. 2048X2=4096,
4096-1=4095, and 4095 —1=4095 Ans.
(4) 41=4194304, 4194304X1=4194304, last term.
4194304X4=16777216, 16777216-1=16777215,
4 —1=3, 16777215 —3=5592405 cts.=$55924.05 Ans.
(5) Ratio=2,239=549755813888, this multiplied by 10, gives
5497558138880, for the last term.
5497558138880X2=10995116277760; from this subtract
10, and divide by 2-1-1, and we get for the
Ans. $109951162777.50
~4 —--- -- - - -----
GEOMETRICAL PROGRESSION. 157
(6) Ratio=3, 33'=617673396283947,
617673396283947X3=1853020188851841
3-1=2) 1853020188851840
$9265100944259.20 Ans.
(7).3X10-3; 10-1=9; 3 —9 — Ans.
(8) 100X1.04=104; 1.04-1=.04, 104_..04-2600 Ans.
(9) Ratio=3, -IX3=1; 3-1=2; 1. 2=- Ans.
(10) Ratio=2, IX2=1; 2-1=1; 1. — =1 Ans.
ARTICLE 309. —PEIUTATIO N.
(1) 1X2X3X4X5=120 Ats
(2) 1X2X3X4X5X6X7X8X9XlOX11X12
=479001600 Ans.
(3) 1X2X3X4X5X6X7X8 —-40320 days
=110 yr. 170 da. Ans.
(4) 1X2X3X4X5X6X7XSX9XlOX11X12Xl3Xl4X15
16X17X18X19X20X21X22X23X24X25X26
=403291461126605635584000000 Ans.
The subject of permutations and combinations properly belongs
to Algebra. (See Ray's Algebra, Part 2d.)
ARTICLE 311.- MENSURATION.
(1) 17X15=255 sqc. ft. Ans.
(2) 2 ft. 3 in.=2.25 ft.. 12 ft. 6 in.=12.5 ft., 2.25X12.5
=28.125=28- sq. ft. Ans.
(3) 15 in.=14 ft.; 1G ft.Xl1 ft.=20 sq. ft. Anas.
4 4 --------— ~
158 KEY TO RAY'S THIRD BOOK.
(4) ft. in.
12 6
1 2 Ans. 14-7 sq. ft.=14 sq. ft. 84 sq. in.
12 6 Or, 1 ft. 2 in.=14 in., 12 ft. 6 in.=150 in.
2 1 150X14-=2100 sq. in.==14 sq. ft. 84 sq. in.
Ans. 14 7
(5) ft. in. $ cts.
5 7 1.50
1 10 10
5 7 15.00
4 7 10 2 in.=-.25
10 2 10 6" =[.0625
4// =/.0416+
Ans. $15.354+
(6) 120rd.X 84rd.=10080 P.=63 A. Ans.
(7) 65 rd.X 65 rd.= 4225 P.=26 A. 1 R. 25 P. Ans.
(8) 35rd.X 16rd.= 560 P.= 3 A. 2 R. Ans.
(9) 693X693=480249 sq. ft.=1764 P.=11 A. 4 P. Ans.
(10) 15X 15=225, 225+225=450 sq. ft., 30X30=900 sq. ft.
900-450=450 sq. ft. Ans.
(11) 80- 10=8 ft. Ans.
(12) 10X144=1440 sq. in., 1440.10=144 in.=12 ft. Ans.
(13) 6>< 5=30sq. yd., 30 —1. —=20yd. Ans.
(14) 21 ft. 3 in.=21.25 ft., 13 ft. 6 in.=13.5 ft., 21.25X13.5=
286.875 sq. ft., 1 yd. —3=3.75 ft., 286.875~. 3.75=76.5
ft.=25.5 yd.=25. yd.
(15) 3X1=4 —- sq. yd., 4'~:3=9>K-4=6 yd. Ans.
0
MENSURATION. 159
ARTICLE 312.
(1) ft. in. (2) ft. in.
25 9 22 7
21 3 13 1.1
540 9 293 7
6 5 3 20 8 5
9)547 2 3 9)314 3 5.Ans. 60 sq. yd. 7sq. ft.+ 34 8 3 5.20
6.80.17777+ =cost 8sq. ft..00555+- -" 3'.00077+- " 5/"
Ans. $6.9841+
(3) ft. in. ft. in. ft. in.
20 6 20 6 6 3
20 6 16 3 4 2
16 3
16 3 328 0 25 0
- 5 1 6 1 0 6
73 6=circum — -
10 4 ference 333 1 6=ceiling. 26 0 6=-fire735 -0 [ofroom. [place.
24 6 Deductions. ft. in.
6 26 0 6 4 2
759 6 —walls. 29 2 7
333 1 6
333 69 0 29 2=door.
1092 7 6
94 2 6=deduc- ft. in.
9)998 5 tions. 3
6
110 sq. yd. 8 sq. ft. and 5/-=5 gsq. ft. 19 6
[Ans. 2
39 0-windows.
4e; —---------------- -
160 KEY TO RAY'S THIRD BOOK.
(4) ft. in. ft. in.
20 0 4 4
20 0 4
14 6
14 6 17 4=fireplace.
69 O=circumference (5) ft. in.
10. 4 [ofroom. ft. in. 35 4
32 83
690 0 32 8
23. 282 8
19 8 10
713 0 19
55 4=deductions. 17 4 9)291
9)657 8 55 4 32 sq. yd. 3sq.ft.
18 [6t
73 sq. yd. 8'
27 256
32
511 32
146 1576
1619.461 3 sq. ft._-! 6
s8/=2-2l 2 6' = 1 1
Ans. $19.73 Ans. $5.83
yd. (6)
21
15 5 ft. 3 in.S — f t.I — yd.
105
21 21><X1-363 sq. yd.
315 =sq. yd. in whole.
36-=sq. yd. in foot-path.
278-1 36
24 36
1112 216
556 108
6 27
$66.78 13.23
66.78
$80.01 Ans.
MENSURATION. 161
(7) ft. in. ft. (s) ft. in.
75 6=75.5 6 ii
12 6=12.5 5 4
4 3
1510 1 6
755
49 6
9)943.75=scq. ft. 8 3
104.8G+ sc. yd _ _7 —=windows in a line.
404 3
1048.6+ cts. 16 cts.
AcS. $10.486+ 2424
404
3 — 1 4
Atns. 64. 68
(9 ) ft. in. ft. t. (, o) ft. in.
36 3.=36.25 35 4 6
16 6- 16.5 30 6
18125 1050=sq. ft. 27 O-fire-place. l
21750 11Ill-deductions..
362 11100)9139 10
1100)5198.125 39s u res 8
5.98125 $5 84 O-stairs.
Ans. $46.95 27
Ans. $17.94375 111 0
(1) ft. in.
18 6
18 6
37 0
40
1100)1.4180 —q. ft.
14.80:squuares.
$3.50
74000
4440
AyasI. $51.8000
KEY TO RIAY'S THIRD BOOK.
A RT iCLE 313.
(1) 19215 —-.290 sq. ft. Ans.
(2) 44X1 5- -2=396 P.=2 A. 1 PI. 36 P. Amzs.
(3) 80X17_. -226S0 P.-16 A. 3 Pi. Aias.
(4) 12 ft. G in.=12.5 f., 16 ft. 9 in.=1!6.75 in.,
12.5X\16. 7'5 —e2=104.6875 sq. ft., 104.6875X12
-1256.5 ctCs.l'o 956 nS.
(5) 13+-14 15= 42, 42- -2=21, 21-1-3=8, 21-14=7,
21 —15=6, 21XSX7X6g7056, /7056=84 sq. ft. Ails.
(5) 2+9-34=9, 9 2 —45;4.5 —2=2.5; 4.5-3=1.5;
4.5 —4=r.5; 4.5x O.5Xlo5>\.58.4375; /8.4375
-2.9047375-.+ sq. ft. A1ns.
A P RT I CLE 314.
(1) 35-26=61; 61X11 —-671; 671 -2=335~ sq. in. Ans.
(2) 19-25=44; 44X1 32= —i 408; 1408 -2=704 P.
— 4 A. 1 R. 2P. Anms.
A rTIC LE, 315.
(1) 48oX3.1416=150 7968 ft. Ans.
(2) 4X301416-12.5664 ft.-12 ft. 6.7968 in. Ans.
(3) 791o2.4>3.116-=4857o59584 mi. Aas.
ARTkICLE 316.
(!) 15 —3.-141i —4.7746+ft. —4 ft. 9.905-in. An2s.
(2) 12 ft. 5 in.=12.416G6+; 12.4166- 3.1416=3.952338 ft.
=3 -ft. 11.428-.in. _ALs.
ARTICLE 3'17.
(3) 42-> -.2=2!; 21\ 921-41; 3.!416X4X41
=1385.4456 sq. f.. Ans
(2) 7 rods( 7.s; 7.5<7.5-56.25;; 3.1416X56.25
=176.715 P.-l1 A. 16.715 P. Anls,
c_~~~ —------------------ X
MENSURATION. 163
ARTICLE 318,
(1) 962.115 —.7854=1225; V1225=35=diameter;
35X3.1416=109.956 Ans.
(2) 160'..7854=203.71785077+; i/203.71785077
=14.2729+; 14.2729 —2 —7.1364+ rods;
or 117 ft. 9+in. Ans.
ARTICLE 320.
(1) 3 ft. 3 in.=3~ ft.; 4 ft. 4 in.=4~ ft.; 12X3vX4'
=169 cu. ft. Ans.
(2) 2 ft. 6 in.=2.- ft.; 1 ft. 9 in.=l- ft.; 6X2AXI
-264 cu. ft. Ans.
(3) 3 ft. 2 in.=3"; 2 ft. 8 in.=2-; 2 ft. 6 in.-=2 ft.
3VX2 —X21-= 1 X19SX =- -'211 cu. ft.
21!-X81 cts.=$17.10 cts. Ans.
(4) ft. in.
4 10
2 11
9 8
4 5 2
14 1 2
2 2
28 2 4/'
2 4 2 4///"'
Ans. 30 6/ 6// 4///
ARTICLE 321.
(1) 97 ft. 5 in.==97.416+ft.; 18 ft. 3 in.=18.25 ft.; 2 ft. 3 in.
=2.25 ft.; 97.416X18.25X2.25=4000.1445 cu. ft.
4000.1445. 24.75=161.6+P. Anls.
(2) 53 ft. 6 in.-53.5 ft.; 12 ft. 3 in.-12.25 ft., 53.5
X1.25X2=1310.75 cu. ft.=52.95+P. Ans.
~ —-— _- __ M —— ~ —---
164 KEY TO RAY'S THIRD BOOK.
(3) 53 ft. 6 in.-53.5 ft.; 12 ft. 6 in.=12.5 ft.; 53.5X12.5X2
=1.337.5 cu. ft.=54.0404+ P., 54.0404X$2.25
8-$121.59+ A4ns.
6 1_ 4x 5X 2 S_ X'4l S.
(i) 48~X16'Xl' —-. X2 — =1196- cu. ft.
1196 X20=23925 bricks.Ans.
(5) 15 in. —14 ft.; 32OX6XlI=2400 cu. ft.; 2400X20
=48000 bricks; 48000X$5.875-. 1000 —S282 Anls.
(6) 120X8Xl —1440 cu. ft. —248S320 cu. in. in wall.
SX4X2.25=72 cu. in. in each brick; 2488320-. 72
=34560 bricks iArs.
(7) 240X6X3=4320 cu. ft. =7464960 cu. in. in wall.
9X4X2=-72 cu. in. in each brick, 7464960 — 72=103680
bricks. 103680-.-1000=103.68; $3.25X103.68
=$336.96 Ans.
ARTICLE 3822.
(1) 2+2+2=6; 6- -2=3; 3-2-=1; 3 —2=-1; 3-2=-1.
3X1XIX1-=3; V3=area of base. 14X/3=141/3
=area of base. 14-/3=-/196XV/3=V/588
=24.2487+cu. in. Ans.
(2) 4X4=16;.7854X16=12.5664-area of end.
12.5664X12=150.7968 cu. ft. Ans.
(3) 18=18.5: 18.52=342.25: 342.25X.7854=268.80315
cu. in.=area of end. 268.80315X8=2150.4252
cu. in. Ans.
(4) 62X.7854=28.2744=area of end. 28.2744X8
=226.1952 cu. ft.=390865.3056 cu. in. 390865.3056
— 2150.4-181.764 bu. An:s.
(5) 15X5X4-300 cu. ft.=518400 cu. in.; 518400 —2150.4
=241+bu. Ans.
(6) 12X3X5=-180 cu. ft.=311040 cu. in.; 311040 —2150.4
=144.6+bu. Ans.
MENSURATION. 165
ARTICLE 323.
(1) 5X5=25 sq. ft.=zarea of base. 25X21 —3
=175 cu. ft. Ans.
(2) 10'2X.7854=78.54, area of base. 78.54X15- -3
— 392.7 cu. ft. Ans.
(3) 37 ft. 8 in.-452 in., 4522X.7854=160460.3616=area of
base. 79 ft. 9 in.-957 in. 160460.3616X957-*-3
-51186855.3504 cu. in.-29622.0227+cu. ft. Anis.
(4) 720X720=518400=area of base. 518400X477 —3
=82425600 cu. ft.=3052800 cu. yd. Ans.
(5) 1 —1.5: 1.52X.7854-1.76715-area of base. 1.76715
X9. -3=5.30145=cu. in. in cup. 231 —-5.30145
-43.57-+times Ans.
ARTICLE 324.
(1) 3X3=9=area of lower base. 2X2=4=area of upper,
base. 9X4=336; 1/36-=6area of mean base.
9+4+6 —19; 19X'2 —76 cu. ft. Ans.
(2) 27X27-729=-area of lower base. 6X16X1625G6area
of upper base. 729X256 = 186624; 1/186624- 432
-area of mean base. 729+256+432-1417; 1417
X ~4-=105802.6666+cu. in.=61.2283950O+cu. ft. Ans.
(3) 2-=2.5: 2.52X.7854=4.90875=-area of mouth.
12X.7854=.7854= area of bottom. 4.90875X.7854=
3.85533225; 1/3.85533225-1.9635-area of imean base.
4.90875+.7854+1.9635=7.65765; 7.65765X,
=:12.76275 cu. in. Ans.
ARTICLE 325.
(1) 133=2197; 2197X.5236-1150.3492 cu. in. 153-3375;
3375X.5236 —1767.15 cu. in. 303=27000; 27000X.5236
-14137.2 cu. in. Ans.
166 KEY TO RAY'S THIRD BOOK.
ARTICLE 326.
(1) 5 ft. 4 in.X3=16 ft.=circumference of base. 7 ft. 6 in.
— 7 ft., 16X7, —2=60 sq. ft. Ans.
(2) 8:1X3.1416-26.7036=circumference of base.
26.7036X25. 2=333.795 sq. ft Ats.
(3) 2 ft. 11 in. —35 in.; 35X3.1416 —109.956-circ. of base.
109.956X55-2=3023.79 sq. in.-area of conv. surface.
352=-1225: 1225X.7854-962.115 sq. in. —area of base.
3023.79+-962.115=3985.905 sq. in.=27.679S95+
sq. ft. Ans.
(4) 79122=62599744; 62599744X3.1416
=196663355.7504 sq. mi. Ans.
ARTICLE 327.
(1) 10 ft.=120 in., 5 ft..-=60 in., 4 ft.=48 in. 120X60X48
=345600 cu. in., 345600. 231 —1496+gal. Ans.
(2) 12 ft.=144 in., 6 ft.=72 in., 10 ft.=120 in.,
144X72X120=1244160 sq. in., 1244160 —.2150.4
=578.57 —bu. Ans.
ARTICLE 328.
(1) 4 ft.=48 in.; 6 ft..=72 in.; 482=2304; 2304X72
=165888; 165888X.7854=130258.4352 cu. in.=564.01.9
+gal.-17.9+bl. Ans.
(2) 6 ft. —72 in.; 9 R.=108 in.; 72'2=5l14; 5184X10S559872; 559872X.7854-439723.4688 cu. in.
=1903.56-4+gal..=60.43+bl. Ans.
ARTICLE 329.
(1) 40X30=1200; 40 —30-10; 10X10=100; 100 —. 3=33;
1200H-33'=1233.; 1233 X50=616662; 616663X.7854
=48433 cu. in.=209 4-209.66+gal. Aws.
U
!ENSUR4ATION. 167
(2) 7 ft. 6 in. -90 in.; 10 ft.=120 in.; 12 ft. 6 in.=150 in.;
90X120-10S00; 120-90=30; 30'2-900; 900. 3
=300; 10800+300=11100; 11100X150-=1665000
1665000><X7854=1307691 cu. in.=5661 gal.
=179$ hi. bis.
ARTICLE 330a.
(1) 40-31=9;' of 9-6; 31+6-=37; 37X37 —i369;
1360X50X.003,4-232.,73 gal. AJzs.
(2) 32 —30=2; 2X-\o-=1.2 3: 0-,1.2~31.2; 31.2X31.2
=973.44; 973O44X40X.0034=1382-.38+gal. Ains.
ARTICLE 330b.
(1) 1 A.=160 P.: 160_-_ 15 —10 rd. Als.
(2) 24X1S8X10O7-4699 cu. ft. Ans.
(3) 144 —.7554=183,34606569; V/13.346065099
-1l3,.5405-+in. lAs.
(4) 1728..5236-3300.229182-+; /3300.229182
=14.88S4+ in. iA)s.
(5) 30+40+50 —120; 120- -2=60; 60-30-=30; 60-40:20; 60-50-10. 60X30X20X10=360000; 1/360000
i=600 sq. ft. —66=( sq. yd.,S.
(6) 12 ft. 6 in.=150 in.; 15+11 —26; 2GX150 —-2
=1950 sq. in.=13i- sq. ft. Amis.
(7) 162X.7854=201.0624, area of greater circle. 102X.7854:78.54, area of inner circle. 201.0624-78.54
=122.5224 sq. ft. Asls.
(8) 6+6+2+2=16=circumference. 16X21-140=area of
sides. 6X2-12=area of bottom. 12+40=52=sq. ft.
of sheet lead. 52XS=416 lb. of lead. 4 cts.X416
—.$16.64 Alns.
(9) 5 ft.=60 in.; 4 ft.=48 in.; 60X4SX48=138240 s. in.;
138240 —. 2150.4=64.28+1-u. 25 cts.X64.28
1.68 KEY TO RIAY'S THIRD BOOK.
(10) 1448-. 3.1416=460.91163+ ft.- diameter; 460.91163+
X1448=667400+; 667400+ —. 4=166850+ sq. ft.=3 A.
3 R. 12 P. 25 sq. yd. 8+sq. ft. Ans.
(11) 37 -=1369 sq. in.= surface of one side: 1369X6
-8214 sq. in. Ans.
(1) 1236- 3=412 Ans.
(2) 675j-225-900: 900 —2-450 Ans.
(3) 292X3 —-876: 876-. — 219 Ans.
(4) 5 lb. 15 oz.-=5I lb..: 1.20\X5} -, 7.12k Ans.
(5) 13'bu. 3 pk.-13'- bu.: $~1.45X13 -$19.93{ Ans.
(6) 81000+$600=$ 600.: j1oQ8o; sToo-; of 320-200
A.=A's share: - of 320 A.=120 A.=]3's share. Ans.
17X2
-(7) 42l - — 3X-..... hr.-=11 hr. 20 min. Ans.
3
(38) 144'. - i z1 hl 108. A.s..
(9) 6 times-3 times the number=3 times thie number=81:
81 - 3=27,Ans.
(10) $ "X/4 — e: $ o 0 __ = 4-
2 3 t23. 1 __-_ 3 I y. Ans6.~i 20
1-1'-~:'of 1 ~ S125X4.i T -—,=$'5~00 Asis.
6 —\/-T — ~-4
(12) 2,-+1-=S21 +1 1 Q-4- jj 2-a 6-l1 1G-1 a: U
O5 1/- 4S 4x-3, 30078125 Ans.
(l3) 60 —19=41 yr.=- tle time since marriage 41+21=
62 yr. Alos.
(14) 2)8 12 21 36 48 65)143(2
2)4 6 21 18 24 130
2)2 3 21 9 ]2 13)65(5
65
3)1 3 21 9 6
13-greatest common divisor.
1 1 7 3 2
2X3X7X3X2X2X2-1008, least common multiple.
100o 8=77-7 Ans.
Q ------------- d
ifMISCELLANEOUS EXAMBPLES. 169
(15) 3 Imi. 5 fur. 110 yd.=233640 in.: 233640 in. —. 39.371 in.
— 5934.317+meters Ais.
(16) S00000000- -250-3200000.min.=14 yr. 223 da. 3 hr. 20
nmin. ns.
(17) 162 —=. 12.625 _. 262X3.7575 Ais
50
(18) yr1. llon. da.
1855 5 10
1854 5 20
11 20=-11- on.- 5- yr.: $250X-. =$243 - A ns.
(19) 1 lb. of each costs 12 —+30+5=47 cts.: $47 —47 cts.100 lb. AAns.
(20) 2 l 9 4 If -74-35 gallons, f of 35 gallons,
which is 5 gallons, will be t4, and the cistern, whlen full,
will contain 5>24 — 120 gallons.
(2 ) C)60X5=*300: $4X23=892: 60-23-37 bl. $300+151
=,351; $351-$-92= —$29; $259- -37:$7 Ans.
(22) 12 cwt. 1 qr. 71 b.-1232 lb. 3. 3 r. 13 lb.=SS lb.: 1232 —
88-14 Ans.
(23) 102X3.1416=314.I6 sq. in. 5cts.X314.16 —$15.708 Ans.
(2 ) If. ox is worth 8 sheep, 3 oxen wrill be worth 24 sheep,
but 2 horses are worth 3 oxen, therefore 2 horses are
worth 24 sheep; hence, 1 horse is worth 12 sheep, and
at $2.50 a piece, 12 sheep are worth $30 (As.
(25) If of $1, buy of a sheep, 1 sheep is worth 5 times
S,2=10, andl q of a sheep is wrorth -$ of $.= -; hence,
-1' of an ox is worth $1~0, and 1 ox is worth 14 times $ -'
=$20; and 10 oxen are worth 10 times $20-=200 Anls.
(26) 19 is 2J times 9, therefore the required number must be
2, times 54: 54X2-=114 Ans.
(27) of: -. — 1; of1; 1 is 3 times ~X3-1 Ans.
(28) 13X5=68-1 hr. required to perform the work. 681- 9
" v273X — 1o"-7- days of 9 hr. each.
(29) $40: $100 $15: $37.50 Anzs.
(30) From Monday noon till 6 P. M. of Thursday is 78 hr.
O9 —-- -------------------- 23
170 KEY TO RAY'S THIRD BOOK.
which, is 3l days of 24 hr. each; hence the clock has
gained 3 ><7 m in. 71 X3 1 5X -195 =
min., therefore the time by the clock will be 6 hrs.
243 lain.
(31) If 7 men can mow 35 A. in 4 days, I man can mow v of
35 A., which is 5 A, in 4 days; and in 1 day 1 man can
mow 1 of 5 A., which is 1- A.; if 1 man can mow 14 4 4
A. in 1 day, 10 men can mow 10 times 1* A., which is
12a A, in 1 day, and in 3A days they can mow 3., times
121 A. A= 25 < 7. 1 -=43 A. As.
(32) 350 —16= y19 d. yd.: $1X6=$20-L5
$ 4X357-=$26:2: $ — X19=8$12':
$20 + 5$12- 7 262 9$5$94-4 —$5.67+ Anso
I0 a 64q-9 ~1 64 g g —~vo ~:/
(33) 11 hr. X6-66 hr.: 10)X66-660 cu. ft.
660 cu. ft.-.(5X2)-66 ft. Anls.
(34) If 4 men can do the work in 20 days, 1 man can do it in
4 times 20 days=80 days; hence 1 man can do Sl, and
3 nen ~-36 of it in one daty. If 6 women can do the work
in 20 days, 1 woman can do it in 6 times 20 days —120
days; hence 1 woman can do 1 and 5 women ~ 5- of
it in I day. Therefore 3 men and 5 women, in 1 day
can do s3 + 5 199 of the work, and therefore they
can do it in 1 ~ 1 9 2o 4 o-12 iL days.
(35) 40+60 —100 yd.: 5 yd.: 40:: $6: $48:
6 yd.: 60 yd.:: $9: $90:
5 yd.: 100 yd.::$7: $140: 90+$4'S-$138
$140 —$138=82 Ams.
(36) After the first drawing;4 or are pure wine, and when
10 gal. more are drawn, 4 of the remaining 40 gal. are
pure wine; 4 of 40 gal.=32 gal. Anis.
(37) In 14 days of 10 hr. each, there are 140 hours. If 27
men can do the work in 140 hours, 1 man can do it in
27 times 140 hr.=3780 hr.; and since the work of a boy
is half that of a man, it will require a boy twice 3780
hr.=7560 hr. to do the work. If it requires 1 boy 7560
es3 ------ -- - - - --- - - ~-~ — ~-~I
MISCELLANEOUS EXAMPLES. 171
hr. to perform the work, it will take 24 boys — I- of 7560
hr. —315 hr. to do it. 315 hr. -4- 45=7 hr. Ans.
(38) 16' 40"/X9-2~ 30': 2 30'-. 15 gives 10 min.
Ans. 10 min. before 9.
(39),0 men: 0O men
g0.4 ft. 1.: ~ft. 1.
$.$ft. w.: 4. ft.w.
X.$ ft. d.: IJ ft. d. [9X1_-1=9 da, Ans.
10 hr.: 9 hr.: 0:. da.
(40) $1+$.20=$z1.20o: 14.$1.20=$112 A_,s.
(41) 9 cows=3 times 3; if 3 cows=2 horses, 9 cows-3 times
2=6 horses: 4+610: 4-2: -=3 2 of $45-=18,
A's share 6 of $45=$27, B's share.
(42) 35 mi., 2 —14 mi. the sumll of the rates: 14-5-9:
9- -: 2=4 mi. one: 4V+5=9-, mi. the other.
(43) $1-$.17= — $.825: $14.85 — $.825=$18, purchase price.
$18X. 12-$2.25: $18d-$2.25-$20.25 Ans.
qt. bu. qt.
(44) 1 bu. —32 qt. 32-4 —28 qt. 28: 2:: 32:
2 bu. 1 pk. 14 qt. Ans.
(45) The provisions of 32 men for 5 mon., will last 1 man 160
mon.: the provisions of 1 man for 160 mon., in 8 mon.
would be consumed by as many men, as 8 mon. is contained times in 160 mon., 160. 8-20 men, hence 3220-12, the number of men that must leave.
(46) Int. for 63 da. at 6 % —$.0105 $1-$.0105=$.989:
$1009.29+*~$.9895=$1020, the face of the note.
Int. for 60 da. at 12 %-$.02: $1+$.02=$1.02:
$1020 —. $1.02 —$1000 Ans.
(47) $12 —20=$.60 gain per yard. The difference betwreen
5 % less, and 10 %o more is 15 o: $.60 —.15-$4 Ans.
(48) 9X9X,!=40, cu. in. 14X11X5-=961 cu. in.:
401: 96:: $3.24: $7.70 Ans.
(49) 120+60-180: 180: 120:: 3 mon.: 2 mon. Ans.
(50) 25-20=5 rd. gain in 25 rd.: 5 rd.: 47 rd.:: 25 rd.:
235 rd. Ans.
172 KEY TO RAY'S THIRD BOOK.
(51) Int. of $1 for 4 mon. 3 da.=$.0205: $1 —$.02)5
-.979 $300$.97 95-3 - 306.27-+Ans.
(52) $L.50X40=$60: 60 —. $1.20=50 gal.: 50 gal.-40 gal.:=-10 gal. Ans.
(53) Int. of $1 for 63 da.=$.0105: $1.26- $.0105 —$120 Ans.
(54) In running 28 rd. the dog gains 3 rd., but the hare has
45 rd.=15 times 3 rd., the start; therefore the dog must
run 28 times 15 rd. —420 rd., to catch the hare.
(55) =20 o: $1+$.20=$1.20: $1.50-.$1.20-= 1.25, the
cost; $2 —$1.25=$.75 $1.25X.01=$.0125
$.75-. $.0125-60 / Ans.
(56) $260+$13=$273 received for those sold: $1.30X.40
=,$.52: $1.30+$.52 —$1.82: $273. 1.82=150 bu. sold:
$260. $1.30=200 bu.: 200 bu. —150 bu.=50 bu. Ans.
(57) A is ahead 8 mi.X5=40 mi.: B gains 10-8=2 mi. per
hour; 40 mi.-. 2 mi.=20 hr. Ans.
(58) A in 1 da. mows ~ A.; B - A.; and both can mow -A
-=1I A. in 1 da.: 9 A —1 A.-.=6 da. Anzs.
(59) Both do 4 times as much as B: hence B could do it in
20X4 —80 da, and A in {- of 80 da.=26' da. Ans.
(60) 1 of the first lot cost 21 cts., and 1 of the 2d 22 cts.;
hence, 1 of each, that is, 2, cost 21-t+2-=5 cts.: If 5
sold for 14 cts., 2 sold for' of 14=55 cts., 5-5
ct.=gain on 2: - ct.: 52 cts.:: 2: 240 Ans.
(61) 2_1_; 31-166; A can do it in $ da. and in' da. he can
do', and in 1 da. -: similarly, B can do -og in 1 da.,:.5
+5 -sa=what both do in 1 da.: 1-. — 1 da. Ans.
(62) As 2 min.: 3 min.:: 11 rods: 161, the space traveled
by A in 3 minutes. Hence B, in traveling 17 rods, gains, rod on A; and the object being to find how many
rounds he must make to gain half a round, we have this
proportion: As 1 rod: 17 rods:: - round: 17 rounds,
the space to be gone over by B: consequently A will
make 161- rounds.
-- ----------— *Is~~ —
MISCELLANEOUS EXArPLES. 173
(63) A does -- in lda., B 1 and C: all can co d3
+, in 1 da.: 1 — — 1 da. s
(64) From the question, it is obvious that in I day A and B
do -, Aa and C andl and C - of the work. By
taking the sums of these fractions, we find tlice -what
they all do in 1 day; that is, -a +-,-b+ I -4-+ 5- - 3-o
i- 0=-, is twice the part of the work they all do in 1
day; here V of - is the part of the work they all do
in 1 day: therefore they all do it in 1 -- 8 days.
(65) 30 lb. at 10 cts. $3.00
25 12 - 3.00
4 15 -.60
50 20 = 10.00
109 )16.60(15 5 eta. Ans.
(66) If 10 men can dig 1 acre in 11 days, then
I man can dig 1 acre in 11X10O110 days,
and 1 man can dig T-'6 of an acre in 1 day;
tl-len 5 men can dig.T a —X;2 of an acre in 1 day,
and 5 men can dig w. X<6=?TL of an acre in 6 days.
G - qS=? a:cres to be dug by boys.
If 18 boys can dig, acre in 11 days, then
1 boy can dig 1 acre in 11X18=198 days,
and 1 boy can dig T 9h of an acre in I day;
then 1 boy can dig T,8X6= —-1 of an acre in 6 days.
But since 8T acres are to be dug in 6 days by boys, each
of whom call dig 3! of an acre, it Will require as many
of whoni c i3 ~ 1
boys as -33 acre is contained times in -'- acres:
-- 33X -=-63X3-189 Ans.
(67) 24 in.=2 ft.: 15+15+12+12=54 ft., the distance
around; 54 ft.X10 ft.=540 sq. ft.: 540 sq. ft.-. 2 ft.=
270 ft.-90 yd. Ans.
(68) A could do it 3 times in I da., B 5 times and C once:
therefore, the three could together do it 9 times in 1 da.,
and do it once in g da. Ans.
(69) 282 —784 7854+50=834 men Ans.
174 KEY TO RAY'S THIRD BOOK.
(70) 402=1600: 1600+1600-3200 sq. ft.: V/X3200-56.56
+ ft. Ans.
(71) Suppose the work to consist of 150 equal parts, then
the person does 3 of 150=90 parts in 30 days, or 3 parts
in 1 day; 150-90=60 parts done by both in 6 days;
hence they both do 60 — 6-10 parts in 1 day; then
10 —3=7, the number of parts done by the assistant in
1 day: therefore 150 +7-213 days.
(72) 1-T-4s=B'share; — th — e of the estate=$7420:
-~=$ of $7420=$1060, and the estate-$1060X15=
$15900 Ans.
(73) 242 —576: 18=324: 576+324-900-square of the
diagonal on the floor, this is the base in a triangle where
the hypotenuse is sought. 122=144: 900+144-1044:
V>1044=32.3+ ft. Ans.
(74) 56 lb. tea —2~X56=140 lb. coffee-=3.X140-490 lb.
sugar: 10 Octs.X490-$49 Ans.
(75) From the question it is evident, that all three do T-'~ of the
work in 1 day, and that B in 1 day does t1, and C,%;
hence A does l f-(- 2 4+3)12 \_
29__ 34 20 & -8- -0 4 0 86 in 1 day; therefore he can do the
4o5-4o2-4o8-4 084day; do the
work alone in 1 - -8 — d'ays. A.ns.
(76) A built 4-4 as much as C: 3!B of 4, that is, as much as
C: 4+1+-1 O?: A built 4 out of Lo, that is, 4-1=-: B
and C each 3 out of 1o0, that is?3: - of 200 ft.=80 ft.
A b- Of 200 ft.-60 ft. B and C also 60 ft. Ans.
(77) 42.432=1800.3049 square of hypotenuse: since the two
sides are equal, -1 of 1800.3049, or 900.15245 is the square
of 1 side, which is the area. 900.15245 P.-5.62+A.
(78) If he had worked the whole time he would have earned
$1.25X(25=$31.25. Now, each day he was idle lie lost
$1.75, that is, 50 cents which he spent, and $1.25 which
he did not earn. But he lost in all the difference between
$31.25 and $19, which is $12.25, and he lost this at the
rate of $1.75 per day; therefore lie was idle as many
MISCELLANEOUS EXAMPLES. 175
days as 1.75 is contained times in $12.25, which is 7;
therefore he was idle 7, and consequently worked 18 da.
An1s.
(79) B, who works 30 days, will do - the work: hence A
must do 2, which will take him 20 days. 20-3-17 days
more: 30-17=13 Ans.
(80) 1 — 5-=-9- of =' 1-(9-5 +-)=?= C's; 14 —
-==$1300. Property $1300X7=$9100 -: of $9100
=$3250=A's; - of $9100-$3900=B'si; - of $9100$1950-C's Ans.
(81) By the first it receives - in the hour, by the second 5;
by the th o i +; by the third 1 — 75 in 1 hour;
1 -— 7' —2-1 hr. Ans.
(82) $l.20X60 —72.00: $72X.10=$7.20: $72+$7.20 —
$79.20 the amount to be received: $1.50X20=$30.00:
79.20 — $30 —$49.20: 60-(20+5)=35 gal.: $49.20 —
35 —$1.404 Ans.
(83) 1 apple cost 4 cts., and sold for cts.: gain on I apple
= - 4-= ct.: 24 cts. — ct.-144 Ans.
(84) $4500X.01S=$,1: $500-(-25+$S1+$134)=$500$240=$260, net income; $4500X.01 —$45: $260 — $45
_5' 7 Ans.
(85) 60X30=1800 sq. ft.: 12 ft. 4 in.X8 ft. 4 in.=104 sq. ft.
10/: 1800 sq. ft.-104 sq. ft. 10'/1695 sq. ft. 2/: 1695
sq. ft. 2'X4=6780 sq. ft. 8' Ans.
(86) $700+$600+$40400=-1700.: -O4-7 —=0-4~:$260= 41 gain:
- 7-4 of $260 —$65 and gain= —$65X17-$1105 Ans.
(87) 120X.10=12: 12+8-20: 20 -.05=400 Ans.
(88) $1200 - 4-30 B's yearly debt: A, spending $350 less,
will save 8350-$30 —$320- his annual income; $320
X3-$960 Ans.
(89) 53X.5236=65.45: 65.45X2=130.9: 5-125: 125X2
-250: 130.9+250=380.9: V380.9-7.24+in. Ans.
|176 I3KEY TO PiAY'S THIRD BOOK.
(90) $1.0 —.!-1i6~-i. 83; o2'. 25- 83- 3-$30=cost of 1st.
$30 —825 =;5 lost on the first, and therefore gained on
the second. 5-5. $.16=$31.25-cost of 2nd.
| (9.) I4- f-=; -8-o2i4000; -7=3000 slain':. of 3000-1500 Antes.
(92) The 3d=-i of 3 times the 2nd= — the 2nd.
113-K,, of whlch - is -4; 1i-, 19; -4d of 95 yd.=20 yd. 1st.: 1 2 of 95 yd.=60 yd.
nd cl: and 3 of 95 yd.-15 yd. the 3d. A42s.
(93) $4X60(=$-2-0: Anmount of $1 for 6 mon. —1.03 ~ o -- 401.03$=233.009-+-: $250-$233.009+ —=,16.99+- As.
(94) As often as he buys one cow, he buys 6 sheep and f=-2
horses; 1 cow cost $30; 6 sheep $30; 2 horses 9240.
While buying 1 cow, he expends $30+$30+$-9240-$300;
$1500 $300-5 cows; 5X6=30 sheep; 30 - 3-10
horses. Ans.
(95) 4 polls cost 94: $328 —4-$324, the tax on his property;
$324..016$ —20250 Ans.
(96) C does 1 — 4; does 3- - As.
(97) B received of A's —8100; toth received.3 of A's+-100,
3 of A's+$100=z1000; A4 A900: A's=$600, B'shf1000-$600=$400 Ans.
(98) 4 of B's= — of A's+$50; - of B's=- of A's-+$25, and
B's-= of A's+875, both have Y of A's+-$75=$500:'7' of A's=- 425:.-25: A's=$200: B's=$500$200=$300 An.s.
(93) -9 of the time past noon is equal to 4 of the time to midnight, and the whole time past noon-g or 3 the time to
midnight; the time from noon to midnight, which is 12
hlr.=-11, 4 the time to midnight: 1 of 12-3 hr.
[Aces.
(100) In thle 4th store $ was her money; hence, on leaRin.g
the 3d sto-re, she had $1; before she spent the $1, she
had 82-half what she had on entering the 3d store;
hence, on leaving 2nd store, she had $4; before spending $ she had $4 -= of $9, her money on leaving Ist
store; $9+$1- -$10-, of $20, her money at first.
TO
RA YS SECOND BOOK,
SOLUTTIO NS TO T.E M'ORE DIFFTICLT OQUESTIONS.
ON TEACHING INTELLECTUAL ARITiC'l ET.MIC.
IT is to be regretted that many teachers are not acquainted
with the great value of mental arithmetic. It is important,
whether it be regarided as a mere mental discipline, or as the'oundation of an accurate knowledgle of arithmetici and the
various branches of mathematical science.
Whenever a pupil is found incapable of anaalyzinlg nlad solving,
mentally, all the questions in Ray's 2d Book, the best course he
can pursue is, at once, to study mental arithmetic, although in
order to do so, it should be necessary to suspend, for a time, the
study even of Algebra or Geometry. Persons more than twenty
years old have studied it, who afterward declared that the time
thus employed was productive of more beneficial results than
twice the same amount of time ever employed at any other branch
of education. It cannot, therefore, be too strongly recommended
to every teacher who desires to render his pupils thorough scholars, to make them well acquainted with mental arithmetic.
It should be taught before written arithmetic. It often happens, however, that this subject has been neglected until after
the pupil lias commenced the study of written arithmetic. In
such cases, the study of both may be attended to at the same
time, one part of the recitation being devoted to mental, and the
other to written arithmetic.
METIIOD or TEACIIINT.-ClcLsses.-Pupils study best in classes;
it is almost as easy for a teacher to instruct 15 pupils in a class,
as one alone.
Keym.-12 (177)
178 KEY TO RAY'S SECOND BOOK.
An important point is to keep the attention of the whole class
directed to each question; one means of doing this is, for the
teacher occasionally to read an example to the class, and then7
having allowed suflicient time for the answer to be prepared, to
call upon some one for it. By this means, each one is obliged
to solve the example mentally, as he may be called upon to
answer it. It is sometimes proper to put the same example to
the whole class, giving all permission to answer it; this plan
succeeds better in small classes than in large, as it has a tendency to produce noise and confusion.
When the class meets to recite, two methods may be followed
in recitation. First, each pupil may have his book open, and,
when his turn comes, read his question, and then proceed to
analyze and answer it. Or, Seconid, the class may close books,
and the teacher read each question distinctly, and then call
upon a pupil to analyze and answer it, taking the pupils in succession.
Both methods are practiced by good teachers; the first, however, is especially applicable to small scholars, while the latter
answers very well with those who are older; it is an excellent
means of teaching pupils to command their attention-a very
important acquisition.
ARITHIIOMIETERt.-Some difference of opinion exists among teachers with regard to the extent to which artificial aid should be
given to pupils in the study of mental arithmetic. It is certain
that visible illustrations may be used, in some cases, to great
advantage, especially to give clear and precise ideas in the commencement of a subject; but, it is also true, on the other hand,
that a frequent use of artificial aid tends to prevent the pupil
from exercising his own intellectual powers, and thus, if carried
too far, is productive of positive injury. It should be recollected,
that the younger the pupils, the more they require such assistance as can be given them from tangible objects.
The superior advantage of the arithmometer consists in this, that
it can be seen by the whole class at the same time, and it is so
simple in its construction that it can be made by any mechanic.
The arithmometer consists of a square frame with 12 strong pieces
of wi.;v passing from one side to the other. On each piece of
wire there are 12 balls, one-half black, the other half white, so
ON TEACHING ARITHMETIC. 179
as to form a contrast, and alternately disposed, so as to be easily
counted. As the arithmometer contains 144 balls, it may be used
to represent all numbers, from one up to this amount, so as to
give pupils an idea of the relative magnitude of numbers.
The operations of addition and subtraction are so easily shown
by it, that any explanation is unnecessary. To represent multiplication, for instance, the multiplication of 4 by 3; take 3 rods,
each having 4 balls on the one end of it, (the other balls being
pushed out of the rway toward the other end,) and the pupils will
easily see that 3 times 4 make 12; in the same manner it may be
shown that 4 times 3 are 12; and hence, that the product of two
numbers is the same which ever be made the multiplier.
Division is shown by taking the number of balls represented
by the dividend, and separating them into parts, each part containing as many balls as the divisor contains units; the number
of parts, into which the dividend is tuus divided, will represent
the quotient: thus, to divide 15 by 3, take 15 balls and separate
them into parts, each containing 3 balls, and it will be seen that
there are 5 divisions, so that 15 contains 3 five times.
The operations in fractions, so far as they can be explained
without dividing a unit, are also easily shown by it. To give
pupils the first ideas of fractions, the division of an apple is preferable to any thing else, it being an operation with which pupils
are generally familiar.
ANALYSIs.-Among the modern improvements in the art of
teaching, the analytic and inductive method, as applied to arithmetic, forms one of the most important. To analyze, means to
separate into elementary or first principles. Intellectual or mental is used to designate that kind of arithmetic, the operations
of which are performed wholly in the mind. This kind of arithmetic is quite as valuable for the habits of close thinking and
accurate reasoning, as for the mere amount of knowledge which
is acquired.
To enable the pupil to realize to the fullest extent the advantages of the study, he should clearly understand every process,
and be able to give a reason for every operation. It can not,
therefore, be too strongly recommended to the teacher, to require
every question to be solved by methods purely analytical, which
0 -0~~~~ —-------—. —"
180 KEY TO RAY'S SECOND BOOK.
the pupil can both comprehend and explain. In this respect, the
pupil will derive great advantage from a careful attention to the
solutions termed ANALYSIS, (Ray's Arithmetic, 2d Book,) which
are intended as MIodel Solutions.
To explain the difference between an Analytical Solution and
one that is not, take the following question.
What is three-fourths of 12?
Solution. 3 times 12 are 36, and 4 in 36 goes 9 times. Ans. 9.
Analytical Solution. 3-fourths are 3 times 1-fourth; 1-fourth of
12 is 3, and 3 times 3 are 9: therefore 9 is 3-fourths of 12.
In analyzing we must generally refer to unity, and the pupil
should always bear this in mind: thus, if 2 yards of cloth cost
6 dollars, what will 5 yards cost? Here we have given the value
of 2 yards, and must find the value of one yard, after which we
can readily find the value of 5 yards.
Again, suppose we have this question: 2-thirds of a certain
number is 8, what is 5-thirds of the same number?
ANALYSIS.-1-third is 1 half of 2-thirds; if 2-thirds are 8,
1-third is 1-half of 8, which is 4; if 1-third is 4, then 5-thirds are 5
times 4, which are 20. Ans. 20.
If learners pursue the method taught in the book, they will have
no difficulty in progressing; but when they depart from it, they
will soon be involved in perplexity. The questions are so arranged, that each one depends, in some measure, on that which
precedes it; consequently, pupils should understand every thing
as they proceed.
When a teacher finds his class unable to advance, he will
always find, that they are not sufficiently acquainted with the
preceding part. In such cases review. The pupils should go
through the first six sections at least two or three times before
entering upon the seventh and succeeding sections. If they then
properly understand multiplication and division, they will find
little difficulty in the remainder of the work.
ORAL EXERCISES IN MENTAL ARITHMETIC.
The following method of exercising pupils mentally, in Addition,
Subtraction, Multiplication, and Division, will be found to have
CI~~Ba~s x 55~3~."aEL~s;.........BPLBLP..
ON TEACHING ARITHMETIC. 181
the happiest effect in concentrating the attention of the class, and
exciting a lively interest in the exercises. It also renders pupils
remarkably expert in the performance of the operations.
The success of the whole, however, depends upon the instructor;
he must propose the exercises orally, and carry on the operations
in his own mind, so as not to give examples that are either too
difficult or improper, and also to be able to decide on the correctness of the result.
The instructor should announce the exercises clearly, distinctly,
and deliberately; while this is being done, each pupil should
carry on the operations in his own mind, so as to be able to give
the result almost as soon as the announcement is closed. To
avoid confusion in the class, when a pupil thinks he has obtained
the correct result, let him hold up his hand. If his answer is
incorrect, let some other pupil who has obtained a different result,
announce it; and so on, until a correct answer is obtained.
Following will be found specimens of various exercises. These
are given to show teachers what kind of exercises may be employed,
but not for the purpose of being presented to the class, for let it
be distinctly understood, that the success of the whole depends
largely on the teachers announcing the exercises orally.
In the following specimens the exact language is given which
the instructor should use. It should always be brief as possible,
no more words being used than are necessary to point out the
exercises to be performed.
1. Five, add 6, add 3, add 4, add 2, add 9, add 7. Result? (36).
The meaning of this is, that the pupil is to take 5, then add to
it 6, then to this sum add 3, and so on. Lastly, for the sake of
brevity, the teacher says Result? instead of TIWhat is the result?
2. Four, add 5, subtract 2, add 7, add 5, subtract 6, add 2.
Result? (15).
3. Seven, add 9, add 8, add 7, add 3, add 2, add 6, add 3, add
9, add 6. Result? (60).
4. Eight, add 5, subtract 3, subtract 4, subtract 7,-class says
can not be done, before the teacher has time to ask for the result,
or to propose any thing additional.
e~~ a~~"~"~~"~l~~~~""~~~~~"~~ —--------------- — ~~~~5
182 KEY TO RAY'S SECOND BOOK.
5. Twelve, subtract 3, subtract 5, add 6, subtract 7, add 9.
Result?
6. Fourteen, subtract 8, add 6, add 7, add 5, subtract 9, add 4,
add 7. Result?
Exercises involving only addition and subtraction, are less interesting than those which also combine multiplication and division. Such exercises should therefore be used as soon as the pupil
is acquainted with these processes. No division should be required
that leaves a remainder.
7. Three, multiply by 2, add 4, multiply by 3, divide by 5.
Result? (6).
8. Four, multiply by 3, add 3, divide by 5, add 6, multiply by
2, add 4. Result? (22).
9. Seven, multiply by 3, subtract 5, divide by 4, add 1, multiply
by 8, divide by 10. Result? (4).
10. Nine, multiply by 6, subtract 4, divide by 10, add 3, multiply by 4, subtract 5, divide by 9. Result? (3).
11. Six, multiply by 5, add 5, divide by 7, subtract 5, add 4,
multiply by 10, add 8, divide by 6, subtract 6. Result? (2).
12. Eleven, add 5, divide by 8, add 5, multiply by 4, add 5, divide by 11, add 5, divide by 4, add 5. Result? (7).
13. Twenty-one, divide by 3, add 3, multiply by 3, subtract 3,
divide by 3. Result? " (9).
14. Sixty, subtract 4, divide by 7, add 3, multiply by 4, subtract
2, divide by 6, add 5, multiply by 6, divide by 9, add 2, divide by
2, subtract 2. Result? (3).
15. Twelve, multiply by 5, add 3, divide by 9, multiply by 7,
add 5, divide by 6, add 1, multiply by 10, add 8, divide by 9, subtract 5, multiply by 6, add 3, divide by 5. Result? (9).
16. Eight, multiply by 8, add 8, divide by 8, subtract 8. Result? (1).
17. Seven, multiply by 7, add 7, divide by 7, subtract 7. Result? (1).
.JA
ON TEACHING ARITHMETIC. 183
The length and difficulty of the exercises must be graduated by
the age, attainments, and ability of the members of the class; if
too easy they will not excite sufficient interest, and if too difficult,
the class will be discouraged. At first, the examples should be
short, afterward they may be longer, like the following:
18. Fifteen, add 5, multiply by 3, add 4, divide by 8, subtract
2, multiply by 7, add 12, divide by 9, add 3, multiply by 10, add
6, divide by 12, multiply by 8, subtract 1, divide by 7, subtract 7.
Result? (2).
By teaching the pupils the meaning of the termss stuare, squazle
7rot, cube, and cube root, the exercises may be made to embrace the
operations of squaring, extracting the square root, etc., as in the
following exercises. For the sake of brevity, the terms square
root, and cube root, are used to denote that the square root, or
cube root, is to be extracted.
19. Three, square, add 7, square root, add 3, square, subtract 1,
divide by 6, add 1, square root, multiply by 8, add 1, square root.
Result? (5).
20. Seven, multiply by 4, subtract 1, cube root, add 5, cube root,
multiply by 12, divide by 3, subtract 4, cube, subtract 1, divide by
9, multiply by 4, subtract 1, cube root. Result? (3).
21. Four, square root, cube, add 1, square root, cube, add 9,
square root, subtract 2, cube, square root, divide by 2, add 1, cube,
subtract 4, square root, add 5, square root, add 5, multiply by 7,
add 1, cube root, square root. Result? (2).
To TnAAClfxtRs. — In the following solutions, fbr the sake of brevity, arithmetical signs are sometimes used; instructors, however,
in explaining solutions to pupils, should always employ language
the most simple and easily comprehended.
A ftull analysis is given to some of the more difficult examples,
while to others less difficult, the solution is as brief as possible.
In all cases the pupil should be required to give a full analysis;
the style and manner of doing this being a better test of scholarship thanl the ability to give merely a correct answer.
184 KEY TO RAY'S SECOND BOOK.
SECTION 10.-LrESSON IV.
(6) 1I each nman received I-sixth, there must have been 6
men; hence each man received 1-sixth of 42 dollars,
which is 7 dollars.
(IO) One-fifth of 20 cents is 4 cents, what she received from
her brother; 1-half of 4 cents is 2 cents, what she received from her sister; hence, she received, in all, 20
cents, and 4 cents, and 2 cents=26 cents.
(12) If he gave 4-sevenths, he had 3-sevenths left; 1-seventh
of 14 cents is 2 cents, and 3-sevenths are 3 times 2 cents,
which are 6 cents.
(1'4) One-fifth of 30 yards is 6 yards, and 2-fifths are 2 times
6 yards, which are 12 yards; if 12 yards cost 48 dollars,
1 yard will cost l-twelfth of 48 dollars, which is 4
dollars.
(15) One-fifth of 25 cents is 5 cents, and 3-fifthls are 3 tines
5 cents, which are 15 cents: at 2 cents each, 15 cents
will buy as many peaches as 2 cents is contained times
in 15 cents, which is 7 and 1-half Ans. 7; peaches.
(16) One-ninth of 54 is 6, and 5-ninths are 5 times 6, which
are 30. To give each girl 1 chestnut would require 3
chestnuts; hence each girl received as many chestnuts
as 3 chestnuts is contained times in 30 chestnuts, which
is 10. Ans. 10 chestnuts.
(18) One yard cost 1-seventh of 7 yards; if 7 yards cost 42
dollars, 1. yard cost 1-seventh of 42 dollars, which is 6
dollars, and 3 yards would cost 3 times 6 dollars, which
are 18 dollars. If 9 barrels of cider cost 18 dollars, 1
barrel would cost 1-ninth of 18 dollars, which is 2 dollars.
(22 One-fourth of 28 is 7; 1-severth of 28 is 4, and 3-sevenths
are 3 times 4, which are 12; 7 cents and 12 cents are 19
cents, which taken from 28 cents, leaves 9 cents. If 3
books cost 9 cents, 1 book will cost 1-third of 9 cents,
whilich is 3 cents.
PL~~~YI i~~~I*I~~.-~~Fl~~~i-P —~:~~~~"I~~(CV~~~L T'~~M~~5nr-.~~~K ov
MENTAL EXERCISES. 185
SECTION I.- LEssoN III.
(13) Since 5-sevenths are apple trees, the remaining 2-sevenths
are cherry trees. One-seventh is 1-half of 2-sevenths;
if 2-sevenths are 12, one-seventh is 1-half of 12, which
is 6, and if I-seventh is 6, 5-sevenths are 5 times 6, which
are 30, the number of apple trees.
(17) Since 25 is 5-eighths, 1-fifth of 25, which is 5, is 1-eighth;
hence, S-eighths is 8 times 5 years, or 40 years, the
father's age. One-tenth of 40 years is 4 years, the age
of the youngest.
(18) Since 45 is 5-thirds, 1-fifth of 45, which is 9, is 1-third
of the first cost, and 3 times 9, which is 27, is the first
cost; hence, he gained 45 less 27, which is 18 dollars.
Or, the gain is evidently 2-thirds of the first cost, which
is twice 9 dollars, or 18 dollars.
(21) Since he gained 2-sevenths, he must have sold it for
9-sevenths of the first cost; therefore, 45 is 9-sevenths,
and 1-ninth of 45, which is 5, is 1-seventh of the first
cost; hence, 7 times 5, which are 35, is the first cost.
SECTION 12. —LESSON I.
('7) One-fifth of 20 is 4, and 2-fifths are 2 times 4, which are
8; 1-third of 15 is 5, and 8 is once 5, and 3-fifths of 5.
The following are the answers to the remaining examples in
this lesson. Although they are explained in the form of common
fractions, the pupil should give them after the manner expressed
in the Arithmetic; thus, the answer to question 8 should be read,
two tines a acnd 2-fifths of 5.
(8) 2. (9) 31. (10) 111. (11) 4. (12) 61.
(13) ].: 21. (14) 4.: 33 (15) 4-.! (16) 7 -.
LESSON I I. I Answers.
(2) A. (3) 18. (4) 15. (5) 9. (6) 40.
(7) 35. (8) 44. (9) 4. (10) 2. (11) ].
(12) 6. (34) 24. (15) 71. (16) 3!- (17) 7.
l(1) *5. (19) 5 (2 0) 7 (21) - 3 (22) 14.
186 KEY TO RAY'S SECOND BOOK.
(23) 1-tenth of 50 is 5, and 7-tenths are 7 times 5, which are
35; if 35 is 5-ninths, I-fifth of 35, which is 7, is 1-ninth,
and 9-ninths, or the number, is 9 times 7, which are 63.
If 8 is 4-fifths, 1-fourth of 8, which is 2, is 1-fifth, and
5-fifths, or the number, are 5 times 2, which are 10. 63
is as many times 10 as 10 is contained tines in 63, which
are 6 times 10 and 3-tenths of 10. Ans. 6 times and
3-tenth times.
(24) Ans. 4 times andc 2-sevenths of a time.
(25) 24. (26) 12. (27) 2-tlirds of 21. (28) 18.
(29) 40. (30) 3 times 7.
8ECTION 13.-LESSoN L
(25) In 42 there are 6 sevens, and Joseph husks as many threes
as his father husks sevens, that is, 6 threes, which are 18
r>OWS.
(28) If the horses eat 16 bushels in 2 weeks, they will eat
1-half of 16 bushels, which is 8 bushels, in 1 week; if
they eat 8 bushels in 1 week, it will require them as
many weeks to eat 56 bushels as 8 bushels are contained
times in 56 bushels, that is 7. Ans. 7 weeks.
It is easily seen, that the rnoumber of horses has nothing
to do with the solution. The question would be the same
if the words, a cerlain nzumber, were substituted for 5.
(30) To eat it in 1 day would require 6 times 12=72 horses;
and to eat it in 9 days, ~ of 72-8 horses Ans.
(31) If 1 lb. of sugar=2 lb. of rice, 1 lb. of sugar and 2 lb. of
rice-4- lb. of rice=] lb. of coffee Ans.
LESSON II.
(22) Ans. 4. (23) To Thomas 1 the most, and he had 6 left.
(24) Ans. 5 dollars. (25) Ans. 54 dollars.
(26) By subtracting 1-third and 2-fifths of 15 from it, there
are left 4 yards. The cost was 30 dollars, and the sales
58 dollars. Difference, 28 dollars.
(27) Ans. 1 dollar.
(28) 23 less 18 is 5, the number of miles per day that B gains
on A. If it requires 1 day to gain 5 miles, it will require
8 days to gpain 40 miles. A'ns. 8 days.
- - ----------— ~ - - - 1 -— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3~
MENTAL FRACTIONS. 187
(29) 10 less 7=3 yd. gain in I min. The hound will gain
90 yd. in 90-.-3=30 min., during which time the hound
goes 30 times 10=300 yd., and the hare 30 times 7
=210 yards.
(30) 7 less 4=3 rods. The hound runs 7 rods to gain 3 rods,
and will run 35 rods, which is 5 times 7, to gain 5 times
3 rods —15 rods Anes.
(31) 3 miles less 1 mile is 2 miles; the first gains 2 miles in
going 3 miles, and will gain 4 miles, or twice two miles,
in going twice 3-6 miles Ans.
(32) Since 1 gains 3 miles on C in traveling 5, and since 15
is 5-threes, it will require him to travel 5-fives, that is, 25
miles, to overtake C.
(33) Since 8 gallons arefilled, and 5 gallons emptied in 1 hour,
it is evident the cistern is filled at the rate of 3 gallons
(8 less 5 equals 3) per hour; hence it will require as
inany hours to fill it as 3 gallons is contained times in 24
gallons, that is 8. Ans. 8 hours.
(34) Since it is filled by the first pipe in 6 hours, it is filled at
the rate of 6 (36 — 6 —6) gallons per hour; and since it
is emptied in 9 hours, it is emptied at the rate of 4
(36 —9-4) gallons per hour; hence when both pipes are
left open, it is filled at the rate of 2 (6-4-2) gallons
per hour; consequently it will be filled in 36-. -218
hours.
(35) 8 is 2-fifths of 20 dollars, the price of the coat, and 1-half
of 8 is 4 dollars, the price of the vest; hence, the whole
cost 8 and 20 and 4 dollars, that is, 32 dollars.
S ECTION 19.-Lzsson 11:I.
(11) It requires 8X11=88 hours to perform the work; 88
hours - 10 hours —=S8 days of 10 hours each.
(12) 8 miles for 37 cents would be 87 cents for 1 mile, and 11
times 37 is 4 —7=507 cents for 11 miles.
(13) If it requires 2 pipes to empty it in 17 minutes, it will
take I pipe twice as long, that is, 34 minutes, and 3
pipes 1-third as long as 1 pipe, that is, 1-third of 34
minutes] l1- minutes.
188 KEY TO RAY'S SECOND BOOK.
(14) 1 horse will require 1-fifth as much as 5 horses, and 7
horses 7 times as much as 1 horse. l-fifth of 18 bushels
is'5and 7 times are 1 2 6, which are 251 bushels.
5.5,
(18) 4 of 18=36=5q feet in the earth; and of 18-90
126 feet in the air.
(19) 4 of 15=30 —31, A's part. l of 15=5, B's part.
5+3 —, and 15_S —S=62, C's part.
(20) 48 dollars. (21) 8 cents.
(22) 12. (23) 14 dollars.
SECTION 20. —LESSON II.
(13) It will require 1 man twice as long as 2 men, and 3 men
1-third as long as 1 man. Twice 3-1 days are 64 days,
the time in which 1 man can do it; and -- of 64- days is
3of o days, which are 2o days, equal to 29 days, the
time in which 3 men can do it.
(14).Ans. 5Tr days. (15) Ans. 93 days.
SECTION 22.-LESSON IL.
(34) Since he gained 1-fourth, he must have sold it for
5-fourths of the first cost. If 18 is 5-fourths, 1-fifth of
18, which is UI, is 1-fourth, and 4-fourths, or the whole,
are 4 times -8, which are 72, equal to 144 dollars.
(35) Ans. 205 dollars cost; gain 4- dollars.
(36) Ans. 44 dollars. (37) Ans. 3- dollars.
LESSON II.
(6) If 4-fifths of a yard cost 8 cents, 1-fifth will cost 1-fourth
of 8 cents, which is 2 cents, and 5-fifths, or a yard, will
cost 5 times 2 cents, which are 10 cents. At 10 cents a
yard, 25 cents will buy as many yards as 10 cents is
contained times in 25 cents, which is 2, times.
An.s. 2- yards.
(7) Ans. 62 dollars.
X- - < —--— ________
MENTAL FRACTIONS. 189
(8) If 6-sevenths of a yard cost 4 dollars, 1-seventh will cost
1-sixth of 4 dollars, which is 4 or of a dollar, and
7-sevenths, or a yard, will cost 7 times 2, which are 14
dollars. 12 dollars will buy as many yards as 14 dollars
are contained times in 12 dollars. But 134 is contained
in 12 as often as J4 is contained in %6; and 14 is contained in 336 as often as 14 is contained in 36; 14 in 36
is contained 2,-_=-24 times. Ans. 24 yards.
(10) Ans. 3} yards.
(11) If he gained 1-third, he must have sold it for 4-thirds of
what it cost. If 12 is 4-thirds, 1-fourth of 12, which is
3, is 1-third, and 3-thirds, or the first cost, is 3 times 3,
which are 9 dollars. If 5 yards cost 9 dollars, 1 yard
cost, of 9 dollars, which is 14 dollars.
(12) If he gained 2-fifths, he must have sold it for 7-fifths df
the cost. If 21 is 7-fifths, 1-seventh of 21, which is 3,
is 1-fifth, and 5-fifths, or the first cost, is 5 times 3, which
are $15. This at 3-fifths of a dollar a bushel, would
buy as many bushels as 3-fifths of a dollar is contained
times in $15, which is 25 times. Ans. 25 bu.
(13) If I cost 3 of a cent, 5 will cost` of a cent, and one
whole apple 1-O of a cent; and 3 apples cost 3a-0=3 cents.
(14) 2= 11 5, 22:: If Vy of a yard cost 4/ dollars, 42,
which is twice -1, will cost twice 4-3-9} dollars.
(15) Then 8 dollars must be - of the cost; of the cost is
2 dollars, and the cost is 5 times 2 dollars, which are
10 dollars; 11 dollars less 10 dollars will be 1 dollar,
which is T1~ the cost.
SECTION 25.-LESSON I.
(3) E of 3-3, 5X =1; X Of - 5 =, and 2X=$5 SAns.
(4) 4 of 24=6, 7X6=42; - of 42=3, 5X3=15 cts. Ans.
(5) 5 of 15=3, 9X3=27;. of 27-$13-.Ans.
(6) ~ of 30=10, 8X10=80; - of 80=5, 7X5=35 trees. Ans.
(7) 15-, of 14-2, 5X2=10; 2 —X10=$25 Ans.
AW-~; —---------— ~ —— ~ ~
190 KEY TO RAY'S SECOND BOOK.
(8) 1 —,3 5_=i-1; of 241= 2 times 7-7 the cost of 1
~ ~L, 4 4 3 4 - -2,
barrel, and 24 times 7-=345=8 dollars Ans.
(9) 331;-~ -Y-, of 20=2, and 3 times 2=6; 25 times 6
=17 cts. Ans.
(10) 3=345, t-L of 30=2, 4 times 2=8 miles, the distance
traveled in 1 hour; 7~ times 8=58 miles. Ans.
(11) -1 of 1I-=, 11 times ~-=13 dollars. Ans.
r=12- dollars. Ans.
(13) 5, 2 9 45, and 32= 1 1 of 46=,and 4 tinmes 2
4-' 5 3 3 5 5,
8 of ancl 11 times =8=5 o dollars. Ans.
T -5 15-15 —I
(14) 1 of w hich is of the number, and 7 times.
3-, ancl 4 of 3.S..9-1 0ls.
(1L5)- 2=8: 3, =: 21= 5 f 17 7o, and 3 times -'
or the number. 21 times 2=1 -21 X -05 -. Ans.
(16) -1 of 4=1la, and 4 times 1`=-6 dollars, the price of a
barrel; - of 6=- and 3 times - s —53 dollars. Als.
(17') of 5-0 and 3 times -o-J-9; - of 9-9 —7, and 5
times'o — dollars. Ans.
(18) D of 30-6, and -1 of 6=2, the wages of one man onze day.
4 times 2=8S, and 7 times 8=56 dollars. Ans.
(19) - of 36=6, and I of 6=-3 5 ties 3 —
and 12 times?1=45 dollars. Anrs.
(20) I of 12=4, S of 4=-; 5 times 1, and 3 times 5
=7- rods. Ans.
(21) G of 36=6, -- of 6=-; 5 times 3-=3, and 9 times 3=
27 bushels. Ans.
(22) If 5 oxen eat 2 acres in 6 days, they will eat 1 acre in 3
days; and 1 ox will eat 1 acre in 5 times 3 days, or 15
days. Then it will require 1 ox 8 times 15 days, which
are 120 days, to eat 8 acres; and it will require 12 oxen
T1: of 120 days, which are 10 days, to eat 8 acres.
Ans. 10 days.
(23) 400 dollars for 5 months is 80 dollars for 1 month, and
if 8 persons spend 80 dollars in 1 mnonth, one person will
MENTAL FRACTIONS. 191
spend 10 dollars in I month. Hence 11 persons will
spend $110 in I month, and in 8 months $880. [Ans.
(24) 10 oxen on 5 acres for 3 months, is
2 oxen on 1 acre for 3 months; or
6 oxen on 1 acre for 1 month; or
90 oxen on 15 acres for 1 month; or
18 oxen on 15 acres for 5 months; or
126 sheep on 15 acres for 5 months. Ans. 126.
LEssoN II.
(1) 5 times 18=90; - of 90=10. Ans. 10 men.
(2) 8 times 15=120; ~ of 120-10. Ans. 10 men.
(3) 9 times 2o-222; 5 of 22-=4.1. Ans. 41, hr.
(4) To do the work in 5 of the time will require 5 times as
many men; 5 times 5 men are 25 men; twice as large
will require twice as many men; twice 25 men are 50. men. _Alzs. 50 men.
(5) 8 times 5-40; o of 40 —8. Ans. 8 days.
(6) If 6 men can do the whole work in 5 days, they will do
one-half of it in 2~ days. The whole work requires 6
times 5-30 days work of one man, and half of it will
require 15 days work of one man. Since 9 men (6+3)
are employed to finish it, it will require ~ of 15 davs, or
1!- days. Hence the whole time is 2;+1-4' days.
(7) It will require 28 days work of one man, and 7 men will
do half the work, or 14 days work of one man in 2 days;
and since 3 men then leave, there are 7 less 3, or 4 men
left, to do 14 days work of one man, which they will do
in - of 14 days, or 3i days. Hence the whole time is
2+3>=51 dclys.
(8) 5 times 8=40, the number of cent loaves. 1 of 40=10,
the number of 4 cent loaves. -j1r of 40-=4, the number
of 10 cent loaves.
(10) 3 times 20-60, the number of cent loaves, and -1 of 60
-15, the number of 4 cent loaves. of 60=12, the
number of 5 cent loaves.
6 h Ace —---------------
192 KEY TO RAY'S SECOND BOOK.
(11) 3 times 8=24, the weight in oz., when flour is $1 a barrel. of 24 of 214=6, the weight respectively,
when flour is 2 and 4 dollars a barrel.
(13) $3=$-: $21 —$5: 7 times 7=49 oz., when flour is $.!
per barrel: 5 of 49=9 4 oz., the weight when flour is
per barrel. Ans. 94 oz.
(14) 1 man could mow it in 6 times 5-=33 days, and 10 men
in' of 33-3- days: 5- -3-2 days. Ans.
LESSON III.
(2) 3+4+5-12. T3 of 36=9, W's; -, of 36=12, T's; -
of 36-15, John's share.
(3) 2 of 71-=5 share of Ist; 9 of 7-=2-1, share of 2nd.
(4) 40-25-15, what B paid. 2_ and of 56-35 dollars, A's share. z} 3 of 5
lar-s, A'S share. 1 =- and ~ of 56=21 dollarls, B's share.
(5) +4-1, and 1- -- C's part. $860-$500=$360::
of 360 —45 dollars, A's loss; 4 of 360=90 dollars, B's
loss; 4 of 360=225 dollars, C's loss.
(6) 7'='T, ancd'2 —94; r4 of 21=-7 dollars,share of1st,
and 9i of 21=13, dollars, share of 2nd.
(8) T0C;: } of 25=15 dollars. Ans.
(9) -15 3 of 80=12 dollars. Ans.
(10) 180 sheep eat as much as 18 oxen. 27-lS-1845. 47
=54. 5 of 25=15 dollars, what A pays. 48, an d
of 25=10 dollars, what B pays.
(11) 15 cows are 5X3 cows, and 5X2 horses are 10 horses;
10 horses eat as much as 15 cows. 14+10=24. -4
v and _5 1
T and U24; 42 of 60=5, - 7 of 60=35, and 7-a of
60-25. Hence A should pay $35, and B $25.
(12) 15 oxen are 5X3 oxen, and 5X2 horses are 10 horses;
hence 10 horses eat as much as 15 oxen. Since I horse
eat as much as 20 sheep, 6 horses eat as much as 120
sheep; hence 15 oxen and 120 sheep eat as much as 16
horses. 8+16-24,, I 6'-2; of 75=$25, A
should pay; and. 2 of 75-$50, B should pay.
% w ---------------------— P —-----.
MENTAL FRACTIONS. 193
(14) $50 for 4 months is the same as $200 for 1 month, and
$60 for 5 -months is the same as $300 for 1 month. 200
4+300=500; A'o_-, A's share, and -~ —, B's share.
2 of $45-$18, A's gain, and 5 of $45-=$27, B's gain.
(15) 3 men for 4 days=1 man for 12 days, and 5 men for 3
days-l man for 15 days. 12+15-27;-, — =;
4 of $81=$36, and' of $81=$45.
(16) $2 for 5 months is the same as $10 for 1 month, and $3
for 4-months is the same as $12 for 1 month. 10+1222, ~ — T; i~ of $55=$25, A's share, and -6of $55=$30, B's share.
(1L7) 4 horses for 5 mon. is the same as 1 horse for 20 mon.,
and 10 cows for 6 mon., as I cow for 60 mon.; but 1 cow
for 60 mon. is the same as 3 cows for 20 mon., and 3
cows eat as much as 2 horses, and 2 horses for 20 mon.
is the same as I horse for 40 mon. 20+40-60, 2=o,
and 0 - of $27-$9, what E should pay; and
of $27-$18, what F should pay.
LEssoN IV.
(2) Of each 7 cents John gets 3, and James 4. In 28 cents
there are 4 times 7 cents, hence John gets 4 times 3
cents which are 12 cents, and James 4 times 4 cents
wrhich are 16 cents.
(3) 4+3+2=9. 9 in 45 is contained 5 times. 5X4-20, A's
share. 5X3-15, B's share, and 5X2=10, C's share.
(4) 5+3=8. 8 in 96 is contained 12 times; 12 times 5=60,
apple trees; and 12 times 3-36, peach trees.
(5) Since there are 2 sheep for each cow, there are 6 sheep
for e.ach 3 cows, or each horse. 1+3+6=10, that is, for
each 10 animals there are 1 horse, 3 cows, and 6 sheep.
10 in 60 is contained 6 times; 6 times 1=6, the number
of horses; 6 times 3-18, the number of cows; and 6
times 6=36, the number of sheep.
(6) 2+3=5. 5 in 35 is contained 7 times; 7X2=14, the
number of boys, and 7X3=21, the number of girls.
e.-~. —! 3
(2 )X,]
194 KEY TO RAY'S SECOND BOOK.
(7) The number and 3 times the number, that is 4 times the
number, is 48, hence once the number is A of 48, which
is 12.
(8) 1+2+3=6. 6 in 42 is contained 7 times; 7 times 1= —7,
A's plums; 7 times 2=14, B's plums; and 7 times 3=
21, C's plums.
(9) 1+4-5; -'1 of 25=5; 4 of 25=20. Ans. 5 yd. and 20 yd.
(10) Since Agnes gets twice as many as Emma, and Sarah
twice as many as Agnes, Sarah gets 4 times as many as
Emma. But 1+2+4 —=7; hence of each 7 cherries Emma
gets 1, Agnes 2, and Sarah 4. 7 in 35 is contained 5
times; 5X1=5, 5X2=10, and 5X4=20; hence Emma
gets 5, Agnes 10, and Sarah 20 cherries.
(11) 3+4=7: of 28 is 4; 3 times 4=12; 4 times 4=16.
LESSON V.
(12) 2+3=5;, of 25 cts.=10 cts., John's; ~ of 25=15 cts.,
George's.
(13) 5+7=12; i~5 of 48=20, tuhe 1st. g7 of 48=28, the 2d.
(14) 3+4-+5= —12; u of 60=15, of 60=20, and A. of 60
=25.
(15) 1+2+3+4=-10; I of 70=7, 2 of 70=14, y6 of 70=
21, and -4y of 70=28.
(17) 1.+21=i4. 4 in 16 is contained 4 times; 4 times 12 —
6, Thomas' share; and 4 times 2>-10, Moses' share.
(18) j4+1+=1 —-. 3 nd in 39 is contained 36 times; 36X-=
18, 36X=12, and 36X= —9. Ans. 18, 12, and 9.
(20) 14-1-=4'.:y in 33 is contained 12 times: 12 times 1
— 12, and 12 times 1 —21. Ans. John 12, James 21.
(21) i+12 —=. 8 in 32 is contained 12 times; 12X1=12,
age of Frank, and 12X12=20, age of William.
(a22) 1+21=3 —. 3o in 30 is contained 9 times; 9 times 1=9,
the number of unsound, and 9 times 21l21, the number
of sound apples.
(23) 1+4=g. ~ in 27 is contained 15 times; 15 times 1- 15,..... J
MENTAL FRACTIONS. 195
the number of feet the 2d can build, and 15 times -_-12,
the number the 1st can build.
(24) Since B has, as many as A, and C 1 as many as B,
therefore C has 1 of _, or x as many as A. 1++1. —7.
7 in 42 is contained 24 times; 24 times 1=24, A's share;
24 times — 12, B's share; and 24 times 1-6, C's share.
(25) Since there are 2 as many sheep as hogs, and - as many
cows as sheep, for each hog there is 3 of a sheep, and 4
of 2- = -1 o a cow. l +, in 104=48;
48X1=48, the number of hogs; 48X —=32, the number
of sheep; and 48X 2=24, the number of cows.
LESSON VI.
(2) 19 —5=14; A. of 14=7, the number of apples James has;
and 7+5=12, the number Thomas has.
(3) 31-7-24; ~ of 24=12, the less number; and 12+7=
19, the greater number.
(4) 32-8=24; 3 of 24=12, the number each had at first.
(5) 20+4+6 —30; ~ of 30=15; 15-4=11, the number
Thomas had left; and 15-6=9, the number William
had ]eft.
(6) 24+7+5-36. Since Mary bought twice as many as
Sarah, they both bought 3 times as many as Sarah, therefore Sarah bought - of 36=12 cherries, and Mary twice
12, or 24 cherries. 24-7=17, the number of cherries
Mary had left; and 12-5-7, the number Sarah had left.
(7) 50-5=45, and I of 45=15, the required number.
(8) 31-10=21, which is 3 of the number; hence 1 of 21,
or 7, is - of the number, and 4 is 4 times 7, or 28.
(9) 21+7=28, 4 of 28=7 is 1, and.535, the numllber.
(10) 4-3=-7. 37-7=30. 4 of 30=10, Henry's age; 10
-t-3=13, Oliver's age; and 10+4=14, James' age.
(11) Since Sarah has 3 cents less than Mary, she has only 5
more than Jane. 8+5=13, 43-13=30, - of 30=10,
Jane's; 10+8=18, Mary's; 18-3=15, Sarah's.
196 KEY TO RAY'S SECOND BOOK.
(12) Twice the age of Frank, plus his age, less 3 years, is 42
years; hence 3 times the age of Frank must be 45
years. - of 45=15 years, the age of Frank, and twice
15, less 3 years, is 27 years, the age of Mary.
(13) The ring cost $5, and the watch $12 more than the chain;
hence 3 times the cost of the chain, and $17 is equal to $62;
therefore 3 times the cost of the chain is equal to $45. 1
of 45=$15, the cost of the chain; 815+5=$20, the cost
of the ring; and $15+12-$27, the cost of the watch.
(14) 30+6=36, which is of his money. a of 36-18, which
is 4 of his money, and 7 times 18=126 cents, his money.
(15) The money of John and James=3 times the money of
James and $3; hence the money of Frank=3 times the
money of James and $10; and the money of all three,
$55,=6 times the money of James and $13, hence 6 times
the money of James is $42. -L of 42- 7, James' money;
7+7+3-=$17, John's, and 7+17+7-$31, Frank's.
LESSON VII.
(2) 4+ =0?. 1 of 60=6, which is 4. =4 and =
18; hence Thomas pays $42, and John $18.
(3) From noon to midnight is 12 hours. Since the time
past noon-' the time to midnight, therefore the time
and a the time, or a of the time to midnight=the time
from noon to midnight, 12 hours. If 3 is 12 hours, -1 is
3 of 12 hours, or 4 hours; that is, one-half the time to
midnight, which is the same as the time past noon, is 4
hours. Hence the time was 4 o'clock, afternoon.
(4) 1+-3=-, 1 of 12 hours=1- hours, which is ~ and 4 —.1
hours; hence the hour was half past 4 P. M.
(5) 143. of 56=8=4 —; 4=32, and 4-24, the distance
traveled each day respectively.
(6) Since the first, plus 4 of the first less 8, that is L of the
first less 8=100, 12 of the first is 108. 1 of 108S9, is
4 of the first part. =-63, the lst; 4=45, and 45 —8=
37, the 2nd part.. ---- --- - -— L
MENTAL FRACTIONS. 197
(7) The first+"+43, that is 9 of the first=45. ~ of 45=5,
of the first. -4=20, the first part; ~ of 20=10, the
second; and a of 20-15, the third.
(8) 2 of 3=twice' of 3= —; hence C has,2 as many as A.
1++2-1t — i of 40-A4, which is. -20, A's;
=12, B's; and 5=8, C's.
3,F 5 ac:,,, 1 5 3 11++ = 5 531 4 4
(9) of 0 — 3 times of 5 - 1+j+5-!. 6
hence J4 of the top part-70; 14 of 70=5, which is a;
6-30, top; =25, middle; and 3=15, the lower part.
(xo) ~ of s —~ —. 1+2 3 _ 4 _ hence
(10) 3 of T= -e% y4. +2+T — -- 0+To+;o-; hence
o- of the cost was $34. r- of 34 is 2, which is o; --
$20, the cost of the coat; 4 =$8, the cost of the hat; and
=- $6, the cost of the vest.
(11) If i the cows=- the sheep, the cows=4 of the sheep.
1.+4-; 1 hence 1' of the sheep is 55. 1l of 55=5,
which is -; — 35; the sheep; and 4=20, the cows.
(12) If 3 of the less — of the greater, the -3, or the whole of
the less=3X2, which are 4 of the greater. 1+-k;
hence 5 of the greater-100; 4 of 100=20 is - of the
greater. 3-=60, the greater number. 2 40, the less.
3 7
(13) If 4 of Mary's age=- of Sarah's, 4 of Mary's=4 of
Sarah's. 1+ -4=. 1 of 14=2, is ~ of Sarah's, 3-=
6, Sarah's age; and 4-8, Mary's age.
(15) If of the first-= of the second, of - or of th
second, and -, or the whole of the first - - of the second.
Hence the second plus 9 of the second, or 7 of the second-51. -T of 51_ —3 is K of the second. 8 -24, the
second part; and 9-27, the first part,
(16) If 2 of the apple trees-4 of the peach trees, 14=, and
or all the apple trees= — of the peach trees. Hence
the peach trees and of the peach trees, that is, 1 365.
T, of 65=5, is. =35, the peach trees; and — 30,
the apple trees.
(17) Since - A's distance= 5 B's, -3 A's=- of 4, or - of B's,
an of =3 of A's 3X or! of B's; hence B's and 6 of
B's, that is l, is 66 miles. IT of 66 is 6 miles, 4 of
d- e a - - - -------------
198 KEY TO RAY'S SECOND BOOK.
B's, and since the difference between B's and - of B's is. of B's, B traveled 6 miles further than A.
(18) Since once the time past noon+3 hours is 4 the time to
midnight, twice the time past noon+6 hours.-the whole
time to midnight; but the time past noon+the time to
midnight is 12 hr., hence the time past noon, with twice
the tinme past noon-+ 6 hr.=12 hr.; hence 3 times the
time past noon is 6 hours, and the time past noon is 3
of 6 hr., or 2 hr.
(19) See the "Explanation." If 12 hours is 4, of 12, which
is 3, is; hence the hour is 3 in the afternoon.
(20) Here 12 hours must ble of the whole time; hence 1
must be ~ of 12 hours, which is 4 hours; that is, the hour
is 4 P. M.
(21) If, the time past noon is ZU of the time past midnight,
the time past noon must be o- of the time past midnight;
then the time past noon is J-a of the whole time from
midnight, and the time from midnight to noon must be
T9~ of the whole time. - of 12 hours is 14 hours, that is,
the hour is 1~ P. M., or 20 minutes past 1 o'clock.
LESSON VIII.
(2) 1+-5 l- of 20=4, is 1 of the number; -3=12, the
number.
(3) I-1=~-4. 4 of 21-3, is 4 of the age; 5-15, her age.
(4) Twice the number and 4 of the number equals'3 of it.
y3% of 52 is 4, which is;.5=20, the number.
(5) Twice the number less its 4, is 14 less 4 or 0; if'O is
40, iT of 40, which is 4, is 4, and o-, or the number is 28.
(6) Three times the number less its 4 is 5 less, or; if
l is 48,'-1 of 48, 4, is and 4 is 20.
1I o 6.6 4_13 3 -12
(7) 1+A- + -;-+ of 26:2, is 4, and -12,
his age.
(8) Her age,' and -4 of her age are ~9 +T- or 1'; but
twice her agre is 24; hence l4 of her age plus 10 years
=4,'d; f.! of her awe is 10 years, and her agre, 24 years.
MENTAL FRACTIONS. 199 |
(9) 1-4-4.. 4 is 30 cents; hence a is 50 cents.
1 1i 31__
(10) 1+ - Q+ 5 +- +(+G=o 1; treble the number is 4,
hence -9- of the number is 27, and the number is 30.
(11) 1 —T=,,- 4 of 40= 5, i-s of the father's age, or 55
years; hence the son's-15 years.
(12) 1+4 -. 3 times her age is 5. HIence -~ of her age is
18 years, and her age is 15 years.
(13) 1 —2=7. of 28=4 yards is B;'=36 yards. Ans.
(14) Twice the distance from A to B is equal to the distance
from C to D; hence twice the distance from A to B is
equal to ~ of the same distance plus 20 miles; therefore
20 miles is 4 of the distance from A to B, which is 15
miles, and from C to D is 30 miles.
( 15) 1+- + - + _9_-+: —4-5-; I of 94 is 1,
which is l.; hence the age is 45 years.
LEssoN IX.
(1) Ans. 4. (2) Ans. 4. (3) Ans. 2.
(4) Ans. as many as - of a day is contained times in I day,
that is, 2'.
(6) He could dig 2 trenches in 7 da., or 4 of 1 trench in 1 da.
(7) Hle could walk 5 times the distance in 16 days, or y of
the distance in 1 day: and in 2 days a 6~o
(8) An~s. -!-+4, which are -3
(9) Ans. 1++ - +, which are ~-.
(11) -+ 1- 3 -i=; 1 in 1 is contained 4 times. Ans. 4 da.! I 2 12 n 1
(12) *+q 4-n'; 43 in 1, 2} times. Ans. 2'4 days.
(13) -F4 -A- 1 —— 1; hence all three can do it in 1 day.
(14) 1+- I +1 =2-o- — 7; O-'in 1, 13 times. Ans. 14 hr.
3 ( ) T' - -T'(l6) I —r i=; in 1 is contained 20 times. Ans. 20 da.
I = z in I d a. -! - th e p art
(17) A and B can do 4+ -454 in1da. 4 —4fjthe part
that C can do in a da.; hence he can do it in 24 da.
(19) The first will empty the cistern 3 times in 1 hour, and
the second 5 times in 1 hour; hence both will empty it
8 times in 1 hour, or once in -~ hour.
200 KEY TO RAY'S SECOND BOOK.
(20) o days in 1 day is contained 1- times. Ans. 1' days.
(21) C can reap 4 of the field in 1 day, and since B can reap
it in 31 days, he can reap 3 such fields in 10 days, orT 3
of a field in 1 day. 3 =4; hence both can
reap V the field in 1 day, or the whole field in 2 days.
(22) A can do I of his piece in 1 da. If B can do a piece 3
times as large in 7 da., he can do - of such a piece in 1 da.;
hence both can do ~+3 6s of it in 1 da., or the whole in
as many da. as 4y is contained times in 1. 4 in1 1, 1
times. To do a piece of work 5 times as large will require 5 times as long. 5X1 -5 da. —6;9 da. Ans.
(23) Tj of 100 gallons=5 gallons to be emptied per minute:
the 1st and 2d empty 2 gallons in a minute; hence the
third must empty 3 gallons per minute, and in 20 minutes
60 gallons.
Or, the lst and 2d empty 40 gallons in 20 minutes and
the 3dc must empty 100 —40=60 gallons.
LEssoN X.
(2) Ans. 3,. (3) Ans 51. (4) Ans. 5`. (5) Ans. 14.
(6) Ans. 1. (7) Ans. 2 2.
(8) Ans. 11. (9) Ans. 20 Thus, if she gave away 4 of
what she received, she had 4 left; X of 3=., and - of 60
is 20.
(10) Since a remains, 35 is {, and 2 is 14 miles. ~ of 14 is
6, and 14 less 6 is 8 miles, the required distance.
(11) Ans. 25. (12) of 27 is 12, -- 8 is 6; 12 and 6 are
18, which is 3 of 30. Ans. $30,
(13) After losing 2v she had 4 left, and after giving of to
5 5 __ 3 3 7
Sarah, she had 4 of left. A of 4=4. If 4 is 6, i is
14. Ams.
(14) Ans. 20 cents. (15) 4 of 36 is 45, and 45 is 9 of 35. Ans.
(16) If Z of B's=2 of A's, then all of B's=4 of A's: 1-4-=
of A's-12: -4 7-=28 cts., A's; and 4-16 cts., B's.
(17) -41 -4+ = =- 8 ——.; hence 32 is 4-, and the whole is 96,
MENTAL FRACTIONS. 201
of which 32 are apple trees, 24 pear trees, and 8 plum
trees.
(18) The whole less 2 is A, and the difference between - and
2 is -; if 5 are 25, g is 5, and' are 10, the pear trees,
and b are 35, the apple trees.
(19) Since the plum trees are -1 of the apple trees, i of the
plum trees —1 of the apple trees; hence the cherry trees.1 and, that is, 5- of the apple trees; and the
whole number of trees=once the apple trees, 3- of the
apple trees, and 7 of-the apple trees; that is, 1+ —+
-, or 4~, of the apple trees=69.' of 69 is 3, which
I is 2. -14 -36, the apple trees; — =12, the plum trees;
and 7T-=21, the cherry trees.
(20) 5 of 12 years is 20 years, which is 4 of the ages of Jane
and Sarah; if 20, is 9,4 - of 20, or 5, is, and 9 —45
years, the sum of their ages. Then Sarah's age and
7 of her age, or!-5 of her age, are 45 years; 5 of 45 is
3, which is A of her age, and 8, or her age, is 24 years;
4 of 24 years is 21 years, Jane's age.
(21) -3 of 55=15; A of 30, or 24, is 4 of 54: twice 54=108;
15 in 108, 71 times. Ans.
(22) ] of John's money=3 of - of Charles':=: 9 9 of
Charles'+-33=Charles' money; hence $33 is Ad, and
Charles has $60, John has of $60$ —36.
LESSON XI.
(2) Since 4 of John's steps are equal to 5 of Henry's, 1 of
John's is equal to 1- of Henry's, and 5 of John's to 64
of Henry's; but while John takes 5 steps, Henry takes
6. Hence John takes 5 steps to gain 4 of one of Henry's
steps on him, and therefore must take 20 steps to gain 1
of Henry's steps on him.
(3) In taking 7 steps, John gains 2 steps on Henry. Hence,
to gain 30 steps, he must take as many times 7 steps as
2 steps is contained times in 30 steps. 2 in 30 is contained 15 times; 15 times 7 steps are 105 steps. Ans.
(5) Since 3 leaps of the hound are equal to 10 of the hare,
202 KEY TO RAY'S SECOND BOOK.
and the hound takes 3 leaps while the hare takes 5;
therefore, in taking 3 leaps, the hound gains 5 of the
hare's leaps on her; and to gain 100 leaps, lie must take
as many times 3 leaps as 5 leaps are contained times in
100 leaps. 5 in 100 is contained 20 times, and 20 times
3 leaps are 60 leaps. An.s.
(6) Since 4 of M's steps=7 of N's, 1 of M's-1 —l of N's,
and 5 of M's -=S- of N's; hence while M takes 5 steps
he gains 1- of N's steps on him, and to gain 35 steps, he
must take as many times 5 steps as 14 steps are contained times in 35 steps. 1, in 35, are contained 20
times, and 20X5 steps=100 steps. Ans.
(7) Since 2 of the hound's leaps-3 of the hare's, and the
hound takes 2 leaps while the hare takes 3, the hound
goes just as fast as the hare, and will not catch it, but
will continue at the distance of 8 leaps firom it.
(9) Since the tail is as long as the head and 1 of the body,
therefore the tail is 4 inches plus X the body, and the
body is equal to 4 inches, plus 4 inches. plus 1 of the
body; hence -2 of the body is 8 inches, and -1- is 4 inches.
body; 3hne
3_-12 inches, the length of the body; 44-4=8 inches,
the tail; and 12+8+4=24 inches, the whole length.
(10) Since C has as many as A+-' as many as B, C has 10
cents+3 of B's; but B's money —A's and C's; hence
B's=10 cents+10 cents+- of B's; or - of B's=20 cents;
hence B has 60 cents, and C 50 cents.
(11) Since the cow- cost as much as the sheep and 4 of the
horse, therefore the cow cost 8 dollars and - of the
horse; but the horse cost twice as much as both the
sheep and cow; hence the cost of the horse was equal
to 16 dollars, plus 16 dollars, plus ~ the cost of the
horse; therefore, the cost of the horse was 32 dollars.
Hence the horse cost $64, and the cow $24.
(12) Since the tail weighs 3 pounds more than the head and
V the body, therefore the tail weighs 11 pounds plus,
the body; but the body weighs as much as the head and
tail; hence the body is equal to 8 pounds plus 1i pounds
e —-- a ----- X —- - ---— ~- ----- -- --- ---- - — ~A
MENTAL FRACTIONS. 203
plus ~ the body; therefore - the body weighs 19 pounds,
and the whole-38 pounds. The tail weighs 11+19-30
pounds, and the whole fish 76 pounds.
(13) If 2 of B's steps=3 of A's, then 4 of B's-6 of A's:
hence, B gains 1 of A's in moving 6 of A's, that is,, of
the distance B travels;. of 30 ft.=5 ft. Ans.
LESSON XII.
(2) The difference between 4 peaches and 2 peaches is 2
peaches, and the difference between having 9 left, or 3
left, is 6; hence giving 2 peaches more to each would
increase the number given by 6. Hence he has as many
children as 2 peaches is contained times in 6 peaches.
2 in 6 is contained 3 times. Ans. 3 children. 3 times
2=6, and 6-+9=15, the number of peaches he had.
(3) The difference between giving 5 and 8 to each is 3, and
between having none left and 21 left is 21; 21 3 — 7;
hence she had 7 playmates, and 56 cherries.
(4) The difference between paying 1 dollar and 1, dollars a
yard is I a dollar, and the difference between having 5
dollars left and nothing left is 5 dollars. Hence she
wished to buy as many yards as - dollar is contained
times in 5 dollars, which is 10 times; therefore she
wished to buy 10 yards of silk, and had 15 dollars.
(5) The difference between 29 over and 6 lacking is 35 cts.;
8 cts.-3 cts.=5 cts. 5 cts. in 35 cts., 7 times. 7X8=
56, and 56-6-50 cts. Ans.
(6) 12 —10=-2,; 2~ in 40, 16 times. Ans. 16 yards.
(8) 3 times the age of James less his age is twice his age,
or 10 years; hence his age is a of 10 years, or 5 years;
and Thomas' age is 15 years.
(9) Since B's age is twice C's, and A's 5 times B's, therefore A's is 10 times C's, and the difference between A's
and C's is 9 times C's, which is 45 years. Hence C's
age is 5 years; B's 10 years; and A's 50 years.
(10) Since A's moneyi- B's, twice A's=B's; and since
(10)l Since A' money=~ -----
204 KEY TO RAY'S SECOND BOOK.
B's money is - C's, twice A's money is -1 C's, and 6 times
A's money=C's. But the difference between A's and
C's money is 815; therefore 6 times A's money less A's
money is $15. Hence 5 times A's money is 15, and A's
money is $3. Therefore B's money is $6, and C's $18.
(171) Since the second contains 4 times as many as the first,
and the third 3 times as many as the second, the third
contains 12 times as many as the first. Both the first
and second contain 5 times the first, and 12 times less
5 times the first, or 7 times the first is the excess of
the third above both the first and second, which is 70
sheep; hence 7 times the first are 70 sheep, and the first
contains 10 sheep, the second 40, and the third 120.
(12) If 3 of C's age=twice A's, C's age must be 6 times A's;
B's is twice A's; the difference between B's and C's
will be 4 times A's-20 years; hence A's age is 5 years,
and B's 10 years, C's 30 years.
(13) A daughter and a son together received 3 dollars, hence
there were as many of each as 3 dollars is contained
times in 18 dollars, which is 6; therefore he had 6 sons
and 6 daughters, in all 12 children.
(15) If he had worked every day he would have received 9
dollars, but as he received only 5 dollars, he lost 4 dollars by being idle; he lost 50 cents each day he was idle,
that is, his wages, 30 cents, and the 20 cents he paid for
his board; hence he was as many days idle as 50 cents
is contained times in 4 dollars, which is S. Ans. 8 days
idle, and therefore he worked 22 days.
(17) The difference between 4 yr. ago and 4 yr. hence is 8 yr.;
3 times B's age (which was A's) plus 8 yr. was equal to
twice (B's age plus 8 yr.); that is, 3 times B's age plus
8 yr. was equal to twice B's age plus 16 yr., hence B's
age was 8 yr., and A's 24 yr.
(18) In 10 years, A's age+10 years-3 of (twice A's+10
years)C — of A's+6 years: hence -5 of A's=10 —6=4:
5=20 years, A's age, and 40 years=B's.
(20) Since the first horse and $12 are worth 1 the second; 3
MENTAL FRACTIONS. 205
times the first horse and $36 are worth the second; but
the second horse and $12 are worth 4 times the first;
hence 3 times the first horse and $36, and $12, are worth
4 times the first horse; hence the first is worth $48;
and the second $180
(2 1) If: cost E pence, A will cost c, and 1, 3 penny: hence,
for 9 pence, you can buy as many as 5 are contained
times in 9 or 3, which is 52 Ans.
LESSON XIII.-MIS CELLANEOUS EXAMPLES.
(1) Ans. 105. (2) Ans. $60.
(3) If A walks 10 miles in 1- days, he walks 8 miles in 1
day, and if B walks 8 miles in 1A days, he walks 5 miles
a day. It takes A 2A days to travel 20 miles; and in 2, A
days B will travel 12A miles.
(4) 2 apples for 3 cents and 2 apples for 5 cents are 4 apples
for 8 cents, or 1 apple for 2 cents, which is the rate of
buying. 3 apples for 7 cents is 1 apple for 2- cents, the
rate of selling; hence on each apple sold he gained - of
a cent, and on each dozen, 12 times 3 of a cent, which
is 4 cents.
(5) For each egg at 2 cts. he bought 2 eggs at 3 cts. a piece;
hence for each 3 eggs he paid 8 cts.= —22 ts. a piece. By
selling them at 3 for 10 cts. he sold them at 31 cts. a
piece; hence on each egg he gained 3A-22, or' cts.,
and on each dozen 8 cts. Had he sold them at 4 for 10
cts., or 21 cts. a piece he would have lost 2 -2-1, or -
cts. a piece, or 2 cts. per dozen.
(6) 2 for 1 cent is l cent a piece; 4 for 1 cent is I cent a
piece. Hence each 2 pears cost +1, or L cent, and
each A cent. He sold them at A cent a piece; A less x
— 49; hence he gained 19 of a cent on each pear sold;
and to gain 18 cents he must sell as many pears as -490 is
contained times in 18 cents, which is 80. Ans. 80 pears.
(7) Each duck cost $, and each 2 chickens $l; therefore 2
chickens and a duck together cost it _.1. Z$ -f5
20 6 KEY TO RAY'S SECOND BOOK.
— $_w what he gained on each lot. Hence he sold as
many lots as l is contained times in $21, which is 30;
therefore he sold 30 ducks, and 60 chickens.
(8) 10X4=40, and 3X40-120, the hours work of 1 man;
of 120-15, the hours work of 8 men; and 15 hours-2.
days of 6 hours each. Ans. 2. days.
(9) Reserving 2 for Mary and 3 for Lucy, there are left 27
to be divided equally. I of 27 is 9, James' share. 9+
2-11, Mary's share; and 9+3=12, Lucy's share.
(10) Ten gal. in 1 hr. is 20 gal. in 2 hr., and 20-17=3, the
number of gal. left in each 2 hr.; hence it will require
as many times 2 hr. to fill the vessel as 3 gal. is contrained times in 15 gal. Ans. 10 hr.
(11) If A can do the work in 4- days, he can do twice the
work in 9 days, or 2 of the work in 1 day. If A and
B together can do the work in 2' days, they can do 7
times the work in 18 days, or.- of the work in 1 day.
T78 2-T-, the part of the work B can do in 1 day.
HIence B can do the work in 6 days.
(12) A, B, and C can do 1 of the work in 1 day; A and B do
of it in 1 day; and B and C I of it in 1 day. ence
l5 _. 3 —, the part C does in 1 day, therefore he can
do it in 13- days. R-g —, the part A does in 1 day,
therefore he can do it in 111 days. 3 the part
4~I~ — aU, the part
B does in 1 day, therefore he can do it in 27-3 days.
(13) Since 5 men, or 7 women, can do the work in the same
time, both together can do it in half the time of either;
that is, a of 35 days, or 17- days.
(14) Since 1 woman can do 4 of a man's work, 4 women can
do 3 men's work; hence the work of 2 men and 4 women
=the work of 2 men and 3 men, 5 men; and the work
of 1 man and 1 woman-the work of 1' men; hence it
will take 1 man and 1 woman as many times 28 days as
1-4 is contained times in 5. J1. in 5-2' times, and
2-X28 days are 80 days. Ans.
(15) Both consumed ts in I day, and f -or C. in 6 days. After
MENTAL FRACTIONS. 207
were consumed;s were left, and if the woman consumed -- in 30 days, she consumed a in 10 days, or the
whole in 50 davs. Bea-a0=l';, the part the man consumed in 1 day; hence it would last him 1 5o_21: days.
(16) 36 at 3 for 4 cents cost 48 cents, 24 at 4 for 3 cents cost
18 cents. 36+24=60, and 80 —60-20, which at 10
for 17 cents bring 34 cents. All the egos at these rates
bring 48+18+34=100 cents. 80 eggs for 100 cents= —
1ij cent a piece, or 15 cents a dozen.
(17) James' money+:- of James' money+S6, or:- James'
money+$G6=$34. IHIence 7 is $28, and,- is $4; 4-I$16,
James' money; and $18-Thomas' money.
(18) HTere A's money plus o of A's money less $5 is equal to
$51; or, 7 of A's money is equal to $56. Hence; is -
of $56, which is $8, and 5, or A's money, is $40; and
B's money is $11.
(19) Since A's age is twice B's, therefore 2-) times A's must
be equal to 5 times B's age; but -- of B's age plus 44
years is equal to 2 toimes A's age; hence 3 of B's tage
plus 44 years is equal to 5 times B's age; therefore 4'~B's age is 44 years; or 2j B's age —44 years; hence -— 2
Hears, and his age is 10 years; and A's age is 20 years.
(20) As many, and half as many more, are 1~ times as many
dollars as he has; hence 11 times A's money plus $2. is
$40, or 1.- times A's money=$37!~, or, of A's money=
87.; hence E is $2;l, or A's money is $25.
(21) a of the number=100 less 2,=97'-=1 ~5;, of the number — and the number-39.
(22) The whole minus - is aI. I of __nd AS - - J
=q —; hence -- of the number at first is G5; therefore ~ is 1 sheep, and -2, or the whole flock, was 72
sheep.
(23) Since - of the silver+$10=- of the gold, of the silver
+$-2==, 7 of the gold, or the silver+$124 —, or X of
the gold, or the silver= — of the gold-$1'. But the
gold and silver=$50, and the gold+- o of the gold!- - 9VBL — 6 —9 -. —--— 1 —C-~
208 KEY TO RAY'S SECOND BOOK.
-$12=0 50, or i' of the gold-$62. Hence'io=$2, and
1-, or the gold=-20; therefore the silver is $30.
(24) Since - of's — 3 of B's+$5, therefore 1 of A'si ji of B'sA-$8, or the whole of A's= — of B's-'I-16. But
since both have $56, therefore B's and - of B's+$16=
$5o5; or 3 of B's+$16 is $56; therefore 5 of B's is $40.
of 40 is $8, which is 1 of B's, and - tht is, the
whole of B's, is $24.,$56 —T24=$32, A's.
(25) Since 3 Of A's distance-4 of B's, therefore vof A's
distance-3 of 4, or 4 of B's, and 4, or the whole of
A's distanCe=7X-A,~ or 01' of B's. I' of B's distance
and B's distance, or 44 of B's distance=S6 miles.
lHence -1~ of B's distance is?-1 of 86 miles, or 2 miles,
and - - is 30 miles. A's distance is 24, or 56 miles.
(26) Since - of A's is equal to 4 of B's, 1 of A's is equal to. of B's, and 43, or the whole of A's, is equal to 5 of B's.
Hence their difference is I of B's, and 4- of 1 of B's is
offthfis5 4 5
$Y15, that is, of B's is $15; hence 1o is I o: $15,
which is $5, and 2o, or the whole of B's, is 20 times $5,
or $100; and A's is 6 of B's, which is $120.
(27) Since the watch and chain are -! of the watch, therefore
the chain is a of the watch. 3 times 4- are,, and twice
the price of the watch plus 4 of its price are h, which
care $100. Hence 1 is -2' of $100, or $5; and 7, or the
price of the watch is $35, and the chain is 2, or $10.
(28) 7- of 24 is 21; if 21 mi. are 7. of 21, which is 7, is
of the distance from B to C; hence'-, or the distance, is
49 mi.; and from A to C is 49+24, or 73 mi.
(29) 4 horses for 2 months=8 horses for 1 month. Since 3
cows eat as much as 2 horses, 9 cows eat as much as 6
horses; hence 9 cows for 3 months=6 horses for 3
mlonths-18 horses for 1 month. Again, since 10 sheep
eat as much as 3 cows, and 3 cows as much as 2 horses,
therefore 10 sheep eat as much as 2 horses, and 20 sheep
as much as 4 horses; hence 20 sheep for 5 months=4
horses for 5 months-20 horses for 1 month. Hence,. A
has the same as 8 horses 1 month, B the same as 18
tS X,-,, --
MENTAL FRACTIONS. 209
horses 1 month, and C the same as 20 horses 1 month,
in all 8+-18+20-46 horses 1 month. 46 for $92 is $2
for 1 horse; hence A ought to pay 8X$2=-$16; B 18X
$2 —$36; and C 20X$2=-$40.
(30) Since the three eat 8 melons, each eats 22 melons: James
eats 2' melons out of the 3 melons he furnished. Of
the 2-l melons eaten by Thomas, James furnished 1, and
John 21, that is, James furnished'a, and John, both
together making s. Therefore, since James furnished
i- of the s, and John 9 of the _, James should have 3 of
the 8 cents, which is 1 cent; and John - of the 8 cents,
which are 7 cents.
(31) + — 617 3-; and 1 3- C's pal't. a
=-p-; hence is $160, and 2'T is - of $160=-SO; f
— 6 times $80=$480, A's share;'7=7 times $80=
$560, B's share; and = —8 times $80=$640, C's share.
(32) Since A's age is 5 of B's, twice A's age is 1o of B's;
hence -- B's age+B's age+- B's age= —0 B's age-2
years; or?1 of B's age=2-0 of B's age-2 years; therefore { of B's age is 2 years, and his age is 12 years; and
-9- of B's age, or A's age, is 20 years.
(33) A and B can do 1I in 1 day, hence in 4 days they can
do -1 or { of it, and 3 remains. If B does A in 36 days,
16 4 4
he does 3 of — 4 in 12 days, or G- of - in 1 day, or
the whole in 48 days. Since both can do -f in 1 day, and
Bcan do j1 in 1 day, A can do 16 48 in 1 day,
and the whole in 24 days.
(34) Since v of B's and C's is $12, q is $6, and, or the sum
of their shares, is $42. Again, since -9 of C's=:~ of A's
and B's, k of C's=T-O of A's and B's; or'~0, that is,
4- of C's=A's and B's; but since A's is $12, therefore
5 of C's=-12 and B's, or 4 of C's less $12=B's. Therefore C's and L of C's less $12 —=$42, or 9 of C's-$54;
hence - of $54, which is $6, is 4 of C's, and A, or the
whole of C's, is $24; and $42 less $24=$18, B's share.
(35) To mowr ~- in 12 days would require 3 men; to do it in,
the time, 12 men; in 4 the time, 4 men. Ans.
Key -14
210 KEY TO RAY'S SECOND BOOK.
(36) During the first 6 months the second has $100 more than
the first; to equalize it, then, the first must dsttingt the
last 6 months have $100 more than the second, that is
$300, and therefore must put in $200 Aims.
(37) After A had given B 2, they each had I of 24=12 cts.;
hence A had 12+2=14 cts., and B 12-2 or 10 cts.
(3$) 3 of 50-30. If 30 exceeds the age as much as 10 is
less, 30-10-20 must be twice thle excess, therefore the
age is 30-10 —20 years. Ans.
(39) T'h difference of 4 cts. in price maknes 30 cts. dif[erence in
aniaount; hence there were 3-~-7I dozenl; 10 cts.-
—:! cs. loss per dozen; 6-+1l —,7 cts. cost per dozen.
(4:0) A difference of 3 cts. in price mnakes -1 50 cents difference in amnount; hence there were $].50- -3=50 lb. Ans.
(41) 12 hoys-=4 women-2 men. If 12 boys or 2 mien can
do the wolrk in 6 days, it would require 6 tim-es 2 men to
do it in I day. Aes. 12 men.
(11) 2 lb. iwould be worth 03, -1 —6 —=10 cts., or 5 Ets. a lb.;
l-b11). is worth l 2 cts., and foxr 1t we must give 20 lb.
(.38) If of the gainu-1- of the selling price, will be -X
and the gain - of the sellinOg price hence the cost will
be 1-2 —- of i:he selling price. 3 ards at 4 cost $ 15
which is - of $25. 42s.
(44) The first package cost 24 3 —21 cts., and contained 21
-:-7-3 lb.; the second contained twice 3=6 lb., and cost
6X8=48 cts., atud must be sold for 43+5 =53 cts.
SECTION P2. —PE rCENNTAGE.
NoTE.Tile greater part of the questions in this section are
of such easy solution, that it is deemed unnecessary to present
either their answers, or tle mnethod of solving them.
L E SON II.
(2) Anis. 25. (3) As. 20. (4) A.s. 50. (5) Ans. 60.
(6) Anas. 30.
PERCENTAGE. 211
(7) If he cleared ~ he must have sold them for a of the cost.
If 6 cents was ~, I is 1 cent, and 5, or the first cost, was
5 cents. Hence, in selling them for 8 cents, he cleared
3 cents on each; 1=60 %.
(8) 6 yards for 3 is $, a yard, and 5 yards for $4 is $4
per yardll. 4'1 is3 1 i, is -15. A is
-'Io, or 60 %.
(9) If he lost 1, 8 cents are 4 of the first cost, and the first
cost was 10 cents. 3 for 25 cents is $8 cents a piece.
10 cents-8 — cents is 12 cents. 12 cents is - of 10
cents; therefore by selling them at 3 for 25 cents, he
would have lost A of the cost, or 16]- per cent.
Again: 2 for 25 cents is 121 cents a piece, a gain of 21
cents. 2- is 4 of 10; hence by selling them at 2 for 25
cents, he would have gained 4 of the first cost, or 25 o.
(10) 2 for 3 cents is 1, cents a piece; 3 for 2 cents is - cent
a piece. 1ie- is ~ loss on each. 14 is 9, therefore 6
is g of 1- cents, or the first cost; hence he lost 7 of the
first cost, or 55 5f.
(12) Aces. 20. (13) Ans. 10. (14) A.es. 40. (15) Ans. 50.
(16) Ans. 12. (17) Ans. 3. (18) Ans. 25.
(19) 1-=2=40 %. At! the loss is l_=121 Yo. At 9 the
loss is -1=10 %. At 47 the loss is =15 %.
PROFIT AND Loss.-LESSON III.
(1) 20 o — 20 Ans. 75 =15 -. Anls.
(2) 100 ~%4-00=1), that is, the cost and gain are equal.
150 o-11- o1- or 4; the gain is 1) times the cost.
(3) 5 25 Ans. 35 o — UT —7. Ans.
(4) The whole, or 1=-I-. Ans. 100 %. = —--
Ans. 300 %.
(6) If: of the gain equals ~ of the cost, 4 equals 3, and
the whole gain equals 9 — = —13%-036,%.
(7) If or, and, or the whole of the gain
-= or 3 of the cost. - o=75o A ns.
~~~4~~~~~"~~ Ans.~""I"""~LII"-~~~P r~~ I
212 KEY TO RAY'S SECOND BOOK.
(8) 25 %=1To=4 —; hence the watch sold for` of its cost;
if $10 is -, 4, or the first cost, is $8. When he sold
it for $12, he gained $4, and 4=1_ -u=50 So. Ans.
(9) 124 whei, when is lost, the muslin is sold for a- of
its cost. Hence 7 cents is s of the cost, and the first
cost was 8 cents. By selling it at 6 cents per yard he
lost 2 cents. — 4==25 So. Ans.
(10) 16B- -—. If 6 was lost the horse was sold for - of
his cost; if $35 are 5, - is $7, and a are $42, the cost.
$63-42=-$21, the gain; - - o -5 % Ans.
(11) 20 2%0i —- —; hence $18 are A of the first cost, or $15.
10 =, and of $15 is $1,. 15-1 —=$13., watch
sold for. He lost $18 —13A=$44 Ans.
(12) 20 o % —o —4; hence $60 are ~ of the first cost, and y
of $60 is $10, which is A, or A's gain. B lost 20 %, or
5 of what it cost him; 5 of $60 is $12, B's loss. Hence
B lost $2 more than A gained.
(13) Since 25 % is 4, he sold one watch for - of its cost,
and the other for 3 of its cost; hence $30 are 5 of the
first cost of the first watch, and 4, or the first cost, is $24,
and he gained on the first watch $6. If $30 is of the
first cost of the second watch, its first cost must have
been $40, and he lost $10 by the sale; therefore, he
lost $10 —$6=$4 Ans.
(14) 4 apples for 3 cents is -3 of a cent each. 50 % is fz;
hence 4 is.3 of the first cost, and the first cost is 4 cent.
5 apples for 4 cents is 4 of a cent each; 4-4= 3, the
gain on each apple. 1=-%, hence 3 is - of the first
cost; -}=60 ~o, the gain Ya.
(15) 5 lemons for 4 cents is 4 of a cent each. 20 o is -, and
he sold the lemons at 4 of their cost. If 4 of a cent is
4 of the first cost, of a cent is I of the cost, and the cost
was 1 cent each. 6 lemons for 5 cents is - of a cent
each; 1 —=- -163 %, the % of loss.
(16) 2 of 10 % of 60 —- of - t of 60 —4; 4 is Z of 8; 8 is 5.
of 40; r=20% A.ns.
PERCENTAGE. 213
(17) ~ of 3 of 50 % of 120=-93 of I of 120=18; 18 is 10
less than 28; 28 is 20 %, or ~ of 5 times 28, which is
140. Ans.
LESSON IV.
(3) Ans. 60 cts. (4) Ans. $2. (5) A2s. $4.80
(6) Ans. $6. (7) Ans. $8. (8) Ans. $9.
(9) Ans. $8.40 (10) Ans. $9.
LESSON V.
(2) Ans. $1. (3) Ans. $2.80 (4) Ans. $2.40
(5) tins. $4.50 (6) Ans. $3.25 (7) Ans. $5.162
(8) Ans. $3.53 (9) Ans. $4.80 (10) Ans. $19.20
(11) Ass. $8.45 (12) Ans. $4.24 (13) Anls. $3.20
(14) Ans. $6.75 (15) Ans. $3.80 (16) Ans. $28.
(17) Ans. $44. (18) Ans. $68.20 (19) Ans. $32.80
(20) Ans. $56.80 (21) Ans. $99.12
LESSON VI.
(2) Ans. $50. (3) Ans. $60. (4) Ans. $75.
(5) Ans. $140. (6) Ans. $240. (7) Ans. $350.
(8) Ans. $4000.
LESSON VII.
(2) Ails. $200. (3) Ans. $500. (4) Ass. $250.
(5) Ans. $300. (6) Ans. $25. (7) Ans. $750.
LESSON VIII.
(2) Ass. 4 years. (3) Ans. 21 years. (4) Aiss. 2-2 years.
(5) Ans. 33 years. (6) Asns. 63 years.
(8) Ans. 50, 33', 20, 162,, 4, 12, 10, and 8 years respectively.
(9) Ans. 40 years. (10) Ans. 25 years, and 20 years.
214 KEY TO RAY'S SECOND BOOK.
LESSON IX.
(2) Ans. 8. (3) Ans. 5. (4) Ans. 7. (5) Ans. 8.
(6) Ans. 7. (7) Ans. 5. (8) Ans. 6. (10) Ans. 8-1.
(11) Ans. 10. (12) Ans. 12~, 20, 25, and 50, respectively.
LESSON X.
(2) Ans. $400. (3) Ans. $25. (4) Ans. $500. (5) Ans. $300.
(6) The amount of $1 for 4 years, at 6 %o, is $1.24; and
the required principal will be as many dollars as $1.24
is contained times in $496: $496. $1.24 —=400.
Ans. $400.
(7) $1, at 5 s, for 4 years, will amount to $1.20; hence the
present worth of each $1.20 of the given sum, is $1;
therefore, the present worth of $24 will be as many
times $1, as $1.20 is contained times in $24: $24__
$1.20=20. Ans. $20.
(8) The amount of $1 for 5 years, at 6 %, is $1.30, and the
present worth of each $1.30 of the amount is $1; therefore, the present worth of $65 will be as many times $1,
as $1.30 is contained times in $65: $65. —$1.30=50;
hence the present worth is $50. By subtracting the
present worth from the debt, we obtain the discount.
$65-$50$o=15.
(9) The amount of $1. for 5 years, at 5 %, is $1.25; hence
the present worth of each 81.25 of the given sum, is
$1. And the present worth of $55, will be as many
dollars as $1.25 is contained times in $.55: $.55-$1.25
=44; therefore, $44 is the present worth. $55-$44=
$11, the discount.
(10) 6 years 8 months:-6- yr. If the interest of $1 for 1 year
is 6-cents, for 6 years it will be 40 cts. $X; hence the
amount of $1 for 62 years, at 6 S%, is $15-7-. Therefore the present worth of $.77 will be as many dollars
as 5 is contained times in 77, which is 55. Hence the
present worth is $55, and the gain by paying it now
$77 —55 —$22.
e, -H e,
MENTAL FRACTIONS. 215
LESSON XI.
MISCELLANEOUS EXAMPLES IN INTEIREST.
(1) 6X4(-=25; i —o =5. Ans.
(2) 5X5=25; _25. 1+-=4, and l is 1- of 5. A.
(3), of — 1F0; hence the interest of 1 year is l1e of the principal; -1 -0 Ans. 10.
(4) 2 years 6 months —2=- years. 5- of and 2 times.j ~U, 4-Z, and 2 tim es
-1 vgg the part the interest is of the principal for 1
year - T-0 —TO%10 o. Axns.
(5) 10 To= ('T j-o; hence for 1 year the3 interest is l-' of
the principal, and at the same rate o, the time for
which the interest is 3 of the principal, will be as
many years as Tl6 is contained times in 3-. -f in 3 is
contained 6 times. Ans. 6 years.
(6) If 3 times the yearly interest is equal to.9 of the principal, once the yearly interest is equal to s- of the principal. 7-1T-l1o 2 %. A'.
(7) - of the interest for 2 years is thle sa.me as 20 Of the interest for 1 yea r.
If 2 of the interest for 1 year equals - of the principal,
of the interest for 1 year equals j of the principal,
and s, or the interest for I yearl equa ls 7 of the princi1' a —% b%-.40%. Ans.
(8), for 3 years is the soae as for I yer. f s of tle
intcerest for 1 vear equals 9 of the principal,. equals
- o 3~ ald g Cclqu'rlS -, fOr T-6o of the principat. 1y' equals 6 /. Ais.
(9) 1 year 4 months= 6 mont1hs, a hiclh Is,twice 8 months;
heince the inte-rest for I yea- 4 moanthfs wrvill be -, of
the principa; of $G'00-m$6.' n".
(10) 1 year 4 mon. are I or ir If rhe interest of yr
is f of the prrincipal, for year it is I{ of — 1T —O, and
for 1 year it is T9% of the principal; that is, the rate
YOUB- ____ _ s --- -
216 KEY TO RAY'S SECOND BOOK.
is 9 %. Hence for 1 year the interest is $9; for 1
month the interest is $4, and for 8 mon. $6; for 18
days the interest is 1, or 3 of $ or 45 cents.
Therefore, the whole interest is $9+-$6. 9=$1529, or
$15.45
(11) 4 years at 10 % is the same as 1 year at 40 s, and 1
year at 40 % is the same as 8 years at 5 So. Ans. 8 years.
(12) 3~ years are,o years. $40 for 1,o years is $4 for'- of a
year, or $12 a year. 5 15 is, hence $12 are I of
the principal, which must therefore be 12 times $20=
$240. Hence B's money, and twice B's money, or 3
times B's money is $240; therefore B's money is $80,
and A's twice $80, or $160.
(13) 1: years are -o years. $49 for. years are $7 for 5 year;
or $35 a year." 7 %o is TiP, hence $35 are Tlo of the
principal, and $5 are T-i, and _1 arec $500, the sum
of A's and B's money. Since 2 times A's equals 3 times
B's, 1 time A's equals 1 B's; therefore B's money and
1, times B's money, that is, are equal to 500; hence
a B's money is - of $500, or $100, therefore B's money
is $200, and A's $300.
(14) 2 yrs. 3 mos.-=2=9= yrs. $18 for - yrs.=$2 for -
year, or $8 for 1 year. 4 o is T4 —; hence $8_=I
of the principal, and -Sn, or the whole principal, is $200,
which is -I of A's, and V of B's money. Since ~ A's
2 B's money, 4 A's=-, B's, and a A's-B's nmoney;
hence B's money and I B's, or. B's=$200; ~ of $200
— 66', which is, B's, and $1331-B's money. Since a
A's —-- B's money, A's money==-4 B's, and -a of $133~
-$177', A's money.
APPENDIX.
TEST EXAMPLES
FOR THE
SLATE OR BLACKBOARD.
SUGGESTIONs. —Almost every teacher, in the course of his instructions in arithmetic, desires to exercise his pupils on the slate
or blackboard with examples not contained in the text-book. The
following are intended for this purpose. A part of the class, or
the whole, if there is room, should be called up together; the examples should be given out slowly and distinctly, all the pupils
working the same example at the same time.
When there is not room for all the members of the class to
work on the blackboard at the same time, a part may work on
their slates at the recitation seat. Where the Mental Arithmetic,
(Ray's Second Book,) does not contain exercises that may be used
in illustrating the rules in the Third Book, there will be found
in the following pages, mental exercises preceding the examples
intended for the blackboard.
AD DI TI ON 1OF STIMPLE lfUMBTElRS.
1. 471 + 724+ 360+117-= how many? Ans. 1672.
2. 258 + 233+ 421 —609= how many? Ans. 1521.
3. 4082+5412+ 340+617- how many? Ans. 10451.
4. 7324+-2465+ 416+- 38= how many? Ans. 10243.
5. 71654. 404+1253+986 —- how many? Ans. 9808.
6. 836 -+1245-+4314+ 980+4207+ 836=12418.
7. 1253+3178+6421-+6893+ 702+1458=19905.
8. 7163+-2461+ 716+3072+ 409+ 516-14337.
9. 1.242+ 563+8000+ 906+1780+- 438=12929.
10. 7362+ 486+3547+1098+ 362+1346=14201.
('217)
218 TEST EXAMPLES.
11. 463 + 246+ 36+ 45+ 37+134+629+89= 1679.
12. 4167+3254+6323+634+705+-248+- 47+89=15467.
13. 78167+8054+11167+9084+ 550+1809+7811+9750
=126392.
14. 2881 +92375+ 648+2437+ 717-+ 2529+2160+360
=104107.
15. 26341+11426+30637+9817+8784+12060+356+1217
=100638.
16. What is the population of the earth, supposing Europe
to contain 230000000, Asia 390000000, Africa 60000000,
North and South America 40000000, and Oceanica 20000000?
Ans. 740000000.
S UBTRACTIO I.
SUGGESTIONs.-The easiest method of exer- 34682
cising pupils at the blackboard in simple 346820 a cipher
subtraction, is to direct them to write down 34682 [added.
any number, such as 34682, then to annex a
cipher to it, as in the margin, and subtract 312138
from this continually the number first set 34682
down; when, after subtracting nine times, if 277456
the operation has been correctly performed, 34682
the remainder will be the same as the origi- 242774
nal number. The reason of this is evident; 34682
adding the cipher is the same as multiplying -
by 10, and the operation really consists in 208092
nothing more than taking 9 times the number 34682
from 10 times. In choosing the first number, 173410
the right hand character should not be a 34682
cipher, for the reason that it gives less variety 138728
to the work. 34682
104046
34682
69364
34682
34682
MULTIPLICATION OF SIMPLE NUMBERS. 219
ULTIPLI CATIONl0.
NoTE.-We have given an extensive list of examples in multiplication and division. Exercises in these rules necessarily
embrace the operations of addition and subtraction, and are,
therefore, well suited to render the pupil familiar with all the
elementary rules.
Multiplicands. Multipliers. Products.
1. 379 75......... 28425
2. 789 57...... e 44973
3. 2728 71.......... 193688
4. 3570 86.......... o 307020
5. 3259 43.o. o 140137
6. 5630 99...... 557370
7. 426 111........... 47286
8. 375 123........ 46125
9. 572 137 a...... o 78364
10. 729 236.......... 172044
11. 973 297.......... 288981
12. 1334 333......... 444222
13. 1779 347.......... 617313
14. 1986 436.......... 865896
15. 2327 572..... 1331044
16. 3377 635......... 2144395
17. 4323 736....... 3181728
18. 5745 892........ 5124540
19. 7963 997..... 7939111
20. 8381. 819.......... 6864039
21. 2764 785.......... 2169740
22. 6392 527.......... 3368584
23. 3231 492.......... 1589652
24. 6798 987.......... 6709626
25. 324 1476......... 478224
26. 135 1246..., 168210
220 TEST EXAMPLES.
Multiplicands. Multipliers. Products.
27. 365 1646 o..... 600790
28. 829 6874..... 5698546
29. 1634 3457..... 5648738
30. 6516 3258..... 21229128
31. 1754 2369..... 4155226
32. 1437 2596..... 3730452
33. 3587 4121..... 14782027
34. 5036 2139..... 10772004
35. 4007 3271.... 13106897
36. 4786 3497.... 16736642
37. 87603 9865.... 864203595
38. 4379 4265.... 18676435
39. 31624 7138...... 225732112
40. 83457 6835..... 570428595
41. 683794 5827...... 3984467638
42. 14278 3236..... 46203608
43. 50361 5267..... 265251387
44. 18014 13184..... 237496576
45. 23930 11491..... 274979630
46. 683794 17481.... 11953402914
47. 54042 39552...2137469184
48. 8735628 35784.. 312595712352
3MULTIPLICATION BY FACTORS. (See Article 33.)
1. 2358 24......... 56592
2. 2596 27..... 70092
3. 4531 28...... 126868
4. 4765 36..... 171540
5. 3589 48..... 172272
6. 5472 64.... 350208
7. 7359 72..... 529848
8. 84622 96.. 8123712
9. 52389 63... 3300507
10. 432234 54........ 23340636
11. 79797 81........6463557
~~ — ----- ------ ~ -- ----- - --------— ~~~~~~~~~~~~~~~~~~
MULTIPLICATION OF SIMPLE NUMBERS. 221
Multiplicands. Multipliers. Products.
12. 98979 63.......... 6235677
13. 1234 44........ 54296
14. 4729 77.......... 364133
15. 13579 108...... 1466532
16. 24682 120.......... 2961840
17. 65423 132..... 8635836
18. 875643 144......... 126092592
IMULTIPLICATION WHEN THERE ARE CIPHERS IN THE
MULTIPLIER. (See Articles 34, 35.)
1. 483 200.......... 96600
2. 4728 40.......... 189120
3. 6823 80.......... 545840
4. 158 350..... 55300
5. 7428 520..... 3862560
6. 9436 870.......... 8209320
7. 369 800. 295200
8. 6230 50..... 311500
9. 8370 240. 2008800
10. 46000 340........ 15640000
11. 5763 3600.2.........0746800
12. 5700 6800........ 38760000
13. 7430 4520..... 33583600
14. 9040 8320.......... 75212800
15. 9651 304........... 2933904
16. 12318 7004........ 86275272
17. 57906 1001......... 57963906
18. 18001 2600.......... 46802600
1 9. 8602 1030..... 8860060
20. 9100 3007.. 27363700
21. 90042 9009......... 811188378
22. 70302 8030..... 564525060
23. 904000 10200..... 9220800000
24. 80307 460001... o. 36941300307
~ —-------------------------— 0
222 TEST EXAMPLES.
D IVI SIO TN.
I. SHORT DIVISION. (See Article 45.)
Divisors. Dividends. Quotients. Rem.
1. 3 3636......... 1212
2. 2 4264........ 2132
3. 4 5372 e s...... 1343
4. 3 7692...... 2564
5. 4 8364....... 2091
6. 6 12798......2133
7. 7 23694....... o 3384,, 6
8. 8 35826....... 4478... 2
9. 9 39875....... 4430... 5
10. 10 42736...... 4273... 6
11. 11 54387.... 4944.. 3
12. 12 98764....... 8230.. 4
13. 9 52345........ 5816.. 1
14. 8 763279........ 95409... 7
15. 7 53262....... 7608.. 6
16. 9 30205...... 3356... 1
17. 7 1234567....... 176366... 5
18. 9 4326721..... 480746... 7
19. 12 7295432...... 607952... 8
II, LONG DIVISION. (See Article 46.)
1. 21 4524.....,.. 215...9
2. 42 9870..... 235
3. 58 87654..... 1511.. 16
4. 61 28609....... 469
5. 29 153598..... 5296.. 14
6. 36 4104..... 114
7. 23 3496...... 152
S. 88 15362..174.. 50
9. 57 44973.... 789
10. 86 307020..... 3570
11. 23 461.25..... 2005.. 10
12. 75 8425..... 379
13. 71 661419........ 9315. 54
@-Gwrausrns~~asa-~l~av.-ru,~~~~,~u~a
SHORT DIVISION. 223
Divisors. Dividends. Quotients. Rem.
14. 53 982700... 18541.. 27
15. 58 4327865...... 74618.. 21
16. 91 764231..... 8398.. 13
17. 92 333222..... 3621.. 90
18. 93 426891..... 4590.. 21
19. 94 5432687.... 57794.. 51
20. 85 444333... 5227. 38
21. 87 71778543..... 825040 63
22. 124 46215...... 372.. 87
23. 136 24374...... 179.. 30
24. 137 78364....... 572
25. 297 288981..... 973
26. 347 617353...... 1779 40
27. 436 865896......1986
28. 472 251104.. 532
29. 223 52135.. 233.. 176
30. 572 1331044...... 2327
31. 635 2144395..... 3377
32. 746 549056...... 736
33. 7'76 297208...... 383
34. 725 5828921.... 8039.. 646
35. 806 328848...... 408
36. 842 70173290..... 83341.. 168
37. 846 353628...... 418
38. 8 76 379308...... 433
39. 855 38923458... 45524.. 438
40. 906 405888... 448
41. 967 48325859..... 49975.. 34
42. 975 8971021304.. 9201047.. 479
43. 1111 142372.. o. 128.. 164
44. 1212 2346521... 1936.. 89
45. 1432 347854... 242.. 1310
46. 2312 453276... 196.. 124
47. 3587 5497630.. o 1532.. 2346
49. 1563 4917361... 3136. 113
j50. 1774 5967854. 3364 118
224 TEST EXAMPLES.
Divisors. Dividends. Quotients. Rem.
51. 2140 4560700........ 2131.. 360
52. 4625 3456745....... 747. 1870
53. 2507 9736205....... 3883. 1524
54.. 6425 23470525....... 3653
55. 7898 3875642....... 490. 5622
56. 7569 57289761....... 7569
57. 8832 45785088....... 5184
58. 5476 29987000...... 5476.. 424
59. 7396 54701000....... 7396.. 184
60. 6787 86167752....... 12696
61. 6315 103415000....... 16376.. 560
62. 7171 142308495....... 19845
63. 23785 237854...... 10.. 4
64. 43587 41258724....... 946. 25422
65. 442211 66335522....... 150. 3872
66. 1.23456 87654321....... 710.. 561
67. 538723 1.00002002....... 185 338247
68. 400203 96700054....... 241 251131
69. 639406 12707070....... 19 558356
70. 4670012 513287321....... 109 4256013
71. 32378 4012301...... 123. 29807
72. 56000 5600001....... 100... 1
73. 111111 2222222...... 20... 2
74. 333771 43723521...... 130 333291
75. 21432 23786543....... 1109 18455
76. 333 999999999. 3003003
77. 414141 462462462..... 11.16 281106
78. 47217 37854321....... 801 33504
79. 785433 785432179..... 99 784612
80. 153269 175432798....... 1144 93062
81. 182766 18276543....... 99 182709
82. 297208 88332595264...... 297208
III. DIVISION BY COGMPOSITE NUMBERS. (See Article 47.)
1. 24 56592....... 2358
2. 1 ]5 735379..... 49025... 4
M 5G -_f. <
DIVISION OF COMPOUND NUMBERS. 225
Divisors. Dividends. Quotients. Rem.
3. 36 171540.... 4765
4. 48 172272..... 3589
5. 72 529848... 7359
6. 63 3300507... 52389
7. 56 249738....... 4459....34
8. 54 23340636.... 432234
9. 81 6463557... 79797
10. 72 84630986..... 1175430.. 26
11. 81 34210390..... 422350... 40
12. 16 3254632..... 203414... 8
13. 49 78787878... 1607915.. 43
14. 108 871992.... 8074
15. 121 5200769... 42981... 68
16. 108 991254......9178... 30
17. 132 491832.... 3726
18, 120 425717.... 3547... 77
19. 144 729814...... 5068... 22
20. 180 29070..... 161... 90
IV, WHEN THERE ARE ONE OR MORE CIPHERS ON THE RIGHT
OF THE DIVISOR. (See Articles 48, 49.)
1, 20 32785..... 1639.. 5
2. 30 437856..... 14595... 6
3. 40 812569... 20314... 9
4. 50 437694... 8753.. 44
5 60 328320....5472
6. 70 830547..... 11864...67
7. 80 545840....6823
8. 90 765932.....8510... 32
9. 100 457834.... 4578... 34
10. 110 363587.... 3305... 3
11. 120 7395173..... 61626... 53
12. 70 67216540..... 960236... 20
13. 450 3613500.... e. 8030
14. 2600 413798..... 159.. 398
15. 3500 4448623.... 1271.. 123
Key.-15
226 TEST EXAMPLES.
COQIPOUND NUMiVBERS.
I. REDUCTION FRO A HIGIIHER TO A LOWER ORDER.
1. Reduce 3 bu. 5 qt. to pints. Ans. 202 pt
2. Reduce 17 bu. 3 pk. to pints. Ans. 1136 pt.
3. In 7 lb. 11 oz. of gold, how many gr.? Ans. 45600 gr.
4. In 5 lb. 17 pwt. of silver, how many grains?
Ans. 29208 gr.
5. At 3 cents a grain, how much will 1ib 5 ~ of musk cost?
Ans. $244.80
6. Reduce 17 lb 7 3 1 gr. to grains. Ans. 98341 gr.
7. Reduce 5 cwt. 3 qr. 12 lb. to ounces. Ans. 9392 oz.
8. Reduce 4 T. 1 cwt. 3 qr. to pounds. Ans. 8175 lb.
9. At 5 cents a pound, what will 5 cwt. 1 qr. 15 lb. of
cheese cost? Ans. $27.00
10. At I cent 5 mills a pound, what will a load of hay cost,
weighing 15 cwt. 2 qr. 20 lb.? Ans. $23.55
11. In 3 mi. 8 rd. how many rods? Ans. 968 rd.
12. In 17 yd. 2 ft. 9 in. how many inches? Ans. 645 in.
13. In 1 mile there are 5280 feet; how many times will a
wheel 17 ft. 6 in. in circumference turn round in running 14
miles? A2ns. 4224 times.
14. H-low many times will a steamboat, 136 feet 3 inches
long, run her length, in going from Pittsburgh to the mouth of
the Ohio, the distance being 981 miles? Ans. 38016 times.
15. In 3 sq. mi. how many perches? Anzs. 307200.
16. In 7 sq. yd. how many sq. inches? Ans. 9072.
17. In 5 cubic yards, how many cubic inches? Ans 233280.
18. In 17 cords of wood, how many solid feet? Ans. 2176.
19. Hlow many nails in 7 pieces of cloth, each containing 5
yd. 2 qr. 1 na.? Ans. 623.
20. Reduce 15 E. Fr. 5 qr. 3 na. to nails. Ans. 383.
21. In 2 hhd. of wine, how many gills? Ans. 4032.
22. How many times can 3 gills be filled out of 31 gal. 2 qt.
of cider? Ans. 336.
23. How many times can a bucket of 3 gal. 2 quarts, be
filled out of a cistern containing 55 hhd.? Ans. 990 times.
24. From 5 vessels, each containing 3 hhd. 12 gal. of cider,
how many kegs can be filled, each containing 2 gal. 3 qt.
I pt.? Ans. 349 and 13 pints left.
25. From a barrel of beer, how many bottles can be filled,
each capable of holding 1 pt. 2 gi.? Ans. 192.
COMPOUND NUMBERS, 227
26. From 14 hhd. of beer, how many kegs can be filled,
each capable of holding 5 gal. 1 qt.? Ans. 144.
27. WVhen a person is 36 years old, how many minutes has
he lived, reckoning a year 365 days 6 hours? Ans. 18934560.
28. How many seconds in a lunar synodical month, which
contains 29 da. 12 hr. 44 min. 3 sec.? Ans. 2551443.
29. How many seconds in a lunar siderial month, which
contains 27 da. 7 hr. 43 min. 11 sec.? Ans. 2360591.
30. How many seconds in 18 centuries, the solar year being
365 da. 5 hr. 48 min. 51 sec.? Ans. 56802475800.
II. REDUCTION FROM A LOWER TO A HIGHER ORDER.
1. In 240 pints, how many bushels? Ans. 3 bu. 3 pk.
2. How many bushels are there in 10 bags, each containing
195 pints? Ans. 30 bu. 1 pk. 7 qt.
3. In 639 pt., how many bu.? Ans. 9 bu. 3 pk. 7 qt. 1 pt.
4. In 12960 grains of gold, how many ounces? Ans. 27.
5. If a piece of gold weighing 341304 gr. be divided into
8 equal ingots, what will be the weight of each?
Ans. 7 lb. 4 oz. 17 pwt. 15 gr.
6. ltow many ounces in 916 pills, each weighing 5 grains?
Ans. 9 3 4 3 1 D.
7. In 14769 ounces, how many cwt. Ans. 9 cwt. 23 lb. 1 oz.
8. What is the weight of 373 bags of salt, each weighing
85 pounds? Ans. 317 cwt. 5 lb.
9. In 647 in., how many yards? lAns. 17 yd. 2 ft. 11 in.
10. In 1745 rods, how many miles? Ans. 5 mi. 3 fur. 25 rd.
11. In 4459 perches, how many acres? Ans. 27 A. 3 R. 19 P.
12. In 12959 square inches, how many square yards?
Ans. 9 sq. yd. 8 sq. ft. 143 sq. in.
13. In 1664 cubic feet, how many cords? Ans. 13.
14. Reduce 5966781 cubic inches to tuns of hewn timber.
Ans. 69 tuns 2 cubic feet 1725 cubic inches.
15. In 432 nails, how many yards? Ans. 27.
16. In 75 English ells, how many yards? Ans. 93 yd. 3 qr.
17. In 17 pieces, each containing 27 ells Flemish, how many
yards? Ans. 344 yd. 1 qr.
18. In 9072 pints of wine, how many pipes? Ans. 9.
19. In 10080 gills of wine, how many hhds.? Ans. 5.
20. In 13248 pints of beer, how many barrels? Ans. 46.
21. In 72015 hr., how many weeks? Ans. 428 wk. 4 da. 15 hr.
228 TEST EXAMPLES.
22. Reduce 900610 seconds to days.
Ans. 10 da. 10 hr. 10 min. 10 sec.
23. How many days from the Sth of May till the 22d of
July?
Solution. 31-8=23, the number of days left in May; then
23+30+22-75 days Ans.
24. From August 10th, 1844, till April 26th, 1845, how
many days? Ans. 259.
25. How many days from January 10th, to July 30th, 1847?
Ans. 201.
26. How many days from January 20th, to April 25th, 1848?
Ans. 96.
27. If the 8th of August be on Saturday, on what day of
the week will the first of November be?
EXPLANATION.-The number of days between these dates is
found to be 85, which, being divided by 7, becomes 12 weeks
1 day; counting 1 day, therefore, after Saturday, we find that
the first of November must be on Sunday.
28. If a leap year begins on Friday, on what day will the
17th of June, the anniversary of the battle of Bunker Hill,
happen? Ans. Friday.
29. If a common year begins on Wde'nesday, on what day
will the 4th of July happen? Ans. Friday.
30. In the year 1830, the 4th of July happened on Sunday;
on what day did the year commence? Ans. Friday.
NoTE. —The eight preceding questions do not properly belong to
reduction; they are inserted on account of their utility.
ADDITION.
1. A man bought 3 bu. 3 pk. of wheat at one time, 5 bu.
1 qt. at another, 2 bu. 1 pt. at another, land at another 9 bu.
6 qt. 1 pt.; how many bushels did he buy in all? Ans. 20.
2. A jeweler has 4 pieces of gold; the 1st weighs 7 oz.
11 pwe. 3 gr., the 2d 5 oz. 9 pwt., the 3d 10 oz. 15 gr., the
4th 5 oz. 19 pwt. 6 gr.; how much do all weigh?
Ans. 2 lb. 5 oz.
3. An apothecary made a composition of 3 ingredients; the
first weighed 73 23 19 10 gr., the second 1lb 23 2D, and the
third 2 lb 4. 2 3 2 D) 18 gr.; what did the whole mixture
weigh? Ans. 4 tb 8 gr.
d" - XI —-~I...-~`. ---------- --------- ~W
ADDITION OF COMPOUND NUMBERS. 229
4. A bre*er bought 5 bags of hops: the first weighed 2 cwt.
1 qr. 10 lb., the second 3 cwt. 20 lb., the third 2 cwt. 2 qr., the
fourth 3 cwt. 1 qr. 12 lb., the fifth 2 cwt. 3 qr. 11 lb.;' what
was the weight of all? Ans. 14 cwt. 1 qr. 3 lb.
5. There are 3 pieces of road; the first is 3 mi. 1 fur. 18 rd.
long, the second 2 mi. 3 fur. 9 rd., the third 5 mi. 3 fur. 20 rd.;
required the length of all. iAns. 11 mi. 7 rd.
6. Add together 5 A. 2 R. 15 P., 4 A. 3 R. 25 P., 6 A. 1 R.
19 P., 8A. 1 R., and 3 A. 15 P. Ans. 28A. 34 P.
7. Add together 15 C. 40 cu. ft., 19 C. 85 cu. ft., 20 C. 34 cu. ft.
12 C. 94 cu. fi. Ans. 67 C. 125 cu. ft.
8. Add together 4 gal. 2 qt. 1 pt. 3 gi., 3 gal. 1 qt. 1 gi.,
5 gal. 1 qt. 1 pt. 1 gi., and 3 gal. 3 qt. 3 gi. An's. 17 gal. 1 qt.
9. Add together 1 da. 5 hr. 10 min. 15 sec., 3 da. 7 hr.
40 min., 5 da. 12 hr. 35 sec., and 9 da. 9 min. 10 sec.
Ans. 19 da. 1 hr.
10. Add together 6~ 3' 12//, 40 7' 15", 8~ 4' 40//, and 1~
45' 13". Ans. 20~ 20".
SUB3 TRACTION.
1. From 12 bu. of wheat, there was taken 2 bu. 7 qt. and
1 pt.: how much remained? Ans. 9 bu. 3 pk. 1 qt.
2. From 2 lb. 5 oz. of silver, there was taken 2 lb. 10 pwt.
5 gr.; how much remained? iAns. 4 oz. 9 pwt. 19 gr.
3. From 3 lb 5 3, take 2 lb 7 2 15 gr.
Ans. 5 X 4 3 2 D 5 gr.
4. From 9 lb. 13 oz. 14 dr., take 3 lb. 13 oz. 15 dr.
Ans. 5 lb. 15 oz. 15 dr.
5. From 50 miles, take 5 miles 1 rod.
Ans. 44 mi. 7 fur. 39 rd.
6. A lot of land contains 11 A. 3 R.; how much will remain
after selling 3 A. 2 R. 20 P.? Ans. 8 A. 20 P.
7. Bought 8 cords of wood; after using 5 cords 87 cubic feet,
how much will remain? Ans. 2 C. 41 cu. ft.
8. If from 11 yd. 2 qr. of cloth, there be cut 5 yd. 3 qr. 2 na.,
how much will be left? Anls. 5 yd. 2 qr. 2 na.
9. If from 31 gal. 2 qt. of cider, you draw 17 gal. 2 qt. 1 pt.,
how much will remain? Ans. 13 gal. 3 qt. 1 pt.
10. There are two brothers, whose ages are 11 yr. 2 mon.
20 da., and 9 yr. 5 mon. and 15 da.; what is the difference
between them? Ans. 1 yr. 9 mon. 5 da.
230 TEST EXAMPLES.
11. The city of Washington is 5hr. 8 min. 6sec. of time
west of Greenwich, and Cincinnati is 5 hr. 37 rmin. 36 sec.
west of the same meridian; what is the difference of time
between Washington and Cincinnati? Ans. 29 min. 30 sec.
12. The latitude of Philadelphia is 39~ 56' 58"M, and that of
Greenwich 51~ 28' 39//; how much nearer to the equator is the
former than the latter? Ans. 110 31' 41//.
MIVULTI PLIC AT I O E.
1. In 5 bags of wheat, each containing 2 bu. 3 pk. 4 qt.,
how many bushels? Ans. 14 bu. 1 pk. 4 qt.
2. What is the weight of 9 silver spoons, each weighing
1 oz. 9 pwt. 13 gr.? Ans. 1 lb. 1 oz. 5 pwt. 21 gr.
3. What is the weight of 3 packages of drugs, each weighing
2 t 11,~ 23 2D 7 gr.? Ans. 8it- 10~ 1D I gr.
4. What is the aggregate weight of 8 bars of lead, each
weighing 6 lb. 11 oz. 2 dr.? Ans. 53 lb. 9 oz.
5. What is the weight of 5 tierces of rice, each weighing
4 cwt. 3 qr. 20 lb.? Ans. 24 cwt. 3 qr.
6. If a man walk 20 mi. 3 fur. 15 rd. in one day, how far
can he walk in 12 days? Ans. 245 mi. 20 rd.
~7. How many acres in 5 lots of ground, each containing
7 A. 2R. 8P.? Ans. 37 A. 3 R.
8. How many cords of wood, in 8 piles, each containing
2 C. 38 cu. ft. 432 cu. in.? Ans. 18 C. 50 cu. ft.
9. Ilow many yards of cloth, in 7 pieces, each measuring
5 yd. 3 qr. 2 na.? Ans. 41 yd. 2 na.
10. If a man drink 3 gal. 2 qt. 3 gi. of wine in a month,
how much would he drink in 12 months? Ans. 43 gal. 1 pt.
11. htow much beer in 11 casks, each containing 6 gal. 3 qt.
I pt. Ans. 2 bl. 3 gal. 2 qt. 1 pt.
12. If it require a man 1 hr. 15 min. 25 sec. to walk 1 league
how long to walk 12 leagues? Ans. 15 hr. 5 rain.
13. If a ship sail through 2~ 10'15// of latitude in one day,
how far will she sail in 8 days? Ans. 17~ 22/.
14. The moon moves 13~ 10' 35/' in a solar day, and the sun
59' 8//; now, supposing the moon at any given time, to be 10~
in advance of the sun, how far will they be separated at the
end of 5 days from that period? Ans. 70~ 57/ 15//
GREATEST COMMON DIVISOR. 231
DIVISION.
1. Divide 12 bu. 3 pk. 4 ct. by 8. Ans. 1 bu. 2 pk. 3 ct. 1 pt.
2. Divide 31lb. 8 oz. of gold into 6 equal ingots; what is
the weight of each? Ans. 5 lb. 3 oz. 6 pwt. 16 gr.
3. Eight equal packages of medicine weigh 16lb 7 3 2 3,
what is the weight of each? Anzs. 2 Db 7 3 15 gr.
4. Divide 92 lb. 10 oz. 4 dr. by 7. Ans. 13 lb. 3 oz. 12 dr.
5. Divide 53 cwt. 10 lb. by 9. Ans. 5 cwt. 3 qr. 15 lb.
6. Divide 40 A. 3 R. 39 P. by 7. Ans. 5 A. 3 R. 17 P.
7. Divide 97 cu. ft. 1032 cu. in. by S.
Ans. 12 cu. ft. 345 cu. in.
8. Cut a piece of cloth, containing 41 yd. 1 qr., into 12 equal
parts, what the length of each? Ans. 3 yd. 1 qr. 3 na.
9. Divide 21 gal. 3 qt. by 6. Ans. 3 gal. 2 qt. 1 pt.
10. Divide 14 da. 5 hr. 6 min. by 12.
_Ans, 1 da. 4 hr. 25 min. 30 sec.
11. Divide 810 30' by 8. Ans. 100 11' 15"
GREATEST COMlMION DIVISOR.
MENTAL EXERCISES.
1. Name the divisors of 6; of 8; of 10; of 12; of 14; of 16;
of 18; of 20; of 22; of 24, etc.
2. What is the greatest common divisor of
2 and 4? 6 and 9? 10 and 15? 15 and IS?
2 and 6? 6 and 10? 12 and 15? 15 and 21?
3 and 4? 8 and 10? 7 and 14? 15 and 25?
3 and 6? 8 and 12? 14 and 21? 18 and 24?
4 and 6? 9 and 12? 21 and 28? 20 and 35?
FOR THE SLATE OR BLACKBOARD.
i. 45, 12.. Ans. 3. 9. 583, 1961. Anls. 53.
2. 120, 70.. Ans. 10. 10. 3285, 9125. Ans. 365.
3. 752, 156.. Ans. 4. 11. 1001, 5291. Ans. 143.
4. 280, 142.. Ans. 2. 12. 693, 763. Ans. 7.
5. 4590, 4410.. Ans. 90. 13. 799, 1071. Ans. 17.
6. 377, 551.. Ans. 29. 14. 966, 1495. Ans. 23.
7. 909,1717.. Ans. 101. 15. 3763, 4757. Ails. 71.
8. 481, 703.. Ans. 37. 16. 33522,41223. Ans. 453.
232 TEST EXAMPLES.
LEAST COMIWMON MULTIPLE.
IENTAL EXERCISES.
1. Name a multiple of 3; of 5; of 6; of 7; of 8;
of 9, etc.
2. Name a comulon multiple of 2 and 3; of 3 and 4; of 4
and6; of 5and7; of 4and8; of 6and9; of 8and12.
3. What is the least common multiple of 2 and 4? of 3
and 4? of 6 and 8? of 6 and 9? of 6 and 10? of 8 and
12? of 9 and 12? of 10 and 12? of 8 and 10? of 10
and 15?
FOR THE SLATE OR BLACKBOARD.
1. 4, 14, 20............ ns. 140.
2. 3, 4, 5, 6, 7, 8....... ns. 840.
3. 2, 3, 4, 5, 6, 7, 10, 14...... Ans. 420.
4. 3, 6, 9, 12, 15, 18..... Ans. 180.
5. 2, 4, 8, 16, 32, 64, 128, 256.... As. 256.
6. 2, 3, 4, 5, 6, 10, 15, 20..... A. Ins. 60.
7. 2, 3, 4, 7, 8, 16, 20, 25, 9... Ans. 25200.
8. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Ans. 27720.
COMMON F 1lAC T IONS.
I, REDUCING FRACTIONS TO THEIR LOWEST TERMS.
1. 75.... 25. 7. 3109. 7
2. 208_ 5* ~ 5 4 ~ 33
9 _ 192 I o 5 S170 235
3 7 6-e o o 3. 8 4 3 4 =. c o 97 q7
4. s2 55 10. 5l28 6=5 547. 18_ 3 -o -4' 1 1T4
4 z*'. - *- 1 1 4 e 10 3'
II. IMIPROPER FRACTIONS TO FWHOLE OR MIXED NUMBERS.
1 2. IS 1. 6.'1'... 711
3. 45=-. 0. 8-. 23 -
27"g..... 23
4. T2.. 7.5... 37-4.
5. -48 M 8 F. T 166 2. 10. 47...2734R.....~~~~~~~ T ~ ~ ~.~
COMMON FRACTIONS. 233
ITI. COMPOUND FRACTIONS TO SIMPLE ONES.
1. of of =.. ~ ~~ *... 4I
2. 5 of 4 of -- 55
2. 9..... 3..
3. 9% of - of t of........ -
4. 4 of of of of=....
7. 4 of of of of......
6. j of,j of -- 11..4...
7. 4 of of of 4 of ~ of of.. 4.
8..-f of 4 of 4 of -of - =.... 3-4.
4 4
IV. FRACTIONS OF DIFFERENT DENOMINATORS, TO EQUIYALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR.
1. 1 1 1 1 A1s. 35 7q4 110 70
2' -8- 1' T T.. ^ An. TOO' 4 D, ol) T7O42. 5 7 1o 1 3. Ans. 30 2.8 33 2
-1':2' TO'4.. 42 427 4'1 48
3., ~ -~ —~4 1. Ans 84'105 Sn 86
3. -2 of 9 o * o Ans.........3.
6 9of7 -. T2 12f f 42_' T
4, o 7 -... S.. 10 200 140
2 5 7 38 60 5 0 6a
CN O' DI F FR ET DNOHATOR S TO' ] OIA6, 7 ~ 1 7 Ans. 40 42 33 34
*' I1 T 4- 42 *4 *', 42', 48
37 8 13 36 42 32.27
7 T -., Ans.
S I 019 1+ 3 1 6 900 6.4 3 30 0 6
V. ADDITION.
a 7- +2 T 4 +q = -o e 3 -;l 3o
3, T-+,'~ 4... 4-, 4,..'
3. +4 +4 +5 7.1,!5 4+ +.n.2
6. 4 +_I 4+ +_4 _..4...... 3.. 4
Q5.; 3 + 7 1 4
8, 4+4,++ 7-4.....
9. i.... -,........2O
3 51 I 1 1_ 3 91 4 a 4,
4' ~%,, I+......... 3.
-+7+1'+21-. ~ iUU - 9 -6, zJ~U,39
4 ~- --...4.
I3...... —--------------
234 TEST EXAMPLES.
10. 1 -+13+17+ l+2-.. S
11. 3+ I + I + +1 1 +=.....
13 3a6 +Tl -1- 7 +~-1-.-+ 3=.....,3-9-,~7.
14. ~+ + + ++ - + -6 -- *+
VI. SUBTRACTION.
4 3 T = — 8. 1~ — 5=..
1 11 of 9 1 6
4 1 7B *
91 1 __ *991i G
1.., 2 14-l-o-. 5.2 -., *4
VII. MULTIPLICATION. (See Article 151.)
1 _ 1 X9 5 1 1 * 7=s * 22
2. 21-1.... 1 2 of. 2=. 6.
5. T7- * 92. 10. 15X12= 7 2 o 1 29
(See Article 152.)
1... 2 1X 9. 5. lX-5 1. 140.
4. 39X * e 14q.. 43X5=. *.. 253.
1 _ 4
2. -- X 12 7-W= 7..2 68.
1 3 1 1 3
2' 1' 5 20 515.
6. o X16-. y.. 134X1 -- 190
2L 48X.... 42. 6. 25x97..4..
~g a...... f
4. 39X16:...6. 9. 134X14....190.
COMMON FRACTIONS. 235
(See Article 153.)
1... 6. 5AX 5' =. 30o.
2.... ). 7. 33X 44... 18.
3. 1. x4.= 1 3. 8. 2 X 14-=.. 3 7
4. 4>X 3=,...3. 9. 12'X184=.. 234g.
8 4 3 ~ 4
5. 5 X -=X.... 3. 10. 52X 63. 3812
IL X'7X 4XI143...X5 =....... 4->.
12. 3-X21.X1 X3 54.
13. 3-X4VX 1 X25=.69.
14. 21X24X9X45X3X=....... 242.
VIII. DIVISION. (See Article 154.)
i. 25 ~ ~... T.
2. -- 10. 6. 3 — 5 -=..
3 4
3. -7... j. 9 7 53.7 *
~ —J7 - - 7. 5ai ~7
4. 5.-4-22=. P' 5 68 5..
(See Article 155.)
1. 36__..2... 54. 5. 6 2B.... 2.
2. 1 —= 11. 6. 1. 3i=..
3, 12+i.. 18. 7. 4 1 1.. 3.
4. 1012=. 8. 9 3.... 2.
(See Article 156.)
1. 9. s 3 9. 2. 2.. o.
_. 9_ 4 4
2. 3. 1= 9.. 14. 1 74 — 3=. *
5. 4 * 1 *
3. Us * = 11. 12.4 —. 2 24... - 1*. 12. 15-3 —. 4 *.
5. ~ 1.4 4- 4
-- 7. - 13. 12-22 -. 5.
4' — = 31. = 5 5.. ~,~. 1-~. 4~'~
236 TEST EXAMPLES.
REDUCTION OF COMPLEX TO SIMPLE FRACTIONS.
(See Article 158a.)
36- 133 28'
31.3 =A. 5 of -- of' of 2- 43
3. -= 5. o f.o. 123.48 144 38]
38
47 2. 4 of 2 32 0 1
9. l of 0 of 2 of.. 273
44(See Article 14 60.)
911. 72-.
4. 44to te faction of an oz. AnS 5=
5. educe of an b. to the fraction of a pwt. Ans.
6. 1 of a. to the fraction of an inch. Ans.
7. g of an acre to the fraction of a perch. Ans. 83, T f acwt.to sthe fraction of an oz. o~. o.
1. Rdc1, of an. to the fraction of a D. Ans. -.
2. 1 of a bu. to the fraction of an inch. Ans. sT.
7. ~i of an acre to the fraction of a perch. Ans.:~.33
8. o of an hour to the fraction of a second. Ans..
NOTE.-The same examples may be applied conversely, in
reducing fractions of a lower, to equivalent fractions of a higher
denomination.
8~ -- — ~~~~~~~~~~~~~~~~~~~~~~~~~~l
COMMON FRACTIONS. 237
II. FINDING THE VALUE OF A FRACTION IN INTEGERS OF A
LOWER DENOMINATION.
(See Article 160.)
1f. What is the value of 44 of a bushel? Ans. 1 pk. 7 qt.
2. Of 4 of a pound troy? Ans. 7 oz. 4 pwt.
3. Of 2ss of a mile? Ans. 6 fur. 105 yd.
4. Of of anacre? Ans. IA. 2R. 3~P.
5. Of ] of an acre? Ans. 3 R. 32 P.
6. Of ~o of a day? Ans. 21 hr. 36 min.
7. Of 4 of a pound apothecaries? Ans. 5 9 1 3 84 gr.
8. Of 3 of a hogshead? Ans. 37 gal. 3 qt. 1 gi.
II. IN REDUCING A QUANTITY, COMPOSED OF ONE OR MORE
DENOMINATIONS, TO THE FRACTION OR PART OF ANOTHER
QUANTITY, COMPOSED OF ONE OR MORE DENOMINATIONS.
(See Article 163.)
1. What part of 35 bushels is 15 bushels? Ans. 4.
2. What part of 3 bushels is 6 quarts? Ans. y1c.
3. What part of 4 bu. 3 pk. is 3 pk. 4 qt.? Ans. -.77
4. What part of 1 yard is 2 feet 8 inches? Ans. A.
5. What part of 3 A. 2 R. 20 P., is 1 A. 1 R. 20 P.?
Ans. s.1
6. What part of 3 pounds 2 ounces 15 pennyweights, is 1
pound 9 ounces 5 pennyweights? 4ns. 31.
7. What part of 5 pounds 3 ounces 3 drams, is 4 pounds 4
ounces I dram? Ans. -1~.
S. What part of 3~ l yards is 24 feet? Ans. 9.
9. What part of 14 hours is 154 minutes? Ans. -W.
10. What part of 1 hr. 63 min. is 4 min. 32-T sec.? Ans. 44
11. What part of 12 cents is ~ of 2 cents? Ans.'.
12. WVhat part of $1.69 is 91 cents? Ans. 73.
IV. ADDITION AND SUBTRACTION.
1.; Add together ~ oz., U oz., and lbs. Troy.:
Ans. 7 oz. 19 pwt. 20 gr.
238 TEST EXAMPLES.
2. Add together - of 3 A. 1 R. 20 P., - of I A., and - of
3 R. 15 P. Ans. 3 A. 2 R. 334 P.
3. From 2 of 5 pounds Troy, subtract g of 31 ounces.
Ans 3 lb. 2 oz, I pwt. 2- gr.
4. From 15T miles, take 7444 furlongs. Ans. 4 fur. 9 yd.
MIISCELLANEOUS EXERCISES.
1. 5723 — 5 — = what number? Ans..7
2. What fraction is that, from which if you subtract sq, the
remainder will be? Ans. 3
3. What fraction is that, to which if you add -31, the sum
will be s? Ans. m301
4. What number is that, which, being divided by 3 -, the
quotient will be 5? Ans. 14
5. What number is that, which, being multiplied by 33, thp
product will be 1? Ans. 1-4
6. Divide T; of 4 by oft. Ans.
7. What number, multiplied by ~, will produce 11 9?
Ans. 264j
S. What number, added to 11-7, will produce 17-r?
Ans. 6':
9. A expends $4 in cloth, at $7 per yard, and B, $5 at $g
per yard. What quantity does each purchase, and how mucI4
does one purchase more than the other?
Ans. A 4 yd., B yd., and B 3 yd. the most.
10. A merchant bought 35 yards of cloth for $25, and sold
it for $30; what was the purchase price, and the selling price
per yard? Ans. $q and $4.
11. If 5 of a yard of cloth cost $2, what will be the entire
cost of two pieces, each containing 5- yards, and a third one
containing 5&- yards? Ans. $574.
12. Sold to A 25 barrels of apples for 564 dollars; to B 12-.
barrels for 40 dollars; and to C 10 barrels for 260 dollars;
what did the whole amount to, and what was the average price
per barrel? Ans. $1231 and $24 8a.
13. Sold to A 54 yards of cloth at $3-1 per yard, and to B
34 yards at $54 per yard; what did the whole amount to, and
COMMON FRACTIONS. 239
what was the average price per yard? Ans. $36.~ and $458
14. At 183 dollars per tun, what will be the cost of 4 of a
tun of iron? Ans. $11-.l
15. How far will a man travel in 5~ days, at the rate of 22
miles per day for 21 days, and the remainder of the time at
the rate of 18 miles per day? Ans. 111 miles.
16. How far will a man travel in 54 days, allowing him to
travel 1 of the time at the rate of 24 miles per day, 4 of the
time at the rate of 30 miles per day, and the remainder of the
time at the rate of 36 miles per day? Ans. 1673 miles.
17. Bought 40 yd. of cloth at $32 per yard. Sold 204 yd.
at 3; per yd.; 93 yd. at $34 per yd.; and the remainder at
$4 per yd. What gain by the sale? Ans. $6 9.
18. Divide 1-9A by v-3 of 10 Ans. 4-.
19. What part of 20 is? Ans..1
20. If 34 yards of cloth cost $144, what is the price per
yard? Ans. $4.07- 5
21. Divide 4 by g of 4. Ans. 54.
22. Find the difference between 54 and of 4 1. Ans. 4- 3'I
23. Divide 4 of 44 by of 2 Ans. 1..
24. Divide 14 by 2 —, and to the quotient-add the quotient
of 5, divided by 34. Ans. 2-0.
25. A man having 4 of a boat, sells ~ of his share for
$96384; at that rate, what is the whole vessel worth?
Ans.. $20080.
26. What part of 31 is 4 of 4? Ans. 4-.
27. Two persons bought a piece of cloth, for which the first
paid 24, and the second 2-1 dollars; what part of it should
each have? Ans. and 2 s
28. Two men bought a lot of wood for $28.35, of which one
got 54 cords, and the other the remainder, which was 8 cords;
what ought each to pay? Ans. $11.55 and $16.80
29. Of an estate valued at $6000, the widow receives 1;,
each of two sons 4 of the remainder; and the residue is divided equally between three daughters; what is the share of
each daughter? Ans. $11424.
240 TEST EXAMPLES.
30. A, B, and C, bought a haystack, weighing 103 tuns, for
$45'. A got 3. tuns, B 31, and C the remainder; what ought
each to pay? Ans. A $145; B $14-1.4; C $164 9
DECI1MAL FRACTIONS.
I. ADDITION.
1. 3.2746+0.38 +2.763 +1.89743 +2.7=11.01503
2. 6.274 +2.34689+-3.004786+2.3478 +3.7+ 2.648329
-20.321805
3. 2.4789+3.26 +4.37897 +2.178643+0.004+1.3478
=13.648313
4. 0.7+0.37+0.2370.1237+0.91237+0.491237
=2.834307
5. 123k4 + 1 #45234 o 5 1240o5+2
+12345 1 —=13717.319616
6. 14 T -+3T'1 o+2 - T +-3 a — +3 ]j-+72TgF % 2 a%$
+T oUo-24.7573
7. 3.7468+2.0374+3.04789-+2.6+324+1.73+2.672898
+3.478-11.00746+2.897304=347.217752
II. SUBTRACTION.. 8.342-2.518........... 5.824
2. 14.273-8.786...... 5.487
3. 9 2......... 6.3778
4. 3 1..... 1.7378
5. 15.27468-10.4879 —.. 4.78678
6. 4.67432-1.894746=.... 2.779574
7. 3.04-1.68=............ 1.36
8. 2.4063-1.8794.......... 0.5269
9. 3T2 o —1 T......... 1.564
10. o ~28-8 23s=.......... 19.77
11. 16.4-8.326=........... 8.074
12. 4.31 -2.3872-..... 1.9228
13. 17-3.472.......... 13.528
^9 __4J —-— ~ —— ~ —~ — ~ —~
DECIMAL FRACTIONS. 241
14. 21.3-2.4796 -....... 18.8204
15. 3-2.2641= 4........ 0.7359
16. 26.37-8.'17341-......... 18.19659
17. 3.4-1.763214-......... 1.636786
18. 3000 —0.003-...... 2999.997
III. MULTIPLICATION.
SoUGESTroNs.-By changing te7e place of the decimal point,
each operation, with the same figures, may be made, both in
Multiplication and Division, %,o give Several examples. Thus,
the product of 32.7 by 12.5 is 408.75; hence, 3.27X1.25=4.0875,
and.327X.125=.040875, and so on.
In like manner, 552.5. 1'7= 32.5; hence,
55.25~ ^ 17= 3.25; 5.525 — 17 =.325;.5525 — 17-.0325; 552.5 — 1.7 325;
552.5.17-= 3250; 552.5 —..017= 32500,
and so on.
1. 8X324-2592. 8X3.24-25.92
2. 23X57.98=..... 1333.54
3. 36X5.769=-..... 207.684
4. 543X7.8906... 4284.5958
5. 12 X,5789...... 6.9468
6. 11OX.23456....... 25.8016
7. 48 X9876 —.. 474048. 4.8X9876. 47404.8
8. 2.34 X123456=..... 288887.04
9. 3.456X28767 =.. e. 99418.752
10. 0.325X50807 =........ 16512.275
11. 3.9875X9804=........ 39093.45
12. 0.0025X375=.. 9.6875
13. 11X3457=38027. 11X34.57=380.27 1.1X34.57
=38.027
14. 5.43X7.8906=........42.845958
15. 1.804X45.607= e.... ~ 82.275028
16. 0.4X57.98....... 23.192
17. 1.7X0.579=......... 0.9843
18. 0.31 X0.579-......... 0.17949
242 TEST EXATIPLE:S.
19. 0.07X0.08....... 0.0056
20. 3s,ts -
20. (-..... 0.0000432
21. - 3 X 2. 0.000000416
22..45X0.3456 —... 0.15552
23. 0.92X0.9>0.5.5-...... 0.414
24. 0.15X0.16XO.05..... 0.0012
25. 1.23X34.5XO.567XO.0789- 1.8983848905
26. 0.0025X0.015XO.16..... 0.000006
IV. DIVISTON.
1. 404.341-47........603
2. 5.76 —- 12-..... 0.48
3. 5.44- -17........ 0.32
4. 0.474048- -48-.. o.. 0.009876
5. 46.5- 6......... 7.75
6. 27.3-:8........ 3.4125
7. 56.7 —:4......... 2.3625
8. 73.091- 16...... 4,619375
9. 1120 —14=.... 80. 112- 1.4. 80.
10. 192.1.6.....120.
11. 924:' 1.32 o.....700.
12. 2769 0.213=..... 13000.
13. 43... o — 13.4375
14. 19 0.128=......... 1.48.4375
15. 32 —1.5 —.......... 21.333-+
16. 42 —0.014 o...... 3000.
17. 53- 0.00008 —...... 662500.
18. 9 —0.562 5........ 16.
19. 7- -0.21875...... 32.
20. 11 0.04296875-..... 256.
21. 4,8 1.2=........ 4.
22. 32.2_1.4....... 23.
23. 5.76_0.4 8....... 12.
24. 449.503_ 6.709=..... 67.
25. 0.343 — 0.049 o. o.. 7.
26. 3.24-: 1.44......... 2.25
DECIMAL FRACTIONS. 243
27. 12.18. 2.9=... 4.2
28. 44.928 —1.3-..34.56
29. 4.94795 — 2.11=......... 2.345
30. 3.8788620.669 —...... 5.798
31. 0.208728. 0.5798-........ 0.36
32. 1.236-.0.48=.......... 2.575
33. 51.2 —1.28 —.......... 40.
34. 85.12 1.216........... 70.
35. 2.4 —.0048. —..... 500.
36. 3386.88 —0.3456-..... 9800.
37. 44.8 2.56. —. 17.5
38. 3.27-.0096.-...... 340.625
V. REDUCTION OF COMMON FRACTIONS TO DECIMALS.
1. =. 0.]36
2. U7...... 0.00875
3............ 0.006875
4... 0.01171875
5.........0.0135546875
6. 1 UDa -a -a. 0.0001
7......0.222464
8.:if~ evv —....... 0.00000048828125
9. 3. 0.085714285+, (Period 857142.)
10. }-. 0.88235+, (Period 8823529411764705.)
11...0.000566+
12...... 0.0000001238+
VI. REDUCTION OF DECI-MALS TO COMMON FRACTIONS.
1. 0.2976_....' 2 a *
2, 0.06125=........... g49.
3 0.19921875=.... -
4 0.0406640625-.......... 04
VII. REDUCING DECIMALS OF ONE DENOMIINATION TO THEIR
EQUIVALENTS IN ANOTHER DENOBINATION.
1. Reduce.03375 of a yd. to the fraction of a qr. Ans..135
2..0225 of a lb. T. to the fraction of an oz. Ans..27
244 TEST EXAMPLES.
3. Reduce.00390625 of a bu. to the fraction of a pt.
Ans..25
4..0025 of a minute to the fraction of a second. Ans..15
5..0006875 of a day to the fraction of a minute. Ans.99
6..7 of a quart to the fraction of a bushel. Ans..021875
7..21 of an in. to the fraction of a yd. Ans..005833+
8..9 of a dram to the fraction of an ounce. Ans. 0.05625
9..3 of a pint to the fraction of a gallon. Ans. 0.0375
10..84 of a 3 to the fraction of a lb..Ans. 0.00875
VIII. FINDING THE VALUE OF A DECIMAL IN INTEGERS OF
A LOWER DENOMINATION.
1. What is the value of 0.865 of a bushel?
Ans. 3 pk. 3 qt. 1.36 pt.
2. Of.325 of a yard? Ans. 1 qr. 1.2 na.
3. Of.00625 of a day? Ans. 9 min.
4. Of 0.78125 of an acre? Ans. 3 R. 5 P.
5. Of.54296875 of a pound?.Ans. 8 oz. 11 dr.
6. Of.34619140625 of a cwt.?
Ans. 1 qr. 9 lb. 9 oz. 14.5 dr.
7. Reduce 12 gr. to the decimal of a lb. T. A2ns. 0.0020583+
8. 15 perches to the decimal of an acre. Ans. 0.09375
9. 3 R. 21.52 P. to the decimal of an A. Ans..8845
10. 3 min. 4.032 sec. to the decimal of a day. Ans. 0.00213
11. 1 foot 1.5 in. to the decimal of a yd. Ans..375
12. 1 pk. 6 qt. to the decimal of a bu. AAns. 0.4375
DiIISCELLANEOUS EXERCISES.
1. What part of a year is a day, reckoning the year at 365
days 6 hours? Ans. 0.0027378507+
2. VWhat part of a year is a day, reckoning the year, as is
right, at 365 days 5 hours 48 minutes 48 seconds, or
365.24222+ days? Ans. 0.002737909+
3. Reduce the following fractions to decimals, and find their
sumn: )-4-, 5 T,' Gus T 63, 630, and I. Ans. 6.0025
4. How many times is 0.0002 contained in 0.01? Ans. 50.
5. What is the product of 0.0107 by 0.00206?.Ans. 0.000022042
6. What will 6 pounds 13.504:ounces avoirdupois cost at
$.3125 per pound? Ans..$2.13875
7. What is the difference between.11 of a pound and.11
of an ounce Troy? Ans. 1 oz. 4 pwt. 4.8 gr.
_b, _> ~_~ -- _0
v ~ --
PROPORTION. 245
8. How many times is 0.0015 of an ounce Troy, contained
in 47 pounds?.Ans. 376000.
9. How many times is 0.03125 of a nail contained in 25
yards? Ans. 12800.
PRO P 0 R T ION.
SIMPLE PROPORTION.
MENTAL EXERCISES.
1. If the product of the means is 24, and one of the extremes is 3, what is the other? If one of the extremes is 4,
what is the other? If one of the extremes is 12?
2. What is the fourth term of the proportion 3 2:::12.
3. What is the fourth term of the proportion 6 9:: 8.
4. What is the fourth term of the proportion 10: 2:: 20.
5. Complete the proportion 7: 4:: -: 8.
6. Complete the proportion 3:-:: 6: 16.
7. Complete the proportion-: 7:: 8: 28.
FOn THE SLATE OR BLACKBOARD.
1. If 6 lb. of tea cost $5, what will 8 lb. cost? Ans. $6].
2. If 5 men can do a certain piece of work in 12 days, in
what time ought 8 men to do it? Ans. 7- da.
3. If a man travels 55 miles in 2 days, how many miles
ought he to travel in 5 days? Anls. 137, miles.
4. If 5 yards of silk cost $4.40, what sum, at the same
rate, ought to be paid for 11 yd. 3 qr. 2 na.? Ans. $10.45
5. If a certain quantity of provisions last 30 men 25 days,
how long ought it to last 20 men? Abns. 37- days.
6. If 12 oxen require 20 A. 2 IR. 30 P. of pasture Sfor a
summer, what quantity ought 30 oxen to have for the same
length of time? Ans. 51 A. 2 R. 35 P.
7. A cistern is filled by 3 pipes, of a certain size, in 32 min.
5 sec.; in what time would it be filled by 7 pipes of the same
size? Ans. 13 min. 45 sec.
8. If a barrel of flour last 7 persons 8 weeks, how long will
it last 15 persons? Ans. 31-l- weeks.,.PQ — - --- - - -- -- --— ~
246 TEST EXAMPLES.
9. If a note at 5 % yield $185 interest, at what % would it
yield $259? Ans. 7.
10. If 3- of flour cost 252 cents, what will 143 lb. cost?
Ans. $1.08'
11. If 6 2 yards of cloth cost $134, what part of a yard will.$. buy? Ans. 3 5
12. If when the days are 121 hours long, a man can perform a journey in 84 days, in how many days can he perform
it when the days ale 74 hours long? Ans. 134 days.
13. Find a number which shall have the same ratio to 7
that 27 has to 3; also, a number to which 39 has the same
ratio that 31 has to 25. Ans. 63, and 31O.
14. Complete the proportion of which the first, second, and
fourth termus are, 35, and 3-; also, that whose first, third,
anfd fourth terms are.35, 1.25, and.0145
Ans. Third term is.a: second term,.0000406
15. How much in breadth that is 15 rods in length, will be
equivalent to an acre of land, which is 40 rods in length and
4 rods in breadth? Ans. 10 rd. 3 yd. 2 ft.
COMPOUND PROPORTION.
1. If 4 men in 7 days, earn $17, how much will 14 men
earn in 10 days? Ans. $85.
2. If 4 men, in 6 days, can build 23 feet of wall, in how
many days can 12 men build 460 feet? Ans. 40 days.
3. If 400 men consume 5 bl. of flour in 12 days, how many
men will consume 15 bl. in 20 days? Ans. 720 men.
4. If 12 men can mow 105 acres of grass in 10 days, howv
many men can mow 7 acres in 4 days? Ans. 2 men.
5. If 3 men, working 10 hr. a da., can reap a field measuring 150 yd. by 240 yd., in 5 da., how many men, working 12
hr. a da., can reap a field 192 yd., by 300 yd. in 4 da.?
Ans. 5 men.
6. If 7 men can build a wall 245 yd. long, 8 ft. high, and
18 in. thick, in 35 days of 12 hours each, what length of wall,
10 ft. high and 27 in. thick, could 12 men build in 43 days of
10 hours each? Ans. 229 yd. 1 ft., ~ —----------------
ALIQUOTS 01 PRACTICE. 247
ALIQUOTS OR PRACTICE.
MENTAL EXERCISES.
1. At 25 cents a yard, what will 8 yards of cloth cost 10
yd.? 12 yd.? 15 yd.? 20 yd.? 25 yd.?
2. At 50 cents a yard, what will 8 yards of cloth cost? 10
yd.? 11 yd.? 15 yd.? 18 yd.? 21 d.?
3. At 20 cents a bushel, what will 5 bushels of oats cost?
10 bu.? 12 bu.? 18 bu.? 33 bu.? 64 bu.?
4. At 12. cents a yard, what will 8 yards of cnusli;n cost?
10yd.? 12 yd.? 15 yd.? 18 yd.? 20 yd.? 24 yd.?
5. At 16-2 cents a yard, what will 6 yards of calico cost?
9yd.? 12 d.? 1 yd.? 2yd.? 20 y?
6. At 12 cents a yard, what will,- of a yard of muslin cost?
21 yd.? y-,d.? 44 yd.?
7. At 20 cents a bushel, what will 3 pecks of potatoes cost?
2 bushels 1 peck? 2 bushels 2 pecks?
F'OR THE SLATE OR BLACKBOARD.
1. At $2.31~ per yd., what will 27 yd. cost? iAns. $62.433
2. At $3.93- per yd., what will 64 yd. cost? BAs. $252.
3. At $3.50 per yd., what will 5 yd. 2 qr. cost.? Ans. $19.25
4. At $1.28 per yd., what will 7 yd. 1 qr. 3 na. cost?
Ans. $9.52
5. At $2.24 per bu., what will 9 bu. 3 pk. 4 qt. cost?
Ans. $22.12
6. At $3.84 per bu., what will 11 bu. 3 pk. 7 qt. 1 pt. cost?
Ans. $46.02
7. What will 11 lb. 7 oz. 12 pwt. of silver ore cost, at $122
per pound? Ans. $139.60
8. What will 3 pounds 11 ounces 19 pennyweights 23 grains
of metal cost, at $5.76 per pound? Ans. $'23.039
9. What will 5 weeks 5 days' work amount to, at $5 per
week, of 6 days each? Ans. $29.16
10. What will 5 pounds 11 ounces of butter cost, at 12 cents
per pound? Ans. 68{ cents.
_
248 TEST EXAMPLES.
11. What will 5 A. 3 R. 12 P. of land cost, at $11.20 per
acre? Ans. $65.24
12. What will 13 pounds 13 ounces 13 drams of spice cost, at
$2.56 per pound? Ans. $35.49
13. At the rate of 216 mi. in 24 hr. how far does a ship sail
in 5 hr. 24 min. 22 sec.? Ans. 48.655 miles.
14. July 1st, 1845, the sun's longitude changed 57' 12".8
in passing from meridian of Greenwich to that meridian
again; supposing the meridian of Cincinnati to be 5 hours
37 minutes 22 seconds of time west of Greenwich, how much
did the longitude change while the sun passed from meridian
of Greenwich to that of Cincinnati? Ans. 13' 24".244+
PE3RCENTAGE.
1. A merchant, with a capital of $2150, increased it 10 %o
the first yr., and the second yr. lost 5 %O of the amount had at
the close of the first yr. What sum had he at the close of the
second yr.? Ans. $2246.75
2. A merchant, with a capital of $2235, gained 20 So the
first year, and the second year lost 16 — S% of the capital at the
close of the first. What sum had he at the close of the second
year? Ans. $2235.
3. If a man gains 25 %o of his investment, and then loses
20 % of both investment and gain, what will be the gain or
loss on the first investment? Ants. Nothing.
4. A and B have each $5000; A increases his 12- %, and
B diminishes his 8, %'b. How much more, then, has A than B?
Ans. $1050.
5. An agent receives $5.65 for collecting $226; what % is
this? Arls. 21,.
6. An agent, after deducting his commission of $15.47,
returns to the owner $348.53: what % did he receive for collecting? Ans 4,'.
7. What %o of any sum is } of it? Ans. 12.
S. A vessel containing 39 gallons, lost by leakage 3 gal.
1 qt.; what:O is this? Arns. 8.
PERCENTAGE. 249
9. What is the commission on $150 worth of groceries at
21 %, and $250 worth, at 11 0o? Ans. $6.50
10. An agent receives $231.84 to purchase goods, at 3. %o on
the amount of his purchases; how much must he expend?
Ans. $224.
11. A factor receives $554.85 to purchase dry goods, at 2'- So
on the amount expended; what will be his commission?
Ans. $14.85
12. What is the annual premium for insuring a brick house
worth $3840, at A %? Ans. $33.60
13. A house worth $5280, is insured on " of its value, at a';i
what is the premium? Ans. $21.12
14. A man pays $15.50 for insurance on his house, at.;
at what sum is the house valued? Ans. $3100.
15. A man pays $29.40 for insurance, on a house valued at
$1680; what is the rate %? Ans. 1.
16. At 1~ o, what sum must be insured on $1580, to include
the premium in case of loss? Ans. $1600.
17. At 2 -%, what sum must be insured on $457.08, to
include the premium in case of loss? Ans. $468.80
18. Bought 25 shares of stock, (par value, $50 a share), at
3 So advance; what did it cost me? Ants. $1287.50
19. The owner of the stock, (Ex. 18), sold it at 2 % discount;
what did he receive for it? Awls. $1225.
INTEREST.
1. What is the interest of $240, for 3 years 9 months, at 1 go
per annum? Arns. $9.
2. What is the interest of $120, for 2 years 6 months, at
21 % per annum? Anrs. $7.50
3. What is the interest of $240, for 6 years 4 months, at
10- % per annum? Ans. $163.40
4. W hat is the interest of $432, for 2 years 7 months 21
days, at 7' %o per annum Ans. $85.59
5. VWhat is the amount of $234.45, for 1 year 4 months
10 days, at 8j3 % per annum? Ans. $262.372+
6. What is the amount of $841.50, for 3 years 7 mouths
20 days, at 9' S per annum? An7s. $1132.401+
2 _... _....,.. hi —---- --
250 TEST EXAMPLES.
7. What is the interest of $319.89, for 2 years 9 months
10 days, at 36 % per annum? Ans. $319.89
8. What is the interest of $243.75, for 3 years 5 months
10 days, at 11 %o per annum? Ans. $96.552+
9. What is the amount of $450.25, for 5 years 2 months
20 days, at 15 % per annum? Ans. $802.945+
10. What is the interest of $625, for 4 years 9 months 18
days, at 6 % per annum? Ans. $180.
11. What is the interest of $781.25, for 4 years 9 months
18 days, at 7 lo per annum? Ans. $262.50
12. What is the amount of $720, for 12 years 11 months
29 days, at 7 Yo per annum? sAns.,1375.06
13. At 6 o, what is the interest of $250, from Jan. 10th,
1840, to Ilarch 10th, 1841? Ans. $17.50
14. At 7 %, what is the interest of $60, from Anay 10th,
1841, to Jan. 25th, 1842? Ans. $2.975
15. At 5 %7, what is the interest of $241, from Feb. 12th,
1840, to April 24th, 1842? Ans. 826.51
16. At 4' 0o, what is the interest of $124.60, friom Nov. 25th,
1843, to Jan. 5th, 1845? Ans. $6.23
17. At 10 %, what is the interest of $344, from June 11th,
1845, to May 1st, 1846? Ans. $30.577+
18. At 6 %, what is the amount of a note for $367.40, dated
Nov. 8th, 1829, paid March 4th, 1834? Anzs. $462.679+19. At 6 %, what the amount of a note for $108, dated Feb.
12th, 1836, paid 3March 1st, 1838? Ans. $121.302
20. At 8 ~o, what the amount of a note for $325, dated Dec.
12th, 1843, paid Jan. 7th, 1846? Ans. $378.805+
PROBLE 1S IN INTERlEST.
1. A't a5 interest, in what time will $600 gain $125?
Ans. 4 yr. 2 mon.
2. At 3 %7 per annum, in what time will anly principal
double itself? Ans. 33- yr.
3. A man pays $15 for the use of $360, for 8 months; what
is the rate %? Ans. 64
PERCENTAGE. 251
4. If $96.25 is paid for the use of $440 for 2 yr. 6 mon.;
what is the rate %? Ans. 8.3
5. What principal, at 6' %, will gain $23.075 in 1 yr. 3
mon.? Ans. $284.
6. What fund at interest, at 7 t, will yield an annual income of $1299.20? Ans. $18560.
DISCOUNT.
1. What the present worth of $500, due 10 mon. hence,
when money is worth 5' per annum? Ans. $480.
2. What the present worth of $415, due 1 yr., 4 mon. hence,
money worth 8 %O per annum? Ans. $375.
3. What is the present worth of $281.26, due 1 yr. 5 mon.
6 da. hence, money worth 10 % per annuma? Ans. $246.
4. What is the discounit of $355.25, clue 4 months hence,
at 4. %? ils. $5.25
5. What is the discount of $690.183, due 9 months hence,
at 3 0? Ans. $15.183
6. A note not bearing interest, dated Jan. 6th, 1849, for
$943.68, is payable March Ist, 1850; what was its value on
the dclay it was dated, reckoning money worth 8 7o per annum?
Ans. $864.
7. A note not bearing interest, dated Jan. 1st, 1845, for
$176.30, is payable July 1st, 1847; what is its value, April
1st, 1846, reckoning moneyworth 6%o per annum? Ans. $164.
8. How much stock, the selling price per share $92], and
par value $100, must be sold to pay $715.85, due 9 mon.
hence, reckoning money worth 4 %o per annum?
Ans. $750, or 71 shares.
BANK DISCoUNT.
1. At 6 %, what is the bank discount of a note of $75, p)ayable 30 days after date? Ans. $0.412. At 6 %, what is the bank discount of a note of' $230,
payable 60 days after date? Ans. $2.413. At 6 1o discount, what is the proceeds of a note of $420,
payable 30 days after date? Ans. $417.69
4. At 6 %o discount, what is the proceeds of a note of $375,
payable 60 days after date? Ans. $371.064
art —---
252 TEST EXAMPLES.
5. If $393.80 be received at a bank, for a note payable in
90 days, discounted at 6 %, what is the face of the note?
(See Art. 241.) Ans. $400.
6. If $592.65 be received for a note payable in 60 da., discounted at 7 %, what is the face of the note? Ans. $600.
7. WVishing to borrow at 6 %, $1000 for 30 days, for what
sum must I give my note? Ans. $1005.53+
MISCELLANEOUS EXERCISES IN PROFIT AND LOSS.
1. A merchant bought cloth for $225, sold it at 20 go advance; how much profit did he make? Ans. $45.
2. A cask of wine, containing 70 gallons, lost 15 So by
leakage; what quantity remained? Ans. 59 gal. 2 qt.
3. A man bought a buggy for $75, and sold it at a profit of
8 s; how much did he get for it? Ans. $81.
4. When hats that cost $3.75 a piece, are sold for $4, what
is the rate oc of profit? Ans. 62.
5. When hats that cost $4, are sold for $3.75, what is the
rate o/ of loss? Ans. 6..
6. Bought sugar at 4 cents a pound, and sold it at the rate
of 18 pounds foi' a dollar: what %/c profit did I make? Alns. 38S.
7. Supposing the same sugar sold at the rate of 7 lb. for
25 ct., what is the rate 5/ of loss? Ans. 10'.
8. A grocer bought a barrel of fish for $11.75, and sold it
for $13.65: what %o profit did he make? Ans. 164s~.
9. Bought 12 bl. of apples, each containing 23- bu., at $2.50
a bl., sold them at $1.20 a bu.: required the whole gain, and
gain So? Ans. Gain, $9.60; rate 32 %.
10. Bought eggs at the rate of 5 for 4 cents, for how much
per dozen must they be sold, to gain 25 So? Ans. 12 cts.
11. Bought 8 barrels of flour, each containing 196 pounds, at
$5.25 a barrel, and sold it at 25 cents for 7 pounds: required
the whole gain, and the gain per cent.?
Ans. Gain, $14; rate, 331.
12. A merchant, by selling cloth at $1.82 per yard, loses at
the rate of 9 5o: at what rate per yard ought it to be sold, to
realize a profit of 7 %? Ans. $2.14
PARTNERSHIP. 253
13. A merchant, by selling cloth at $2.43 per yard, gained -
of the first cost; what 6 would he gain by selling it at $2.64
per yard? Ans. 22g.
14. Sold quills at 11 cents a bunch, of which - was clear
profit; afterward raised them to 13A cents a bunch: what 1
did I clear at the latter price? Ans. 96 4T.
15. A merchant, by selling 4 yards of cloth for $13, gained
25,o: what % would he lose by selling 4 yards for $10?
Ans. 31l.
16. By selling bacon at 12' cents per pound, a retailer makes
25 So profit: what % profit will he make by selling it at 13
cents a pound? Ans. 30.
17. A merchant, selling flour at $3.22 a barrel, loses 8 %o;
when he bought it, he expected to sell it at $4 per barrel:
what %o profit did he expect to clear? Ans. 147.
18. A trader purchased $856 worth of goods, which he sold
at a loss of 12,1 SO; he invested the proceeds in goods, which
he sold at a profit of 20 %: by the two transactions, what %
profit did he make on his first investment? Ans. 5.
ASSESSMIENT OP TAXES.
1. A tax of $2441.25 is to be levied on a corporation, of
which the taxable property is valued at $420000, and which
has a list of 525 polls, each taxed 25 cents: what ro is the tax
on property? Ans..0055, or 5' mills on the dollar.
2. What is A's tax, whose property is valued at $1200, and
who pays for 1 poll? AAns. $6.85
3. What is B's tax, whose real estate is valued at $8500, his
personal property at $260, and who pays for 3 polls?
Amzs. $48.93
PARTNERSHIIP.
MENTAL EXERCISES.
1. James has 2 cents, and John 1 cent, with which they
buy 9 apples: what is the share of each?
2. A has 2 cents, and B I cent, with which they buy 20
chestnuts: what is the share of each?
IC, --— ~~ --------------
254 TEST EXAMPLES.
3. A has 2 cents, and B 3 cents, with which they buy 10
peaches:'what is the share of each?
4. A has 3 cents, and B 4 cents, with which they buy 14
pears: what part of them must each have? What is the share
of each?
5. C has 5 dollars, and B 3 dollars, with which they buy 12
yards of cloth: what part of it should each have? What is
the share of each?
6. A has 3 cents, and B 2 cents, with which they buy 12
peaches: what part of them should each have? WVhat is the
share of each?
FOR THE SLATE OR BLACKBOARD.
1. Two men hired a pasture for $24.09. A pastured in it
5 horses, and B, 6 horses: what ought each to pay?
Ans. A, $10.95; B, $13.14
2. Three men together perform a job of work for $57.50. A
worked upon it 20 days, B, 17 days, and C, 13 days: what
should each receive? Ans. A, $23; B, $19.55; C, $14.95
3. Three partners put into business the sums of $300, $400,
and $500; they gained $375: find the share of each.
Ans. $93.75, $125, and $156.25.
4. Divide $1000 among three persons, so that their shares
shall be as the numbers 2, 5, and 9.
Ans. $125, $312.50, and $562.50
5. A person bequeathed by will the following legacies; to A
$1500, to B $875, to C $525, and to D $350: but when his
property was sold, it produced only $2437.50: how much did
he really leave to each? Ans. to A, $1125, to B, $656.25, to
C, $393.75, and to D, $262.50
6. Divide $60 between A and B, so that A shall have $ — as
often as B has $a. Ans. A $24, B $36.
7. Divide $115 between three persons, so that the parts shall
be to each other as l, 4, and 3. Ans. 4$40, $30, and $45.
8. Gunpowder is composed of 76 parts of nitre, 14 of charcoal, and 10 of sulphur: how much of each is there in 112 lb.
of powder? Acns. 85;% lb., 15 7 lb., and 111 lb.
~ ~ 1~ lb. —-— e
EQUATION OF PAYMENTS. 255
PARTNERSHIP WITHI TIBIE.
1. Two men finished a job of work for $45. A had 5 men
upon it 3 days, and B 3 men 7 days: what is each man's share
of the money? Ans. A's, $18.75; B's, $26.25
2 A, B, and C traded in' company; A put in $100 for 3
months, B, $90 for 5 months, and C, $35 for 10 months; they
gained $198: what was the gain of each?
Ans. A $54, B $81, C $63.
3. Two men hired a pasture for $14; in it A pastured 4
horses for 5 months, and B, 5 horses for 6 months: how much
should each pay? Ans. A $5.60, B $8.40
4. A advances $150 for 9 morn., and B, $120 for 6 mon.; they
gain $115: what the share of each? Ans. A $75, B $40.
5. Four men hire a pasture for $50.25, in which A pastures
5 cows 9 weeks, B, 4 cows 10 weeks, C, 6 cows 10 weeks, and
D, 7 cows 8 weeks: how much should each pay?
Ans. A $11.25, B $10, C 15, D $14.
6. Three persons with a joint stock, gain $183.60: the first
furnishes: of the capital for 4 of the time; the second, r of
the capital for 4 of the time, and the third, the remainder of
the capital for the whole of the time: find their shares.
Ans. $24.48, t36.72, and $122.40
EQUATION OF PAYMIENTS.
1. What will be the equated term of payment of $200, due
in 3 months, $300, in 8 months, and $500, in 12 months?
Ans. 9 months.
2. If $75 be due in 4 months, $125 in 5 months, and $150
in 7 months; what is the equated time? Ans. 5 9T months.
3. A owed B $750, to be paid in 15 months, but at 12
months he paid himn 8250: at what time was the remainder
due? Ans. 164 months.
4. Find the equated time of payment, when.) of a sum of
money is due at 3 months, 5 at 8 months, and the remainder
at 15 months. Ans. 7 — months.
256 TEST EXAMPLES.
5. If a person owing $1000, payable in 7 months, pay $300
down, how long after the 7 months may he delay the payment
of the remainder? Ans. 3 months.
ANALYSIS.
1. If 9 barrels of flour cost $29.25, what will 11 barrels
cost? Ans. $35.75
2. If a railroad car can run 609 miles in 29 hours, how far
will it run in 13 hours? Ans. 273 miles.
3. If $79.25 will pay for 25 barrels of flour, how many
barrels can be purchased for $60.23? Ans. 19 barrels.
4. If a man walk 3 miles in X of an hour, how far can he
walk in 63 hours? Ans. 30,4 miles?
5. If 4 of a pound of silver ore is worth $4.05, what is a
pound worth? Ans. $6.75
6. If 42 of a bushel of pecans is worth 714 cents, what is C)
of a bushel worth? Ans. 1.389
7. If 4 of a steamboat is worth $3750, what is -'-& worth?
Ans. $6875.
8. If 24 bushels of wheat cost $2;04, what will 54 bushels
cost? Ans. $5.46
9. If 3~ bushels of wheat cost $2.16, how many bushels
can be purchased for $7.50? Ans. 11.9.
10. If 11 - bl. of flour serve a family (of 15 persons) 45. wk.,
how many bl. will serve them 17- wk. Ans. 4,.
Observe that the answer is entirely independent of the numzber
of persons in the family.
11. If a barrel of flour serve 5 men 17 days, how long will
it serve 3 men? Ans. 284 days.
12. HIow much wheat, at $4 a bu., must be given in exchange for 5 yd. of cloth, at $1~ per yd.? Ans. 8-3 bu.
13. If 44 bu. of corn cost $3, what will 8$ bu. cost?
Ans. $7'3.
14. If 34 lb. of coffee cost 3564~ cents, what will 251 lb.
cost? Ans. $2.55
- --- ------- -----------— e
ANALYSIS. 257
15. If 4-g acres of land cost $27-, how much will $50 buy?
Ans. 7-A A.
16. If 12 yards of muslin cost $3$, what will 11 yards cost?
17. The distance from A to B is 28 miles, or 1 of the distanuce from B to C; what the distance from B to C?
Ai, 49 miles.
18. A has 88%-; anLd of A's mloney is equal to $ of B's:' lwhat is B's money? Ans. $7.35
19. A has $36, and 3 of A's mon ey is equal to - of B's, and
g of B's is equal to -~s of C's: required C's money. A ns. $55
i 20. If 5 men can do a certain piece of work in. of a day,
in what time ought 3 men to do the same work? Ans. 1~ days.
21. If 4 men eat 32 pounds of bread in 2 days, how many
pounds will 9 men ea-t in 5 days? AlZs. 180 pounds.
22. If a man can travel 195 miles in 5 days, when the dclays
are 12 hours long, how far can he travel in 6 days, when the
days are 1 hours long? A's. 2141 miles.
23. If 9 men can build 54 rods of wall in 8 days, how much
would 12 men build in 5 days? AIs. 45 rods.
2-14. If 5 men mow 20 acres of gross in 6 days, how mnany
men nwould mow 12 acres in 9 da.ys? Als. 2 men.
25. If 10 bushels of oats supply 18 horses 20 days, how
many bushels will supply 60 horses 36 days? Ais. 60 bushels.
26. How many men will it reqcuire to dig a trench i35 yards
long in 8 days, when 16 men can dig 54 yards in 6 days?
LAs. 30 men.
27. If 20 bushels of whealt are sufficient for a family of 8
persons 3 months, how much will be required for 7 persons 2
months? -A4s. 11:- bushels.
2o8. If' 3 barlrels of flour sserve'14 persons 5,. weeks, how
long will 5-/ barrels serve 7 persons? Ails. 19; weeks.
29. If 5 len earn $26.2c5 in 7 davys, llo how much xwill 7 men
earn in 11 days? Lt[s. $57.75
30. If 9 horses eatr 28.8 bushels of ota-s inr 8 days, how
many bushels will 13 horses eat in 15 days? AZns. 78 bushels.
31. If $24 will buy d of an acre of land, what part of an
acre will $15 buy? A us. -,.
258 TEST EXAMPLES.
32. A man can do a piece of work in 32 days; what part
of it can he do in 1 day? Ans. ]3,
33. A man can mow a certain meadow in 3- days; what
part of it can he mow in 27 days? Ans.:26.
34. If 24 yards of cloth cost $34, what part of it, and how
much, will 3- quarters cost? Ans. 2-7, and $1711
35. A cistern, containing 1500 gallons, can be filled by one
pipe in 4 hours, and emptied by another in 5 hours: in what
time can it be filled, if they are both left open? Ans. 20 hr.
36. A cistern, containing 800 gallons, is filled by one pipe
at the rate of 100 gallons in 3 hours, and emptied by another
at the rate of 144 gallons in 5 hours; in what time will it be
filled, if both these pipes are left open? Ans. 176-8 hr.
37. A can do a piece of work in 2 days, and B in 4 days;
in what time can they both do it? Ans. 1i- days.
38. A and B can dig a trench in 20 days; A can dig it
in 34 days: in what time can B dig it? Ans. 484 days.
39. A can dig 32 yards of a trench in 6 days; B, 29 yards
in 5 days, and C, 54 vards in 10 days; in what time can
they finish 100 yards, if' all work at the same time?
Arns. 6~3 days.'40. If A can finish a certain piece of work in 6 days, of 7
hours each, and B can do 4 times as much in 15 days, of 9
hours each, what is their comparative strength? Ats, The
strength of A is to that of B, as 45 to 56, or as 1 to 141.
41. A, B, and C, dig a trench in 4 days; A can do it by
himself in 7 days, and B in 14 days; in what time can C
perform the work alone? - Ans. 28 days.
42. A and B can perform a piece of work in 2 days; A
and C in 3 days; and B and C in 5 days: in what time
can each do it by himself?
lAns. A in 31-, B in 5 T, and C in 60 days.
43. A, B, and C, can do a piece of work in 10 days; B, C,
and D, in 12 days; C, D, and A, in 14 days; and D, A, and
B, in 16 days; in what time can each do it by himself?
As.A, in44 B 293, C 23-l3o, D 173;- days.
44. A leaves Cincinnati for Dayton at 7 in the morning;
and B leaves Dayton for Cincinnati, at 9 the same morning;
ANALYSIS. 259
A travels 3V, and B 44 miles per hour; how far will each
have traveled when they meet, reckoning the distance from
Cincinnati to Dayton 59 mi.? Ans. A 30T~, B 28 —l miles.
45. A cistern has 3 pipes; by the first, it can be filled in
1- hours, by the second, in 34 hours, and by the third, in 5
hours: in what time can it be filled by all these pipes running
together? Ans. 48 minutes.
46. After paying away 4 and 5 of my money, I had $33
left; how many dollars had I at first? Ans. $60.
47. After paying away 1 of my money, and 4 of the remainder, I had $57; how much had I at first? Ans. $95.
48. A father's age is 4 times that of his son, and the sum of
their ages is 40 years: what is the son's age? Ans. 8 yr.
49. The greater of two numbers is 24- times the less, and
their sumn is 14: what are the numbers? Ans. 4 and 10.
50, The smaller of two numbers is - of the greater, and
their sum is 25: required the numbers. Ans. 74 and 17k.
51. The smaller of two numbers is 2 of the greater, and
their difference is 21: required the number. Ans. 6 and 27.
52. In a certain s.chool there are 32 pupils, and there are 12
more boys than girls: required the number of each.
Ans. 10 girls, 22 boys.
53. Three-fourths of a certain number exceeds 4 of it by 25;
what is the number? Ans. 140.
54. The 4,,, i, and 4 of a certain number, added together,
make 127; what is the number? Ans. 84.
55. The 4, -, and 4 of a certain number, added to the number itself, make 175; what is the number? Ans. 56.
56. There are 195 sheep in 3 flocks; the second contains 10
more, and the third 20 more than the first: required the number in each flock. Ans. 55., 65, and 75.
57. At what time between 7 and 8 o'clock, are the hour and
minute hands of a watch opposite? Ans. 5 T min. past 7.
At what time are they together? Ans. 38y min. past 7.
58. A number of men, each at 75 cents a day, and half as
many boys, at 25 cents a day, together earn daily $10.50: required the number of each. Ans. 12 men, and 6 boys.
59. A, B, and' agreed to dine together. A furnished 2
260 TEST EXAMPLES.
loaves, B, 3 loaves, and C, for his share, contributed 20 cents,
to be equitably divided between A and B3: required the share
of each. A's. A's, 4 cents; B's, 16 cents.
60. If 30 mnen, in 40 hours, can dig 80 cubic yards; how
many men, who are stronger in the ratio of 4 to 5, would
it require to dig 120 yards in 90 hours; supposing the ground,
in the latter case, is harder than that in the former, in the
ratio of 8 to 9? Ans. 18.
EXCHIIAITGE OF CiURRENTIES.
1. Reduce ~130 10s. sterling, to U. S. money, exchange
being 9 %o advance. AAns. $632.20
2. Reduce ~318 7s. 6d. sterling, to dollars, exchange being
7 % advance. Ans. $1514.05
3. Reduce $1106.35 to pounds sterling, when exchange is
9 %0 advance. Anzs.: 228 7 s.: 6 d.
4. Reduce $2119.67 to pounds sterling; exchange being 7 7
advance. Arls. ~445 14 s. 6 d.
5. At $1 for 5.'30 francs, what will be the cost of a bill of
exchange on Paris for 1457.50 francs?; Ans. $275.
6. At $1 for 5.45 francs, what will be the cost of a bill of
exchange on HIavre for 2633.44 francs?:Ais. $483.20
7. Reduce $385.25 to French money, reckoning 1 worth
5.30 francs.. Ans. 2041.825 francs.
8. Reduce $217.25 to French nmoney, reckoning $1 Worth
5.44 francs. iAns. 1181:.84 francs.
DUO D E C I Mv A L
Find the product of the following lengths and breadths.
Lengths. Breadths.
ft. /'.'ft. /' ft. //
1. 7 9 by 3 6... Ans. 27 1 6.
2. 8 5 by 4 7... Ans. 38 6 11.
3. 8 1 by 3 5.. Ans. 27 7 5.
4. 7 5 9 by 3: 5 3.. Ans. 25 8 6 2 3.
5. 10 4 5 by 7 8 6... Ans. 79 11 0 6 6.
6. 7 6 by 5 9. Ans. 43 1 6.
TABLE. 261
A Table of Squares, Cubes, and Roots,
TO EXERCISE PUPILS AT THE BLACKBOARD.
1 1 1 1.009000 1.000000 43 1840 79507 6.5 7438 3.5033.08S
) 4 8 1.414213 1.259921 44 1936 85184.63 349 3.53:0348
3 9 1 7 1.732050 1.442250 45 2025 91.125 6.708203 3.556893
4 16 04 2.000000 1.587401 46'2116 97336 6.782323,0 3 58018
5 25 125 2.236068 1.709976 47 2209 103823 6.855654 3.608826
G | 6, 216 2.449489 1.817121 48 2304 110592 6.928203 3.634241
7 49 343 2.645751 1.912933 49 2401 1117649 7.000000 3.659306
8 G64 512 2 828427 2.000000 50 2500 125000 7.071067 3.6840:31
9 81 729 3.000000 2.080084 51 2601 1:32651 7.141428 3.70S430
10 100 1000 3.162277 2.154435 52 2704 140608 7.2[1102 3 732511
11 121 1331 3.316624 2.223980 53 2809 148X77 7.2801.09 3.756286
12 144 1 7,2 3.4641.01 2.289.12 5-1 2916 157464 7.348469 3.779763
1:3 169 2197 3.605551 2.;351335 55 3025 166375 7.416108 3.8029-53
14 196 2744 3.741657 2.410142 56 31:36 175616 7.483314 3.825Q62
15 225 3375 3.8729S3 2.466212 57 3249 185193 7.541 834 3.849501
16 256 4096 4.000()00 2.519S42 58 3364 195112 7.615773 3.870877
17 289 4913 4.123105 2.571282 59 3481 2015379 7.681 145 3.892996
1S 1321 58312 4.242640 2620741 60 3600 3 16000 7.745966 3.914867
19 3611 659 4.358S98 2.066102 61 3721 2o6981 7.810249 3.936497
20 400 8003 4.472136 2.714418 62 3844 238328 7.874007 3.957892
21 44t11 1261 4.582575 2.758924 63 3969 250047 7.917253 3.979057
22 4854 1068l 4.690415 2.802039 64 4096 2621.44 8.000000 4.000000
23 529 2167 4.795831 28438'67 65 4225 274625 8.062257 4.020726
24 576 13524 4.S98979 2.884499 66 4356 287496 8.124038 4.041240
25 625 15625 5.000000 2924018 67 4489 300763 8.185352 4.061541S
26 (i76 17576 5.099019 2.962496 68 4624 314432 8.246211 4.081655
27 7-29 19683 5.196152 3.00(0000 69 4761 328509 8.306623 4.101566
28 784 21952 5.291502 3.036589 70 4900 343000 8.366600 4 121255
29 841 24389 5.38516,1 3.07231.7 71 5041 357911. 8.426149 4.140818
30 900 27000 5.477225 3.107232 72 5184 373248 8.485281 4.160168
31 961 29791 5.5677641 3.141381 1 3 5:329 [389017 8.544003 4.179339
32 1024 32768 5.656854 3.174Q802 74 5476 405224 8.602325 4.198336
33 1089| 35937 5.744562 3.207534 75 i 5625 421875 8.660254 4.217163
34 1156 39104 5.830951 3.2396L2 76 5776 438976 8.717797 4.23582-1
35 1225 42875 5.91.6079 3.271066 77 5929 456533 8774964 4.254321
36 1296 46656 6.000000 3.301927 78 6084 474552 S.831760 4.272659
37 1369 50653 6.082762 3.332222 79 6241 493039 8.888194 4.290841
38 1444 54872 6.164414 3.361975 80 6400 512000 8.944271 4.308870
39 1521 59319 6.244998 3.391211 81 6561 531441 9.000000 4.326749
40 1600 64000 6.324555 3.419952 82 6724 551368 9.055385 4-'344481
41 1681 6S921 6.403124 3.448217 83 68S9 571787 9.1104133 4.362071
42 1764 74088 6.48(0740 3.476027 84 7056 592704 9.16s5151 4.379519
262 TEST EXAMPLES.
INTVOLUTION AND EVOLUTIOiN.
NOT-E.-The preceding table may be used to extract the square
root of any number that can be separated into two factors, one
of which is a perfect square, and the other not greater than 84.
Thus, the square root of 700=-/100X7_10/ 7; hence, we find
the square root of 7 in the table, and multiply it by 10, and the
result is the square root of 700. In a similar manner, we may
find the cube root of any number composed of two factors, of
which one is a perfect cube, and the other a number whose cube
root is given in the table.
MISCELLANEOUS EXERCISES IN EVOLUTION.
1. The base and perpendicular of a right-angled triangle
are 42 and 5G feet respectively: what is the hypotenuse?
Ans. 70 feet.
2. The base and hypotenuse of a right-angled triangle are
21 and 35 feet respectively: what is the perpendicular?
Ans. 28 feet.
3. Two men start from the same point; one travels 60 miles
west, and the other 50 miles south: how far are they then
apart? Ans. 78.1+ miles.
4. A ladder, 32 feet long, stands against a house, the foot
resting on the ground, on a level with the foundation of the
house, and 18 feet distant from it: how high on the wall will
it reach? Ans. 26.45-+ feet.
5. There is a room 32 feet long, 16 wide, and 10 high;
what is the longest straight line that can be drawn on the
floor; and what is the length of the line joining the opposite
upper and lower corners of the room?
Ans. 35.77+ ft., and 37.14+ ft.
6. How far apart are the diagonal corners of a cubical block,
each side of which is 6 inches? Ans. 10.392+ in.
7. HIow long is the diagonal of a square, whose side is 10
feet? Ans. 14.14+ feet.
8. HI-ow long is the diagonal of a rectangular block of marble, whose length, breadth, and thickness, respectively, are 10,
4, and 2 feet? Ans. 10.954+ ft.
MAISCELLANEOUS. 263
MISCELLANEOUS EXABIPLES.
1. If each of the side walls of a house contains 80325 bricks,
and each of the end walls 47975, and 1 partition, 46560; how
many bricks are there in the house? Ans. 303160.
2. A has $5231 in cash, and property worth $8427; B owns
a farm worth $20265, and owes $6987; how much is A worth
more than B? Ans. ~380.
3. How many pounds can be carried in 729 wagons, each
capable of holding 2793 pounds? Ans. 2036097 lb.
4. If 327 horses cost $28449, how many can be bought for
$69861? As. 803.
5. What is the difference between $97.03 and $3.97?
Ans. $93.06
6. A man bought 640 acres of land at $217.28 per acre, and
a mill for $47000; how much did the land cost more than the
mill? Ans. $92059.20
7. Dividing $2323 equally among 25 men, what is the share
of each? Ans. $92.92
8. What will be the cost of 11 bu. 3 qt. I pt. of salt, at 2 cts.
a pint?.Ans. 214.22
9. What cost 11 lb. 11 oz. 11 pwt. 11 gr. of gold, at 3 cents
a grain? Ans. $2067.45
10. How long will a star be in moving 830 15', if it moves 9'
in 1 mlin.? Ans. 23 da. 3 hr.
11. What cost 3 lb 2 5 2 3 2 D of Iodine at 2 cents a scruple?
Ans. $18.40
12. If a man breathes 20 times in 1 min., how often will he
breathe in 3 wk. 3 da. 3 hr.? Ans. 694800.
-13. How many cannon balls, weighing 2 qr. 5 lb. each, can
be made from 16 T. 10 cwt.? Ans. 600.
14. If a man takes 6 steps to each rod, how many steps will
he make in 18 mi. 7 fur.? Ans. 36240.
15. What cost 3 hhd. of ale at 3 cents a pint.? Ans. $38.88
16. How many acres in a field 4 fur. 16 rd. long, and 3 fur.
10 rd. wide? Ans. 143 A.
17. If a wine glass holds 3 gills, how many times can it be
filled from 2 p. 27 gal.? Ans. 2976.
18. What will be the cost of a pile of wood, 288 ft. long,
8 ft. high, and 6 ft. wide, at $4.57 a cord? Ans. $493.56
19. If 39 lb. 7 oz. 15 pwt. of silver be divided among 5 men,
what will each lman's share be worth at 6 cts. a pwt.?
Ans. 114.18
264 TEST EXAMPLES.
20. The longitude of a certain place is 84 VW.; what is the
longitude of a second place, where it is 26 min. past 1 P. M.
when it is 10 A. M. at the former place? Ans. 320 30/ W.
21. Two pieces of cloth, one containing 35, and the other
55 yd. are to be cut into pieces, the longest that will admit
of the same number of yards in each; what will be the length
of the pieces?
Ans. 5 yd., the greatest common divisor of 35 and 55 yd.
22. Three piles of wheat, corn and rye, containing respectively 42, 54 and 78 bu. are to be put into sacks, the largest
that will contain the same number of bushels in each: what
number of bushels must each sack contain? Ans. 6 blu.
23. What is the least sum, for which I can buy, either cows
at $12 each, oxen at 620 each, or horses at t.40 each?
Ans. $]20.
24. A can build 4 rd. in 1 day, B, 6 rd., and C, 15 rd.; what
is the least length of fence that will employ either an exact
number of days? Ans. 60 rd.
25. What are the denominators of all the fractions that can
be reduced to twenty-fourths? Ans. 2, 3, 4, 6, 8 and 12.
26. What -wil ~Tv of a mi. of railroad cost, at, 312.50 a ft.?
-Ans. $42000.
27. H-low nmany apples, at the rate of 21 for 31 Ct8s., can be
bought for $? Ans. 60.
28. What fraction of a mi. will i38 rd. 4 yd. i ft. be? Ans. 4-3.
29. At $-15.08 per yd., what will eight-thousandths of a yd.
of cloth cost? Ans. 6.36064
30. What decimal of 2 bu. I qt. is 1 pk. 5 qt.? Ans..2
31. If 11 men in 12 da. plow 300 A. of land, how much can
S men plow in 22 days? Ans. 400 A.
32. If 3 bu. 5 qt. of corn cost 62,02, what will be the cost of
17 bu. 3 pk.? Ans $11.36
33. If there are 2000 bricks in a pavement, 20 ft. by 30 ft.,
how many will there be in another, 57 ft. by 119 ft.?
Ans. 22610.
34. A man bought a house for,4800, and sold it for 85700:
what %o did he gain? Ains. 18 -5
35. A ii.an bought stock in a railroad at 4- Jv advance, and
MISCELLANEOUS. 265
sold at 9 ~% discount; what } on his investment did he lose?
An s. 12.1 %o
36. WThait cost 327 T. S cwt. 3 qr. 17 lb. of sugar, at $150 a
tun? Ans. $49116.90
37. When railroad stock is at a discount of 18 %, how many
shares can I buy for,$3690, par value of each share being
$50? Ans. 90.
38. AWhat must be paid to insure 2 of the value of a house
worthl $3000, at -9 o, t'he policy costing $1.25? Auts. $31.25
3o). What is the interest on $873.27 for 4 yr. 11 mon. 22 da.,
at 8 %? Ans. $347.75540. By selling goods for $400, I lost -- 7, what had they
cost? Ans. $402.01+
41. Wfhat would be thle proceeds of a note for 8575, due in
77 da., discounted in bank at 9 %? Ans. $563.50
42a Sold flour at $5.85 per bl. gaining 17 %; wha;t % would
have been lost by selling 12 bl. for $57? Ans. 5 %
43. A put in $600 for 2 yr., B $900 for 8 mon., C $1000 for
4 mon.; they gtin $320; how should it be divided?
Ans. A $18(, B $90, C $50.
44. If 17 lb. at 12 cts., 14 lb. at 14 cts., 23 lb. at 11 cts.,
and 211 lb. at 10 cts., be mixed, what will 1 lb. of the mixture
be worth? lAns. 11+ ects.
45. If 2 of a pt. cost 7- cts., what will " of a bu. cost?
AnRs. $4.
46. At what time between 4 and 5 o'clock, is the min. hand
9 min. behind the hr. hand? Ans. 12 min. past 4.
47. Divide $37 between A and B, giving A 1-I cts. as often
as B receives 1~ cts. Ans. A $16, B $21.
48. How many gal. of beer at 15 cts. per gal., together with
an equal quantity of water, must be mixed with 43 gal. of
beer at 23 cts. per gal., and 35 gal. at 21 cts. per gal. to make
a mixture worth 16 cts. per gal.? Ans. 28 gal.
49., of a numnber exceeds its 5 by 14;,what is the number?
Ans. 114
50. Reduce ~47 16 s. 9 d. to U. S. money. ins. $231.5335
51. How many cu. ft. in a room 26 ft. 3 in. long, 21 ft. 4 in.
wide, and 10ft. 5 in. high? so-?. 5833 — cu. ft.
C~~ —------------- ~~X~s "'~"~~ —---------
266 TEST EXAMPLES.
52. A ladder 46 ft. long reached as far up a wall as its foot
was distant from the bottom of the wall: how high did it
reach? Ans. 32 ft. 6+ in.
5.2 4-d
53. T of of =what decimal? Ans..675
54. A ball is 6 in. in diameter; what is the diameter of a
ball twice as large? Ans. 7.559+ in.
55. A man whose property increased $578 annually, was, in
the year 1856, worth $22471; what was he worth in the year
1842? Ans. $14379.
56. A man commenced business with $100000, lost $327
each mon., and closed with $88228; how long was he in business? Ans. 3 yr.
57. A piece of cloth 1 E. Fl. 2 qr. 2 na. wide, contains 55 sq.
yd.; how long is it? Ans. 40 yd.
58. The sides of a triangular field are 5 fur., 12 fur., and 13
fur.; what will be the cost of fencing and plowing it; the
fencing at 23 cents a rd., and the plowing at 10 cts. a R.?
Ans. $396.
59. In a field in the form of a trapezoid, the parallel sides
are 80 rd. and 76 rd. long, and are 36 rd. apar.t; what is the
value of the field at $50 per A.? Ans. 8877.50
60. The circumference of a circle is 20 ft.; what is the
radius? Ans. 3 ft. 2.197+ in.
61. A tree exactly round, is 7 ft. in circumference; what is
the width of the widest board that can be sawed from it?
Ans. 2 ft. 2.73+ in.
62. What is the diameter of a circle, whose area is 42 %o of the
area of a rectangle 17 rd. long, and 11 rd. wide? Ans. 10 rd.
63. What is the solidity of a pyramid 8 ft. high, whose base
is 2 ft. 3 in. square? Ans. 13. cu. ft.
64. If a cube 4 in. in length weighs 6 lb., what will be the
weight of a globe of' the same material, 5 in. in diameter?
Ans. 6.13+ lb.
65. What is the area of a circle whose circumference is 10
ft.? Ans. 7.9577+ sq. ft.
~~~~~~~~~~ ------ - - - - - --- --.- - — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4~~~~~~~~
MISCELLANEOUS. 267
66. A circular measure 8 in. high, contains 1 bu. or 2150.4
cu. in.; what is its diameter? Ans. 18.495+ in.
67. If 47 yd. of cloth 4 ft. 2 in. wide, cost $29.375, what will
37 yd. cost, 5 ft. 4 in. wide? Ans. $29.60
68. A crib 15 ft. long, 8 ft. wide, and 7 ft. high, was filled
with grain as follows; 120 bu. of corn at 31~ cts. a bu., 150
bu. of wheat at 87' cts. a bu., and the remainder oats at 20
cts. a bu.; what was the value of the entire contents?
Ans. $249.75
69. HIow many apples must be bought for $5, so that 20 S
may be gained by selling them at the rate of 2 for 3 cents?
Ans. 400.
70. How many must be bought for the $5, so that 20 % may
be gained by selling them 3 for 2 cents? Ans. 900.
71. How many sq. ft. of boards, 9 in. thick, can be sawed
from a log 15 ft. long, and 3 ft. 4 in. in diameter, allowing j-3
for waste? Ans. 1523.2 sq. ft.
72. How high must a triangle be, whose base is 24 ft., to be
equal to a rectangle 18 ft. long, and 14 ft. wide? Ans. 21 ft.
73. 20 % of -=what? Ans..1S
41
74. How many lots, each 33 ft. by 112 ft., can be laid out
from 10 A. of land, allowing 16 %o for streets? Ans. 99.
75. How many bars 8 ft. long, 5 in. wide, and I in. thick,
are equal to 85 bars, 4 yd. long, 4 in. wide, and -1 in. thick?
Ans. 68.
76. A borrowed $300 July 21st, 1850; 8400 Sep. 7th, 1851;
$500 May 5th, 1852, all at 9 o: how much does he owe Jan.
1st, 1857? Ans. $1774.90
77. What sum will amount to $665.50 in 2 yr. at 10 rO compound interest? Ai71s. $550.
78. The time till noon to-day, is ~ of the time till noon tomorrow; what is the time? Ans. 9 o'clock A. AM.
79. An American borrowed 800 francs in Paris, Jan. 20th,
1852: what will it cost him to pay the debt in Cincinnati,
July 20th, 1853, interest being 5 15, exchange 2 % above par,
and $1=5.375 francs? Ans. $163.20
268 TEST EXAMIPLES.
80. A book contains 400 octavo leaves; if the same paper
had been folded in duodecimo, how many leaves would it have
made? Ans. 600.
SUGGESTrOn.-See table, page 100, 3d Book.
81. HIow many leaves can be made fromn a quire of paper,
folded in octavo?;Ans. 384.
82. Wha.t is the bank discount on $120, payable in 40 da.,
at 9 %C? Ans. $1.29
83. Wlhat %o of any number is 433 Yo of 3 of it?
84. A. person who spent $200 more than 4 his yearly income, had left $50 more than 4 of it; what was his income?
A)zs. $1000.
85. A owns - of a house worth $3600, B,:, C tihe remainder; it is insured for $2160; what will each lose by its destruction? Ans. A $,q180, B $360, C $900.
86. A note for $300, dated Jan. 15th, 1856, at 60 cIa., was
discounted in bank at 6 %, March 6th; what -was the proceeds? Ants.,299.40
87. A man having lost - of his property, and afterward
gained ~ as mluch as the remainder, found that he then had
$2520; how much had he at first? Anzs. $3000.
88. A ship can move 14 mi. per hr. by' steam alone, or 17
by steam and sails; how long will it be in going 3410 mi.,
using the sails half the time? Ans. 220 hr. or 9 da. 4 hr.
89. If 3 apples are worth 2 peaches, and 3 peaches are
worth 5 pears, and 2 pears worth 3 plums, how many plums
should be given for 48 apples? Ans. 80.
90. Divide $60 among A, B and C, giving A half as much
as B, and B $5 more than C. Ans. A $13, B $26, C 821.
91. What is the least floor, that can be exactly covered
either with marble slabs 1 ft. 6 in. square, or iron castings 2
ft. 8 in. square? Ans. one 24 ft. square.
92. How many cans 6 in. high, 4 in. long, and 1 in. wide,
can be made from 26 sq. ft. of tin, making no allowance for
joints? Ans. 48.
93. If 4?1 lb. of flour cost 15:} cts., what will 5 bl. cost?
Atzs. 33.60
MISCELLANEOUS. 269
94. What will 180 windows cost, each having 2 sashes, and
each saslh, 1.2 panes of glass, 8 in. by 10 in.; the sashes costing 37 - cts. each, the glass at $4 per box of 100 sq. ft., and
the glazing 1' -cts. a pane? Ans. $303.
95. How many bu. of coal will a boat 100 ft. long, 12 ft.
wide, loaded 3 ft. 6 in. deep, contain; counting the coal bu.
at 5 pk. or 2688 cu. in.? Ans. 2700 bu.
96. C lent D $500 Jan. 1st, 1852; when must he lend himn
I8500 more, so that he may receive $1200 on settlement, July
19tfi,:1856, money;:worth 6 % per annum?
AnOs. June 7th, 1854.
97. Wfhat is the least number of lb. that would make an
exact number of bu., either of wheat, corn, barley or oats?
(See note 2, page 83, 3d Book.) Ans. 3360 lb.
98. How many yd. of carpeting - yd. wide, will cover 2
floors, each 22 ft.' 6 in. square? An&S. 150 y d.
99. Cincinnati, is 7~ 18/' 30" west of Washington, at what
time could an event; be announced in C., that happened at
noon in W., allowing 4 min. 14 sec. delay in telegraphing?
Ans. 25:min. before noon.
100. If I lose 10. by selling cloth at $3.33 per E. En., what
w will I gain by selling the same cloth at $2.96 per E. Fl.?
Ans. 33~ o%.
101. If it:require 600 square tiles to- cover floor 10 ft. by
15:ft.; how large are the tiles? Ans. 6 in. square.
1:02. A man bought as follows, at'6 rmen. credit: Jan. 1.5th,
$234; Mar. 13th, $2M05; June 1sti, $128;- What does he owe
Dee. 31st? (6 %o) Ains. 8577.804
103. HIow many yd. of carpet, 32 in.a wide, will be required
for a 1roln 22 ft. square, leavinfg 4 in.: around each side uncovered? A ns. 565 yd.
104. A- wheel is 6 ft. in diameter; how many revolutions
will it make in going 10 mi. Anls. 2801.1]2-+
105. Hiow many cans 4 in. long, 4 in. wide, and 6 in. high,
can be filled from- 1 bu. 1 pk? Ains. 28.
106. What is the difference between Tv and.15? Ans. lQ,
107. If 317 mi. 3 fur. 2 rd. 2yd. 2 ft. 3 in. of railroad is
built by 1 000 men, how much does each build?
iAns. 2 fur. 21 rd. ~ yt. 3. in..o os in.s
TEST EXAMPLES.
108. I pay $2100 for a note having 2 yr. 8 mon. to run;
what is the face of the note? Ans. $2436.
109. A street 60 ft. wide crosses another 40 ft. wide, at right
angles; what is the distance between the opposite corners?
Ans. 72.11+ ft.
110. A note dated Jan. 5th, 1855, and running 90 days, was
discounted in bank, Mlar. 15th, 1855, and the holder received
$398.40; find the face of the note. Ans. $400.
111. At $28125 a mi., how much railroad can be built for
$100000? Ans. 3 mi. 4 fur. 17 rd. 4 yd. 10 in.
112. In how many ways could a class of 8 boys be arranged,
if one particular boy keeps the first place the entire time?
Ans. 5040.
113. Divide 2.24 ft. by 6~ in. Ans. 4.32
114. Having mixed 7 gal. at 43 cts., 9 gal. at 55 cts., and 14
gal. at 46 cts., how many gal. at 32 cts. must I add to make a
mixture worth 47 cts. per gal? Ans. 2 gal.
115. A man bought some apples 2 for 1 ct.; and twice as
many pears 5 for 1 ct.; he sold them 7 for 2 cts.; did he gain
or lose, and what So? Ans. He lost 4 1).
116. What is the cube root of 7 units of the 7th order?
Ans. 191.293+
117. A, B and C took in succession from 30 marbles; B took; of what A left, and C who took the rest, had 12 less than
A, how many did each take? Ans. A 16, B 10, C 4.
118. Howmany wk. in the 19th century? Ans. 5217 wk. 5 da.
119. How would 3 lb 3 ~ 3 3 3 gr. be expressed in Troy
weight? Ans. 3 lb. 3 oz. 7 pwt. 15 gr.
120. Divide 1 T. of sugar among 5 men in the ratio of their
ages, which are 57, 53, 51, 49, and 46 yr. respectively.
Ans. 1st. 4 cwt. 1 qr. 20 lb. 5 oz., 2nd. 4 cwt. 141 b 1 oz., 3d.
3 cwt. 3 cqr. 23 lb. 7oz., 4th. 3 cwt. 3 qr. 71b. 13 oz., 5th. 3
cwt. 2 qr. 9 lb. 6 oz.
121. Divide 5 T. 3 hhd. 51 gal. by the square root of the
ratio of 1 bu. 5 qt, 1 pt. to 42 bu. 6 qt. Ans. 3 hhd. 61 gal.
122. A bankrupt whose debts are $30000, can pay 20 o; if
he pays one of his creditors, in full, $2000, what % can he
pay the others? Ans. 142 %
MISCELLANEOUS. 271
123. A field containing 102 A. is 6 fur. in length; how wide
is it? Ans. 1 fur. 28 rd.
124. What is the area of a triangle, whose sides are 12 ft.,
35 ft., and 37 ft., respectively? Ans. 210 sq. ft.
125. How much would the fences around 4 separate square
fields, each containing 10 A., exceed in aggregate length, a
fence around a square field containing 40 A.? Ans. 1 mi.
126. A had $10 more than B; he then wvon $20 of B, when
he had twice as much as B had left: how much had each at
first? Ans. A $80, B $70.
127. For how much must I give my note at 60 da., so that
when discounted in bank, the holder may receive $237.48?
Ans. $240.
128. What do quills cost per thousand, on which I gain 20
% by selling at 9 cts. per dozen? Ans. $6.25
129. Which is worth the more; $600 due 5 yr. ago, or $1200
due 5 yr. hence, reckoning 10 So? and how much more?
Ans. the first, worth $100 more.
130. The time since Tuesday noon is y57 of the time since
Sunday noon; what is the time?
Ans. 8 o'clock Wednesday morning.
131.' of A's money —-7 of B's, and together, they have
$215; how much has each? Ans. A $105, B $110.
132. When must a note for $300 have been given, so that it
may be worth $501.20 on Oct. 1st, 1857? Ans. July 27th, 1846.
133. The time to Wednesday noon is i-4 of the time to Saturday noon; what is the time? Ans. 4 o'clock Tuesday morning.
134. What sum of money, when divided between A, B and
C, in the proportion of a,, and 4, will give C $22.50?
Ans. $66.25
135. The solidity of a sphere is 65.45 cu. in.; what is the
surface? Ans. 78.54 sq. in.
136. A town of 10000 inhabitants increased annually 10 %
for 4: years; what was its population at the end of the 4th
year? Ans. 14641.
137. Two cubes have their sides in the ratio of 4 to 5; the
solidity of the first is what o of the solidity of the second?
Ans. 51 0.
272 TEST EXAMPLES.
138. If 833 bl. of flour cost $402, what will 93-1 bl. cost?
IAns. 844S.
139. William spent 20 o of his money: he then spent 43i'
Yo of the remainder, after which he found that -- of all he had
spent was $24 less than the amount he had left; how much
had he at first? Ans. ~$200.
140., of A's money is $3 less than B's, alnd together they
have $28; how much has each? znls. A $15, 1 $13.
141.'What is the area of a floor, whose length is 18 ft. 2'
3//, and breadth 16 ft. 5' 4//? Ans. 299 sq. ft. 1/.
142. A lost 5 % of his money; he then gained 6 % on what
he had left, when he had $14 more than at first; what sum
had he at first?.Azis. $2000.
143. A dog ru-ns 6 rd., while a rabbit is making 21 leaps,
and the rabbit runs 4 rd. while the dog is making 9 leaps; if
the rabbit is 48 rd. ahead, and makes 7 leaps while the dog
makes 3; how many rd. must the dog run to overtake the
rabbit? Ans. 144 rd.
1.44. The driving wheels of a locomotive being 15 ft. in
circumference, the tender wheels 6 ft. 8 in., and the car wheels
10 ft. 5 in.; how far must the train run, in order that each
may mlake an exact number of revolutions? Ans. 500 yd.
145. A man owed $600, payable in 4 yr. 2 mon., he pays
$300 now; what sum should he pay at the expiration of the
time? Ans. $225.
146. What number is that, whose half is as much less than
30, as the number itself is greater than 30? Ans. 40.
147. x of the distance from A to B — 5 of the distance from
B to C; and 2 of the distance from B to C ( — of the distance
from C to D; from A to D is 88 miles; how far is it from B
to D? Ans. 52 mi.
148. John and William built a wall; John built 10 rd.
more than - of it, and William 5 rd. less than I of it; how
long was the wall? Ans. 75 rd.
THE END.