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1
2
3
1
2
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4
5
6
GRAFTON'S
GRADED ARITHMETIC
BOOK IV.
BY
E. W. AliTHY,
SUPERINTENDENT OK CITY SCHOOLS,
MONTREAL.
LOGARITHMS
BY
S. p. ROIUNS, LL.D.,
PRINCIPAL OF THE Mc(ilLL NORMAL SCHOOL,
MONTREAL.
Authorised for use iu, the Froviuce of Quebec.
E'RIOE, 25 OEOSTTS.
MONTREAL :
F. E. GRAFTON & SONS, PUBLISHERS
1897.
<p4 loL
i" ti.e Office o, the Minister of AgHcuIt.1 ' '™-^' * '°"^'
NOTE TO TEACHERS.
tlie year
'^' & 8o\s,
This book will be found to cover the requirements of
both the preliminary and advanced A.A. examinations
and of the examination for Model School diplomas.
The examples have been carefully graded, beginning
with easy, straightforward exercises, and leading up to
others which require some thought and ingenuity besides
the mere accurate manipulation of figures and tables.
It is hoped that the sections on mensuration and
logarithms will be found especially useful, not only for
preparing students for examinations in which a know-
ledge of mensuration and logarithms is required, but also
as furnishing an instructive sequel to the higher parts of
arithmetic.
No Teachers' Manual accompanies this book, as teachers
of advanced work generally prefer to use their own
methods. Ansvvers to the examples will be found at the
end of tlie book.
CONTENTS.
L.C.M. AND H.C.F.
Common Fractions
Recurring Decimals .
Ratio and Proportion
Proportional Parts
Compound Proportion
Averages
Work and Time
Distance and Time
Exchange .
Commission .
Insurance .
Taxes and Duties
Stocks
Business Exercises
Weights and Measures
Metric System .
Powers and Roots
Mensuration of Plank Fig
Mensuration of Solids
Test Exercises in Mental
Test Problems
Tables
Logarithms
Lcgarithmic Tables .
Answers . , ,
URES
Arithmetic
page
1
4
10
15
IS
20
22
23
24
26
29
30
32
34
39
46
49
, 53
56
61
. 70
. 73
. 76
. 78
. 109
. 125
b3
cU
itJ
di
til
di
di
FRACTIONS.
A number is divisible by 2 if its units digit is divisible
by 2.
A number is divisible by 3 if the sum of its digits is
divisible by 3.
A number is divisible by 4 if the number expressed by
its two hivest digits is divisible by 4.
A number is divisible by 5 if its units digit is 5 or 0.
A number is divisible by 8 if the number expressed by
its three loivest digits is divisible by 8.
A number is divisible by 9 if the sum of its digits is
divisible by 9.
A number is divisible by 11 if the difference between
the sums of its digits in alternate places is either or
divisible l)y 11.
Tell at gight whether the following numbers are
divisible by 2, 3, 4, 5, (J, 8, 9, 10 or 11 :--
1. 288. 4. nr>. 7. 4851. 10. 16500.
2. 960. 5. 048. O. 3520. 11. 40040.
3. 912. 6. 495. 9. 2200. 12. 33165.
FRACTIONS.
Find the L.C.M. of :— -
1. 7, 3, 56, 49, 21.
2. 35, 45, 55, 65.
3. 19, 38, 15, 20.
4. 12, 15, 16, 21.
I.
5. 120, 144, 24, 96.
0. 10, 14, 15. 21, 28.
7. 22. 26, 30, 32, 39.
8. 96, 110, 120, 121.
Find by factoring the H.C.F. and L.C.M. of :-,
9 66, 99. 13. 720, 840. 17. 60, 75 90
10. 56. 140. 14. 37, 518.
11. 51. G8. 16. 205, 287
12. 45, 81. 16. 230, 414
Find the H.C.F. of :—
21. 14112 and 22176.
22. 18776 and 49287.
23. 735 and 1176.
24. 1065 and 16614.
18. 26, 52, 65.
10. 24, 60, 72, 108.
20. 19, :iS, 57, 114.
25. 8874 and 17697.
26. 2079 and 33638.
27. 567, 1674 and 4041.
28. 354, 876, and 3645.
Reduce to lowest terms without using H.C.F. :
29. t^.
35. r\\.
41.
7 6.-}
1 4 1 T-
47.
1 9 5 0'
2 2 100 0-
30- tW.
36. iff.
42.
111
48.
1 5 <»0
18 SF"
31 « 3
37 i'5'J
*i(. yep
43.
4 5 5
49.
9 9 it
10 IT.
3 Q -J •■*
3ft •** '»
44.
17 7 5
2 ;f 5 •
60.
_8 7
1 7 4 IT'
33 i'''0
45.
9 2 5
10 2 5'
51.
-JUL-'
3715 0.
34. 2 12
4.n 7 14
^^- ¥18«
46.
4 8.'}
9 2(r.
62.
116 5
15 8 50.
Heduce to lowest terms,
using
H.C.F.
•
53.
ti^. 59.
r> » 1
T5T6"-
65.
i
119 7
ITTT*
54.
infV 60.
17 7 5
T9 2^¥
66.
7 9
4 8 19"
66.
tVtV 01.
tH-
67.
8 30 7
T2 9 IFO".
66.
UH- 02.
7 03
4 lOT.
68.
124 23
67.
.'J i» 1 ;$ «Q
4 7 2
90 6 a-
69.
13 6 9
68.
T5T¥. 04.
!> 9 9
2 8 a 5".
70.
9 664
¥F¥T'
Define ])7
<me factor, H.C.F., L.C.M.
f
FRACTIONS.
, 96.
n, 28.
52, 39.
, 121.
90.
65.
72, 108.
57, 114.
697.
638.
d 4041.
\ 3645.
Liis 0.0.0'
2 2 1 IT'
I 5 <«
18 5F*
_i> St
lITTT-
AT
I 7 4 U"'
_S 7 5
JTTr. oo-
LJ-fi 50.
: 5 S 50'
3
If-
O
IT-
II.
MENTAL PRACTICE IN SIMPLE RULES.
(Where J or J or j occurs in the miiltiplicand, instead of adding two
ciphers, add 50, or 25 or 75 respectively. )
1. Multiply by 5 :—
76, 384, 9875, 671, 86|, 91i, 167J. 860^.
2. Divide by 5 : —
16, 37, 84, 356, 685, 769, 876, 687i, 769f.
3. Multiply by 25 : —
100, 350, 4750, 6845, 65J. 84^, 32f, 76f.
4. Divide by 25 : —
75, 375,^875, 764, 879, 989, 2375, 5875.
5. Multi})ly by 12i :—
840, 845, 1845^ 1840, 78i, OJ, 5}, 9i.
6. Divide by 12i :—
75, 750, 1250, 25, 125, 78, 85, 68, 79.
7. Multiply by 125 :—
60, 56, 560, 565, 1565, 76i, 9^, 75f.
8. Divide by 250:--
1250, 3450, 4750, 8975, 2525, 932, 6845.
0. 19 X 18 ; n X 36 ; 48 x 17 ; 49 x 19 ; 68 x 25 ;
7x Jof81; 8xiof 80; 9x ^rof 117; 12 x f of itself;
168 X I of 36 ; 8i x 17 ; 9J- x 28 ; 8J x 25.
10. Give tlie prime factors of: — 84, 85, 261, 117, 143,
176, 484, 210, 441, 378, 963, 1605, 292.
1 1. Find the H.C.F. of :— 6, 12 ; 9, 15 ; 84, 91 ; 49, 259;
85, 51; 165,300; 16,84; 29, 261.
12. Find the L.C.M. of :— 3, 4, 5 ; 6, 8, 10; 6, 10, 15;
7,14,21: 7,11,13; 9,10,12; 20,16,24; 25,50,75;
5,6,7; 7,8,9.
I
* FRACTIONS.
1 . To add or subtract common fractions reduce them to
similar fractions, and then add or subtract the numerators.
2. To multiply a common fraction by an integer, either
njultiply the numerator or divide Mic denominator. To
divide a common fraction by an integer, either divide the
numerator or multiply the denominator.
3. To multiply a common fraction by another common
fraction, multiply the numerators to find tl)c new numera-
tor, and the denominators to find the new denominator.
4. To divide by a common fraction, invert the divisor
and multiply.
5. To change the form of a common fraction without
altering its value, multiply or divide both numerator and
denomiiiator by the same number.
Find the value of :-
III.
2- ;: + ;o + i?+i^ 4. TiD-f-iyi-M + isii.
6. 8i + ->G;j + 7J-f:574+12;'4-47^+0S,V-h20.
6- 68,^-1- U4- 724- il +0;, + /v+ 11)^ + 12^
7. 8-(] -v. 10. 8i;|-4,V,. 13. li:{,^^-5(),V..
8. ;^-i^,y. 11. io]ii-(Hv- 14. 1(;|^.;-i:;i;m;
O. 50;^,»10. 12. 132ii-8J^. 15. ll8Hi~«^j.
16. 3-2,'V + i.
17. 4,^,-;;<;,-H|.
18. 8J-2-i-;],V
19- i-j4-^"-A.
or»
21.
22.
+
I rt
I H-
23. lO-HJ-i + J.
24. ri-f.;;,i-4-f-v-5.
26. 8^-f ^4.^_:5j^.
27. 7+i-fl-i-J-
I fi-
I M
+ h-!^^- 28. A + ,\-(j + ^.).
10
-f
■1 —14-4
I r,
29. 7-(m--'i)-(i;-})-
FRACTIONS.
3 them to
30.
Tierators.
31.
er, either
32.
itor. To
33.
vide the
34.
35.
common
42.
iiuinera-
43.
luttor.
44.
e divisor
45.
46.
47.
without
ator and
48.
49.
50.
51.
52.
53.
54.
Hh 1
55.
20. 1
56.
1
57.
■
58.
^ill'f 1
59.
60.
fflJ. 1
61.
1
62.
^1
63.
1
64.
1
65.
1 tj ■
1
1 1
WX^X
12
X
1 4
X
G^x
/ o
1
1 If
X 5,
i 1 5 X - X J 1 ,y X 1
51 xl:!xl*^xi.
36. lfof^ofVx6Jx
37. lJof]Jx#of2.
38. 3].Vof|X.">lXT,V
39. lGJx§x23|xll.
40. 8ixUx2lx-.\.
It 1'
1 7
41
1 .s
V of n of 3GM X u.
JO
a4'
f !f X U X i X U X t', X 2§ X 14 X 12^: X 21 X J^
41 X ^ T X 1 1 i X U X .V*Y X Uf- X A X li^ X SI.
(;! + Kx(r + /.)-
69. (HS)^T.
70- (li}-U)^o.
(2iof^)-(Uofli). 71. (5-i)-
(Uxjiof-)
72. (IK^^li)^
li)iof-i»^of:5-15i. 73. ri| of f;-^(i + f).
5s of ^a + A of
74. (j + V+ 1 )
- 1 'J
ffufiof:]-.-,. 75. (U + V)-fiof24.
(Ux,Vof7)-fof5L 76. 12i-^(19i + 7).
2si^lGi
1:51
O.T-r.
-"^■2
1 4- i)
-f-14
Ttr-
TT-
7. lOi^-fU^
TT*
:. i
77. (fxe)~(|-i;).
78. (^x/,)^(a + ,y.
70. (11-f;J)-^(a + Hl^).
81. (A-^^)^(lxHxi)-
o*^- ' :i(I • :)0/ • Uo ao/'
83. (
,w
••in
14^1
I- _:_
1:i'
iXi-r4.
^>n-rGj.
i\\ of 1^4-2
1 J
11)
01
84 2^^(l]^l^+:!3).
86. (r.^^;J^)ofIof3.
86. (S^-^V)^-^of-i
87. (l-Mi)^(n ofG).
88. (7i-f:>^)^(^xt).
89. 1 -f 5; of 3.
90. ll.i of A -fUxi.
T(T
f
00. i
67. 1
68. Oiof Jij-r
91,
f 2 of I ^ 1 1 of
HT'
1 n
X i o
92. l.V.r-fMof-l
'lli
T¥*
«yTT'
93.
r 1
o
f "■> - .:-
ii'
FRACTIOXS.
Simplify: —
1. 10
3
-4
12^
2. -0
7
IV.
3. 1-^
1 «
1'
2
6. — » ::^-
7.
8-34
4_!L -^ 1 A
13f^
8.
6. -1-1
81 — 1-1-
_3 _. ThT
01 Toi
- ;3 T- - 4:
7-3i
« . - - 4- *>! ^ 11" 1 4
U X 2^ 1-^V • U- U 1 O;! 1
3
12. J of 31- " +/
•^ - 4i 1 1
>
14. f of ?|+f of 2f
t
2Gi^- 1^
16. -- ' -1
i- -L 1 1 _ 1
13.
33 . 1
15. ^«+ +^Lofli
1 4 4. :'• 4. O 7 _ ;}
;Ui- y J i
'*i2 -^ 3 J
1 " 'i' "^A
^11 1 J 1 oTo ^ r
18. ' " V '^i V -'•I >• ••ST
19.
1
I .-I •_'
1 T^ (f ^T
.'» 4
'5 1
20 '':i^ AX^^t
1251 --17
22. (;:- 5)^(^-5).
21 1;'. '^ 1 <r -^ ' 2 • ' A X 1 4'
~l'Hx/, + 2ii-M|
23 ^.fliof4i
* ,Vof lJof3i
24. 3+ ,^f"^y^-of 4^of 3f
oil 1 |» *■ •• '
-If — -l^iiO-
26.
A-?^fi
. il of i + #of 5
A+iVof ;''i -(i of i'i - i) ■ 9j - IS
26. (i+i'_ui + /.)_(RS) (J + A) , (J + s) (j + j)
(h-!,H\-l) (A-D(i-]) (.l-iHi-J)
FRACTIONS.
V.
1 . A post is }r in the ground, i- in the water, and 14
feet out of the water ; find the length of the post.
2. One-sixteenth of a certain number exceeds ^V of it
by 11 ; find the number.
3. Find the number whose eighth part and twelfth
part together make 10.
4. After giving away f, yV and /^ of my money, T
had Sol left ; how much had I at first ?
5. lieduce -^Ml to its lowest tarms.
6. What number, divided by 3f, will give 8 J for
quotient ?
7. Subtract half the sum of 5 and ^j- from S+J + f.
8. After spending y\ of my money, I lost i of what
remained, and then had S28 ; how much had I at first ?
9. If 1 _|_ ^i_ of a farm were worth $uO, what would
half the remainder be worth ?
10. Divide the sum of :]\ and 2?j by their difference.
11. Ex})ress § of f of 12 acres as a fraction of 17
square miles.
12. Find the difference between the sum and the
product of i; and 2 /o.
13.
41
Difference between ^ and 2^ ?
14. What number must be added to 4 J that ,V of the
sum niay be 47 j ?
16. K.\])ress .*5J inclics as a fraction of 4A yds.
16. Take i of I of £12. 12s. from 'j of J of £5.
17. Divide tlie diflerence between /v and 'l by the
product of 1 A and |.
18. I own 3 of a sliip and sell f, of my share for S914
how much is the whole ship worth ?
8
FRACTIONS.
19. A yacht wortli $900 and insured for $540 is lost;
i of the yacht belonged to A, l to B and the rest to C.
How much did C lose ?
20. Add 23|, li and 4 of 3^.
_4-_ **
9
21. Find the difference between the sum and the pro-
duct of §, -j'y- and -^%.
22. If a certain number were added to J of 12 J, the
result would be equal to J of 50^ ; find the number.
23. :\Iultiply the sum of J, J, J, ^^ and j-^ by the
difference between If and 4J ; and divide the product
by "'r
•^ 9 3
24. Which is greater, ^V of ^i or f of 8^%, and by how
much ?
* * 1 ^2
25. AVhich is greater, ^ or if and bv how much ?
26. Divide the sum of U\ and -- by their difference.
1*
27. Multiply the sum of i, «, § and V^ kv the difference
between \l and } i, and divide the product l)y f of |4
of 8J.
28. AYhat is the smallest increment which will convert
]^ + llyV + *, ^ iiito an integer ?
29. Find tlie product of two numbers whose sum is
19y'\r and dilfercnce o|.
30. Find two numbers whose sum and difference arc
5^ay and Ij; respectively; and divide the greater by the
less.
31. If I is worth $270, what is the value of ^\ 1
32. If ',] be added to each term of the fraction
value increased or diminished ? bv how mucli ?
r. »
FKACTIONS.
VI.
MENTAL PRACTICE IN COMMON FRACTIONS.
1. Eeduce to improper fractions: — 6^, T-J, 8/^, 9^V,
17i 18i 19|, 21J, 761, 68f, llli 142f. "
2. Reduce to mixed numbers in lowest terms : — V-,
XI S 1J0J5 ejUL SJlA -1^^ 2.0.0 14ii 4 4i ^_j_3. 1 «. 2.A
T J la'' 10' 1U>"140' 1 1 1 1' Tl 7 > VlO» 1 HJ lOOO-
3. Reduce to equivalents having a CD. : — h, §, -| ;
14 7.1 _."> _!!_ ■ 1 1_ 1 n XI. ' _J» _4_ _?_ • H. _7_ 1 1
3» "tT» U J '8' lU' li 4 ! lii' 15' li J 11' 2 2' .'J ;i ) 9» 24' 'SG'
A. Find llip «mTi nf • 6 I 5 i i . i s 9 1 Qn . i i !>
4. 1 Ilia Liie bum oi . — -y, §■, -g, ^ , --, j, ^~-, o- , ^j, -j j,
7 11- « « r»f j!* • H ^L nf ^ • S il 4 . _7_ i_ . _i_ _i_ .
TT' ^T i ' l>' 'J '• ' '/':}'-"•'.»? 3' 4' .■)"? 11' 12 * 12' 14 '
2 r. . 2 3_ 5_. t5_ fi_ _6__ J5_. 'T 1 Oi Ol • 1 fji 1 Q 1. OQ^-
r»' "2 7" 5 TTj 1 s' "2 7 5 i j' .j 0' 4 j' 7 5 J < w, <Jo, .^2 ; -"-^l' -'•^2' -'^4 '
17i 22f , 40f.
5. Find the difference between: — -J, Vt 5 iV' iV5 iV'
1 . 1 _1 • 'J 2 . 2. _2_ . .'L _••>_ • JJi 1 « • 71 '^1 • fi3 J.1 .
2 7' ) -iT' s 1 r :',> V. 5 7' 17 5 "14' 1 '.» 5 2 0' 1 Yt J • 2' "^3 J ^4' ^2 '
11!) ni . r.i 1 A 4 . 90 s 11
6. N ahie or : — .. X -^ X ,0 '■> -j ^ 1! '^ if '■> it ^ T8 ^ -^5 j
4A X (U X y^^ : ( H - ^) X 14 ; ^ of U of iV ; M of iVf of 1^ ;
Gx8x^] 18xl()xi4 'ix^X''><I^_2 1-1 xi^^
12x16' 5Gx:l6 ' " 6:3 X 36 ' 14:^x143'
7 V'lliiP nf • 2 _:_ 1 • 1 _!. 2 • ►")_:_ 'J • •". _:l a • r»l — '^1 •
• • > '^lillG 01 . y — .- , -5 — 5 J •-> — 1: , rr — d J U;;, — -t; ,
'> 1 -1. r> 1 • 7 1 _:_ S 1 • 14^1 nf ■"• • J nf - -:. - "•'
- i; ~ O ^ , < ;. — O ;- ; f 5 ~ 1' OI y , ;, 01 y — yr^.
8. Reihice : — $'do to the fraction of SI ; 12s. 6d. to
fraction of £1; 7^.11. to fraction of Is.; 1 hr. 30 min. to
iniction of a day; 3 (|ts. 1 pt. to fraction of a gal. ; 1 yd.
1 ft. to a fraction of a mile ; 4 lbs. 8 oz. to fraction of a
cwt. ; 3 oz. 5 dwt. to fraction of a lb.
>. Vahie of :— 4 of Sii^l; f of Is.; ^'V of S3; fV of 2s. 6d.;
I- of a ton ; f of a mile ; -jV of a lb. Av. ; iV of i^ Ih. Tioy.
10.
1. — 1
> 3
h of 16-^. ut
13; iof63J; 26HM-2i) + 3i.
10
Reduce to decimals :-
DECIMALS.
VII.
1.
2.
3.
4.
5.
6.
1
7.
1
"'J-
4
i)'
8.
1
9 0*
7
1 •^'
9.
1
It 0"O'
4
1 !•
10.
1
o
f5-
11.
I
'.) !• It i)
5
7 '
12.
TrTTTT-
I -*—
13
14
15. m
16. 2^V-
17. 1}^
18. 20"
•■J 0'
19. -4^.
20 ^'
^* 1 1
2 1 -^U.
22. -S-4.
23.
24.
)1 7 •
It it "
25. i^t.
26. yVV-
27. G^V-
28.
29.
30.
1 1 J.
:.' u •
- ■"* />_
Express as vulgar fractions in their lowest terms :—
(Examples 31-50 at sight.)
■6(i. 51. 3-(J0.S.
Oo. 52. 3-Gd8.
•OOo. 53. ;;5-G0cS.
•00a. 54. 10-G8i.
•6O0. 55. 3-02083.
■24. 56. O-2307G9.
-^4. 57. 4-71428;i
Oi. 58. -31200.
01. 59. -ooii^d.
Ooi. 60. 11-0G94.
Express at sight as non-recurring decimals :—
( ■!?= 1, and may be replaced l.y 1 in the next place up. )
VI. -0. 7 3. 5-9. 75. -49. 77. -G189. 79 -399
72. -09. 74. -009. 76. -029. 78. -G199. 80. 3-999.
31.
•1.
41.
•6g.
51.
3-G08.
61.
20^70S3.
32.
•oi.
42.
Oo.
52.
3-Gd8.
62.
-00d5o.
33.
..1
0.
43.
•005.
53.
3-G08.
63.
•03Gi.
34.
■6.
44.
•00a.
54.
10-G8i.
64.
21-003.
35.
•03.
45.
•005.
55.
3-02083.
Q5.
5-4iG.
36.
•23.
46.
•24.
56.
5-2307G9.
QQ.
] 7-009.
37.
•1:!.
47.
•24.
57.
4-714285.
67.
G-382.
38.
■{}(\
48.
•oi.
58.
-31200.
68.
8-643.
39.
•81.
49.
■61.
59.
•0012(i.
69.
17-316.
40.
•009.
50.
•ooi.
60.
11-0G94.
70.
2-005.
VIII.
Find the value correct to 5 decimal places of :—
1. 7-5 -f-G-8i + -908 + -2134. 4 -l--()9
2. 7-90 + -3410 + 3-245 + 1-8. 5. 1-13--5874G:;
3. 11 + 7-2 + -0814 + -0021. 6. 1-2-M709
DECIMALS.
11
7. -Sia + O-OG + T-OSi + 'OOTU. 9. -oiGt- •28634
. -829603 + -5632 + o9-037 + -06'92. 10. -6632 --0785196.
8
11. -Oox-dS.
12. -069ix-Ti63.
13. -o906x-07.
14. 1-284 X -0307.
15. -714280 X -361.
16. 11-072x5-086.
17. -11216 X -0637.
19. 3-03 + -58.
20. 1-27 + -037.
21. -03142 + -067.
22. 2-124+ -302.
23. -026 + -7890.
24. -3 + -1156.
25. -207 + 5-294.
26. 11-063 + 3-21
18. -0041 X -725.
Find the recurring decimal which is eqiual to
27. -936 + -71.
28. •7312 + -89.
29. -4187 + -306 + -12o.
30. 4-6 + '25i + -02ol4.
31. 2-001 + -1818 + -O.
32. 6-6-4-8.
33. 4-35-2-7.
34. 2-4O--08.
35. 5-314-4-67.
36. 11-213-4-689.
37. -5975x18.
3R. 75'19tx5-2.
39. 9-7x2-4o.
40. 3-6x4-09.
41. 3-:i7x 12-83.
42. -617 + -16.
43. 5-8 + 7-06.
44. 7-3 + 2-93.
45. 3-1.8 + 1-136.
46. -758 + 4i.
IX.
Find the value in compound quantities of
1. £7-83.
2. 2-1393 days.
3. 256-073 yds.
4. 4-45 miles.
5. -08;-) of an hour.
9. 6-878 of 2 sq. yds. 2 sq. ft
10. 5-83 of 1 lb. 8 oz. (Troy.;
1 1. -706 of 5 tons 11 cwt.
12. -3865 of 7^-sq. yds.
13. 3-998 of 1 yd. 1 ft. 6 in.
6. 5-2578125 of 8 wks. 14. 375 of 65 rods.
7. 31 of 90\ 15. -0764 of 9f days.
8. -06 of a rod. 16. 15-7] 96 of 3^ miles.
12
DECIMALS.
17. -6 of £4. 4s. 9(1 + -JG of £2. 5s. lOd.
18. -073 of a day + 3-75 hours.
19. -3 of a yd. + -3 of a ft. + -125 of 1 ft. 4 in.
20. -714280 of a week- -6 of an hour.
21. -09 of 40 rods +'01136 of a mile.
22. -03 of 1 11). Troy +-41(; of 1 oz. Troy.
23. -0710 of 554-4 tons +-428571 of b(j\hs.
Reduce : —
24. 3s. 4-kl. to the decimal of 19s. 6d.
25. 2 tons 3 cwt. to the decimal of 10 tons.
26. 3 oz. 2 dwts. to the decimal of a lb. Troy
27. 3 yds. 1 ft. to the decimal of -} mile.
28. 2 ft. 6 in. to the decimal of a yard.
29. 12 cwt. 56 lbs. to the decimal of 4 cwt. 50 lbs.
30. 2'} pks. to the decimal of 3 busliels.
31. 7} gal. to the decimal of 18 gallons.
32. 5° 13' 40" to the decimal of 12' 30'.
33. 124i ac. to the decimal of 121 sq. rods.
34. 29 ac. 120 sq. rods to the decimal of a sq. mile.
35. llf sq. ft. to the decimal of 4 sq. yds.
X.
, o. ,., 5-8-4-916
1. iMmpliry ~ ^-
2. Simplify
5-375-2-94
2-6 of -81
3-714285+4-125
o c- re 6-75 — 1-8
3. Simplify - — ; r
•583 of 1-6
4. Simplify \S-^
6. How many pence in -583 of a shilling ?
0. How many cwt. in -649 of a ton ?
DECIMALS.
13
7. Express 42 rods as the decimal of half an acre.
8. After spending f and -125 of my income, I am able
to save $117 a year ; wliat is my annual income ?
9. Express a day as the decimal of a year.
1 0. Express as a vulgar fraction the difference between
•(30;3 and -003.
11. What fraction having 24: for a denominator is
equivalent to '625 ?
12. Find by vulgar fractions sum of '98, -1)8 ^nd -08.
13. What fraction having 27 for numerator is equiva-
lent to •00375 ?
14. Express in lbs. the difference between -034375 of a
ton and -90025 of a cwt.
15. Value of -0324 of a mile.
16. Divide 1-02 by ^-i of -144.
17. Which is the greater, -765 or -760, and by how
much ?
18. What number multiplied by the sum of -(io-i and
•654 will give 1 for the product ?
19. How many times is 9-037 sq. rods contained in
244 ac. ?
-^ ^. ,.„ -00281 X -0625
20. Simplify — -1.^0- -—
21. Simplify (•5H--75)x(2-5--4)^(125 + -^J.
„^ ^. ,., 2-791()X 3-237
22. Snnplify - . .-,
1-861 X -80934
23. Subtract '0523 of 11 weeks from -932 of 6 davs,
and 2-ive the answer in minutes.
24.
^. ,.. -iof-3, 9-25
Simplify 1-^-,^ + ^^,.
25.
26. '^
xprcss in vards r03125 mi. --292-5 rods.
1- V
14
DECIMALS.
XI.
MENTAL niACTICE IX DECIMALS.
1 . Reduce to decimals : — I, }, }, i, f , f, t, to, fV' tV<
16? 13' »' T 'J> 5 if 5~ > 1150' 40*
2. Keduce to vulgar fractions: — '5, -25, -75, -125, -025,
•875, -3, -SIJ, -G, -606, '66, -78, ll, -li, -li, -864, -SOi
3. Add:— 6, -6, -OG, -OOG; 67, '67, G-7 ; 16, I'G, -016:
834, 8-34, 83-4 ; 7-G, 8-4, 9-3 ; 8-5, 15, -65 ; ^, -}, -8.
4. Find the difference between : — '6 and "6 ; '7 and -7:
•18 and -18 ; -06 and '06 ; 78-91 and 36-74 ; 81-17 and 34-34.
5. Multiply :— G-8 x 5 ; 7-4 x 2*5 ; 68 x "75 ; 7-5 x 7*5 ;
•66 X -6; 80 X -125; 72 x -625 ; 7Sx-13; 3-3 x -6 x -75.
6. Divide:— 6-75-^5; 6-75^-5; 32 -f -4; 25-^-05:
27^-000.); -056 -f--7; -4 -f- 1-6; -006 -f-015; 22-5 -f-09;
-0451 -^-ll; -0049 -^-035; 1-7-^5; 4-64 -f 20.
7. Value of :— 3-775 of SIO ; -1235 of 81000 : 4-625 of
4c\vt. ; 3285 of 4 mi.; 1-0325 tons; 13-775 cwt. ; 8s. -^
9-6 ; -75 of a lb. ; -72 of a ro<l ; '375 of a lb. Troy.
8. Iieduce: — 4 lirs. to decimal of a day; 2s. 7.U1. to
decimal of a guinea ; 3 pints to decimal of a gal. ; 3 oz. to
decimal of a 11). Troy ; 4 in. to decimal of a yard ; 4^- in. to
decimal of a foot ; 330 yds. to decimal of a mile.
9. One-tenth + one-hundredth + one-thousandth.
7-tenths -f 92-hundredths +'one-millionth.
7-tenths — 7-tliousandths.
17 — 17-tenths + 17-ten-thousandths.
10. i-f-2-fi + ll; 3x_3-125; 67*8 x ^- of ^- : 16^^
of itself; S;^ +S-33?,- + S^ ; 61-f 7-75-f 8^ ; 10-8-^3/;; -l-i-
84; lx-33^^2; 1-5x1-25; ^of 3x-3xV; •9x-9xH;
•3x'6x-7;'$15-i-2-25; S120-M-44; 8-5^2-5.
RATIO AND PROPORTION.
15
XII.
RATIO AND PROPORTION.
The relative magnitude of two numbers is called their
ratio, lialio is expressed by the fraction which the first
number is of the second.
The first term of a ratio is called the antecedent, and
the second term the consequent.
When two ratios are equal, the four terms are said to
be in proportion, and are called proportionals.
The first and last terms of a proportion are called the
extremes, and the two middle terms are called the means.
Test of a proportion. — Four numbers are propor-
tionals when the i:>7vduct of the extremes = the ijroduct of the
means.
If any three terms of a proportion are given, the
missing term may be found, for tlte product of the extreme?^
divided hif either mean gives tlie other mean ; and tlie x>rO'
duct of the means divided hy either extreme gives the other
extreme.
Find at si^ht the ratio of the following : —
1.
21 : 7.
7.
2 . T
3 • if
13.
1 gal. : 1 qt.
2.
7 : 21.
8.
5 . 1 r>
8 • 1 •
14.
30 rods : 1 acre.
3.
20 : 5.
9.
T • TT-
15.
1 c. yd. : 15 c. ft.
4.
21. : 10.
10.
.9 . .-L
- • 10-
16.
4 yds. : 9 in.
5.
r:2i.
11.
•5 : 2.
17.
6 dvs. : IG hrs.
6.
12.
U : 50.
18.
2s. Gd. : £1.
19. Is the ratio alwavs an abstract number ?
20. What is the effect of multiplying the antecedent?
the consequent ? Of dividing the antecedent ? the conse-
quent ? (Show by examples.)
21. What is the effect of multiplying or dividing both
terms by the same number ? (Show by examples.)
16
RATIO AND I'UOPOKTIOX.
Find the missing term in the following proportions
(Examples 22-30 at sight.)
22. X
6 : 10.
37. a; : 6 :: 31 : IS.
23.
24.
25.
26.
27.
28.
29.
30.
31.
/ :
5 :
(J :
8 :
4 :
1-4
26
32
,/; :
:
11
13
r'J !
38.
39.
T) , %Xj
23
: 20
50 :
40. 81
41. 8J
42.
43.
^,1 . . 1 '\
7i
44. 4-59
45. 1-02
: 4i : 17.
• l'^ . 1 •» •
9i : : lOti- : ->'.
•089 : ./J : : 24 : -0984.
1-333 : 172 w x -. -012.
10-8 : : -00300 : x.
•1
32. 51 :2^
33. 63 : X
34. 3 : X :
46.
47.
48.
49.
50.
51.
£10.
i 01 -g-
1 of 4i : X
1590 : 53
15t : 12 :: 46 :^
2-07 : -051 :: -69
17-15 : 6-32
: X.
S914 : X.
; X.
: : S75 : ./;.
^n;
X.
1-03 :./■
21.
:65
1710 : X.
1 :5
1 .->•
t>'.
1-87 \x.
21 : 12.
a: : 18.
: X : 65.
16 -. ^%.
: 42 : ^.
78 : 54.
: 8 : a;.
: 9 : 10.
::,.;: 9.
:9:7.
4 : 11.
35. X : 17 : : 5 : 9.
36. 12 : 7 : : 8 : :/;.
52. £1. 10s. : £1. 15s.
53. 5 (Ivs. 3 hrs. : 61 dvs. 12 hrs. : ;
54. llyV inches : hi}\ yards : : 854
55. 18 bus. 3 pks. : 2116 bus. 1 pk.
tQ. 19 J ac. : 11 ac. 60 sq. rods : : 79 tons : .'•.
57. 60-^ 37' : 360' : : 151 dys. 13 hrs
58. X : 3 qts. 1 pt. : : 84 : 80-10.
59. 8 oz. 6 dwt. : 13 dwt. 20 c^ra. \\x\ S0.44.
60. 34 dys. 16 hrs. : 17 hrs. 20 niiii. : : 83.60
61. Assuming the earth's tixcuuiicjrence to be 25,000
miles find its diameter, the ratio of the diameter to the
circumference being 113 to 355.
62. Find a fourth proportional to {ll) the sum, (//; the
d'ffcrence and (>•) the product of f and ^.
63. What would 225 yards of clotli cost at the rate of
£1. 17s. 6d. for 7i yds. ?
KATIO AND PROPORTION.
17
X.
64. If a loaf of "bread weighs 1 lb. 4 oz. when tlour is
S6.50 a barrel, what should a loaf of the same price weigh
when Hour is S4.25 a barrel ?
65. A workman digs out f of a cubic yard of earth in
\ hr. ; how long would he be occupied in excavating a
cellar 5 yds. square and 5 yds. deep ?
66. If the first-class railway fare from Montreal to
Quebec (180 m.) is S450, what should be paid from
Montreal to Toronto (333 miles) ?
67. A cubic foot of water weighs 62i- lbs. ; what weight
would a vessel 6 in. long, wide and deep contain ?
68. Pure lead is 11*3 times heavier tlian water ; find
the wei.dit of a block of lead 2 ft. 6 in. long, 2 ft. 5 in.
broad and 18 in. thick.
69. A train travels 7^- m. in 12-58 min. ; how far will
it travel in 5 hours ?
70. Find the yearly wages of a coachman who was
paid S78 for services from 10th April to 14th June.
7 1 . A clerk was engaged on 5th March at a salary of
S574I a vear. When he left he was paid SllSi ; on what
day did he leave ?
72. If I lent a friend S350 for 52 days, how long
ou"ht he in return to lend me $280 ?
73. On 20th Apr. a friend lent me $560 until 6th Aug.
I repaid the debt by lending him a certain sum from 11th
Oct. to 3rd Jan. How much did I lend him ?
74. 3000 soldiers were supplied with provisions for 87
days ; after 26 days so many men left that the provisions
lasted 300 days more. How many men went away ?
75. Find a fourth proportional to -98, -98 and '98.
76. Four-fifteenths of a ship's crew are able to do a
piece of work in 22 days ; how long would one-third of
the remainder of the crew take to do it ?
18
KATIO AND PROPOKTION.
XIII.
rr.oroRTioNAL parts.
Divide at siglit : —
1. S-1-8 into parts having the ratio 3
' >.
2. S?()0 into parts luiving the ratio '1 : 1 '^.
3. S2r> into ])arts having the ratio 2 : 3.
4. ??44 into parts luiving tlie ratio 7 : 4.
6. S«SO into parts proportional to the nnrnbers 0,7 and S.
6. J?28 into parts proportional to the nnniljers 4, 3 and 7.
7. SIO into parts proportional to h and J.
8. S(>4 into parts proportional to I and I.
9. S^IT) into parts proportional to "o and ^-'i.
10. S<)0 into parts having the ratios: {a) '1\:\\ {h)
5:1:4: (r) 7:8; (>/) J : J ; (/) 11 : : 10 : (/) "7 : -3.
11. Divide S140 among 3 persons in the })roportion of
(«)5:7:8: (/.) 10:11:14: (r) ^ : J : §.
12. Divide S06 between A and B so tliat
(«) A sliall liave 814 less than B.
(?>) A sliall have $20 more than B,
(c) A shall have i^o more than twice as much as Tl
(d) If A had S7 more lie would have twice as
much as B.
13. Divide .S4r> amongst A, ]> and (' so tliat
(a) A shall have twice as much as B, and C S?."t
more than ]>.
(h) A shall have twice as mudi as )'>, and C J^fl
less than ]>.
(c) A sliall have $10 more than B, and C ludf as
much as ]>.
((/} A shall have S.' more than twice B's share and
C iialf of .V's share.
RATIO AND PROPORTION.
19
1 4. Divide $105 among 4 men, 7 women and 5 children,
so that a man may have as much as a woman and child
tocrether, and a women three times as much as a child.
15. Divide 8155 among 8 men, 2 women and 10
cliildren, so that a man has half as much again as a
woman, and a woman half as much again as a child.
16. Two persons contribute respectively 81010 and
811 50. If the profits are 8852, what should each receive ?
17. Two copyists are employed in transcribing a manu-
script, and they copy respectively 224 and 288 pages. If
8140 is paid for the work, what should each receive ?
1 8. In l)uilding a wall one mason works -i of the time,
another J, and a third full time. What ought each to
receive out of 8340 ?
19. If 8 men or 12 boys can do a piece of work in 20
days, hi what time can 12 men and 8 boys do it ^
20. How long would 15 boys be doing a piece of work
wliicli 8 men can do in 21 days, the work of 7 men being
eiiuivalcnt to that of 10 boys ?
21. Divide 078 into three parts, sucli that tlie second
shall be ;q of tlie first and tlie last }, of the second.
22. A and W hire a pasture for 880. A puts in 4 yoke
of oxen and li i)uts in 40 slieep. If an ox is e(|ual to 8
sheep, liow nnich ought each to pay ^
23. A, 1) and (" enter into partnership with a comlmied
cai)ital of S(;0,()00. At the end of a year A's share of tlie
l)rolits is 82()00, IVs share 8.'U00 and ("s share 84000.
Wliat capital did each invest ?
24. If 15 men c;..i do as much work as 21 boys, how
long will 25 nu'u take to do what 30 boys do in 11 hrs. ?
25. A's rate of working is to IVs as 4 to :5, and IVs is
to C's as 2 to 1, How hni"' will it ta.ke C to do what A
would do ill davs i
20
COMPOUND PROrORTION.
XIV.
COMPOUND PROPORTION.
Eesolve by cancelling ; —
6 :
4 : : 20
5.
17:
51::S13
8 :
9
40 :
135
/ :
1 2 : : G
6.
18^
t
: 49 : : o lbs. o§ oz. Tro
:
14
28
:43
ft
• 4 • • 1(7
7.
l;3 :
24 : : 52 weeks
o
' ')
3G :
1 1
1 -J
■~i
3G5 :
lOi
•9
: -8 : : '02
8.
: 22 : : 35 yds. 1 ft.
•8
It^^
: 2
"/
: -G
1^
taw
: t
9. If 15 horses consume 3G0 bushels in 252 clays, how
many horses will be kept with 220 bushels fur 154 days?
10. If 15 liorses consume 3G0 bushels in 252 days, how
many l»ushels will 8 horses eat in 210 days ?
11. If 15 horses consume .'{GO bushels in 252 days, how
long will 12 l)ushi'ls last 7 hcu-ses ?
12. AVorking lU hours a day, 14 men can finish apiece
of work in 12 davs: how manv nuMi working G hours a
(lav will be rcfiuired to finish it in .')5 davs ^
13. AVorkinif 2 hrs. a dav, 180 men can Iniild 480 yds.
of wall in 8 dys. : how many men, working 10 lirs. a day,
can build 45(1 vds. of the wall in 18 davs?
14. A man can do a certain ])iece of work in 18 days
of 9J hours each; in how many days of l()[ hours each
could 3 men do foui- times as much ?
16. If J^24n <'-ain<'d .^5:M in .".(i5 dnvs. in wliat time
would S225 njuu S3() at the same rate ?
COMPOUND PROPORTION.
21
16. 12 men can dig a trench 108 ft. long, 6 J ft. deep
and 3 ft. broad in 5 days; how long will 15 men take to
dig one 100 ft. long, 3^ ft. deep and 2 ft. broad ?
17. The cost of carpeting a room 19 ft. by 15 ft. with
carpet at 84 cents the yard is $67.20 ; what will be the
cost of carpeting a room 24 ft. by 20 ft. if the price of the
carpet is $1.14 a yard ?
18. A lends B $400 for 15 mos. at 4%; how long in
return should B lend A $1500 at 3% ?
19. A locomotive making 162 strokes per minute
travels 90 miles in 2 liours ; how many strokes per minute
must it make to travel 200 miles in 4:h hours ?
20. If 3 men or 5 women or 8 boys can weed 18 acres
in 9 days, how long would it take 5 men, 8 women and 3
boys to weed 109 J acres ?
21. If 42 lbs. of raisins cost £1. 16s., what would 35
lbs. of a liigher grade cost, 5 lbs. of the former being
equal in value to 3 lbs. of the latter ?
22. An ollicer wished to convey 80,000 lbs. of pro-
visions in 9 days ; at the end of 6 days, IC men having
been employed, 15 tons only luid been carried. How
many men would be required to carry the remainder
within the time specified ?
23. 12 women make 7 dresses in 8 days; in what time
could 15 girls make 5 such dresses, the work of 2 women
being equivalent to that of 3 girls ?
24. The travelling expenses of 7 touriots for 5 weeks
amounted to $1505 ; a second party of 18 made tlie same
tour in 6 weeks, tlieir average weekly expenditure })er
man being ^ of that of the first party. What were the
total ex]>eiises of the second party?
25. Kiflcen men do a piece uf work in eight days ; how
'HI
ni
nuy men could dn .-| of the work in } of the time ?
1
! 1:,
22
AVERAGES.
XV.
AVERAGES.
1. Ill a class of 24 boys 4 are 14 years old, 6 are Vol,
7 are 12 J- and the rest lU. What is tlie average age ?
2. A woman's income for 3 years is S250 a year ; for
the next 5 years it is $294, and for the next 4 years $307.
What was her average income for the 12 years ?
3. In an exercise set to 35 pupils, 1 has 7 mistakes,
2 liave 5, 4 have 3, 6 have 2, 8 have 1, and the rest none.
Find their average number of mistakes.
4. If a tradesman sells on the first 5 days of the week,
243, 117, 112, 195 and 207 yards respectively, wliat must
he sell on Saturday that the daily average may be 179 yds. ?
5. In the month of April a man sle])t 7 hours on each
of 16 nights, 6.1 hours on each of 8 nights and 5 hours on
each of 5 nights. How long must he sleep the last night
that his average may be 6 it hours ?
6. A candidate answers two examination papers, the
first of which is valued at half as many marks again as
the second. He gains 58% of the maximum marks on tlie
first, and 43% of the maximum on the second. What
percentage of the total marks does he gain ?
7. To 112 gallons of spirits worth 21 francs a gallon
a grocer added as much water as re(hiced the value to 16
francs a gallon ; what quantity of water did he add ?
8. A merchant has teas worth 54 cents and 44 cents
a lb. respectively, wlii'h lie mixes in projuirtioii to 3 lbs.
of tlie former to 2 lbs. r.f the latter, and sells the mixture
at 52 cents a lb. What does he gain per cent. ?
AYOUK AND TIMK.
9. In a wholesale business a certain number of clerks
receive $^)0 a week, twice as many receive $31.50, and
clever, times as many receive SU. The weekly pay-sheet
amounts to $1809. Find the number of clerks.
10. A man bought a herd of cattle, 136 in number, for
$3230; on the way home he lost 4, and 12 others, being
unable to complete the journey, were sold for $o a head
below cost. At how much per score must he sell the
others so as to gain $215 on the whole transaction ?
XVI.
WORK AND TIME.
1 . A can do a piece of work in 5 days, B in 6 days.
How long will they take if they work together ?
2. One pipe empties a cistern in 5 hours, another in
8 hours. In how long will the cistern be emptied if both
are open ?
3. One pipe fills a cistern in 4 hours, another empties
it in 8 hours. If both pipes are open, in how many hours
will tlie cistern lie filled ?
4. Two men together can do a piece of work in 5
hours, and one of tliem alone in 8 J hours. How long
would the other take to do it ?
5. Two pipes cim fill a cistern in 3^ and 4J hours
respectively, and a tliird empties it in 20- hours. In how
long will the cistern l)e filled if all the taps are open ?
6. The hot-water tap can fill a bath in 10 min., the
cold-water tap in 12 min., and the waste-pipe can empty
it in 8 min. If all three are opened, how long will it take
to fill ? And how long to empty again, if the hot water is
then turned oil'?
m
24
DISTANCE AND TIME.
7. A and B separately can do a piece of work ia 4i
hrs. and 2| hrs. A, B and C together can do it in 1 /jAj
hrs. How long will C take to do it alone ?
8. A and B together can do some work in 8 days, B
and C together in 6 days, A and C together in 6^ days.
How long would each take by himself ?
9. A and B together can do a piece of work in S-^^^
days, B and C together in 9^^ days, A and C together in
8f days. How long would each take by himself ?
XVII.
DISTANCE AND TIME.
1 . A and B are 6 miles apart, and walk at the rate of
4J and 3^ miles an hour respectively. How long will
elapse before they come together (1) if they walk towards
each other, (2) if they walk in the same direction ?
2 A walking 5 miles an hour starts from Bcaconsfield
for Montreal (16 miles) at the same time as B walking 4 J
miles an hour starts from Montreal for Bcaconsfield. How
far from Montreal will they meet, and how long after the
start ?
3. In the preceding example, if A walked 3J and B 5
miles an liour, where and when would they meet ?
4. A walking 5^ miles an hour gives ]i walking 3 J
miles an hour one hour's start. How long will A take to
catch B, and how far will he have walked ?
5. At 10 A.M. a train starts from Montreal to Quebec
(180 miles) at the rate of 48 miles an hour, and another
at 12.30 M. from Quebec to at the rate of 44 miles an hour.
How far from Montreal will they meet, and at what time ?
6. A takes 9 steps while B is taking 8, but 10 of B'.s
are ecpuil in length to 1 1 of A's ; which is the faster walker ?
DISTANCE AND TIME.
Zi}
How
7. A train starts from a terminus at 9 A.M. travelling
25 miles an hour. An express starts at 10.30 a.m. and
travels 43 miles an hour. At wliat time and how far
from the terminus will the express overtake the slow train?
8. In a mile race A runs at the rate of 6 yards a
second, and gives a start of 140 yds. to B, who runs hi
yds. a second. How far from the winning post will A
overtake B, and by how much will he win ?
9. An express starting at 3 p.m. stops first at a station
77J miles distant at 4.27 p.m.; the whole journey is 104
miles, and 15 per cent, of the time is expended in stop-
pages. At what time is the train due at the terminus ?
10. An express runs 303 •} miles in hours, making
one stoppage of 30 minutes, three of 5 minutes eacli, and
one of 3 min. What is its average speed wlien in motion ?
11. A man rode a hicvcle from A to I), a distance of
54 miles, at an average rate of 8 m. an hour. Another
nuin started from A on liorsehack h hour j>fter the
bicycHst, and arrived at B 15 min. before him. Find the
ratio of their speeds.
12. A hare pursued by a greyhound was 87 yards in
advauce at the start; l)Ut for every 8^ yds. whicli the
hare ran tlie dog ran 10 yards. How far liad the dog run
when the liare was caught ?
13. Wlien will tlie liands of a clock be together (o)
between 2 and 3, (/*) iK'twccii <» and 7, (c) Itetween 10 and 1 1 ?
14. When will the hands of a clock be opposite one
another {a) between 3 and 4, (Ij) between 7 and 8, (r)
between and 10 ?
15. When wid the hands of a. clock be at right angles
to one another (a) lietwcen 4 and 5, (//) between 10 and
11, (<•) l)etween 1 and 2.
16. Two clocks point to 2 o'clock at the same instant
26
COMPOUND PROPORTION.
on the afternoon of Christmas Day; one loses 8 snc. and
the other gains 7 seconds in 24 hrs. When will one be
half-an-hour before the other, and what time will each
clock then show ?
17. A watch which at 9.30 A.M. on Tuesday is 4 min.
8 .yW sec. too fast, loses 2 min. 45 sec. daily. What time
will the watch indicate at 5.15 p.m. the following Friday?
18. Two clocks, one of which gains 1 min. 12 sec. and
the other loses 1 min. 28 sec. daily, are set right at 11
A.M. on 1st May; on what day and at what hour will the
times indicated by them differ by 30 minutes ?
XVIII.
EXCHANGE.
1. If 23-85 francs are exchanged for $5, how many
francs will be obtained for $51.25 ? and how many dollars
for 17 -49 francs ?
2. If $4.95 are exchanged for £1 and £1 for 25-05
fr., how many francs will $33 yield ?
3. A person goes to France with £56, which hd
exchanges at the rate of 25 J- francs for £1. He st " '^0
days, spcniling 37). fr. a day, and cluuiges what he ha>
at the rate of 1 fr. for ^hd. How much English moii
will he liave ?
4. If 5 fowls are worth 3 ducks, 14 ducks worth 5
(feesc, and 3 L'eese worth 2 turkeys, what is the price of a
fowl when a turkey costs $2.10 ?
5. If 2 lbs. of tea were worth 3 lbs. of coH'ee, and 4
l])s. of coffee worth 21 lbs. of cocoa, and 7 lbs. of cocoa
worth 9 lbs. of sugar, and 20 lbs. of sugar worth 45 lbs. oi
raisins, how many lbs. of raisins would be worth 24 lbs,
of tea ?
MENTAL PRACTICE.
T
XIX.
MENTAL PRACTICE IN AVERAGES, PROPORTION, ETC.
1. Find the average of:— (a) 11, 17-25, 18i, 19f ;
(Z.) i, 100, 1000 ; (e)i, i,ii; (^) 6-3|, 4-U, 3-^.
2. What is the average attendance for a week in a
school which has the following for each day: 80, 82, 81,
83, 85 ? If the number of pupils on the roll is 90, what
is the average percentage of absence ?
3. The temperature at 6 A.M. was 39° Fahr. ; at noon
47"5 ; and at 6 p.m. 3G. What was the average for the day?
4. The average of three numbers is 7J ; the largest
and smallest together make up t| of the whole ; what is
the middle number ?
6. If 27 out of 1000 die annually in one town, and 2G
in another, what is the average death-rate per cent, per
annum in the two towns ?
6. At an examination there were 4 candidates at the
age of 19, 3 at 20, 2 at 21, and 3 at 23. Find average age.
7. Find the missing term in the following: — (a)
3 : 5 : : U : a;; (Z>) t : I'V : : f : ^; W 6 : ^ : • *^ : «•
8. iJiVide :—[a) $350 in the ratio of 3:4; (?>) £10 in
the ratio of 2 : 13 ; {c) $75 into parts proportional to 2, 5, 8;
{d) $21 into parts proportional to '25 and J.
9. One candle lasts 4 hrs. 20 min., another lasts 3 hrs.
15 min. What is the ratio of the first to the second ?
1 0. A man can do a piece of work in 4 1 days. What
])art of it can he do in 1] dys. ? What decimal ? What
per cent. ?
11. A window is G ft. 4 iu. high by 4 ft. 2 in. wide.
What is the ratio of the height to the width ?
12. Wliat i.s tlie ratio of 3 quarts to t gal ? G pks. to
5 bus. ? Is. to $1 ? ^v to •;? ? $3G.r,0 to $18.25 ?
28
I'ElUlKNTAdK.
' t
XX.
MENTAL I'llACTICE IX VRRCEN TACIEf^.
I. How much per cent, is 17 <>f 2;"); lU of ^'•oj ; 19.
of 20 ; :^a of 40 ; 21. of 24; U of ^ ; ^ of 12i ; 18 of 63;
*59of T:;; -O^of 1; 1 ?. of 5X ?
2 Express as a comniou fraction in lowest terms :—
2% ; 41% ; ^% ; G% ; 10% ; 20% ; 25% ; :M% ;^ 3:U% ; T5% ;
001%; l<is-%; 121%; 374%; 02i%; «T1%; !)-09%; 18-18%.
3. Tea is bought at 4s. and sold at :'.s. 4d. ; wliat is
the loss per cent. ?
4. Tea is bouglit at 4s. and sold at 4s. 8d. ; what is
the iLjain per cent. ?
6. In a school of 100 children 12r/, are absent; how
many are present ?
6. In a scliool of 78 pupils 20 are girls ; percentage
of boys ?
7. In a school of 88 pupils .Or. cannot write; what
percentage can ?
8. In a school of 27.0 pupils, per cent, learn Latin, 8
per cent, mathematics, and 20 ])er cent, geograpliy; how
many in each branch ?
9. The population of a town was 00,000 in 1881, and
and 70,000 in 1801 ; required increase per cent.
, 10. A bankrupt's debts are $000, his effects are $27)0 ;
how much per cent, can he pay, and how much per dollar?
I I. If every sliilling's worth of goods yield 2d. of gain,
what is gained on £77) ?
12. Sold goods for $200, by whicli $2o was gained;
find the gain per cent.
13. Sold at $0r and gained $12 ; find gain per cent.
14. What was lost on £25 worth of sugar bought at
r)d. a lb. and sold at 4.\d. a lb. ?
!:*
COMMISSION.
i>0
XXI.
COMMISSION.
(For Percentage, Interest, Discount, Present Worth, etc.,
see Book III., pp. 36-60.)
A Comxuision Merchant or Agent is a person who
buys or sells goods for auotlier. What he receives for his
services is called his commission.
A consignment is goods sent to an agent to sell.
The net proceeds are the amount left after the com-
mission and other charges have been paid.
1. A sold B's farm for $G750. He bought him a new
farm for $4825. The commission for selling was 4°/^ and
for buying 2"/^. How much should A receive ?
2. Wlio is the J^rincipal and who the Agent in the
above transaction? Is he a buying or selling agent?
AVhat are the net proceeds of each transaction ?
3. Find the commission {a) at 4A°/^ on sales amount-
ing to S34G8, {h) at oT/o o^^ '^'^'"^ barrels of apples at $2.25
a barrel, {c) at C|°/^ on a ton of wool at 87i cents a lb.
4. What is the amount of the sales {a) when the
commission at C^'Yo i^ ^^1^0, (h) when the commission at
3r/^ is $294, (c) when the commission at 1^/^ is $270 ?
5. Find the amount of the sales {a) when the net
proceeds are $4845 and the commission 5^/^, (/;) when the
net proceeds are $229.80 and the rate 37o, {c) when the
net proceeds are $15,250, the rate of commission lJ:7o>
with additional charges amounting to $G2.40.
6. A merchant remits to Ins agent a sum of money to
be invested after deducting his commission. Wliat sum
will be left to be invested {a) when the remittance is $7098
and the commission 4°/ , (A) when the remittance is $4908
oO
INSrUANCK.
i f
! ll
and the commission 4^;;. (c) when the remittance is $4454
and the commission 2V'//?
7 In the uh<.ve prohlems wliat represents t/,r base, the
pcrcfntar, the rate 7.. the ^nnomU, the dijjerenee ? ^
8. Make rules or formuUis for finding tlie coramimon,
the amount of sales, the sum invested.
0. A real estate agent receives $95 for selling a house
for $47r.O. What is his rate. of commission ?
10 AMiat was the cost of printing 500 copies of a hook
which was sold at $0.90 a copy, if the hookseller's com-
mission and charges were 34 per cent, of the gross receipts,
and the author's profit $135.90 ? ^ ^
11 An aoent, selling goods at 21 per cent, commission,
sent the consignor $1207.50 as the net proceeds ot days
sales. AVliat were the average daily sales ?
12. An agent sold goods for me amounting to $10 «G0.
He charged 2}. 7. commission for selling, and 2,^ tor
ouaranteeing liayment, and $37.50 for freight and storage.
How many barrels of Hour at $5 a barrel can he buy witli
the net proceeds, if he charges r/, commission for buying ?
XXII.
INSURANCE.
Insurance is security against loss.
The premium is the sum paid for insurance.
The policy is the written contract between insurer and
insured. . . . ,
1. What premium must be paid for insuring propeit)
(ft) worth $5000 at I per cent., (h) worth $800 at 1| per
cent., (c) worth $05,000 at f per cent. ? . ., ,
2 What is the rate of insurance (a) if $lo is paid for
insuring'$1000, (h) if $420 is paid for insuring $18,000 ?
INSURANCE.
31
3. What amount of insurance can be obtained (a) for
S40 at 2/^, (h) for $157.80 at U"/^, (c) for $187 at 2f /„ ?
4. In each of the above problems jwint out the base,
rate and pereentage.
5. Make rules or formulas for finding the 2>'>'^'>niiim,
the rate, the ammtnt of the jioliey.
6. Pind at sight tlie premium in tlie following cases :
Amount of Policy. Rate.
{a) $100 \ (Wj^
(h) .$1200 ' '2^7^ (9 examples.)
(e) $2000j [6='/^
7. Find at sight tlie amount of ])olicy in the following:
Premium paid. Rate.
(a) $15 1 ay
{h) $2.50 I '•>'
00 $10.50j
8. Find at sight the rate
Amount of Policy.
(a) $50
|--Vo
il2iV
({) examples.)
/o
in the following :
(b) $200
(e) $1000j
Premium paid.
($0
m
y
(9 examples.)
9. AVhat sum must be insured at 4 /,, so that tlie
owner may receive, in case goods worth $7.'j5 are lost, the
value of the goods and the premium ?
10. A dealer shi[)ped 1000 barrels of Hour worth $0.50
a barrel; for what sum must he take out a policy at 2.y'/^
to cover the value of botli Hour and premium ?
11. Find the premium for insuring 4840 busliels of
wlieat wortli $1.20 a bush, at 1^^/^ on •; of its value.
12. After 20 years' insurance, a mill worth $48,000,
and insured for J of its value at 2.^/, is destroved by tire.
Find the owner's loss, not counting interest.
M
TAXES AND DUTIES OR CUSTOMS.
XXIII.
A.— TAXES.
A Tax is money assessed upon the person, property or
income of citizens for public purposes.
1. What is meant by real estate, personal property,
assessors ^ t •
2 The rate oE taxation on real estate in Montreal is
17 for nn.nicipal purposes and 1% additional for school
pm-poses. Find at sij^ht the an>ount levied on properties
assessed at $1000, S2000, !?;:!000, up to *1 0,000 rospeet.vely.
3. When the rate of assessn.ent is l-^« '"'"^ °" ^J';
dollar, what will he jmid on a property valued at $U 000 ,
4 The schools cost S:!nVl'.r>0 a year, and the rateable
value of real estate is $81,0oT.50. What is the rate of
scliool tax ? • • i-i
5 A tax of $^>000 is to be levied tor repairin- tlie
road.. Tlie assessed value of the distnct is $2,242,000.
What is the tax on a farm valued at $07. >0 ?
6 Wliat is tlie valuation of a piece of property that
pays a tax of $273 at the rate :\\ mills on the dollar ^
7 If a tax of 1^2850 is to be raised, and tlie collector
receives 5°/, commission for collecting the taxes, what
sum must be levied ?
B.-DUTIES OR CUSTOMS.
Duties arc taxes levied by a government upon -nods
hiip<»rled from foreign countries.
1. Define t'lir, Irahn/r, hmtkage.
_ _ io duty on 1^-. caoPH n..i._ , .t-
tainin<M50 1))S., at 75 cents a lb. ?
DUTIES
OK CUSTOMS.
7^
38
3.
What
is the duty on GIO gallons of olive oil,
at
25
cents
a galloi
I, allowing 2°
'^ tor leakage
?
4.
What
is the duty
at r» cents a
lb. on 400 sacks
, of
coffee,
each coiitaiiiiiiiT Go
lbs., the tare
being 2°/ ?
the
that
LToods
6. Find the ad valorem duty at 2.V7o ^^^ ^^^ boxes of
raisins, 25 lbs. in a box, invoiced at SO. 12 a lb., the tare
being 3.V lbs. a box.
6. A lady brought from France 2 dozen pairs of gloves,
for which she paid G francs a pair. The duty was $2.25 a
dozen and 50°/^ ad mlorehi. What did the gloves cost her
in Canadian money, the franc being reckoned at %W)?i ?
7. A Montreal fruit deahsr received from Florida 8
boxes of oranges at $3, and 15 cases of bananas at $2.50.
Each box of oranges contained 250 and each case of
bananas 20 dozen on the average. If the dutv is 15°/ and
other charges $12.50, how much will be gained by sellinj"-
the whole at $0.25 a doz. ?
3. Find the duty at oQi"!^ ad ralornii on an invoice of
English goods amounting to ^1500. 10s. Gd. (£1. = .$4*.SGG.'.).
9. What is the duty on an invoice of china from
Vienna, the value of which is G420 liorins, at .'mS°' , a llorin
being worth $'407 ?
10. l{e([uired the duty and total cost of a case of
French silks, value 3500 francs, duty 50°/^ ad valorem, •diid
a case of velvets, value 28,000 francs, duty 50"/ , other
charges being 025 francs and commission 2.V7o» ^^ *^ ^*'^"c
is worth $103.
11. I)uty on a ea.se of woollen goods from Germany
invoiced at 8437 marks at 457o, a mark being worth $-2:58.
12. Wliat is tlie invoice cost of goods upon which $300
is paid in duty, if the duty is 257^ ad valorem f
34
STOCKS.
XXIV.
STOCKS.
An incorporated company consists of several persons
who are authorised by law to transact business as a single
'"tlolkt the name given to the capital ot incorporated
companies. The capital stock ot a con.pany is usually
divided into shares ot $100. The stock is said to be
at par when a shave ot *100 sells tor $100 m money; at
a premium, ^vhen it sells for more tlian $100 in money ;
Z at a discount, when it sells t.,r less than $100 in
money. Tl>e pi-miuui or discount is the dilierence
between the (|uoted price and 100.
Consols are Kn.^Ush government securities.
1. What IS meant by thm- per cent. >:lock at 6,:- .' is
the stock at premium or discount >
3. Kind (a) the market value of, (h) the annual income
arising from : —
{a) $01^0 T) per cent, stock at 80.
{h) $204 21 per cent, stock at 90.
(<•) $0000 o'l per cent, stock at 87^-.
(^/) $ir.r>0 ;» per cent, stock at Tr)^
(r) $1200 4 i)er cent, stock at 92^
3. Find tlic amount of stock obtained and the annual
income derived from investing: —
{a) $910 in 4 per cent, stock at 104.
ih) $ir.(j0 in 5 per cent, stock at 81 1- ^
(c) $1991.7'. in n per cent, stock at 9G'^.
4. Find tlie juice of stock when
(a) $4900 stock can be bought for $:i907.75.
(h) $2000 stock can be bought for $252;).2o
((') X089r. stock can
be bou'dit for £7050. 2s. 9d.
STOCKS.
35
5. Find the quantity of stock lield
(a) In the 5 per cents., if the income is S43.75.
(b) In the 3 per cents., if the income is $7.70.
(c) In the 5^ per cents., if tlie income is $91.
(r/) In the 4 per cents., if tlie income is £115. 4s. 2(1.
6. Find wliat sum of money must be invested to
derive an income of
{a) $400 from the Al ])er cents, at 75i'.
{h) $500 from tlie 4 per cents, at 00 L
{c) $:)00 from tlie 2.V per cents, at 57;.
{(1) £72. 15s. from tlie *v>^ per cents, at 05.
7. Find price of stock, if there is derived an income of
{(() $106.50 by investing $2560.4:5^ in the 3 per cents.
{},) $107.50 by investing $4374.62^ in the 4 per cents.
{r) $154 by investing $3432 in the 3.^ per cents.
(f/) £07. 10s. by investing £2302. l()s. 3d. in the 3 per cents.
8. Dctennine the rate per cent, paid by stock
{(() AVhen $3720 of stock yields an income of $139.50.
{})) When $2975 of stock yields an income of $133^.
(r) When $11 is obtained by investing $204 at 81.
(r/) AVhen $40.20 is obtained by investing $924 at 105.
9. What is the rate of interest on money per cent,
per annum
{a) AVhen 3» per cent, stock sells at 80 ?
(/>) When 3 per cent, stock sells at 84 ?
{!') AVhen 3.y per cent, stock sells at 75 ?
{(J) When A\ per cent, stock sells at 135 ?
10. Which is the better investment : —
{a) The 3.\ ])er cents, at 77 or 4 per cents at O:].} ?
(/») The 3 per cents, at 72 or 4 ])er cents. \\[, 90 ?
(r) The 3 per cents, at 82.1 or '.\\ per cents. 93.V ?
cents, at 90 ui
{(i)
V
P
((') The 4.V per cents, at 120 or 3..^, per cents, at 90 ?
36
STOCKS.
1 1 Find the chan^o in income caused hy transferring
(a) Sr.OOO from the -A per cents, at 90 to the o.
per cents, at 81. „
(/>) $7300 from the 3 per cents, at 00 to the o
per cents at 100 H. ^
(r) ^:>300 from the 3 J. per cents, at 89^ to the 3^
per cents, at 94^hrokerage i per cent, on eacli tra.^actiom
(,/) $880 from the 4| per cents, at 106 j: to tlie 4
««»fa .,<- O'V' hrokera^e I on each transaction.
per cents, at J-*^, lunivci.i.,^ j, ^
(.) $4900 from the 3^. per cents, at 109 i to the o
per cents, at 91^ brokerage i on each transaction.
12. How much 4° „ stock nmst be l>ought to give an
income of $320 ?
13. If !?niL'5 is i„vcsW in (i° „ stock at 1021, wliat
income will Vic olitained ? , , , ^ f
14 If a person buys .V,^ slock at I -O, what rate ot
interest does lie receive on bis money invested >
15. yind the sun. renuired for an investment m 4/„
stock at OS.', to produce an income of ^'MO a year.
16 AVbat must be tbc ,>rice of a 5\ slock m order
that a buyer may receive 0° „ on his i,.vcstmeut ?
17. AVhen :!°„ Onsols are .luoted at 101, what sum
nuist be invested°to yicl.l an income ot .EHOO ?
18 AVhat is the e.-cact interest on an mvestn.ent ot
$-,000 in 4.', per cents, at U+i fron, dan. I to March - .
10 If a' n,au buys stock at 177. l"'^"""'"' '"^f 1'"
cent, does he receive o« his investment, if the stock pays
a dividend of Sr„ on its par valued „,,.., , •,,
20. If S4 shares of stock, i-ayin;,' a 0°; dividend, yield
lOy on the money invested, what did tb.> slock cost ?
21 (;overnmeut bo.ids yieldin- *2-10 a year at 4,„
interest wee sold at H% ,,.em,um and the vroeeeds
invested in land :U *7r. an ac. Mow many acres bought ,
STOCKS.
37
sum
'ieUl
22. }\y selling 3 per cent. Consols at 102^] and invest-
ing the proceeds in a railway stock which pays dividends
of 7% per annum, a man finds tliat he can double his
income; what is the price of the railway stocl^ ?
23. One company pays 5.V% on shares of $100 each;
another pays 31% on shares of $10 each ; if the sluires of
the former sell at 151% premium, and the shares of the
latter at 22^7 discount, compare the rates of interest
- /o ' •■
which the shares return to purchasers.
24. By selling out £4500 in the India o"/^ stock at
112 J and investing the proceeds in Chinese 7% stock, a
person finds his income increased by £168. 15s. ; what is
the price of the latter stock ?
25. When the price of a 3 per cent, is 90, a person can
obtain an annual income of $1 more than he can if the
price is 07 ; how much has he to invest ?
26. A person has $2950 in 3 per cent, stock at 83 1 ;
wlien the stock has fallen 2i, he transfers his capital into
5% stock at 107| ; find the alteration in his income,
brokerage in each case being |.
27. I buy 3% stock at 89 J ; after receiving one half-
year's dividend, I sell the stock at 94^, and find that I
have gained $54 ; what sum did I originally invest ?
28. A person having 1,0:5:), 200 francs in French 3 per
cents, at 744 transferred to English 3| per cents, at 98|;
liiul the change in his income, the rate of exchange being
£1 = 25*2 francs.
29. Which is the better investment, stock at 25// dis-
count wliicli ])ays a half-yearly dividend of 4%, or money
lent at 10%, interest payable annually ? What % better ?
30. A ])orson sells out of the :» i)er cents, at 90 and
invPstK hi;-, money in 5 per cent, stock at par: by how
much per cent, is his income increased ?
38
STOCKS.
XXV.
MENTAL ri!4CTICE IN STOCKS.
1 . If the o per cents, are at 82, required iiu-onie for $574.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
o
O
• >
O
«->Tr
0:!
-4'
4
4
4
84,
81.
80,
89,
93,
96,
91,
87i-,
85f,
89,
" $270.
" $250.
" $924.
" $979.
" $465.
" $624.
$500.50.
" $350.
- $686.
" $1157.
Kequired the price of the stock when the
13. 3 per cents. i,nun $12 for $300 invested.
14. 2f " " $30 " $1000
15. 4 '' '' $27 " $600
16. o\ " " $42 " $1000
17. 3i '•' '• $18 " $560
18. 4i " " $20 " $400
What quantity of stock can be purchased for :-
19. $300 at 75.
20. $729 at 81.
21. $656 at 82.
22. $684 at 90.
23. $760 at 91.
25. $765 at par.
26. $6:')0 at 210.
27. $700 at 175.
28. $606 at 151?..
29. $806 at 100^.
30. $733 at 150.
31
24. $382.50 at 85.
1 receive 3=?- p.c. interest on my money by investin
m
4^- p.c. stock. At what price did I buy
BUSINESS EXERCISES.
39
XXVI.
BUSINESS EXERCISES.
A.— BILLS OF ACCOUNT AND INVOICES.
A bill of account is a detailed statement of merchandise
sold, or of services rendered.
An invoice is a detailed statement of merchandise sold
hy one dealer to another.
1. As clerk for Arthur Fitts and Co., you sell Mr.
Alfred Brown, on 15th June, 1897, 3 cases of torpedoes at
$2.20; 12 boxes of fire crackers at $1.62^; 3 gross pin
wheels at $1.35 ; 5 gross sky-rockets at $3.25 ; 2 dozen
balloons at $2.25 ; 45 Chinese lanterns at 9,^.
Copy and fill out the following bill of account, and
receipt it for the firm : —
Montreal, 15th June, 1897.
Mil. ALFRED BROWN.
To ARTHUR FITTS & CO., Dr.
2. Mr. R. W. Stuart has bought the following goods
of J. R. Bradley & Co. :—
Jan. 3rd, 1897.— 9.^ yds. flannel at 32^^ a yd.; 26 yds.
calico at A\y a yd.
Jan. 7th.— 23 yds. muslin at 81/; 18 yds. linen at 64/.
Jan. 18th.— 15 yds. ribbon (w. T.K ; J- do/, pairs socks
@ 42/ a pair.
Rule and make out a bill dated Feb. Ist, 18i)7.
Receipt the bill on 5th Feb.
iO
BUSINESS EXEliClSES.
3 Make out a hill for the following articles bought
duriiic. ]\larch aud April. Supply the names of buyer and
seller ; also the dates : —
23^ yds. silk ® SOf ; IJ yds. lace ® $2.40; G4 yds.
muslin @ ^^ : 8 spools silk @ 7f' ; 4 pairs stockings @
05/ ; 6 yds. linen at S:h^ ; i doz. collars @ S2.10.
4. An upholsterer charges S:>.V5 per day for repairing
some furniture. He sui,plies G lbs. hair g r>0<' a lb. ; 1 .
yds. plush (B S1.75 a yd. ; 3 papers tacks at 10^^ a paper;
cord, gimp, etc., Onc He woiks 4 days. Make out his
bill, supplying names and dates.
5. Boston, IVIass., 10th May, 1897.-Messrs. F. E.
Grafton & Sons, Montreal, bou-ht of Cinn & Co., 12 sels
New Headers (a $2.2^ ; in sets Kindergarten Ihawmg
r,ooks (« $1.75; 25 Science Keaders (a $0.00; ".O IMnc-
tical Arithmetic @ $0.90; IS Brand's IMiVsiology di
$0.50; 18 Standard Elocution (a, $1.20. Less discounts
of 257^ and 10"/^. Make out the invoice.
6 Ottvwa, 10th Oct., 1897.— Messrs. Bulmer & Co.,
Montreal, bought of ( lilmour & Co., 293,500 ft. pine at $42
per M • 132,000 ft. pine, third (luality, di $10 per M. :
425 250 ft.hendock fe: $22 per M. : 83,750 ft. basswood
(01 $20 per M. ; 48,G50 ft. elm fe, $72.50 per M. Less a
regular discount of 37 r/. and 57, additional for cash.
Make out the invoice.
7 Montreal 18th July, 1897.— Messrs. H. Morgan &
Sons boiudit of A. Chisholm & Co., 3 bbls. granulated
sugar @ $7.50 ; 17 boxes raisins 'ii $1.75 ; 4 kegs lard @
$2!l5 ; 15 lUs. spice at 1G<' ; 24 boxes oranges fe $1.40 ; 2
bacTs Java coffee C« $27.20; 2 bbls. syrup @ $18.50; 12
lbs. nut mess
for cash. Cartage, $1.50. Make out the invoice.
BUSINESS EXERCISES.
n
and
8. Examine the following form of account with
James Parker : —
Mr. James Parker, Dr. Mr. James Parker, Cr.
1897. I ! i 'I ^^97. I !
June 3 To 10 lbs. sugar @ 10c . 1 ' 00 ^ June 5 Uy 2 days' labour. . 4
" 2 lbs. tea (floSc.; 1 10
June 6 •' 3 ll)s. coffee (5) 30c . | | 90
00
What does "Dr." mccin ? What does " Cr." mean?
What does the item on the Cr. side of the account mean ?
Did he work for you or did you work for him ?
9. Eule and make out the following account with
John Wallace : — June 1 : He owes you $2.85. June 5 :
You sell him 2 qts. berries @ 12/', peas W, and 2^
lbs. steak @ 2^/. June 8 : He and two of his men work
for you 6 hours, each at 25;^ an hour. June 1) : You sell
him 2.V lbs. cliops @ 28f', potatoes 20/-. June 12 : He and
one man work for you ol hours, each at 25f' an liour.
June 15 : You sell him 7 lbs. lamb @ 22;^ and berries
28f'. June 21 : You sell him l^ lb. steak @ 28<^ and he
])ays you on account $o. June 24 : Yo4i sell him 3 pks.
])otatoes @ 30^', peas 12<', olives 35<', and 2[ 11)S. steak @
28<^ June 28 : You sell him 61 lbs. beef at ?Af, and
l)erries 30/'. How does the account stand at the end of
the month ?
10. Charles Harrison, gardener, owes you rent for the
month of May, $25 ; but during the montli he lias done
3.V days' work for you @ $3.25 per day, and has furnished
3 rose-bushes @ 75f^ 4 grape-vines (w 50<', 11 fuchsias @
30^^ 25 pansies @ lOf'. How does the account stand on
Ist June ?
11. You employ a plumber to put in a new kitchen
sink, '^[ake out liis bill for l.ibour done and materials
furnished, supplying names and dates.
42
BUSINESS EXERCISES.
12. Rule and make out the following account with
Benjamin Smith :— Oct. 1 : You owe him $6.24. Oct. 3 :
You buy of him 2 doz. apples @ 25^, I lb. coffee 19f, and
10 lbs. sugar @ ¥. Oct. 5 : You buy 5 gal. oil @ 10c,
gelatine 15^, 2 lbs. rice @ 9f^. Oct. 6 : You sell him 6
bush, potatoes @ 68^. Oct. 8 : You buy 1 lb. tea @ 60;^,
spice 40;^ wicks of. Oct. 10 : You buy 10 lbs. sugar @
Qf, cocoa 24^, and biscuits W. Oct. 15 : You buy coffee
W, and flour $1.40, and you pay $o an account. Oct.
19 : You buy 1?. doz. lemons @ 25^, 2 lbs. raisins @ 13^,
biscuits 20,^, 2 lbs. brown sugar @ 1f> and sell him ISJ
lbs. butter @ 28 f. Oct. 24: You buy 5 gal. oil @ 10f^
walnuts 20/', and 3 lbs. oatmeal @ 6/^. How does the
account stand on Oct. 31st ?
$730.
B. — NOTES, DIJAFTS, CHEQUES.
A note is a wriUen 2mmiisc to pay a specified sum at a
certain time.
Montreal, 3rd August, 1897.
Three months after date I promise to pay
Henry Webster or order
seven hundred and thirty dollars, at the office
of the Bank of Montreal, for value received.
John Cox.
Tlv^ above note is payable at the Bank of Montreal 3 months and 3
days a.ter 3rd Aug., or Nov. 6th. It is negotiable, but needs Henry
Webster's enilorsenient to make it transferable. It does not bear
interest.
1 . Explain the terms negotiable and endorser. What
does an endorser do by his act ?
2. Distinguish between bearer and order.
3. Jolin Cox discounts the note on Sept. 15th at 9%.
Whnt will the proceeds be ? (See Book 111., p. 56.)
BUSINESS EXEKCISE3.
4:3
4. A note for S4000, dated 3rd July, payable in GO
days, is discounted 1st August at 12%.
Write out the note, supplying names.
Find the day of maturity, the discount and the proceeds.
5. E'-amine the following note; —
S250.
Halifax, Feb. 10th, 189^
Sixty days after date we jointly and severally
promise to pay to the order of Mr. ]t*etcr Smith, tiro
hundred and fifty dollars, for value received, with interest
at six per cent. Mortimer Jack.
Richard Shaw.
If the words " and sevtrally " were omitted, the above note would
be a joint note, and eaoh of the makers would be responsible for one-
half of the amount. Eitlier of the makers of the above note could be
sued for the full amount. The note bears interest from its date at 67o
per annum. If tlie note is discounted, tl»e discount will be computed
on the face of the nots with the interest added.
6. James Allen borrows $337 from Eichard Lee, and
gives liim a note at 3 months, which for liis better
security is signed not only by himself, but by his friend,
Fred. Harris. The note is to bear interest at 7%.
Write the note supplying dates, and make it payable to
order. When will it be due and what amount will be due
at maturity ?
7. Eichard Lee (after holding the note for 18 days)
has it discounted at 87, and receives cash in return.
Endorse the note before discounting.
How much cash should Lee receive ?
8. A draft is a written order hy one permn on another
for the payment of a specified sum.
44 BUSINESS KXKHriSES.
SIOO. Montreal, IGth Oct., 1897.
At sif/hf pay to the order of
Mr. irUliam Duvuhou
ojic hundred dollars,
for value received, and charge to account of
To Messrs. ])avid Law & Co., Herbert Bond.
Toronto.
Tliis is a sight draft, and is paid on presentation. It nii'.st be
endorsed by \Yilliam Davidson.
^.QQ Montreal, IGth Oct., 1S97.
Thirty days after sight pay to the order of
Mr. William Davidson one hundred dollars, for value
received, and charge to account of
m A/T ^^ -IT . e n . Herbert IJond.
lo Messrs. Davul Law & Co.,
Toronto.
This draft must be presented to David Law & Co. for acceptance.
They accept the draft by writing across the face the word "accepted,"
with the date and their signature. When accepted the draft l)econies
a note, payable SO days after tlie date of acceptance.
9. A cheque is a draft on a haul', /layahle on dniiand.
Sir.O.To. Montreal, 2nd March, 1897.
THE BANK OF MONTREAL.
Pav to Thomas Nicholson, Esq or order
one hundred and jifty tVo dollars.
Eobert Linton.
Tills che(|ue must be endorsed by Nicholson before being presented
for payment.
10. Draw a cheque on the Molsons liank for $r)7.80,
supplying names and dates.
11. Montreal, March 18ih, 1897.— Messrs. Henry
Gold & Co. buy of E. I[. Dods 320 barrels of Hour at $r».20,
and give in part payment their cheque for $1000.
BUSINESS KXKKCISKS.
4:.
Make out Henry Gold's & Co.'s bill, and credit the
amount paid. Draw the cheque.
12. Montreal, ;5rd April, LSOT. — E. H. Dods draws a
draft at 30 days after sight on Henry Gold & Co. for the
balance of his account. The draft is in favor of Messrs.
R. Green & Sons, Toronto.
Write the draft and accept it for Henry Gold & Co.,
date Sth April.
13. Toronto, 8th April, 1897.— Messrs. R. Green &
Sons have the draft discounted at the Imperial Bank at
10^ discount, and receive cash in return.
Endorse the draft. What are the proceeds ?
1 4. Draw a draft, at 20 days sight, in your own favor,
and have it discounted at Molsons Rank at 8^, supplying
names, dates and amount. See that it is properly accepted
and endorsed. Find the proceeds.
15. Montreal, May 10th, 18!)7.— Charles Wood buys
of Messrs. Gillespie & Sons 10 hhds. sugar, each kn'tO lbs.,
at 4J/^, and 20 chests tea, each G2i lbs., at 53<^. On July
r)th Wood gives his cheque on Bank of Commerse in
payment. Write Gillespie's invoice and Wood's cheque.
16. Quebec, Aug. 19th, 1897. — Mr. S. Brush has on
deposit in the Quebec Bank $198.84. He draws from his
account $25.30 for himself. On the same day he deposits
^422.85, and issues a cheque in favour of James Hall for
$14.90. Write both cheques. How does his account stand ?
17. Three Rivers, Apr. 24tli, 1898. — J. Ross borrows
$425 from H. Bird, and gives his note at 90 days, drawing
interest at 8°/^, in payment. At maturity Ross pays $132
to Bird, and gives a new note at 2 mos., drawing interest
at 10°/^, for the balance. Ross pays the second note l)y a
cheque on the Quebec Bank, (a) Write the first note.
(h) Write the second note, (c) Write tlie cheque.
4G WEIGHTS AND MEASURES.
XXVII.
WEIGHTS AND MEASURES.
(See tables on p. 76, with notes.)
1. How many loads of coal, each 14 cwt. 96 lbs., are
contained in 5 trucks, each weighing 10 tons 8 cwt. ?
(Long ton.)
2. A dealer bought 600 tons of coal at S5.25 a long
ton, paid 75^ a ton for freight, and sold it for $5.75 a
short ton. Find his profit.
3. Sea water contains 2^7o of salt; what weight of
water would be required to yield half a ton of salt ?
4. Find the weight of a nnllion cent-pieces, each
weighing a quarter of an ounce.
6. Twelve tons (gross) of tobacco were sent to an army
of 1M,000 soldiers; how many lbs. should each receive?
6 A silk-worm produces 28 grains Troy of silk ; how
many must be kept to produce 112 lbs. Avoir. ?
7. Express in Troy weight the dilVerence between 140
lbs. and 2120 oz. Avoir.
8. Reduce the sum of :^00 lbs., o06 oz., 306 dwts. and
oOO grs. to Avoirdupois weight.
0. Find the dilit'erence in grains between the weight
of a pound of feathers and a pound of gold.
How many lbs. Troy are there in a long ton of gold ?
10. Express in lbs. Avoir, the weight of a nugget of
gold weighing 18 j- lbs. Troy.
11. A watch gains 13 seconds each hour; what will it
gain in a fortnight ?
12. How long would it take to count ten millions,
twenty thousand, three liundred, at the rate «•! uiio
huiulred and fifty per minute i
WEIGHTS AND MEASURES.
47
eacli
13. A workman goes to work at 6.30 a.m. each morn-
ing and leaves at 8.15 p.m. What does he earn in 4 days
if he is paid 18 cents per hour till 6 p.m., and after that
time 24 cents per hour, but loses Ih hours of working
time each day for breakfast and dinner ?
14. A train travelling 45 miles an hour continues its
journey for 2h Ins., stopping twice for 7 min. 30 sec. each
time. What distance is traversed ?
15. A train leaving at 8.05 A.M. arrives at its destina-
tion at 2.45 P.M., travelling on an average half a mile per
minute. What is the distance traversed ?
16. If sound travels at the rate of 1000 ft. a sec, and
if a gun is discharged 5 J- mi. away, what time will elapse
after seeing the flash before hearing the sound ?
17. If light travels 186,000 miles a sec, how long
would it take to pass from the sun to the earth, a distance
of 02,000,000 miles ? How long to go round the earth, a
distance of 24,800 miles ?
18. From a rod a yard long, pieces each 057 of an inch
long are cut off; how many such ])ieces can be cut off, and
what will be the length of the remaining piece ?
10. How many roi)es, each 24 yds. 1 ft. 6 in. will
reach to a depth of 204 fathoms ?
20. Ho AT often is a chain contained in a mile and a
half ? What is the length of a chain in yards ? In rods ?
21. How many inches long is a link ?
22. The distance between two places is found to be
1000 chains. Express the distance in miles, etc
23. The four sides of a garden are 25 chains 10 links,
15 chains 8 links, 24 chains 08 links, and 16 chains 4
links. How many yards arouud the garden ?
24. Find in acres and sip chains the area of a rectan-
[^\\\nr field 17<>0 links lonr^ and 1200 links broad.
i !
48
WEIGHTS AND MEASURES.
W
25. A path is 15 links wide and 88 chains 40 links
long. Find the area of the path in sq. rods.
26. From a field of 6 acres, a rectangular piece 4*25
chains long and 2 4 chains wide is fenced off. Row much
is left ?
27. One field contains 4 sq. chains and another is 4
chains square. How many acres in both together ?
28. Take *432 of an acre from 6 J sq. chains, expressing
the result in square yards and a decimal.
29. A school-room having in it 45 pupils is 26 ft. wide
and 10 ft. 6 in, high. How long must it he to give 250
cub. ft. of space to each pupil ?
30. How many loads of loam, each a cubic yard, will it
take to cover a quarter of an acre of land 2 in. thick ?
3 1 . Find in gallons the measure to four decimal places
of a cub. ft. of water, and its weight in ounces Avoir.
32. Find tlie weight of water in a cistern 10 ft. long
4 ft. wide, the water standing 18 inches deep.
33. What weight of water does the Suez Canal contain,
if it be 100 miles long, 100 ft. wide and 25 ft. deep ?
34. Find the weight of air in a room 24 ft. long, 20 ft.
wide, 10^ ft. high, if 100 cub. in. of air weighs ;»1 grains.
36 Find the value, at ],<' a lb., of the ice taken from a
pond half an acre in extent, if the ice is 10 in. thick and
one cubic foot weighs 58 i lbs.
36. A floating body displaces it.s own weight of water.
How many cubic feet of water will be displaced by a ship
and cargo weighing 10,000 tons ?
37. A ship sails 2" DV o!ic day, 2" 35' the next, 2 So
the next, and 2 20' on tbe fourtb. Wbat (bslnncc is
traversed, reckoning fiO mibis to a degree in that latitude^
38. A ship sails \-)^, 192, 1^7, 245, 241?, 203 and 22f>
knots (2000 yds.) in a week. Kind tbe distance in miles,
MKTUIO SYSTEM.
49
XXVIII.
METRIC SYSTEM.
(See Book III., pp. 13-23.)
Length.
M
m.
Km.
H
m.
])
in.
m.
1 = 10 =- 100 =:r 1000 .:= 10,000
m.
hn.
cm.
mm.
1 = 10 = 100 = 1000
the
Construct after
unit of measure beiuir tlie lih
instruct tl
tlie above model the table of capacity,
he table of weight, the unit being the gram.
The theory of this system is that a metre is the
10,000,000tli of a quadrant of the earth through Taris:
the litre is a cubic decimetre in volume; the gram is the
weight of -J ^,\,(j^ of a litre fdled with water at 4 C. ; and
the franc weighs 5 grams. Hence
1 litre=l cubic decimetre=l kilogram. (Book III., p. 22.)
1. A wagon is loaded with 5 boxes weighing respec-
tively 102;ur. kilos, :\7-9:] kilos, lUl^-Ol kilos, 9S49 kilos
and l^o'G kilos ; what is the weiglit of tlie whole load ?
2. A peison paid 24 francs for 15 m. oi cloth; how
much will he pay for To cm. ?
3. How many litres of vinegar can l)e put into 500
bottles, if 25 of them will hold 025 cl. ?
4. j\[;ike out the f:»lb)wing bill : — G doz. dinner plates
at i'i't centimes each, 84 cheese plates at 4*45 fr. a doz., ))
dislies at 1-25 fr. each, 5 doz. glasses at •(J5 fr. each.
6. Two bicyclists start from opposite ends of the same
journey, which is 102-5 Km. long, and meet in 7 hr. 42
min. One rides Vfi) Km. an hour faster than the other;
what is the speed of the faster ?
;o
METRIC SYSTEM.
KfVl
Wn 'il
6. Two trains start from Paris with an interval of
2 hrs. 21 rain. The first runs 48 24 Km. and the second
54-27 Km. per liour. At wliat distance from Paris will
the second overtake the first ?
7. A lawn 27 m. long and ir> m. wide has a walk
round it 2 m. wide ; what will it cost to cover the walk
with gravel at -oO fr. a sq. m. ?
8. Tf a pile of wood is 2 8 m. higli and 4 m. deep, how
lon^ must it be to contain 112 steres ?
9. Find the weight of water in a cistern 4 m. long, 3
m. vide and 2 m. deep.
10. How many HI. of water will a tank hold whicli is
880 m. long, G m. wide and 4.40 m. deep ?
11. A bin 4-5 m. long, 2 m. wide and :) m. deep will
hold how many litres of grain ?
. 1 2. Eacli ediie of a cube of br.ass is o cm. long ; find its
weight assuming brass to be 8 times as lieavy as water.
13. Find the weiuht of oil in a tank 5 m. x 4 m. X o m.,
tlie weight of oil being 92% of the weight of water.
14. What is the weight of a bar of iron 4-00 m. x 7
cm. X 1-80 cm., specific gravity iM'ing 7*8 ?
15. How many franc pieces will weigh as much as a
cubic foot of water ?
16. A swimming-bath contained r»2-70 HI. of water:
water was tlien allowed to run for 2.V hrs. from a tuj.
which supi)lied it at the rate of .*-.2 1. i)er min. Find the
(piantity and weight of water then in the balh.
17. A family burned GOO kil(»s of c«)al in 40 days. If
a HI. of coal weighs 10 kilos, and 28 HI. cost oG fr., what
was the cost of a day's fuel ?
18. A cubical vessel 40 cm. on an edge is full of water.
n 14 1. are drawn oil and re}»laced ^.y a li(|uid wiiose weight
!
IS
I tliat of water, what is the weight of the mixture ?
METRIC SYSTEM.
51
walk
walk
Metkic Equivalents.
1 laetre = 39-o7 inches. 1 litre =1-76 pints.
1 kilometre = -6214 mile. 1 liectolitre =22-01 gal.
1 S(|. metre =1'106 sq. yds. 1 gram =15*4.'>2 grains.
1 hectare =2471 acres. 1 kilo = 2-2046 lbs. Av.
1 oil. metre =l-o08 cii. yds. 1 metric ton = 1-1023 tons.
AlTROXIMATE EQUIVALENTS.
Metre
Kilometre
S(|, metre
Hectare
= 1-1 yds.
= ^ mi.
= 11 sq. yds.
Cu. metre =l-o cii. yds.
Litre = 1^ pints.
Hectolitre = 22 gal.
— 01
acres.
Gram
= 1
grains.
Cu. centimetre =j^g- cvl inch. Kilogram =21 lbs. Av.
19. Express a metre in yards; an inch in cm.; a yard
in metres ; a mile in Km. {S places of ihrimnh.)
20. Express a sq. metre in sq. inches; a sq. inch in sq.
cm. ; a sq. yd. in sq. m. ; an acre in hectares.
21. Express a cu. cm. in cu. in.; a stere in cords; a
cu. inch in cu. cm., a cu. vd. in cu. m.
22. Express a pint in litres ; a gal. in HI. ; a bu. in 1.
23. Express a mg. in grains; a grain in grams; an oz.
Av. in grams ; an oz. Troy in grams ; a lb. Av. in kilos.
24. How many nules and rods are there in 30-675 Km. ?
25. Change 3 cu. yds. 4 cu. ft. to cu. m.
26. Express 10 cu. ft. 156 cu. in. in litres.
27. Ex})ress 1*S00 gal. in cu. m.
28. Change 1525 eg. to oz. dwt. grs.
20. Change CI 2-5 dl. to cu. ft. ami cu. in.
30. Find as tlu^ decininl of an inch the dillerence
between 12 inclics and 30 cm.
31. If a cul)ic foot of cj1j»ss weish 156 lbs,, find in 11)S.
and also in kilograms the weight of a pane of glass 24
inches long, 20 iuclies wide and I inch thick.
52
METRIC SYSTEM.
32. xV train goes at an average rate of 5 Km. in 6
minutes. How long will it take to go from Montreal to
Ottawa, a distance of 1 20 miles ?
33. Find the weight in kilograms of a rectangular
block of gold 8 inches long, o inches wide and 2 inches
tliick, gold being 19| times as heavy as water.
34. A metre being oOoTl inches, show that a kilo-
metre is -621385732 of a mile.
36. Express a millimetre in in. and a kilometre in ft.
36. A kilogram is ecpial to 2-2046 lbs. Av., and a
cubic inch of water weighs 252-0 grains. Hence find the
number of cubic inches in a litre.
37. A litre of air weighs 1-3 grams; how many grains
will a cubic foot of air weigh ?
38. in 25 kilograms how many lbs. Troy ?
39. A vessel full of water weighs 5*25 Kg. ; the weight
of the vessel when empty is 250 g. How many litres will
the vessel hold ?
40. A piece of iron weighing 100 lbs. is made into a
bar 5 cm. wide and 2-5 cm. thick. What is its length if
the specific gravity of the iron is 7*5 ?
41. A bottle empty weighs 10,000 grains; full of oil it
weighs 1075 grams. What part of a litre will the bottle
hold if the specific gravity of the oil is -9 ?
42. Mention the standard units in the metric system,
and explain how each of the other standards is derived
from that of length.
43. How was the standard of length determined i
44. Express the metric standjirds in British denomina-
tions, correct to o places of decimals.
45. Wliy is the system called INIetric ?
46. What is its great superiority over every other
system of weights and measures i
POWERS AND ROOTS.
53
'
XXIX.
POWERS AND ROOTS.
A. — SuUAKES AND CUBES.
Find the S(|uare of
1. 41.
o
4. 3. 025. 4. 01. 6. 24.U 6. 50-24.
Find the cube of : —
7. :u. 8. l-i:j. 9. 1005. 10. 1:1. 11. iv". 12. ;;i
1 'J-
5 'J 8-
'O.
Find the value of :-
13. 51*'. 14. (21)'.
15.
17''.
16. (;;/. 17. {l\f.
18. 2:;-+l^' + ^'. 19. l-O^P-l-O^;-. 20. ^572-5^-36^.
21. ;;8''+17'-18l 22. 39"^x4S=\
23. (307H 307') -f 30700. 24. (1 5'^ - 1 -iW) -^ 15.
25. 1-03 (4-07 + 3-l())'. 26. 3000 (M x •031)''.
K — Square Koots.
Extract the square root of: —
1. 576. 2. 1681. 3. 1156. 4. 4761.
5. 18760. 6. 18;;iS4. 7. 10404. 8. 266256.
9. 1-502644. 10. 234-00. 11. -0005,3361. 12. 1073-741824.
13. 42|. 14. 30iV 15. 5021. 10. 113!-^.
17. \ii. 18. A^'V. 19. .^.^yVV. 20.
1 .•{ 8 4
(5" j'j'y (T*
^81-4.
v^-OOOlH).
Find to three decimal places the value of: —
21. V2. 22. VO. 23. V706- 24.
25. V4;:|. 26. x/8.V 27. s/ i. 28.
29. ^328 + ^7 + ^5. 30. V 1 142^44 -fVl 1-628 L
31. 5v/3 + :;>/.5. 32. V-07~V00.
Find a mean proportional, conect to '> ])la('es, to :— r-
33. 25 and 0. 34. 8 and 18. 36. 28 and 6:; 36, 7 and 13.
37. j and I 38. 25 and 81.39. /, and .',. 40. -12 and -48.
54
POWERS AND HOOTS.
41. "What is the side of a square whose area contains
106,929 sq. yards ?
42. A rerrinieut consists of 15,876 foot-soldiers ; how
many must he phiced in rank and tile to form a solid sq. ?
43. A square farm contains o6-l acres ; how many rods
square is it ?
44. Find in rods the length of a side of a square piece
of land containing 262 ac. 105 sq. rods.
45. A rectangle is 81 yds. long hy 64 yds. wide. Find
tlie side of a sijuare that has the same area.
46. A rectangular field containing 80 acres is twice as
long as it is hroad. Find its length and breadth.
Hypotenuse -= x/Base' + rerpendicular'.
Perpendicular = -v/Hypotenuse>' - Vynsei
}^n.se = /v/Hypotenuse^ — rerpendicidarl
47. If the sides of a right-
angled triangle be 2 in. and 3 in.
in length, find the hypotenuse
correct to j^y part of an inch.
48. If the hypotenuse be 8 ft.
and base 4 ft., find the perpendi-
cular correct to yoVo ^^ ^ ■^*^'^^-
49. Find the height of a window
reached Ijyaladder 25 ft.long.whose
foot rests 15 ft. from the house.
50. The height of a tree standing on a riverside is 100
ft., and a line stretched from its top to the opposite bank
is 144 ft. ; find the width of the river.
51. A tree was broken .")5 feet from its root, and
strack the crround 21 feet from it- base. Find the height
of the tree.
52^42+32
POWERS AND ROOTS.
55
C. — Cube Roots.
how
Extract the cube root of : —
13
1. 2197.
2. 12167.
3. 19683.
4. 493039.
14. 15i
39651821.
e 4 •
6. 7077888
9. 1-225043.
10. 27270-901.
7. 134217728. 11. -000300763.
8. 228099131. 12. 233-744896.
15. 37oV. 16. 13]^f. 17. 51641^^.
Find, correct to 3 places of decimals, the value of : —
18. 4/73. 19. ^108. 20. v^-lT2. 21. 4^-00.3.
22. 4/^)00714 + ^32. 23. x/wi + ^301 -v^ 1607^1^ +
v/24T-i40625.
24. Find the cube root of the fourth power of 112.
25. What is the length of the side of a cube which
contains 9 c. yds. 11 ft. 64 in. ?
26. Find tlie content of a cube, if the diagonal of one
of its faces is 12v^2 inches.
27. The areas of similar surfaces are to each other as
the squares of their like dimensions.
(a) If a pipe one inch in diameter will fill a cistern
in 60 minutes, how long will a pipe 2 in. in diameter take?
(h) If one side of a triangle containing 36 sq. yds.
is 8 yds., what is the length of a corresponding side of a
similar triangle which contains 81 sq. yds. ?
28. The contents of similar solids are to each other as
the cubes of their like dimensions.
(a) If a globe 4 in. in diameter weighs 32 lbs., what
is the weight of a globe 5 in. in diameter ?
(b) If a sphere 3 in. in diameter weighs 4 lbs., what
is the diiimeter of a sphere that weighs 32 lbs. ?
tr\
Ml
|t%4
I't"
56
MENSUUATION.
(liooklll., p. 27.)
area
What
MENSURATION.
XXX. 3F»X«..^ JM lEJ
A. — rAllALLELOGRAMS.
Area = length xjKrpcndUudar height.
Area — hase x altitude.
If h stands for altitude and h for base,
(1) area = A x h, (2) h = %;"% (?>) h -
If base and altitude are equal, h = h, then
area = h"^, h = v area ; or area = Ir, h = v area.
1. The area of a square is 544^ s(j[uare yards,
is the lentith of its si<le ?
2. The area of a rectangular field is 14 acres 91 sq.
rds. ; what is its breadtli, its length l)eing ir)7r) links ?
3. What is the pijrpendicular breadth of a rhomboid
n acres in area and 20,000 links in length ?
4. The paving of a square court cost £1. 7s., at the
rate of Is. 4d. per square yard. How long was it ?
5. 900 paving stonee each 2 ft. by 1 ft. .'5 in., are
required for a pathway 100 yards in length. What is the
width of t'le pathway ?
6. AVhat wid be the cost of painting the walls and
ceiling of a room 17 ft. long, 13 ft. o in. broad and 11 ft.
high, containing two windows, each G ft. by 4 ft. ,*> in., and
two doors, each 8 ft. by 4 ft., at the rate of SI. 08 sq. yd. ^
7. Two pathways, 5 ft. wide, at right angles to one
another, and parallel to the side, run across a rectangular
courtyard 79 ft. by 63 ft. Find the cost of paving them
at 99 cents per sq. yd. (Draiv diagram.)
8. How many links lomr is a square field containin'j
9 ac. 81 sq. rods ?
MENSURATION.
57
B. — Triangles.
1. Area = ^ (base x altitude.) (Book III., p. 28.)
(1) area = ^-f , (2) A = «^^f^2^ (3) b = ^^'^-^\
2. Area ui terms of the sides = >/s {s ■— a) (s —■ b) {s — e)
where s is half the sum of the three sides.
Rule. — F7vm half the sum of the three sides subtract each
side separately ; midtiply the half sum and t, j three remain'
dcrs together, and extract the sq\iare root of the product.
(For hypotenuse, base and perpendicular of right-angled triangles,
see p. 54.)
1 . Find the areas of the following triangles : —
(a) Base, 150 ft. ; altitude, 42 ft.
(b) Sides, 12, 15 and 18 yards.
(c) Sides, 1200, 1450 and 1500 links. (Ans. in ac.)
{(l) Sides, 4*5, 6*2 and 7 8 inches.
{e) Perimeter, 120 ft. ; sides proportional to 5, 9, 10.
(/) Perimeter, 27 yds. ; triangle, equilateral.
2. Base, 128 ft. ; area, 298fy sq. yds. Find altitude.
3. Area, 144 acres ; altitude, 60 rods. Find base.
4. A board 16 feet long is 22 inches wide at one end
and tapers to a point ; w^hat is its value at 4\ cents a ^(\.
foot ?
5. Find the area of a square field whose diagonal is
380 yards.
6. A square field contains 35 acres ; find its diagonal.
7. The area of a triangle is 6 ac. 88 sq. rods, and a
perpendicular from one angle on the base measures 524
links. Find the length of the base in chains.
8. The sides of a triangular field are 300, 400 and 500
vards : if a belt 50 yds. wide is cut ofl* tlie field, find the
sides of the interior triangle and the area of the belt.
5
68
MENSURATION.
C. — Tkapezoid and Trapezium.
A trapezoid is a four-sided figure having two of its
opposite sides parallel.
A trapezium is a four-sided figure having no parallel
sided.
Trapezoid.
Trapezium.
1 . The area of a trapezoid is equal to half the sum of its
parallel bases 111)01112)1 led bt/ its altitude.
If b and b^ stand for the parallel bases,
area = -^ {b-\-b^)xh.
2. The area of a traitezium is equal to the diagonal
multiplied by half the suvi of the two p)(^Tpendiculars falling
upon it from the opposite angles.
If // and h^ stand for the perpendiculars, and d for the
diagonal, area = ^ (A + h'^) x d.
3. Prove the two rules given above by showing that
both are equivalent to the following : —
The area of a trap)ezoid or trapezium is equal to the sum
of the areas of the two triangles into which the figure may
he subdivided.
4. A field is in the form of a trapezoid ; its parallel
sides are 10 chains 30 links and 7 chains 70 links iu
length, and the distance between them is 7 chains 50
links. Find the acreage.
5. Find the area of an irregular piece of ground, the
diagonal of which is 320 yards and the perpendiculars
35*5 yds. and 42.} yds.
MENSURATION.
50
6. Find the acreage of a trapezoid whose parallel
sides are 1964 and 1250 links respectively, and whose
altitude is 250 links.
7. Find the acreage of a field ABCD, right-angled at
B, if AB = 525 links, BC = 440 links, CD = 875 links, and
DA = 260 links.
8. In a trapezium EFGH, EF = 586, FG = 1068, GH =
766, HE = 964, and EG = 1468. Find the area.
9. A four-sided field has two sides parallel and the
other two sides equal to one another ; if the parallel sides
are 370 and 250 links long, and each of the equal sides
100 links long, find the area.
1 0. The opposite sides of a quadrilateral are parallel,
and the distance between them is 7 chains 50 links ; if
the area is 675 acres and tlie length of one of the parallel
sides is 10 chains 30 sides, find the length of the other.
D. — Circle.
1 . The circumference of a circle — diameter x 34 or 3-1416.
(Book in., p. 30.)
The letter tt stands for the ratio of the circumference
to the diameter. Hence tt *= y- or 3vl416.
If r stands for radius, 2r will stand for diameter, then
circumference
TT
p. 30.)
(1) circumference = 2xr, (2) 2r =
2. Area of circle = ^ {circumference x radius). (Book III. ,
= ^(2 7rrxr) = 2^
= 7r?'l
Rule. — The area of a circle is found (a) hy multiplying
the circumference hy half the radius, or (b) hy multiplying
the square of the radius hy ^ or 31416.
3. Find the area of the circle whose radius is (a) 40
feet, (h) 1760 yards, (c) 5 ft. 9 in.
* In the exercises of this booic tt = •^f\ unless stated otherwise.
60
MENSURATION.
4. Find ilie area of a circle whose circumference is
(a) 200 yards, (h) 10 feet.
5. Find tlie radius of the circle whose area is (a) 50
sq. ft., (h) half an acre.
6. The radius of a circle is 10 ft., and a square is
inscribed within it ; find the difference between the area
of the circle and that of the square.
7. How many times will a roller, whose breadth is 3
ft. 6 in. and diameter 2 ft. 6 in., have to revolve to roll a
cinder-path a quarter of a mile long and 7 ft. broad ?
8. A circular field contains 3 acres, and a walk round
it contains ^ acre ; what is the diameter of the field, of the
field and tlie walk, and tlie breadth of the walk ?
9. The circumference of a circle is one mile ; find its
area in acres.
10. Find the circumference of a circle whose area is
equal to tliat of a square the side of which is 320 yds. long.
1 1 . Find the number of sq. rods in a roadway 5 yds.
wide rouml a circular pond 120 yds. in diameter.
12. A circle, an equilateral triangle and a squctre liave
tlie same perimeter, 120 feet. Find tlieir areas.
13. The diameters of the wheels of a bicycle are 52 and
15 inches respectively; determine how many more revohi-
tions the small wheel would make than the large wheel
in a distance of 13 miles.
14. The area of a square is OS acres ; find in yards the
circumference of the circumscribing circle.
15. The radius of the inner boundary of a circular ring
is 14 inches ; the area of the ring is 100 sq. inches. Find
the radius of the outer boundary.
16. An as^ is tethered in the midst of a field so that it
can feed on an acre of grounil ; what is the length of th<'
tether ?
MENSURATION.
61
XXXI. SURFACES AND VOLUMES OF SOLIDS.
A. — Cube, Prism, Cylinder.
Cube.
General Rules. — (a) The area of
the lateral surface of a cube, prism or ^*'"^'*'"' Cylinder.
cylinder is the product of the perimeter of the figure by its
altitude.
• (b) The area u/' the total surface of a cube, prism or
cylinder is the sum of the areas of its lateral surface and its
two bases.
(c) The volume of a cube, prism or cylinder is the product
of the area of its base by its altitude. (See Book III., p. 31.)
From the above rules we derive the following:—
1. Cube. — Tf X stands for the length of its side,
Lateral surface = 4:.t^ (1), Total surface = ijji^ (2).
Volume = x^S).
2. Prism. — lij) stands for perimeter of huse,
Lateral Surface = pxh ( 1 ).
Total Surface — (p x h)-^-- areas of two bases (2).
Volume = area of base x h (.'»).
3. Cylinder.
Con rex surface^ circumference of base x altitude =
2 7r;-x//(l).
Total surface = {'2 IT r X h)-\- arras of two basest
2Trrxh-{-2Trr^(2).
Volume = area of base x altitude = x r x h (.'> ).
4. Find the total surface of a cube whose edge is {a) l.'
iu.,(/>)n ft. 10 in., (0 ft. n in., (W) 5 ft. 1 in.. 0)10 ft.;) in.
62
MENSURATION.
5. Find the total surface of a parallelopipedon which
measures (a) 10 in., 15 in., 18 in. ; (h) 6 ft. 1 in., 2 ft. 10
in., 1 ft. 8 in., (c) 6 ft. 3 in., 5 ft. 4 in., 4 ft. 9 in.
6. Find the total surface of a triangular prism, the
sides of whose base and lieiglit are (a) 2 in., 3 in., 3 in. and
1 ft.; (b) 2 ft. 3 in., 1 ft. 5 in., 1 ft. 5 in. and l.V ft.
7. Find the total surface of a cylinder wliose radius
and height are (a) 2 in. and 1 ft. 4 in. ; (h) 1 ft. G in. and
9 in. ; (c) 9 in. and 1 ft. 6 in.
8. Find the lateral surface of a liexagonal prism, eacli
edge of base being 2 ft. 3 in. and height 5 ft.
9. If 30 cubic inches of gunpowder weigh a lb, what
weight of gunpowder will be required to fill a cylinder
whose internal diameter is 8 in. and length 2.} ft. i
10. Find the length of the edge of a cubical block of
stone containing 46 cub. yds. 513 cub. in., and the number
of s(|. inches in its entire surface.
11. AVater is poured into a cylindrical reservoir 20 ft.
in diameter at the rate of 400 gallons a minute. Find
the rate per minute at which the water rises in tlie
reservoir.
12. The length of a trough is 15 ft. and its ends are
equilateral triangles; it contains 17.".,2U0 cuV)ic feet of
water, liequired the depth.
13. Find the volume of a prism on a triangular base,
the sides of the base being 51, 40, 13 in., and the lieiglit
58 in.
14. The .surface of a cube contains 337 sq. ft. 72 sq.
in. ; find its volume.
16. The external measurements of the sides of a box
are 3, 2-2, 1-52 ft. ; find its volume. Find also the culacal
space inside tlie box (closed by a lid), the thickness of the
wood being ^V ft.
MKNSUnATION.
63
B. — PtEGULAR Pyramid and PiIght Circular Cone.
pyramid. Cone.
A pyramid is a solid, (he base of uhich is any plane figure,
and its lateral faces are triangles meeting in a point called the
vertex of the pyramid.
Pyramids, like prisms, are named triangular, square, etc.,
as tlieir bases are triangles, squares, etc.
A cone is a round pyramid with a circle for its base.
Oeneral rules. — (a) The area of the lateral surface of a
pyramid or cone is half the product of the perimeter of the
base by the slant height.
(b) The area of the total surface of a pyramid or cone is
the sum of the areas of the lateral surface and of the base,
(c) Tfie volume of a pyramid or cone is one-third the area
of the base multiplied by the altitude.
From the above we get the following formuhe : —
1. Pyramid.
Lateral surface = .V (perimeter x slant height).
Total surface = sum of triangular faces -\- area of base.
Volume = } (area of base) x altitude.
2. Cone. — If r — radius of base,
Latere^ surface = 1 (circumjerence of base) x slant
height— tt r\ lant height.
'Total surface ^(ir rx slant height) + tt rl
Volume — }i (area of base) x altitude = ?, tt ;•- x h.
r''
64
MENiSUIfATION.
3. Calculate the entire surface of a square pyramid
whose slant height is 18 inches, the area of its base being
144 sq. ill.
4. Calculate the entire surface of a triangular pyra-
mid whose three lateral faces and the base are equilateral
triangles, each side measuring 2 inches.
5. l)raw the developed convex surface of a cone, the
diameter of whose base is 4 inches, and whose slant
height is inches. Find this surface.
6. How many square inches of paper would be required
to cover the side and base of a cone inches in diameter
at the base, and having a slant height of 10 inclies ?
7. Find the slant height of a cone whose altitude is
11^ inches, the diameter of its base being 10 inches. What
is its convex surface ?
How is a right circular cone generated ?
NoTK. —The altitude of a cone or pyramid, the slant height and the
peipendimilar from the centre of the base to the aide, form a right-
augl"" \ triangle.
8. liequired the volume of a square pyramid lo ft.
1) in. high, the side of its base being 2 ft. G in.
9. AVhat is the solidity and surface of a cone 5 feet
in diameter and 12 feet high ? (x =)V1416.)
1 0. A triangular pyramid has each side of the base 5
ft. ; the iieight is 10 ft. ; find its solidity and surface.
(tt =ai410.)
11. Find the volume of a cone whose altitude is 18
metres and diameter of base 6 metres.
12. A conical tent whose slnnt height is 12 ft. requires
132 sq. ft. of canvas to cover it. Find how many feet of
ground the tent covers.
13. It is requireil to cover a piece of ground 80 feet
squnre with a pyramidal tent uO feet in perpendicular
I
#
MENSURATION.
65
I
lieight. Find the cost of the requisite quantity of canvas
at 4^d. per square yard.
14. How many square yards of canvas will l)e neces-
sary to form a conical tent whose perpendicular heiglit is
10 ft. and diameter on the around 21 f t. ?
15. Find the slant height of a cone whose volume is
19| cub. ft. and diameter at base o} ft.
16. Find the volume of a cone, the height of which is
15 ft. and circumference of base 14 ft.
17. The diagonal of the base of a square pyranud and
the diameter of the base of a cone are each 16 ft. ; their
altitudes are equal, but the volume of the cone exceeds
that of the pyramid by 281 cub. ft. Find the altitudes.
1 8. The Great Pyramid of Egypt stands on a square
])ase, each side of which is 764 feet, and its height is 480
ft. Find its volume in cubic feet.
C. — The Sphere.
A sphere or globe is a solid hounded
hy a curved surface,
all points of which
are equally distant
from a point within
called the centre.
Sphere.
Ilemisphere.
Cut a sphere (a round apple, for instance) into a
number of small pieces, passing tlie knife in each case
through tlie centre of the sphere.
Eacli i)iece is a triangular solid having for its l)ase .;
portion of the surface of the sphere, and for its altitude
the radius of the sphere.
C6
MENSURATION.
■4:3'
When the pieces are very numerous the base of each
may be considered a plane and the solid a triangular
pyramid. The volume of each pyramid is equal to the
base X ^ altitude, and the total volume of all (which is
the volume of the sphere) is equal to the total surface of
all the bases (which is the surface of the sphere) multi-
plied by } altitude, that is } radius.
1 . Surface = circumference x diameter = 4 tt r'^.
2. Volume = 4 TT r^ X ]^r = * x r^.
3. Find the surface of a globe 8 inches in diameter.
4. Find the surface of the earth, its diameter being
SOOOmi. (tt =3-1416.)
5. Find the solidity of a 10-inch globe.
*3. Solidity of a cannon-ball 9 inches in diameter ?
7. If a heavy sphere o inches in diameter be im-
mersed in a pail full of water, how much water will run
over ?
8. What is the weight of a cannon-ball 12 in. in
diameter, iron being 7"5 times as heavy as water ?
9. Find the ratio between tlie volume of a sphere 1
ft. in diameter, and that of a cube whose side is 1 ft.
10. Find the weight of water in a hollow sphere whose
internal diameter is 3 ft. 6 in.
11. A cube and a sphere being of equal volume, find
the ratio of the radius of the sphere to the side of the
cube.
12. What is the solid content of a sphere wljcn its
surface is equal to that of a circle 4 feet in diameter ?
13. Find the ratio between ihe volume nf a sphere
4 in. ill diameter and that of a cylinder 4 in. high, radius
of base 4 in.
14. Weight of a hollow spltere, specific gravity l)eiug
7'77G, inside diameter 18 in., and tliickness 2 in. ?
MENSURATION.
67
5f each
uigular
to the
liich is
'face of
multi-
leter.
r being
ycr ?
be ini-
ill run
in. in
pliere 1
ft.
3 wliose
ne, find
I of the
}]cn its
er?
spliere
, radius
Frustum of
Pyramid.
Frustum of
Cone.
1). — The Frustum.
AVhen the top of a cone or
pyramid is cut ofi' parallel to the
base, the part remaining is called a
frustum.
The distance between the paral-
lel bases is called the altitude of
the frustum.
General Rules. — (a) The lateral
Hirrface of a frustum is equal to half the product of the sum
of the perimeters b(/ the slant height.
{b) The total surface of a frustum is the sum of the areas
of the lateral surface aud of the two bases.
(c) The volume of a frustum is equal to the sum of the
areas of the two bases and the square root of their product,
multiplied by one-third of the altitude.
From which we derive the following formulte : —
1. Frustum of Pyramid. — If b and b^ stand for areas
of the l)ases, andj?; and;?! for perimeters of bases,
Lateral surface = .V {p +2)^) X siant height.
Total surface — i {p -\-p^) x slant height + (h + /;,).
Volume = \h (b + b^ + Jbb^).
2. Frustum of Cone. — If ;• and rj stand for radii
of bases.
Lateral surface = tt (>• -f- r,) x slant height.
Total surface = tt (>• + >'i) x slant height + tt ( r" -f rf ).
Volume = I TT h (r^ + r^^, + rr^).
3. The altitude of a frustum of a pyramid is 2^^ ft.,
tlie ends are 5 ft. and 3 ft. scjuare ; what is its solidity ?
4. Find the lateral surface of tlie frustum of a S(]uare
pyramid, one side of the upper base measuring 2 feet, of
the lower base 3 feet, and having a slant height of 4 feet.
Find tlie entire surface.
68
MENSURATION.
5. Draw the patcern of a small shade for a candle.
Make the upper opening 11 inches in dia-
meter, the lower 2 5 inches in diameter, and
the slant heicdit 2 inches. Find its convex
surface.
6. How many square inches of tin will l)e required to
make a circular j^^ii (open at the to])), its diameter at
top being 9 inches, at bottom G inches, and slant height 4
inches. Draw the pattern.
7. Find the cost of a piece of marble in the form of
the frustum of a cone, the diameters of tlie two ends beimr
11 ft. 8 in. and 8 ft. 2 in., and the slant height 18 ft. 5
in., and the ^alue of a cubic foot of marble being 12s.
8. Find the convex surface of the frustum of a cone,
the circumferences of the bases being 15 inches and 20
inches, and the slant height 10 inches.
9. How many square yards are there in the entire
surface of a frustum of a cone, the radius of the upper
base being 3 yards, of the lower base 5 yards, and the
slant heidii vards ^
10. Find the volume of the frustum of a cone, the
circumferences of wl\ose bases are G6 and 56 feet, and
whose height is 4 feet.
11. Find the w-'ght of water in a tank in the form of
a frustum of a cone, the radii of the ends being 10 feet
and 8 feet, and tlie distance between beinsj: 5 feet.
12. How many sq. feet of tin will be required fur a
funnel if tlie diameters at top and bottom are to be 28 in.
and 14 in., and the slant heifdit 24 in. ?
13. How often can a conical wine-glass, 2.V in. in
diameter and 2 in. deep, be idled from a cylindrical
tumbler, 4 in. in diameter and ?>l in. deep ?
MENSURATION.
CO
candle.
lired to
eter at
ei"lit 4
form of
.s being
.8 ft. 5
2s.
a cone,
and 20
entire
upper
nd the
we, the
it, and
"orni of
10 feet
1 fur a
I 28 in.
in. in
ridrical
FORMULA IN MENSURATION.
I.-PLANE FIGURES.
Ji b stands for base, h for altitude, r for radius, (/ for diagonal,
formulae will be as follows : —
Parallelogram. Trapezium.
area -- bh. area - h{h + h,)x(L
Triangle.
area --■- \bh. Circle.
ana
Trapezoid.
circumference ---^
\{h + b,)x
h. area
r=
V'2 ■--- 1-414.
?/ —
l-46o.
V.S 1-732.
ir
- -3183.
VV - 1-772.
1
- -5642.
II.
-SOLIDS.
J TT r
TT r'
If b stands for area of base, j) for perimeter of base, r for radius of
base, h for altitude, and I for slant height, formula' will be as follows :
Prism.
Sphere.
lateral surface
ph.
total surface
ph-h2b.
volume
bh.
Cylinder
,
lateral surface = 2 t rh.
total surface --^ 2 t
rh + 2Kr-.
volume ----^ TTr'
M.
Pyramid,
lattral surface
I pi.
total surface
\pUb.
volume
hbh.
Cone.
lateral surface
TT rl.
total surface
7Trl + -Tr\
{■olnme
l^-r-h.
surface
volume
4 rr >•
TTV
Frustum of Pyramid.
lateral surface ^- h {p + ])^^)xL
total surface —
volume -^ ^h (b + b^+ \'bb ^ ).
Frustum of Cone.
lateral surface ■■- ^/(>"+>"i).
total surface ~
volume -■= ;\ ~ h {r- i- /■,- + rr^ ).
70
MENTAL ARITHMETIC,
xxxn.
TEST EXERCISES IN MENTAL ARITHMETIC.
1. 72-) articles
2. 846 articles
3. 397 articles
4. 807 articles
5. G48 articles
6. 296 articles
7. 588 articles
8. 588 articles
9. 588 articles
10. 588 articles
at 15 cents each,
at 3."»>; cents each,
at 12^ cents each,
at GG| cents each,
at ?i'7\ cents each,
at 62} cents each,
at 87.V cents each,
at 8i cents each,
at 16 1 cents each,
at $1.25 each.
B
1.
2.
3.
4.
5.
6.
Multiply 68 by 75.
Cost of 68^1 articles at 8 cents each.
9 acres 90 sq. rods at $10 a sq. rod.
Express -594 as a vulgar fraction.
Difference between "16 an-' 'l^ in common fractions.
Interest on $ooo..33^ for 5 mos. at 6 per cent.
7. L.C.M. of 1,2,3,4,5, 6.
8. 6^+7^ + 8^-21.
9. |(13 + 7) + 8; xl6 =
C
1. 365x41 =
2. School of So pupils ; 45 boys. Percentage of girls i
3. $13ixl2 + (T?^ + J^). •
13x2_x8x7xJ
■ " "26 x"4S
1
lETIO.
MENTAL ARITHMETIC. 71
5. Reduce 3 qts. 1 pt. to fraction of a bushel.
6. Reduce 7s. 8^d. to fraction of £1.
7. One cent and a lialf a day; liow nuicli per year ?
8. Amount of $100 for 2 yrs. at 41%.
9. $9 a year; how much a day ?
10. U : 7i :: -5 :^.
1. Value of 4-15 oi$0.o^l
•^- 7 114 -*-4 i» —
' 3. Rent of 5 acres at 12^ cents a sq, rod ?
4. (V)3t price, $1.;- ; selling price, $2. Gain per cent.?
5. Difference between -J', and -^J^ in decimals.
6. What decimal of 1| lbs. is 1 oz. ?
7. Average of 13, IS, 913, li, a, 3J.
8. Total value of an article if ^\ is worth $13.
9. Sum of 3.V%+ 10^/^ on $50.
10. Number of telegraph poles 96 ft. apart in 10 miles?
actions,
nt.
)f girls i
1. Sum of all the odd numbeis between 20 and 40.
2. A square contains 570 sq. inches. What is the
length of a side in feet ?
3. Cost of paving a yard 36 yds. long, 12 yds. wide,
with osphalt 3 inches deep, at $2 per cub. yd. ?
4. A man runs 220 yards in 40 seconds. What is his
rate per hour expressed in miles ?
5. Number of yards of carpet, 15 inches wide, required
for a room 30 feet long by 17^} feet wide ?
6. Lost 10% on $50 and gained 12% on the remain-
der. Required total gain.
7. 10% lost by selling for $65.25. Cost price = ?
8- If i-hl + ^. = £2. Os. Od., what is the whole sum ?
9. Value of 31 of $15. lo. L.C.M. of 16, 24, IK
MENTAL ARITHMETIC.
15
1. Simple interest on $277 for 3 yrs. at 4 per cent.
2. Divide S27.30 between A and B in the ratio 9 : 4.
3. From seven thousand take fifteen hundred and 15.
4. Add five times nineteen to four times sixteen.
5. Find the fourth term of 36 : 63 : : 20 : x.
6. Express 3s. 6d. as the decimal of £1.
7. A man rows 4 miles down the stream in 20 min.,
and up the stream in 40 min. "What rate in smooth water ?
8. Piequired the time for two trains to pass eacli other
in opposite directions, each train being 220 yards long
and uoimi at the rale of 40 miles an hour.
9. A man gave $30 for 44 cwt. and sold at $'72 per
cwt. What is his gaui ?
10. Amount of annual income secured by investing
$504 in 4 per cents, at 96 ?
1. 90x
;068
1-24
2. A starts at 6 mi. an hour, and in half-an-liour ]>
follows at 9 mi. an hour. When will B overtake A ?
3. The surface of a cube is 150 sq. inclies ; what is
the length of its edge ?
4. What sum at 7% will gain $84 in a year ?
6. A mother is half as old again as her daughter, and
their united ages = 95. Find their ages.
6. Cost of 59-^- lbs. of cheese at 9.} cents a lb.
7. What are the 3 equal factors of 343 ?
8. 9 doz. hats at $1.33^ each.
9. A can do a job in 4 days, B in 6 days. How long
will they take together ?
10. Present worth of $1080, due 5 years hence, at 4'
per annum simple interest.
0/
c
TEST I'HOHLEMS.
Ho
cent,
io 9 : 4.
[ and 15.
een.
20 min.,
h water ?
ich other
fds long
$-72 per
nvestinir
1-1) our ]*.
A?
what is
hter, and
low long
3e, at 4'J'
XXXIII.
TEST PROBLEMS.
A
1 . Value of # of h^^J"^!}
12^ 2v of If
and express the result as a decimal.
2. A man hought a piece of ground containing OlUG
ac. at r,:]^ a sq. foot. Wliat did he ])ay?
3. Find the compound interest of $320 for ?»}, years
at 3 per cent, half-yearly.
4. A tradesman marks his goods at 25% ahove cost,
but allows a customer a discount of 12% for Cixsh. What
per cent, of profit will he make on a cash sale ?
5. A rupee is worth 2s. O.^d. and a dollar 4s. 4kl.
Find the least number of rupees which makes an exact
number of dollars.
B
2 2 I _3_ " 1 I s
1. Simplify l]^rj^^i ^-^^±31^
4 1 r 1 9 4 ;V -f- vj "4
2. Add together 7G95 sq. inches, -GOl of an acre, and
] *; of a s(i. rod, giving answer in sq. yds. and decinud.
3. If 4 men mow 15 ac. in 10 dys. of 7 hrs., in how
many dys. of Gl hrs. can 7 men mow 19.} acres ?
4. A man receives 4% on one-third of his capital, 4.1%
on one-sixth, and 5% on the remainder. 'vVhat perceiita'ge
does he receive on tlie whole ?
6. Find the difference between the compound interest
on $3840 for 3 years at 5 per cent, and the true discount
^^w $12,710.87 due G montlis hence at 10
o/
/
1. Divide -723905 by 2-17 and express the difference
between l-5384Glo and -070923 as a common fraction.
II
74
TEST I'RDBLEM.S.
2. It' three pipes separately can fill a cistern in I7i,
19i and 21 fV minutes, how long will they take when
running togetlier ?
3. Divide £105 between A, B and C, so that as often
as A gets 4s. 6d. B may get 7s. 6d., and as often as B gets
OS. C may get Gs.
4. Find hy practice the value of ^.7° IG' 30" at G3 mi.
780 yds. for eacli degree.
6. If the :> per cents, may be bought at 88 i, what
should be the price of the 4 per cents. ?
1. Simplify
D
•071 + -0:38r) . 3-14ir>.3-f70
7 1 _ -007 ■ -041 X -015 •
2. At what time between 2 and o o'clock will the
hands of a watch intercept an angle of 90 ?
3. Find the time in the foll(»wing capitals when
Greenwich mean time denotes 12 noon: — St. Petersburg
30^20' East longitude; Berlin 13° 24' East; Duldin (J'
10' West ; New York 73" 58' West. (See Book III., p. 35.)
4. If the discount on a Dill due five months hence at
3;^ per cent, is $774^, what is the amount of the bill ?
6. ]l(tw many tiles 5 inches S(]uare woidd be required
to line tl»e bottom and sides of a swimnung-bath which is
UK) ft. by 30 ft. and 5 ft. deep ?
1. Express V''*'.^o5«i^^ correctly to the nearest integer.
2. A and li are in partnership. A invests $4700 for
12 montiis and receives $440.50 as his share of the proiits.
What shouUl li receive who invests $2300 for months (
3. Find change of income from transferring $5400
stork fr(>i!i tlio 4H Tier cents, at 98 to tlio 3-^ per cents.
V
at 1 5.
TEST PROBLEMS.
<y
4. A straight plank is ol inches thick and 6| inches
broad ; what lengtli should be cut off so as to contain i3\
cubic feet of timber ?
6. Find the surface and diagonal of a cube of granite
containing 162,144 cub. inches.
)or cents.
1. Reduce 120 grams to ounces Av. ; 63f yds. to m.
2. A grocer buys sugar at 18/^ a kg., and sells it at
one cent, per 50 grams. Find his gain per cent.
3. When gold ife quoted at $1.12 J, what is a $1
greenback worth ?
4. What income should I derive from an nivestment
of $6990 in 3% stock at 87;^? How much must be
invested in 4/^ stock at 112 to yield an equal sum ?
5. The distance between two towns is 54 miles, and
the distance between their places on a map is 6-^' inches.
What area of country is represented by a circle on a map
of one inch radius ?
G
1. Find the sum of 3], 6|, TW. ^^^»\ express the
result as a decimal and extract the cube root.
2. How many litres in 6 gallons of water ?
3. If a gallon contains 277} cub. inches, find appro.xi-
niately the number of gallons of water which would cover
a s(|uare mile to the depth of an inch.
4. If 8000 metres are e(iuivalent tb"^ miles, and if a
cubic fathom of water weighs tons (2240 lbs.), and a
cubic metre of water 1000 kg., find the number of lbs.
Av. in a kilogram.
5. Ha room is 40 feet long by 20 feet broad, and
contains 12,800 cubic feet, what addition will l)e made to
its cubic contents by throwing out a semicircular bow at
one end ? ( tt =.)14I6.)
70
TABLES.
WEIGHTS AND MEASURES TABLES.
Time.
60 seconds (sec.) - 1 niimite (min.)
60 niimitcs —1 hour (lir.)
24 liours - 1 clay (dy. )
7 days — lwcek(wk.)
36.1 days = 1 common year (yr. )
100 years - 1 century (C. )
Capacity.
2 pints (pt.) - 1 (|Uart {qt.)
4 quarts - 1 gallon (gal. )
2 pints (pt.)
S quarts
4 pecks
1 (juart (qt. )
1 peek (pk.)
1 bushel (1)U.)
1 pint (water) weighs 1 j 11)3. Av,
1 gidlon contains 277 "274 cul»ic
inches, and weighs JO ll>s. Av.
1 cuhic foot (water) contains 6j gal-
lons (nearly), and weighs 1000 oz.
or 62i lbs. Av.
Wekjiit.
16 ounces (oz.) — 1 pound (lb.)
lOOpounds— 1 hundredweight (cwt.)
20 cwt. or 2000 lbs. =- 1 ton.
2240 lbs. = 1 long ton.
Knislish Monfcv.
4 farthings - 1 penny (d. )
12 pence I shilling (s.)
20 shillings - 1 p<tund (f.)
LENdTlI.
12 inches (in.) -= 1 foot (ft.)
3 feet ^ 1 yard (y<l.)
fl.i yards - 1 rod (r«l.)
320rods 1760y.rds_^j,,,.j .
or 5280 feet ^ '
Surface.
144 square inches — 1 sq. foot.
9 square feet - 1 square yard.
30^ square yards = 1 square rod.
160 square rods or _^ j ^crc (ac.)
4840 scjuare yards
640 acres = 1 square mile.
10,000 scjuare links ^ 1 S{{. chain.
10 square chains = 1 acre.
Volume.
1728 cubic inches — 1 cubic foot.
27 cubic feet — 1 cul)ic yard.
16 cubic feet - 1 cord foot.
8 cord ft. or 128 cubic ft. - 1 cord.
Miscfllaneous.
12 units - 1 dozen.
12 dozen
20 units
24 sheets
20 <|uires
4 inches
6 feet
1 gross.
1 score.
1 (|ture.
1 ream.
1 haiul.
1 fathom.
100 Hnks
80 chains
= 1 chain.
= 1 mile.
Tr(»y Wkhjht.
24 grains 1 j)cnnyweight (dwt.)
20 dwt. - 1 ounce.
12 ounces - 1 pound.
1 lb. Troy -^ r>760 grains.
1 lb. Av. - 7000 grains.
CiKCt'i,.\K Mkasi'iu:.
60 seconds (") 1 ndnute (')
60 minutf'a - 1 dogiee (').
360 degiees - 1 circumference.
60/i iiiih's
-- I degree.
TABLES.
77
Money.
100 centimes = 1 franc (fr.)
A franc weighs 5 grams.
METRIC SYSTEM.
Capacity.
10 millilitres(ml.):^l centilitre (cl.)
10 centilitres = 1 decilitre (dl.)
10 decilitres — 1 litre (1.)
10 litres = 1 decalitre (Dl.)
10 decalitres = 1 hectolitre (HI.)
Wekjht.
Length.
10 millimetres (mm.)=:
1 centimetre (cm.)
10 centimetres = 1 decimetre (dm.) 10 milligrams (mg.) =
10 decimetres - I metre (m. ) °i centigram (eg.)
10 metres = 1 decametre (Dm. ) 10 centigrams^ 1 decigram (<lg. )
10 decametres ^1 hectometre (Hm.) 10 decigrams =1 gram (g.)
10 hectometres^ I kilometre (Km.) 10 grams ^ 1 decagram (l)g.)
10 kilometres^ 1 myriametre (Mm.) 10 decagrams ^1 hectogram (Hg.)
10 hectograms ::= 1 kilog^ram (Kg. )
1000 kilograms make a metric ton.
SU K FACE M E ASURE.
100 sfjuare millimetres = I square centimetre (.s(i. cm.)
100 s.(uare centimetres = 1 square decimetre (sq. dm.)
100 square decimetres — 1 square metre (s({. m.)
Land Measure.
100 centiares (ca.) = I are (a.) 100 ares=l hectare (Ha.)
A centiare is the same in size as a sq. metre.
Solid Measure.
1000 cubic millimetres = 1 cubic centimetre (en. cm.)
1000 cubic centimetres = 1 cubic decimetre (cu. dm.)
1000 cubic decimetres = 1 cubic metre (cu. m.)
In measuring wood the cubic metre is called a stere.
Metric Kquivalents.
I metre -^
I kilometre
I 8(j. metre
1 liectare
1 cu. metre
Metre
Kilometre
i^q. metro
Hectare
Cu. centimetre
39 "37 inches.
•62N mile.
11% sq. yds.
2 ••471 acres.
1 308 cu. yds.
1 litre
1 hectolitre
1 gram
1 kilo
1 metric ton
Al'fKOXIMATE KgtriVAKKNTS.
1*1 yds. Cu, metre
a nii. Litre
I.' ftq. yds. Hectolitre
1«
ifl
rii'i'pa
<:•
ani
CU. inch.
I<tgram
r76 pints.
22 01 gal.
1 5 432 grains.
2 204()lbs. Av.
M023 tons.
1 -3 cu. yds.
1"/ pints.
22 gal.
15^ giains.
2i lbs. Av.
ill
'8
LOGARITHMS.
XXXVI.
LOGARITHMS.
The expression 10 x 10 x 10 x 1 = 1000 is usually writ-
ten 10^=1000, where o indicates the number of factors
each equal to 10 which must successively multiply unity
in order to give lOoO. Each number in the latter form of
the expression has a name indicative of th^ relation it
sustains to the others. Thus 1000, derived from unity by
three successive multiplications by ten, is called the natural
number; 10, the repeated factor, is called the base; and
o, which indicates the number of successive factors used,
is called the logprithm ;— fully staged, 3 is the logarithm
of the natural number 1000 to the base 10. Similarly,
since 2*= 16, 4 is the logarithm of tlie natural number 16
to the base 2.
PARTIAL TABLE OF LOGARITHMS.
Natural Their ■„
Kunihers. Logarithms. Because
1000000 6 1x10x10x10x10x10x10 = 1000000
1 X 10 X 10 X 10 x 10 X 10 = 100000
1x10x10x10x10 =10000
1x10x10x10 =1000
1x10x10 =100
1x10 =10
1 needs no multiplier to give 1
= 1
= 01
100000
10000
1000
100
10
1
1
•01
•001
•0001
•00001
•UOOOOl
5
4
3
2
1
I
2
5
4
6
TIT
10X10
1
i 6 A 1 (» K 1
I
I * 1 A 1 o X 1 o
1
10 » 10 • 10 ' 16*10
10*10*1 n ■(.I OMOKIO
= 001
= 0001
= 00001
= 000001
LOGAHITHMS.
79
An unlimited number of systems of logarithms might
be used ; but we shall concern ourselves only with that
system whose base is 10.
In the foregoing table it is easy to see the significance of
the first seven logarithms and the manner in which they
are found. The consistency of the rest of the table appears
when we reflect on the relation of the successive natural
numbers to each other and to their logarithms, and mark
the meaning of the minus sign, here, as in all logarithms,
printed above the number to which it belongs.
One factor of those which compose eacli natural number
ill the table, is removed from it when we divide it by 10 ;
consequently the number of sucli factors indicated by its
logarithm will be diminished by jne. In other words, —
dinding a natural numher h// tni tahs one from its logarithm.
For example, 1000 = 10 x 10 x 10 x 1 = 10^ if divided by
10 will give ;'^; =10^-^ = 102=100. The logarithm of
1000 is 3, of 100 is 2. AgJn
10, a single 10, a fact indicated by saying that the logar-
1000 ^10"_..,_2_.^,__
10x10 10- -lu -
ithm of 10 is 1. Further
10^-^=100=1
1000 _10»
10 X 10x10 ~10''
a immber whicli has in it no ten at all as a factor, a fact
iii'Hcated by saying that the logarithm of 1 is 0.*
Pushing the division still further, **^
10^
10 X 10 X 10 X 10
^^ = lO*^-* = 10-'= - = 1, we reach a number which
requires to be multiplied by 10 in order to be brought up
' » unity; it is one factor short of being unity; its logar-
i'hm is 1 less tiian the logarithm of 1, one less than 0;
this fact is indicated by writing a minus sign over 1,
* That 1 contains no factor of any sort, whether 10 or 2 or 7 or any
"Ihcr U expit'sai'd by saying 10' - I, 2" - 1, 7'^ 1, etc. In fact the
logarithm of 1 is whatever be the base of the gysteui of logarithma.
80
LOGAKITHMS.
■
thus 1. Similarly when we say that the logarithm of -01
is 2, we are merely stating the fact that 01 has been
derived from 1 by two successive dU-islons by 10 ; just as,
on the other hand, when we say that the logarithm of 100
is 2, we state the fact tliat 100 is derived from 1 by two
successive multiplications by 10.
If any two of the first seven numbers of the foregoing
table be multiplied together, it is evident that the number
of tens, which successively multiplying u:iity give the
product, is equal to the sum of the numbers indicating the
tens contained in each of the two numbers multiplied
together. Thus, 1000 X 100 = 10 x 10 x 10 x 1 x 10 x 10 x
1 = 10x10x10x10x10x1; that is, the product contains
5 tens as factors; because one of tlie quantities multiplied
tugetlier contained :], and the other 2 tens as factors, and
the sum of 2 and 3 is 5. Tliis fact may be thus stated,—
the lofjavithm of the produd of two iiumhcrs rquals the mm
of the logarithms of the two vuwhers. To revert to the
preceding illustration, the logarithm of 1000 is 3, the
logarithm of 100 is 2, and the logarithm of the product of
100 and 1000, that is of 100000, is 5. If proper account
be taken of the signs of the logarithms the same statement
applies to all the numbers in the foregoing table. It must
be borne in mind that the addition of a'negative number
to a positive diminishes the latter, may even feduce it to
zero, or, indeed to a negative quantity. So while the sum
of 2 and 3 is o the sum of -2 and 3 is but 1, of -2 and
2 is 0, and of -3 and 2 is -1. So the logarithm of 1000
X 100 is the sum of 3 and 2, ;"., as before shown ; the
logarithm of '01 x 1000 is the sum of -2 and 3, 1 ; of 01
xlOO is tiie sum of -2 and 2, 0: of -001 and 100 is
the sum of - 3 and 2, - 1. All this is easily veriried and
understood }>y inspection.
LOGAIUTHMS.
81
It is evident that the statement above made of two
quantities multiplied together, may be extended to any
number of quantities multiplied together, and the state-
ment may be made and remembered in this more com-
preliensive form,— /Ac logarithm of the j^roduct of any
mmhcr of factors equals the sum of the logarithms of the
factors. Thus : What is the logarithm of 1000 x 100 ?
Answer; It is the sum of tlie logarithm of 1000, o, and of
100,2. Thatis, iti8r, + 2, f).
1. What is tlie logarithm of 10x100000? of 100 x
10000 ? of 1000 X 01 ? of 100000 X 0001 ] of -01 x '0001 ?
of 100 X 10 X -0001 ? of -01 X -001 X 10 ? of -01 x 10000 x
•001 ?
If the factors are alike the logarithm of their product
may be as easily obtained by multiplying the logarithm of
one of tliem by their number as by setting down the
logarithm of each and adding them toiietlier. The lo^ar-
ithm of 100^ is as easily got by saying 3x2 are 6 as by
saying 2 + 2 + 2-6. 1 1 is evident that the logarithm of a
power of any numhcr is found hy multiply i^ig the logarithm
if the number hy the index of the power.
2. What is the logarithm of 10'? (jf -r' ? of 100'-? of
•OP? of -OOl-^xlOO^? of -Ol'xlOO^x-l? of -Ol'x
•oor'xiooo*?
Since the logarithm of a product is the sum of the log-
arithms of the factors, it is evident that if the logarithm
of a product and that of one of the factors be known, the
logarithm of the other factor will be found by sulitracting
the logarithm of the given factor from the logarithm of
the ])roduct. Tfcncc the logarithm of a quotient is the
logarithm of the dividend less the logarithm of the divisor.
Because a fraction is ;»n exiuession indicating the division
of the numerator by the denominator, the above ]»rinciple
82
LOGARITHMS.
may be thus enunciated,— ^/te logarithm of a fraction equals
the logarithm of the numerator less the logarithm of the
denominator. Thus : What is the logarithm of 1000000 -^
100 ? Answer : It is the logarithm of 1000000 less the
logarithm of 100; that is, it is 6-2 = 4.
3. What is the logarithm of 1000-^10? of 100000 -f-
100? of 100^10000? of -01 -^ -0001? of 100 -f -001? of
i\%%% ? of -,U', ? of AVt ? of ,^Uy ? of jO^V ?
Since the logarithm of the square of 100, that is of
10000 is found by doubling tlie logarithm of 100, obviously
the logarithm of the square root of 10000 is found by
halving the logarithm of 10000. So the logarithm of the
square root of any number is found by dividing the logar-
ithm of the number by 2 ; the logarithm of the cube root
of any number is found by dividing the logarithm of that
number by 3 ; and, generally, — the logarithm of any root
of a number is found hg dividing the logarithm of that
number by the inde.c of the root. For example : What is
the logarithm of the sixth root of 1000000? Answer:
It is 1 ; because the logarithm of lOOOCOO is 6, and the
logarithm of its 6th root is the sixtli part of 6, is 1.
4. What is the logarithm of the square root of 100 ^ of
10000 ? of -01 ? of -0001 ? of the cube root of 1000 ? of
1000000? of -001? of -000001? of the fourth root of
10000 ? and of the fifth root of -00001 ?
Recapitulatory Questions.— When the logarithms of
the numbers are known, liow can we find the logarithm of
the product of any nundier of factors ? of any power of a
number ? of a fraction ? of the quotient of any dividend
divided by any divis(»r ? of any root of a number ?
Note that the logarithm of the sum or of the diH'erence
of two numbers cannot be found by any simple process
frnni f.fio L^/vov^if )i*k>c< r^f 4l,«-. ..i,.w. !.,.». 4I i _
LOGARITHMS.
no
The table given above may be used not merely to find
tlie logarithms of products, quotients, powers and roots,
luit the products, quotients, powers and roots themselves.
For when tlie logaritlims are found they will serve to
point out in the table tlie numbers of which they are the
logarithms; so indicating the answer sought. For
example, if I wish to know what the quotient of 1000
times -01 divided by the cube root of -001 is, I may find
the logarithm of the result first, which I know, from what
has gone before, to be the logarithm of 1000 4- the logar-
ithm of -01 —one third of the logarithm of -001. This will
give 3 - 2 - (-/) = 3-2 + 1 = 2. But J is the logarithm of
100 ; therefore 100 is the answer soui'ht.
5. Solve all previous examples, substituting the word
" value " for the word " logarithm " in each question.
Thus the first question will read : What is the value of
1000 X 100 ? of 10 X 100000 ? etc. Many of the questions
can be solved more easily without reference to logarithms,
but the value of the exercises will appear later.
6. By the table of logarithms given on page 78 find the
vahie of 1000 X'l, of 10 x 100 x 1000 x -0001, of 10 -j-
•001, of -0014- -00001, of -01 X -001 + 100, of Vioooo;
of ^^00001, of 4/r6x>^ioo, of 10s/-l^^-0i, of
'^lOxs/YoOO s/'lx^''0i I 10
^7^ , of - -— ^--, of
^•1 '"^ ^ooi ' "' vio^ioo-^^10
Although the exercises just given, which appear difficult
to one unaccustomed to the manipulation of surds, can all
be readily solved by the preceding table, yet the table is
much too small to be of practical value, and must be
leplaced by others of nmch greater extent.
It does not fall within the scope of this little work to
sliow how the logarithms of such numbers as 2, 3, 5, 7, 11,
I
84
LOCJAIUTHMS.
in short of all prime numbers, are calculated ; but, these
being calculated, it is easily seen tliat the logarithms of
all numbers that are composed of these and of ten, in any
combinaLions of multiplication and diw.sion, can be found.
Thus, if the logarithm of 2 were known, the logarithm of
5 could be easily found by subtracting the logarithm of 2
from the logarithm of 10, which is 1; for V- = 5. The
logarithms of 4, of 8, of 16, would be respecth^ely twice,
three times and four times the logarithm of 2 ; for 2^ = 4,
2" = 8, and 2^=16. The logarithms of 20, 200, 2000, etc.i
would result from the logarithm of 2 by adding to it 1, 2,
3, etc. ; for these are the logarithms of 10, 100 and lo'oo'
the respective multipliers of 2 in each case. In like man-
■ ner the logarithms of -2, -02, -002, etc., would be found by
subtracting, from the logarithm of 2, 1, 2, 3, etc. ; for
these are the logarithms of tlic divisors 10, 100, 1000, etc.,
by which -2, -02 and -002 are derived from 2.
Because 2, 3, 7 lie between 1 and 10, and the logarithms
of 1 and 10 are and 1, the logarithms of 2, 3 a'lid 7 lie
somewhere between and 1 ; that is they must be proper
fractions, or liOn-terminating, non-recurring decimals
within these limits. They are indeed non-terminaling,
non-recurring decimals, as is the case with all logarithm^'
to the base 10, except 10 and its integral powers.^ Conse-
rpiently the logarithms of all other numbers than those
last mentioned are, and can l)e, only approximately given.
7. Tiie logarithm of 2 is approximately -30103. Find
the logarithm of 5, of 4, of 20, of 25, of 200, of 32, of 3-2
of 8, of 250, of 50, of 512, of 51-2, of 512, of 125 of l->5'
of 400.
The logarithm of 2 being as given, the logarithm of -2
which is the tenth part of 2, is equal to the h)garithm of
2 less the logarithm of 10 : the logarithm of -2 then equals
LOGARITHMS. 85
30103 — 1. This might of course be written as —.69897 :
hut it is found more fonvenient in mimy ways to make
cliano-es of si.ou only in tlie integral part of tlie logarithni,
and so the logarithm of '2 is written i-30103,in which tlie
integral part alone is negative, the decimal part remaining
positive. In like manner the logaritlini of -02 which is
the result of dividing 2 by 100, and of which, consequently,
the logarithm is -30103-2 is written as 2-30103. Sub-
joined is a table of logarithms which will prove instructive.
Tlie integral part of each logarithm is called its charac-
teristic, and the decimal part is called its mantissa.
Natural 'N'umber. Logarithm.
2000000 C-30103
200000 5-30103
20000 4-30103
2000 3-30103
200 2-30103
20 1-30103
i J30103
€ 1-30103
•02 2-30103
•002 3-30103
•0002 4-30103
•00002 5-30103
•000002 6-30103
It will be observed :—
1st. That tlie numbers differ from one another only in
tlie position of the decimal point.
2nd. That the mantissa of the logarithm remains the
same throughout, notwitlistanding the change of position
of the decimal point in the natural nuni])ers. This is
always true of common logarithms ; no change of position
of the decimal point changes the mantissa.
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86
LOGARITHMS.
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3rd. The characteristic changes with every chancre of
position of the decimal point in the number ; every move-
ment of the decimal point one place further to the ricrht
diminishes the characteristic by 1. This is in accordance
with what we have already learned.
4th. The characteristic indicates in each case the dis-
tance of the 2 from the units place, being positive when
the 2 is to the left and negative when it is to the right of
the units place.
• If in any numbers the digits follow in the same order,
the mantissa of the logarithms of all the numbers is the
same ; the logarithms will difler only in the characteristic.
Thus the logarithms of lOO^Ono, 1009035, -0001009035,
1009-035 are 000390025, 100390G25, 400390G25 and
300390625. The numbers differ from one another in the
position of the decimal point only, and the logarithms
differ in characteristics only.
TJie characteristic, indeed, indicates the distance of the
highest jdace in the numher from the unifs pl(ice, the char-
acteristic Uing positive if that highest ^tace he to the left of
the decimal point, and heing negative if that highest place he
to the right of the decimal 'point.
Thus in the example just given 1 occupies the highest
place in all the numbers. In the first number it is in the
sixth place to the left of the units place, accordingly 6 is
the characteristic of the logarithm ; in the second instance
1 is in the first place to the left of the unit and the char-
acteristic of its logarithm is 1 ; in the third instance 1 is
in the fourth place to the right of the units place and the
characteristic is 4 ; finally in the last instance 1 is in the
third place to the left of the units place, and the charac-
teristic of the logarithm is 3.
Hence, /(? /?<«? the characteristic of the loaarithm of anv
i \
LOGARITHMS. 87
number, count from the units place to the place of the highest
fiyure in the member, the number got by this count ivill be the
characteristic ; the characteristic will be positive if you have
to count towards the left, negative if you have to count towards
the right.
Examples. — Find the characteristic of the logarithm of
743-6859. Since 7 is of the highest denomination in the
number, and is in the second place to the left of the units
place, 2 is the characteristic. Find the characteristic of
•001392. It is 3, because 1, the figure of greatest value in
the number, is in the third place to the right of the units
place.
8. Find the characteristic of the logarithms of 4973,
8654-227, 3987, 000642, -0007, 00892, 1-7368, 10009,
9-538, 9000, -000008, -G271.
As the characteristic of the logarithm of any number is
so easily reckoned, it is not usually printed in tables of
logarithms ; but it is printed in the table of logarithms,
page 109, which gives the logarithms of all integers from 1
to 100. Thus the logarithm of 59 is given as 1 770852,
where 1 is the characteristic and '770852 is the mantissa
of the logarithm.
Examples on this First Takle.
Multiplication. — Multiply 17 by 3. Process: Add
the logarithms of 17 and 3 together and find of what
number the result is the logarithm.
Logarithm of 17 1-230449
" 3 477121
The sum 1 •707570
is the logarithm of 51, which is the product of 17 and 3.
9. Bv tlie loiiarithmic table. paL'e 100, find the vhjiu' of
f I
W4
f:|
It
88 LOGARITHMS.
3x7, 5x9, 7x8, 2x3x7, 5x9x2, 3x3x8, 11x2x3
and 2 X 7 X 7.
Division.— Divide 99 hy 11. Process: Subtract the
logaritlim of 11 from the logarithm of 99, the remainder
will be the lo^rjirithm of the quotient.
Logaritlmi of 99 1-995635
" 11 1041393
Difference, "^954242
Which is very nearly tlie logaritlmi of 9, the answer.
Here it will be well to note, as was Ix'fore stated, that
the logarithms are at best only approximately correct, and,
consequently, some little doubt always hangs over the last
figure of the logarithms which we find.
10. Divide by this table 21 by 7, 94 by 47, 68 by 17,
95 by 5 and 98 by 2.
Involution.— IJaise 3 to tlie 4th ]M)wer. I'rocess:
Multiply the logaritlmi of 3 by 4, the result wiil be the
logarithm of tlie answer.
Logarithm of 3 '477121
4
1-908484
which is very nearly the logarithm of 81, the answer.
11. Find the S(|iiiirc of 7, the cube of 4, the squjire of
G, the liflli power of 2.
Evoij rioN.—Find the sixth root nf 64. Process:
Divide the logarithm of 64 by G the (luotient will be the
logarithm of the answer.
Log. 64, 1-80G180 divided by 6 is -3U1U30.
Cut this is the logarithm of 2, the answer.
12. Find the sqiuire rcxtt of 64, the culte root of 64 and
the fifth root of 32.
Two or niorci of these processes may lie combined in one
LOCJAHITHMS 89
exercise. Thus, find the square of the cube root of the
thirtieth part of oo times 14 times 21. •
Logarithm of of) 1-544068
14 1-146128
21 1-322219
4-0T2415
no 1-477121
:{)2^i5294
•845098
2
1-G90m
wliicli is the h)garitlim of 49, the answer.
13. Divide 9 times 49 by 21. MuU-iply one seventy-
fifth of 95 by 15 times '^. Muhiply one thiid of one
eleventh of 9 by one tliird of 22. Wliat is the product of
Iq, 24, Gi and l^V ? What is the cube root of the fourth
l)ower of 27 ^ What is the cube root of the square of five
sevenths of the product of 1^'.^ x 2t, x 2j\ ?
Tlie table of logarithms of numbers from one to one
hundred is too snudl to be of mucli practical value ; but
the table which is contained in the next 15 pages, pp.
1 10-124, is much nu)re serviceable. It gives the mantissa
corresponding to every possible sequence of four digits,
that is, of all numbers from 1000 to 9999. The three
highest digits of the nund)er whose logarithm is required
are printed in the colunni headed N, and the fourth di<ut
in the same line as N at the top of the page. The corres-
]»onding mantissa, the decimal i)oint of which is omitted
in printing, is found opposite the first three digits and
under the fourth digit.
Thus, if we wish to find the mantissa corresponding to
T945wesiiaH find on page 121, rather more than lialf-way
r!
,lt!
• 1,
90
LOGAKITIIMS.
down the column N", tha three first digits 794 ; opposite
tliese digits, under 5, at the top of the page, we have
000094 ; this nundjer, when the decimal point is supplied,
•900094, is the mantissa corresponding to the digits 7945.
Again, on page 116, in column K, about one fourth of the
way down, we have 473. Erom 473 trace the line across
the page till we come under 8 at the top of the page.
Here we find the number 5595 ; but 07, printed above,
are the first figures of the mantissa, so that the mantissa
corresponding to 4738 is '675595.
Hence, to find the mantissa corresponding to any
sequence of four digits, find the first three digits in one of
the columns headed N, and trace a line from them across
tlie i>age until under tlie fourth digit as printed at tlie top
of the page; the number so found is the mantissa, which,
however, nii.^L have a decimal point supplied at its left-
hand side.
14. Find the mantissas corresponding to the sequences :
3928, 4765, 9827, 5903, 4790, 6800, 7495, 5300, 7370,
1952 and 1007.
16. Bearing in mind that the mantissa is the same for
the same sequence ui digits, no matter where the decimal
point may be, find the mantissa^ for 469,, 46.91, -04691,
8800, 880, 88, 8.8, -88, -088, -0088, 7000, 700, 70, 7, 5-3,
•0867, 954000, 48-76, 9-482, -0016, 1000, 100, 10, 1, -1, -01,
•001, -0001, -5408, •0006'-3, -09821.
16. ]>y su])])lying the characteristic before each man-
tissa find the logarithms of the preceding 31 numbers.
17. Find the logarithms of 28.37, -9546, -48, 5-694,
7-862, 97-34, 428-7, -006832, 59-71, -05982, -5907, •0058:5,
•062, •OT, -8, 95-42, 968-3, 79540, 146800, 967000.
To find the sequence of digits corresponding to a given
muHlissa it is, of course, nccessar
^7
})r(
LOGARITHMS.
01
jn.st described. For c.van.ple, to thnl the four di-its cor-
.espomlmg to the „mnti.s,sa -.SCO.Vl.S, the numti.ssa i-ivcn
"ill be fomid „eur the middle of page ll'O, u.uler 3 nnd
opposite /25; therefore 7253 i.s the group of di,.-t,s
required. "^
•l090Q0^^^'',^oo!?"*"'"' <=«>-re.spond to the mantis«e,
102289. -415808, -563718, -700444, -780369, -S7326'>
■903090 and -937810. '
If the whole logarithm be given, characteri.stic as well
as nmntLssa, the natural number corre,si,onding to it can
le founo. For the mantis.,a will give the sequence of four
digits, an.l the characteristic will determine the position
7«o.' nfo''"""' ^"'^ ''"'""S ""="'• Thus to the logarithm
•8-3018 determines the four digits 6653, and the eharnc-
leristic 4 shows that the highest figure is 4 places to the
lett of the units place.
19. Find the luitural numbers corresponding to the
logarithms 5-763428, 4-830011, 3-906443, 2-070858
1-189209, -330819, 1-440594, 2-527243, .3-560385, 4-597476'
5"6o5081. '
In actual work two difficulties which have not vet
been discussed will be encountered. Virst, the table c-ives
mantissa, corresponding to four succe.ssive di^nts o'nly-
liow .shall the mantis.sa corre.spondiv.g to five o"r si.v sue'
ocssive digits be found ? Secondly, how is the natural
number corresponding to a manti.s.sa not in the table to be
lound ?
Both difficulties are met by the application of tlie prin-
ciple that when numbers, and consequently their locrar-
itlims also, are near one another, the diflerences betvv'een
the logarithms are nearly proportionate to the diffmvnr.p«
between the numbers. This may be illustrated by refer-
1)2
LOGARITHMS.
enee to tlio table as given. Thus, 9984, 9987 and 9991 are
three numbers whose successive differences, 3 and 4, jire
small in comparison with the numbers themselves. Now,
the logarithms of these numbers are respectively 3-999:505,
3-999435 and 3-999609, of which the successive ditter-
ences are -000130 and '000174. But as 3:4: : -000130:
•000173^, which is very nearly the same as '000174.
The smaller relatively the differences are, the more
nearly will the proportion give the correct result, and as
the differences in practice are always relatively less tlwin
those in the illustration, the approximation is always
nearer than in the illustration.
The proportion asserted may be applied in two ways :
1st, to iind the nmntissa of the logarithm of some number
not found in the table ; or 2nd, to find the number corres-
ponding to a logarithm of which the mantissa is not found
in the tal)le.
Thus, 1st, if we need the logarithm of 598732, the near-
est numbers to this of which we can find the logarithms
from the table are 598700 and 598800. The logarithms
of these are 5'777209 and 5'777282. Now, the difference
between the numbers 598700 and 598800 is 100, and the
difference between their logarithms is '000073 ; also the
difference between the numljers 598700 and 598732 is 32.
Therefore by the principle stated above
100: 32:: -000073: the difference between the
logarithms of 598700
and 508732.
•000146
•00219
100)'002336(00002336
Therefore the logarithm of 598732 equals the logaritlmi
of 59S700 increased by -00002336, i.e., 5-777209 +
"X
LOGARITHMS.
93
•00002336, i.e., 5-77723226. However, as we do not know
fioiii the table the logaritlmi of 598700 to more than 6
(Iccini.'il places, it is a mere affectation of accuracy to carry
the logarithm of 598732 further, so that its logarithm
rii^lit to the sixth place is 5*777232.
The labour of finding the difference between the logar-
ithms of 598700 and 598800 is saved by the table. Under
the heading 1), which stands for difference, the differences
between the successive nuintissjc are giv(in, 8o, opposite
the line in which the logarithms of 598700 and 598800
are found, tlie difference, 73, standing for -000073, is
printed under I) ; and, instead of nuiking the subtraction
for ourselves, we might have copied the difference from
the table.
Again, as this difference is the difference between two
successive mantissa', it is tlie difference in logarithms cor-
responding to a difference of 1 in the lowest of the 4
places for wiiich the mantissjc are calculated in the tabic.
Tiiercfore one tenth of this, 7*3, would be the difference of
mantissa) corresponding to a difference of one in the lowest
of five successive digits, and twice, three times, four times,
etc., this 7-3 would be the difference corresponding to the*
difference of 2, 3, 4, etc.,' in this fifth place of the natural
number. In like manner -73 would be the difference in
mantissie corresponding to a difference of one in the low-
est of 6 digits. The difference thus corresponding to a
difference of 3 in the fifth place would be 3x7*3 = 21-9,
and of 2 in the sixth place would be 2 x '73 = 1-46 ; so that
the difference of 32 between 598700 and 598732 would
correspond to a difference between their logarithms of
21 -9 4- 1-46 = 23-36, it being understood that 23 is of the
same denomination as the 73 from which it was derived,
namely, so many millionths of unity. Tlie logarithm of
T
Mi
94 LOGARITHMS.
598732, then, would be us before 5-777209 4--000023-;-;6 =
5-777232, omitting tlie last two figures as useless, because
they transcend the limits of accuracy of the table as gi\cn.
A table of the amounts to be added to the mantissic for
eacli possible difterence in the fifth place of the natural
numl)er, counting from its highest figure, when the differ-
ence between the successive mantisste given in the table
is 73, follows.
l-)iffereiice in the Corresponding Nearest number
iifth place of tlie difference of within the limit of
natural numherp. mantiss;e. the table as calculated.
I 7-3 7
2. 146 15
3 21-9 22
4 29-2 29
5 36-5 37
6 .....43-8 44
7 51-1 51
8 58-4 58
9 65-7 66
With such a table before us, the calculation of the amount
to be added to the mantissa of 5987 in order to get the
mantissa of 598732 would be extremely easy. The
amount corresponding to the additional 3 would be found
in tlie table as 22; that corresponding to the 2 one place
lower down would be found from the table to be the
tenth part of 15, i.e., would be one, and the total amount
to 1)6 added for 32 would be 23 as before found.
Such a table, modified to bring it within the limits of
the table as calculated, is found in the first column of the
table of mantissa'. The valu3s given above are found on
page 118 under PP, which stands for proportional parts,
opposite the 1, 2, 3, 4, etc., printed under 590 in the column
headed 1.
LOGAllITHMS.
ri
L>
Three ways of extending the table to include more than
four digits in the natural number have been discussed.
They all agree in this, that tliey furnish a method of cal-
culating a correction which, added to the mantissa of the
highest four digits of the natural number, shall give the
mantissa for one or at the utmost two additional dibits.
1. The first method of calculating the correction is to
use directly the principle on which all three methods are
founded, viz. ; as the difference between two numbers
near together is to the difference between one of these
numbers and a tliird near it in value ; so is the difference
between the logarithms of the first two numbers to the
difference between tlie logarithms of the .second two
numbers, wliich is the correction required.
Here remark that the first two numbers whose differ-
ence is the first term in the proportion, must be natural
numljers in the table, for their logarithms must be known
in order that their difference may be known, as it is
required for the third term of the proportion. Remark
also that they must be as near the number whose logar-
ithm is required as possible ; for the principle employed is
only approximately true, and is more nearly exact the
nearer the numbers dealt with are to one another. Hence
it is best to take the numbers in the table that differ only
in their lowest or fourtli digit by unity, one being less and
the otlier greater than the number of which the logarithm
is required.
As an additional illustration let us discuss the question
of what correction must be added to the mantissa for 7981
to give the mantissa for 79812345. As the mantissa
remains the same, wherever the decimal point may be, we
will try to find the mantissa corresponding to 7981 '2345.
The number whose logarithm we can find in the table
9G
LOGARITHMS.
II
next less than 7981-2345 is 79S1, of which the logarithm
is 8-902057 ; and the number next greater than it is 7982,
of which the logarithm is :;-902112. Therefore by our
principle
7982 - 7981 : 7981-2345 - 7891 :: 3-902112 - 3-902057 :
the logarithm of 7981-2345 ~ 3902057 ;
that is 1 : -2345 : : -000055 : the correction required
•000055
Tl72^
11725
•0000128975 the correction required.
Tf the logaritlims use<l in the calculation were exact the
correction so found niight be used to the last decimal -
but as the logarithms given in the table are only correct
to the sixth decimal place, the correction is of value only
to the same extent, and is, therefore, -000013, which,
tliough a little too great, is nearer the calculated correc-
tion than -000012 would be. So then the logarithm of
7981-2340, as nearly as these tables will mve it is
3-902057 + -000013 = 3-902070.
If in the same way we found tlie logarithm of 7981-234
our proportion would be
1 : -234 : : -000055 : the correction required
-000055
~ii7o
117^
•0000128^0
which is practically tlie same C4)rrection as before. Ac^ain
tlie correction needed for 7981-23 sixnilarly found would
be -00001265, which to the sixth decimal place is the same
correction as before. We may conclude that our table as
given will not enable us to distinguish, the logarithms of
lOGAllITHMS.
97
series of digits extending to more than 6 in number. In
other words, up to six places of decimals the logarithm of
57328-4659, for in.«tance, is the same as the logarithm of
57328-4862. All digits after the 4 may be cancelled, as
they will not affect the logarithms to be used, when as in
the tables before us they are not calculated beyond the
sixth decimal place. As, however, 57328-5 is nearer to
both the numbers given tlian 57328-4 would be, the lo<^ar-
ithm of 57328-5 would be found as tlie nearest practical
representative of the Icgarithnis of 57328-4659 or of
57328-4862. In actual work, of course, the difference
between the mantissa' in tlie table is not written down
with a long array of ciphers as in the example just
worked, -000055, but more simply 55 ; and the work to be
done is simplified into multiplying 55 by the difference of
the numbers, -2345, so tliat the correction is thus found :
-2345
55
11725
11725^
12^977) = 13 practically,
which of course is added in its proper place at the end of
the mantissa of 7981.
20. Find the mantissa' corresponding to the following
sequences of digits: 13796, i;5847, 265778, 394876,
429948, 568872, 5088723, 5688724, 5688719, 56887198,
93462, 789546, 597989.
21. Find tlie logarithms of 3-6871, 45-382, -0067935,
-0798984, 958276000, 964-7399, 11-7184, -0005949259,
79-8463, 2758-458.
2. The second method of calculating the correction is
like the tirst^ except that instead of subtracting one man-
-:::»"■
r'-. L\.
98
LOGARITHMS.
m
tissa from the next to nnd the difference which is the
second term of the proportion, the difference is taken from
the last column on the page headed D. This method is
not quite so exact as the first method, because the differ-
ences in a few cases are not quite right. Thus the differ-
ence between the mantissa? corresponding to 1061 and
1062 is 410, but the difference given in t'ne table is 408.
This arises from tlie fact that the 408 given under D is
the average difference for the line of mantissjc opposite
v/hich it is printed, and is a little too small for those at
the beginning of the line and a liti^le too great for tliose at
its end. The error resulting from the use of the differ-
ences given in the table under D is insignificant.
It is obvious that we may find the logaiitlim of 736854
thus : Find the logarithm of 736800, which is 5-867350,
and add to it as the correction sought the product of the
tabular difference, given under 1) as 59, by "54, tlie t\V(j
digits in the fifth and sixth places. The product, 50 x •r»4,
is 3r86. The nearest integer to this, 32, is to be added
as llie correction ; it makes the logarithm of 736854 lo
be 5-867350-1-32 in the fifth and sixth places, whicli is
5-867382.
22. Find the mantissa- corresi)oudiug to tlie f(dlowiug
sequences of digits : 357897, 682483, 705962, 7998089,
8632157, 8878981, 95342, 96878215, 9736857, 99.S7119.
N.IV — Since yuir table will not enable you to extend
your sequence of digits more than to six digits, take the
nearest se(juence of viix digits when the mantissa of a
longer sequence is rccjuired. Thus, in the last two of tlie
])receding exercises, substitute 973686 for 9736857, and
998712 for 9987119.
3. The correction for each digit ii! the fifth place is
calculated in the column ntanding first on the page and
LOGARITHMS.
99
headed PP. That correction divided by ^ w 1 b t •
correctiou for each digit in the .sixth place. Accordmgl)
the process by which the .nantissa tor the sequence .,.91
cC J i"^ the n>antissa for 379178C2 by tl>e a.d of
, colu.nn of proportional parts, Pl>> j-- f 1°-^ =
Mantissafor.....3791 is -078704
Proportional part for 7 "
^ a « 8 " 9S
a « G " 70
,, u '« 2 " 2^
Mantissa for 37917862 " •5788-15023
„i which the last throe figures are "seless, and the man-
tissa reciuired to six places of decimal is -o , 884o.
The uselessness of going beyond the s.xth place in h
se,|ue.,..e is evident froni this exau.ple also. Substitute
„ •;79178G2, :i79l79, ^vlnch is the nearest sequence of
1;: ;:,aces to the sc.nonce given. The mantissa for
:;79179 is then found as before : ,^„^^ .
Mantissa for :^.791....^ '578704
Proportional part for 7. . . • •
« " 9. ... lO'^
Mantissa for ^79179 -5788454
which, dropping the useless figure in the seventh place, is
the same mantissa as tor .. , n : «0..
23. Kind mantissa, for the s«q"'"'""f ; j^^ ^^^^^^^ [^ ^ '
588077, 589734, 01S1937, 02.548391, 78S09o4, 893881--,
(U 3.84209, 937S80.5942. ^
24 Find the logarithms of 308271, ''-l^i.'', 'es84o,
898732 Vo378145: -0059897, -0003.082479, -OOOooooo.
100:hS42 100004. ,. . .
The pupil who has mastered the preceding exercises .s
, , ; ' ' , „e<.|lv !-» the tal.l-s given wdl serve, the
;rShm!;::;m.".>>er.-.l.illb^^^
100
LOaARlTHMS.
i
able to use any of the more extended tables of logarithms
which may be reciuired in his work at college. It remains
by an inversion of one or other of the three methods just
described to find tlie sequence of digits corresponding to
a nuintissa not found in the table. So we may find the
sequence of digits corresponding to the mantissa -902070
thus : In the table find the mantissai nearest to '902070.
These are -902057 and -902112, corresponding to the
sequences 7981 and 7982, between wliich the sequence
souglit must lie. Then say, as the difference between the
two mantissa^ in the table is to the difierence between the
given nuintissa and the one next below it in the table, so
is the dill'erences between the sequences in the table to the
dilt'erence between the lower sequence taken in the tal)le
and the sequence sought. In tliis particular instance the
proportion becomes -902112 - -902057 : '902070 - -902057
::l:the correction. That is, 55: i:'.; : 1 :the correction,
whicli is tlierefore \^ = -2oiJ^-^. But as these calcula-
tions cannot be relied on for more than two additional
places in the sequence, and -24 is nearer -2303 tlian -23
would be, the se(pience sought may be written as 798124.
1*)V conn)arinfr this result with the inverse process on page
90 tlie pupil will understand more clearly liow it is tliat
tlie finding of sequences not in the talde cannot be safely
extended to more than six places in all, when a six-figure
table of logit^'ithms is used.
The metliod given altove is an inversion of method 1 for
finding the mantissa corresjionding to a se(iuence of more
than four figures, liy an inversion of the second method
we may find the sequence corresi)onding to the mantissa
•807382. Tlie mantissa next below it in tlie table is
•8()7.".50, to which corresponds the sequence 7308. The
dilVerence between successive mantissn- at this pait of the
LOGAUITIIMS.
101
taV)le is given under D as 59. The difference between the
civen mantisna and the mantissa of 7308 is 32. Therefore
Uie di^its to be annexed after 7308 will be given as H
that is -542, which cannot, however, be trusted after tjie
second figure. The corrected sequence reads 730854.
C^ompare with tlie inverse work on page 98.
Finally, the development of the sequence may be effected
by an inversion of the third method of correcting a man-
tissa criven above. Thus, to find the sequence correspond-
\u<r to the the mantissa -578845, find the neai-est lower
mantissa in the table, which is '578754, corresponding to
the sequence 3791. Subtract the latter from the former ;
the remainder is 91 ; the nearest proportional part below
it is 81, corresponding to 7 and leaving 10 ; to this annex
a cipher, making it 100 ; to this the nearest proportional
part is 104, corresponding to 9. Therefore the corrected
setpience is 379179.
25. In each of the three ways given al)Ove find tlie
seduences, to six places only, corresponding to the mantissa'
•i:>r,478, -111111, -2:54507, -345078. -450789, -507890.
•07890l' -789012, -890123, -90123.4. Note particularly any
discrepancies of result arising from dilferencc of method
of working. . , . i
26 Of what numbers are the f.»llowing the logarithms :
3019876 2109870, 1-210987, 0:i21098, 1-432109,
2-54321o! 3054321, 4-705432, 5-870543, 0987054. Use
the last method.
llmiintidiUorii StatancM.-A^xe logarithm of the pro-
,lu,.t .)f any number of factors is the sum of the logarithms
of the factors. The logaritlim nf a .ptotient is the logar-
illmi of the dividcn.l less the logarithm of the divisor.
Tho Incrnritlnn of a power of a number is the logarithm of
thut number multiplied by the index of the power. The
102
LOdAUITH.M^.
lo^Tarithm of a root of a number is the logarithm of the
number, divided by the index of the root.
Multiplication.
Remember that mantissa^ are always positive, and that
the carrying from the sum of any number of mantisste is
always positive, although some or all the characteristics
may be negative.
27. What is the product of 1-0378 X 1*946 x I'OOOS x
3-006 X 5-00082 x 7-48139 x 6-8735 ; of 16-437 x 286845 x
357-642 X -086824 x 00961 ; of -0632 x -07954 x 068689 x
•00074368 X 7329-688 ; of 10084 x 7-3695 x -000086957 x
22-46 X 37-89 x 058276 ; of 06 x 073 x -0892 x 6934 x
•82659 X -717876; of 179-63 x 482-7 x 795-687 x 555-55 x
•00000687324 ; of 17 x -18 x 19 x -21 x 23 x -64 ; of 4951
X 5-368 X 7-965 x 83837 x 0001 ; of 10112 x 1-0234 x
1-1378 X 1-6654 X 1-78877 ; and of 637842 x 97918 x
654837 X -0079892 x 00004632 x 000005948 ?
Division.
When you have in the same exercise several quantities
multiplied together, divided in any order by several
divisors, add the logaritlims of all the multipliers, and from
the sum subtract the sum of the logarithms of all the
divisors. In subtracting remember tliat your numtissa'
are always positive; that taking away a negative (luantity
is etpiivalent to the addition of a positive ([uantity ; and
that taking away a negative quantity is equivalent to
adding a positive quantity. Each of the points in this
paragraph is illustrated by an example below.
! );..Mo -ti ^ r, H V 7.« r, ]jy 07:V«4 v 5(i-67.3.
. « 7 .•» 1 H 4 "^
Here 00673, 100084, 97384 and 56673 are all divisors,
LOGARITHMS.
103
tlierefore add their logarithms and subtract the sum from
the sum of the logarithms of 31458 and 765.
Log. 00763 3-882525
" 100000 5-
PP. of 80 331
4 17
Locr. 973-8 2-988470
Log. 3-145 0-497621
PP. 8 110
Log. 7-65 0-883661
Sum T-381392
Subtract... 7-624737
PP. 4.
18
Log. 56-67 1-753353
23
PP. 3.
um i 0J4/O/
7-756655 which is the logarithm of
000000571025, the answer.
Divide 7-6348 by 0172.
Log. 7-634 -882752
PP. 8 JS
Log. 7-6348 •882708
« -0172 2-235528
2-647270 .-.log of 443-885, answer.
2-r47187 log. of 443-8
83
78 PP. of 8
50 " " 5
Divide -0172 by 763-48.
Log. -0172.. 2235528
" 763-48.. 2-882708
5-3527:{0 log. of -0000225284, answer.
VV. 8
PP. 4
5-352568 log. of 00002252
162
154
80
77
104
LOGAlilTHMS.
i
:!'.'
Divide -0076348 by '0172.
Log. -0076348 .... 3-882798
" -0172. 2-235528
1-647270 = log. -443885, answer.
Divide 0172 by -0076348.
Log. -0172. 2-235528
" -0076348.... ;-5-882798
0-352730 = log. 2-25284, answer.
Divide -172 by -00076348.
Log. -172 1-235528
" -00076348... 4-382798
2-352730 = log. 225-284, answer.
28. Divide 59-6834 by 47-9218, 736-68 by 795*887,
768000 by 754, 32788 by 6742-11, -76957 by -38G21,
•5948 by -038797, "008255 by -018173, -0006866 by
495-371, -0068524 by -0000798, and -0000798 by -0068524.
29. Find the reciprocal of 11, of 14-682, of 97-381, of
3754-68, of -01, of -0001, of -0036827, of -0098286, of
556789. of -556789.
30. Give the value in decimals of the following frac-
. 1 .'5 2_Ji.' 1 !>^t!.H Trt_2.SS.
.n H
.H « H
1 4 . !l « .'.
tlOnS: -^i}, toT' T's&i> 0THijy» liT-TT* .Vii' .0 12> 17.su 7.
- ** L*iAiL . • « >" ■-
a.-> It.WO 1' .00 7 !» 1 4 s" .
31. Find the values of the recinrocals of the fractions
in the preceding exercl3e.
32. Divide the product of 67 X 39-58 x -067879 by tlie
product of 112x7-8868x59-473. Find the vabie in
decimals of ^^^ x ,V/h x "^^i X ,-^,r. l^i"^ tlie vabie ..f
•3'_«J X "'•'^" X "-''^^ — -■'''"-. Find the value of «•««'•
1 .0 » 7 -i 1 4 . U H -J It V . 7 4 • H !• 5 7 1 . • U J 1
J4 _i./ -.1 1 « 7 X 1 •' • ).
15-»87'vi41lsy 1U7.'J01>'
Involution.
Hero observe cnrefullv that the carrying from any mul-
LOGARITHMS.
105
tiplication of luantissie is always positive, even although
the characteristic and its product are negative^ Thus,
we wish to get the cube of 9874, its logarithm is 1'994493
which wheu multiplied by 3 gives 1-983479, for the car-
rying from the multiplication of^ -9 by 3 is + 2 which
added to the three times 1 gives 1. As 1-983479 is the
logarithm of -902672, this last is of course the cube of
•9874, right to the fifth decimal place.
33. Find the square of 19-G82, and of -07975, the cube
of 7-6864 and of -0777777, the fourth power of 1-897, the
5th power of 1-4158, the 13th power of -77816, the 19th
power of 1-12345, the Uth power of the square of -02468
and the 9th power of the cube of 1-0124.
Evolution.
It will be remembered tliat to find the logarithm of the
root of a number the logarithm of that number is to be
divided by the index of the root. Tliis is easily done
when the characteristic and the mantissa of that logarithm
are positive; also when the characteristic although negative
is exactly divisible by the index of the root. So there
will be no dithculty in finding the logarithm of the sixth
root of that number whose logarithm is 5-739874, the quo-
tient, which is the logarithm of the root, being -956646.*
Xor is there any difhculty in finding the logaritjun of the
sixth root of that number whose logarithm is 12-358726,
the nearest quotient being 2059788, and being the logar-
ithm of the root sought, But if we are findi^ig the sixth
root of that number whose logarithm is IU'476793 we
meet the difficulty that 6 is contained in 10, 1 with a
remainder 4. How can this negative 4 be so blended with
* The pupil will notice that the last figure is a little too large, but it
is nearer the truth than 5 in that place would be.
lOG
LorjAKiTinrs.
: i
the positive mantissa that follows it, as to give a positive
quotient, and this we need because every mantissa is to be
positive. Perhaps at first sight the difficulty seems
insuperable ; but it may be readily overcome by saying
that 6 goes into 10, 2 times with a remainder of +2; for,
while it is quite true that if G be multiplied by 1 and the
result be subtracted from lO, 4 will remain, it is no less
true that if G be multiplied by 2, and the result be sub-
tracted from 10, + 2 will remain. The necessary division
is thus accomplished :
6)10-476793
T-4T2799
the quotient being the logarithm of the answer.
One example more will suflice. What is the cube root
of -0007585 ? The logarithm of -0007^85 is 4-879956 of
which the third part is found as above to be 2-959985.
This last is the logarithm of -091198 ; and this again is
the cube root of -0007585, right to the last digit.
34. Find the square root of 933156, of 8335*69, of
28-4089, of -383161 and of -00168921. Extract the cube
root of 804357, of 50-053, of 166-375, of 9, of 574. What
is the fourth root of -938, the fifth root of -0007, the sixth
root of 4-3928, the seventh root of -018195, the eighth
root of 1-0789, the ninth root of 548-733, the tenth root of
•0078145, the one hundred and nineteenth root of 67'7349,
the two hundred and fifteenth root of -000068782, and the
three hundred and seventeenth root of 115-317 ?
The preceding examples demonstrate the great assist-
ance that logarithms may be in effecting intricate calcula-
tions that involve multiplications, divisions, involutions
and evolutions only. It must never be forgotten that
additions and subtractions of natural nuniljers cannot be
effected logarithmically ; hence additions and subtractions
LOGAUITIIMS.
107
must be done either before logarithms are introduced in
the example, or after return to natural numbers has been
made. Thus, to solve the following problem, which
cannot be easily done witliout the aid of logarithms, the
following order of operations must be followed with but
slight variation :
s/'
/■
Loe.j> _ ^5 4 -J t -s ^ -1 n 61^)20 which is log. of 1-36873
•' l'243o8
a <.'
Log. ]^^ ij) 4 i.".j»ji^. 094672 '•
\ 11
>/r-3G873 -T-24358 = >^-12515
^"g- llllA = Ii"-l'l--* •'' ^ = 1 -81 0486 = log, of -659911
LoK^^ _ 1 . 1 4 «} 1 2 8 _ -229226 = log, of 1-69521
LoK% _ .0 1. H !. 7 _ -049926 = log. of 1-12185
^h - i/5 = 1-69521 - 1-12185 = -57336
LoK^.5rj5;5jj = r^_.VH4^^H^|.879214 = log. of -757205
•659911 + -757205 = 1-417116, answer.
;5. Find the value of (^/98?6 - ^49-59)
(V^n89->^1200-G8); of (V^8-l + x/37-695) (V'4;M-
V37lll)5); of ^7im~^0069~^'^b^8r^~^^^^
of VVl)^'-87xv^4877-V^6iF7^ and of
(^987327 -x/683-4\''
3i
X
()83-4\''
683-4/
!>v/'.)873!w+x/6J
36. By l()'4inithms solve the following examples from
'earlier pages: P. 5,-42, 43, 64, 67, 68, SI, 83, 86, t)l, 93;
p. 6,-4, 5, 7, 10,11, 14,18,29,21,23; p. 16,-42, 43,44,
46, 47, 48, 50, 51, 54, 61 ; p. 48,-25, 26, 29, 30, 31, 33,
34^36; pp. 50-52,-13, 14, 17,31,33,36,40,41 ; ])f. 53,
5.1 _10, 11, 16, 19, 20 'N, 32,39, 41, 43,44; i». 57,-2,3.
108
LOGAKITHMS.
ill
iH
G, 7 ; p. 50 --G, 8, 9, 10 ; p. 60—7, 8, 9, 11, U, IC ; pp. 64,
6-),—;!, 9, 10, 11, 14, 16, 18 ; p. 66,-8, 10, 11, 14.
37. What is the ainoimt at simple interest of $494*36
for 11 years at 41^ per annum ? What would it be at
compound interest ? What at compound interest is the
amount of 1 mill for 500 years, at 3^ per annum ? What
is the compound interest of $728"36 at 4^% for 23 years ?
38. What is the acreaije of a trianj^le whose base is
74'35 chains and altitude 49-77 chains ? of a triangle
whose three sides are 56'38 chains, 49-71 chains and 52-31
chains ? of a parallelogram whose base is 38-94 chains and
perpendicular height 42-17 chains ? of a rectangle whose
con-terminous sides are 15-69 chains and 14-48 chains?
of a rectangle of which one side is 13-87 chains, and the
diagonal 15-93 chains ? of a parallelogram of which two
adjacent sides and me diagonal are respectively 13-47
chains, 19-63 chains and 1618 chains ? of a circular
enclosure o£ which the diameter is 9-36 chains ? of a walk
one rod wide around the outside of this enclosure ?
39. If 45*37 were accurately raised to the one millionth
power, how many volumes of 320 pages each with 60 lines
on a page and 60 digits in a line would be required to
hold the answer ?
LOGAllITIIMIC TABLES.
109
MATHEMATICAL TABLES.
LOGARITHMS OF NUMBERS FROM 1 TO 10,000, WITH
riFFERENCES AND PROPORTIONAL PARTS.
Numbers from 1 to 100.
No.
IiOg> No.
liOfi.
No.
liOg.
No. i liOg.
No.
Ltog.
1
0-000000
2
0-301030
3
0-47;i21
4
O-COJOCO
5
6
0-698970
0-77Sl-)l
7
0-S-l.'>0'.)8
8
0-003090
9
0-;t->4243
10
11
1-000000
1-041393
12
1-0791S1
13
1-113043
14
M4i;i28
15
16
M7'WJ1
1-204120
17
1-230449
IS
1-255273
19
1-27875 1
SO
1-301030
21
23
24
1-322219
1-342423
1-361723
1-380211
L'j 1-397941)
20
27
23
29
30
31
32
33
34
35
36
37
3S
39
40
1-414973
l-4313o4
1-447158
l-4()2398
1-477121
1-491362
1-5051.">;I
1-518514
1-531479
1-544068
l-55630;5
1-56S:;02
1-579781
1 -591065
1-B02060
41
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
56
57
53
59
60
1-6127S4
1-623249
1-633468
1-643453
1-653J13
1-662758
1-672098
1-681241
1-690196
1-698970
1-707570
1-716003
1-724270
1-732394
l-7403ti:<
1-74SI88
1-755875
1-763428
1-770852
1-T78151
CI
62
63
64
65
66
67
63
69
70
71
72
73
74
75
76
77
73
79
80
1-785330
1-792392
1-799341
1-806180
1-812913
1-819544
1-826075
1-832509
1-838849
1-845098
1-851258
1-857332
1-863,323
1-869232
1-875061
1-880814
1-S86491
1-892095
1-897627
1-903090
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
93
99
100
1-908485
1-913814
1-919078
1-924279
1-929419
1-934493
1-939519
1-944483
1-949390
1-954243
1-959041
1-963783
1-968483
1-973123
1-977724
1-982271
1-986772
1-991226
1-995635
2-000000
110
LOGAKITIIMIC TAliLES.
pr
N. O
.'I
G
9
D.
41
Ki
12 J
Kit)
207
24.S
200
■SM
37.'}
3S
7()
11,H
151
1W»
227
2()5
302
340
100
1
2
3
5
C
7
y
no
1
2
3
4
6
C
7
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190051
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LOGAKITII.MIC TABLES.
Ill
vv
N.
:{
9
26
63
79
105
132
15S
184
210
237
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1
2
3
4
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118
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165
188
212
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67
89
112
134
156
178
201
21
421
64
85
106
127
148
170
191
230449 230704 230960
180
1
2
3
4
5
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7
8
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204934
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6
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255273 255514 255755
7679 791SI 8158
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264
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301030 301247
20
40
61
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101
121
141
162
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6
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3106
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2.50
249
248
246
245
243
242
241
239
238
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2;i5
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2.33
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32.3046
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2236
42.53
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206
205
204
203
202
202
201
200
199
198
112
LOGARITILMIC TABLES.
1.!
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1 ■■:
1
1
i 1
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1
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— 1 •*
1
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1
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6178
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7135
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8889
9083
9278
9472
9000' 9800:a''^0054
194
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4
350248
350442
3500.30 350.829
351023 35I2I6
3514:0
351003 351790
1989
193
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2.3751
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113
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195
194
193
193
192
191
190
189
189
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184
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LOGARITHMIC TARLES.
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127
38
3
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6.300
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126
60
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101
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123
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7775
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123
49
4
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LOGARITHMIC TABLES.
115
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128
3899
127
5167
127
6432
126
7693
126
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126
020-1
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1328
122
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3762
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4973
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6182
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120
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120
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573(1
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116
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116
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115
8525
115
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C
7193
7272
7352
7431
7511
7590
7670
7749
7829
7908
79
ftfi
t
7937
8067
8140
8225
8305
8384
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8022
8701
79
64
8
8781
8860
8939
9018
9097
9177
9256
9335
9414
9493
79
72
9
550
9572
9051
9731
9810
9889
9968
740047
740126
740205
740284
741073
79
79
740303
740442
740521
740000
740078
740757
740830
740915
740994
8
1
1152
1230
1309
13S8
1I()7
1646
1624
1703
1782
1860
79
Ifi
o
1939
2018
2090
2175
2254
2332
2411
2489
2568
2047
79
23
3
2725
2804
2882
2901
30.39
3118
3196
3275
3:553
3431
7ft
31
4
3510
3588
3007
3745
3S23
3902
3980
4053
41.-56
4215
7ft
39
5
4293
4371
4449
4528
4006
4084
4762
4840
4919
4997
78
47
5075
5153
6231
6309
63S7
5405
6543
5021
5099
6777
7ft
55
/
5855
6933
0011
6089
6107
6245
6323
0401
6479
6550
7ft
r.2
8
66:54
6712
0790
0808
6945
7023
7101
7179
7250
7334
78
70
9
560
7412
748188
7489
7507
7045
7722
7800
7878
748053
7955
8033
748808
8110
78
748200
748343
748421
74.S49S
74.8.570
748731
74,8.885
77
8
1
8963
9040
9118
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9272
9350. 9427
9504
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9(559
77
15
2
9736
9814
9S91
9908
750045
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750200
750277
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77
23
3
750508
750580
750003
750740
0817
0894
0971
1048
1125
1202
77
31
4
1279
1356
1433
1510
1.587
1064
1741
1818
1895
1972
77
39
5
2048
2125
2202
2279
2.350
24:53
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2893
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51
1
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6040
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76
76
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750484
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1
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6712
6788
680 J
6940
7'I16
7092
7168
7244
7320
76
15
2
7390
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7700
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8079
76
23
3
8155
82:50
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8533
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4
8912
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76
38
5
9r;<i8
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9894
9970
760015
760121
700190
760272
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75
4(1
6
700422
760498
760573
760010
760724
0799
0875
09.50
1025
1101
76
63
7
1170
1251
1326
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1477
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1027
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1778
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76
61
8
1928
2003
2078
215.1
2228
2:50a
2378
2453
2529
2604
76
(kS
9
2679
2754
2829
2904
2978
3053
3123
1 3203
3278
33.53
76
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118
LOGARITHMIC TABLES.
lUf
m
pp
N.
1
1
2
3
4
*
6
r
8
D.
75
580 ,'63428 1
763503
763578
76:^653 763727
763802 7638771
763952
764027
764101
7
1
4176
4251
4326
4400
4475
4550
4624
4699
4774
4848
75
15
2
4923
4998
5072
5147
6221
5296
6370
6445
6520
6594
75
??.
3
5669
5743
5818
5892
6966
6041
6115
6190
6264
6338
74
,S()
4
6413
6487
6562
6636
6710
6785
6859
6933
7007
70,S2
74
37
5
7156
Tm
7.'W4
7379
7453
7527
7601
7675
7749
7823
^
4i
6
78it8
7972
8046
8120
8194
8268
8342
8416
8490
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74
W
7
86;i8
8712
8786
8860
8934
9008
9082
9156
9230
9303
74
r>'\
8
9377
9151
9525
9599
9673
9746
9S20
9.S94
9968
770042
74
67
500
770115
— .
770852
770189
770263
77o:«6
770410
771146
770484,770557
770631
770705
0778
74
770926
770999
771073
771220 771293
771367
771440
771514
74
7
1
1587
1661
1734
1808
1881
1955
2028
2102
2175
2248
73
IT)
?.
2322
2395
2468
2542
2615
2688
2762
2835
2908
2981
73 1
^•^
3
3055
312.S
3201
3274
3348
3421
3494
3567
3640
3713
73
?')
4
3786
3860
39;w
4006
4079
4152
4225
4298
4371
4444
73
37
5
4517
4590
4663
4736
4809
4882
49551
6028
5100
6173
73
44
6
5246
6319
6392
6465
5.538
6610
5683
6756
5829
5902
73
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7
5974
6047
6120
6193
6265
6:j:J8
6411
6483
6.')56
6629
73
SS
8
6701
6774
6846
6919
6992
7064
7137
7209
7282
7354
73
66
11
600
7427
7499
7572
;644
7717
7789
7862
7934
8006
8079
778802
72
72
778151
778224
778296
778,368
778441
77a513
778585
778658
778730
7
1
8874
8947
9019
9091
9163
9236
9308!
9380
9452
9524
72
14
2
9596
9669
9741
9313
9885
9957
780029
780101
780173
780245
72
22
3
780317
780389
780461
780533
7»}605
780677
0749
0821
0S93
0965
72
20
4
1037
1109
1181
1253
1324
1396
1468
1540
1612
1684
72
36
5
1755
1827
1899
1971
2042
2114
2186
2258
2329
2401
72
43
6
2473
2544
2616
2688
2759
2S31
2902
2974
3046
8117
72
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7
3189
3260
ai;}2
3403
3475
3546
3618
3089
3761
3832
71
68
8
3904
3975
4046
4118
4189
4261
4332
4403
4475
4546
71
65
9
610
4617
4689
4760
4831
4902
4974
6045
6116
6187
785899
6259
71
71
7&5a30
785401
785472
785543
785615
785686
785757
785.828
785970
7
1
6041
6112
6183
6254
6325
6396
6467
6538
6609
6680
71
14
2
6751
6822
6893
6964
7035
7106
7177
7248
7319
7390
71
21
3
7460
7531
7602
7673
7744
7815
7885
7956
8027
8098
71
28
4
8168
8239
8310
8381
8451
a522
8693
8663
8734
8804
71
36
5
8,S75
8946
9016
9087
9157
9228
9299
9369
9440
9510
71
43
6
9581
9651
9722
9792
9863
993;i
790004
790074
790144
790215
70
50
7
790285
790356
790426
790496
790567
790637
0707
0778
0848
0918
70
57
8
(1988
1059
1129
1199
1269
1340
1410
14,80
1550
1620
70
64
9
620
1691
1761
1831
1901
1971
2041
2111
2181
2252
2322
70
70
792392
7924<)2
792532
792602
792672
792742
792812
792,S,82
792952
793022
7
1
3092
3162
3231
3301
3371
3441
3511
3rM
3651
3721
70
14
2
3790
3.860
3930
4000
4070
4139
4209
4279
4349
4418
70
21
3
4488
4558
4627
4697
4767
4836
4906
4976
6045
6115
70
28
4
6185
6254
5324
6393
6463
6532
6602
667i
6741
6811
70
35
5
588<)
5949
6019
6088
6158
6227
6297
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6436
6505
69
42
6
6574
6644
6713
6782
6852
6921
6990
7060
7129
7198
69
49
7
7268
7;«7
7406
7475
7545
7614
76.83
7752
7821
7890
69
56
8
7960
8029
8098
8167
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8;W5
8374
8443
8513
8682
69
63
630
8651
799341
8720
1 8789
1
8868
8927
8996
9066
9134
9203
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69
69
799409
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799547
799616
799685
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799961
1
800029
800098
800167
800236
800305
800373
800442 800511
8005,80
800648
69
14
2
0717
0786
0854
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1061
1129
1198
1266
133,5
69
21
3
1404
1472
1541
1609
1678
1747
1815
1884
1952
2021
69
28
4
2089
2158
2226
2296
2363
2432
2500
2568
2637
2705
69
!V>
5
2774
2842
2910
2979
3047
3116
3184
3252
3:J21
a^89
68
41
6
3457
3525
3594
3662
3730
8798
3867
3935
4003
4071
68
48
7
4139
4208
4276
4344
4412
4480
4548
4616
4686
4763
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55
8
4821
4889
4957
6()25
6093
6161
6229
6297
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64.33
68
62
9
65U1
6569
6637
67U6
6773
6841
6908
6976
6044
6112
tia,
LOGAPJTIDIIC TABLES.
119'
D.
34101
75
4848
75
65;»4
75
6338
74
7082
74
7823
74
mm
74
9303
74
mi2
74
0778
74
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74
2248
731
2981
73 1
3713
73
4444
73
6173
73
6902
73
6(529
73
7364
73
8079
72
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72
9524
72
S0246
72
0965
72
1684
72
2401
72
8117
72
3832
71
4546
71
6259
71
85970
71
6680
71
7390
71
8098
71
8804
71
9510
71
30215
70
0918
70
1620
70
2322
70
93022
70
3721
70
4418
70
6115
70
6811
70
6505
69
7198
69
7890
69
8582
69
9272
69
999«51
69
O0(;48
69
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69
2021
69
2705
69
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4071
68
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69
6433
68
6112
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1
2
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4
3
6
r
8
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640
806180
806248
806316
806384
806461
806519
806587 8066551
806723
806790
63
7
1
6858
6926
6994
7061
7129
7197
7264
7332
7400
7467
63
13
2
7535
7603
7670
77;i8
7806
7873
7941
8003
8076
8143
63
20
3
8211
8279
8346
8414
8481
8549
8616
8684
8751
8818
67
27
4
8886
8953
9021
9088
9156
9223
9290
9358
9425
9492
67
34
6
9560
9627
9694
9762
9829
9896
9964
810031
810098
810165
67
40
6
81023,3
810300
810367
810434
810501
810569
810636
0703
0770
08,37
67
47
7
0904
0971
10,39
1106
1173
1240
1307
1374
1441
1,508
67
64
8
1575
1642
1709
1776
1843
1910
1977
2044
2111
2178
67
60
9
2245
2312
2379
2445
2512
2579
813247
2646
2713
2780
2847
67
67
6,50
812913
812980
813047
813114
813181
813314
813381
813448
813514
7
1
3581
3648
3714
3781
3848
3914
3981
4048
4114
4181
67
13
2
4248
4314
4381
4447
4514
4.581
4647
4714
4780
4847
67
20
3
4913
4980
6046
6113
6179
6246
6312
5378
6445
6511
66
26
4
6578
6644
5711
5777
5843
6910
6976
6042
6109
6175
66
as
6
6241
6303
6374
6440
6506
6573
6639
6705
6771
6838
66
40
6
0904
6970
7036
7102
7169
7235
7301
7367
74.33
7499
66
46
7
75()5
7631
7698
7764
7830
7896
7962
8028
8094
8160
66
63
8
8226
8292
8358
8424
8490
8556
8622
8683
8754
8820
66
59
9
660
8835
8951
9017
9083
819741
9149
9215
9281
9346
9412
9478
66
66
\
819544
819610
819676
819,807
819873
819939
820004
820070
820136
7
1
820201
820267
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320399
820464
820530
820595
0661
0727
0792
66
1,
13
?.
o858
0924
0989
10.55
1120
1186
1251
1317
1382
1448
66
20
3
1514
1579
1645
1710
1775
1841
1906
1972
2037
2103
65
26
4
2168
2233
2299
2364
24;iO
2495
2560
2626
2691
2756
65
33
6
2822
2887
2952
3018
3083
3148
3213
3279
3344
3409
65
39
6
3474
3539
3605
3670
3735
3800
3865
3930
3996
4061
65
46
7
4126
4191
4256
4321
4386
4461
4516
4581
4646
4711
65
62
8
4776
4*11
1906
4971
6036
6101
6166
6231
6296
6361
65
69
9
6426
5491
5556
826204
6621
6686
6751
826399
6815
6880
6945
6010
65
65
670
826076
826140
826269
826334
826464
826528
82f..593
826658
6
1
6723
6787
6852
6917
6981
7046
7111
7175
7240
7305
65
13
2
7369
7434
7499
7563
7628
7692
7767
7821
7*86
7951
65
19
3
8015
8080
8144
8209
8273
8333
8402
8467
8531
8.595
64
26
4
8660
8724
8789
8853
8918
8982
9046
9111
9175
9239
64
32
6
9;i04
9368
9432
9497
9,561
9625
9690
9754
9818
9882
64
38
fi
9947
830011
830075
830139
830204
830268
830.332
830396
830460
830525
64
45
7
830589
0653
0717
0781
0845
0909
0973
1037
1102
1166
64
61
8
1230
1294
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1422
14,86
1550
1614
1678
1742
180(i
64
68
9
1870
1934
832573
1998
2062
2126
2189
2253
2317
2381
2445
64
64
680
832509
8326.37
832700
832764
832828
832,892
8329,56
83;W20
833083
6
1
3147
3211
3275
3;i38
3402
3466
3530
3593
3657
3721
64
13
2.
3784
3848
3912
3975
4039
4103
4166
42;w
4294
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64
19
3
4421
4484
4548
4611
4675
4739
4802
4866
4929
4993
64
•
25
«
6056
6120
6183
6247
6;J10
6373
64.37
6500
6.564
6627
63
82
6
6691
6754
6817
6asi
6944
6007
6071
6134
6197
6261
63
38
6
6324
6387
6451
6,514
6.577
6()41
6704
6767
6a30
6894
63
44
7
6957
7020
70,83
7146
7210
7273
7,3;i6
7399
7462
7525
63
60
8
7688
76;72
7715
7778
7841
7904
7967
8o:w
8093
81.56
63
57
9
8219
8282
8345
8403
8471
8534
8597
866U
8723
8786
63
"63
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838849
838912
838975
8390;i3
839101
839164
839227
839289
839,352
a39415
' 6
1
9478
9.541
9()04
9667
9729
9792
9855
9918
9981
840043
63
13
2
840106
840169
840232
840294
840;«7
840420
840482
840545
840608
0671
63
19
3
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0796
0859
0921
0984
1046
1109
1172
1234
1297
63
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4
ia59
1422
1485
1547
1610
1672
17a5
1797
1860
1922
63
32
6
19S6
2047
2110
2172
2235
22i»7
2360
2422
248-1
2547
62
38
6
2609
2672
2734
2796
2,859
2921
2983
3046
310S
3170
62
44
7
323:i
3295
3;«7
3420
3482
3514
3()0«
3669
3731
3793
62
60
8
3855
3918
3981J
4042
4104
4166
4229
4291
4355
4415
62
67
9
4477
45;«
4<)01
4664
4726
4788
4860
4912
4974
60.3f
02
120
LOGARITHMIC TABLES.
i p '
If
I, I
1
pp
N. O 1
'2 :i
4
"i
« 7
H
D
700
m5093
845160 8452221
846284
845346
845408
845470
846532
845594
845656
62
6
1
6718
5780
6842
6904
6966
6028
6090
6151
6213
6275
62
12
2
6337
6399
6461
6523
6535
6646
6708
6770
6832
6894
62
19
8
6955
7017
7079
;i41
7202
7264
7326
7388
7449
7511
62
2.1
4
7573
7631
7696
7758
7819
7881
7943
8004
8066
8128
62
:i\
5
8189
8251
8312
8374
8435
8497
8559
8620
8682
8743
62
V
6
8805
8866
8928
8989
9051
9112
9174
9235
9297
9358
61
4.3
7
9419
9481
9542
9604
9665
9726
9788
9849
9911
9972
61
M
8
850033
850095
850156
550217
880279
850340
850401
850462
850524
8505a5
61
56
9
0646
0707
0769
0830
<:891
0932
1014
1075
1136
1197
61
61
710
851258
851320
851.381
851442
851503
851564
851625
851686
851747
&51309
f)
1
1870
1931
1992
2053
2114
2175
2236
2297
2.358
2419
61
12
2
2180
2541
2602
2663
2724
27J5
3394
2846
2907
2963
3029
61
1«
3
.3090
3150
3211
3272
3333
34.55
a5i6
3577
3637
61
24
4
30!»3
37.59
3820
3881
3941
4002
4063
4124
4185
4245
61
81
5
4.306
4367
4428
4488
4549
4610
4670
4731
4792
4852
61
37
6
4!»I3
4974
5034
5095
5156
6216
5277
6337
5,398
5459
61
43
7
.W19
55,80
5640
6701
6761
6822
5882
5943
6003
6064
61
49
8
6124
6185
6245
6306
6366
6427
6487
6543
6608
6668
60
65
9
720
6729
6789
6850
6910
8.57513
6970
7031
7091
7152
7212
7272
60
8.^>7.332
857.393
8.574.53
857574
857634
857694
8577.55
857815
857875 60
6
1
7it:«
7995 i
80.56
8116
8176
8236
8297
83.57
8417
8477, 60
90781 60
12
2
8.537
8.5971
8657
8718
8778
8838
aS98
8958
9018
18
3
9138
91981
92.53
9318
9379
9439
9499
95.59
9619
9679 60
24
4
9739
9799'
9859
9918
9978
8601».38
860098
8601.5,8
860218
860278 60
0877 60
1475 60
80
5
HGO;«S
860.398
.8604.58
860518
860578
06.37
0697
0757
0817
86
6
0937
01W6
10.56
1116
1176
1236
1295
1355
1415
42
7
1534
1594
1654
1714
1773
18.33
1893
1952
2012
2072 60
48
f
2131
2' 91
2251
2310
2370
2430
24.89
2549
2608
2668
60
64
9
7'M)
2723
80;tt23
r787
2847
2906
2966
3025
3085
3144
863739
3204
86.3799
3263
60
S6;i382
86.3442
86.3501
863.561
863620
863680
8&3.S68
59
6
1
3'.'ir
3977
4036
4096
4155
4214
4274
433;i
4.392
4452
59
12
2
4y.i
4.570
46;i0
46,89
4748
4,808
4867
4926
4985
5045
59
18
8
5104
6163
5222
.5282
&341
5400
64.59
5519
5578
5637
59
24
4
5696
57.55
.5814
5,874
5933
6992
6051
6110
6169
6228
59
80
5
62.S7
6.146
6405
6465
6524
6583
6642
6701
6760
6819
59
a-)
6
6878
69.C
6996
7055
7114
7173
7232
7291
7350
7409
59
41
7
7467
7526
:.>s5
7644
7703
7762
7821
78,80
7939
7998
59
47
8
8056
8115
8i/4
82;i3
8292
8;«o
8409
8468
8527
8586
59
63
9
740
6644
8692:«
8703
8762
8821
8879
8938
8997
869584
9056
9114
9173
59
59
869290
869319
869408
869466
869.525
869642
869701
869760
6
1
9818
9877
99.35
9994
8700.53
870111
870170
870228
870287
870345
59
12
2
870404
870462
870.521
870.579
06;i3
1 0696
0755
0813
0872
0930
53
17
8
0983
1047
1106
1164
1223
1281
1.339
1398
1456
1515
58
28
4
1573
1631
1690
174H
1806
1865
1 1923
1981
2040
2098
58
29
5
21.'>6
2215
227.3
2;i3l
2.3,89
2448
2.506
2564
2622
2681
58
85
6
2739
2797
2.855
2913
2972
30,30
30,83
3146
3204
3262
58
41
7
3321
.3379
.3437
8495
3.5.53
3611
36ti9
8727
3785
3844
53
46
8
3'.K)2
3960
4018
4076
41.34
4192
42.50
430a
43(56
4424
68
52
9
7.')0
4482
4540
4598
4656
875235
4714
4772
4^311
4838
4945
5003
58
8750611 sri^l 19
876177
875293
875351
875409
875466
875.524
875.582
58
6
1
.5640 )69S
6756
6815
5,871
6929
6987
6045
6102
6160
53
12
2
6218 6276
63.3:
6:191
6449
6507
6561
6622
6681
6737
53
17
3
6795 68.5;J
6910
6968
7026
70a3
7141
7199
7256
731^
63
23
4
7.37' 742t
7487
754^
7602
76.59
7717
7/.'4
7835
788£
68
29
5
7947
800-i
8062
6111
817<
82;n
8292
8349
840;
846^
57
85
6
8522
857J
863;
869J
8752
8809
8866
8924
8981
9039
67
41
7
9096
91.5;
9211
926}^
9325
9.38S
9440
949]
9.5.5.1
9612
67
46
8
0669
972(
978-1
9841
989S
99.56
880013
88007U
880127
88018!
67'
1^1 '
8H0242
880299
88086e
I8804U
1 880471
880528
I U689
W43
t
069S
•
0756
i
67
D
5«
62
7S
62
94
62
11
62
28
62
43
62
58
61
72
61
85
61
97
61
09
61
19
61
29
61
37
61
45
()1
52
61
59
61
(54
61
68
60
72
60
75
60
77
78
60
60
79
60
78
60
60
60
77
75
72
60
<m
60
63
60
)68
59
52
59
45
59
>37
59
•28
59
no
59
(H)
59
m
59
m
59
73
69
'60
59
145
59
)30
58
)15
58
)98
53
m
58
J()2
58
^44
58
124
58
J03
68
58
58
58
68
68
67
67
67
57;
571
LOGAUITHMIC TABLES.
121
PP
N.
3
G
9
D.
760 880814 880871
13S5
11)55
2525
3093
3GGI
4229
4795
530 1
5926
1442
2012
2581
3150
3718
42,S5
4852
5413
59S3
SSC131
7054
7017
8179
8741
9302
9802
890421
0980
1537
880928
1499
2069
2038
3207
3775
4342
4909
5474
C039
880985
1556
2126
2695
3264
3832
4399
4905
6531
609G
6
11
17
Of*
27
33
38
44
^9
.0
22
27
32
38
43
49
•80
1
2
3
4
6
6
7
8
9
790
1
2
8
4
6
6
7
8
9
892095
2051
3207
3702
4310
4S70
5423
5975
0526
7077
880547
7111
7074
8230
8797
9358
9918
890477
1035
1593
892150
2707
3262
3817
4371
4925
6478
6030
6581
7132
88GG04
7107
7730
8292
8853
9414
9974
890533
1091
1649
S80G60
7223
77
8348
8909
9i70
890030
• 03S9
1147
1705
881042
1613
2183
2752
3321
3888
4455
6022
6587
6152
88G71G
7280
7842
8404
6905
9520
8900S0
0045
1203
1700
881099
1670
2240
2809
3377
3945
4512
6078
5044
6209
881156
1727
2297
2866
3434
4002
4509
5135
5700
C2G5
881213
1784
2354
2923
3491
4059
4625
6192
6757
6321
881271
1841
2411
2980
3548
4115
40S2
6248
5813
C378
897627
8170
8725
9273
9821
900307
0913
1458
20'>3
254.
89220G
2702
3318
3873
4427
4980
6533
6085
6030
7187
800
1
2
S
4
"5
6
7
8
810
6 1
903090
3033
4174
4716
5256
5796
0335
6874
7411
7949
897682
8231
8780
9328
9875
900422
09681
15131
2057 1
2G01|
903144
3087
4229
4770
6310
6850
6389
6927
7405
8002
892202
2S18
3373
3928
4482
6030
6583
6140
6092
7242
88G773
7336
7898
84G0
9021
9582
890141
0700
1259
ISIG
892317
2873
3429
3984
4538
6091
6G44
6195
6747
7297
88G829 8808S5 886942
7392,
7955
8516!
9077
9038
890197
0750
1314
1872
7449
8011
8573
9134
9094
7505
80G7
8029
9190
9750
8902:)3 890309
0812 1 0308
1370 1 1420
192S, 1983
881328
1898
2468
3037
3005
4172
4739
5305
6S70
6434
88G99S
7531
8123
8085
9240
9800
8903C5
0921
MS2
2039
67
67
67
67
67
57
57
57
57
56
11
2
16
3 S
21
4
27
6
82
6
37
7
42
8
id
9
9084S5
9021
9556
910091
0624
1168
1690
2222
2753
897737
8286
8835
9383
9930
900470
1022
1567
2112
I 205:;
903199
3741
4283
4824
6364
6904
6443
6981
7519
8056
897792
8341
8890
9437
9985
900531
1077
1022
2106
2710
897847
8390
8944
9492
900039
05SG
1131
1676
2221
2704
903253
3795
908539
9074
9610
910144
0078
1211
1743
2275
2806
892373
2929
3184
4039
4593
6140
5099
0251
6802
7352
897902
8451
8999
9547
900094
0010
1186
1731
2275
2818
903307
3849
433V 4391
4878 4932
6418 6472
6958
6497
7035
7573
8110
3284 S337
908592
91281
9663
910197
0731
1264
1797
2328
2859
3390
908646
9181
9716
910261
0784
1317
1850
2381
2913
8443
6012
6551
7089
7026
S163
908699
9235
9770
910304
0838
1371
1903
2435
2966
3496
903361:
3904
4445
4986
6526
6006
6604
7143
7680
8217
924i;9'
2985
3510
4094
4048
5201
6754
6300
0857
7407
897957
8500
9054
9002
900149
O095
1240
1785
2329
2873
903416
3958
4499
6040
6580
6119
6658
7196
7734
8270
892484 892540
3040! 3030
3595
4150
4704
5257
58091
6301
6912
7462,
3051
4205
4759
6312
6804
6116
C907
7517
898012 898007
85011 8015
9109 9104
9050' 9711
900203 900258
56
56
56
50
56
56
56
56
50
892595
3151
3700
4201
4814
6307
6920
6171
7022
7572
07491
1295
1840
2384
2927
903470
4012
4553
6094
6034
6173
6712
7260
7787
8324
56
50
50
55
55
55
55
55
55
55
908753
9289
9823
910358
0891
1424
1956
2483
3019
3640
0804
1349
1894
2438
2981
903524
4006
4607
6148
6688
6227
6766
7304
7841
8378
898122
8070
9218
9700
900312
0859
1404
1948
2492
3030
908807
9342
9877
97.0411
0944
1477
2009
2541
8072
8602
908860
9396
9930
910404
0998
1530
2063
2594
8125
8665
55
55
55
55
55
55
55
64
54
64
903578
4120
4601
620;
6742
6281
682(t
7358
78i;j
8431
908914 908967
94491 9503
9984 910037
910518
1051
1684
2116
2647
3178
3708
0571
1104
1637
2109
2700
3231
3701 63
54
54
54
54
64
54
54
64
54
54
54
64
63
63
63
6H
63
63
63
-I !
199
LOGARITHMIC TABLES.
PF
N.
!
1
3
»
4
5
G
7
8
9
D.
8?0
D138H
913867
913920
913973
914020
914079
914132
914184
914237
914290
63
5
1
4343
4396
4449
4502
4555
4608
4660
4713
4766
4819
63
11
?
4872
4925
4977
6030
6083
6136
5189
6241
6294
6347
63
If)
3
6400
6453
6505
6558
6011
6664
6716
6769
6822
5875
63
?1
4
5027
6980
6033
6085
6138
6191
6243
6296
6349
6401
53
?7
5
6454
6507
6559
6612
6664
6717
6770
6822
6875
6927
53
3'^
fi
C9S0
7033
7085
7138
7190
7243
7295
7343
7400
7453
53
37
7506
7558
7611
7663
7716
7768
7820
7873
7925
7978
52
4?.
8
8030
8oa3
8135
8188
8240
8293
8345
6397
8450
8502
52
48
9
s-^o
8555
8607
919130
8659
6712
8764
8816
8809
8921
6973
9026
919549
52
52
919078
919183
919235
919287
919340
919392
919444
919496
5
1
OGOll
9653
9706
9758
9810
9862
99141
9967
920019
920071
62
10
?.
920123
920176
920228
920280
920332
920334
920430
920489
0541
0593
52
l(i
3
0045
0697
0749
0801
0853
0906
0953
1010
1062
1114
62
?.\
4
110(5
1218
1270
1322
1374
1426
1478
1530
1582
1634
62
?(■)
5
1086
1738
1790
1842
1894
1946
1998
2050
2102
2154
62
31
(\
2206
2258
2310
2302
2414
2466
2518
2570
2022
2674
52
31)
7
2725
2777
2829
2asi
2933
2985
3037
3089
3140
3192
52
41
8
3244
3296
3348
3399
3451
3503
3555
3607
3058
3710
52
47
840
3702
3814
3865
3917
3909
4021
924538
4072
4124
4176
4228
52
52
924279
924331
9243&3
924434
924486
924589
924641
924693
924744
5
1
4790
4848
4899
4951
6003
5054
5106
5157
5209
6201
52
10
?
5312
6364
6415
5407
5518
6570
6621
6673
5725
6770
52
ir.
3
5828
6879
6931
6982
6034
6085
6137
6188
6240
6291
51
?0
4
0342
6394
6445
6497
6543
6000
6651
6702
6754
6805
61
?(1
5
0S57
6903
6959
7011
7002
7114
7106
7216
7268
7319
61
31
f)
7370
7422
7473
7524
7576
7627
7078
,7730
7781
7832
51
3(i
7
7883
79;j5
7986
6037
8083
8140
8191
8242
8293
8345
51
41
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10
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20
25
29
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pp
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ANSWERS.
125
ANSWERS.
I. Page 2.
1.
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1440.
2.
6.
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420.
3. 1140.
7. 68640.
4. 1680.
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15. 41, 1435.
19. 12, 1080.
12. 9, 405.
16. 46, 2070.
20. 19, 114.
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III. Pages 4, 5.
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180/.iV-
02 9
2S'
1.
6.
11.
16.
21.
26.
31.
36.
41.
46.
51.
56.
61.
66.
71.
76.
<-» -t
o i. .
86. h
91. *.
1
B-
liTff'
2i.
33§.
I
,'ihV
A-
2.
7.
12.
17.
22.
27.
37.
42.
47.
52.
57.
62.
67.
72.
77.
82.
87.
92.
1 '^^
4.1 19
129
1 ;»7
* 9 •
7i -Vt)'
9 il 5
1 o;i7 !•
Its-
1 1
laJi*
1 2 5 2 •
10.
a (> •
1 ;13
r2 0-
5tV
20.
6^.
7
;( 2 •
18,
3.
8.
13.
18.
23.
28.
33.
38.
43.
48.
53.
58.
63.
68.
73.
7 8.
83.
88.
93.
28Af's.
2S.4
^15'
■^liO*
2 :>
t i O •
1200.
Hh
16i.
26i.
4^i.
Of,.
1
1 1 f
U-
n-
\ 2
Co-
1.0 71!
1 »XL>
T2tT0'
20Ai
46.A-
4.
9.
14.
19.
24.
29.
34.
39.
44.
49.
54.
59.
64.
69.
74.
79.
84.
89.
SOi?^
2 H 5 •
Ol 9
•> I 1 y .
5
»'
t .1 r, 'J .
14.
2654 1'rt.
1|-
a ••.9
1 5
lO"
1
48'
13
IH-
5.
10.
16.
20.
25.
30.
35.
40.
45.
50.
55.
60.
65.
70.
75.
80
85.
90.
^90*
1187,.
1 7:>
2 1
S O"
'J 1 9 •
•J 2
4H.
7 '• U
ij o'
193^.
5
V 0'
2 9
srs'
oi r.
14/f.
Q I ')
.26
ANSWERS.
IV. Pauk 6.
1.
3f^.
2.
m-
3. 14. 4.
17
5. 7[g
6.
2^.
7.
H-
8. m- 0.
/o.
lO. H.
11.
2-
12.
u.
13 I'i.V 14.
IH.
15. I.
16.
mu.
17.
4ii^
18. A\. 19.
12,V
20. n'».
21.
t> 7 <i
T4 7 2'
22.
3|.
23. 11 24. 28.
25.
f 26.
V. Pages 7, 8.
3. 48. 4. 1540
1. 36 feet. 2. 880.
7. llil 8. $360. 9. $32i5>^.
14. 279i. 15. /A.
19. ^5225. 20. 43i'V
24. The latter by hU.
13. 2|.
18. $1371.
23. 6Ui^.
26. 5. 27
31. $110 A,
10. 6A.
16.
5o4
• TOt*
1 1 ^
£1. 188.
21. 1
6. 27/1.
12. m
I'T 44
22. 35^.
^il^?^. 28.
.."1
29. 90H^I
32. Increased by ^^q
25. The former by IjVs
30 Q257.12 8 3.91SU
VII. Page 10.
1. 3. 2 -4. 3. -58.3. 4. 36. 6. 13. 6. 714285. 7. 1*.
8. Oi. 9. OOi. 10. 01. 11. 6001. 12. 023.
13. 5-230769. 14. 738. 15. 67083. 16. 2 037. 17. 168i.
18. 20138. 19. '108. 20. 4-.376068. 21. 34. 22. 106.
23. 26074. 24. 8-20. 26. 1 0099. 26. -315476196.
«
27. 60571428. 28. 5045. 29. 295138. 30. 072.
61.
3D5{i!'
52. 32
U' 53.
3ie?..
54.
10
u.
55. 3^',.
56.
•'i.'i-
57. 4j
58.
i» fli i{
59.
7.'.
% •
60. HA-
61.
20.u^
62. ,,V,«- 03.
;iti 1
ft ft rt •
64.
21
,1 .1 ;i •
65. 5.;;;''.
66.
17, 5o.
67. 6;;
U' 08.
tt 4.1
^ftOO'
69.
mi
70. 2^0.
VIII. Pages 10, 11.
1.
15-44049.
2.
13 38482.
3.
18 28.360
4.
-02020.
5.
•54586.
6.
-02302.
7.
16 41805
8.
60 49949.
9.
•23036.
10.
•52468.
11.
•01952.
12.
•00652.
13.
•04594.
14.
•03948.
15.
•25793.
16.
.')6 31427.
17.
00041.
18.
00298.
19.
5 r 50>)4.
20.
34.5.
21.
•46359.
22.
7 03411.
23.
•03293.
24.
2 -88 184.
25.
03911.
26.
3 44424.
ANSWERS.
127
VIII. Pages 10, 11 {continued).
27. 1-654108. 28. l-63022ri. 29. -8502588315. 30. 4-94332696.
31. 2-738437.
32. 1-7. 33. 1-57.
34. 2-365. 35. -64431.
36. 6-5234416. 37. 10756. 38. 3910284. 39. 24. 40. 15.
41.42. 42.3-86011. 43.83. 44.25. 45.28. 46. 18213.
IX. Tages 11, 12.
1 £7. 16s. 8(1. 2. 2 dys. 3 hr.«. 20 .nin. 38-4 sec. 3. 250 y^^«^2-64 m.
4' 4 mi. 800 yds. 5. 5 min. 6. 42 wks. 10 lus. 30 miu 7 280 .
a 1 ft. 9. 15 sq. yds. 2 ft. 80.V^; in. 10. 9 lbs 8 oz. 13 chvt. S grs.
11. 3tous 18evvt. 36 6 lbs. 12. 2 s.,. yds. 8 ft. 12-. 8 m.
13.5 yds. 2 ft. 1 1 -892 in. 1 4. 243| rods. 15 1. In s .>2 ..uu
39-36 sec 16. 58 mi. 303 52 rods. 17. tl Lis. M.
18. 5 ilio ndn. 36 sec. 19. 1 ft. 6 in. 20. 4 dys^ 23 hrs. 20 nun.
2 1 . 40 yds. 22. 16 dvvL. 8 grs. 23. 39 tons 14.^2 lbs.
24 172+ 25. -215. 26. 2583. 27. Oii 28. '8.3.
29. 275+ 30. -22916. 31. 416. 32. 4182. 33. 168595 +
34. 0464+ 35. 3263 +
X. Pages 12, 13.
1.
8.
13.
17.
20
25.
•4. 2. 2-3625. 3. 5. 4. 'iU-
$728.
6. 7d. 6. 13cwt. 7. -525.
9. -0027 +
14. 215 U>«-
The former by -0001,
10
;l ;l 5
11
1 5
2 1'
12. ml
•-•7
15. 10 rods 2 yds. 2= in. 16. 10.
18.
Hi'^-
19. 4320 times.
•000125.
21
0:1
;i (I i
22. 6. 23. 2253-456 min. 24. 1010.
1150. 26. 2061 yds.
XII. Pages 16, 17.
37.
42.
47.
62.
56.
60.
64.
67.
70.
74.
101
38 21
39. m
40.
49.1
003649. 43. 09
3. 44. 0072. 45. 5 15.
:i 1 -i
r :t i S
48
or
49. 3(i.
50. 017
£11. 139.
45 tons 10 owt.
4d. 53. S 10,968. 54. $9720
41. 12iV
46. 51A.
51. -6891 +
55. $8465.
57. 900 days.
58. 35 gal. 59. S5.28.
7i cents,
lib. 1411! o/..
7 lbs, !3 oz.
$438. 7 1
2390 men.
61. 7957 J,' mi
lea. 62. ,h
65. 140 hrs. 37A mn
68. 3 tons
19th May.
75. -979.
400 ll)s. 6i o/,.
72. 65 days.
76. 24 days.
63. €56. 5s.
66. *8.32i.
69. 178 855 mi.
iH. $<"*0.
128
ANSWERS.
XIII. Paoks 18, 19.
11. (a) $35, m, So6. {b) $40, ^44, |56. (c) $20, $40, $80.
1 2. (a) A $26, B $40. (/>) A $43, B $23. (c) A $45t, B $20^.
id) AUn, B$24i
13. (a) A $20, B $10, (' $1.5. (h) A $25.50, B $12.75, C $6.75.
(c) A $24, B $14, C $7. (<0 A $23.75, B $9.37 V, C $1 1.87^.
14. Each man $10, each woiiiau $7.50, each child $2.50.
15. Each mail $11.25, each \v( iiaM $7 50, eacli child, $5.
16 $497, $355. 17. $61.25, $78.75. 1 8. $106|, $100, $1.33i
19. 9,\ days. 20. 16 days. 21. 339,226,113. 22. A $49tV,
B$30i;|. 23. A $15,600, B $20,400, C$24,000.
24. 12 hrs. 26. 16 days.
2010
TIT, 7 5
1. 15. 2. 16. 3
7. 5^ wks. 8. 130 yds. 2 ft.
1 1. 18 days. 12. 8 men.
16 219 days. 16. r\ days.
19. 160 strokes. 20. 15 days
XIV. PAOE.S 20, 21.
4. 1^5- 5. $34.12i. 6. 14 lbs. 2^6^07.
9. 15 horses.
13. 15 men.
17. $15.3.60.
10. 160 bushels.
14. 221;' *Ws-
18. 57'i months.
21. C2. 10s. 22. 60 men.
23. 6f days. 24. $2064. 25. 50 men.
XV. PAOE.S 22, 23.
1. 12yr3. S^moa. 2. $287?v. 3. Ig mistakes. 4. 200 yds.
5. 6 hrs. 6.52%. 7. 35 gal. 8.4%. 9. 98 clerks. 10. $536^
XVI. Pages 23, 24.
1. 2 A days. 2. 3j\\ns. 3. 8 hrs. 4. 12i hrs. 5. G.J hrs.
6. 17) min. ; 24 min. 7. 3] hrs. 8. A 19» dys. ; B 13f dys. ;
C \0s dys. 9. A 15 dys. ; B 18 dys. ; C 20 dys.
I» ^ *4
AJ^SWKU.S.
129
XVII. Pages 24-26 (continued).
15. {a) 4 hrs. S/V i»i»- ; 4 hrs. 38i\ min. (6) 10 hr". 5i\ min. ;
10 hr.s. 88i"r "»«• (<^) 1 l^i'- 21iV "!>"• 5 1 I"- ''^*i"i """•
16. In 120 days (2 p.m. April 24tli) ; 1.44 p.m. and 2.14 p.m.
17. 5.10 p.m. 18. 12th May; 5 p.m.
XVIII. Page 26.
1. 244-4625 francs ; S3.68t*,,.
4. 30 cents. 5. 56 ^^^ lbs.
2. 167 fr.
3. £11. 3s. 3d.
XXI. Pa(je8 29, 30.
1. $366.50. 2. $6480 ; $4728.50. 3. («) $156.06 ; (/>) $46.41 ;
(c) $109.37i. 4. (a) $2400 ; (b) $8400 ; (<•) $18,000. 5. (a) $5100 ;
(/>) $2.36.9r; (o) 15,506.2.3. 6. (a) $6825 ; (/>) $4696.65 ; (r) $4,345..37.
0. 27„. 10. $161.10. 11. $260. 1 2. 2035 barrels (nearly).
XXII. PAGE.S 30, 31.
1. (a) $25 ; (h) $10 ; (r) $487.50. 2. (a) 1.^ p.c. ; (h) 2^ p.c.
3. (ft) $2000; (^>) $10,523; (r) $6800. 9. $765.62.1. IQ. $6666.66^.
11. $40.84. 12. $.30,000.
XXIII. Pages .32, 3.3.
A. 3. $177.60. 4. 4J p.c. nearly. 5. $17.76. 6 $84,000. 7. $3000.
B. 2. $20,475. 3. $149.45. 4. $1234.80. 5. $7 095. 6. $46.19.
7. $.33.44. 8. $2190.69. 9. $994.31. 10. $3039.75; $9394.88.
11. $903.60. 12. $1440.
XXIV. l»AOE8 34,37.
2. (a) .$527 ; $31. (l>) $183.60 ; $5. 10. (r) $5257.50 ; $225.
(</) $ 1 1 68. 3 1 i ; $4(5. 50. (r ) $ 1 111 . 50 ; $48. 00.
3. (a) $875 ; $.35. (I>) $1920 ; $96. {r) .$2066^ ; $62.
4. (a)79|. {h)'Mk- ('•) in2.i. 5. (a) $875. (/;) $256.66.,. (<) $173.3. .33.^.
(./) f28S0 4^. 2d. 6. (a)$673.3.V (A)$l 1.312.50. (.). $6945. (./) £1843.
7. (<i) 72i. (f>) HSl (<•) 78. (</) 73^. 8. (a) ^TL- (M *V/o- ('')'^r/..' OOr^R-
9. (a) Sf/v (M ar/.. ('■) 4,r/.. ('0 ^Ho- 10- (") ^^'^ '"^i 1"^'' ^•'^"<''-
(h) the same. (<•) the 3^ per cents. (.7) the 3',' per cents, (r) the 3,^
1 i. (a) h>sH of .>».;5;».^. I'fl LMttJ of .$109.50.
V
'V cent
>S-
(r) lo.sa of 75 cents, (f/) less nf $1 .80. (r) gain of $3.50.
130
ANSWERS.
XXIV. Pages 34-37 {confirmed).
12. $8000. 13. poo. 14. 4n. 15- ^4925. 16^^831.
17. £26,933. 6s. 8d. 18. $32.41. 19. 7aVP-«- ^O. $^obO.
2 1 . 86j acre.. 22. £1 19^. 23. Rates 4.V'^ and 4^ P-c. ; as lo5
to 147. 24. £90. 25. $3104. 26. $22i gam. 27. ?f/15.
28 £222. lOs. loss. 29. The former by 'OO^. 30. 607,.
A.
B.
XXVI. PA«iEs 39 45.
1. $54.95. 2. $49.22. 3. $35.96. 4. $48.98. 5. $88.60.
6. $17,520.81. 7. $171.69. 9. Balance due from W . $1^91.
1 O. Balance due from H. $3.57.V. 1 2. Balance due to S. $0.2b.
3. $720.64.
(93 days).
15. $1360.
4. 4th Sept. ; $44.71 ; $39-55.29. 6. $343.01
7. $337.36. 11- $664 balance. 13. $658.54.
16. $576.49. 17. (c) face of cheque $307.03.
XXVII. Pages 46-48.
5. Uh lbs.
1. 70 loads. 2.1264. 3. 20 tons. 4. 7 tons 162., lbs.
6. 28,000 silk-worms. 7. 9 Ib... I o/. 7 dwt. 12 gvs. 8. 2.3 lbs.
6090 grs. 9. 1240 grs. ; 27221] lbs. 1 0. lo/, lbs.
1 1 1 hr. 12 min. 48 sec. 1 2. 6 wks. 4 dys. 9 hrs, 22 nun.
1 3. $9. 36. 14. 1011 miles. 15. 200 miles. 1 6. 25 43 4- sec.
17 494f]!? sec. ; a little more than | of a scond. 1 8. 631 pieces ;
rem. '0.33 inch. 19. 24 ropes. 20. 120 times; 22 yds. ; 4 rods.
7-92 in 22. 12 mi. 880 yds. 23. 1788-38 yd.. 24. 21 ac.
2-28 s,.. chains. 25. 21216 sq. rods. 26. 498 acres.
2 acre. 28. 93412 s.i. yds. 29. 41 ft. 2^,' in. 30. 67H loads.
6-2321 gal.; 997136 oz. 32. 37501b,. 33. 41,250,000 tons.
'>624-64oz. 35. $2637.43. 36. 512,000 cub. ft.
21
27
31
34
37. 599 miles. 38. 1694 mi. 560 yds.
XXVIII. ]'AGE.s 49 52.
1.618-334 Kg. 2. 1-20 f.. 3.125 1. 4- 113 50 fr. 5 14^5 K,„.
6 1020 276 Km. 7. 92 francs. 8.10 m. 9. 24,000 Kg.
-^ .11 11 o7 nrwn 12 1 Kir. 13 55 2 metric tons.
14. 45^2088 Kg. 16. 5670 francs. 16. 10,070 1. ; 10,070 Kg.
17. 3 francs. 18. 62 25 Kg.
ANSWEUS.
131
19.
24.
27.
30.
33.
37.
40.
XXVllI. Pages 49-52 {continued).
1093 yds. ; 2539 cm. ; 914 n». ; 1609 Km. 20. 1550016 sq. in.;
6-451 sq. cm. ; -836 sq. m. ; "404 Ha. 21. '061 cu. in ; "275
cord ; 16 386 cu. cm. ; -704 cu. m. 22. 568 1. ; 045 HI. ;
36-36 1. 23. 015 grains ; -004 g. ; 28349 g. ; 31103 g.; 453 Kg.
19 mi. 19-6 rods. 25. 2 4068 cu. m. 26. 285 71 1.
8-178 cu. m. 28. 9 dwt. 19 343 grs. 29. 2 <m. ft. 1502 cu. in.
•189 in. 31. 5/j lbs. ; 2457 Kg. 32. 3 hrs. 517 min.
1516 + Kg. 35. 03937 inches ; 3280[| ft. 36. 61 02 cu. in.
567 grain! 38. 66-93 lbs. Troy. 39. 5 1.
484 cm. (nearly). 41. -474 1.
B
XXIX. Pages 53-55.
855625. 4. 39t^g. 5. 592J.
8. 1 -44289;. 9. 1015-075125.
12. 31'255-875. 13. 17596287801.
A. 1. 1681. 2. 1156. 3
6. 2524-0576. 7. 29791.
10. mi 11. ^nmih
14 21^,V,. 15. 118587876497. 16. ?) = ;!. 17. mUni-
18 781. 19. -035012791. 20. 9210025. 21. 525.
22. 1632104.32. 23. 1580iV^\v 24. 14-885593.
25. 53-841087. 26. 118955463.
1 24 2 41 3. 34. 4. 69. 5. 137. 6. 428. 7. 102.
o' 516 9. 1-262. 10. 15 3. 11. 0231. 12. 32-768.
13. 6.1. 14. 61. 15. 2Ah. 16. 10§. 17. i- 18. I
19. ^V 20. i
21. 1-414. 22. 2 449. 23
27. -894. 28.
32. 1038.
26. 2-886.
31. 15-368.
28 213. 24. 9-022.
•031. 29. 22-992.
25. 2 091.
30. 37-21.
33. 15.
C.
37.
41.
45,
48
1. 13.
8. 611.
14. 2.!i
•16.
327 yards
72 yards.
6-928 ft.
34. 12.
38. -45.
35. 42. 36. 9539.
39. -03. 40. -24.
42. 126 soldiers. 43. 76 rods. 44. 205 rods.
46. 160 rods ; 80 rods. 47. 36 inches.
49.20 ft. 50. 103614 I ft. 5 1. 75-816 I ft.
2 23 3. 27. 4. 79. 6. 341. 6. 192. 7. 512.
9.1-07. 10.30-1. 11.067. 12.616. 13. J.
15. 3^. 16. 21. 17. 17|.
18. 4179. 19. 4-762. 20.
oo inn 94 539-'*< neiirly.
•4S2. 21. -149. 22. 3264.
25. 70 inches. 26. 1 cub. ft.
27. (a) 15 min. (M 12 yds 28. (a) 625 lbs. (/>) 6 in.
132
ANSWERS.
11
XXX. Pages 56 60.
A. 1. 70 feet. n. 37ro(lt3. 3. Glinka. 4. 4i yainls. 5. 7 ft. 6 in.
B.
8. 97ii links.
6. $93.09. 7. $75.35.
1 . [a) 350 sq. yards. (/>) 89-3 sq. yds. (c) 8 ac 1 rod.
((/) 13-69 s<i. in. {e) 150^14 sq. ft. (/) 70-14J8 sq. yds.
2. 14 yds. 3. 768 rods. 4. $0.66. 5. 14 ac. 4440 sq. yds.
6. 582-06 yds. (nearly). 7. 25 chains.
8. 150, 200, 250 yards ; 45,000 sq. yar J ■
C. 4. 62-ac. 5. 12,480 sq. yds. 6. 4 ac. i: . ,ds. 7. 1 ac. 134 rods.
8. 5 ac. 155.VtH »«^l8- ©• '^^^ ac 10. 7 chains 70 links.
1). 3. (a) 5028* sq. ft. (h) 2011? ac. (c) 10311 sq. ft. 4. (a) 3181 A
sq. yds. ' (h) 8 sq. ft. nearly. 5. (a) 3-98 ft. (/>) 27-7 yds.
6. 114-^ sq.ft. 7. 336 times. 8. 13596 yds. ; 14686 yds.;
5 44 yds. 9. 50-92 ac. 10. 640V 7r'-= 1134 08 yards.
1 1 . 641 i- s,i. rods. 1 2. 1 145-45 s(i. ft. ; 6928 s«i. ft. ; 900 sq. ft.
13. 12^432 revolutions. 14. 968 yds. 15. 15 093 in.
16. 39-25 yards.
XXXI. Packs 61-68.
A 4. (a) 9g sq. ft. (ft) 204i s(i. ft. (c) 2341 sq. ft. (d) 155^^ sq. ft.
(e) 630':^- s<i. ft. 5. (a) 8h s^l- ft. ('') G4Jo sq. ft. (c-) 176.Vi sq. ft.
6. (a) (96+4V2^)-=10r656 sq. in. (/>) 9^ sq. ft. nearly.
7. (a) 1^ sq. ft. {h) 21 A sq. ft. {r) 10.\^ sq. ft. 8. 67.^ sq. ft.
9 16 7r = 50? lbs. 10. 10 ft. 9 in. ; 99,846 n- »»• 1 1- 245 in.
perniin. 12. lOOy' 2^ - Hl'^ ^t- 13- eu. ft. 408 cu. in.
14. 421 cu. ft. 1512 cu. in. 15. 10032 cu. ft. ; 7-392 cu. ft.
B. 3. 4 sq. feet. 4. 4V 3 - 6928 sq. in. 6. 37? sq. in.
6 122t sq. in. 7. 13 in. ; 204? s-i. in. 8. 321;} cu. ft.
9. 2 cu. yds. 24-54 ft. ; 12 sq. yds. 7-846 ft. 10. 1 eu. yd. 9-083
ft : 9 s<i. yds. 5-59 ft. 11. 169? cu. ni. 1 2. 38^ sq. ft.
13. £16. 13s. 4d. 14. 5Sh sq. yds. 15. 6i ft. 16. 77B cu. ft.
17. 11 H 1} ft. 18. 93391 360 cu. ft.
C. 3. 2011 sq. in.
6. 381? cu. in.
10. 1403 11)3. nearl>
4. 20106-2400 scj. mi. 5. 523.H cu. in.
7. 14J cu. in. 8. 245.1§lb8. 9. 11 : 21.
oi i,. r.n 1 O. 4r * 'Ml ft.
iv. 11
\
L to 1 o
_ oi f,. r.n 1 Q 4
13. 1 :6.
14. 709', 11»H.
ANSWERS.
1 o*)
XXXI. Pa(!Es (Jl OS (conthmed).
1,3 441 cub fl. 4. 40 K,,. ft ; 51^ s, ft.
5 lois4. in. G. 122isM.i". V. £8o8. lo.. Sd. (nearly .
11.39tl3usl8S0^;ylhs. 12. lls.i.ft. 13. Betimes.
A. 1. -428571. 2.
XXX III. PAfiKS 73-75.
?;7295.43. 3. §73.50. 4. 107,. 5- 1^'' rupees.
li. 1
5
C. 1
1). 1.
3.
1. 2. 2942-2775 sii- yai
The same, .^(505.28.
h; IMI.'.-^)- 2. GMuin
C
.Is. 3. 8 day:
4. 4i".. p.
3. A £22. 10s. ; P. £37. 10s.
£45. 4. 2:104 mi. 1480i yd.s. 5. 117.1
•0021+. 2. 27, ■"', minutes past 2.
St. Tetersburg, 2 In.s. 1 n.in. ,2igeo. P.M.
Belli
l)ul)l
in.
Olu
s. iJ.>
mill. 30 sec. r.M.
11 hrs. 34 min. 5(5 sec. A.M.
New York.
4 $50,350i.
71
U'S.
4 mill. 8 sec. A.M.
5. 24,708 tiles.
K. 1. 101.
2. .S 1 03.87 i.
ft.
3. S12.60gain.
F. 1.
4.
a. 1.
5.
5. 123-888 sq. ft. ; 7 "87
4-23 }<)/.. ; 58-293 111.
$240 ; $0720.
2. 11?^ p. c.
5. 0t-=--201f s(i. 1
a
lies.
4. 4Uft.
3. S0.88:
4-38
2.
7.27 1. 3. 14,479,074 gal. 4. 2 23008 lbs.
2513-J8CU.
it.
134
ANSWKll.S.
XXTX. Pa(iks 81 101,
«•■■
•01 ; 10;
2-.3010:i ;
1-70927;
•94448:^ ;
•845098
•688064
•000000
6; 6; 1; 1 ; 6 ; T ; 4 ; T- 2. 4 ; :5 ; 4 ; (J ; .. ; ..; 1. 3. 2, 3,
2;2;5;I;2; 1 ; 5 ; 6. 4. 1 ; 2 ; 1;2; 1;2; 1;2; 1; .
6 1000000; 1000000 ; 10; 10; OOOOOl ; 'l ; 0001 ; "l ; 10000; '001 ;
10000 • -OOOOOl ; -00001 : 'OOl ; 'l ; 100 ; 1000 ; 01 ; 100 ; -00001 ;
•1- -01 • 10; 100000; -OOOOOl; 10; 100; -1 ; '01 ; 10; 100; •! ;
•01 . 10- -1. 6. 100; 100 ; 10000 ; 100 ; OOOOOl ; 100
lO-'lOo'^ 1-1. 7. -69897; 'OO'iOO ; 1-3010:); 1-39794
1-^515; 50515; -90309; 2-39794; 1-69807; 2-70927: ^
•70927 ; 2-09691 ; -09691 ; 2-60-206. 8. 1 ; 3 ; 1 ; 4 ; 4 ; o ; ;
• • 3 • 6 • 1. 9, 1 O, 1 1 , 1 2 and 1 3. The answers to these
exeroise/arJ easily verified hy orclinary arithmetic. 14. '5941 71 ;
•678063; •99-2421; -771073; -680336; •83'2509 ; -874/ .2 ; -r242.6 ;
•867467- -290480; -003029. 15. -671265; 671265; -6>126o;
•944483; -944483; 944483; '944483; 944483; 944483;
•845098; -845098; •84,-.09S; -7'24-276 ; 938019; -979548;
•976902- -204120; -000000; -000000; -000000; -000000;
■000000; 000000; 000000; -7330.37; -828015; -992150.
16 3-671265; 1-671265; '2-671265; 3944483; 2 944483;
1-944483 0-944483; T-944483; 2-944483; 3 944483; 3 845098;
iMms' 1-845098 0-845098; 0-724276; 2 938019; 5 979548;
?-688064; 0-976902; 3204120; 3000000; 2 000000; lOOOOOO ;
0-000000; 1000000; 2000000; 3 000000; 4^000000 ; 1 •73303. ;
4-828015; 2992156. 17. 1-152859; V 9.9821; -68124 ,
0-755417- 0-895533; 1-988291; 2-632153; 3-834o48 ; 1//604. ,
l^\ r771367; 3-765669; 2792392; 2845008 ; H)03090 ;
1-979639; 2-986010; 4900586 ; 5166726 ; ^;^''- ^^■,;^\
2605: 3662; 5017; 6157; 7469; 8000; 8666. 19. 580000,
67610- 8062- 9.351; 1546; 2142; 2758; 03367; 003634;
«8 00004316. 20. 139753; -141356; -424519; -596461;
•fi'^'til 5- •755015- -755015; -755015; -755015; •75;)015; 9.0b.,a;
• 3 7- -to' 3.' 21. 0-566685; 1-656883; 3 832094; 2 902538;
S;;0; ;-934410; 1-068869; 4-774463; 1 •.K)2255 ; ») •
22. -553757; -834092; -900893; -003034, -930119; 948.>63 ,
-979284; 986226; 988420; -999440. 23. •rK,4443 ; OSO.,1- ,
-769877- -770656- -791126; -706217; -896009; •9.)1280; -0608^ ;
- ;.' 2^ 5-56 .68; 4-734984; 5 885830; 5 953631 ; 2-804695;
i^^,, 4-566U1 ; 4-744723; 001 649 ; 0^00002^ 2..^ 3 ;
129153 or 5; 161620; 211655 or 6; 281.2.8 ^^'J^ '^ ^f^'f^;'^!'
477421 ; 615193 or 4 ; 776466 or 7 ; 7i)6587 or 9. 26. 1046 83 ,
ANSWERS.
135
XXIX. Pages 101-108 {continued).
,28-785; 10-2351; 2 0««7 , •2-'««* ' J^f ?« ' .^^.^Id ^
•000582684; 0000752562; -OOOOOS-'OT. 27. "^^ | ; ^f « « ;
r;r';oor"'28-~i;^^=;mt
^tofit ISSSn- 454246; -00000 1 :W03 ; 85 8094; -0116455
Lr^OO '^«B''05- -0.02.^1; -O0O20O3-2S;10O;.O«0; 2^^
I0V743, -00000179602; 1-79602. 30. 68421; -^<bo6-; 2,4892,
'804446 -000268455; -7-293; 7-23.^35; -836177; 1^''' j^.^^'J^]
f ^0-88 ;-255. H6 4. 32. -1««4« '; -049.8.8 ; -2,7178 ;
.^4 .' 33 387-3^1; -00636007; 454-„7; -000470505; 12^^^;
^ ~4.»S4,rs:i:^:-o8;'^::^;
^ nnnnnfi^n4M493 36. See answers in proper places.
•758415 ; -000000004444 J3. f"^' \ o o j s -502 acres ;
37. 1719.65; $772.64; $2621.20; ^1 -12. a. 38 18 oO,
\^f^ ifti-90S lores • 22-719 acres ; 13m00/ acies , -i iou«
37.8729 acres ; 164 208 '^^^^ '^^^ ^ ,j ^.^^..^es would be
acres; 6-88086 acres; 3 2o4b9 ^c^es ^ ^^^^
needed, but the last volume would have 4 blank pages,
^Ige 3 blank lines, and the last Une 50 blank spaces.