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{(Sff- ^^ 



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THE Girr OF 
Urs. Slancb* Harley 










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A Key 



to Rosens Explanatory 

and 

Practical Arithmetic, 

by 
John Rose, 



-■< » 

u * rf -■ » 

» J ■> * * 

• 4 

■i J 



Philadelphia, 

Thos. Sutton, and Denny 

and Walker, 

1835. 



'A 






w:> . ^' •^< ■■"< 



^C'f3 -4/ 



'"I 0-\ tc- 



CONTENTS- 



Page 

Division 5 

Problems and Miscellaneous 

(Questions 10 

Reduction of Federal Money. . 12 

Addition of Federal Money. ... 13 

Subtraction of Federal Money. 13 
Multiplication of Federal Mo- 

ney #.'1 »• .. 14 

Division of Federal Money. ... 15 

Reduction 17 

. Addition and Subtraction of De- 
cimal Fractions 28 

Multiplication and Division of 

Decimal Fractions 29 

Compound Subtraction 30 

Compound Division 31 

To find the Value of a Decimal, 

&c 34 

Multiplication and Practice... 35 

Common Measure 40 

Multiple 41 

Ratio and Proportion 42 

The Single Rule of Three 42 

The Double Rule of Three 50 

Interest : 52 

Discount 61 

Compound Interest by Decimals 65 

Cubic Measure 67 

Square Measure 70 

Oblong Square 72 

Paving and Plastering 72 

Shingle, or Roof Measure 73 

Circle Measure 73 

•Round Timber, &c 74 

Admeasurement of Stone in a 

Well „.... 74 

Fellowship 75 

Loss and Gain 80 

Equation of Payments 85 

Barter 85 

n 






Pag» 
Custom-House Allowances .... 88 
Method of Assessing Taxes. ... 89 
Reduction of Vulgar Fractions 90 
Addition of Vulgar Fractions. . 97 
Subtraction of Vulgar Fractions 100 
Multiplication of Vulgar Frac- 
tions 102 

Division of Vulgar FractioAs. . 103 
Promiscuous Q,uestions in Vul- 

gar Fractions 103 

Single RuliB of Three in Vulgar 

Fractions 106 

Double Rule of Three in Vulgar 

Fractions ■. 109 

Involution..... .-.- 311 

Square Root Ill 

Cube Root 114 

Extraction of the Roots of all 

Powers 117 

Further use of the Square Root 122 
Further use of the Cube Root. . 126 

Exchange 133 

Cubic and Square Measure .... 141 

Multiplication Contracted 144 

Alligation Medial 146 

Alligation Alternate 147 

Permutation 148 

Combination 148 

Arithmetical Progression 148 

Geometrical Progression 151 

United States' Duties 152 

Single Position 154 

Double Position ]56 

Ships' Tonnage '. 160 

Gauging 161 

Annuities at Compound Into- 

rast 163 

Perpetuities at Compound Inte- 
rest 166 

Promiscuous Examples 167 



€3 



^.! 




EXPLANATION OF CHARACTERS. 



Signs. Significations. 

= The sign of equal ; as, 100 ceiits=tl, 
+ The sign of addition ; 38,6+4=10. 
■^ The sign of subtraction; as, 8 — 2=6. 
X The sign of multiplication i aa, 4x3=12. 
-i- The sign of division ; as, 12-i-3=4.,or ^=4;. 
■ : : : The sign of proportionality ; as, 2 : 4 : : 6 : 12. 
v' The sign of the square root ; as, ^36=6. 
•^ The sign of the cube root; as, ^64=4. 
^ The sign of the fourth root j as, ^64=2. 

'he sign of dollais ; as, 98., or 8 dola., or 8 D. 
"he sign of a decimal part of a dollar ; as $.75, 
or, 75 cts., or 7B c, 

-6=13. A vinculum connects all the num- 
bers over which it is drawn ; 9 less 2 more 
6 equals 13. 
5=4 denotes 12 lees 3 and S equal 4, &c. 
le third power of 6 ! 6x6x6=216. 



A KEY 

TO 

ROSE'S EXPLANATORY AND 
PRACTICAL ARITHMETIC. 



LONG DIVISION- 
EXAMPLES. 

(2) 35)875(26 Ans. or, 5)875 (3) 52)1248(24 Ans. 

70 104 

7)176 

176 208 '^ 

175 quotient 25 208 

(4) 14)1624(116 Ans- or, 2)1624 

14 

7)812 

22 

14 Ans. 116 

84 
84 

(5) 82)6880(215 Ans. or, 4)6880 

64 

— 8)1720 

48 

82 Ans. 215 

160 
160 

a2 



DIVISION. 

(6) 24)288(12 Ans. (7) 56)504(9 quotient 
24 504 



48 
48 



(8) 16)2176(186 Ans. or, 2)2176 

16 

8)1088 

57 

48 quotient 136 

96 
96 



(9) 45)270(6 Ans. (10) 18)2556a42 Ans. 
270 18 

75 
72 

36 
86 



(11) 22)8514(387 Ans. (12) 807)46015(57 quotient 
66 4035 



191 6665 

176 5649 



154 16 remainder. 

154 — 



DIVISION. 7V^ 



(13) 18)12532(696 Ans. or, 3)12532 ;' 

108 

6)4177 + 1 



173 



162 Ans. 696 + 1 

112 

108 1x3 + 1=4 



remainder 4 When we use two such 

- numbers, as being multi- 
plied together, produce the divisor, and a remainder oc- 
curs, multiply the first divisor by the second remainder, 
and to the product add the first remainder, for the true 
remainder. 

(14) 15)3120(208 Ans. (15) 14)1526(109 Ans. 
30 14 



120 126 

120 126 



(16) 32)19520(610 Ans. (17) 125)176000(1408 Ans. 
192 125 



32 510 

32 500 



1000 

- 1000 



(18) 16)2368(148 quotient, or, 2)2368 

16 

8)1184 

76 

64 148 Ans- 

lis 

128 



8 , DIVISION. 

(19) 04)17484(186 Ans. (20) 225)16875(76 Ans. 
94 1575 



808 1125 

752 1125 



564 
564 



(21) 56)18144(324 Ans- (22) 36)792(22 Ans. 
168 72 



134 72 

112 72 



224 
224 



(24) 24)25020(1042^ miles, Ans. 
24 

(28) 13)598(46 Ans. Or, 4)25020 

52 102 * 

96 6)6255 

78 



78 60 1042.5 Ans. 

— 48 

ii" 

(26) 321)231120(720 Ans 
2247 

(25) 236)708944(3004 Ans. 

708 642 

642 

944 

944 



DIVISION. 9 

(28) 62^0)131812^0(2126 Ansu 
124 



(27) 666)309136(556 Ans. 

2780 78 

62 

3113 

2780 161 



124 

3336 

3336 372 
372 

(29) 56)24864(444 Ans. (30) 72)11376(158 Ans. 

224 72 

Or, 7)24864 Or, 6)11376 

246 417 '- 

• 224 8)3552 360 12)1896 

224 Ans. 444 576 Ans. 158 

224 576 

(31) 144)979776(6804 Ana. 

864 (32) 25;(K))525^00(21 Ans. 



1157 

1152 25 



50 

2^ 
25^ 



576 
576 



(33) 15)26250(1750 Ans. (34) 135)505845(3747 Ans. 

15 405 



112 Or, 3)26250 1008 

105 945 

5)8750 

75 634 

75 Ans. 1750 540 

945 

945 



■» «» « ■ •■ I ■•^■V^^M^^^^w^^^^pq^B^^I 



lO DIVISION. 

(35) 84)14700(175 Ans. (36) 175)14700(84 Ans- 
84 1400 

Or, 7)14700 . 

630 700 

588 12)2100 700 

420 Ans. 175 
420 

(37) 144)2592(18 Quot. (38) 18)2592(144 Ans. 
144 18. 

Or, 3)2592 

1152 79 

.1152 72 6)864 

72 Ans. 144 
72 • 

(39) 425)218875(515 Ans. 

2125 * (40) 8675)5257050(606 Ans. 

62050 



637 

425 52050 

52050 

2125 

2125 



PROBLEMS AND MISCELLANEOUS QUESTIONS. 

PROS. 4. ^EXAMPLES. 

(1) Sum 48-^2=24 half the sum. Or, sum 48 
Diff. 14h-2= 7 half the difference. diff.l4 

Ans. 31 larger number. 2)34 

Ans. 17 less number. 17 less. 

+ 14 diff. 



31 larger. 



ADDITION OF FEDERAL MONEY. 



13. 



(12) $35 and 4 cents =3504 cents. Ans. 

(13) 1000 mills =$1. Ans. 

(14) 7512 cents =$75.12 cents, or 75 doUwrs and*12 
hundredths. • 



ADDITION OF FEDERAL MONEY. 

EXAMPLES. 

(2) Thus, $52.65 (3) $132,125 (4) $16.35 



8.16 
4.21 
7.01 



1.250 
.75 



3.40 
10.00 



— ^ Ans. $134,125 Ans. $29.75 c. 

Ans. $72.03 c. 

(5) Thus, $19.00 (6) $1.06 (7) $75.00 

9.00 5.50 15.75 

4.25 50.00 9.25 

.75 

Ans. $56.56 c. Ans. $100.00 



Ans. $33.00 



SUBTRACTION OF FEDERAL MONEY. 

EXAMPLES. 

(1) A hat cost $4.25 c. (2) $22.25 c. 

Boots — , 3.75 c. — , 18,00 



Ans. diff. .50 c. 



(3) $50,000 
— , 16.455 



Ans. $4.25 c. 

(4) From $25,000 
Take 12.375 



Ans* $33.54,5 m. 

(5) $319.00 
— , 47.56 



Ans. diff. $12.62,5 m. 

(6) $3.00 
— , .75 



Ans. $271.44 c 



B 



Ans. $2.25 c 



14 BfULTlPLICATION OF FEDERAL MONET. 

(7) Thus,*6.00 (8) t7.07 c 

—,1.16 —,6.06 



Ans. S3.85 c Ans. 82.01 c 

(9) 8911.06 0. (10) From 8110.00 

—,626.00 Take—, 18.10 



Ans. 8286.06 c Remainder 891.90 

Add + 22.56 



Ans. 8114.46 c. 



MULTIPLICATION OF FEDERAL MONEY. 

EXAMPLES. 

(1) Thus, 86.45 c. (2) 81.126 (3) 4726 A. 

5 20 2.26 c. 



Ans. 832.25 c. Ans. 822.50^0 23630 

9452 
(4) Thus, 88.36 c. 9452 

225 c- 



Ans. 810633.50 c. 

Gain 86.11 per week. 

52 (5) 85.87^5 m. 

12 yd. 



1222 — 

3056 Ans. 870.50^0 



Ans. 8317.72 c 

(6) 16 lb. or, 16 (7) 32 lb. (8) 75 c 
8 c 8.08 8.04 8 



Ans. 81*28 c 81.28 c 81-28 cw' Ans. 86.00 



DIVISION OF FEDERAL MONEY. . 15 

(9) $.876 m. (10) Thus, ••125 m. 

175 bu. 96 ]b. 



4375 750 

14875 1125 



Ans. 8153.125 Ans. 812.000 

(11) .25 

.25 (12) 125. 



.8 



125 

50 Ans. 100.0 product. 



Ans* .0625 product. 



DIVISION OF FEDERAL MONEY. 

EXAMPLES. 

(1) Thus, 4)636.96 (3) 7)3213 

(2) 8)8230.00 



6)159.24 12)459.00 

Ans. 828.75 c. 

Quotient 826.54 c. Ans. 38.25 c 

(5) 125)8181.25(81.45 c. Ans. 
(4) 3816)95400 0.(25 c. Ans. 125 

7632 

^ 562 

19080 500 

19080 



625 
625 

(7) 6)8234. (8) 6)87.60 c- 

(6) 4)344 c. 

12)39.00 Ans. 81*25 c. per day. 

Ans. 86 

Ans. 83.25 c. 



16 DIVISION OF FEDERAL MONEY. 

(10) 100 bu. X 87'c.=8700 c 
(9) 26)«14.00(S.56 c. Ans. Then, 5)8700 c. 
125 



150 



5)1740 



150 Ans. 348 lb 

(11) Thus, 81.05 c. 

55^0 bu. 

(12) 1.25)53.75(43 days. Ans. 

526 500 
625 



376 



7^0)5775^0 c. 376 

Ans. 825 bu. 
(13) 9.)8.1 (14) .9)8.1 

Ans. .9 Ans. 9« 



(15) 761b. 
at A 


Or, 8.1876 
76 


2)228 
8)114.00 
Ans. $14.25 c. 


11250 
13125 


Ans. 814.25 

■a 
m 


(16) 


2)89.06 c. 


Ans. 84.53 c « 






REDUCTION. 17 

REDUCTION. 
TROY WEIGHT. 

EXAMPLES. 

(3) Thus, 2 lb. 10 oz. 6 dwt. (4) 4)184320 grs. 
12 



34 
20 

686 
24 



2744 
1372 



6)46080 
2^0)768^0 dwt. 
12)384 oz. 
Ans. 32 lb. 



■J 



Ans. 16464 grs. 

AVOIRDUPOIS WEIGHT. 

EXAMPLES. 

(3) 3)10240000 dr. 



(1) Tha8,16T. 
30 

Am. 330 H. 

4 

Ans. 1380 qr. 
25 

6400 * 
3560 



Ans. 33000 lb. 
16 



Ans. 513000 os. 
16 



Ans. 8183000 dr. 



8)5130000 
3)640000 oz. 
8)330000 
5)46nDH>. 



(3) 5)67300 lb. 
5)13440 
4)3688 qr. 
3y0)67^3 H. 
Ans. 33 T. 12 H. 



5)6000 (4) 765T. 8H. Oqr. 161b. 

' 30 



4)1600 qr. 
8/))«I^H 
Ans. 30 T. 



15306 H. 

4 

61233 qr. 
25 



306176 
133464 



Ans. 1530616 lb. 



b2 



#^ 



18 REDUCTION. 

APOTHECARIES' WEIGHT. 

EXAMPLES. 

(1) 17 lbs. (2) 2;0) 133200^5 grs. 



12 



204; 



3)66600 scruples 



8 8)22200 drams. 

16325 12)2775 oz. 
3 



48969 



Ans. 2311bs.3?03095grs. 



(3) 5 lbs. (4) 2;0) 18849,^6 grs. 
12 



60 



3)9424 16 



8 8)3141 1 

Ans. 

16)480(30 parcels. 12)392 5 

48 • 







Ans. 32 lbs. 8^ 55 19 16 grs. 



CLOTH MEASURE. 

EXAMPLES. 

(1) 127 yds. (2) 4)9173.na. (3) l^^e. E. 



4)2293+1 na. — 

508 qrs. Ans. 35 qrs. 

4 Ans. 573yd. Iqr.lna. — . 



Ans. 2032 nails. 



? 



t 

c 

■A 

I 

ft 



in. Ans. 
(4) 27)729(27 
54 

189 
189 



REDUCTION. 19 

e» Fl. e. Fr. e. Sc. 

(5) 1 (6) 3 (7) 27.2 in. . 



54 



25 



3 
4 



Ans. 162 in. 



1360 
544 



Ans. 12 na. 



Ans. 680.0 in. 



LONG MEASURE. 

EXAMPLES. 

(1) 273 miles (2). 12)34594560 in 



528^0 

2184 
546 
1365 

1441440 feet 
12 



Ans. 



528^0)288288^0(546 m, 
2640 



2428 
2112 



Ans. 17297280 in. 



(3) 360 dega. 
69.5 



3168 
3168 



(5) i)86400 turns 
(4) Imile 18* 

820 — — ^i^'" 



1800 
3240 
2160 

2502^00 m. 
176^0 



32^0 p. 
25 



160 
64 



15012 
42534 



1555200 

28800 

3)1584000 feet 

Ans. 

176^0)52800y^0(300m. 
528 



An9. 8000 links. 



00 



44035200 yds. Ans* 



20 REDUCTION. 



TIME. 



EXAMPLES. 

(I) 28 days (2) 366,25 days 
24 24 

(3) 4)84 

112 146100 — 

56 73050 Ans. 21yrs. 



Ans. 672 hrs. Ans. 8766^00 hrs. 
60 



Ans. 525960 min. 
60 



Ans. 31557600 sec. 



(4) 365.25 days (6) 6;0)18410703840^0 

11 . 

weeks 6,0)306845064^0 



4017.75 (5) 14 

24 7 4)51140844 



16071 00 Ans. 98 days 6)1278521 1 

80355' — Ans 

865.25)2130868.50 (5834 

hr. 96426 00 Ans. 182625 



304618 ** ^ 
292200 \. 



124185 
109875 



146100 

14610* 1^ 



I 






i 



f 






REDUCTION. 21 

days d. h. m. sec. 

(7) June - - 25 (8) 365 5 48 57 

July - - 31 24 

August - 31 

Sept. - - 30 1465 

October - 9 730 



Ans. 126 days. 8765 
6,^0 

525948 



6^^0 



Ans. 31556937 sec. 3=1 sol. yr« 



sec. sec. 
(9) 31556937)126227748(4 years. Ans. 

126227748 



• 










MOTION. 







EXAMPLES. 




(1) 4 sigs. 23deg 
30. 


. 15ffl. 


34sec. 


(2) 12 sigs. 
30 


143 deg. 
60 


• 




360 deg 
60 


8595' 


21600' 


60 




■ 


60 


Ans. 515734 sec. 


Ans. 1296000" 



22 KEDUCnON. 

(3) 6^0)113138^^5 seconds. (4) 6^0)16680,^0 

6^0)1885^6 25 6^0)278^0 



3^0)31^4 16 Abs. 46^^20' 

Ans. lOsig. 14deg. 16min. 25 sec. 
SQUARE MEASURE. 

EXAMPLES. 

(1) 29a. 3r. 19p. (2) 4;0)19Vp. 

4 

4)49 37 p. 

119 — 

. 40 Ans. 12 a. 1 r. 37 p. 

Ans. 4779 p. 

(3) 80.25)89763. yd. (296j,7p.-5-4^0 
6050 — 



4)74 7 



29263 — 

27225 Ans. 18 a. 2r. 7 p. 11yd. 2ft. 36in. 



20380 
18150 

22300 
21175 



yds. 11^25 
9 

feet 2^^. r 

' 720 

288 

in. 36.00 



REDUCTION. 23 

(4) 5)1299600 p. (5) lm.=r32,,0p. 
^ 32,0 



5)259920 



64 



4^0)5198,^4 96 



4)1299 24 4n9. 102400 p. 

Ans. 324 a. 3 r. 24 p 






SOLID MEASURE. d i 

^ EXAMPLES. 

c. in. c. in. 

t. 1728 (2) 1728)622080(36v;0-j.4^0 v 

(1) 15x50=75,0 ft. 5184 



8640 1036^ 

12096 10368 



. 9t. Ans. 



Ans. 1296000 c. in. 



c. in. • cm. 

rs) 221184 (4) 221 184)5529600(25 cds. 
25 442368 



A 



1105920 1105920 

442368 1105920 



kna. 5529600 c. in. 



c. in. • 

t. 1728 in one c. foot. 
(5) 12x50=600 c. feet in 12 ton« 



Ana. 1036800 c. in: 






24 REDUCTION. 

LIQUID MEASURE. 

EXAMPLES. 

hhd.gal.qt8. pts. or thus, pts. 

(1) 9 15 3 (2) 2)19152 8)19152 



63 



1864&gals.Ans. --A 

(3) 31,6 gals, (4) 1 gal. ^at. lpt.=llpts. 
11 165s£^s. 

346.5 gals» \ 

4 1 1)1320 pts\ 



Ans. 1386.0 qts. 12)120 botdes. 

■ ■ 

Ans. l^ doz. botUes. 

DRY MEASURE. * . 

EXAMPLES. 

(1) »bu. 3pe. (2) 2)5054 pt. 



4 



111 



;f8>527 

9 ' 4)315 7 

888 Ans. 78bu 3p6.7ql. 



Abs. 1776 pL 









4)9576 7)2394 gal. ^^ 

582 gals. Ans. * 

4 63)2394(38 hhd. 9)342 

189 . 

2331 qts. ^ Ans. 38hhd. 

8 .504 

504 



* , REDUCTION. ' ^ 

(3) llbu. 3p. * (4) 2)518 pl«. 

4 ' ' , 

— 8)259 qts. 
Ans. 47 p. 

— • 4)32 3 

Ans. 8 bu. p. 3 qts. 



PARTICULARS. 

EXAMPLES. 

(1) 444 doz. (2) 34716 buttons. 

12 6 



Ans. 1728 buttons. 12)208296 single buttons. 



(3) 2y0)82^0 



12)17358 doz. 
12)1446 6gT0. 



5)41 score. 
— Ans. 120g.gro.6gfo.6do2. 

Ans. 8 bund. 1 sco. .^.^ 



PAPER. 



EXAMPLES. 



(1) Thus, 10 reams. 
20 

200 qr8.x24=:480a«heets. 
Deduct 10x2s=20x4=80 do. cassies. 

■* * 

Ans. 4720 she#ts. 






<*!«■■ 



96 REDU^ON. 

(3) Firit find t)ie arerage ntimber of sheets in a 
quire* 
Thuf, 1 ream x20x24sb480 sheets. 
Deduct for 2 cassies 8 do. 



2j0)47\2 sheets in are. 

2^0 

sheets 23.6 sheets per qr. 

Then, 23.6)10722.0 ( 88i^5qr. 
1888 



■ 2)41 15 



842 — 

708 Ans. 20 bua. 1 re. 15 qr. 16 shts. 

184.0 
118.0 



16. sheets. 
ENGLISH MONEY. 

BXAMPLIS. 

(9) 86£ 12s. lOd. Iqn 

20 (8) 4)24616qr. 

782 s. 12)6158d. 8qr. 

12 

2^0)51.2s. 9d. 3qr. 

8704 d. — 

4 Ai». 2|£ 12s. Od. Sqr. 

(4) 12)10200 d« 



Ans. 36177 qrs. 



2^0)160^0 s. 
80 Ans. 



ft 



REDeCTION. 27 

REQUCTION hF GOLD COINS. 

EXAMPLES. 

(1) Thus, 1 lb. 

12 gr. 

— (2) 25.8)5760.0(«223.255Jf Ans. 
12 oz. 516 

20 ' 



600 

240 dwt. - 516 
24 



840 



960 774 

480 

Ans. 660 



23.2)6760.0gr.(«248.275|f 516 
464 



1440 

1120 1290 
928 



1500 



1920 1290. 
1856 



.210 105 

640 2)— = — = If 

464 ^258 129 



1760 
1624 

1360 
1160 

200 



— if 

232 



». '.. 



^ REDUCTION 

(8) 12)5760 gr. 
180 



23.2)5280.0($227.586A Ans. 
464 



640 
464 

•— "^■~"" 

1760 
1624 

1360 
1160 

2000 
1856 

1440 
1392 



^232 



ADDITION AND SUBTRACTION OF DECIMAL 

FRACTIONS. 

EXAMPLES. 

(1) Thus, 46.75 A. ' (2) From 1.6 
25.50 Take .9 . 

36.25 _ 

28.50 Ans. .7 remain^. 

Ans. 137. acres. 



i 



DECIMAL FRACTIONS. 29 

(3) Thus, .6 (4) From 1.000 (5) 825.00 

.4 Take .001 6.75 

.2 10.20 

•8 Ans. .999 9.05 

5.0 

Ans, $50.00 

Ans. 7. 



MULTIPLICATION AND DIVISION OF DECI- 

MAL FRACTIONS. 

EXAMPLES. 

(1) Thus, 8.125 m. (2) Mul. 2.68 

.5 bu. by 25 

Ans. 8<0625= 6i c. 1 340 

536 



Product 67. 

(3) .8).8 (4) .8)8.0 (5) 25.5 days. 

— 8.75 c. 

Quotient 1. Quotient 10. 



1275 
1785 

819.12,5 m. 



o2 



# 



.* 



30 



COMPOUND SUBTRACTION. 



COMPOUND SUBTRACTION. 



y. m. d. 

(4) 1828 8 10 

1820 9 16 

Ans. 7 10 24 



EXAMPLES. 



(6) 



y. m. d. 
1828 2 18 
1789 6 20 



Ans. 38 7 28 



Operation. 
10+30^16=24 d. 

8+12—10=10 m. 

8— 1«= 7y. 



Operation. 
18+30—20=28 d. 

2+12— 7= 7 m. 



y. m. d. 

(6) 1830 3 4 

1767 3 15 



(7) 



Ans. 62 11 20 

Operation. 
.Pirst4+ai— 15=20 d. 



1492 



■ then3 + 12— 1— 3 (or 4)=11 m. 

0+10— 7—1 =2y.&3+ 
10— 6— l=6y. 



m. d. 

10 12 O.S. 

+ 11 days add, 



1492 10 23N.S. 



y. m. d. 
1776 7 4 
1492 10 23 

Ans. 283 8 12 

Operation, 
4+31^23=12 d. 



7+12— 11=8 m. 



COMPOUND DIVISION. 



y. m. d. y. m. d. 

(8) 1799 12 14 (9) 1820 9 14 O. S. 

1732 2(11 + 11)=:22 ' X12 d.ad. 

Ans. 67 9 21 1820 9 26 N. S. 



Operation, y. m. d. 
14+29—22=21 1821 1 13 
1820 9 26 



12— 3=9 



Ans. 3 17 

Operation* 
13+30— 26=17 d. 



1 + 12—10= 3 m. 



COMPOUND DIVISION. 

EXAMPLES. 



(1) Thus, 26)$32.50($1.25 c. Ans, 

26 



— (2) 2)69A. IR. 24P. 

65 

52 8)34 2 32 



130 Ans. 4 1 14P. 
130 

(3) 48)$528($11. Ans. (4) 28.5)$6.84 c.(24 c Ans. 

48 570 

48 1140 

48 1140 



32 COMPOUIO) DIVISION. 

■ REDUCTION OF VULGAR FRACTIONS TO 
THEIR LOWEST TERMS- 

EXA3IFLES. 

(2) 91)119(1 91 13 

91 7) = — Ans. 

119 17 

28)91(4 
So 7 Is the largest 84 
tonatnon measure, or 



divisor. 7)28(4 

28 

(3) 195)468(2 195 5 

390 *39) = — Ans. 

468 12 

78)195(2 
156 



Divisor *39)78(2 

78 

(4) 417)973(2 417 3 

834 *139)— = - Ans. 

973 7 

^139)417(3 
417 

REDUCTION OF A VULGAR FRACTION TO 

A DECIMAL. 

EXAMPLES. 

(2) Thus; 2)1.0 (3) Thu3, J)3.00 

.5 Ans. .75 Ans. 

So i or .5 (tenths) are So | or .75 hundredths 

x= the same in value. are equal. 



^ . COMPOUND DIVISION. 33 

^ .) i)1.000 

r (5) 1)3.000 

] .125 Ans. 

I i or .125 thousandths .375 Ans. ' 

are = the same. 



if 



\ 



« 



REDUCTION OF COMPOUND NUMBERS TQ 

A DECIMAL. 

EXAMPL^IS. 

c c 
(2) Thus, «1 =100)50.0(8.5 Ans. qr. qr. 

50 (3) 1E.E.=5)4.0 

.8yd. Ans. 
(4) Thus, 2p. 4qt. 

8 Or,' 8)4qt. 

1 bu. = 32)20.000{.625bu. Ans. 4)2.5p. 
19 2 



80 
64 

Too 

160 



Ans. .625bu. 



(5) Thus, 2R. 20P. 

40 Or, 40)20.0(.5R 

29 

1A.= 16^0)1 00.00^0 

96 Then 4)2.5R. 

40 Ans. .a25A. 

32 



80 

^ 80 



34 COMPOUND DIVISION. 

(6) Thus, 14h. 45mr36s. 

AQ f» ft A US 

(7) Thus, $1 = 100)5.00(8.05. 

885 5 00 

60 



lda.=86400)531.36.0(.615da. An&. 
518 40 

(8) Thus,2qr. 141b.=701b. 

12 96 00 Ans. 

8 64 00 Then lcwt.= 112)70.0(.625. 

4 32 000 
4 32 000 



To find the value of any decimal, in the terms of an 
integer of the inferior denomination. 

EXAMPLES. 

(1) Thus, .78751b. Troy. (3) Thus, .125gal. 

12 (2) Thus, .625 A. 4 

_ 4 

Ans. oz. 9^45^00 .5. 00 

20 Ans. R. 2.5^00 2 

40 • 

Ans. dwt. 9.00 Ans. pt. 1.0 

Ans. P. 20.0 

(4) $.635m. (5) Thus, 365^5D 

1000 .BY. 



Ans. 625.000nv Ans. 109.575D. 

^ 24 



2300 
> 1150 



v.* 



;.'4r Ans. 13.8v00H. 

60 



Ans. 48. M. • 



MULTIPLICATION AND PRACTICE. 35 



(6) Thus, .875yd. 

4 

1 


(7) Thus, 9 .1125 

840bu. 

• 


Ans. qr. 3.5^^0 
4 


45000 
9000 


^ — 

Ans. 2 na. 


Ans. $94.50^00 


MULTIPLICATION AND PRACTICE. 

EXAMFUSS. 

(3) By Notation. Fractional, or, Multipli- 
If Icwt. cost $4.75c. cation and Practice. 

$4.75c. 

1 Then, i=$2.375m. % 
i= 1.1875 ^ 


i= $3.5625 Ansc 

Practice. 
2qr. = i)$4.75c. 


Ans. 3.56^25m. 

Or, Thus, $4.75c. 
Iqrs.— .75 


i « = i) 2.375 
1.1875 


2375 
3325 


3qr. Ans. $3.56y25m. 

(4) Thus, 65.75hu. Or, 
$1.16 


Ans. $3.56y25 

Thus, $1.16c. 

65bu. 3p. 


► 39450 

72325 


580 
696 

. 58 half of 1.16. 
29 quarter. 

An«. $76.27c. 


Ans- $76.27^00 * 



36 MULTIPLICATION AND PRACTICE. 

(5) Thus, 40)20.0(.6 Or, $10. 

20 20A- IR. 20P. 

* 4)1-5R. 



9200 the 20A. 



20.376A. - 2.5 the IR. 

1.26 the 20P. 



Ans. $203.75^0 f203.75c. Ans. 

(6) Thus, 84.25c. Or, $4.25c 

5i 5.5 



21 25 2 125 

2 125 21 25 



Aotf. $23.37 /5m. Ans. $23.37^m. 

(7) Thus, 224 Or, Thus, 224 

$.015 l^c. 



Ans. $3.36^0 224 

112 

Ans. $3. 36c. 

(8) Thus,«64lb. Or, 2641b. 

12ic. 125m. 



3168 1320 

132 3168 



Ans $33.00 Ans. $ 33.00^0m. 

(9) Thus, $2.25c. Or, 56 hats, 

hats 56 $2^ 



1350 112 

1125 14 



Ans. $126.00 Ans. $126. 



.£J 



i^ 



V MULTIPLICATION AND PRACTICE. $7 

riO) Thus, 28-6Jb. (11) Thus, 64=i)#8.26c. 



Ans. <^2.28i*>0 * 



Or, 72 -» 128=. 6626 
12.5626a 



1005000 
3140626 



■••- 



Ans. $103.640626 
(13) Thus, 4)2na. 
4)3.5qr. 

.875yd. 
60c. 



Ans. 9 .625^00 

Or, 2qr. =i)60c. . 

1« =i30 
2iia.=| 16 

76^ 

Ans. 9 .626 



9900 
8=^=4126 

616626 



Ans. 1(103.64,0626 



(12) Thus, 4)»2.60c. 
Ans. $ .62,6m. 



Or, Sqr. 2na. 
4 

14 " 7 
2)— = -yd. 
16 84# 

Then, 60b. 
7 

8)420 

Aas. $ .52,6 



D 



1 



MULTIPLICATION AND FRACTICEL ; 

Or, 5in.^ic.)I84 

Ana* 9^ 



(^^ Thus, 184yds. - 

5m. 



Ans. • .92,0 
(15) Thus, $1.25c. 

876 
626 



^ * 



' Aos. 94.37,5m. 

(16) Jhua, 626.6bu. 

30c. 



Ans. 1157.66^0 
(18) Thus, 12)45books. 

3.75doz. 
$4.5 " 



1876 
% :: 1600 ^ 

m* ■ 

Aiis.1|l6.87,6m. 

(19) Hius, 18 Z 

— = -qr. 
24 4 

Then, 25c. 
3 



4)76 
AS. • U875 ^^ 




Or, $1.25rT 
t.5 

625^ 
375 



Ans. $4,375 
(17) Thus, 12.61b 

crC. 



Ans. $1.125ni« ^ 

Or, 9c 
12i 



'I 



108 
44 

$l.l2i- 



0|^ 4).76qr. 
Ans. • .1875 



I 



^ J 



/ 



* 1# " •' 

* aiXJLTIPLICATION AND PRACTICE. S? 

(90) Thus^e- . • Or, 64ft.=l)»6.- / 

,, ' • — 32 =i)$3. 
' 8)42 16 =4) 1.5 
V 75 



i 



1% 






* Ans. 85.25c. — 

:- 112 Ans. $4.25c. 

(21) Thus, 10.6yds. X «1.25=8l3.125m. 

8.5 X 75c. = 6.375 
5 X 85c. = 4.250 

4.5 X 65c. = 2.925 

3 X 625in.=ii 1.875' 



Ans. $28.5dc. 

(22) Thus, 16.51bs.x8c.=$1.320 " 
•5fcs.Xl25m.= .625 

8.5yds. X28c.= 2.380 

■ 

'•Ans. $4.32,5 

(23) Thusr, IWrbu. at 47c.=$9.40 

i « at 47c.= .235 
2bbs. at $1.25= 2.50 
8.251bs. at 6c. = .495 



Thus, $1.75c. 
41 


Ans. 

« 
Ae 


$12.63c. 

Or, 4.75doz, 
$1.75 


7.00 

875 
4375 " 


2375 
8075 


1^. $8.3125 


Ans. $8.3125 



40 



COMMON MEASURE. 



To find the value of articles sold by the 100 or lOOO. 

NotB, — ^To divide any numbe#4)ylOO, point off two figures to 
the right hand ; and to divide by 1000, point off three figures to 
Um right hand. 



(1) Thus, 4.25 



(2) Thus, 6.423^ 



Ans. $17.00 



Ans. $51.40 



(3) Thus, 8 -f- 100= .08 (4) Thus, 75 -f- 1000 =.075 

$30 

Ans. $2.25;0' 



Ans. $ .32c. 



COMMON MEASURE. 

* EXAMFLSS. 

(2) Thus, 323)425(1 (3) Thus, 2310)4626(2 



323 

102)323(3 . 
306^ 



iViii=ii 



4620 



Ans. 6)2310 



385 



Ans. 17)li)2(6 

1 OS And 6)11^ =:^^ Ans. 




(25) Thus,$5«25c 

12.5CWU 


(26) Thus, lOOlbs, 

90c.' 


2625 
6300 


Ans. $W).00 

(27) Thus,$1.68c. ^ 

3 

4)5.04 

Ans. 1.26<S. 


Ans. $65.62,5m. 

(28) Thus, 1.75yd. 

$6 


Ans. $1 0.50c. 



•, - '» ._. 






Jt* 



T 



/ ^ 





MULTIPLE. M 


k < 


(4) Thus, 135)165(1 


(S) T]| J 


mi^4s»(X 


135 




1 


BRl092 


30)135(4 


• 

r • 


W 336)1098(3 


Then, 15)235(15 


120 


•• 


• • 1008 


15 




-» . . 


\ 




15(30)2 




84)336(4 


85 


30 


« 


336 


75 




And 84)1197(14 


._ 






84 


10)15(1 






._ 


10 




A 


357 


_ 




t 


336 


ns. 5)10(2 















21)84(4 






» 


84 


(6) Thus, 135)180(1 








135 




Then, 21)805(3» 


« 


1 — » 




63 




45)185(3 


• 


..^ 


And 45)285(6 


135 




175 


270 






168 



41 



Ana. 15)45(3 
45 



Abs. "J^US 



MULTIPLE, 

" EXAMPLES. 

(4) Thus, 2)1 23456789 

2 . 

(^ Thus, 8)16 24 (3) Thus, 5)3 5 8 10 3 3)1 1 3 2 5 3 7 4 ^ 

^— . ..___ - * 

23 2)3182 6 2)1 11251743 

2 



3 14 1 — 
And, 12 

8X2X3s=48Ana. Then» 5 

5X9rx3,^csl20An8. — 

, > 60 

^ So 

840 

' Ans.2520 

. d2 

• ft 



111151723 



/^ 



^mmati^Krmm^i^m* 



43 THE 9iirGLE RULE OF THREE. 

RAiTIO AND PROPORTION. 

EXAMPLES. 

(3) Thus, 84)336(4 Ans. (3) Thus, 12)96 

336 — 

4) Thus, -8 X 1 2 = 96 Ans. Ans. 8 

PROPORTION.— EXAMPLE. 

(1) Thus, 3yds. : 9yds. : : 1 2M. 

9 Or, 3)3 : 9 : : 12 

' Ani. 

3)108 1 : 3 : : 12 : 3d ^ 

Ans. 36 



THE SINGLE RULE OF THREE. 

EXAMPLES. 

(2) State the question, thus : 

• lbs. lbs. c. V \ 

As 5 : 13 : : 75 : $I.95c. Ans. 
For 13x75=975, which ^5=$ 1.95c. 
lb. cwt. m. 

(3) As 1 : 1 : : 35 : 3.92c. Ans.* 

For lcwt. = ll2lb. Wl^ichx35=392 which-j-l = 
$3.92c. , 
oz. cwt. c. 

(4) As 1 : 6 : : 8 : $860.1 6c. Ans. 

- For6xll«Xl6=10752oz.x8«86016-7-l = 
$860. 16c. 



(5) As 100 : 57 : : 6 : $3.42c. Ans. 
For 57x6=342, which -r- 100 =$3. 42c 

lb. lbs. c. 

(6) As 1 : 75.5 : : 8 : $6.04c. Ans. 
For 75.5x8=604, which-5-l=:$6.04c. 



• 



THE SINGLE RULE OP THREE. 43 

bar., bar. cwt. cwt.qr. cwt. 

(7) As 24 : 1 : : 42 : 1 3 or 1.75. Ans. 
For 1x42=42, which-=-24=5l|cwt. :». ; 

yd. yd. $ c. 

(8) As 1 : 26 : : 1.25 : $32.50c. Ans. 
For 26x125=3250, whjch-f-l=$32.50c. 

g. g. Ib.oz.dwt. dwt. dwt.g. 

(9) As 380 : 1 : : 8 3 15 : 5.25=5 6. Ans. 
For8lb. 3oz. 15dwt.=1995dwt.;Klj3vl99&dwt., 

which-^380=5dwt. 6gr. '"^ 

bu. bu. $ c. 

(10) As 15 : 7 : : 14.25 : $6.65c.»Ans. 
For 7X14.25=99.75, which-*- 15*=$6.65c. 

lb. cwt.qrs. lb. c. 

(11) As 2 : 1 2 14 : : 25 : $22.75c. Ans. 
For Icwt. 2qTS. 14.1b.=182lb.x25c.=4550c., 

which~2=$22.75c. 

yds. yds. $ 

(12) As 8 : 96 : : 2(i : $240. Ans. 
For 96x20=1920, which^8=$240. 

yds. yds. $ 
Cancelled* 8)8 : 96 : : 20 

1 : 12 : : »0=»240 



bu. 1)U. $ c. 

(13) As 135 >. 25 :•: 74.25 : $13.75c. Ans. 
For25x74.25=185625,which-^135=$13.75c. 

men. men. days. 

(14) As 25 : 20 : : 15 : 12 days. Ans. 

For 20x15=300, which-4-25=12days ; or can 
eel it. 
d. hrs. hr. $ 

(15) As 365 6 : 1 : : 25000 : $2.85c.+ Ans. 
For 365 6=8766 h. div., and 2,500,000 c. divi- 
dend, gives a quotient«of $2.85|^f^. Ans. 



'tm.- 



.1 



» 



44 THE .SINGLE RULE OF THREB. 



i 









grs. lb. $ 

(16) As 371.25 :. 8.25 : : 1 : $128. Ans. 8.251b. 

=^47520 grs. 
For 47520 grs. xHl =47520, whichn- 371.25=* 

$128. Ans ' jl 

lb. cwt. m. 

(17) As 1 : 10.5 : : 15 : il7.64.Ans. (10.5 X 

1.12^=1176 lb.), 
•For 1176xl5=1764c., which-*- l=til7,64c. 
cwt. lb. $ c. 

(18) As 4 : 1 : : 13.44 : 3c. Alis. 4xll2» 

4481b. , 
For 1 X 1344sl344, which -f- 448 » 3c. 

yd. yds. c. . . 

(19) As 1 : 120 : : 30 : $36. Ans. * 
For 120x30==:3600c., which^l=$36. * 

hrs. hrs. d. ' , * 

(20) As 12 : 16 : : 3 : 4 days. Ans. 
For 16x3=48, which-5-12^4 days. 

yds. m. steps. . 

(21) As 4 : 64 : : 5 : 140800 steps. Ans. 

64x1760=112640 yds. 
For 112640x5=563200^ which-f-4=140800s. 

bu. bu. i c. * 

(22) As 425 : 1 : : 263.50 : 62c. Ans. 
For 1 x263.50=26350, wliich-^425=62c.' ,| 

d. d. c. • 1 

'23) As 1 : 365.25 : : 8 : $29.22c. Ans. 
For 365.25 x8=2922, which-5- 1 =$29.22c. 

6z. cwt»qrs. lb. ^ c. 

(24) As 1 : 17 3 17 : : 1 : $320.80c. Ans. 
For 17 cwt. 3 qrs. 17 Ib,=82080 oz.xl c.= 

$32 0.80c. 

ft. .ft. ft. 

(25) As 4.5 : 6 : : 186 : 248 feet. Ans. 
For 6x186=1146, which ^4. 5 =248 feet, 

« 

T 






't 



♦ ^ THE STNGJ^E rule OF THREE. 45 

h. * h. d. 
(26) As la* : 15 : : 4 ^ 5 days. Ans. 
. '\' For 15x4«60, which-7-12=5 days. 

^ m. m. $ 

f (27) As 8 : 12 : : 200 : $300. Ans. 

For 12x200=s2400, which-^8=$300; or can- 
T eel it. 

(28) Thus 9 in.=| or .25yd., or .75fSBet, which x3 ft. 

=2.25 ft. in 1 yard of nankin. And lOft.x 

9ft.x3=270ft., or-^9=30yds. of yard'widp 

wante^ 

yds. yd. yds. yds. 
Stated thus, As 25 : 1 : : 30 : 120. Ans. 
Or, As 2.25 ft. : 9 ft. : : 270 ft. : 108 ft 

=120 yds. 
Or, As 9 in. : • 36in. : : 30yds. : 120yds. 

or as Iqr. : 1yd. : : 30ydl3. : 120yds, 
: For 1 yd. X 30yds. =30, which-7-.25=120yd«. 

of 9 in. wide. Ans. 

c. c. ' gal. gials. old. g.b. 

(29) As 6^,0 : 780,^0 : : 1 ^ 130, which— -120 
«. sslOgals. water. Ans. 

For 780X1=780, which —6= 130 gals, sold, 
which — 120= 10 gals, water. 

cwt. cwt. qr. lb. $ c. 

(30) As 1 : 33 1 22 :: 4.25 : $142.14.7^. 
For lcwt.=112lb., and 33 cwt. Iqr. 22 lbs. = 

3746 lbs. prepared for work. 
Then, 3746x4.25=1592050, which-5-112= 
$142.14.7^V. 

c. c. $ 

(31) As 100 : 65 : : 1256 : $816.40c. Ans, 
For 1256x65=81640-T-100=4816.40c. 

lb« lbs. c. 

(32) As 1 : 112 : : a : $8.96c. Ans. 
For 112X8=896, which -f-l=W. 96c. 



i 



■r 



46 THE »D!fGL£ RULE ffF THREfiT • • , 

h. h. $ ' I 

(33) As Q : 3 : : 3^4 : $194.40<. Ans. 
For 324x3»972, which-7-5»=:$194.40. 

yd. Vds. * c. 

(34) As 1 : Bo : : 30 : $36. Ans. 1 
For 120x30=3600, wliich-4-l=$36. ** ^ 

ft. m. sec. 

(36) As 1142 r 12 : : 1 : 55+8ec. Ans. 
For 12m. X 1760x3=63360 ft. prepared; we 

then 63360 x 1 ^ 63360, which -i- 1142= 
^5|^f- sec. Ans. 

(37) First find the time ; thus, 

p. p. sec. 

As 75 : 10 ; : 6p : 8 secorids. 
Then state ; thus, ' . • * . 

sec* sec. feeC *' ' 

As 1 : 8 : : 1142 > 9136 feet Ans. 
For 8x1142=9136, which ^^ 1=9136 ft., or 
Im. 5fu. 33p. 3.5 yd. 1ft. 



^'^*.. 



m. h. p. *. .^4 \ 

(38) As 24 : lO : : 1 : 25 pipes. Ans. ' '' . I 
First 10h.X"60=600m., then 600x1 — 600, .^ 

nt' '. which -^24=25 pipes. ".• ,. . 

h. ox. s. h. ox. s.^ c Ans. 

(39) As 3+4=7 : 6+8=14 t: 9 : ' 18 cows. 
For 14x9=126, which-T-7=18 cows. 

deg. min.sec. deg. do. 

(40) State, As 13 10 '35. : 360 : : 1 i 27 do 

7h. 43-//yV s®^* ■^P'^v 
For 13deg. lOmin. 35see.=:47435sec. divisor, • t 

and 360deg:=l296000sec.xl=129«aDO di- 

vidend, will give a quotient of 27jiays 7h. 

43.g^'ysec. An«. 

s. ck dS s. oz. Ans. 

(41), As 5 4 : 10 12 : : 1 : 31b. 3oz. 15dwt. 
Orirst, 5s.xl2+4=64d., and 10/.x20+12= 



THE SlNGiP RUXE OF JTHREB. 47 

2r2s.Xl2^544d., then 2544x1=2644, 
* which4-64=39 oz. 15dwt., which 59-^12*= 
3lb. Soz.y and 15dwt .annexed, the work h 
finished. 

lb. lb. $ • 

(42) As 5 : 75 : : I : $15. Ans. 

For 75x1=75., which-^5=:$15. Ans. 
c. c. d. 

. (43) As 10 : 900 :: 9 : 810d, Ans. 

Fdr 900x9==8106, which^l0=810d. Ans. 
a. • a; s. 

(44) As 1 : 547 : : 7.75 : 211/. 19s. 3d. Ans. 

• (15.68.-^2=7.758. third term.) 

For 547a. X 7.758.==4240.25s., which -i-20=r 
211/. 19s. 3d. The^25s.»i or 3d., the first 
term being 1, dividing by it was omitted, 
pt. p. c, 

(46) As 1 : 1 : : 10 : $100.80c. Ans. 

*. Fur Ip.x2x63x8=-1008, which X 10, or annex 

I ' ■ for the Ans. 

f *(40) First, 4x35=140 miles A travels before B sets 
out, aad 40~i-35s=5 miles B gains upon A in 
^ day's travellii^. V- 

m. m. d. 

Then say, A» 5 : 140 : : 1^ : 28 days, 
d. d. m. 

and A$l 1 : 28 : : 40 : 1120 miles. 
For 140x1=140, which-f-5=28 days, and 28 
*X40=1 120 miles. Ans. 

m. m. gsl. 

(47) ,4s 765: : 125000 : : 1 : 163 gals. 1 qt. 

I 3 9 p^^ Ans. 

Fot 125000x1=125000, »diich-T-765»i 63^ 

iq^-'iyV^pt- 

. yd. yds $ ^' 

* *' 1 ; 18.75 : : 1875 : $3.ai; 5825. Ans* ^ 



i£ 






48 THE SINGLE RULE jOP THREE. 

For 18.75 x 1875 = 3SU562j5, wftich -5- 1 = 
$3.515625. Ans. 
p. ~p.. a« 

(49) As 10^0 : 1 : : 6y^0 : 96 poles. Ans. 

First, 6a.xl66=960p.; then, 960x1=^960, 
which-f-10s= 96 poles. Ans. 
qrs. qrs. yds. * 

(50) As 3 : 5 : : 30 : 50 yds. Ans. 

Or, 3qrs.x4qrs. = 12qr., and 4qrs.x 5qrs.=ss 
20qr.; then. As 12qrs. : 20qrs : : 30yds 
: 50yds. 

For 5x30=sl50yds., whichs-3=:50yds. » Ans. 
c. c. $ ' . • 

(51) As 10 : 1 :.: 78 : ..$7.80c. Ans. 
For$78xl=$78jwhfch-^10«:$7.^0c. Ans. 

(52) First 1/. J0s.=:30s., or 360d. 7 ^ J 520d. 1 load 

and 40 groats, at 4d.^= 1 60d. 5 c of hay cost. 
L k d. . ' 

Then, As 1 : 20 : : 52^. : 43?. 6s. gd., 
which — 19c., or 4/. 15,s.=38/. lis. 8d. Ans. 

(19 crowns sterling x€>=95s.s=34f. 15a.) 
or 520 x20;» 1 0400-4- 1 =«1 0400d, =r 43/.68.8d., 
' whidi— 4/. 15s.=a%^ lis. 8d. Ahs.* 

(53) First, assume any tinmber, say 80, which -^10, 

20, &c. Thus, 80 -r- 10^8 

80-4-20=4 
80—40=2 
• •80-t-80=4 ' 
In 80 inin. the 4 spigots > ^^ ^.^^^ 
would empty - - J 
c. c. m. 

Then, As 15 : 1 : : 80 : 5i min. Ans. 

(54) First, 1 8ft. -i-B'^ 6yds., and 30A.-^3=S10yd8. 
Thei, state ; thus, As .6yd* : 6yd. : : 10 yd* 

: 120 yards. Ans. 
For 6x10=60, whichH-.6ssl20^yarda« Ana. 



. % 



n, : * 



<n^Fi>|»»« «■« III 



49 



THB SINGLE RULB OF THRBE. 

«- a. a. bar. 

(56) As 1 r 50 : : 15 : 750 barrels. AW 
. For J5x 50=750, which-^ 1=750 bar. 
c. c. lb. 

(56) As 16 : 800 :: 1 : 50 lbs. Ans. 
For 800x1=800, which-r-15=501b. 

(57) First, 30ft. -f- 3= 10yds., and 1 8ft. -i- 3= 6yds. 

yd. yds. yds. 
Then, As 1 : 6 : : 10 : 60yds. Ans. 
yds. yds. yds. 
• .As 75 : 6 : : 10 : 80yds. Ans. 
yds. yds. yds. 
As .5 : 6 : :' 10 .: 120yds. Ans. 

men. men. day^. 

(58) ,As85 : 20 : : 15 : 12 days. Ans. 

For 15m. x20m.=300d., whicbii-35= 12d.Ans. 

. h. m.fiec. h« m. 

(59) As 23 56 4 : 1 : : 19150 : 800 m. 32p. 

3yds» 2ft. 2jmjin. Ans. 
Then, 2^hr. 59m. 4sec.=86164sec. and lhr.= 
=3600 sec, and 19150x3600=68940000, 
which «^ 86164 s^ 800 m. 32 p. 3 yds. 2 a. 

c. ' c. d. 

(60) As 21 : 6 : : 91 : * 26 days. Ans. 
For 91 x6=546, which-4-2 1 =26 days. Ans. 

V. V. WW » 

(61) As 8)8^0 : 1200^0 : : 1 : Ans. 
Cancelled, 1 : . 150 : : 1 : 150w.=37.5m 

Or, As 8 : 150 : ^ 25 : 37.5 m. Ans. 
d. d. m. 

(62) As 4)8 : 4 : : 4 : 2 men. Ans. 



2)2 



1 : *: 4 cancelled. 



I : : 2 men. Ans* 
E 



St 
50 



(63) 



(64) 

(66) 
(66) 

(67) 
(68) 



(69) 



THE DOUBLE RULE OF THREE. 

e. e. 'k , * 

lis 10 : 1 : : 3 : 48min. .Ans. 
For 3hrs.i=180inin.xle.=180m., which-f-lOe. 

=18mln. Ans. 
h« h. bu. 

As 20 : 6 : : 70 : 21 bush. Ans. 
For 70x6=420, which-r.20=2l bush. Ans. 

$ $ bu. , 

As 1.12 : 81.76 : : 1 : 73btish. Ans. 
For 81.76x1=81.76, which-^1.12=73 bush. 

Say 16 : :24 : : 36 : 54 Ans. ' « 

For 24x36=864, which-^16=54. Ans.' 

w. w. $' * * 
As 1 : 52 : : 21 : $1092, which — from 

$1200^111 leave a balance oril08. Ans. 
For 62^21=1092, which-7-l=$1092, 

First, As 4 : 3 : : 1 : .Tf, The woman's 

labour is in proportion to the man's. 
Then, As 1+. 75 =1.75 men. : 3 men. : 

56 days : 96 days. Ans. 
or 56x3=168, which— 1.75 =96 days. 

y. y. men. 

As 1 : 43 ; : 30000 : 1290000 men. An«. 
For 43 X 30000 = 1290000, which -f- 1 = 

1290000 men. 



(2) 



THE DOUBLE RULE OF THREE. 



EXAMPLES, 



84acr. 
10 dy. 



lOOarr. 
24 dy. 



men. 



As 84y^0 



: 240^0 : : 7 : 20 men. Ans. 
For 240X7=»1680, which -^84«=2a. Ans. 



t 



JthM 



iMta 



m^^^mamm 



THE DOUBLE RULE OF THREE. 



51 



(3) 12 men. Si meu. * 

8 days. 15 days. * 

As 96 : 315 : : 8/. 89., or 168s., : 27/. 

lis. 3d. Ans. Ans. 

For 315x168=529208., which -=-96 =55 1^2 5s. 

(4) 20 hor. 50 hor. • * 
30 days. ^6 days. 

As 6y,00 : 8^00 : : 225 : 300 hush. Ans. 
, For 225x8=1800, which -^'^ss 300. Ans. 

(5) ■ 8p. 18p. 

9in. 12ni* 



w 



(7) 



(8) 



12)72 t 216 : 



6)6 



216 : 



As 1 



216 : 



$360 cancelled. 



30 do. 



5 : $1080.' Ans« 



$8. 68c. ' $43.40 c» 
14 men. 4 men. 

days. 

As 121.52 : 73.60 : : 7 : lOdays. Ans. 
For 73.60x7=121520, which-f- 12152= 10 Ans. 
$12.8 $24 
12cwt. 8cwt. 



As 153.6 : 192 ': : 128m. : 160m. 
For 192 cwt:xl28m.=2457eto.,' which -■ 
153.6=160 miles. Ans. 

3v^00 pa. 9^00 pa. 

6y^0 dys. 4^0 dys. 



18)18 
As 1 



36 : : 5 men. 



: : 5 V 10 men. 






<Jw 



sa 

(«) 



INTEREST. 

First 150x48x50x6x8 (or 48) =1728^0000 
first term, and 500x72x45X3x10 (or 30)=a 
4860^0000 second term, and the 16 compositors 
are the tliird term. 

h. h. comp. 

Thus, As 1728 : 4860 4:16 : 45comp.Ans. 
Or contracted ; thus, 

36x3=^108 , 
108)1728 : 4860 : : 16 cancelled. 



16)16 



14)28 



2)2 



45 






16 



(10) Thus, $4^00 il^OO 

• 7 m. 12 m. 



12 : : 14' 



12 : : 1 



do. 



45 : : 1 : 45 Ana* 



6 : : 1 : t6.Ans. 



INTEREST. 

EXAMPLES. 

{4) TboB, £5SI.S (5) Thtu, 9100. Or, $10.00 (6) ThuB, 3mo.-f-3»1.5mo. 
6 10 100. 950 



32.25.0 
5 



Ani. £161.25 
20 

Ans. 8.5.^00 



Int. 910.00 91ia AoB. 
Prin.lOO. 

Ans. 9110. 



Ang. 9 .75c 

(7) Thus, 96.5 rate. 
400 



(8) Thus, 98560 (9) Thus, 9125 

35 100^0 



42800 . 
25680 

AUk 92990 



926.000 
2Y. 

52 Int. 
400Priii. 



Ans. 91^.00 — 

Am. 9iS2. 



J*., 



INTEREST. 



53 



(10) Thus, 365.25-^6=60.875 

$35.25x 327da.= 11526.75 
60.875)1 1526.750($1.89iJ| Ans. 
6087 5 



Or, $35.25 
da.327 



24675 



5439 25 
4870 00 


• 
• 

»- 

Or, I960, 
da. 10 

3^0)260^0 

* 


10575 




6)11526.75 

Ads. 91.92115 
Bank Int. 


569 250 

547 875 

• 


21 376— 


60 875— 

(11) Thufl,$360. 

30 •« 
c. m. 

60^5)5900.000(85,4j}f AnB. 


10 
Or, — — fof$260c 

•90 — 

86|c. 



86ic. 

(12) Thus, $1. (13) Thus, 2cwt. 3qr. 181bs. 

.06 4 * 

— (14) Thus, $225 

•06 ^1 mo. 2 

12 28 

Ans. $5.50ct 



Int. 72c. 
Prin. $1.00 



326Ihs. 
2.5 



Ans. $1 .72c. 



1630 
652 



Ans. $8.15^0 
e2 



54 INTEREST, 

(15) #2I6.B5c. 
6 



4 m. is i) .12.98§v0 interest f year. 

5yr. 5m. 6d. 

^64.695=s5yrs. int&rtsst. 
Im. is i) 4.313=4 months* int. 
6da.=| (nearly) 1.07825=1 month's int. 

' .215653s^6da. interest; . 



♦70.3019^0 Ans. 

»■ ■ 

(16) $.85c. (17) * $.75c. (18) $66.6c. 

3 m. 2m. /05 - 

ft, 

$.025.5 Ans. $.01.5^6 or l|c. Ji^3.325 int. 
■ ' +66.5 prin^. 

Ans. $6^.825 amo. 



(19) $146x 146 days=af 316, which -f- 7305 = ♦?. . 
91c.87^§7m. Ans. 
(20) $125 

30 (21) 1461 X 100. or annex 00= 

H6100-5-60fii7.5=«$24.Ans. 

6b87.5)3750.00($,61.6m+Ans. 
365250 



97500 
60875 

(22) 10 X lei: IjOO, which -^ 

366250 6087.5=.01.6Jf|f^. Ans. 

365250 



1000 =:'2000 

^^6087.5= 12175 






5. 



.^Llbad. 



INTEREST. 



(23) *16.5 
6 






55 




intere&t. 
incipal. 






c. Ans. 



(24) 1x19=19, which^60^87.5=$.00.312+ Aas. 
and a'*»emaiilder of 7000. 

25) $174||^x6x3.75=if391.59c. Ans. 

26) $20^r5~.5 discount, which subtract from $20 
s^l9.50c. Ans. 

(27) $2195.5 (29) $12.08c. 



\ 



125 



109775 
263460 



6 



).7248 inter6««^1(|fe^, 
2i 



27744.37.5m. Ans. or 
add \ to the principal. 

(28) $44 
4 



14496 
1812 



• ■» 



.^'':; 



$1.76c. Ans, 



1*6308 interest. , 
12.0S principal. 

$13.71.08 Ans. 



Note $240 
6 



7m.-^2=3.5m. 
$100 



3|n.-T?2=sl.5m. 
80 



ly. in.$14.40 
prin. 240. 



amo. $254.40 
paym. 184.70 



int. $3.50<0 
do. $1.20 
prin. $100. 

do. $ 80. 



int. $1.20,^0 



Bal. $ 69.79c. Ans« 



$184.70c. payments and int.i \^ 



56 INTEREST. 

"^ * $240 note. 

aim. 

4.80 "f^^ • 
1.201 



$6. int. to first pa]f ment. 
+240. principal. 

$246 amount. 
— 100 pajiiident. 

$146 balance. *■ 

2 m. 

$2.92 interest. 
+ 146. principal. 

$148.92 amount. 
•—80. payment. 

68.92 balance. ^. . 
1.5m. 



$1.0338^0 interest. 
-1-68.92 principal. 

$69.95.38 Ans. 



A difference in the two last examples of $.25.38 



1 



. i 



« ■*■ 



INTEREST. 57 



CASE 2. 
EXAMPLES. 

(2) iioo 

X5 



5.00 
• 4 y#ars. 

■ 

$20 interest. 
+ 100 principal. 

As $120 : $57 1.20c. : : $100 : $476. Ans 

100 



12;0)571.2y,000 
$476. Ans. 



(3) $100x6=:6 interest ope year. 

8 

$48 interest for eight years. 
+ 100 principal. 

As $148 : 925 : : 100 

100 



148)92500(8625. Ans. 

888 



370 
296 



740 

740 



« •* 



58 INTEREST. 

(4) $100 

X6 per cent. 

$6.00 interest 1 year. 
2.5 years. 



♦15v^0 interest 2i years. 
$+100. principal. 

As $115 : 718.75 : : $100. 

100 ^ 

115)71875.($625. Ans. 
690 * 



287 
230 



575 
575 



(5) 



$100 

7 



$7y^00 inter6st4 year 
16 interest 16 years. 

112 interest. 
+ 100 principal. 

As $212 : 65.72 : : $100 

100 



212)6572.($31. Ana. 
636 



212 
212 



■■■■■■iHVMPaawavBMiv^"^^"^^^^^^^^^^^*^*'"'"''^*'^**'"^ 



N 



INTEREST. 

CASE 3. ^EXAMPLES 

(2) $820 $1078.30 amount. 

X45 years. 820. principal. 

Ajs $3690 : $100 : : 258.3 

100 



369^0)2583^0(7per cent. Ans, 
2583 



(8) $837 $1029.51 amount. 

X4 years. — 837. principal. 



As $3348 : $100 : : 192.51 

100 



-Ans. 



3848)19251.(5.75 per cent 
16740 



25110 
23436 



• 
* 

• 


16746 
16740 


(4) $225 
X4i 


$285.75c. amount. 
225v principaL 


1012.5 


: $100 : : 60.75 

100 



1012.5)60750(6 per eent. Ans. 
60750 
lit ■ 



6D 



(5) tSl 
Xl6 



INTEREST. 



$65.72 amount. 
31. pniicipal. 



As $496 t $100 : : 34.72 

100 



496)3472(7 per cent. Ans. 
3472 



CASE 4.<-— EXAMPLES. 

(2) $1029.51 amo. $837x6.75 ==448. 1275 int.ly. 
--837. principal. 

$192.51 whole interest. 



As $48.1275 : $192.5L : : ly. 

1 



48.1275)192.5100(4 years. 
192.5100 



^"^ 



(3) $1600 $2752 amount 
X6 — 1600 principal. 



^ . An $96.00 : 1152 : : 1 y* 



1 



8)1152 
12)144 



12 y. Ans. 






' » 



I 



(4) 1500 
6 



As $90.00 



DISCOUNT. 

2332.5 
1500. 



61 



832.5 : : lyr. 
1 



• 00)832.5(9.25 years the guardmn 

* 810 had the money, w^zch 
deduct from 21 years 

22.5 for the ans. required ; 
18.0 Thus, 21.^ 
9.25 



4mo 

4.50- 



Ans. 11.75 



DISCOUNT. 



EXAMFLBS. 



(2) Assume $100 

X6 



$6.00 interest 1 jr. 
2 



^ 



$12 interest 2 ynC 

• present worth. 

As $112 : $75 : : $100 : $56.9642+. 
For 75 X 100»=7500, which-^ 1 1 2 ===$66.9642 + present 
worth. Then (the given sum) $75 — 6d^96#2 =» 
$8.03,68 discount. 



64 



DISCOUNT^ 



(8) I assume $100., which «: 10000c. x 93d. » 
930Q00C., which-f-6087.5=$1.52.772+inter. 
est, wlpicb add $100, 'which=$101. 52772+ 
the amount, then state ; Thus, 
As $101.52772 : $10000 : : $100 : $9849.5268+ 
For 10000x1 00= 100000ft, which -^ 101.52772 = 
$9849.5268 the equivalent value, or present worth, 
which suhtracted from $10000»$1 50.4732 the dis- 
QDunt required. 
• /' $10000 ' 

9849.5268 + present worth. 
'■■ « 

^ • $l80.4732+ discount. '' 



■^hen $1000x93=i$930000, which-=-6=$155 the 
discount, which Subtracted from $10000i=$9845 th« 
ready money value at the bank. 

Then the interest found as above ; Thus, 
^ $10000==rl000000c. 
1 V +93 



t 



6087.5)93000000.0($152.77.2+intereat. 



Assume $1.00 
8 



.00 int. 1 yr. 



+ 100 



cance/Zc(i 9)108 : $432 : : 100 • 



12)12 




v^ \ 



4 : : 100=$400prest. worth. 
+500 do. do. 

$900 Ans. 



■ml -^■.^— » ■>, - ' i I ■ ■ .. ^ _L i -JT I <_. x: 



COMPOUND INTEREST BY DECIMALS. 65 



^ 



Assume $100 
8 



F 



8.00 
2 . 



16 interest 2 years. 

foo 

As 4)116^ : 580 : : 100 
20)29 : 145 : : 100 



5 : : 100ss$500 present wor1h» 



aaps 



9 



COMPOUND INTEREST BY DECIMALS, 



^ EXAMPLES. ^ 

IP 

(2) First 3.5-*-100=.035 ratio, or interest of $1 for 
1 year, and add $1.*^ to the interest. 

il and int. lyr. 1.035 amount. 
Then 1. 035 x 1.035= 1.071225==:2d. power, and 
1.071225xl.071226=:l. 147523^000825= 4th 
power, and 1.1475fe3 x 1.147523=* 1.316809^ 
035529 = 8th power, (and the 8th,) 1.316809 
Xl.l47523(the4th)=1.5110686vl4107=»12th 
power. Then the 12th power 1.51 10686 x 
$1500«:i2266.60.29. Ans. 

f2 '^ 



k*— • ■L.*.:^*^. 



r ' 



■66 COMPOUND INTEREST BY DECIMALS. 

(a) ' -First 6-5-100==.06 int. of fl. for lyr. 

$1. 

1.06 the amount of $1 for 1 yr. 
1.06 . 

6.36 
1.06. 



1.1236=2d. power. 
X$100 principal. 



V, 



$112.36,^00 amount of $100 for 2 yrs. 
— -100. the principal. ' 

Ans. $12.36 interest. 



(4) First 5-f-100ia=.05 int. of $1 for lyr. 

+$1.00 principal. 



% 



$1.05 amount of $1 for lyr. ^ 

X105 ..^ p.; 



525 

105 

1.1025=2(1. power. 
Xl.05slst do. 

55125 
11025 



# 



.1 57625 =3d. power. 
X450 principal. 



5788125 
4630500 



Ans. $520.93.125i0 



at^i. 



mmm$mmw ■ - f " m « - .y i 



CUBIC MEASURE. 



(/ 



CUBIC MEASURE. 

EXAMPLES, 
ft. ft, ft. C. ft. C. fL 

(2) Thus 8.5x20x4.25=722.50, which H-24.75=« 

29.191 ph Ans. ■, 

ft. ft. ft. c.in. 

(3) 12.25x6.5x4.75=378.21875, which -^1.24446 

a=s303 bu. 3p. 5qt. lpt.+(a remainder of 854.) 
ft. ft. ft. eft. 

(4) 15.5x3.5x3=102.75, which-r-1.47779=110bu. 

4qts. + 
ft. ft. f. 
(6) 1.5x30x.5=22.5c.ft. Ans. Or, 1.5=30=45, 

which-5-2=22i. 
ft. ft. ft. 

(6) 22.5:5c30x8.4=5670c.f., which-5-128=44ic.d., 

and 6 ft. Then state, 
eft. eft. $ 

As 128 : 5670 : : 12 : $531.56.25 value, 
ft. ft. ft. 

(7) 48.25 X 1.5 xl2 =:= 868.50^0, which -^ 24.75= 

35.099 p. + Ans. 
ft. ft. ft. c.f. 

(8) 64.5x4x4=1032, which— 128=8TVcd., or 8cd. 

and 8ft. eft. 

(9) 25x12.5x5.5 = 1718.75, which -^ 24.75 = 69. 

444ph.+ Then 69.444 x75=$52.08.3m. Ans. 
ph. ph. e 

Or, As 1 : (59.444 : : 75 : $52.08.3m. 
in. in. in. c.in. 

(10) 147 X78 x57=653562, which -^2150.4252= 

303 bu. 3p. 5qt. lpt.+ Or, 12.25ft. x 6.5ft. 
X4.75=378.21875c. ft., which -f-1. 24446 « 
303 bu. 3p. 6qts. lpt.+ 
(11 ThiOi, 15.5x3.8 X*«23d.6ft.*, whichx6|« 
•14.72.5. 



^ . '^ -f 



■i— w^—i > ■ 



«8 CUBIC MEASURE. 

(12)' , Uhus, 3.75 (13) Thus, 1 side=16 ft. 

8 1 do. =16 

^ 1 side=16— 1.6— 1.5=al3 

4m.=i)3e,00 1 do,=:16— 1.5— 1.6=13* 

4.4 ^ — 

f^ •^-^— Whole length of wall 68 ft. 

120 WaU of 1.6 ft. thic. 

10 

87.0 



Absw 130ft. 6ft.high. 

. — — ' Ans. • 

(14) Thu^, 22x4.5=99 24.75)622.00(2 l^V- 

JA 

594 
83 



16.5)027.0(38. Ans. 

(15) Thus, 5.75 (16) Thus 16 

6 12 

34.\0 192)1728(9. Ans 

4.5 1728 



1725 
1380 



1.47779)1 55.25^000(105 bn,. 
147 77 9 

747100 

738895 






8205 
4 



)fe820( 

V. 8* 






)262560(l+^i 



■ —'^ ^m -X :. 1 ' 


m 
CUBIC MEASURE. 69 




(17) Thus, 8 
4* 


130 . (18) Thus, 1 80 X 60 X 10=108000 
3 ' Then, 128)108000(843.75 Ajf». 


I 


33 

2| 


104)390(3ft. Ans. 
312 




341 
3 


78 
12 




104 


)936(9in. Ans. 
936 





(19)ThTis,30 (20) Thus, 64ft. 

20 4 (21) Thus, 221184)231184(1 Ans. 

221184 

6v00)36^,00 256 

- J 

Ans. 6ft. 

Ans. 64ft. 

(22) Thus, 625 (23) Thus, 4ft. 

9 .5 



5625 2. 

3.5 .5 



2812 5 Ans. 1. c. ft. = 1728c. in. 

16875 



27)19687.5(72914 Ans. 

(24) Thus, 5280ft. (25) Thus, 21ft. 

20 lift. 



105600 232 

6 ^ 16ft. 



27)633600(28466} Ans. Ans. 3696c. feet. 

6c. 



$1408.00 Ans. 



SQUARE MEASURE. 



SQUARE MEASURE. 


BUKFLSa. 


(25 Thus 16ft. 
X3m. 
48 
X5iii. 

12)240 


Or, 16 
6 

3in. ifl i)io 
Aus. 20ft. 


Am. 20ft. 




(8) 22.5 
6 


Or thiu, 22.5fl 
6in.=.5ft 



40B 


3375 


13)472.5 
Aus. 3S.376=i=30ft. .64iii. 


Am. 39.375fi 



(4) 4iji,Ui)30ft. ^, 

^ 2,4:11. "nen 400+i=500rt. 

jf and 500+i=625ft. 

t 60 . -^ 

10 

Ani. 70ft. 



— -51 



CIRCLE MEASUBlL -^1^3 

CASB 2.— EXAMPLES. 

Cl) 4050x8x4=129600, which-^1296=100sq.ydi. 
Or, d|||s, 12)129|00 in. • ' « 

12)10800 

f>)900 square feet. 

• m I I 

Ans. 100 square yards. 

m 

SHINGLE, dR,<RO©F MEASURE. 

(1) First 1 6ft. X 12= 192in..which-r-5=338.4 shingles 

4n each course ; and 16f\.xl2=192in., which 
-^8=24 the nymher of courses. Then 38.4 x 
24=921.6 shingles. Ans. 

(2) »First22ft. x 12==:264in., which -^ 6.5 «= 48 shingles 

in each course; and 26ft.xl2=312in., which 
-1-8=39 courses. Then 48x39=1872 shin- 
gles. Ans. 

(3) First 20ft. X 12=240in., which-s- 5.5=43.6363+ , 

shingles in one cotirse ; and 28ft. xl2=336in.« 
* which -5-8=42 courses. Then 43.6363x42=3 
1832.7246+ , nearly 18321 shingles. Ans. 



CIRCLE. MEASURE. 

EXAMPLES. 

(1) The diameter of a circle, 154, being given, to 

find the circumference by rule 1st, state. As 

7 : 22 : : 154 : 484 Ans. For 154x22 

=3388, which-7-7«=:4S4. Ans. 

By rule 2d.— As 113 : 355 :: 154 t 

483.8+ Ans. 

• G 



7* lioUND inMBER, Ac. 

(2) As 22 : 7 : : 154 : 484. An 

For 154x7=3388, which^22 (or 2 and 11)== 
154 Ans. 

To find the Area of a Circle. 

EXAMPLE. ' 

(1) The diameter 42-t-2=s21 half diameter; and 

circumference 131.946^2=65.973 half cir- 
cumference... Then 65.973x^1 — 1385.433 
area required. 

■- 1 > i^ ' — g 

ROUND TIMBER, &c. 

EXAMPLES. 

(2) First, As 7 : 22 : : 21 in., or liC5ft. : 

5.5ft. circumference. Then the diameter 1.75f%. 
-i-2=.875 half diameter ; and the circumference 
5.5ft. -f-2=2.75ft. half circum. Then .875 X 
2.75=24.0625 areax20ft.=48.125c.fl. Ans., 
which X 12=577.5 square feet. 

(3) As 7 : 22 : : 2 : 6.285+. Pw 22x2= 

44, which -T- 7=6.285+ ; and the diameter 2 
-f-2=sl half diameter; pad the circumference 
6.285 -T- 2 = 3.147 + circuntference. Th8n 
3.147xl=3a47,whichx6.5=20.4555c.f.An8. 
N. B. — ^The above rule is applicable to the ad- 
measurement of stonf in a well. 



ADMEASUREMENT OF STONE IN A WELL. % 

EXAMPLE. 

(2) First, the diameter in the clear 4ft. 9in.=4.75ft., 
which +15in., or 1.25ft., thickness of the wall 
ss6ft. the true diameter. Then state, As 7 : 
22 2 : 6 : 18.8$7+feet circum. or leng^ i 



/ 



• S' 



\^ 






FELLOWSHIP. ^ 75 

and 18.857 X 1.25x20 = 471.4te, which -*- 
24,75=19.047 perches +, or computed at 16.6 
square feet, pir perch=28.5Tl perches + 



FELLOWSHIP. 

CASE 1.-— EXAMPLES. 

(fe) D's Stock $280 
E's do. 600 
F*s do. 32(r 

As $1200 : 280 : : 120 : $28 D's share. 
For 280x120=336,^00, which -s-12^00=5$28 D's 

share. Ans. Ans. 

Then, as 1200 : 600 : : 120 : $60*'s share. 
And, as 1200 ; 320 : : 120 : $32 F's share. Ans. 
(3) A's $639.00— $134.75=$404.25 A's loss. Ans. 
B's= 756.&0— 189.2 = 667.60 B's do. Ans. 
C's= 854.16— 213.54= 640.62 C's do. Ans. 
D's=1200.00— 300 '= ftOO. D's do. Ans. 



As $3349.96 : 539 : : 837.49 : $134.75 

As share 
For 837.49x539=451407.^1, which -r.*3349.9« 

■=$134.75. 
As $3349.96 : 766.8 : : 837.49 : $189.2 

B's share. * * . 

t'or 837.49 x756.8=633812.432i which-5-3349. 

96= 1«9.2 B's. 
As $3349,9.6 : 854.16': : 837.49 : $213.54 

C's share. 
For 837.49 X 85'4.lft = 715350.4584, which -5- 
, 3349.96=e$21^.54 C's. 

As 3349.96 : 1200 : : 837.49 : $300 D's share. 
For 837.49 /lt200=100498800, which~3349.9& 



76 FELLOWSHIP. 

• 

aBs$300 D's share. Then subtract each man's 
share (the sum he does receive) from the sum 
due to him, and you h|ve each one's loss. 
Answer required, as you se^ before. 
(4) A 25 t. 

B36t. 

C 40 t. 

D43t. 

As 144 : 25 : : 36 : 6.25 1 A^s proportion 
of the loss. Ans. 

For 25x36=900, which-f.144 (or 12 and 12)= 
6.25 tons A's loss. Ans. 

As 144 : 36 : : 36 : 9 tons B's proportion 
of the loss. Ans. - Ans. 

For 36x36=1296, which-5-144=9 ton^ B's loss. 

As lA : 40 : : 36 : 10 tons C's proportion 
of the loss. Ans. Ans. 

For36x40=1440,which-f-144=10 tons C's loss. 

As 144 : 43 : : 36 : 10.75 tons D's pro- 
portion of the loss. Ans. 

For 43x36=;1548, Which-^ 144=10.75 tons D's 
loss. Ans. 

(5, Eldest son $184 
Second =155 
Third = 96 

As $435 : $184 : : $184 : $77.8298855^ 

eldest son's share. Ans. 
For 184x184=33856, which-T-435 =$77.82988 

55+* Ans. 
As $435 : 155 : : 184 : $65.563218+ second 

son's share. Ans. Ans. 

For 155 X 184=^28520, wh.-v435=$65.563218+ 
As $435 : 96 : : 184 : $40.606896+. 

third son's share. Ans. Ans. 

For 18 1x96= 17664, wh.-*-435=$40.606896+ 







FELLOWSHIP. 


'i 




(6) 


A 6 
B7 ^ 

C8 ' 




m 






8 21 : 6 : 


: $3050.25 
6 


•1 

a 




3)18301.50 










7)6100.50 






: 7 


Ai 

: : $3050.25 

7 


as. $871.50 A's. 


Afl21 


As 21 : 


8 : : 3050.25 
8 




7)21351.75 


3)24402.00 






3)3050.25 


« 


7)8184. 




Ans. $1016.75 B 


's. Ans. $11 fe C's. 



77 * 



(7) First 800x2=1600+40=$1640 gained, 

and 800-^140A,— 260B,— 300C,=100 D's 
stock. Ans. 

Then, as $800 : $140 : : $164^0 

14^0 

8^00)2296y^00 

Ans. $287 A's gain. 



g2 



78 



FELLOWSHIP. 



As 800: 260:: 164,0 

26v0 



(3) 



As $800: $300:: $1640 
' ^ 300 



, 984 
328 



8^00)4264^00 



8^00)4920^00 
C's gain $615 Ans. 



Ans. $533 B's gain. 



As $800 



$100 : : $1640 

100 



8^00)1640^00 
Ans. $205 D's gain. 



CASE 2.- 

M 80x35=2800 
N 75x50=3750 
-f 90x45=4050 



-EXAMPLES. 



As 10600 : 2800 : : 120 : $31.69.8/^ 
M's rent. Ans. 
As 10600 : 3750 : : 120 : $42.45.24^ N's rent. 
As 10600 : 4050 : : 120 : $45.84.9^ P's rent. 

(3) A $1000x12=12000 
B 1200x10=12000 
C X 7.5= 12000 



36060 



It 18 evident, if the profit or gain be equal, the pro- 
ducts (of each man's stock and time) are equal. 
Now, to find C's stock, we have the product of two 






FELLOWSHIP. 79' 



numbers ; and one of them giyen to find the other, 
or dividend and juiotient given to find the divisor : 
thus, the prodiiHor dividend 12000-^7.5=1600 
C's stock. Ans. 

And the whole gain $1200-h3=$400 each man's gain. 

Or, as 12)36^000 : 12^000 : : 1200 



3)3 



Cancelled 1 



1200 

[gain. 



400s=f400 each man's 



Then A's stock $1000+^00 gain=ss$1400 A's share. 
B's do. 1200+400 gain=:>1600 B's share. 
O's do. 1600+400 gain=$2000 C's share. 

(4) D*40x3==$120 ^ 

E 75x4 =$300 



As $42^0 : $120 : : $7^0 

7 



42)840($20 D's share. Ans. 
84 







Then, as $42^0 : $300 : : $7^0 

7 



6)2100 
7)350 « 

Ans. $50 E's share. 



80 LOSS AND GAIN. 



LOSS AND QMP^. 

EXAMPLES. 

(2) 3.00— *2,5(y=.50, which Xl00=$50, which -^ 
K $2.5=:$20. Ans. 

Or thas, 3.00^ sold at. 

•—2.50 prime cost. 



2.5/))50.0^0(20 per cent. gain. Ans, 
50 



Or, as $2.5 : $100 : • $5 : 20. Ans. 

(S) 50 prime cost. 
-—45 price sold at. 

6;0)50y^0 two cyphers annexed. 

Ans. 10 per cent, loss. 



(4) Assume $100 
Half No. of mo. =1 m. 



1.00 interest. 
+ 100. principal. 



As 101 : 50.50 : : 100 

100 
Then, $50 



k ^ — *^ 101)5050.00($50 the present worth 

► « — — 505 of $50.50 

4;0)100^0 



Pr.c.gain 25 Ans. 







LOSS AND GAIN. 81 



(5) Assume dSlOO 
Half No. nio.e=4 m. 



4.00 interest. 
+100. principal. 

As £104 : 13s. : : £100 : 128. Ad. 



So, 1 3s. s= 12.5s. prime price. 
—12. sold at. 

Ans. . 

12.5)50.0(4 per cent loss. 
50.0 



(6) 10.5c.+2=lS5ro^ prii^e. 

105 sold ^. 



• 




125)2000(16 per 


cent. Ans. 


(7) l«.+2id. 
Is. = 


s=s57 qrs. sold at. 
48 qrs. prime. 








6)900 
8)150 
£18.75 Ans. 


» 


• 

(8) Assume $100 
Half No. ma: 1.5 






• 

Then, $7^ 
• •"6. 


1.5 Int 
Prill. 100. 




1.2 

XlOO 


# ■ 9 101^ : 


7.308 : : 100 : 7.2 


6)120 


•• 






Ans. 20 



82 LOSS AND GAIN. 

CASE 2. — EXAMPLES. 



(2) 100 


» 




+26 




• 


As $100 : $.08 : : 125 

.08 


Ans. 
(3j 100 


-^ 


10^0)1^0.00 




$.1:==:10C. 


• . 


Or, $100 


+25 


—10 




125 As $100 : 


$5 : : 90 


- 


8 c. ' 


.5 
100)45.0 

45 c. Ans. 




10^00 Ans. 




(4) 1100 




• 


+ 100 


y 


. 


Aff$100 : $.04 : 1200 






.04 






m 

100)48.00 


« 


• 



2). 48 c. per lb. Aris. 
8)24 
Ans. do. per oz. Avoirdu|i€HS.f* 



I» 



LOSS AND GAIN. 83 

(6) 50 gals, at 75c. =$37. 50- whole prime cost; and 
60 gals. — 10=40 gals. So, $37.50-f-40= 
$.9375 prime cost per gal. 

Then, as $100 : $.9375 : : 100+10 

^^11^0 

10^0)10^3.125^0 

Ans. $1.03125 per gal. 

CASE 3. ^EXAMPLES. 

(2) $100 t (3) $100 
—12.5 —10 



Kb $87.5 : $.875 : : $100 As 90 : $.45 : : $100 

100 * 100 

Ans. 

87.5)87.5^^00($l. per lb. 9^0)450^0 

Ans. 50c. bus. 



(4) Assume $100 Then, $100 

Half No. of mo. 1 i^*- -' +2^ 

m. 

Interest $1.00 As $125 : $50:: $100 

Principal +100. 100 

— — — Ans. 

$101 : $50.5.: : $100 125)5000($40 

100 500 



101)5050.0($50 pres. worth. 
505 



^- -H 



84 LOSS AIifD GAIN. 

5) tlOO 



As $95 : $.19 : :^$100 

100 



95)1900(20c. Ans. 
190 



CASE 4« ^EXAMPLES. 

(2) lOfi 
+ 12.5 



Astl : $1,125 : : 112.5 
112.5 



5625 
126000 

Result $1^6.5625 
—100. 



Ans. $26.56.|[^ gj^ 



(3) $100 

+10 

Afi $2.23 : $2.75 f : 110 

11^0 



2.23)3$250($135.65 result. 
—100 



$36.05+ Am. 



. . (4) •36-i-48B:$9 the secotid price of the sugar. 

' $100 
. —10 

Theft* as $8 : $9 : : 90 

9 • 

8)810 

f 101.85 result. 
—100. 



Ans. $1.25 per cent. gain. 



EQUATION OF PAYMBNTS. 

; EXAMPLES. 

(2) $200 •» (3) $420x6=2520 
400 X 5»2000 —60 
400x10=4000 



36;0 )«ft52^0(7m. Ans. 



1^000 ) ft;000 252 

Ant. 6 month . 



BARTER. 

CASE 1. ^EXAMPLES. ^^j^ 

(2) First ♦44x35=$1540 value of A's cloth. ^* 
then, as $1.42 : $1540 : : lib. t 1084.51b.+. Ans. 

(3) 7i cwt.=840lb. X 12c.=$100.80c..yal. A's sugar. 
■ Then, as 12.5cwU, or 14001b. : 1 lb. : : $100.80 : 

$07.2m: Ans. 
{4) First, 50 bu. at 70c. =$35. value of the rye. 
Then, as $1.25 : $35.00 : : Ibu. Or, $35.00-i» 
1.25=28 bu. 1 

1 .96)35.00(28 bush, wheat, Ans. 
H 



86 ^ BARTER. 

(«) First, 189 gal. x 80c. =$15 1.20 val. B's brandy, 
which -f- 126=$ 1.20. Ans. Or, as Ig. : 189g. 
: : 80c. : $151.20. Ans. And^ as 126yd. : 
1yd. : : $151.20 : $1.20. 

CASE 2. EXAMPLES. 

(1) $1.25 $.62.5m. 

150 bu. 65 bu. 



$187.50 A's wheat. 3125 

40.62.5 B's barley. 3750 



375m.)146.d7.5(391§bu.oat8.Ans. $40.62.5 m. B's 



(2) 5x95x23c.=$109.25c. value E's muslin. 
32x$2.50= 80.00 value F's sheep. 

1.50)29.25(19.5 cwt. Ana. 
150 

1425 
1350 



# 



750 
750 

CASE 3. EXAMPLES. 

(1) As 2)22 : 25 : : 88 



11)11 : 25 : : 44 



Ans. 1 : 25 : : 4 : $1.B'8 casinet. 



Itm, 30 yds. at $1=$30, and $30-^$.25=120 yds. 
muslin. *^8* 



BARTER. ^^ 87 

(2) As $1,375 : $1.60:': 44c. Then, 15^0 gals.atl 60c. 

1.50 15v;6 



1.375).6600(48c. 4)22500-^48c. 

12)5425 
Ans. 468.75 gals. 

(3) Is. 9d.=1.75s., and ls^'6d.=^l;|fe* 

200 200 



A's linen sold for 350s.=:4200d. 300.0s 

Prime cost 300s. subtracfed. " — 

A gained 50s. 

Then 4200d.-r-7.5d.==560 gross buttons. Ans. , 
B sold 560 gross at 7.'5d.=^^0s. sold for. 
B'sss560 gross, but 6d., or . 5s., =^^B0sr prime value. 

B gained in the trade 70s. (balance.) 
Subtract A's gain 50s. (balance.) 



** 
* 



•a 



B gained more tham A20s.=dSl. 
. 7.5d. '^r, 3509, IN&I^^ 



6. 280 



* 



1.6 28^0)700^0(25 

100 56 

6)150^0 140 

140 



B gaan^25 per cent. 

A <j|Vl6 13 4 per cent. 

Difference ^8 6s. 8d. 









«0 REDUCTION OF VULGAR FRACnON& 

(4) Thue, 200 poUs. 12140 the whole tax. 

at 70c. —140 pdl tax. 



$1400 



$140.00 $2000 balance. 

Then, $5^00000 : $2000 : : $1 

$1 



4m. 5^00.000m. 



$5.60,0 for A*s. real estate. ' 4m. per $1. 

•.70 thelpolL 



AnflW 



LdOc. A's. tax. 

(5) Thaa,540poU8. To whole tax, i2350.90c 

atf.eO TbepoUtaz, —384.00 

$3M.OO poll tax. • Balance, •1935.00e. 

Then, $64530 : •1035.90 :: $1 

fi«l30)ld35.90(«.03e. per $1. Ana. 
1935.90 
And $1340 real estato^ 
+ 874 personal. 

$2214 A's. n^ole estate. 
X $.03c. 

66.42 
60c. X 2= liJO 

Ans. $67.62 

1 




REDUCTIO]>rOF VULGAR FRACTIONS. 

CASE 1 . EXAMPLES. 

(2) First 91)119(1 91=IS 

9L Then 1) ^ Ans. 

119=17 

. 28)91(3 
84 

Common measure ^7)28(4 

28 



REDUCTION OF VULGAR FRACTJ0N8. 91 

(3) First 195)468(2 196 5 

390 Then 39) =— Ans. . 

468 12 

78)195(2 
156 

Common measure s» 39)78(2 

78 . 



(4) First 417)973(2 417 3 

834 Then 139) «- Ans. 

973 7 

Com. measure 139)417(3 

417 



(5) First 1770)1887(1 1776 16 

1776 Then 1 1 1) «— Ans. 

^ 1887 17 

Com. measure 11^776(1 <. 

Ill 

666 
666 



(6) Fifs( 896)1152(1 896 7 

896 Then 128)—=- Ans. 

1162 9 

W 266)896(a 
768 



'4*. 



Common measure 128)256(2 

256 



92 REDUCTION OF VULGAR FRACTIONS. 

CASE 2. EXAMPLES. 

(2)*127A (3) 653^ (4) 15^ 

Xl7 Xl9 11 



2163 12410 172 

Ans. ' Ans. Ans. 

17 19 ■ 11 

CASE 3. — ^EXAMPLE. 
(2) 15X12=: WO 

— Ans. 
12 

CASE 4. — ^EXAMi^ES. 

(2) 17)2163(1273V Ans. 
17 

— (3) 5)45 

46 •— 

34 9 Ans. 



•** 



123 
119 



4 

CASE 5. EXAMPLES. 



(2) Thus, 3x4x6xll_s3_,. .„. 

4X6X6X12~^~"" *° 

(3) Thus, lXlx3x25_-,_,,, ^^ • 

4X2X4X 2-«-'"- ^^• 

(4) Thus, 15X17X4 . ^ 

17x30x5 ^^ * 






REDUCTION OF VULGAR FRACTIONa 93 

CASE 6. EXAMPLES. 

(2) Thus, Ix3x4x6x8_ --. _, .^ 
2x3x4x6x8^TTryT— s- -^s* 



Or, 1x3x4x6x8.^1 and2x2x4x6x8^^^«' 
2x3x4x6x8 ^'^^^^.'^^'^^ 

• • • • 4 

3x2x3x6x8=iVTVAi«.5x2xax4x8=yVTV-^ns- 
• And7x2x3x4x6=|f^|. Ans. 

(3) First I of f =|f , or f , and 7|== V» it is, then, 

• • • 

Then, Ix9x4xl3_^,_, . 

2x9x4xl3~Tjy— 5- -^ns- 

5x2x4xl3=|4f. Ans, 

And3rx2x9xl3=^T^y- ^ns. 
And 3x4x9x2=^f. Ans. 

(4) First I of 1^, or 1=1 prepared ; and is thus, -f^, 

I, f^, and |. "^ 



V 



. Theii, 11x8x12x8 •,^^, 

15x8x12x8 TTT"^' ° T«- ■*"*• 

• • ■ 

• And0xl6xl2><8==ffJ|f Ans. 

Ahd'7xl5x8x8=pVV2V Ans. 
- And5xl5x8><:i2F=T%%V Ans. 

CASE 7. EXAMPLES. 

(2) i, f, I, f 2 3 4 5^2c=l 3 2 5, and 2x3 
X2x5=60, the least common denominator. 



^-^ 
-•|» 



m • 4 

»^ REDUCTION OF VULGAR FRACTIONS. 



I 



Then, 60-7-2x1=30, the first numer ator. 60-J-3 
X2=40 t he se cond do. 60-7-4x3=45 the • 
third do. 60-5-5x4=48 the fourth do. 

So tha required fractions are | J, f ^ |f , |f . Ans. 



* • 



CASE 8. ^EXAMPLES. • 

2). Thus, Id.XiV^^^-rAo -=4*0^- Ans. , 

3) Thus, idwt.K2VXiV«FAolb* Ans. 
(4) Thus,4w.XA=Th^=%Jt^ 

CASE 9. EXAMPLES. * 

[2) Thus, ji^l. X V X V =Hf =?d. Ans. 

>) Thus, ■^\\b. X V =if =tOz. Ans. 

(4) Thu8,^yCwt.xH^=HI=Hlb. Ans. 

CASE 10. EXi^MdH^?. 

1^) Thus, 4^. 
• 5 
20 

• 

7)100 

Ans. i4s.-|-2 • 
12 

■ 7)24 
Ans. 3d; -1-3 
4 

* 7)12 

* Ans. 1^ qreb 

(3) Thus, |lb.x V = V=* 12oz. -f 4 X V = V =^ 
12|df. Ans. 



/ 



V.V 



• 



# 



« 



r 



REDUCTION OF VULGAR FRACTIONS. 95 

(4) Thus,fm.x|=V=6fur.+4xV='r=26pt.+ 
4xl6.5=V=:ll ft. Ans. 

(6) Thus, T^dayxV-=W = 16h.+ 8xV = #=« 
36m. + 13.xV=W=^5i^. Ans. 

• (6) Thus, fa. 6 ^ 

4 " 

T)» + 3 
— 40 
• Aqs. 3 r. — - 

• 7)120 

Ans. 17-J p. 

CASE 11. ^EXAMPLES. . 

(2) Thus, 14s. 3d. Ifqrs. reduced to Tth's of qrs.s 
- 12 4800 

— -^— =f/, An^ 

171 1/, redu. to 7th'g qrs.=::6720 
4 






685 



•48QP 



(3) Thus, 12oz. 12|drs. reduced to 5th's ofdr.ss 

1024 

=ilh, Ans 

lib. do. to do. ==1280 ^ 

(4) Thus, 6 fur. 26 p. 11 ft.=4400ft. 



1 mile=d280fl. 



=f m. 



($) TfiU99 16h. 36m. SS^^^^sec. reduced to the 13th of 

day =777600 

... , =Aday. Ans. 

Iday do. =1123200 '^ "^ 



96 REDUCTION OF VULGAR PRAOTIONa 

(ft) Thus, 3r. n^p.= 960 

s=sf a. Ans. . 

• la.=1120 



^ CASE 12. — ^EXAMPLES. 

^) estate, as 7 : 42 : : 8 : |^f . Ans. 

8 

7)836 

Ans. 48 dedmmnator. 

(3) As 8 : 27 : : 9 ': 27. 

9 Ans. 



1^ 



"8)243 

Ans. 30| denominator. 

(4) As 5 : 36 : : 16 : 36 

16 Ans. 

U6i . 

5)676 

Ans. 115} denominator. 



CASE 13. ^EXAMPLES. 

(2) State, as 9 : 5 : : 45 : f{. Ans. 

3) As 17 : 4 : : 68 : }f . Ans. 

4) As 4 : 3 : : 46 : 34i 

Ans. 

46 

(5) Aall : 7 : : 20 : 12^^ 

■ Am. 
20 






( 



ADDITION OF VULGAJR-FItACTIONS. 97 



ADDITION OF VULGAR FRACTIONS. 

EXAMPLES. 

(2) First 4|=V» and 9|=V» and ft of y=Y/» 
and I of |»f. 
Then the fractions are V» > ft and y, ^ 

So 259x8x4«= 9288 
3x80x4=1 960 
37x8x80=23680 



32928 1029 

=-^-or 12U. Ans. ' 

80x8x4=2560 80 

(3) First tjx ' V*=^ V®» ^^^r™- 

|c.,x V==V» or ym. Ac.»X V«f|' or '/«»• 
and fm. 

Then, 200x4x8x8=51200 

25x1x8x8= 1600 

15X1X4X8= 480 ^ 

7X1X4X8= «24 

• 

53504 

— 4i^tB20c. 9m. An«. 
and 1x4x8x8= 25d ^ 

(4) First i/.xV xV=H**d. sfs.xV=«Vd., 
Then, 240x7x9=15120 

36x9x9= 2916 

4x7x9= 262 




18288 



>32d. ^. Ans. 



9x7x9= 567 Or, 2s. 8d. l^qr. 



\ 



98 ADDITION 6F VULGAR FRACTIONS. 

Or thus, J/.xY=Vs.+?s.=140 

167 

9x7= 63 

+^d.=4428 
+252 



4680 



•«8ifd. 



567 



(5) Thus Jw. xi=Jda.+ida.=21 +4=f f «2d. and 
• ^xV=Tt==2h.,andih.xV=V=3am.,and 

|m.xV = *l®=^5s®<5. Ans. 

(6) Tims 4e.x^7<^c.=«V<»c. ^y'yX'^^'c.zs^ofc. 

Then, 4000 x U x 1 5=660000c. 
300X7X15= 31600 
70x11x7= 5390 



# 



7X11X15= 1155 



696890 

:$6.03ifVi<^. -^i^"- 



Thus, 3x4=12 
3x8=24 

— • 
86 d 

8x4=32 8)9.000 

Ans. •1.12c.5m. 



ADDITION OF VULGAH FHACTI0N8. 99 

Or, •l+ti^.ia Or, »f =».375 

+24 and 1= .75 

36 t •l.lS.Dm. 

—=}=»!. 12c.5m. 

8x4=32 

(8) ThuB, ilb.xV=V*' or ^. «' *>z- ^^• 
And ^oz.xV='tV=35 

' . 3)36 

Ans. lldwt.+2x24=V,or 
S^.oriegrs. 

(9) Firit^t.xV=V''w'- 

Then, V'+/i=800 
+63 

863 cwt.qr.lb.oz. dr. 
7j(10="70)863(12 1 B 12 12f. Am. 

(10) First Jm xf=V=f, or 6 fur. Ana. 

Then, ■,^fur.xV=VT = poles, Ans 

(11) First ^y.xf=fft. 

Then, |ft.+f ft.=9 



2x3=6)!^ 

Ana. 2ft.+l 



6)12 



100 SUBTRACTION OF VULGAR FRACTIONS. 

(12) First {m.x»V''="l*''=1540yds. Ann. 
Aiid|y.xf«f«f, or 2 ft. Ans. 

Then, |ft.x¥«V«»ia- Ans. 
(la) First }+/y=48 

+40 

88 

=«|^. Alls. 

8xl6«128 
Then H of •^••=*f#*=^2760. Ans. 
Or, •4000xll-*-16i=$2TfilO. Ans. 



SUBTRACTION OF VULGAR FRACflONS. 

EXAMPLES. 

(2) Thus, ^»— f «441 (4) If— A-A- Ans. 

—250 Or, fj_^«130 

—62 

191 

Ans. 78 Ana. 

50x9«460 «<f 

"% 13x13=169 

(6) Thus, I— |=it9 (6) First 7|=31, and 

'—8 5i=sV. 

Then, V— V=62 
1 ;— 44 

•*- Ans. ^ 

4x3«ifl2 18 Ans. 

~«}=2i 
'* ' 4x2s8s 8 

(7) First 8i=»V, and 5f«V. 
Then, y— ^=51 

—84 



2x3«6 



17 

— «;2|^Ans. 



SUBTRACTION OF VULGAR FRACTIONS. 101 

(8) First 5i=V- Then, y— J =33 



29 

. ^=4J-. Ans. 
2x3»6 

(0)' Thus, 12 We subtract the numerator from its 

— f denominator, and place the re- 

raainder over the denominator, 

Ans. Hif^ and carry, &c. 

Or thus, Y— «=84 

—3 

81 

— =11^. Ans. 
1X7«7 

(10) Firstf of Js=:|=i, andlofi^A- 

Then i— A=20 
—9 

11 

— Ans. 
3x20=»60 

(11) First 3|c.=sVc.,and ^of *2|,arf=:^s=Wc. 

Then, Vy*c.— Vc.=2100 

—150 



1950 

aa43ic. Ans*;* 



16x3= 45 

(12) First J/.xV=V=fs- 

i2 



4 



102 MULTIPLICATION OF VULGAR FRACTIONft 

'Then, |s. — ,^s.ss50 

—9 . 

41 

lXlO==rlio)41(4s. l^d. Ans. 
40 

12 
10)12 

"TA=i|d. 

(13) First 6w.x7=V days, and 19f =V ^y^- 

Then, 3^— y=175 

—99 

76 

— =«15da., and |x V=V' 
1X5»«6 4h.,andfxV==4''' 

48m. Ans. 



MULTIPLICATION OF VULGAR FRACTIONS. 

EXAMPLES. 

(2) First 6|«V. Then, yxf«^H'«=64|. Ans. 

(3) First 9J=y, and i of i=r|. Then, V'xJ= 

fj=3-ft. Ans. 

(4) First i of f =4, and i of ^^^r=z^. Then, | 

X^3ss|f=i^. Ana, Or, cancel it; thus, f x 

•A-^Syx* Ans. 

') Thus, 5i=Vxi=H- ^^' 



w « 
1 • 



Miiwi^ii 



* PROMISCUOUS QUESTIONS 103 

(6) First 7f =.V» ^i-=f » ^i'^h ^^ f of i|=ffr. 

Then the fractions are V» l» J» tW- Cancelled 

thus, V. I, frV- Then, '-ixiX-f^^mV 
s=39. Ans. 



DIVISION OF VULGAR FRACTIONS. 

EXAMPLES. 

(2) Thus, f xf = If = 1^. Ans. Or, f)K« « 

1^. Ans. 

(3) First 5 J =» V » and 7| = V • Then, /^ x V =lli 

=1^. Ans. 

(4) First i xi X§= A» or ^, and J xi= A- ' Then, 

¥xA=lf»orf. Ans. 

(5). Thus, TWV(fA=i*- Ans. 

(6) Thus, J)KA. Anfi. 

(7) Thus, |xf=V=4f Ans. 



PROMISCUOUS QUESTIONS IN VULGAR 

FRACTIONS. ^ jjL 

EXAMPLES. V* Z 

(1) Thus, 384)1152(3, then 384) 384 

1152 «!. Ans. 

1152 

(2) State, as 1 : 8 : : 6 : 48 num. Ans. y. 



(3) Thus, 2s. 6d.=: 30d. 

a=J/. Ans. 

jei.==:24 



104 IN VULGAR FRACTIONS. #•. 



(4) Thus, 36x8+5=293 

Ans. • V 

8 

(5) Thus, V=12, or72 

"6)72 

12 Ans. 

(6) Thus, Jxf Xj«^=j. Ans. 

(7) Thus, Ix2x4a8, so the answer is ^. 

. 1x4x4=16 do. if. 

3x2x4=24 db. ||. 

4x2x4=32 

(8) Thus,^a.x4r.=V~3r. 17^p. Ans. 
Or thus la. =160 poles. 

6 

7)960 



4^0)13 Jl poles. 

3r.l7|p. Ans* 

■ ■ 

(9) Thus, 3x15=45 
6x15=90 



« 



135 

=f. Ans. 

.15x15=225 

(10) First 71 = y , and 5f = V 
Then, 31x3=93 
17x4=68 

161 
4x3=12)161 

13^. Ans. 



'V —rl 



• • 



PROMISCUOUS QUESTIONS 105 

(11) Thus, f xl=28 

+24 

52 
8x4=s32(52. ($1. 62p. 5m. Ans. 

(12) Thus, i lb. X S* = V =^oz- Then, ^oz. x V 

. =:V5«= 11 dwt. And tV <>' |xV=V« 
16 gr. 



(13) Thus, $!=«£•. 

Then, 3joxj=1200 

—12 



1188 

=:aj7«|>.7425. Am. 

4x4= 16 



(14) Thus, i cwt.'XH*«H^ = V. 

Then, \«xf5=672 

—7 

665 

1x12=12)665 



55^ lb. Ans. 



ftX V==lt«V=6oz., and f X V = ¥ = 10|dr. 

So, 551b. =1 (Jr. 271b., and the 6oz. lOfdr. com- 
pletes the Ans. 

^16) ■ Thus, i of f =^=T^ the part sold. 



106 IN VULGAR FRACTIONS. 

Then, |x/j=60 
—40 

20 

— =/:j left. An0. 
8X12=96 

^X*f®=*ff*=*lS''-5 worth. Ans. 

(16) Thus, I of &=V» »nd j of f^.^. 
Then, 24x35=840 

=21. Ans* 

5x8=40 

(17) Thus, •259TV=Hr- 

Then, TV)H*KWo'=»n.29^. Ans. 

Or,TVXHr=4151 

«40)4i51($17.29tV. 

(18) Thus, i of f=t. Theo, 4)^(=f Ans. 

(19) Thus, J of |=^V- 

Then, VxH*=*^V®=^533 6s. 8d. Ans. 
Or thus, £100 
Xl6 



8)1600 
J533. 6s. 8d. Ans. 



SINGLE RULE OF THREE IN VULGAR 

FRACTIONS. 

EXAMFLES. 

(a) Thus, 3651=^^1 xV=^^r*'=^V' <lays in 
20 Julian years. And 6is=^-£c, 
Then state ; thus, 

Asjda. : 'f\^^da,. ; : ^c. : i?V3*'Aiiij. 

f 






'" r 



THE SINGLE RULE OF THREE 107 

For, 1^7305x25=182625 



=$456.56c.25. Anff. 



1X1X4= 4 

(3) Thi»,f of|off=^=V^jlb. 

Then state ; as fib. : y\lb. : : $^ : $|1. 
For, 9x5x9=405 

=$}X.=b2700c. 

8x 12X 10=960 Ans. 

64 )2700(42i|, or 
256 . 42^. 

140 
128 

12 

(4) State, as Jyd. : lyd. : : Vyd. : ^f. Ans. 

^ 4x3x20=240 

'— =sl2 yards. Ans. 
5x4x1=20 

(6) Thus, 7i=V» aJi^l U=f. 

Then state; as | : f ^ : y : y. 
Thus, 4x3x15=180 

=sV'' ^^ 15yds, Ans. 

3x2x2= 12 

(6) Thus, 5J=V- l5=f 27|=4S and ^X 
iii=«iij9, or =>f«yd. Ans. 

Then, as |yd. : '^»yd. : : •} : t^S or $55.5. 
For, 1x148x3=444 111 



1X1X8= 8 2 )111 



$55.50. Ans, 



(7) Thus, 6f=Vh. l|=Ja. 9i=V^- 8|=Va- 
7i!he fractions beinf prepared, find how mucb **^ 

5ip 






108 IN VULGAR FRACTIONS. 

will mow in yh. (the same time A ib mowing 
1 acre.) 
As *ih. : yh. : : Ja. : ^^ 

For, 8X17X7=357 

=b|^ aeres^ or la. and lOp 

28x3x4s336 B will mow in yh 

Then, A}a.+|J=xl6 

33 

— acres A and B mow in ||n. 

1X16=16 
Therefore, as ^|a. : V^. 5 : yh. : y, Ans. 

or, 22fh. Ans, 
For, 16X33X17=8976 68 

33X4X3^396 3)68 

22h. 40m. Ans. 

(8) Thns, 3|=y, U=f, and y Xf=Vyd- 
Then, fyd. : ^yd. : : V • f yf^- 
For 8 xl>f 45=360 

_ ssQydil. Ans. 
5X1X8=40 

(9) Thus, 7J=V- 

Then, as yin. : H*in- : : H"- • ISH^n- A^* ' 
For 9x144x1=1206 



70x1x1*= 70 )1206(18i|in. Ans. 

(10) Thus, l«i=V. 13==}- 

Then, as Jc. : Vc ; : jib. : 7lb. Ans* 
For 4x49x1=196 

7 X4 x 1 » 28 )196(7lb. Ans. 

a. 



/, 



THE POUBLE RIFLE OF THREE. 109 

(11) Thus, I of ^=#:,. 

Then, as ^% i | : : $«-f2 : $1300. Ana. 

For, 25X1X312=7800 

=$1300. Ans. 

a^xixi=» 6 

(12) Thus, ft. x4 X V - ' V = *T "g- 

Then, as ^g.^ ^f »g. : : $ j : $140. Ans. should 
be caaceUed. 

(13) Thus, 3in.=fmen, 45=|h., lm.=jm. 

• Then, as jm. : -Jm. i : |h. s 13|hrs. Ans» 
. Forlx3xd=:27 

1X1X2=1=2)27 

13|hr8« Ans. 



* 



DOUBLE RULE OF THREE IN VULGAR 

FRACTIONS. 

EXAMPLES. 

(2) Thus, 17^da.s= V<i" 26id. = »f Ma., 5m.={men, 
16m.=Vmen, and $32|J=$*^^ 

Then, fm. : ym. reduced to the Single Rule, 
or three terms. 

Vd. : ^J«d. : : $«^. 
Thus, fm.X V^'==^l* days, first term, 
Ym.x^£*d,=^y* days, second term. Ans. 
Or, as '|«da. : ^^^^dz. : : $V-/ : $147.65.«25. 
For 2x1575x525=1653760 4725 



175x4x16= 11200 320 )4725($147. 

65c.625m.ADii 
K 



no IN VULGAR FRACTIONS. 

Or thus, 17,5 . 26.25 

5 15 



87.5 days. . 393.75 days. 



r|T 



Then, as 87.5d. . 393.75d. . . $525 . ^ ^ 
"IT" i 1? ,. ' 

For 1x393.75x525=206718.75 Ans. 

.==$147.65.625. 

87.5x1x16= 1400. 

(3) Thus, 9fda.««Vdac 12i hrs.=V^rg., and Vx 
V = i|J«=i4ohrs., 10|da.=Vda., 76|m. == 
6|«m., 204m.=*|*m. 

Then, *f *, •!■*» reduced to three terms. 

Thus, 294x3=9498 



V I '-^ 



lX3i 



As 9408 82320 1 686x120=82320- 






3 9 1 1X9» 9 

For 3x82320x1=246960 



i2}|-days. Ans. 



9408x9x1== 84672 ^ 

^* 
(4) Thus, 2671yd. = ^ V 'y^.. »7f = $ V ., miSi = 
$30^9 3, 3jyrs.=J years. 

Then, $>V* : $V. i^ V?/ 
A : i As^H^ : >sV - : *V^' : 

For 48x273x3093=40530672 

* =$140.4.An»r 

7217x10x4= 388680 



SQUARE ROOT. Ill 

(5> Thus, $2-^j7^ = $3a5, $I3i==:$V, and $1^=* 

Then, $s-jP : f ^o. 

♦tH • if- As $VW • ^'^ : : fra. : 9m. 

For 144x520x5=a374400 

. =9m. Ans. 

1625x36x1*= 58500 



INVOLUTION, 
OR THE RAISING OF POWERS. 

¥iXAMPLT!S 

3) Thus, 549x549=301401. Ans. [Ans. 

3) Thus, 64.36x54.36x64.36=160634.321856. 

4) Thus, .16x.l5x.l5x.l5=.00060625x 

.00050625 =.0000002562890625: Ans. 

(5^ Thus, ^X|x|xix|=^\V Ans. 

(6) Thus, 11=1. 

Then, JxJxJxJxJ=VV\V=16^\Vt- Ans. - 

2d. 4th. 8th. 16th. 

(7) Thus, 2x2=4x4x16x16=256x256=65536 

4th. 

X 16=1048576. Ans. 



SQUARE ROOT. 

EXAMPLES. 

(2) Thus, 106929(327. Ans. 
3 

62 169 
2 124 

647 4529 
4629 



\ 



112 



SQUARE ROOT. 



(3) Thus, 152399025(12345. Ans. 
1 



22 
2 



52 
44 



243 
3 



839 
729 



2464 
4 



11090 
9856 



24685 
5 



123425 
123425 



(4) 119550669121(345761. Ans 
9 



64 
4 



295 
256 



685 
5 



3950 
3425 



6907 

7 



52566 
48340 



69146 
6 



421791 
414876. 



601521 




'*m 



m 



i 



EXTRACTION OF THE . 117 



For, 729(9. Ans. numerator. 
729 



And, 1331(1 U Ans. ife deilominator. 
1 . 



331)331 
331 



lXlx3(K)=300 trial divisor. 

80,1831-5-300=1 second quotient figure. 

And, 1x1x30=30, which add the square of 1, 
and we have 31, the second part of the divisor, 
then their sum 300-|-31=:331 complete divisor. 



A GENERAL RULE FOR EXTRACTING 
THE ROOTS OF ALL POWERS. 

EXAMPLES. 

. Ans« 
(4) Thus, 782757789696(96. C6th root.) 
96=531441 



9* =59049x6=254294)2512167(6, the 2d ^g. of the 

^ quotient 

96« =782757789696 proof. 



(5) Thus, 2916(54." Ans. (2d, or square root.) 

5x5=25 

— t^ 
5x2=10)41(4, second fig. of the quotient. 



54x54^=2916 proof. 



118 ROOTS OF ALL POWERS. 

• • • 

(6) Thus, 15625(25. Ans. (3(l,or cube root. 
23=8 

2x2x3=12)76(5 second fig. of the quotient. 

•*^ 

253=15625 proof. 

/ 

s 
... 

(7) Thus, 133225(365 

3* =9 first figure of the root. 

3x2=6)43(6 second do. do. 
36x36=1296 * ' ' i 

36x2=72)322(5 third do. do. 1 

— •— .^ 

365«= 133225 proof. ' » 



(8) Thus, 5.(2.23606+ 
2x2=^4 

2x2=4)TL0(2 

2.2x2.2=4.84 



2.2x2=4.4). 160(3 
2.23^=4.9729 



2.23x2=4.46)2710(6 
2.236*=5f:4.999696 



2.2«6x2=4.472).0003040(0 



2.2360 x2=4.4720).000304000(6. 



2.23606^ =4.9999643236 ,j 

"""^ 356764 



EXTIACTIOIJ OF THE ll!) 



(9) Thus, 180 V36 6 



45 ^/49 7 



• • • • 



(10) Thns, .00032754^01809+ Ans. 

1 

1X2=2)22(^=8 

.018x.018=.000324 



.018x2=.036).0000035,4(0 
.0180 X2;=0360)000003540(9, 



■ ' ¥ 



•01809x.01809=.0003272481 

•i. 00000029 19 



. • 



^11) Thus, 1092727(103 
1 

1X1X3=3)09(0 

10x10x3=300)927(3 



103^=109^2727 



. Ans. . . Ans. 

(12) Thus, 729(9 numer. * And 1331(11 denomin. 
9x9x9=729 1 

1X1X3=3)T'(1 

113«1331 



4 ^ 



• * 



< « 



J 20 



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EXTRAGTION 9F THE 

(16) Thus, 30x30= 900 

40x40=1600 



121 



Sum of the squares 2500(50. Ans, 

25 • 




West 40 leagues. 



(17) Thus, 50x50=2500 
40x40=1600 



Difference of the squares 900(30. Ans. 

9 




Base 40 feet. 



124 USE OF THE SQUARE ROOT. 

PROB. 4. EXAMPLES. 

• • • 

(I) Thus, 82ft. 8|in.=82.6875ft.(9.09326+squ. rt. 

81 



1809 
9 

1818?" 
3 

181962 
2 

1819646 
6 



16875 
162^1 



59400 
54549 



485100 
363724 

12137600 
10911876 

1225724 



.«t 



Then, 9.09326x1. 12837= 10.2605617862ft. Ans. 

12 .. 



3.1267414344+ in. 



(2) Thus, 2 acres=9680y<is.(98.3869+ square root. 

81 



188 
8 



1580 
1504 



1963 
3 



7600 
5889 



19668 

8 



171100 
157344 



196766 
6 



1375600 
1180596 



1967729 
9 



19500400 
17709561 



1967738] 1790839 



USE OF THE SQUARE ROOT, 



135 



Then, 98.3869 x 1 . 1 2837 =111.11 6826353 yards the 
whole diameter, which-r-2=s55.55+ydfi. Ans. 

PROB. 5. ^EXAMPLES. • 

(1) Thus, 12(3.4641+. Then, 3.5449x3.4641 + = 
•- 9 ^ * 12.27988806+. An*. 



641300 
4 256 



68614400 
614116 



6924128400 
4 27696 



69281] 70400 



4 • 



(2) Thus, 160(12.649+. 
- 1 



22 
2 


60 
44 


246 
6 


1600 
1476 


2524 
4 


12400 
10096 


2528! 

1 


d 230400 
» 227601 



252981 I 2?99 
l2 



Then, 3.5449x12.649+ 
=44.8394401+. Ans. 



126 USE OF THE CUBE ROOT. 

PROB. 6. EXAMPLES. 

(1) Thus, 2000-^2=100 sum of the number, and 
,and 100-t-2=50 half sum, and the difference 

20-7-2=10 half difference. 
Then, 50+10=60 the larger number, and 50— 
10=40 the less. Ans. V 

(2) Thus, 288~12=24.sum of the numbers, and 24 

-4-2=12 half sum, and 12^2=6 half diff. 
Then, 12+6=18, the larger number, and 12 — 6 
=6 the less. Ans. 



Further use of the Cube Boot. 

PROS. 1. ^EXAMPLES. 

(1) Thus, 189-^7=27, which^27=3 cube root. 
Then, 7x3 = 21 less mean, and 21x3=63 

larger. Ans. 

(2) Thus, 256-r-4=^64=4 cube root. 

Then, 4x4=16 less mean, and 16x4=64 the 
larger. Ans. 

PROB. 2.— ^EXAMPLES. 



(1) Thus, 10648(22, Ans, 
2x2x2=8 



2x2x300=1200)2648 



• . 



Or thus, 10648(22. Ans. 
2x2x2=8 



2x2x3=12)26(2 



USE OP THE CUBE ROOT. 



127 





(2) Thus, ?31x80= 

23^ 


18480(26.43+ 
8 




2x2x3000=1200 

6x2x30=360 

6X6=36 

• 

Complete divisor 1596 


10480(6 


• 


10480 
9576 • 


26x26x300=202800 

26x4x30=3120 

4x4=16 


904000(4 


Complete divisor 205936 

m 


904000 
823744 



264x264x300=2090880(l»ft0256000C3 
264x3x30=23760 
3x3=9| 



20932569 



80256000 
62797707 



17458293 

PROB. 3. — EXAMPLES. \ 

. AnK 
(1) Thus, 12x12x12=1728x3=5184(17.306+ 

1 



1X1X300=300 

7X1X30=210 

7X7=49 



4184 ; 



Carried forward — 559 1 4184 



^i. 



. » 



128 



USB OF THE CUBE ROOT. 



Brought forward — 569 



4184 
3913 



17X17X300=86700 

17x3x30=1530 

3X3=9 



271000 



882391271000 

1264717 



175x173x300=8978700 1 6283000 



17302 X 300 =897870000 

1730X6X30=311400 

6x6=36 



6283000000 



898181436 



6283000000 
5389088616 

893911384 



. • Ans. 
(2) Thus, 125xl25xl25x3=5859375(180:28+. 

1 length of keel. 



1X1X300=300 

1x8x30=240 

8X8= 64 

604 



4859 



4832 



18x18x300=97200 



180x180x300=9720000 

180x2x30=^ 10800 

2X2= 4 



Cbrricrf/on^arJ— 9730804 



27375000 



19461608 



USE OF THE CUBE ROOT. 
Brought f orward— 9730804] 194G160S 



129 



18022 x300: 
1802x8x30: 

8X8: 



=974160600 

432480 

64 

974593144 



7913392000 



7796745152 



116646848 



. [breadth of beam. 
Then, 25^=: 15625 x 3 =46875(36.05 +ft. Ans. 

3»=»27 



3x3x300=2700 

3x6x30= 540 

6x6^ 36 



3276 



19875=6 



19656 



362x300=388800 

3602x300=38880000 

360x5x30= 54000 

5x5= 25 



38934025 



219.000=0 



219.000.000=5 



194 670 125 



24 3d9 875 



• > 



130, 

^• 



USB OF THE CUBE ROOT. 

• • 
And, 15' x3=10125(21.6+ft.Ans. depth of hold. 
23== 8 



2x2x300=s=1200 

,2X1X30= 60 

1X1= 1 



1261 



2125=1 



1261 



21»x300=132300 

21x6x30= 3780 

6x6= 36 



864000=6 



136116 



816696 
47304 



FROB. 4. — EXAMPLES. 

(1) Thus, 5x5x5=125in. 

Then, as 201b. : 1601b. : : 125 in. 
1 160 



2^0)2000^^0 



1000(10 in. Ans. 

1 

v^ 

000 -^'^ 



I 



(2) Thus, 75X75X75=421875 lb., and lOOxlOOx 
100=1000000 ib. 



•K* 



USE OF THE CUBE ROOT. 131 

Then, as 421875 : 1000000 : : 300t. 
» 300 ^ 



421875)300000000(7 nitons. Ans. 
2953125 



* 4*8750 
421875 

468750 
421875 

46875 

46875) =■} 

421875 



(3)* Thus, 12x12x6=864 in. in a half solid foot. 

6x6x6=216 in. in a solid half foot. 

216)648(3 half feet. Ans. 
648 

Or, lxlX.5=.5 
.5x.5x.5=.125 

.125)^375(3. Ans. 
'375 • 



{4) Thus, 12x12x12=1728, dividend. 
6x6x6=216 div.)1728(8 of 6 in. 
4x4x4= 64 doi ) 1728(27 of 4 in. 
3x3x3= 27 do. )1728(64 of 3 in. [Ana. 
2x2x2= 8do'. )l728(216of2in. 
1 X 1 X 1 •* I do. )1728 of 1 in. J 



132 



u6b op the cube root. 



• • 



. . Ans. 
5) Thus,=*2150.425200(j^907+, 



1X1X300=300 

1x2x30= 60 

2x2= 4 



1150=2 



364 728 



12x12x300=43200 

12x9x30= 3240 

9x9= 81 



46521 



422425=9 



418689 

4> 



12.9 X 12.9 X 300=4992300 



12.90x12.90x300: 

12.90x7x30: 

7X7: 



:499230000 

270900 

49 

499500949 



3736200=0 



3736200.000 



3496506643 
339693357=7 



(6) Thus, 1728(12 Ans. 
1 



1X1X300=300 

1X2X30= 60 

2x2= 4 

364 



728=2 
728 



EXCHANGE, 133 



EXCHANGE. 



WITH GREAT BRITAIN. 

To reduce Federal Money to Sterling, or English. 

EXAMPLES. * 

(I) Thus, $371.75 (2) Thus, $756 

9 9 



4^0)334^5.75 

je83. 64375 
20 


4^0)680^^4 
Ans. jei70 2s. 




8.12.875^'&0 
12 

* 




d.10.5^00 




(3) Thus, $888.88 (4) 

9 


Thus, $4536 
3 

Ans. 

14)13608(972 gui 
126 

100 
08 


4^0)799^^9.92 

Ans. dei99.998 

20 


s.l9.96y^0 
12 


28 
28 


d.11.52 
4 


*>o 



qrs.2.08 



M 



p . -»—--. 



134" . EXCHANGE. 

# 

- To reduce English, or S^^rling, Money, to Federal 

Money. 

« 

EXAMPLES. 

(1) Thus, £83 12s. 10id.=J83.64375 

• . 4,0 

• 2_ 

9)3345.75^000 
Ans. $371. 75c. 



(2) Thus, dei70 2s.=£170.1 

40 



9)6804^.0 

Ans. $756 



(3) Thus, '£1020 12s. (4) Thus, £25 
. 40 



Or, £1020.6 



40 9)1000 



9)40824^0 Ans. $Ul.ll|c, 

Ans. $453e • ' 






BILLS OF EXCHANGE. 

m 

To find the value of Bills aboye pan 

EXAMPLE. r - 



(1) Thus, 750xl.04=«$780. Ans. 



•• 



EXCHANGE. J35 

To find the value of a Bill of Exchange bel6w;par. 

EXAMPLES. 

(1) Thus, $780. So, 780.— 31.20=*$748.80. Ans. 

.04 31.20 



31.20 intr. $748.80. Ans. 



(2) Thus, £1000 

40 And, * •! • X* V'" * V ® ^^* 

9)40000 
Federal value $4444.444. 

Then, 4ojoo_i6^o^s84oo^ or 4266.66| paid. 
So, from the Federal value $40ooo take $3«|oo 
sum paid, and we have $>V® =»$177.77|. 
saved. Ans. '/ 

(3) Thus, .025x5000=$! 25 interest or gain. 
So, $5000 — 125=4875 the price paid for the 

$5000. 
*And, 1.025x5000=:$5125 sum received. 
Then, $5125 — 1875=250 gaifli. Ans. 

EXCHANGE WITH FRANCE. 

EXAMPLES. 

(n Thus, $.ia;Jx50000=$9250. Ans. 
(2) Thus, $.1873125 

9275.25 



/ 9365625 
3746250 
9365625 
13111875 
3746250 
16858125 

Aim. $1737.37'.0265625 



ltd 



EXCHANGE. 



t 



(3) Thus, 4444.441=* oj«». 

Then, VX*°r^=^2V^^c.=$800. Ans. 

(4) Thus, 4444.44^=4*<^|-«o. 

Then, 8of«»c.-T-'*®-J^<*c.=;'?j2, or 18^, Ans. 



" EXCHANGE WITH SPAIN. 

To reduce tpals vellon to reals plate. 

EXAMPLES. 

(1) Thus, 8.5' ' • (2) Thus, 24-^4=6 



800' 



16)6800.0(425. Ans, ^ 



Then, 5740 
8i 



.«■ 



(2) Or thus, rls.v. qils. mrs. 
5740 24 

8.5 



28700 
45920 

48790^0 
4 



45926 
2870 

4)48796 

, 4)12199 / V 

Ans. 30491 rls. pit. 



0) 



64)195184(30491. Ans. % 

To reduce reals plate to reals vellon. 

EXAMPLES. 

reals, quartos, mar.plt. 
Thus, 6450 20 

16 






8.5)103210.0(12142 rls.Vki. and 3 quartos. Ans. 



mmr. 



EXCHANGE. ^" ^^^ 

(2) Thus, 425 '^'^^' 

^•' talea. 

8.5)6800^0(92^ 

^^ ^1568 * 
90244 



flsT. $9540.08 



IX 



To reduce S/'aND SQUARE MEASURE. 

x2.5ft; (2) Thus, .75ft 
8.0 1.5 

(1)'. \^ 

625 1.125 

375 6ft. 



43,75 :i3368J6.75000(50fl4gals. Ans, 
8ft. 6 6840 



Ans. 350. c. ft . 6600 



13368 

Or, 6.75 X 1728= 11664c. in-s which divided 
by 231 gives -SO^fgals; Ans. 

(4) Thus, 15ft. (5) Thus, 24.25ft. 
^ 12,8in. 48,5 

(3) Thus, 16.5ft. 

1.5 180 12125 
7.5 19400 



Ans.!24.75ft. 2.5 • 9700 



lin.= 12)190 Ans. Iir6.125ft. 
• 15* 

Ans. 205f ft. 

« 



142 CUBIC AND SQUARE MEASURE. 

(6) Thus, >i7X \^ or ^=^^ or 209is. yd. Ans, 

Then, e|? X ac.=$6.28c. Ans. ^ 

(7) Thus, 63.5ft. (8) Thus, 15in.= 1.25ft. 

' 10.25" 18"= 1.5ft. 



JL 



2675 1.875 • 

LO 
535 



1070 7^0 



548.375 

2ft. 



Ans. 131.25ft. 



Ans. 1096.75c. ft. 

(9) Thus, 14.5ft. -12.5 3.5ft. 

8.5 8.5 6 



725 625 21ft. door. 

1160 • 1000 



3.5ft. 





123.25 
123.25 
106.25 
106.25 


106.25' 


4 

14. one windoM 
4 

56 four do. 
21 door. 




459. c. ft. 

—77 




382Ce ft. 




77 


Uk 


Then,^ ^^ X 


1.0=: 3^90 = 


:84.24f Ans. 


1 


(10) Thus, 18ft. 
16in. 


Or, 18 

n 


(11) Thus, 20ft. ^ 
1 




12)288 


18 
6 


Ans. 20ft.- 

« 




Ans. 24ft. 


Ans. 24ft. 


« 








* • i 



CUBIC AND SQUARE MEASUJIE. 143 

(12) Thus, 22in. Or, 22 (13) Thus, 4.5ft. 

30ft. 30 10 

2in.= 1)660 660 45 

2 3ft. 

Ans. 110ft. 



12)1320 Ans. Id5c.ft. 

Ans. 110ft. 



(14) Thus, 3.75ft. (15) Thus, 3.5ft. 

28 16 •• 



i 



8000 56. 

750 ' 4.ft . ' 

' bu. Ans. 

105. 1.47779)224.00000(151.57 + 
4ft. 10 



420ft. 915.15.7 Ans. 



128)416(3.25 cords 
384 



320 

256 Then, 3.25 X 4=913. Ans. 

640 
640 

(16) Thiu,4^in. 35]n. 

7 18 



31^ 630 
J5 IS'ai. 



DiTiflor, 15.75) 9450.00(600 books. 
94S0 

00 



144 BfULTIPLICATION CONTRACTED 

MULTIPLICATION CONTRACTED- 

EXAMPLES. 

(2) Thus, 2414 (3) Thus, 24851 

16 19 



Ans. 88624 Ans. 462669 

To Multiply by any number of Nines. 

^ EXAMPLE. 

(Q) Thus, 72031000 
—72031 



Ans. 71958969 

Having the Longitude of two places given, to find the 

difference of time. 

EXAMPLES. 

(1) Thus, 6° 40' (2) Thus. 1^ 7'.25 

4 4 



Ans. 26' 40" past 12 o'clock. Ans. 4' 29" . 

MISCELLANEOUS EXAMPLES. 

(2) Thus, 92x70=864.40<? 
And 1.375x40= 56. 

(1) Thus, 15).75(.05 Ans. 

75 4^0)9.40 . 

Ans. 23.5bu 

(3) Thus, 82475 Or, i)2475 

.005 



Ans. 812.37,5m. 



Ans. 812.37^5 



MULTIPLICATION CONTRACTED, 145 

(4) Thus, I of 884 Or, 1)84 (5) Thus, 36 

6 * —12 9 



7)504 872 Ans. 324)1 8ft Ans, 

Ans. 872 

28)224 
224 

(7) Thus, 8)1728c. in. 



(6) Thus, 8 X 4 X 2=64c. in. Ans. 8)216 

Ans. 27 bricks. 

(8) Thus, 1728c. in. (9) Thus, 7 : «2 : •* 14 

40 14 



69120 7)ad6 
12 



, 8)829440 By Rule 1st. 

8)103680' 

Ans. 12960 bricks. 

(10) Thus, 7911 miles. (11) Thus, 24853 miles 
355 7911 do. 



39555 2; «5383 

39555 223677 

23733 173971 

Ans. 

113)2808405(24853^ Ans. 196612083 3. miJies. 
By Rnle 2cl. 

N 



146 ALLIGATION MEDIAL. 

(13) Thus, 4 of 7=3.5 and i of 
(12) Thus, 7 : 22 : : 3.5 • 22=11 

3.5 



38.5 area; 

11.0 2y^0 



66 

7)77. 
11 ciicumference. 



12)770 



12)64^8. fl. 



Then, 3.5 X 1 1 = 38.5 Ans. Ans. 5|fc. ft. 



ALLIGATION MEDIAL. 

EXAMPLES. 

(1) Thus, 6gal. at 125c.=750 
9 " 80c. =720 
5 " 40c. =200 



20 2^0)167^0 

Ans. 83i 

(2) Thus, Ibu. at 75c. = 75c. 
5 " 80c. =400 , 
15 " 30c. =450 



21 divisor. )92.5(44jV Ans. 

84 

"is 

84 

T 

(8) Thus, 12gal. at 75c.= 900c 
24 " 90c.=2160 
16 " 110c=1760 

Diyisor 52* )4820($.928|| Ans. 



(2) Thus, $1.75 < 



ALLIGATION ALTERNATE. 147 

ALLIGATION ALTERNATE. 

EXAMPLES. 

$ Ans. C« C. 

' 1.20n. =75gal. at 120= 9000 
1.50>vA=25 « 150= 3750 
2.00>'y=25 " 200= 5000 

L2.50'^ =55 " 250=13760 



18^0 2)3150^0 

9)1575 



•1.75c. 

Proof. 

c. Ans. 

80^=5+75=80gal. rum. =6400 
(3) Thus, 75c. -J 70^ 5" do. 350 

r 5 " water. 



c. 
(80< 



9^0 )675^0 



75c. 1 
Proof. 



CASE 2. EXAMPLES. 



\c. gal. c. c. gal. 

'o\ Th„o ft! in $ 130\=110 Then, 110 : 120 : : 20 
v2) Thus, 81.10 I q)^ 2^ 20 

11^0)240^0 

%• 

Ans. 21 ^j^ gals. 



1 



148 ARITHMETICAL PROGRESSION. 

PERMUTATION. 

EXAKPLES. 

(1) Thus, 1 X 2x 3X 4X 5=120 days. Ans. 

(2) Thus, 1 X 2x 3x 4X 5x 6X 7X 8=40320changes 

Ans. 

(3) Thus, 1X2X3X4X5X6X7X8=40320 posi- 

tions, or days ; — ^and 40320-7-365.25=110 
years, and 142.5 days. 

COMBINATION. 

EXAMPLE. 

(1) Thus, 10X9X8X7X6=30240 



=252 Ans. 



1X2X3X4X5= 120 



ARITHMETICAL PROGRESSION. 
To find the last term. 

EXAMPLES. 

(1) Thus, No. terms 100 (2) Thus, No. terms 41 
Subtract — 1 — 1 

No. terms less 1=99 No. terms less =40 
Com. diff. X 3 Com. diff. 'x 2 

297 80 

Add first term +4 Add first term +1 

Last term. Ans. dOlc. Ans. 81 hilfs 



ARITHMETICAL PROGRESSION. 149 

(3) Thus, No. terms 12 (4) Thus, 18 (5) Thus, 19 
Subtract — 1 —1 —1 

No. terms less 1 = 11 17 18 

Com. diff. X 4 X 12 X 2 

44 204 36 

Add first term +20 +4 +3 

No. miles. Ans. 64 Ans. 208 Ans. 39 

Last term. 

To find the common diilference. 

EXAMPLES. 

(l) Thus, 605 (2) Thus, 45 

—5 —10 



150—1 = 15^0)60^0(4 Ans. 8— 1=»-.7)35 

60 — 

Ans. 5 years. 

To find the sum of the terms. 

EXAMPLES. 

(2) Thus, 605 

2)610 sum of the extremes. 

305 half sum do. 

Xl51 No. terms. 

305 
4575 



Ans. 46055 sum of the terms. 

N 2 



'— ^ 



150 ARITHMETICAL PROGRESSION. 



(2) 


Th^is, 12/ 

+ 1 




(3) 


Thus, 24 1 

+^ 1 




2)13 








2)25 




6.5 
X12 

• 








12.5 

X24 '■' 




Ans. 78 times. 








500 . i 
250 < 










Ads. 300 times. 




To find the first term* 


1 




EXAJIPLES. 




1 

I 


(1) 


Thus, 19 • 
—1 


(2) 


Thus, 8 
—1 




• 


18 
X2 






7 
X6 


« 



39— 36= 3 Ans. 54 — 42=*12 Ans. 

To find the number of terms. 

EXAMPLES. 

H) Thus, 63 (2) Thus, 51 Then, 61 

,-2 -7 -KJ 

3)51 4)44 2)58 

17 11 29 

-f-l -f-1 X12 

Ans. 18 terms. Ans. 12 days. 348 miles, the 

sum^ of all the 
terms. Ans. 



GEOMETRICAL VnOGBSSSSDlft. 151 

GEOMETRICAL PROGRESSION. 

EXAMPLES. 

18 8 4 5 e 

(8) ThlU, 10 K 100 K 1000 M 10000 M 100000 MJOOOOOO 

lOOOOOO 



1000000000000 

Thii fum c«n be proved by addition. —1 



999909999990 

Ml 



10-1-^ 



Ans. •llllllllll.lle. 

(3) Ttnu SH 4 H8M 16 H 22X64 Mi38MS$6the8tb power. 

S56 

1536 
1280 
S12 

0S536 the 16th power 
^ 65536 



993216 
196606 
337680 
337680 
373816 

4294967290 
—1 

Ans. $4394967S.95c. 



1 2 S 4 6 6 
(4) TbOB, 3H9HS7H81H243M739 

729 

656f 
1458 



531441 the I2th power. 
—1 



531440 
Ml 



3-l»2)53144D 

Ans. 9265720 



159 



UNITED STATES' DUTIES. 

(5) Thus, 1.5=1 Or, 2)81 000=$1 000 
1.5 500 



2.25=2 
1.5 



2)1500= 1500 
750 



3.375=3 
1.5 



2)2250= 2250 
1125 



5.0625=4 
1.5 



2)3375= 3375 
1687.5 



7.59376=5 
— 1. 



85062.5=5062.5 



6,59375 

X lODO first term. 



Ans. 813187.5 



1.5— 1 = . 5)6593.76 
Ans. 813187.5 



UNITED STATES' DUTIES. 

EXAMPLES. 

(2) Thus, 1 franc=8.1873125 

265,^0 

93656250 
11338750 
3746250 



10)496.3781250 actual cost. 
-7-49 6378125 ten per cent, added. 



546.0159376 
*• 20 



Ans. 8109.20.31875^^00 duty re luired. 



< j> 



BAITED STATES' DXJTIEa 168 

(3) Thus, 2500 rupees. 

50 



20=i 



1250.00 actual cost. 
+250. 



1500. 
X25 



Ans. $375.00 duty required. 



(4) Thus, 640 piast. 4 reals. 28 marv. 

8 



5124|freals=8VT<>XVc.«87|fooc. ac- 
tual value 
10)871200 
+87120 



958320 

xV=3«3^2 8.^^225.48,7tV. Ans. 

17 



CASE 2. EXAMPLBS. 



(1) Thus, 11250 lbs. 
—3000 


150 
20 


€250 
40c. 


3000 tare. 


Ans. $3300.00 





(2} Thus, 2520 gal. x48c.»:$1209.60c. Ans. 



''^ 



154 SINGLE POSITION. 

(3) Thus, 25x7=175 lb. draft or scalage. 

And, 437501b.— 175lb.=43575lb. X 12=5229,^00. 
Then, 43575 — 5229=383461b, neat. 

X3c. 



Ans. $11 50.38 c. 



SINGLE POSITION. 



EXAMPLES. 



(2) Thus, suppose A's age to be 20 years. 

Then, 20 and 20-7-2=30 do. B'fl. 
And 20x30=50, whichx2= 
100, which €idd 5(t-V of 50), 
and we have - - - 105 do. Cs. 

Result 155 
As 155 : 93 : : 20 
20 



155)1860(12 years A's age. Ans. 
155 



310 
310 



Then, 12 and i of 12=18 B's. 
» And, 12+18>«»+tV of 30=63 Cs. 

(3) Thus, suppose 30 As 2.25 : 6 : : 30 
Then, I of V =26.25 30 

And, 4 of V =24. Ans. 

2.25)180.00(80 

Result 2.25 180 



SINGLE POSITION. 155 

(4) Thus, sup. 60 A's. As lOOy. : 140y. : : 60y, 
Then, 60h-2= 30 B's. 60 



And, 30-=-3= 10 C's. 
Result 100 



1^00)84^00 
Ans. 2)84 A's age. 
3)42 B's age. 
14 C's age. 



As $47 : ^94 : : $60 
60 

(5) Thus, suppose $60 Ans. 

47)5640($120. 
Then, i of 60=20 47 

i of 60=1 5 

} of 60=12 94 



94 



Result $47 



(6) Thus, assume - - $100 
Int. at the given rate and time as 72 

$100 Result 172 : 860 : : 100 
6 100 . 

— Ans. 

$6.00 int. 1 year. 172)86000(500. 

12 860 



$72 int. 12 /ears. 00 



156 DOUBLE POSITION. 

(7) Thus, assume $ 40 harness cost. 
Then, 40x2= 80 horse. 
And, 40+80x2=240 chaise. 

4^0)36,^0 : 100 : : 4^0 

Cancelled 9 : 100 : : l=$ll.llf 
' the harn. 

And, $11.U|X2=$22.22J the horse. Ans. 
Then, $ll.ili+$22.22|x2=$66.66f cha. Ans. 
• Or, »^<>xf=^t®==$22| horse. Ans. 
And, $»$»+$»^»xf=*f?^®=$66| chaise. Ans. 



DOUBLE POSITION. 

EXAHBLES. 

(«) Suppose $400. Then, 400+ J =500 A's. 
And, 400—225 =175 B' s. 

Therefore, 600—175x2=150 error+A. or— B. 

Suppose $500. Then, 500+1=625 A's. 
And, 500—225 =275 B's . 

Therefore, 626—275x2=76 error+A. or— B. 

Then thus, 500x150=75000 
And, 400 X 75=30000 



Diff. 75 ) 45000($600. Ans. 
450 



(3) Suppose 20 oxen. 

Then, $24 x 20=$480 oxen cost. 
And, $16x20=$320 cows cost. 
And, $6x20x4=$480 cakes cost. 

Sum 1280— 820»960 error-t 



•* . 



DOUBLE POSITION. 

Suppose 10 oxen. 

Then, $24xl0=$240 oxen cost. 

And, $16xlO=$I'80 cows. do. 
And,'4xi0x$6»$240 calves do. 



157 



Sum 640 — 320=320 error+ 



Then^thus, 10x960=9600 

20x320=6400 * 



64;0 )320^0(6ox.&cowsea.An«. 
320 



And 6x4=20 the number of calves. Ans ^ 



w 



Thus, assumeasBO 
i more=xl5 
i do. = 7.5 
do. — 5. 




Assume 24 

12 

• 6 

5 


Sum as 57. 5 




Sum 47 


Thenr 30x2=60 




24x2=48 


Diff. 2.5 errors- 


Biff. lerr.+ 


Then, 24x2.5=60 
30x1. »30 






1.5 ) 30(20. 
*80 


Ans. 


■ 



158 DOUBLE POSHTON. , 

• * ■ f 

(5) Assume 8 ladies. 

Then, 8x3«24, yfhic\fiak& — 10afel4 above ten. 
So, 10 — 8 assumed number a 2 under ten. 

m 

12 error + 






Assume 6 ladies. 
Then, 6x3=18, wHich subtract ^0=8 above ten. 

So, 10 — 6=4tinde];ten. 



4 error + 



And, 6x12=72 
8x 4=32 



8)40 

5 ladies. Ans* 



(6) Suppose 30 body. 
And 10 h«ad. 
Then, i of 30+10=25 tail. 

■ •-• 

Sum 65 
And 30btidy+30 head and tail=60 ^ 

Diff. 5 enror — 

Sup. 36 body. 
+ 10 head. 
Then, i of 36+10=28 tajiL' 

* Sum 74 , 
Body 36+36 head and tail 7% 

*^ 2 error— 



>»*^ 



• « 



II 

DOUBLE POSITION. % 159 "^ 

36x5=n80 
30x2^ 60 



»} 



120 



\ 



, ^ • 40 body required. 

• '10 head. 

40^2+ 10=.30 tail. 

80 whole length. Ans. 



fy) Thus, suppose 200 A. Suppose 150 A. 

Then, 200+15=215 B. 150+15=165 B. 

J)415 1)315 

83 C. 63 C. 

Sum 498 378 

—324 324 ' 

174 error+ 9i er. + 



Thai, 150x174=26100 
200 X 54=10800 



I 

\ 

\ 

i 

i 






^ 



12^0 )^530,^G 



— 1 

•127.5 A got. Ans. 
+ 15 



$142.5 B got. Ans. 
1)270= ^um of A and B. 
$54=C*got. Ans. 



l«0 ^ 8HIPS' TONNAGE. % 

SHIPS' TONNAGE. 
By Carpenter's measure. 

EXAMPLES. *- 

(1) Thu»,60x20x8=9500,which-s-95=101^^^ns. 

2)26 

<2) Thus, 26 X 80 X 13 = 187040, ^hich -5- 95 == 
284.631H* Ans. 

2)21.6 

(3) Thus, 21.5x66xl0i75=:15023.125, whicJi^95 
Bl58.138fV tons. Ans. 
And, 15023.125x«16 = 240370, which -i- 95«: 

•2530.21tV- Ans. 
Or, »*<>|fi"x¥=''*f?^ •=•2530.21^. val. 

Government tonnag6. 

EXAMPLES. 

(3) Thus, length 87.5 
I of 29.2 breadth:== 17.52 

69.98 x20.2iX 14.6»29833.8736. 
"fhen, 95)29833.8736(314.0407^- Ans. 
(4) Thus, 66 
I of 20==— 12 

54 
20 



1080 
9 



y 



95)9720(102.3111. Ans. 



''^.. , Ships of War. 

EXAMPLE . 

(5) Thus, 97x31x15.5 = 46608.5, which -s- 100 
466.085. Ans. 



•■jf^hw*^""*****^^^^^*" 



m^m 



mmi^'i^mtmmm 



■mmuvMvMI 



GAUGING. 



IBl 



To find the length of the mast o^ ahijK 

EXAMPLE. 

(1) Thus, 108 ft. 

2 

3)216 

"^ 

+40 

t Ans. 112 feet. 



GAUGING. 

EXAMPLES. 



(1) Thus, 36in. buDff. 
->SI7in. head. 



9in. difference. 
f 



(3) Thus, 35in. bang diam. 
—35iik. bead. 



lOin. difl&ience. 
N.68 



318 

I of d-«6iii. 

'^ 27in. bead. 

33in. mean diameter. 
H33in. 

99 
99 

1089in. Bqaare, Sec 
H 45in. the length. 

5445 
4356 

49005 product. 
H.0034 



196020 
147015 



Am. 166.617vQga]i. 



02 



6Mn. product. 
<4*S5in. head diameter. 

31.8ln. metti do. 
H31.8in. ^ ' do. 

3544 
318 
954 



1011.34 square diam. 
H 40iK length. 

40449.6 
>4.0034 



1617984 
1313488 



Ans. 137. 



.538^H| 



y. 



^m^mmt 



"^"■* 



■■^ 



■ ■■• ■ 



IGS 



GAUGING. 






(4) Thu8,36in. 
^24in. 


(5) 


Thus, 30in. 
—24 


12iTi. 
X.7iiL 




6 
X.62 


8.4in. 
+24 


« 


3.72 

+24 


32.4in. 




27.72 
X 27.72 


1296 
648 
972 


5544 
19404 
19404 
5544 


1049.7.6 
X40m. 




768.3984 
X38 


41990.4in. 
.0034 


61471872 



1679616 
1259712. 



Ans. 142.76736ga]s. 



(6) Thixs, 31m. bmig. 
— ^26m. head. 

5in. difference. 
X.6 

3.in. 
+26m. head. 

29iii. mean diam. 
X29 

261 

58 

841in. square d. 



23051952. 

291991392 
X.0034 

1167965568 

875974176 

Ans. 99.27707328 gals. 

841m. 
X36in. length. 

5046 
2523 



30276 
X.0034 

121104 
90828 



Ans. 102.9384gals. 



ANNUITIES AT COMPOUND INTEREST. 163 
A SHORT METHOD OF GAUGING. 

EXAMPLE. * 

(2> Thus, SOin. 
30in. 



900 
.SOin. 



27000 
.00272 

1904 
544 

Ans. 73.44gals. 



^ >%^ 



ANNUITIES AT COMPOUND INTEREST. 

EXAMPLES. 

(2) Thus 1. first term of series. Or thus, 1.04 
1.04 second term. 1.04 

1.0816 third term 

1.124864 fourth term. 416 

1.16985856 fifth term. 104 

1.2166529024 sixth term. 

1.265319018496 seventh do. 1.0816=:» 
1.0816=2 



7.898294480896 



50 173056 
86528 



•394.91.47240448^00 10816 



1.16985856=»4th. 
1.0816=:s2 



1871773696 
935886848 
116985856 



Camcrf/on^ar(?— 1.265319618496 



^ iWl I I I il l ! ■ 

164 ANNUITIES AT COMPOUND INTEREST. 

BrougMfommrd— 1.2653190 I8496s:6th. 

1.04=1 



5061276073984 
1265319018496 * ' 

1.31593177923584 =:7th. 
— 1. 



(3) Thus, 1. 

1.06 
1.1236 
. 1.191016 



.0^.31593177923584 

78.98294480896 

50 

Ans. $394.91.47240448v00 



4.374616 
50 



Ans. •218.7308v,00 the amt. for yearly payments. 

' Then, 218,7308 x 1.014781 =*221. 96.38599648. 
Ans. amount of the half yearly payments. 
And, 218.7308 x 1.022257 »f 223.599091 4 156. 
Ans. the amount for the quarterly payments. 

CASE. 9.*— EXAHPLES. 

Present worth. 
(2) Thus, 1.05)20.00($1 9.04761+ 1st yr. 
1.05x1.05=1. 1025)20.0000($18.14058+ 2d yr. 
&c. l,157625)20.060000($17.27675+ 3d yr. 
, 1.21550625)20.00000000($1 6.45404+ 4th yr. 
1.2762815625)20.0000000000($15.67052+ 6th yr. 
1.34n095640fi26)20.00000000($14.92408+ 6th yr. 

A'atedie. A^s.SlOl.dlSS + Sum. 



I 



* 



ANNUITIES AT COMPOUND INTEREST, 165 

Then, $101.5138xl.012348=$102.76.72+ B's.Ans. 
And, $101.5138xl.018559=*f 103.39.77+ C's. Ans. 

(3) Thus, 1.06)f300.0a($283.01.88+ 
1 .06 X 1 .06= 1 . 1236)300.afi00($266.99.«9 + 

i . »- — 

Ans. 650.01.77+ 



4r 






► 7 



(4) Thus, 1.06)$1.00.0(f.9523+ An». 

CASE 3. ^EXAMPLE. 

(.1) Thus, the 5th power of 1.06=1.3382255776, whi<A 
subtract the 4th power 1.26247696 



1.26247Q96— 1.=.26247696).0757486176(.28869 
Then, .28859x207.904=59.99.9+ Ans. 
* Or, .0757486176 207.904 

X =s»60 nearly. 

.26247696 1 

CASE 4.— -EXAMPLE. 

: (I) Thus, $2132.34+$300.=$2432.34, which 

'• subtract 2132.34 Xl.05=$2238.957 



193.383 
Then, $300—193.383=1.55132, &c.=^, the 9th 
power of 1.05= 1.55132, &c.=9yrs. Ans. 

ANNUITIES, LEASES, <feb. tAKEN IN 
REVERSION AT COMPOUND INTEREST. 

EXAMPLES. 

(2) Thus, the 4th power of 

1 .04= 1 . 1 6985856)50.00900006(42.7402 + 
Then, the ^so.oOOO 

—42.7402 / . ^ng 



5th power df 1.04= 
1.2166529024X.04 



048606116096) 7.2598.6d6(WDddrei47.12+ 



\ i 



166 PBRPBTUrnES AT COMPOUND INTEREST. 

(3) Thus, 1.05x1.05x1.05x1.05 $300. 

X 1 .05= 1 .27628156253300.000.000.000.0(235.0578+ 

div. $64.9422 



And, 1.05xl.05xl.05xl.05x.a5«^.0607768125div. 
Then, .0607753125)64.9422000000(1068.563+. Aas. 

PERPETUITIES AT COMPOUND INTEREST, 

dASS 1. ^EXAMPJ.£S. 

(1) Thus, .07)»I40.00 (2) Thus, .04)$29t).00 



Ans. $2000 Ans. $7250 

CASK 2. ^EXAMPLES. 

1) Thus, $2000x.07=s$140. Ans. 
2} Thus, $7250x.04«=$290. Ans. 

CASE 3. ^EXAMPLES. 

(1) Thus, $2000 present worth. 

140 annuity. 



s 



2;000)2v^l40 sum. 

1.07y^0 amount of $1 for 1 year. 
—1 



Ans. $.07 per t^ent. or ratio. 

Thus, $290 
7250 

■ [cent. 

725^0)754^^0($1.04 amo. of $1. for 1 yr. at 4 per 

—1. 



Ans. $ .04 per cent, or ratio. 






' PROMISCUOUS EXAMPLES. 167 

A 

ft 

PERPETUITIES IN RJEVERSION. 

example's^ 
(1)^ Thus, 1.07xl.07xI.07x.OI7=.08575301. 
Then, $ 1 40 -j-. 08575301 =$1632. 595+. Ans. 

(2) Thns, 1.04j<i.t)4xl.04xl.04x.04= * 

$.04679434224 divisor. 

Ans. 
Tlben, .64679484224)$2p0.000.000.000.0($61 97.83 -f. 



PROMISCUOUS EXAMPLES. 

(1). Thus, 16)576(36. Ans. Or dius, 2)576 . 
.^48 

,r 8)288 

96 

96 , .36 Ans. 

(2) ' Thus, 56r;. s(ild for. (3) Thus, 17 

—50 prim6 cost. • ' . 29 

23 

6y0.)60^0 two cyp. annex, to diff. — 

^^ — Ans. 69th yr. 

Ans. , 12*per cent gain. 



• 



^^ acre per. ft. . ft. 

(4) Thus, 1 = 160x272.25=43560, Which^33, or 
3 <& 11. 
3)43i560 

«) 14520 
Ans. 1320ft. in length, and 1320-^10.5=80 poles. 



-». 



168 



PROMISCUOUS EXAMPLES. 



(6) Thus, 12 b. 
2 

14 b. 



: 12 b, : : 7 da. 

7 

14)84(6 dftjs. Ans. 
84 



(6) Thus, 7 : 22 : : 40 : 125.714+ ciroum* 
And, 125.714-^2=^2.867+ half circum. 
And, 40-4-2=20 half diam. 
Then, 62.857x20x5=6285.7 in* 
So,1728)6285.7(3.63+c. feet. Ans. 

$1.50 



Ans. $5.44.5 -f 



(7) Thus, J of I of i«=ff. A's share of the veiwl. 
And, f of A of fi=/^/^=/^. E's share bought. 
A«, I : j%% : : $^« V »-^ : $^o V^y*''-i =$2032.5942 

cost B. Ans. 
So. »iA.—g\%B,=^^%%, or 3ft^A's share after 

the sale to B. - ^ 

Also, A of 7VA=?WWr P'« sh«e bough^. ^ 
Then, as \ : ^^^ ; . ley 1.7 ; #1012.5591^1 P 
paid. Ans. 



(8) Thus, $456 

.06 



(9) Thus, 729(27 Aii8< 
4 



Ans. $27.36c. 



471329 
1 329 



.^ 



(I 



•*» 



> ■ 



. PROMISCtrOUS EXAMPLEa 

(10) ihlb9,*i36x 10=^1^0 

©6X 7= 672 
fteOX 4^1040 



169 



4iv. 4i2 



)3a72(6 months. 
2952 ' 



365.25-^12^ 



120* 
30.4375 



492)3652.6^000(7 days. 
3444 



208.5 
24 

8340 
4170 



)5004(10^ hours. 



ff . 



(11) Thu§, 8x12=96 
And, 6xl2«=72 



Then» 96x96»9216 
And, 72x72^5184 



• • • 



14400(120 Ans. 



22 



44 
44 



00 



170 



PROMISCUOUS EXAMPLES. 




6xl2»72£a8it. 



(12) Thus, as 100 

(13) rhus, 9per. 14 per. 
5 m. 8 in. 

45 : 112:: $450 
450 

5600 

448 



100 

+ 10 

20 : : 110 
.20 



1^00)22^00 
Ans. 22 c. 



45)50400($1120. Ans. 



(14; Thus, 28 



L 



71b. 
1 qr. 25 



(15) Thus, 75 
12 



-Ans 



Ans. 3125 a cwt. 3x6=18)900(60 
t 4 90 



Ans. $1.25^00 value* 



-'*^'-" 



ribk. 



/I 



PROMISCUOUS EXAMPLES. 171 

(16) Thus, 10|P)1224(1 . 

1080 

• 

144)1060(7 
So, 72\m^\i. 1008 

Ans. 72)144(2 
144 



• • 



(17) Thus, 1728(12 Ans. 
1 



1X1X300=3001728 
2xlX 30= 60 
2x2=r 4 



364 



728 



(18) Thus, I of 1= j share that B bought of A. 

Then, i of 4=^ B's and C's each. Ans. 
And, i A— i B=rlx8=8 

2x3=6 

(19) Thus, $1 2 

•06 — =J A's share. Ans. 

2x8=16 

int. .06 

prm. 1. 

— the 4th power 
Amount 1.06x1.06x1.06x1.06=1.26247696 

X5,00 • 



Ans. $631.23.848^000 



(20) Thus, 14.6x3.8x2.3=137.75 feet. 

10 



^■_ 



128)1377.50($10.76fi 



ft 

* 



172 PROMISCUOUS EXAMPLES. 

(21) ThuB, $75 - (22) Thus, 30 ft. 

25 8 in. 

^W5 6=J)240 

160 

Ans. 120 ft. 

6087.5)1875.00($.30.8+in. Ans. 

1826 25 



487500 
487000 



500 



(23) Thus, 1. first term. 
(The amount of $1) 1.06 second term. 
1.06xl.06»:1.1236 third' term. 

^ ' 3.1836 sum of the series. 

X75=annuity. 



159180 
222852 



Ans. 238.77^,00 



(24) Thus, I of i=y\, or } A's share. 
And, 4 of |=tV S's share. 
Then, as ^ : | : : $*f ». 
12xlXl00t=1200 

=^$1200. Ans. 

ixixi = 1 

(2S) Thus, 4628 

41 



189748 
^ Nbt0^-^ee Case 5j(JVIultipiication Contracted. 



'• 



• w 



■ ■"■ ■ J 



PROMISCUOUS EXAMPLES. 173 

(26) In this example we l^e the amount, rate, and 
time given to &nd the principal. 

Rule. — ^Divide the amount by The amount of 
one dollar for the given time and rate. 

Thus, $1.1 amount 1 year. 
1.1 

1.21 asdo. 2d do. 
1.21 



1.4641 sssQo 4th do. 
1.21 



$1.771561=do 6th do. 
1.1 



$1.9487171 =do. 7th do. 



Then, 1.9487171)19487.1710000($10000. Ans. 

19487.171 



0000 



(27) Thus, suppose the hound to take 120 leaps. 

Then the hare has ii\ the start =50 leaps the hare. 
And, as 3 : 4 : : 1210 : =160 do. do. 

210 leaps the hare. 
So, as 2 : 3 : : 120 : (hound)=180 of the hound*. 

Erro^— 30 diff. 



174 PROMISCUOUS EXAMPLEa 

Second, suppose tl^ hound to take 180 taftps* 

Then the hare has in the start 50 leaps. • 
And,4M jS^ : 4 : : 180 : (hare) 240 do. 

290 do. hare. 
So, as 2 : 3 : : 180 : (hound) 3=:270 do. hound. 



Error —20 diff. 



180x30=5400 
120x20»2400 



10)3000 

Ans. 300 leaps. 



Error 30— 20s«10 



(28) Thus, suppose the eldest to be 49 years. 

Then, as 7 : 5 : : 49 : 35 years youngest. 

Then« 49—30=: 1 9 eldest. And 35— 30= 5 
As 1 : 2 : : 5 : 10 youngest. 

Er.-f OdifF. 

Second, suppose the eldest to be 56 years' 

As 7 : 5 : : 56 : 40 youngest. 

And, 56— 30«26 eldest. And 40 — 30«10 

Then, as 1 : 2 : : 10 : 20 56x9=504 

— 49x6=294 
Er. 6 diff. ^ 

— 3 ) 210 



Ans. 70 eldest. 



As 7 : 6 : : 70 : 50 jroungest. 



'-"«.^ 



•^^^p^^p— »^ 



» 
•> 



fROMffiCUOUS EXAMPLBa 175 

(29) Thus,fx|X|Xf=i|*'*J»' 

■ 

(30) Thus, A $500 x4 nio.«=2000 -j 

.^ Ul0800,A. 

A 1100x8mo.=:8800j 

B $750x4 mo.»3000 

X8 mo.a* 

The sum of A's product $10800— B's $3000^= 
7800. We now have the product (7800,) and 
one factor (8,) to find the other factor. 

Thus, 7800—8 = 975, which subtract 750 = 
$225. Ans. 

(31) Thus, 22.5x8.3x4=747 ft., which -f. 24.75= 

30.18fy. Ans. 

(32) Thus, 126 gals, at 10s.=1260 126 

—16 8. or ^s. 

no 11 

110 

100/. 
12.5 



Then, as 100/. : 1268. : : 112.5 

1 11 1 

1x126x112.5=14175 

/.=12s. 10/yd Ans. 

100 X20=2000s.xllx 1=22000 

• (33) Thus, 4)4875 

—1218. 




1^00)36.56.25 cattas. 
Carried forward — 86.5625 piculs. 



176 PROMISCUOUS EXAMPLES. 

Brought forward-^Z%.^&5tb piculs. 

44 



1462500 
1462500 



1608.75^00 tales. Am. 
1.48 



1287000 
2252250 



Ans. t2380.95y^00c. 



(34) Thus, lm.=:528^0ft. 



15,^00 



7920000 
20 

158400000 
20 



24.75)3168000000.00(128000000. Ans. 
2475 



6930 
4950 

1980 
1980 



000000 




PROMISCUOUS EXAMPLES. 177 

(36) Thus, lm,=:528^0ft. 

X62y^0m. 

1056 
3x9=27 divisor. 3168 



9c. multiplier. 3273600 ft. in length. 

X50 
The two nines are — 

cancelled. 3)1636800000 one cypher annex- 

■ • ed for the 10. 

Ans. $5466000.00 



(36) Thus, 160 bu.x*1.28=:$187.50 A 's wheat. 
66bu.x625m.=f 40.625 B's barley. 

• bush. 

♦.375)146.875(391f. Ans. 



(37) Thus, 1 share to daughter. 
2 do. to mother. 
4 do. to son. 

7 shares in the estate. 

f-f=14 
6 

^=82400 

/r : I : : a V« : $6300 the whole estate. 

I of 6300=$2100 ; and | of 6300=«1800, which 
the widow received. 



178 



PROMISCUOUS EXAMPLEa 



(88) Thus» suppose 80 

Then, 4 of 80»»32 

I of 80=30 

I of 80=50 



As 192 : 80 : : 216 

80 



192)17280(00 Ans. 
1728 



(39) ^ Thus, 54x64=2916 
46x19= 874 

Ans. 2042 



(40) Thus, 80X80=6400 
And, 60x60=3600 



^"10000(100 m. Ans. 
1 




4 



""y^ 



*^ ^ PROMISCUOUS EXAMPLES. * 179 

• • Ans 

(|1) Thus," 1785793(104896(34 ' 
3«= 6561 

[figure. 

3x3x3p<3x3x3xaK8=17496)112969(4seco.quot. 

. The 8th power of 34=1785793904896 prf. 

f42)',Thu8, as 15° : 31° 27.5' z : Ih. 

90 60 



900 1887.5 
1 



900)1887.5(2h. 5' 50" Ans. 

(43) Thus, the interest of $1 for 1 year at the given 
rateaB.06)$600.00 

$10000 Ans. 



(44) Thus, as 7 : 22 : : 4.75 : 14.928+ length, or 

»V ' length. 14.928 x 1.25 x 20.25 =- 
377.865, which -4-24.75 =15JM»+ Ans. 

Or, >0^.5;^afL35xl»5=:3e4^^f||5, ^hichX 

TiT7="HViH'*=lS.267VV perch. Ans. 

(45) Thus, 999000 (46) Thus, jei25.5 or ^126 lOs. 

—999 4 20 



Ans. 998001 Ans. $502.0 2510 

12 



6;0)3012^0 
Ans. 502 



ry \ 

/80 PROMISCfbOUS EXAMPLES. f ^ 

(4T) Tbus, 76 quol. (48) Tj^us, f of 48 =32broken off. 
X21 divison +48 stump. 

Ans. 1596 dividend. . Alls. 80 sum. 



(49) Thus, il.06xl200^*l272. Ans. 

(50) Thus, 3)7600 (51) Thus, 55x^l5«^jea25 

+2.55P •; . 3<3 



-i-b>i^ 



Ans. 10000 lbs. ^ . Ans. $!9475 



(52) Thus, i of 6»2, and i of 20»:5. 

Tfaeoi as 8^ : B : : 5 : 7.9 Ans. 

(53) Thus, 198. ' (54) Thus, 35x3«=105, thrice 35 

+ 127 Thrice five, 5x3=15 
— And thirty, +30=45, thrice 5, 

Ans. 325 — & 30 

Ans. 60 diff. 

(55) Thus, Vxl=t/V=iAr. Ans. 

^ of y a=J|=| proof. Ans. 

f66) Thus, 16.5x4.5x4=297. which-=.16.^«:18. 

(57) Thus, 4.25 ft. 
** . X8 



34.00 
XlO 



1.47779)340.00000(230 bu. 2qt.+ 






■hr»<r^ . 



•\ 



PROMISCUOUS EXAMPLES.. 183 



.:>^ 



(67) Thus, i^a V X Y -^f = ^ V " ==S8888J 
Then, $8 0^^X»i* = 3yo::^$355.555 gave. 

(68) 'Thus, ^175.25 

. 120 



6087.5)21030.00($3.45.4Vt*- Ans, 

(69) Thus, assume $100 

.06 

4= J) 6.00 

T 

* ' $2 intr. for 4 mo. 

lOa prill. 

\ • As 102 : 125 : : 100 

loa 

125.000 [worth. 

102)12500($122.549^V present 

"* - $2.450/y discount 



(70) Thus, 32x32x32x.00S72=98.12896, ^r 89 

gal. Ipt. 
Or,32x32x32=32768-4-368=89+gals.Ans. 

(71) Thus, 2 
• • 3 

As 9 : 2 : : $234 cancelled. 



1:2:: 26:$52A'sgain.t(2~4^13 
As 1 : 3 : : 20 : $78B'sgain. 78-5-5=15.6 
As 1 : 4 : :26 : $104 C's gain. 104-^13^ 8 

Sum 3^ ^ 



f 



184 PROMISCUOUS EXAMPLEa 

CanceUed,.zs 36.6 : 13 : : 4392 



1 : 13 : : 120 : $1560 A's stock. 
As 1 : 15.6 : : 120 : $1872 B's do. 
As 1 : 8 : : 120 : $960 C's do. 
Or thus, 

Ratio ^ 3 

)234 

r x4=104 C's gain. 

26^ x3=78 B's gain. 
(^ X2=52 A's gain, 
mo. mo. da. $ 
As 4 : 13 : : 52 : 169 what A would gain in 13mo. 
5 : 13 : : 78 : 202.80 do. B. do. do. 

104 C's gain do. 

475.80 total gain in 13mo. 
The times being changed to an equality, and the 
gains for such time being ascertained, it is evi- 
dent that the stocks will bear the same ratio to 
each other that the gains do. We then find 
them thus; 

Total gaiin in ISmo. t A's gain in ISmo. : Total stoclu. : At ahure of ttock. 

As $475.80c. : $169 : : 4392 : 1560 
In like manner with the others : 

A Aryfi QA S 202.80 > . . . oQo . S 1^72 B's stock. 
As 475.80 : j j^4 ^ : : 4392 : J ^^ ^,^ ^^^^^ 

(72) Thus, 3.75 x 4 X 64=960, which -=-128=t7i 
• cofds received. 

So, 8 — ^7.5=55 cord lostf. 

(73) Thus, 20+20 + 17+ 17=74, length of the 

whole wall. 



\ 



PROMISCUOUS EXAMPLES. 



185 



Then, 74x1.5x8=888 cubic feet Ans. 
And, 888-^24.75=36.87|f perches. Ans. 
888-T-16.5=s 58.8l/^ perches, of 16^ per peroh. 

NoU. 20— 1.5 + 1.5=17 Ifc. two of the walls 
each. 



(74) Thus, 38 
32 



(75) Thus, $1.06 

1000 



32)600(18.75 Ans. 



$1060.00 



(7#) Thus, $1. 



1.06 

1.1236 ^ 
1.191016 

4.374616 

100 



(77) Thus, .06)240.00 
Ans. $4000 



Ans. $437.4616^00 

(78) Thus, 20ft. X 12in. =240 in. 
10ft.xl2in.=120in. 



8x4=32)28800(900 Ans. 



(79) Thus, $766 

9 

4^0)6804 

Ans. ^6170 2s. 



(80) Thus, $1060x.06=63.60 intq^st. 

Then, $1060— 63.60=$996.4Dc. Ans. 

42 



186 PROMISCUOUS EXAMPLES. 

(81) Suppose the mule's load to he 4, at first 

Then, f — 1=3, which x2=6, the ass's load, in- 
cluding 1 cwt. of the mule's ; therefore, 6—1 = 
6 the ass's lo^d at first. 
Then, 4+l=S*the mule's load, and 5 — 1=4, 
which x3=12 ass's load, which — 5=7 error- 
Second, suppose the mule's load 3, at first. 
Then 3 — 1=2, which x 2 =4, the ass's load, in- 
cluding 1 cwt. of the mule's ; therefore 4 — 1 = 
3 the ass's load at first. 
And, 3+1=4 the mule's load, with 1 cwt. of 

the ass's. 
Then, 3 — 1=2 ass's. So, 2x3=6, which — 4 
(mule's) =2 error. 

3x7=21 So, 2.^—1 = 1.0, whichx2=3.2, 

4x2= 8 which — 1=2.2, the ass's load. Ans. 

— Or thus, 2.6 

6)13 +1. 

Ans. 2.6 the mule's 1 ad. 3)3.6 

1.2 

+ 1. 



2.2 Ans 



(82) Thus, as 1 hr. 20' : 1' : : 8 m. 

60 5280 



«t 



SO 4224i 

1 



* 



8;f^4224^a: 



• 



^ tf 528 pei^ min. afainst 

— the wind. 



• » 



PROMISCUOUS EXAMPLES. 187 

Thei^ 08 32 I 1 : : 42240 

1 So, 1320 
—528 



4)42240 
8)10560 



702 diff. Ans. 



1320 ft. with the wind. 



(83) Thus, 1 acre.=4 r.^s 160 p., which X 272.25 ft. 

S343560 square feet, which divide hy the square 
of 6 (36). Thus, 43560-1-36=1210 trees. Ans. 

y. m. w. d. h. m. 

(84) Thus, from 14 0^ 

take 11 II 11 II 11 11' 



Or, take 12 I 4 11 11- 
Ans. rem. 1 11 3 2 12 49 diff. 



11 years =xll 








11 months =s 11 








11 weeks sa 2 


3 





11 days = 


1 


4 


11 hours s 




11 


11 taijgftea^ 




All 



Sum 12 1 4 11 11 



JV€>^«.— 'When once borrowing is not sufficient, we 
must continue the operation until it is, and carr 
•ne for every time we borrow. 



•<( 



f 



138 PROMISCUOUS EXAMPLES. 

(85) Thus, 63 (86) thuB, 4429 

7s. 4(1. SB 103 

• 43 

441s. 

21s. Then, 240—103=137. Ans. 

Or thus, 4429-h43rr=:103. 

56)462(8s.3d. Then, 240— 103=137. Ans. 

^7) Thus, suppose 400 A. Suppose 41 a A. 

^ Then, 400+72=472 B. Then, 410+72*=482 B. 

And, 472+112=684 0. And, 482 +112=5940. 

1456 1486 



So,1500— 1456=44er.— So,l 600—1486=14 er.^ 

410x44=18040 
400x14= 6600 



3^0)1244^0 



414| A's share. Ans 
+72 



486| B's sliareu#Aii$. 
+112 ^ \ 

598| O's share. Antf. « 



(88) Thu8,!^0x80=8400 ^'^ ' 

||^x6flr=3600 



. a • 



,10000(400 miles. Ans. 

. :." V 1 

0000 



9*» 



PROMISCUOUS EXAMPLES. 
C * 60 A 



189 



4 • 




(89) Thujas 2a. : 120a..: : Ic. : =^60c 
as 3 : 120 : : 1 : »40c. 

100 cost. 
And, as 5 : 240 : : 2 : » 96 sold for. 

Ans. 4c. loss. 



(90) Thus, 16 gal. at $1 per gal.=s$16. 
Theo^ 16 
+4 



|0 I 1 : : 1600c. : 80c. pet ^. Ani^ 



(91) Thus, 113+649+24=786 divisor. 

Then, 786 x 1 1 3 + 649 =89467 dividend. Ans. 

(92) Thus, as I : i : : f. : .^f^lS.6. Ans. 
Or, suppose, 12. Then, f of 12=8. . 
As 8 .r 9 : : 12 : 13.5. Ans. 



190 



PROMISCUOUS EXAMPLES. 

(93) Thus, i of: 40s.^ldiB. 
i of 40s. =10 
i of 40s. » 8 

I of 40s. = 6| 



19 
To 



Or, Y + V+4+\^«388. 
Then, as y : ^ — V 

^d. A's share. Ans. 
As 38 
As 38 
As 3 



38s. 



VrV=W=14s, 



38 : 40 : : 10 : 10s. 6^^^. B*fl.l 
38 : 40 : : 8 : 8s. 5yVd. C's* I 
V : *? : : V •' 7s. ^\d. D's. J 



Ans. 



(94) mus, $350 
* .04 



Asstime (100 
.04 



1 year's int. $14.00 

8 

8 jears' int. $112. 

—84.8411 

4 — ^-^ 



4.00 int. 1 year. 
X8 years. 

32 int. 8 years. ^ 
100 principal. 



Aqs.> 27.1^ dtif. $132 amount. 

Then, as $132 : $350 : : $100 : 265.15/^ 
present worth. 
^>. So, faSD-r-$265.14^taE=$84.84|f dLiscount. 

(95) Tkus, al 2a. : 125a. : : Ic. : 62^0, 
;- As ^. : 125a. : : lo. ; 41|c. 

. The price. 250^1es sold for $1.04|. 
Xhc^rice gwapples cos^lrlJQ. 



<•■ "^ * 



The amount gained on 25# w^ .04f««5y . 



PROMISCUOUS EXAMPLES. 191 

C C & A 

Then, as V : u> .• : s^o . i5_o^oo_-6oo Ans. 

men. men. days. 

(96) Thus, 6 : 3 : ; 16 (97) Thuy, 8 

3 

2:1:: 16=8 Ans. — 

24)128(6ift. Ahfe. 
120 

(98) Thus, 240fl.=2880iu., 6ft.=72in., and Ifl. 6in. 
= 18in. 
So, 2880 X 72 X 18=3732480in. dividend 
And, 8x4x2.25=72in. divisor. 
Then, 12)3732480 



6)311040 

Ans. 61840 bricks. 

(99) Thus, 30 + 30= 60fl. in length. ft. in. 

20in.=ljorfft. (100) Thus, 1 3=1.25ft. 

7 4= 7ift. 

3)300 

' 8.76 

S^0)10^0ft. 411 

Ans. 2 tons. Ans. t9ml63c. 

: (101) Thus, 100 X 100«=10000fl. (102) Thus, 65 

—lOOft. . . 42 

Ans. 9900ft. diff. 130 

260 

Ans. 2730 



193 PROMISCUOUS EXAMPLES. 

(104) Thus, 26X25X24X2 
(103) Thus, i — J=i =7893600 wor 

Then, } : { :: '^ : i8 Ans. 
{105} Thus, 24853X7911 = 1966120831 



088060415 
1572896664 
198612083 
589836249 



Aos. 259,333,081,435.5 



(107) Thus, 36x9=334ft.{18 Ans. 
(106) Thua, lacre=160ro(l8. 1 

40 

. 28)224 

5^0)640,^0 224 

Ans. 128 rods. 
(108) ThuB, 175X 100=17500 dividend. 
\ndi, 365.25-^6=60.875 divisor. 
Then, 60.875) 1750 0.0 00(82. 87 ja^ Ana. 
121750 



PROMISCUOUS EXAMPLES. 193 

(109) Thus, 78Glb. at 6c.=$4^80 

250 at 8c. = 20.00 
154 atl5c.= 23.10 



Amount $89.90c. 

601b. at 10c. =$6.00 
15gal. at 42ct= 6.30 
'i barrel fish = 3.75 
4bu. at 1.25= 5.00=21.05 bought. 

Ans. 868.85c. in cash 

(110) Thus, 2.75ft. (Ill) Thus, 15° 

3.6 4 





1375 
825 


60 
So, 12- 

• 

Thus, 34320 


1' minutes, or 1 
hour difference* 

•1 — 1 1 - A na« 




9.625 
4 


J>— •XJ>« ^XU9. 


Ans. 


38.5c. ft 
(112) 




. 


346)34254(99 
3114 


An8> 




8114 
3114 

THE END. 





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