I
RATIONAL ARITHMETIC
COMPLETE
BY
GEORGE P. LORD
n
• ■>_ > ;« J
> O 5 . , » ° I ' '
THE GREGG PUBLISHING COMPANY
NEW YORK CHICAGO BOSTON SAN FRANCISCO
LIVERPOOL
1
.copyright, 192 0, by the
'•/:.•': *:geegg publishing company
' •'.
1 •
• •» • •
A5a
PREFACE
Rational Arithmetic is intended for use in business colleges^
and in commercial high schools, by pupils who have com-
pleted the equivalent of the eighth or ninth grade in the
public school system.
While deficiencies of early training may be remedied by
its use, it is not intended as a textbook for those who are
approaching the subject for the first time. Neither is it
intended to take the place of any of the many excellent
works now in use in the grades for the purpose of develop-
ing a general understanding of mathematical principles.
Such books, while they have satisfactorily discharged this
function, have failed to develop the accuracy and facility
so vitally essential in commercial calculations.
Other commercial arithmetics have tried to overcome this
weakness by following similar plans of instruction in abridged
form. Rational Arithmetic follows a very different plan.
It is purely a vocational work and aims to teach the "how'*
rather than the "why." It is a reference book of com-
mercial operations, rather than a method of presentation,
and should be so used.
Part One is a collection of practice exercises arranged
along the lines of the generally accepted order of presentation.
Part Two contains illustrated solutions covering the entire
range of commercial arithmetic as generally understood.
The methods used are those of business. The explanations
are expressed in language which may be understood easily,
rather than in the more scholarly language usually employed.
iii
460956
iv PREFACE
References throughout the book are by paragraph num-
bers, which will allow the pupil to ascertain for himself the
best method of solving any desired problem.
The aim has been to produce a book so elastic that the
teacher may arrange a course of study to suit himself. The
author has found it advisable, however, to start pupils with
the subject of balancing accounts which arouses their in-
terest and gives them something new and practical.
Drill on decimals should immediately follow this, for the
purpose of developing accuracy in locating the decimal point.
The advisability of work on the subject of fractions de-
pends entirely upon the attainments of the individual pupil.
The writer has found that fully 75 per cent of his pupils are
greatly benefited by taking up this subject before beginning
the strictly commercial work which commences with the
subject of aliquot parts.
It is suggested that the problems in addition at the be-
ginning of Part One be used as drill problems throughout the
course.
Teachers will not find it necessary to use all the problems
provided for each subject. The aim has been to give enough
problems to meet any demand that may arise.
No claim is made for originality in any of the methods
presented. Every method that appears in this book may
be found, in some form, elsewhere. To give credit to the
sources from which the author has obtained assistance in
the compilation of this book would be to name all the text-
books consulted by him in an experience of nearly thirty
years as arithmetic teacher.
George P. Lord
Salem, Mass.
CONTENTS
PART ONE
PAGE
Preliminary Problems 1
Addition 1
Subtraction ^
Decimals 1^
Multiplication 1^
Division ....•••••• 1^
Fractions ^^
Addition of Fractions 16
Subtraction of Fractions 17
Multiplication of Fractions 17
Division of Fractions 18
Practice Problems — Fractions and Decimals . . .18
Denominate Numbers 2^
Aliquot Parts ^"^
Exercises in Billing 30
Percentage ....•••••• ^^
General Problems in Percentage 38
Profit and Loss ....••••• "^^
General Problems in Profit and Loss . . . .47
Trade Discount ^^
General Problems in Trade Discount . . . .55
Commission ^^
General Problems in Commission 66
Time 69
V
vi CONTENTS
PAGE
Interest 73
Ordinary Interest ... .o ... 73
Accurate Interest .77
To Find Time 77
To Find Rate 78
To Find Principal 79
General Problems in Interest . . . . . .81
Partial Payments . 83
Bank Discount 86
Compound Interest ....... 89
Periodic Interest ........ 90
Averaging Accounts 92
Taxes 99
Customs and Duties . . . . . . .100
Insurance 102
Life Insurance . . . . . , . .103
Exchange 105
Domestic Exchange . . . . . . .105
To Find the Value of a Sight Draft . . . .105
To Find the Value of a Time Draft . . . .105
To Find the Face of a Draft 106
Foreign Exchange . . . . . . , .107
Stocks and Bonds 109
PART TWO
Definitions « . o o 1
Notation 2
Common Processes 5
Addition — Integers ....... 5
Subtraction — Integers ....... 7
Multiplication — Integers ...... 9
Division — Integers . , . . . . .11
CONTENTS
Vll
PAGE
Decimals 13
Addition — Decimals . . . . . . .13
Subtraction — Decimals . . . . . . .13
Multiplication — Decimals . . . . . . 14
Division — Decimals . . . . , . . 1 .5
Factoring . . . . . . . . .17
Least Common Multiple . . . . . . .18
Greatest Common Divisor . . . . . .19
Fractions 22
Changing to Lower Terms ...... 24
Changing to Higher Terms . . . . . .26
Changing an Improper Fraction to a Mixed Number . 27
Changing a Mixed Number to an Improper Fraction . 28
Changing a Decimal Fraction to a Common Fraction . 29
Changing a Common Fraction to a Decimal Fraction . 30
Addition of Fractions . . . . . . .31
Subtraction of Fractions ....... 33
Multiplication of Fractions ...... 34
Division of Fractions ....... 37
Denominate Numbers 40
Reduction of Denominate Numbers .... 40
Addition of Compound Numbers ..... 43
Subtraction of Compound Numbers .... 43
Multiplication of Compound Numbers .... 44
Division of Compound Numbers ..... 44
Computing Time . . . . . . . . .45
Aliquot Parts ......... 46
Percentage 49
Profits and Losses 57
Discount .......... 63
Trade Discount ........ 63
Commission and Brokerage 70
viii CONTENTS
PAGE
Interest .... ..o ... 76
Accurate Interest . . ... . . .77
Ordinary Interest ........ 79
. Explanation ......... 79
Sixty-Day Method — Ordinary Interest Rule ... 80
Sixty-Day Method — Accurate Interest .... 83
Commercial Papers 94
Partial Payments 96
The United States Rule for Partial Payments . . 96
Merchants' Rule 98
Bank Discount 100
Compound Interest 104
Periodic or Annual Interest . . . . . .105
Averaging Accounts . . . . . . . .108
General Principles of Average . . . . . .108
Taxes 116
Duties and Customs . . . . . . .117
Insurance .......... 120
Life Insurance . . . . . . . .122
Exchange .......... 124
Domestic Exchange . . . . . . .124
Foreign Exchange .125
Stocks and Bonds 127
Tables 130
Index 147
, , 5 J 3 3
1 3' 3 3 V> 3
1 3 -, -, , ,
' •■ '^ 3' » ' *•> '3 ' '
RATIONAL ARITHMETIC
PART ONE
The following problems are intended to afford
sufficient practice to develop a thorough working
knowledge of practical business arithmetic.
An effort has been made to confine the problems as
far as possible to actual business conditions and to
present only problems similar to those met in actual
business experience.
References are by paragraphs to Part Two and are
sufficiently copious to allow ready solution by the
pupil of all problems.
PRELIMINARY PROBLEMS
ADDITION
The following exercises in addition afford opportunity
for frequent drills. The pupil should practice upon
them and similar problems provided by the teacher
throughout the course, or until he is able to add in
the time specified, or in less time.
Study carefully paragraphs 96 to 101 inclusive.
1
^'•/RATIONAL
ARITHMETIC
Practice' -tintil
you ;
are able to
add each
of
the
following
in 15 seconds or
• less :
1
2
3
4
5
6
324
196
596
287
812
285
436
289
321
422
263
635
243
781
284
389
426
149
429
423
675
674
529
728
182
317
329
263
198
463
327
262
918
721
984
824
148
425
786
416
623
296
283
348
465
129
467
179
2. Practice until you are able to add each of the
following in 25 seconds or less :
7
8
9
10
11
12
324
463
247
472
289
521
642
721
962
749
394
347
^85
567
721
638
672
625
763
289
463
236
416
262
297
143
265
429
781
729
425
264
789
642
186
453
642
721
496
187
237
642
193
168
721
346
421
287
721
459
453
721
563
472
438
672
624
254
464
296
267
284
289
689
789
563
193
596
198
746
462
189
3. Practice until you are able to add each of the
following :n 45 seconds or less :
RATIONAL ARITHMETIC
3
13
14
15
16
17
2864
3829
4962
8426
5479
4233
6471
2794
7195
6294
7185
1687
6171
3824
1781
1679
2762
2437
6271
3326
2763
4463
8263
2617
7182
4638
9174
2819
5409
4963
9162
2896
6279
6276
7126
2746
4789
2854
2830
4413
8427
6217
1962
3418
7824
1679
2834
7148
9016
1671
2634
9753
6523
2468
5607
4791
2891
3764
1695
2896
4. Practice until vou are able
following in 60 seconds or less :
to add each of the
18
264118
428307
711695
386472
369143
642785
617192
548237
167589
294462
162781
146229
382716
19
528563
742896
478132
264389
1462^27
584296
817529
428127
362419
780962
278438
261971
446236
20
428137
298461
541672
832744
167182
322907
541891
851693
395816
724594
280790
642031
451682
21
252763
376329
167251
146327
421791
573619
287513
324671
271293
348162
912872
268047
634918
22
427183
284562
711456
378275
462871
146265
551681
287354
851762
718319
440892
632757
642819
4 RATIONAL ARITHMETIC
5. Practice until you are able to add each of the
following in 75 seconds or less :
23
24
25
26
27
416342
487902
284062
614385
812716
913457
226439
713345
422716
341675
296731
914362
167182
312814
472386
284562
167943
421671
567583
611743
811706
208209
284371
642217
296342
273468
613317
176327
551638
542138
296329
672438
420416
281954
617516
284672
719243
798296
371621
271642
542983
264738
146329
560932
182133
718296
194513
817043
174837
162904
287981
382761
241671
165329
816238
273468
280642
146329
241671
271609
28
29
30
31
32
24627
93281
27682
14632
91387
48231
44638
38225
71136
26785
62783
71289
16781
28483
54321
24167
45642
91483
43729
28654
85262
71483
82675
52847
43832
71843
19721
46721
37625
71819
29636
54163
53482
54783
44623
71083
27386
27624
29654
48729
27642
16721
14729
18729
17453
29827
28294
61453
47387
37529
16429
26783
27185
26475
62745
54540
54296
16291
83267
71258
68296
78287
54385
29453
54183
RATIONAL ARITHMETIC
33
34
35
36
$6743.76
$3462.78
$4527.82
$7345.60
2846.75
5287.95
3675.18
2847.29
8421.62
6379.86
2643.89
5640.36
7329.44
7429.80
1796.97
7281.28
3780.50
5463.29
4238.54
1267.43
2894.62
7128.42
2879.36
3629.75
7481.13
3864.19
7481.29
4678.37
2563.27
7133.64
3279.81
8126.42
6247.16
4461.72
4782.63
3726.42
3729.42
9562.70
2871.54
4671.38
8427.16
4683.49
7627.18
6275.29
2945.71
7216.30
2375.62
5463.72
4453.75
3200.50
4671.16
2963.47
37
38
39
40
$3678.44
$5617,81
$2896.75
$8297.50
2917.53
2976.37
4429.36
6384.96
6279.45
3728.44
1785.29
7185.16
1468.71
1671.38
6271.38
4183.94
2783.94
5418.75
1862.45
5467.28
7162.85
2763.48
5483.92
4392.40
1671.36
6378.27
2861.57
5671.39
2768.29
5423.75
4675.38
2716.42
5419.75
1627.30
9863.75
3895.16
1683.27
4862.19
5728.50
5482.95
4462.91
7143.84
6275.81
7436.81
2789.65
3627.52
3829.62
2918.15
3601.82
5462.79
7426.85
4183.95
4618.79
2183.96
5387.09
6343.19
4844.60
4528.17
6425.70
2874.67
6 RATIONAL ARITHMETIC
6. Practice until you are able to add each of the
following in 90 seconds or less :
41
42
43
44
45
283
487
275
438
219
347
365
381
622
736
462
791
456
563
432
728
384
179
729
275
529
429
527
247
863
633
186
623
384
179
387
795
819
862
327
472
324
942
721
453
694
432
163
903
618
458
175
209
415
721
182
293
725
541
153
791
764
483
287
286
453
379
342
453
723
645
453
671
628
429
286
186
538
143
186
729
791
827
517
791
452
428
113
386
917
628
384
452
295
283
139
292
938
672
418
255
574
286
286
726
386
387
721
295
814
675
198
453
721
428
341
421
618
193
297
296
502
721
382
548
416
218
453
453
353
209
721
618
182
287
381
347
721
763
186
RATIONAL ARITHMETIC 7
7. Pupils who have acquired proper facihty in the
preceding problems will have no difficulty in per-
forming the following with sufficient rapidity :
$1283.64
$ 284.16
$ 94.43
$3728.54
785.30
1728.32
2168.75
6241.57
2721.83
4278.19
1413.80
287.63
1763.20
2674.19
287.60
729.42
487.32
905.16
782.19
98.17
9.05
728.40
3187.42
46.54
87.62
7.14
4163.27
19.72
44.33
287.64
973.29
83.71
1286.75
4239.17
4871.30
129.40
343.06
2476.28
2972.43
987.60
428.06
287.19
187.62
2763.42
71.37
458.16
98.43
1486.71
289.70
6724.13
642.19
4938.27
2729.40
8.60
3894.16
9132.38
2876.42
278.40
721.32
764.20
29.00
4291.62
193.64
381.65
427.40
71.85
4287.16
28.53
9.90
13 63
1328.72
9.19
727.62
2842.75
46.84
2.85
3478.24
4871.20
13.16
642.38
9288.75
389.45
9.68
94.73
2471.05
642.16
171.24
2652.81
287.63
71.00
283.95
453.17
274.28
9 80
90.60
287.29
297.62
2621.13
468.13
4182.19
48.39
38.79
71.24
78.50
184.19
178. '3
457.62
1246.53
8
RATIONAL ARITHMETIC
8. Complete the following statement by finding the
totals of columns and the totals from left to right :
Mar. 1
2
3
4
5
7
8
9
10
11
13
14
15
16
17
18
20
21
22
23
24
25
27
28
29
30
31
Totals
Corn
284.65
78.39
164.70
348.62
175.29
98.40
467.28
298.70
587.64
298.63
728.45
687.44
285.90
429.18
697.65
597.67
738.42
914.16
576.80
894.48
1098.27
384.62
725.30
456.82
689.63
521.16
782.73
Wheat
487.62
629.34
587.19
984.63
729.40
1180.68
960.27
450.18
829.30
486.90
752.83
647.92
473.85
738.24
287.16
587.90
699.48
826.16
487.28
745.60
952.75
397.26
678.40
1238.50
927.24
627.42
795.38
Oats
125.30
326.45
217.16
98.43
454.20
90.12
72.15
248.00
316.90
109.09
402.14
48.30
178.85
148.16
242.90
371.42
416.78
212.13
98.42
248.16
500.00
187.19
238.16
71.45
122.58
108.12
314.60
Apples
285.64
171.80
219.17
287.60
58.09
171.23
264.12
175.11
147.16
209.04
348.17
250.60
658.12
347.60
795.80
721.80
424.70
368.40
519.67
295.48
487.29
368.12
563.27
448.12
219.60
348.75
218.90
Beans
697.60
721.80
450.67
385.90
287.40
458.62
721.30
318.90
427.65
386.37
516.80
428.54
719.80
579.60
387.40
473.20
517.68
429.90
343.80
624.39
387.26
419.72
611.41
516.75
218.19
365.40
409.08
Flour
48.60
75.60
308.00
9.50
24.72
317.80
240.16
421.75
48.90
97.50
105.12
215.70
84.95
198.60
385.00
95.16
171.20
384.60
142.80
218.70
140.68
214.09
190.54
138.20
97.42
164.83
120.95
Totals
SUBTRACTION
While no practice seems necessary on simple sub-
traction of integers, the pupil should read carefully
102 to 109 inclusive, and then balance the following
RATIONAL ARITHMETIC
9
accounts according to the method explained in 108
and 109.
9. Balance the following accounts :
1. Dr. QUAKER OATS CO.
Cr.
<2A--e-^<-
$76.50
16.75
39.75
2. Dr.
D. T. AMES & CO.
Cr.
$185.25
$603.75
8.
215.50
2.10
73.94
121.50
- 24.
3. Dr.
INTEREST
Cr.
$1.
$ .68
7.70
9.21
2.52
10.
3.70
1.40
1.12
10
RATIONAL ARITHMETIC
4. Dr.
F. B. SMITH
Cr,
$ 6.75
$33.45
3.75
6.
14.50
10.25
12.25
5.40
13.25
5. Dr.
SALES
Cr.
$12.20
$ 6.60
4.65
1143.75
5.10
843.19
4.85
6.
8.
5.50
6. Dr.
PARK & STEWART
Cr.
$1021.65
$589.05
964.41
56.40
558.72
236.25
280.
661.50
RATIONAL ARITHMETIC
11
7. Dr.
JAMES CARTER
Cr.
$ 1.91
$278.60
716.
956.20
733.13
8.20
588.03
9.16
436.
833.70
47.31
8. Dr.
BLANCHARD & CO,
Cr.
$ 16.50
$ 3.60
7.64
10.80
18.50
7.75
219.21
19.60
12.50
4.23
9. Dr.
F. G. MILLER
Ci\.
$63.60
$ 4.17
25.
.60
14.50
140.
25.
207.50
21.15
55.
16.67
9.
61.50
12
10. Dr.
RATIONAL ARITHMETIC
CHARLES SMITH
CV.
$ 76.
$ 51.01
98.55
60.03
83.02
171.
59.20
80.61
105.08
2.10
43.20
210.33
13.10
29.76
Dr.
H. A. THOMAS
Cr.
$1240.50
$ 500.
876.
750.
453.35
1250.
96.73
2575.
1000.
354.56
Dr.
SAWLER & HASKINS
Cr.
$1246.34
878.14
2543.65
3746.32
$2354.56
3143.42
1245.15
1873.94
DECIMALS
MULTIPLICATION
The object of these exercises is to acquire accuracy
in locating the decimal line.
References 133, 134, 135.
10. 1. .974X.35 =
2. 8.2X9.6 =
3. 284X.75 =
4. 346.5X10.02 =
5. 27X.38 =
6. .1655X18 =
7. .4355X16.66 =
8. 844.5X7.404 =
9. 1263X8.04 =
10. .44X8.05 =
11. 65.410X.585 =
12. 75000X.0098 =
13. 5.9X26.7362 =
14. 75X6.0053 =
15. 3.926X464 =
16. 4.872X.386 =
17. 84672X8.4 =
18. 2.8973X. 80806
19. .94X100.82 =
20. 446.8X3.044 =
21. 287X9.0104 =
22. .9634X58 =
23. 283.86X.396 =
24. 94.652X4.87 =
25. 84X. 000238 =
DIVISION
Study 136 to 141 inclusive.
In solving the following problems make strict applica-
tion of the rules given in 137 and 138.
13
14
RATIONAL ARITHMETIC
Paragraphs 139, 140,
method of sokition.
11. 1. 14.875^3.5 =
2. 338.52^8.4 =
3. 1385.128-^21.6 =
4. 3.456^12 =
5. 654.5 H- 11 =
6. 2.464-7-1100 =
7. 43.4172^74.6 =
8. .01581 -^ .255 =
9. 6.305 -=-3.25 =
10. 8.63-^3.84 =
11. 79.896 H- .53264 =
12. 372.012-^58 =
13. 14.157-^2.6 =
14. 14.157-^.26 =
15. 1284.7^.3875 =
16. 246.9^23 =
17. 1640.625^1875:
18. 286.996 -^ .914 =
19. 17.408-^12.8 =
20. 6264-^.348 =
21. 14.825^-8.29 =
22. 286.327 -^ 156 =
23. 16.38^.284 =
24. 284.62^84 =
25. 29.728-^8.4 =
and 141 fully illustrate the
26. 1.75 -^ 23.5765 =
27. 437.8675^23.8 =
= 28. 23.183^19.7463 =
29. 246.-^7.875 =
30. 75-^125 =
31. 1.8675^5.75 =
32. 124.56^15.8 =
33. 48567.75^4.875 =
34. 76.50^1250 =
35. .5863^12.52 =
36. .9875 -^. 584 =
37. 23.45675 -^ 1.375 =
38. 23.-^27 =
39. 125.^56 =
40. 324. -T- 678 =
41. 12.875-^4.25 =
42. 125.6^2^7.75 =
43. 415.875^.1275 =
44. 234. 15 -r- 5.875 =
45. 56.625^-128.50 =
46. 153.8756^53.962 =
47. 346.4278^8.4695 =
48. 16.4584-^3.4565 =
49. 125.4632-7-18.4965 =
50. 4356.4589-^27.4875 =
FRACTIONS
Study carefully IGO to 180 inclusive.
References 181, 18^2, 183.
12. Reduce the following to lowest terms :
1.
7 20
12 00
6.
2.
6 30
2 8 35
7.
3.
9 1
104
8.
4.
105
2 3 1
9.
5.
2 1
A 5 5
10.
35
45
16 08
_6JL5_
12 00
LJ.5.
'7 8 2
2 3 2
11.
12.
13.
5 5 1
u
9
12
Q2
8
71
14. ^
25
15. ^
18
References 184, 185, 1 80.
13. Change the following fractions to the denomina-
tions designated :
1. ttol25ths. 6. tto88ths.
2. I to 56ths. 7. tV to 1524ths.
3. i to 72ds. 8. f to 126ths.
4. A to 195ths. 9. A to 352ds.
5. f to 84ths. 10. i to 27ths.
References 187, 188, 189.
14. Change the following to mixed numbers :
1 2J7
3.
1-6
9
2.
1 23
5
A. 124
5.
6.
15
5 1 5
16
3 2 3
1 5
7.
8.
4 27
9
] 2 5
16 RATIONAL ARITHMETIC
References 190, 191, 192.
15. Change the following to improper fractions :
1. If 4. 9t 7. 81i 10. 214f
2. 4i 5. 23f 8. 371 11. 34 A
3. 8| 6. 46| 9. 123* 12. 43|
References 193, 194, 195.
16. Change the following to common fractions or
mixed numbers :
1. .375 ' 5. .15625 9. .66f
2. .875 6. 8.25 10. .272t\
3. .0625 7. 17.875 11. 1.77J
4. .125 8. 9.28125 12. .384t'^
References 196, 197, 198, 199, 200.
17. Change the following to decimal equivalents :
1.
3
4
5.
4
9
9.
1 1
17
13.
8i
2.
4
5-
6.
5
6^
10.
9
2^3"
14.
9i3
3.
5
8
7.
7
8
11.
3f
16.
12|
4.
1 1
1 3
8.
9
1 3
12.
4f
ADDITION OF FRACTIONS
References 201, 202, 203, 204, 205.
18.
1.
3 1 5 _
4 1 8 ""
2.
7 14 1 _
8 1 5 2 —
3.
1 3 5 _
13 2 4 1 8 ~~
4.
12S+8f+27| =
RATIONAL ARITHMETIC 17
6. 21t+46M+29t =
7. 628A+56f+16i+30A =
8. 34i+26f +91x^0 + 631 =
9. 4f+27oV+33i+12iJ =
10. 7|+9i+15A+5i =
SUBTRACTION OF FRACTIONS
References 206, 207.
19. 1. 1-1= 5. 146^-791 =
2. 12f-7|= 6. 1246^^-9831 =
3. 214H-185i= 7. 25f-14f =
20.
4. ^5i-5i= 8. 3461 -217 A
MULTIPLICATION OF FRACTIONS
References 208 to 218 inclusive, and 135.
5
1.
4 v 5 —
11.
319tXl8f =
2.
12iX8 =
12.
2.15iX24 =
3.
124X161 =
13.
24X.12i =
4.
218x29f =
14.
246X.16f =
5.
289fXl26 =
15.
814X.44i =
6.
1^2"X03" =
16.
31.62iX8f =
7.
124iX27f =
17.
312x24.08i =
8.
128fVX23i =
18.
459X3121 =
9.
5122^X137 =
19.
12461X34.52 =
10.
461X43 =
20.
4.16fX.12i =
18 RATIONAL ARITHMETIC
DIVISION OF FRACTIONS
References 219 to 225 inclusive, and 139, 140, 141.
21.
1.
4-5
5 • 8 ~
11.
246f^l3f =
2.
1 2 • 8 _
1 9 • 9 ~
12.
12461 : 41 =
3.
7 • 5 _
8*6
13.
4181-151 =
4.
125^1 =
14.
2161-^19 =
5.
24 : 1 =
15.
1248^ : 27^ =
6.
346 --1 =
16.
456fV-M25 =
7.
1246^151 =
17.
5. 141 --23 =
8.
482 --171 =
18.
4246 : .171 =
9.
8461 -^ 26 =
19.
128.571 : .12^ =
10.
5321 ^ 18 1^ =
20.
43.55f^.l6f =
PRACTICE PROBLEMS INVOLVING THE USE OF FRACTIONS
AND DECIMALS
The following problems are intended to show the
application of the general principles of common frac-
tions. Their proper solution involves a knowledge of
paragraphs 160 to 225 inclusive.
22. 1. Four pieces of cloth measure respectively
311 yd., 43i^ yd., 5Q^ yd., and 44i yd. What is the
total length.^
2. What is the sum of 23.8t%, 32.35f, 56|, 194, and
i? Carry to the fourth decimal place.
3. I am about to ship a box containing 20f lb.
coffee, 3^6 lb. tea, 23^ lb. ham and 16f lb. bacon to my
camp. If the box weighs 2f lb., what is the total
weight of the shipment?
RATIONAL ARITHMETIC 19
4. A grocer bought six bags of coffee, weighing
respectively 132f lb., 128^ lb., 127f lb., 136| lb.,
134f lb., and 128| lb. Allowing H lb. for the weight
of each bag, what would be the total net weight of the
coffee ?
5. I bought 5 barrels of sugar. The net weight of
each respectively was 275t lb., 283i lb., 2711 lb.,
293f lb., and 2851 lb. Find the total net weight.
6. I bought a f interest in a bowling alley, and sold
my brother a -re interest. How^ much do I own ?
7. An automobilist on a tour completes -^e of the
trip on the first day, i on the second day, and i on the
third day. He then finds himself 350 miles from his
destination. What is the total length of the trip and
how far has he alreadv advanced ?
8. I bought a house for $7500. I paid i of the pur-
chase price in cash, f of the remainder was paid in six
months, and I am now ready to make the final pay-
ment. For what sum must mv check be WTitten .^
9. I can do a piece of work in 5 da vs. Mv brother
requires 7 days to do the same thing. If we work
together how long will it take to complete the job ?
10. If coffee loses yq of its weight in roasting, how
many pounds of green coffee must be roasted to pro-
duce 375 lb. ?
11. A farmer bought a cow for $52f and a ton of hay
for $29f . How much change would he receive out of
Si one-hundred-dollar bill ?
12. A bookkeeper's pay envelope contains three
$10's, one $5, one $2, and one $1. He paid for board
20 RATIONAL ARITHMETIC
$1H, a bill amounting to $8f , and bought a hat for $3, a
pair of gloves for $lf , and two pairs of stockings at $.50
a pair. What part of his week's pay did he have left ?
13. What will \l-r2 dozen eggs cost at $.58f per
dozen ?
14. If 71 tons of hay cost $182^, what will llf tons
cost ?
15. I bought 4375f bushels of corn at $.80f a bushel,
and 2350^ bushels of oats at $.61f a bushel. What
was the entire investment.'^
16. Find the total cost of the following: 350 lb.
Rio coffee at $.47| ; 450 lb. Mocha coffee at $.41f ;
900 lb. white sugar at $.10f ; 900 lb. brown sugar at
$.09f; 970 lb. granulated sugar at $.08f; 172 lb.
butter at $.56|.
17. A merchant sold 80 lb. of butter at $.57f ; 43
dozen of eggs at $.61f per dozen ; 32^ gallons of milk
at $.60 a gallon. What was the total amount of sales ?
18. A piece of cloth containing 47f yd. was sold for
$9. 94 J. What was the price per yard ?
19. One- third of a firm's capital is invested in
merchandise, three-eighths in real estate, and the rest,
$18,200, is cash. What is the capital of the firm.^
How much is invested in merchandise, and how much
in real estate ?
20. A farm yields 96.08 bushels of potatoes to the
acre, 36.625 bushels of oats per acre, 15.52 bushels of
wheat per acre. 156 acres were planted in potatoes,
214 acres in oats, and 19.3 acres in wheat. What is the
total number of bushels harvested ?
Rx\TIONAL ARITHMETIC 21
21. I have withdrawn J of my money from the bank
and have $376.40 remaining. How much did I with-
draw?
22. A partnership consists of three members who in-
vest respectively i, i, i of the capital, and agree to share
losses and gains in the same proportion. How much will
be each partner's share, if there is a profit of $13,416.75 ?
23. A business man finds himself unable to meet his
entire obligations. He owes $12,360 and has $10,300
available with which to pay. What part of his lia-
bilities can he meet? How many cents on the dollar
is this ?
24. A invested i of the capital of a firm, B i, C i,
and D the remainder. D's share is $1690. What
was A's, B's, and C's investment?
25. A house and lot cost $6600. The house costs
i more than the land. What was the cost of each?
26. How manv bushels is .75 of 640 bushels?
27. A merchant sold 162 barrels of flour which is f
of his stock of flour. How much flour had he at first ?
28. A merchant sold 480 barrels of flour which is .625
of his entire stock. Hov/ many barrels had he at first ?
29. A man, at his death, left $30,000 to his wife, son,
and daughter ; .5 of this sum went to his wife, .375
to his daughter, and .125 to his son. How much did
each receive ?
30. I have just learned that one of my customers has
failed and is able to pay only $.525 on the dollar. My
claim against him amounted to $134.40. How much
will I receive?
DENOMINATE NUMBERS
Reference ^29.
23. 1. Reduce £34, 8^, 7d to pence.
2. Reduce 4 T., 5 cwt., 85 lb. to pounds.
3. Reduce 14 gal., 3 qt., 1 pt. to pints.
4. Reduce 1 cwt., 24 lb., 3 oz. to ounces.
5. Reduce 1 da., 3 hr., 25 min. to seconds.
6. Reduce 14 yr., 5 mo., 3 wk. to days.
7. Reduce 1 m. 25 rd., 4 yd., 2^ ft. to inches.
8. Reduce 2 hhd., 14 gal., 3 qt. to pints.
9. Reduce 14 bu., 3 pk. to pints.
10. Reduce 3 A., 2 sq. rd., 10 sq. yd. to sq. ft.
Reference 230.
24. 1. Reduce 3462 sq, in. to higher denominations.
2. Reduce 14>6Sd to higher denominations.
3. Reduce 17696 lb. to higher denominations.
4. Reduce 32625 gr. to higher denominations.
5. Reduce 12760 in. to higher denominations.
6. Reduce 18428 sq. in. to higher denominations.
7. Reduce 3896 cu. ft. to higher denominations.
8. Reduce 4843c? to higher denominations.
9. Reduce 120615 sec. to higher denominations.
10. Reduce 633 pt. to higher denominations.
22
RATIONAL ARITHMETIC 23
Reference 231.
25. 1. Reduce .327 m. to lower denominations.
2. Reduce .35 hr. to lower denominations.
3. Reduce .875 yd. to lower denominations.
4. Reduce .135 yr. to lower denominations.
5. Reduce f mo. to lower denominations.
6. Reduce .125 m. to lower denominations.
7. Reduce £2.3456 to lower denominations.
8. Reduce £12.456 to lower denominations.
9. Reduce f m. to lower denominations.
10. Reduce tt yr. to lower denominations.
Reference 232.
26. 1. Reduce 4 yd., 2 ft. to a decimal of a rod.
2. Reduce 3 gal., 2 qt., 1 pt. to gallons.
3. Reduce 2 pk., 3 qt., 1 pt. to a decimal of a
bushel.
4. Reduce 35 min., 18 sec. to a decimal of a day.
5. Reduce 18 rd., 4 yd., 2 ft. to rods.
6. Reduce 4 cwt., 85 lb. to a decimal of a ton.
7. Reduce Ss, lOd, 2/ to a decimal of a pound.
8. Reduce 18 sq. rd., 4 sq. yd. to a decimal of
an acre.
9. Reduce 14 hr., 35 min., 10 sec. to a decimal
of a day.
10. Reduce 185 lb., 12 oz. to a decimal of a ton.
ALIQUOT PARTS
Study paragraphs 240 and 241, memorizing the table
and noting appHcation as explained in note a.
Reference 241.
27. Find the cost of :
1. 720 1b. at 50^; at 33J^ ; at 25^.
2. 120 lb. at 33i^; at 25^; at 20^; at 12i^.
3. 360 lb. at 6U \ at 6f ^ ; at 10^ ; at 12i^.
4. 840 yd. at 10^; at 12^^; at 14f ^ ; at
25^.
5. 4800 lb. at 8^^ ; at 6i^; at 12^^; at IGf^;
at 10^.
6. 240 yd. at 8i^ ; at 6f ^ ; at 10^ ; at 12^^.
7. 2480 yd. at 25^; at 50^; at 33i^ ; at 20^.
8. 480 yd. at 6i^ ; at 8^^ ; at 6f ^ ; at 10^ ;
at 12i^.
9. 560 yd. at 8J^ ; at 6i^ ; at 6f ^ ; at 10^;
at 12i^.
10. 204 yd. at 50^; at 33^^; at 25^; at
11. 4200 yd. at 10^; at 12^^; at 14f ^ ; at
16f^; at 25^.
24
RATIONAL ARITHMETIC 25
12. 1800 lb. at nW. at 16f^; at 20^; at 25^;
at 33i^.
13. 1500 yd. at $1 ; at 12J^ ; at 14f ^ ; at 16f^;
at 25^.
14. 490 doz. at 12i^ ; atlOf^; at 20^; at 6f ^.
15. 960 yd. at 8i^ ; at 6i^; at 10^; at UW,
at 6f ^.
Reference 241.
28. Find the total cost of :
1. 38 lb. at 25^ 2. 63 yd. at 28^^
84 lb. at 37ijzi 81 yd. at 33i^
72 lb. at 75^ 18f gr. at 52^
48 lb. at 41|^ 28 doz. at 50^
96 lb. at 33i^ 58 yd. at 14f ^
24 lb. at 12i^ 235 yd. at 40^
3. 61 lb. at 48^ 4. 25 bu. at 96<^
480 lb. at 16f ^ 20 bu. at 88jZ^
25 lb. at 44^ 31i bu. at $2
161 lb. at 48^ 50 bu. at $11.50
240 lb. at 18f ^ 12i bu. at $3.60
72 lb. at 37i^ 25 bu. at $1.64
5. 25 yd. at 76^^ 6. 37i bu. at 72^
37^ yd. at 96^ 75 bu. at $3.20
750 yd. at 12^^ 62^ bu. at $1.36
168 doz. at 12i^ 14f bu. at $1.54
420 yd. at 33i^ 50 bu. at $5.85
176 yd. at 31i^ 12i bu. at $.64
26
RATIONAL ARITHMETIC
Reference 241.
29. Find the cost of :
1. 6i A. land at $192.
2. 125 lb. tea at 48^.
3. 34 lb. tea at 50^.
4. 25 lb. coffee at 44^.
5. 25 T. coal at $10.80.
6. 72 pieces lace at $1.25.
7. 44 yd. velvet at $2.50.
8. 2i bu. potatoes at $1.48.
9. 12i bu. turnips at 74^.
10. 12i yd. silk at $1.04.
11. 84 tables at $12.50.
12. 36 sets chairs at $125,
13. 12i yd. linen at 56^.
14. 25 pieces lace at $6.60.
15. 62i T. coal at $9.50.
16. 375 T. coal at $11.50.
17. 264 A. land at $37.50.
18. 320 bu. potatoes at $2.12^.
19. 810 T. coal at $12.50.
20. 1250 bbl. pork at $24.
21. 1280 lb. rice at 12^^.
22. 366 yd. silk at $1.66f.
23. Hi yd. duck at 36^.
24. 474 gal. cider at 33i^.
25. 1680 qt. vinegar at 16f ^.
26. 648 lb. sugar at 10^.
27. 208 yd. tape at 2i^.
28. 176 lb. tea at 50^.
29. 1742 yd. silk at $1.50.
30. 560 gal. oil at 12i^.
Reference 241.
30. Find the total cost of the following :
RATIONAL ARITHMETIC 27
1. 180 \h. at SSU 2. 138 lb. at 33i^
760 lb. at 25^ 7^28 lb. at 62^^
54 lb. at 37i^ 224 lb. at 25^
144 lb. at 33i^ 960 lb. at 66f ^
72 lb. at 37i^ 72 lb. at 12^^
150 lb. at 66f f!^ 904 lb. at 87^^
3. 196 yd. at 16f ^ 4. 147 yd. at 55|^
180 yd. at 66|^ 24 yd. at 66f ^
288 yd. at 33i^ 28 yd. at 75^
459 yd. at IH^ 84 yd. at 25^
72 yd. at 25^ 56 yd. at 12^^
48 yd. at 16S^ 48 yd. at 75^
183 yd. at 33^^ 246 yd. at 25^
5. 66 gal. at 33i^ 6. 441 gal. at 55U
64 gal. at 87i^ 3248 gal. at 6U
63 gal. at llU H^ gal- at 22f ^
144 gal. at 83^^ 266 gal. at 28^^
91 gal. at 71f ^ 384 gal. at 62^^
96 gal. at 62^^ 368 gal. at 31i6
16 gal. at 87i^ 248 gal. at 87^^
945 gal. at 55U ^^^ gal. at 83ij^
7. 750 yd. at 33^^ 8. 648 yd. at QU
427 yd. at 42f ^ 684 yd. at 33i^
87i yd. at 50^ 496 yd. at 75^
52 yd. at 62^^ 186 yd. at 83i^
450 yd. at 6f ^ 125 yd. at 18^
2112 yd. at SU 144 yd. at 37i^
240 yd. at SU 297 yd. at 444^
174 yd. at 16f ^ 287 yd. at UU
249 yd. at 25^ 918 yd. at 33i^
28 RATIONAL ARITHMETIC
9. 144 lb. at 83i^ 10. 144 lb. at 16f ^
480 lb. at ^lU 216 lb. at m^
282 lb. at 83i^ 872 lb. at 12i^
312 lb. at 33i^ 348 lb. at 25^
427 lb. at 71f ^ 72 lb. at SU
184 lb. at 12i^ 186 lb. at 87^^
940 lb. at IH 138 lb. at 334^
462 lb. at 66f ^ 96 lb. at QU
342 lb. at 16f ^ 384 lb. at 50^
11. 84 yd. at 91f^ 12. 84 lb. at 58^^
288*^ yd. at 12^^ 960 lb. at 16f ^
345 yd. at 66|^ 728 lb. at 62i^
192 yd. at 37^^ 36 lb. at 43f ^
423 yd. at 33i^ 64 lb. at 41f ^
280 yd. at 12i^ 96 lb. at 12^^
324 yd. at 411^ 72 lb. at 41f ^
284 yd. at 25^ 348 lb. at 75^
396 yd. at 33^^ 246 lb. at 33i^
64 yd. at 5Q\i 344 lb. at 37i^
13. 87i yd. at $2.48 14. 176 yd. at $1.12i
192 yd. at 87^^ 75 yd. at 16^
28 yd. at 75^ 27 yd. at 75^
Hi yd. at 18^ 5Q yd. at 83^^
144 yd. at lli^ 17 yd. at 25^
25 yd. at 44^ 12^ yd. at 39^
75 yd. at 24^ 72 yd. at 41|^
87i yd. at $2.88 344 yd. at 37^^
270 yd, at \\U ^4 yd. at 8^^
24 vd. at 75^ 87i yd. at 88^
RATIONAL ARITHMETIC 29
15. 11511 lb. at 20^ 16. 156 lb. at 66f^
960 lb. at 16f ^ 284 lb. at 25^
728 lb. at 62i^ 396 lb. at SSU
32 lb. at 43f ^ 64 lb. at 5^^
64 lb. at ^U 384 lb. at SlU
96 lb. at 12i^ 84 lb. at 58^^
72 lb. at 41f ^- 960 lb. at 16f ^
348 lb. at 75^ 728 lb. at 62^^
246 lb. at 33i^ 96 lb. at 43|^
344 lb. at 37i^ 98 lb. at 37^^
17. 96 yd. at SU 18. 594 lb. at 66f^
96 yd. at 12^^ 963 lb. at 83^^
348 yd. at 75^ 312 lb. at 37i^
72 yd. at 41f ^ 251 lb. at 50^
246 yd. at 33^^ 603 lb. at 11^^
344 yd. at 37^^ 552 lb. at 66f ^
156 yd. at 66f ^ 133 lb. at 14f ^
132 yd. at 91f ^ 528 lb. at 61^
84 yd. at 50^ 273 lb. at 83^^
328 yd. at 25^ 368 lb. at 31i^
•
19. 146 gal. at 37^^ 20. 200 yd. at 37i^
245 gal. at 42f ^ 384 yd. at 18f ^
672 gal. at 16f ^ 288 yd. at 83i^
18 gal. at 87i^ 294 yd. at mU
162 gal. at 16f ^ 918 yd. at 44^^
332 gal. at 18 J^ 459 yd. at 11^^
369 gal. at 33i^ 18 yd. at 12i^
828 gal. at 311^ 111 yd. at 33i^
693 gal. at 16f ^ 164 yd. at 62i^
918 gal. at 44|c^ 8 yd. at 87^^
30
RATIONAL ARITHMETIC
EXERCISES IN BILLING
Reference 241.
31. 1. Copy and extend the following bill :
Salem, Mass., July 20, 1919
Mr. J. A. Brown,
3 Leach St.,
City.
To S. S. Pierce & Co., Dr.
Terms Cash
8 bu. beans
@ $3.75
108 lb. butter
@ .50
84 lb. cheese
@ .33i
129 doz. eggs
@ .50
150 lb. lard
@ .42f
25 bu. potatoes
@ 2.16
72 lb. rice
@ .161
12 lb. Japan tea
@ .5Q\
360 lb. granulated
sugar @ .10
128 lb. coffee
@ .43f
2. Feb. 1, A. W. Smith & Co., Boston, Mass., sold to
Jones & French, Marblehead, Mass., on 30 days' credit :
36 boxes oranges at $3.66| ; 12 chests T. H. tea, 840 lb.,
at 50e; 14 chests Japan tea, 980 lb., at 37^^; 12 bbl.
St. Louis flour at $9.10 ; 4 bags coffee, 576 lb., at 33^^ ;
54 boxes lemons at $4.60 ; 14 bbl. pineapples at $7.50 ;
44 bunches bananas at $5.45. Write the bill.
3. R. C. Adams, Danvers, Mass., bought of Davis &
Bicknell, Salem, Mass., on account 60 days: 25 bbl.
RATIONAL ARITHMETIC 31
St. Louis flour at $11.48 ; 35 boxes apricots, 25 lb. each,
at 23^ ; 20 boxes apples, 25 lb. each, at 114^ ; 10 boxes
peaches, 25 lb. each, at 374|Z^ ; 6 boxes raisins, 25 lb.
each, at 25^; 8 boxes prunes, 25 lb. each, at 16f^;
13 boxes currants, 25 lb. each, at 16^ ; 15 cases Quaker
Oats at $3.75 ; 12 cases canned corn, 24 doz., at $2.12| ;
18 cases canned tomatoes, 36 doz., at $1.87^; 15 chests
Japan tea, 70 lb. each, at 54^; 1600 lb. gunpowder
tea at 43i^. Write the bill.
4. Jan. 31, K. R. Good & Co. sold to Lewis W. Sears,
Middleton, Mass., on 60 days' time : 6 pairs men's kid
gloves at $2.48 ; 5 doz. napkins at $5.50 ; 4 doz.
children's hose at $2.75 ; 9 pr. blankets at $6.50 ;
2 pieces jeans, 80 yd., at 25f^; 2 pieces point, 80 yd.,
at 12^^ ; 5 doz. towels at $3.50 ; 15 doz. spools thread
at 87i^ ; 7 doz. ladies' collars at $2.12^; 7 doz. ladies'
cuffs at $3.25 ; 14 robes at $2.33i ; 4 pieces Irish
linen, 156 yd., at 66f ^. Write the bill.
5. March 1, James K. Broderick & Co., Boston,
Mass., sold to Henry T. Lewis, Peabody, Mass., on
30 days' time: 2 pieces gingham, 63 yd., at 37^^;
1 piece blue denim, 31 yd., at 28^; 1 piece brown
denim, 30 yd., at 28^^ ; 1 piece duck, 31 yd., at 50<z^ ;
1 piece shirting, 35 yd., at 22f ^ ; 1 piece sheeting,
29 yd., at 43|^ ; 2 pieces cottonnade, 76 yd., at 33^^;
1 piece jeans, 34 yd., at 30^ ; 1 piece Irish linen, 40 yd.,
at 62^^; 2 pieces jaconet, 84 yd., at 25^; 5 doz.
children's hose at $2.75 ; 8 pr. ladies' kid gloves at
$2.50 ; 4 doz. spools thread at 62^^^. W^ite the bill.
PERCENTAGE
Study 242 to 254 inclusive.
Reference 244.
32. Express the following, both in decimal and
common fractional forms :
1.
H%.
9.
38%.
17.
137i%.
24.
3 9C7
4 /O*
2.
2t%-
10.
431%.
18.
1431%.
25.
15C/
16 /O*
3.
3i%.
11.
681%.
19.
156i%.
26.
.2%.
4.
31%.
12.
92i%.
20.
162i%.
27.
.4%.
5.
7i%.
13.
931%.
21.
1%.
28.
.5%.
6.
7f%.
14.
112^%.
22.
fV%.
29.
.12%
7.
3H%.
15.
1181%.
23.
25/0'
30.
.25%
8.
O/C-^ /q'
16.
13H%.
References 255, 256, 257, 258.
33. Find:
1. 23% of 460 acres.
2. 34% of 60 cords.
3. 15% of $75.
4. 55% of 280 feet.
5. 45% of 360.
6. 62i% of $.50.
7. 48% of 175 gallons.
8. 77% of 430 bushels.
9. 85% of 1270 pounds.
10. 1661% of 480 tons.
11. 225% of 699 bushels.
12. 625% of $4560.
13. .2% of $1750.
14. .4% of $1825.
32
RATIONAL ARITHMETIC 33
15. .6% of $156.25. 21. 6|% of lUh
16. i% of $86,424. 22. 7^% of 87^.
17. 3.5% of $1250. 23. |% of $1260.
18. 4.4% of $875. 24. 36|% of $1214.
19. 7i% of $1560. 25. |% of 33i gallons.
20. 6i% of 8|.
26. A man having $1960 spent 23%o of it. How
much did he have left ?
27. A gentleman having $^5,5Q5 invested 18%
of it in city lots, 22% in railroad stock, 30% of it in
bank stock, and the rest in a truck farm. How much
did he invest in each kind of property ?
28. 13% of a grocer's bill of $1665 was for coffee at
45^ per pound. How many pounds did the grocer buy ?
29. I bought 200 little pigs at $7.50 each ; 25% of
them died. At what price per head must I sell those
that are left in order to incur no loss ?
30. A farmer who raised 1880 bushels of corn sold
37i% of it at 87^^ a bushel. How much did he receive
for what was sold ?
31. An agent collected $2430 for his client whom he
charged a 5% fee for his services. How much did he
receive and how much did he remit to his client ?
32. C. E. Bates has failed owing me $1645. He is
able to pay only 43% of his debts. At that rate, how
much should I receive ?
33. I bought 27 bales of cloth, 12 pieces to the bale
averaging 47 yards to the piece, at $1.25 per yard.
For what must I sell the entire quantity to gain 16f % ?
34 RATIONAL ARITHMETIC
34. A private bank that has failed declared a
dividend of 87i%. A's balance on deposit was
$6,437.50, B's $3,856.56, C's $872. How much did
each lose ?
35. A man, at his death, left an estate valued at
$150,000. He left 10% of it to organized charity, 12^%
of it to his college, and 4i% to his church. He divided
the remainder among his family so that his wife received
62^% of it and his son and daughter each 18f %. What
did each receive ?
References ^259, 260, 261.
34. 1. 18 is 9% of what.^
2. 38 is 5% of what ?
3. 54 is 12i% of what ?
4. 84 is 15% of vv^hat ?
5. $460 is 23% of what .?
6. $1143 is 35% of what. ^
7. $650 is 42%o of what ?
8. $9420 is 96% of what ?
9. $3150 is 3.5% of what?
10. $48.60 is 7.5% of what. ^
11. $148.20 is 16|% of what.?
12. $1375.50 is 33i% of what.?
13. $1198 is 55.5% of what?
14. $6570 is 234% of what ?
15. $1254 is 675% of what?
16. Of my flock of pigeons 16|% died. If 215 are
left, how many pigeons were there in the original flock ?
RATIONAL ARITHMETIC 35
17. I have just paid a bill of $135.45, which repre-
sented 18f% of my available cash. How much did I
have before paying the bill ? How much is left ?
18. A broker received $^6.25 as his fee for selling a
certain piece of property on a commission of 2^%.
What was the value of the property ?
19. An analvsis shows that a merchant's costs and
fixed charges amounted to 87^% of his gross sales for a
certain year. If his total expenses and costs amounted
to $246,400, what were his sales ?
20. A merchant sold 15% of his stock of goods for
$45,350. What was his entire stock worth before he
sold any ?
21. A farmer bought 87 acres of land which is 37^%
of what he previously owned. How much did he own
after the purchase ?
22. A sold a carriage at a profit of 16f%, thereby
gaining $43.25. What did it cost and what did it
sell for?
23. If it takes 60 days to complete 16f% of a con-
tract, how long will it take to finish the job ?
References 262, 263.
35. 1. What per cent of 260 is 13 ?
2. What per cent of 480 is 72 ?
3. What per cent of $2.40 is $32 ?
4. What per cent of $188.50 is $22.62.^
5. What per cent of $640 is $131.60 ?
6. What per cent of 28 bu. is 7 bu. ?
36 RATIONAL ARITHMETIC
7. What per cent is $314.50 of $1850 ?
8. What per cent of $.95 is $.70.^
9. What per cent of $2664 is $826.04 ?
10. What per cent of 1^ is
3.5
8 •
11. What per cent of 77| is 66f ?
12. What per cent of $456.75 is $219.24?
13. W^hat per cent is $1414.80 of $5240?
14. What per cent of 324 is 64.8 ?
15. What per cent of $1940 is $9.70?
16. A man owing a debt of $1680, paid $940.80.
What per cent remains unpaid ?
17. In a school of 480 pupils, 24 were absent on a
certain day. What was the percentage of absence?
18. A merchant purchased goods for $425 and sold
them for $510. How many dollars did he gain? This
was what per cent of the cost? What per cent of the
selling price ?
19. A lawyer charged $14.19 for collecting a claim
of $473. What rate per cent did he charge?
20. I sell a house that cost me $2500 for $2125.
The loss is what per cent of the cost ?
21. Of an army of 45,000 men 5625 were killed in
battle and 10,125 were wounded. What was the per-
centage of loss ?
22. An insurance company with a capital stock of
$250,000 declared an annual dividend of $21,250.
The dividend was what percentage of the capital?
23. A miller keeps one quart out of every bushel
he grinds. What is the percentage of his toll ?
RATIONAL ARITHMETIC 37
24. A merchant invested $34,395.30 in business.
At the end of the year he finds he ha: gained $3821.70.
This gain is what per cent of the money invested ?
25. If I sell f of a quantity of goods for what f of
them cost, what is the gain per cent ?
26. John Brown, failing in business, owes $3650.
His entire resources are $2920. What per cent of his
indebtedness can he pay ?
27. The assets of an insolvent concern are $23,450 ;
its liabilities are $33,500. What per cent can it pay
and what will A receive on a claim of $1350 ?
28. I paid $300 for apples bought at $5.40 a barrel.
I sold 38 barrels for $220.40. What percentage did I
gain on the quantity sold ?
29. The enrollment in a certain High School is 846.
102 are enrolled in the Classical Course, 228, the
General Course, 246, the Manual Training Course,
and 270, the Commercial Course. What percentage
of the school could each course claim .^^ Carry the
result to the fourth decimal place if necessary.
30. At the close of the baseball season of 1915, the
first four teams in the American League stood as
given below. Figure the percentage of each, carrying
to the fourth decimal place.
WON LOST
Boston . . . .
101
50
Detroit . . . .
100
54
Chicago ....
93
61
Washington . . .
85
68
38 RATIONAL ARITHMETIC
GENERAL PROBLEMS IN PERCENTAGE
The following problems cover the entire range of
simple percentage. A thorough understanding of the
matter covered by paragraphs '24'2 to 263 inclusive
will enable the student to solve them with facility
and accuracv.
These problems are not graded so as to present
increasing difhculties. They are rather arranged so
an average lesson of ten problems will offer the varying
conditions of ease and difficult}^ that are usually found
in actual experience.
Every problem should be solved in the easiest
possible way.
36. 1. After experiencing a loss of $5000, a business
man has $30,000 left. His loss was what per cent of
his original capital.^
2. The assets of a bankrupt were $27,179.38. His
liabilities were $43,487. What per cent could he pay ?
How much would be due A whose claim is $3540.75 ?
3. One of our creditors who met with financial
reverses agreed to pay our claim in annual install-
ments of 16f%. He has made four payments. How
much does he still owe, the original claim being
$1847.38?
4. A owns i interest in a business, B f, and C J.
A sells 33i% of his share for $1874. What is the entire
value of the business? What is B's share? What is
C's share?
RATIONAL ARITHMETIC 39
5. A and B are partners. A's investment is
$36,783.60 and B's is $26,636.40. To what percentage
of the profits is each entitled ?
6. Out of an inheritance of $17,500, I invested 45%
in real estate, 25% in United States bonds, and put
the rest into a mortgage. What sum does the mortgage
represent ?
7. A owTis i of the stock of a corporation, B 25%,
C i, D 37i%, and E the remainder. B's investment
is $3500. What is E's investment ?
8. A and B engage in business as partners. A
invests $6980 and B $5584. Each partner's share
represents what per cent of the total investment?
9. A regiment went into battle with 938 men and
came out with 804. What percentage was lost.^^
10. Aly holdings in real estate are worth $8500 ; my
personal property is worth $4350. If, during the
coming year, my real estate increases in value 23%
and my personal property 9%, what will be the total
value then and what will be the percentage of increase
of both ?
11. An operator bought a large tract of land and
sold 40% of it to one customer, 20% of the remainder
to a second customer, 25% of what still remained to a
third customer. If 234 acres remain unsold, how many
acres were there in the original tract of land .^
12. In 1918 a merchant's sales amounted to $234,520.
In 1919 they amounted to $213,413.20. If they
increase during 1920 at the same rate that they de-
creased in 1919, what will be the sales for 1920^
40 RATIONAL ARITHMETIC
13. On January 17, 1916, merchandise was bought
for $1475.85 on three months' credit, subject to a dis-
count of 3% if paid within ten days. What sum was
required to settle the bill on January 27, 1916.^
14. Two railroads, one 300 miles long and the other
500 miles long, carry 340 barrels of potatoes at a
through rate of 23^ a barrel. This freight is to be
divided in proportion to each railroad's per cent of the
total mileage. How much does each road receive?
15. An agent sold 160 bales of cotton, averaging
240 lb. each, at 32i^ a pound; 75 hhd. of tobacco,
averaging 360 lb. each, at 26f ^ a pound ; 130 bbl.
sugar, averaging 180 lb. each, at 10^^ a pound. He
charges ^i% of the amount received. What was his
charge ?
16. A speculator bought a farm of 175 acres of land
for $11,900 and sold 64 acres for $5440. What per
cent did he gain on the part sold ?
17. By energetic effort, the sales department of a
certain business was able to increase the sales 20% each
year for three successive years. The total increase
amounted to $36,400. What were the sales the year
before the first increase was effected ?
18. A 9% dividend on stock amounted to $873.
What was the face value of the stock ?
19. A second-hand dealer sold two automobiles for
$640 each. On one he gained 20% and on the other he
lost 20%. Did he gain or lose by the transaction?
How much?
RATIONAL ARITHMETIC 41
20. A speculator invested $5340. He gained 10%
the first year, 13% the second year, lost 18% the third
year, and gained 5% the fourth year. What is his
capital at the end of the fourth year ?
21. Several years ago 176,834,300 pounds of fish
passed through the Boston market. Of this quantity,
Gloucester furnished 110,637,829 pounds. At a more
recent date the total amounted to 215,643,330 pounds,
of which Gloucester furnished 120,418,563 pounds.
What per cent was furnished by Gloucester in each
year?
22. An automobile manufacturer decided to reduce
the price of his cars 10%, and called upon his sales
department to increase the sales a sufficient amount to
counteract the reduction in price. What per cent
increase would the general manager of the sales depart-
ment be obliged to show ?
23. A certain piece of property having depreciated
$2355, is now worth $3925. What was its original
value ? What per cent has it depreciated ?
24. A leather manufacturer owning 75% of a factory
building sold 16f% of his share and received $1555.
Find the value of the factory.
25. Hoyt's stock of goods is worth $9462, which is
15% more than Taylor's, and 15% less than Ashton's.
What is the value of the stock carried by each ?
PROFIT AND LOSS
Study 265 to 273 inclusive.
References 274, 275.
37. What is the profit or loss and seUing price of
goods costing :
1. $43.42 and sold at a profit of 9% ?
2, $87.54 and sold at a profit of 12^% ?
I. $175.89 and sold at a profit of 27% ?
4. $21.43 and sold at a loss of 8i%?
5. $312.51 and sold at a loss of 13f%?
6. $15.07 and sold at a profit of 371% ?
7. $102.73 and sold at a profit of SSi% ?
8. $240.81 and sold at a loss of 15%.^
9. $181.03 and sold at a profit of 28|%)?
10. $37.56 and sold at a profit of 20%?
11. Property valued at $3750 increased 8^% in
value. It was then sold. What was the profit?
What was the selling price ?
12. If I buy cloth at $5.40 and sell it at 161%, loss,
what is my selling price ?
13. Bought an automobile for $1145. It was then
sold at a loss of 16|%. What was the loss and what was
the selling price ?
14. Paid $240 for a pair of horses and sold them
at a profit of 31^%. How much did I gain?
42
RATIONAL ARITHMETIC 48
15. Bought hats for $36 a dozen and sold them at
a profit of 33^%. What was the seUing price of each
hat?
16. My balance sheet shows that advertising, rent,
clerk hire, etc., commonly called overhead charges ,
amount to about 8^% of the total amount of my pur-
chases. To leave a proper margin of safety I have
decided to figure these overhead charges as 10% of
the cost. In order to provide for this, how much
must we mark goods costing $13.40 to clear a profit
of 15% ?
17. Allowing an overhead of 10%, what must the
following goods be marked to show a profit of 10%?
of 15% ? of 20% ?
Refrigerators costing $ 24.50
Chamber sets costing 125.80
Persian rugs costing 163.49
Pianos costing 560.
Buffets costing 61.18
Dining-room sets costing 242.30
Tea- wagons costing 21.73
Couches costing 75.50
Parlor sets costing 407.92
Veranda sets costing 195.60
References 276, 277, 278.
38. Find the cost :
1. Loss $151.20, rate of loss 10%.
2. Loss $107.91, rate of loss 2^%.
3. Loss $205.78, rate of loss li%.
44
RATIONAL ARITHMETIC
4. Loss $456.38, rate of loss 2^%.
5. Loss $220.15, rate of loss 6i%.
6. Loss $117.50, rate of loss 8^%.
7. Gain $34.23, rate of gain 21%.
8. Gain $79.85, rate of gain 25%.
9. Gain $12.73, rate of gain 33i%.
10. Gain $19.05, rate of gain 22|%.
11. Gain $22, rate of gain 16^%.
12. Gain $17.98, rate of gain 20%.
13. Selling price $1056.80, rate of gain 5%.
14. Selling price $2435.28, rate of gain 20%.
15. Selling price $3672.25, rate of gain 16|%.
16. Selling price $1434.75, rate of gain 12^%.
17. Selling price $1806.75, rate of gain 33^%.
18. Selling price $2584.82, rate of gain 2%.
19. Selling price $950.28, rate of loss 5%.
20. Selling price $42.00, rate of loss 50%.
21. Selling price $245.75, rate of loss 16f%.
22. Selling price $5042.80, rate of loss 22|%.
23. Selling price $550.25, rate of loss 6|%.
24. Selling price $64.80, rate of loss 1|%.
25. By selling goods for $140 I lose 12i%. At what
price must I sell them to gain 12^% ?
26. Sold a row boat that cost $80 at a gain of 12^%,
and with the proceeds I purchased another boat which
I sold at a loss of 10%. How much did I gain by both
transactions ?
27. A cow was sold for $67.50 which was 10% below
cost. What was the cost and what was the loss ?
RATIONAL ARITHMETIC 45
28. I bought a safe for $118.80. This was 10%
higher than the manufacturer's price. What was the
retailer's profits ?
29. A merchant marked his goods at 37^% above
cost. What is the cost of an article that he marked
at $156.64.^
30. I sold goods to a customer at a price which would
have netted me a profit of *'25% had the customer paid
his bill. He failed, however, and was able to pay me
only S5(ji on the dollar. In spite of this, I netted a
profit of $314 on the transaction. What was the
amount of my bill ?
31. A manufacturer gained 25%; the wholesaler
made a profit of 20% ; the retailer made a profit of
33^%. What was the actual manufacturing cost of
an article the retail price of w^hich was $100 .^^
32. Suppose, in the above question, that the manu-
facturer should sell direct to the customer, thereby
increasing the cost of the article 10%. At what price
could he afford to sell at retail and still make the same
per cent profit that he now enjoys .^^
References 279, 280.
39. Find the per cent of gain or loss :
1. Cost $105.50, gain $21.10.
2. Cost $165, gain $49.54.
3. Cost $40.75, gain $7.34.
4. Cost $140.75, gain $14.08.
5. Cost $200, gain $75.50.
46 RATIONAL ARITHMETIC
6. Cost $103.40, loss $17.23.
7. Cost $755.90, loss $60.47.
8. Cost $64.80, loss $19.44.
9. Cost $21.70, loss $1.74.
10. Cost $75, loss $23.25.
^11. Cost $220, gain $125.40.
12. Cost $115.20, gain $63.36.
13. Cost $1256.40, gain $527.68.
14. Cost $750.48, gain $202.63.
15. Cost $90, gain $14.85.
16. Cost $15.24, gain $8.08.
17. Cost $420.50, gain $50.47.
18. Cost $50.17, gain $9.03.
19. Cost $430.85, gain $12.93.
20. Cost $59.84, loss $17.95.
21. Cost $2412.50, loss $627.25.
22. Cost $650.75, loss $221.26.
23. Cost $108, loss $54.
24. Cost $542, loss $338.75.
25. A milliner bought hats at $27 a dozen and sold
them for $3 each. What was the gain per cent ?
26. What is gained by buj^ing paper at $2 a ream
and retailing it for 1 ^ a sheet ?
27. I bought a horse for $350 and sold him for $400.
What was the gain per cent.'^
28. If a horse cost $400 and was sold for $350,
what per cent was the loss .^
RATIONAL ARITHMETIC 47
29. A stationer bought 2 bundles of paper for $1.75
a ream and sold it at retail at the rate of 3 sheets for 2^.
What per cent did he gain and how much did he gain
in aW?
30. A dealer bought 150 crates of fruit for $1 a
crate. He sold 35 crates at $1.25 a crate, 30 crates
at $1.15 a crate, 60 crates at $1.20 a crate, 10 crates
at cost, and threw the rest away as worthless. What
did he gain or lose and what per cent ?
31. What per cent profit will be realized from the
sale of peaches at 3^ each if they cost $1.25 a hundred
and 10% of them are lost by decay ?
32. I sold C. S. Chase goods that cost $430 so as to
make a profit of 60%, on 30 days' credit. Before the
account was due, Chase failed, paying only 37^^ on
the dollar. What was my per cent of loss ?
33. A merchant bought gloves at $8 per dozen
pairs and sold them at $1.25 a pair. What was the
per cent gained ?
GENERAL PROBLEMS IN PROFIT AND LOSS
References 274 to "280 inclusive.
40. 1. By selling goods at a profit of 37^% I made
$215.45. What do the goods cost and what do they
sell for ?
2. Goods were sold at a profit of 20%. If the seller
received $36.54, what was his profit.^
3. I bought tea for 53^ a pound. What price must
I sell it for in order to gain 32% ?
48 RATIONAL ARITHMETIC
4. What is the loss on goods sold for $4348.50,
which is 19% below cost?
5. Goods are sold for $436.28 at a loss of 15%.
For what price should they be sold to gain 15% ?
6. A dealer sold hats at retail for $3.50 each, and
at wholesale for $33 a dozen. At retail, his profit was
40%. Does the wholesale price show a profit or loss.^^
How much per hat ? What per cent "^
7. A barrel of flour sold for $9.45 nets a profit of
35%. At what price could we sell it if we were content
with a profit of 20% .?
8. A coal dealer buys coal for $9.75 by the long ton
and sells it for $11.75 by the short ton. What per cent
profit does he make ?
9. A sold a factory building to B for $8621. By
so doing he lost 10%. B expended $2300 installing
a sprinkler system and then sold the factory for 20%
more than A paid for it. How much did B gain and
what per cent did he gain on his investment ?
10. How much should be asked a pound for fish
costing $6.50 a hundred to net a profit of 10% and
allow for 10% waste?
11. A book agent sells two books for $5 each. On
one he loses 20%, and on the other he gains 20%. Does
he gain or lose, how much and what per cent ?
12. Goods bought at $4 a gross and sold at 40^ a
dozen yield what per cent profit ?
13. I sold goods to a retailer at a profit of 40%.
Before settlement he failed, paying 25^ on the dollar.
What was my loss on goods costing $320 ?
RATIONAL ARITHMETIC 49
14. Bought a horse from A at 20% less than it cost
him. Sold it for 25% more than I paid for it. I
gained $15 in the transaction. What did the horse
cost A ; what did it cost me ; how much did I sell it for ?
15. I buy oranges at $2.50 a hundred. What price
must I mark them a dozen to gain 25%, allowing 10%
for decay and 15% overhead expenses ?
16. A dealer buys 6 bags Rio coffee, 218 lb. in a
bag, at 4 If ^ a pound ; 12 bags Java coffee, 75 lb. in a
bag, at 25^ a pound. After mixing the two kinds, he
sells at a profit of 75%. What is the price a pound ?
17. What price can I afford to pay for property
that rents for $50 a month in order to make a net
profit of 5% a year, allowing $200 annually for neces-
sary repairs, taxes, etc. .^
18. Find the gain or loss per cent of each of the
following, carrying to the fourth decimal place :
A. Purchases $13,502.10
Sales 12,786.50
Inventory at closing . . . 4,983.70
B. Inventory at beginning . . 3,908.00
Purchases 10,680.20
Sales 10,450.50
Inventory at closing . . . 6,400.75
C. Inventory at beginning . . 3,100.85
Purchases , 8,900.50
Sales 11,500.00
Returned to us 540.30
Returned by us 560.50
Inventory at closing . . . 4,550.00
I
TRADE DISCOUNT
Study 284 to 290 inclusive.
References 291, 292.
41. Find the net amount of the following bills :
1. List price $340.25, discount 35%.
2. List price $1256.35, discount 27%.
3. List price $438.40, discount 23%.
4. List price $750.50, discount 33^%^.
5. List price $755.75, discount 37^%.
6. List price $351.20, discount 20%^.
7. List price $1050.30, discount 16%.
8. List price $978.80, discount 42%.
9. List price $127.70, discount 21%.
10. List price $2040.50, discount 2%.
11. List price $1434.25, discounts 20% and 10%o.
12. List price $760.20, discounts 12i % and 10%.
13. List price $126.34, discounts 20%o and 25%.
14. List price $285.40, discounts 27i% and 20%.
15. List price $1244.18, discounts 25% and 15%.
16. List price $556.30, discounts 27%) and 12%).
17. List price $112.80, discounts 17% and 10%.
18. List price $680.12, discounts Q^% and 5%.^
19. List price $120.40, discounts 22% and 20%).
50
RATIONAL ARITHMETIC 51
20. List price $450.75, discounts 16% and 12%.
21. List price $845.12, discounts 20%, 10%, and 5%.
22. List price $360.70, discounts W%, 25%, and 10%.
23. List price $850.20, discounts 25%, 25%, and 10%.
24. List price $351.65, discounts 10%, 10%, and 5%.
25. List price $1201.30, discounts 18%, 7%, and 3%.
26. List price $700.50, discounts 16f%, ni%, and
6i%.
27. List price $325.40, discounts 12%, 10%, and 5%.
28. List price $970.60, discounts 15%, 12%, and 4%.
29. List price $1010.25, discounts 15%, 10%, and 8%.
30. List price $242.50, discounts 25%, 25%o, and 5%.
31. List price $1068.20, discounts 50%, 50%, and 10%.
32. List price $978.35, discounts 25%, 40%, and 35%.
33. List price $326.30, discounts 50%^, 30%, and 25%.
34. List price $829.20, discounts 40%, 50%, and 10%.
35. List price $2048.80, discounts 60%, 25%o, and
15%.
36. List price $1218.75, discounts 62^%, 37^%^, and
10%.
37. List price $650.50, discounts 66|%, 40%, and
H%.
38. List price $450.20, discounts 50%o, 30%, and 20%).
39. List price $360.20, discounts 65%o, 15%o, and 10%).
40. List price $520.80, discounts 30%, 50%, and 20%.
52 RATIONAL ARITHMETIC
Reference 293.
42. What single discount is equal to a discount
series of :
1. 20%, 10%, and 5%? 11. 25%, 15%, and Q%?
2. 30%, 10%, and 5%? 12. 50%, 25%, and 10%.?^
3. 10%, 5%, and 2% ? 13. 35%,37i%, and 12i% ?
4. 12%, 5%, and 2% ? 14. 40%, 25%, and 10% ?
5. 20%, 8%, and 5% .? 15. 16f%, 5%, and 2%.?
6. 15%, 10%, and 6%? 16. 15%, 12%, and 4%.^
7. 10%, 10%, and 5%? 17. 30%, 10%, and 8%?
8. 20%, 12i%, and 10% ? 18. 10%, 5%, and 10% .?
9. 33i%, 10%, and 10%? 19. 10%, 20%, and 20%.^
10. 50%, 20%, and 10% ? 20. 25%, 25%, and 10% ?
21. 50%, 50%o, and 20%?
22. 35%, 25%, 20%, and 10%o?
23. 50%, 20%,, 20%, and 10%^ ?
24. 60%, 40%, and 20% ?
References 294, 295, 296.
43. Find the gross amount of the following bills :
1. Discount $125.30, rate of discount 25%.
2. Discount $104.15, rate of discount 20%.
3. Discount $275.50, rate of discount 12^%.
4. Discount $340.75, rate of discount 5%.
5. Discount $210.42, rate of discount ^IWc
6. Net $134.80, rate of discount 10%).
7. Net $84.60, rate of discount 60%.
8. Net $234.50, rate of discount 15%.
RATIONAL ARITHMETIC 53
9. Net $119, rate of discount 12^%.
10. Net $218.^6, rate of discount 30%.
11. Discount $125, rate of discount 35%.
12. Discount $215.50, rate of discount 25%,
13. Discount $36.75, rate of discount 12^%,
14. Discount $105.15, rate of discount 15%,
15. Discount $341.70, rate of discount 35%.
16. Net $85.50, rate of discount 25%, 10%o.
17. Net $128.40, rate of discount 40%, 25%, 10%.
18. Net $275.75, rate of discount 25%, 10%, 10%.
19. Net $187.26, rate of discount 50%, 10%, 5%.
20. Net $150.50, rate of discount 10%o, 5%, 2%.
Reference 297.
44. What rate of discount shall we allow on a bill of :
1. $346.40 to net $259.80?
2. $780 to net $530.40 ?
3. $112.50 to net $93.75?
4. $218.40 to net $136.50?
5. $187.26 to net $112.37?
6. $360 to net $288 ?
7. $450 to net $393.75?
8. $234.50 to net $195.42?
9. $312.90 to net $208.60?
10. $520 to net $374.40?
11. $760 to net $585.20?
12. $365.75 to net $219.45?
13. $1600 to net $1400?
54 RATIONAL ARITHMETIC
14. $2340 to net $1521 ?
15. $298 to net $172.84.^
16. $169.75 to net $105.25.^
17. $2359 to net $2064.13.^
18. $250 to net $182.50?
19. $3296 to net $2307.20 .^
20. $265.50 to net $199.13?
Reference 298.
45. At what price must the following goods be
marked in order to allow the prescribed discount and
still make the designated profit?
Cost
Required Profit
Discount
1.
$100
37i%
20%
2.
$250
24%
35%
3.
$304.75
12%
20% and 10%
4.
$480.20
35%
'371%
5.
$1050.50
40%
25%, 10%, and 5%
6.
$785.40
33i%
23%
7.
$570
18%
32%
8.
$921.20
27%
50%
9.
$893.68
12i%
25% and 10%
10.
$723.85
18%
30%, 10%, and 2%
11.
$5200
22%
21%
12.
$225.20
50%
10%
13.
$720.30
27%
5% and 2%
14.
$850
25%
40%
15.
$365.50
30%
17%
RATIONAL ARITHMETIC 55
Cost
Required Pkopit
Discount
16.
$180
40%
10%, 10%, and 59
17.
$127.50
21%
32%
18.
$150.25
15%
12i%
19.
$900
10%
5%, 5%, and 10%
20.
$675.80
20%
23%
GENERAL PROBLEMS IN TRADE DISCOUNT
References 291 to 298 inclusive.
46. 1. I bought a bill of $546.30 at a trade discount
of 20%, 10%, and 5% with an additional cash discount
of 2%. I took advantage of the cash rate and then sold
the goods at the original list price with a flat discount
of 25%. How much did I gain ? What per cent ?
2. If I buy goods at a discount of 30% from the list
price and sell at the list price, what per cent profit do I
make ?
3. Goods bought at $6 a gross at a discount of 20%,
and sold at 75^ a dozen yield what per cent profit .^^
4. Which is better and how much on a bill of $240,
a discount of 20%, 10%, and 5%, or a discount of 35%?
5. By selling goods at $1.50 a yard I make 25%
profit. What must I mark them in order to deduct
10% and still make the same profit .^^
6. Goods costing $345 are marked up 30% and are
then sold at a discount of 20%. How much is gained
and what is the gain per cent ?
7. Goods are marked up 20%. What discount can
the seller allow on this price and still net the cost.^
56 RATIONAL ARITHMETIC
Make out bills for the following :
8. Use current date, your own locality. Adams
and Baxter bought of the Boston Hardware Company,
terms net 30 days, 2% 10 days : 2 dozen chisels at $3.20 ;
5t2 dozen 10-inch drawing knives at $5.50 ; 1x^2 dozen
ratchet screw drivers at $8.75, subject to a discount
of 10%, 10%, and 5%; 12 steel shovels at 87i^ ; 9
spades at 70^ ; 3 dozen garden rakes at $5 ; 4 dozen
trowels at $1.35, subject to a discount of 20%, 10%,
and 10%; 4 dozen cans prepared paint at $3.15; 480
feet of f-inch garden hose at 10^ ; 5 lawn mowers at
$4.50, subject to a discount of 16f%, 5%, and 5%;
8 dozen boxes of 4-inch bolts at $2.35. Make out bill
showing the total due at the expiration of credit and
the amount due on the cash terms.
9. Use current date, your own locality. Naumkeag
Company bought of Childs Provision Company, terms
2% for cash and net 10 days : 150 barrels Baldwin
apples at $5.50 ; 25 barrels greenings at $4.50 ; 45
bushels of beans at $4.75 ; 90 bushels of potatoes at
$1.95, subject to a discount of 10% and 10%; 4 firkins
of butter, 65 pounds each, at 50^^ ; 5 firkins creamery
butter, 50 pounds each, at 48^ ; 6 boxes American
cheese, 50 pounds each, at 30^ ; 4 boxes Young America
cheese, 33 pounds each, at 36^, subject to a discount of
10%. 5%, and 2% ; 4 sacks of Rio coffee, 140 pounds
each, at 48^ ; 2 bags Java coffee, 215 pounds each, at
46^ ; 2 barrels rice, 320 pounds each, at 16^ ; 1 barrel
New Orleans molasses, 45 gallons, at 35^; 1 barrel
Porto Rico molasses, 45 gallons, at 33^^, subject to a
RATIONAL ARITHMETIC 57
j
discount of 10%, 10%, and 5%. Make out bill showing
the total due at the expiration of credit and the amount
due on the cash terms.
10. Use current date, your own locality ; seller,
L. A. White, Wholesale Company ; purchaser, Parker
Clothing Company ; terms, net 60 days, 2% 10 days :
45 girls' rain capes at $4 ; 75 girls' sweaters at $3.75 ;
125 children's rompers at $1.50 ; 75 boys' rubber coats
at $4.25, subject to a discount of 15%, 10%, and 2% ;
35 voile waists at $2.25 ; 40 organdie waists at $3.50 ;
70 pairs women's gloves at $1.60 ; 50 pairs of women's
silk gloves at $1.25; 14 dozen women's handkerchiefs
at $2 ; 12 dozen men's handkerchiefs at $2.25 ; 25
men's bath robes at $5; 75 men's shirts at $1.50;
120 white skirts at $3.24; 60 dress skirts at $5.75;
75 serge dresses at $14.50; 50 ladies' belts at 75^,
subject to a discount of 10%, 10%, and 10%. Make out
bill showing the total due at the expiration of credit
and the amount due on the cash terms.
11. Use current date ; your own locality ; you are
the seller ; your teacher the purchaser ; terms, net 30
days, 2% off for cash : 3 #124 phonographs at $85 ;
2 #062 phonographs at $125 ; 4 #68a phonographs at
$75 ; 1 #300 phonograph at $250 ; 2 doz. Operatic
Records at $2 each ; 2 doz. Standard Song Records
at $1.25 each; 2 doz. Band and Orchestra Records
at $1.50 each; 1 doz. Comic Monologue Records at
S5i each ; 1 doz. Records for Dancing at S5^ each.
COMMISSION
Study 299 to 310 inclusive.
References 311, 312.
47. Find the commission and net proceeds :
Sales
Commission
Charges
1.
$4342.50
2i%
$214.30
2.
$356.40
3%
$85
3.
$1256
5%
$13.25
4.
$8784.50
34%
$43.50
5.
$788.40
4%
$38.56
Find the commission and gross cost :
Purchases Commission Charges
6.
$435.25
3%
$48.56
7.
$1840.50
2i%
$128.50
8.
$843.35
3%
$83.87
9.
$1258.40
5%
$324.50
10.
$5643.50
8%
$125.40
References 311, 312.
48. 1. An agent sold a farm for $3690 at 2i% com-
mission. What was his commission ?
2. I sold 128 barrels of sugar, each weighing 350
pounds, at 9f ^ a pound. What was my commission at
n% ?
58
RATIONAL ARITHMETIC
59
3. A commission merchant sold 250 barrels of sugar,
each weighing 350 pounds, at 9^^ a pound and 158
barrels of molasses, each containing 48 gallons, at 77^^
a gallon. Find his commission at 2%.
4. A real estate agent sold a house for $4450 at 1^%
commission. What sum did he send the owner?
5. Rule a sheet of paper and copy the following
account sales, making the necessary extensions :
ACCOUNT SALES
Albany, N. Y., August 14, 1919
Sold for the Account of
C. F. Adams, Troy, N. Y.
By W. H. Smith, Commission Merchant.
1919
June
July
June
July
13
28
10
15
10
8
12
100 bbl. G. M. flour
150 bbl. G. M. flour
200 bbl. G. M. flour
100 bbl. G. M. flour
Charges
$10.50
10.75
10.25
10.50
Freight $125
Storage 15.50
Guaranty 1%
Cartage
$27.00
Insurance 5.20
Commission 4%
Net Proceeds
6. Prepare an account sales under date of January 10,
for 5000 bushels of wheat, sold by J. J. Campbell & Co.,
Springfield, Mass., for the account of Walter Bros.,
North Adams, Mass., Sales, No. 25 ; 500 bushels at
$2.08, December 30 ; the remainder at $2. Charges :
freight, $91 ; cartage, $15 ; storage, $15.50 ; insurance,
i% ; guaranty, 1% ; commission, 2i%.
60 RATIONAL ARITHMETIC
7. Prepare an account sales under date of October 15
for the account of J. C. Brown & Co., sold by Thomas
Moody & Co., both of Chicago, 111., October 3 ; 50
barrels of W. R. flour at $9.25 ; 100 barrels of K. A.
flour at $9.80. Charges : freight, $50 ; cartage, $14.50,
both under date of Oct. 1 ; Oct. 11, storage, 150
barrels at 4^ a barrel; insurance, i%; commission,
8. Put the following in the form of an account sales :
William C. Jones, St. Louis, Mo., sold for account of
Charles W. Franklin, Chicago, 111., the following goods
August 4, 7 pieces of summer silk, 284 yards, at $1.85
August 13, 5 pieces black silk, 216 yards, at $1.50
August 17, 16 pieces calico, 798 yards, at 19^; Sep-
tember 3, 19 pieces alpaca, 548 yards, at 38^; Sep-
tember 10, 25 pieces diagonals, 587 yards, at 75^.
Charges : August 1, freight and cartage, $65.48 ; in-
surance, i% ; commission, 4%. Find the net pro-
ceeds.
9. Arrange the following in the form of an account
sales : E. P. Clark & Co., Peekskill, N. Y., sold for
account of John Mason, the following : November 1,
300 bushels potatoes at $1.95; November 16, 200
bushels at $1.85; December 1, 240 bushels at $1.90.
Charges : November 1, freight, $85.45 ; cartage, 2^^
per bushel ; storage at 2^ per bushel ; commission, 4^%.
Find the net proceeds.
10. Rule a sheet of paper and copy the follow-
ing account purchase, making the necessary exten-
sions.
RATIONAL ARITHMETIC
61
ACCOUNT PURCHASE
Utica, N. Y., May 10, 1919
Purchased by F. J. Bowen & Co.
For the Account of E. L. Green, Rome, N. Y.
1919
April
May
24
29
7
9
3 half-ch. J. tea, 165 lb.
4 half-ch. O. tea, 240 lb.
5 half-ch. J. tea, 350 lb.
8 mats Rio coffee, 600 lb.
Charges
Cartage
Commission 5%
$7.50
Amount charged to your accoimt
11. In accordance with the foregoing form prepare
an account purchase of tea purchased by W. L. Thomas,
Feb. 21, for the account of Jones, White & Co., both of
Boston, Mass. ; 10 half-chests of J. tea, 600 lb., at ¥l(ji ;
5 half-chests O. tea, 250 lb., at 45^; 5 cases C. tea,
250 lb., at 50^ ; 8 half-chests E. B. tea, 480 lb., at 49^2^.
Charges : cartage, $8.80 ; commission, 4%.
12. Prepare an account purchase for the following :
David Carey, New York City, bought of the account
of Henry Grant & Co. of Newark, N. J., August 16,
1916, 68 yd. ^ancy prints at 25^; 42 yd. colored silk
at $1.25; 1 dozen ladies' felt hats, $30; 18 yd. black
cassimere at $1.50; 3 suits boys' clothing at $10.
Charges : packing and cartage, $2.40. Find entire
cost, commission being 5%.
62 RATIONAL ARITHMETIC
13. Prepare an account purchase for the following:
March 18, 1916. A. B. Morse & Co., Trenton, N. J.,
bought for account of Harris & Price, of Philadelphia,
Pa., March 18, 150 bbl. Dakota flour at $9.75;
80 bbl. buckwheat flour at $12.40 ; 480 bu. ground
feed at 60^; 500 bu. bran at 30^; 20 bbl. G. M.
flour at $11. Cartage, $7.90, commission, 4%. What
is the entire cost ?
14. A merchant in Boston shipped to his broker in
New York a carload of potatoes, 967 bushels, which
were sold at $2.25 a bushel. What was realized on the
sale if the broker charged 4^% for selling and the
freight was $67.96? How many pounds of Java
coffee could be purchased with the proceeds of the
sale of potatoes if coffee is 45^ a pound and the broker
charges 2% for buying ?
References 313, 314, 315.
49. 1. A commission merchant working on a 4%
commission earned $240.50 for selling a consignment
of flour. What did the flour sell for ? •
2. My commission for selling goods at 2% amounted
to $250.50. What was the selling price of the goods ?
3. I bought goods, receiving $75.30 as my 5% com-
mission. What did I pay for the goods and what was
the total cost to my principal ?
4. A commission merchant whose charge is 1^%
finds that his total receipts for commissions for 3
months amount to $4856. What was the value of
his sales for the same time.^
RATIONAL ARITHMETIC 63
5. A collector charges 5% for his services. In order
that he may clear $3000 a year, what must his collec-
tions amount to ?
6. I receive $736.96 as the net proceeds of the sale
of goods through a commission merchant, the only
charges being 2% for commission. What was the gross
sales ?
7. My lawyer sends me $77.80 as the proceeds of a
claim which he has collected for me. What was the
claim, his commission being 5% ?
8. We have received a check for $344.75 as the net
proceeds of a sale on which the commission was 1^%.
What was the total sales and what was the commission ?
9. I sent my commission merchant $138.65 to pay
for goods purchased by him, including commission of
3i%. What did he pa}^ for the goods ?
10. I received $185.40 from J. C. Bryan to cover
the amount of goods purchased and my commission
of 3%. What was the amount of the purchase?
11. I received $564.20 with instructions to purchase
certain supplies, my commission to be li%. What
amount will I invest in supplies and what will my com-
mission be ?
12. A commission merchant received $1606 to in-
vest, after deducting a commission of |%. How much
can he invest ?
13. We received $145.50 as the net proceeds of a
consignment. The rate of commission was 2% and
other charges $2.50 ; what was the selling price of the
goods ?
64 RATIONAL ARITHMETIC
14. A commission merchant remitted $704.29 after
deducting his commission of lf% and other charges
amounting to $3.20. What was the selhng price of
the goods and what was the commission?
15. My commission merchant has just sent me an
account purchase showing gross cost to be $668.60.
The commission w^as figured at 4% ; the other charges
amounted to $8.20. What was the prime cost of the
purchase ?
16. I paid a real estate broker $225 for selling a
house and lot. This sum included his commission of
5% and other expenses amounting to $50. What sum
did the house sell for ?
17. A commission merchant received $12,500 to
invest in cotton after deducting his commission of 5%.
What sum does he invest .^^ How many bales of 400
pounds can be bought at 35^ a pound?
18. I shipped my agent in New York 950 tons of hay
which he sold for me at $22.50 a ton. Charges were : for
freight, $950; cartage, $327.50; storage, $.85 a ton;
his commission, 2^%. I instructed him to invest the pro-
ceeds in wheat for me. If he charged me at the same
rate for investing that he did for selling the hay, how
much did he invest and what was his entire commission ?
19. I bought a lot of apples for $6.50 a barrel on
3% commission. If my commission amounted to
$81.51, how many barrels did I buy?
20. I received $893.03 with instructions to invest it
in apples after deducting a commission of 5%. How^
many barrels can I buy at $5.50 a barrel?
RATIONAL ARITHMETIC 65
Reference 316.
50. What is the rate of commission :
1. If the prime cost of merchandise is $480 and the
commission for buying is $4.20 ?
2. If the net proceeds is $944.40 and the commission
is $15.60?
3. If the first cost is $3264 and the commission for
buying is $4.08 ?
4. If the gross proceeds is $3200 and the commission
for selhng is $128?
5. If a commission merchant receives $37.02 for
selHng $1234 worth of goods ?
6. If a commission merchant receives $85.75 for
selling $3430 worth of goods ?
7. A real estate dealer bought a house for a client,
paying $8750 for it. His charges were $262.50. What
was his rate of commission ?
8. A commission merchant bought 346 barrels of
apples at $5.75, receiving $65.74 for his commission ;
other charges amounted to $43.13. What is the gross
purchase and what is his rate of commission ?
9. A lawyer earned $37.40 for collecting a claim on
5% commission. What was the claim ?
10. A merchant sent his principal $325.23 as the
net proceeds of a consignment which he sold for $343.43.
What was the rate of commission ?
66 RATIONAL ARITHMETIC
GENERAL PROBLEMS IN COMMISSION
References 311 to 316 inclusive.
51. 1. My agent sells $1400 worth of goods for me
at 3% commission. What amount must he remit .^^
2. An agent sells goods for $2468. His charges are :
commission, 2^% ; storage, insurance, etc., $125.
What are the net proceeds?
3. I sold for a Chicago firm as follows : 950 bu.
corn at 85^ a bushel; 120 bbl. of pork at $15.50 per
barrel. My commission on the corn was 1^^ a bushel
and the commission on the pork was 2^% . Charges
were : freight, $134 ; storage, $67 ; advertising, $63.
What were the net proceeds due my employer ?
4. I purchased for a South American firm goods
valued at $3750. My commission was to be 2%. For
what sum must I draw.^^
5. I have just received $4168 as the net proceeds
of a consignment. The figures in the Account Sales
are blurred and I am unable to read either the amount
of the sales or the rate of commission, which is $145.84.
Ascertain both.
6. I paid a real estate dealer $215 for selling a house
and lot on 5% commission. Advertising and other
expenses amounted to $50. What amount does the
sale net me ?
7. I have purchased for a Southern firm 45,620 feet
of pine lumber at $16.25 a thousand ; 34,257 feet of
hemlock boards at $15 a thousand ; 37,250 feet
of spruce at $18.60 a thousand. My commission is
RATIONAL ARITHMETIC 67
Si% and my other charges amounted to $214. For
what sum shall I draw on the firm to cover the cost of
my purchase and charges ?
8. A commission merchant received $648.40 to in-
vest in wool after deducting all his expenses. How
much did he pay for the wool if his commission for
buying was 2% and his other charges amounted to
$13.50?
9. A commission merchant received a consignment
of Q'i5 barrels of flour on which he paid $84 for freight ;
$12.75 for cooperage ; $22.50 for storage ; $19.50 for
cartage. He sold 110 barrels at $9.25 a barrel; 175
barrels at $10.75; 123 barrels at $11.25, and the re-
mainder at $10. His commission for selling was 2%.
What was the net proceeds ?
10. The gross cost of goods purchased through an
agent was $1221. If the commission was $6 and the
other charges $15, what was the rate of commission.'^
11. I have received $325 to invest in apples after
deducting all expenses. How many barrels could I
buy at $5.75 a barrel ; charges being 3% for commis-
sion ; 2^^ a barrel for dray age ; 12^ a barrel for freight ;
$5 for advertising ? What was the unexpended balance,
if any ?
12. I sent $1547 to my agent in Chicago with in-
structions to buv wheat after deducting his commission
of 3%. How much did he invest in wheat?
13. A commission merchant received from a specu-
lator $2091 to invest in corn after deducting his com-
mission of 2^%. He was instructed to hold the corn
68 RATIONAL ARITHMETIC
subject to the purchaser's order. After an advance
in value he was ordered to sell, and did so, obtaining
$1.50 a bushel. After deducting his commission of
2|% and $20 for storage, he paid the speculator $2220
as the balance due him. What did the commission
agent pay a bushel for the corn ?
14. A commission merchant's regular charges were
3% for selling and 2% for guaranteeing the purchase
price. If he remitted $6842 to his principal as the net
proceeds of the sale, what did the goods sell for.^
15. A commission merchant sold 625 barrels of
potatoes at $11.25 a barrel and invested the proceeds
in wheat at 85 j^ a bushel, first deducting a commission
of 2% for buying and 2% for selling. How many
bushels of wheat could he buy and what was the un-
expended balance, if any .^
Find the missing quantities in the following :
Sale Commission Rate of Com. Net Proceeds
16. 1246.50 5%
17. 234.40 2i%
18. 5450. 272.50
19. 47.49 1535.51
20. 6254.50 6004.32
21. 4% 3317.76
22. 8564.50
23. 1327. 79.62
24. 33.87 6%
25. 856.40 834.99
TIME
Study carefully 317 to 325 inclusive.
Reference 238.
52. By compound subtraction find the time from :
1. Jan. 3, 1910 to March 5, 1916.
2. Dec. 28, 1912 to June 11, 1914.
3. May 13, 1909 to Dec. 2, 1911.
4. June 28, 1910 to May 12, 1911.
5. April 3, 1912 to Oct. 14, 1913.
6. Feb. 28, 1912 to Jan. 1, 1914.
7. July 4, 1914 to April 14, 1916.
8. May 13, 1905 to Dec. 9, 1915.
9. March 9, 1908 to Feb. 6, 1912.
10. June 11, 1909 to April 3, 1916.
11. Dec. 6, 1914 to Aug. 7, 1917.
12. Nov. 28, 1913 to June 28, 1917.
13. Sept. 8, 1907 to Aug. 21, 1915.
14. May 30, 1905 to July 20, 1906.
15. Oct. 9, 1909 to June 11, 1916.
16. June 17, 1913 to May 16, 1916.
17. Feb. 6, 1908 to May 9, 1917.
18. July 8, 1911 to Aug. 2, 1912.
19. April 4, 1910 to March 17, 1913.
69
70 RATIONAL ARITHMETIC
20. July 4, 1908 to May 22, 1916.
21. Dec. 13, 1909 to Nov. 12, 1917.
22. Feb. 3, 1907 to May 11, 1915.
23. June 18, 1911 to Aug. 14, 1917.
24. April 21, 1916 to Dec. 16, 1917.
25. Aug. 17, 1909 to June 11, 1916.
26. Jan. 30, 1912 to Feb. 18, 1912.
27. Oct. 27, 1908 to July 14, 1915.
28. Nov. 13, 1905 to Jan. 26, 1909.
29. April 5, 1910 to Oct. 9, 1917.
30. March 3, 1906 to June 8, 1916.
31. May 30, 1913 to March 12, 1914.
32. Sept. 24, 1906 to April 30, 1915.
33. Nov. 27, 1909 to Sept. 28, 1917.
34. Sept. 13, 1911 to May 11, 1914.
35. June 20, 1905 to Oct. 14, 1916.
36. Dec. 11, 1912 to Feb. 4, 1917.
37. Nov. 13, 1914 to Aug. 24, 1917.
38. May 11, 1908 to July 31, 1912.
39. Feb. 28, 1906 to June 30, 1913.
40. Nov. 8, 1913 to Sept. 3, 1917.
41. Jan. 1, 1908 to July 11, 1915.
42. Aug. 3, 1913 to June 11, 1916.
43. Nov. 21, 1908 to Oct. 9, 1917.
44. March 3, 1911 to June 16, 1912.
45. April 13, 1910 to Feb. 22, 1916.
46. Dec. 31, 1909 to Feb. 22, 1916.
47. Aug. 19, 1912 to April 3, 1917.
RATIONAL ARITHMETIC 71
48. Oct. 4, 1913 to June 8, 1915.
49. June 11, 1914 to May 3, 1917.
50. Sept. 3, 1909 to June 21, 1916.
Reference 239.
53. Find the time in exact days from :
1. Jan. 13, 1908 to Nov. 12, 1908.
2. April 6, 1915 to June 29, 1915.
3. Dec. 14, 1908 to Mav 12, 1909.
4. May 8, 1916 to Nov. 30, 1916.
5. June 11, 1915 to Oct. 27, 1915.
6. Aug. 12, 1916 to July 3, 1917.
7. Nov. 16, 1915 to Jan. 18, 1916.
8. Aug. 17, 1914 to Dec. 31, 1914.
9. July 30, 1913 to Jan. 1, 1914.
10. May 27, 1906 to April 28, 1907.
11. Feb. 12, 1908 to Nov. 13, 1908.
12. Dec. 6, 1916 to Aug. 6, 1917.
13. May 12, 1916 to Oct. 28, 1916.
14. June 11, 1909 to Jan. 3, 1910.
15. Oct. 24, 1912 to May 16, 1913.
16. July 16, 1911 to Dec. 25, 1911.
17. Sept. 3, 1908 to May 27, 1909.
18. Aug. 18, 1915 to Nov. 26, 1915.
19. Sept. 27, 1908 to May 13, 1909.
20. Feb. 13, 1914 to Dec. 6, 1915.
21. June 17, 1915 to April 8, 1916.
22. July 4, 1916 to Dec. 25, 1916.
72 RATIONAL ARITHMETIC
23. May 2, 1914 to April 18, 1915.
24. Dec. 16, 1913 to Sept. 29, 1914.
25. May 28, 1916 to Jan. 2, 1917.
26. Sept. 21, 1914 to Dec. 1, 1914.
27. April 13, 1915 to March 30, 1916.
28. Oct. 27, 1914 to June 3, 1915.
29. Nov. 28, 1906 to Dec. 13, 1906.
30. Jan. 31, 1914 to Jan. 3, 1915.
31. May 6, 1911 to April 14, 1912.
32. Dec. 25, 1915 to July 4, 1916.
33. June 22, 1909 to May 23, 1910.
34. April 28, 1916 to March 17, 1917.
35. Sept. 8, 1916 to March 17, 1917.
36. Dec. 4, 1913 to Feb. 12, 1914.
37. Dec. 25, 1916 to July 4, 1917.
38. Aug. 31, 1914 to May 12, 1915.
39. Dec. 30, 1915 to Sept. 9, 1916.
40. June 11, 1914 to Jan. 1, 1915.
41. April 3, 1915 to Jan. 4, 1916.
42. Dec. 28, 1914 to May 19, 1915.
43. Jan. 7, 1915 to Sept. 8, 1915.
44. Feb. 28, 1916 to Jan. 12, 1917.
45. June 3, 1912 to May 21, 1913.
46. May 14, 1911 to Nov. 28, 1911.
47. Sept. 9, 1912 to June 11, 1913.
48. Oct. 28, 1916 to July 14, 1917.
49. May 28, 1914 to April 7, 1915.
50. Dec. 8, 1915 to Sept. 29, 1916.
INTEREST
Study carefully 317 to 352 inclusive.
References 333, 334.
54. Find the interest on :
1. $462.40 for 2 mo. 6 da. at 6%.
2. $385.60 for 3 mo. 18 da. at 6%.
3. $460.75 for 1 mo. 8 da. at 6%.
4. $200 for 1 mo. 3 da. at 6%.
5. $260.70 for 72 da. at 6%.
6. $650 for 4 mo. 9 da. at 6%.
7. $450 for 6 mo. 15 da. at 6%.
8. $124 for 8 mo. 14 da. at 6%.
9. $285.50 for 93 da. at 6%.
10. $450.65 for 1 yr. 3 mo. 13 da. at 4%.
11. $562.30 for 5 mo. 13 da. at 4%.
12. $287.95 for 9 mo. 16 da. at 8%.
13. $396.40 for 1 yr. 3 mo. 14 da. at 8%.
14. $756 for 8 mo. 18 da. at 5%.
15. $468.40 for 2 yr. 5 mo. 6 da. at 7%.
16. $400 for 29 da. at 7i%.
17. $216.80 for 1 yr. 5 mo. 6 da. at 4^%.
18. $375.90 for 7 mo. 28 da. at 4^%.
19. $219.76 for 1 yr. 3 mo. 18 da. at 9%.
73
74 RATIONAL ARITHMETIC
20. $468.75 for 10 mo. 13 da. at 3%.
21. $240 for 3 mo. 22 da. at 12%.
22. $375.80 for 1 mo. 10 da. at 4%.
23. $265.72 for 3 mo. 7 da. at 4i%.
24. $486.85 for 4 mo. 3 da. at 6%.
25. $265.72 for 1 yr. 7 mo. 14 da. at 8%.
26. $380.60 for 8 mo. 15 da. at 7%.
27. $450 for 3 mo. 17 da. at 7i%.
28. $264.40 for 5 mo. 11 da. at 9%.
29. $942.60 for 2 yr. 9 mo. 16 da. at 5%.
30. $384.32 for 3 mo. 12 da. at 4^%.
31. $295.73 for 1 yr. 6 mo. 15 da. at 3%.
32. $389.60 for 11 mo. 18 da. at 6%.
33. $560.35 for 2 mo. 14 da. at 8%.
34. $387.60 for 6 mo. 5 da. at 4%.
35. $418.62 for 3 mo. 28 da. at 4i%.
36. $560.32 for 7 mo. 19 da. at 6%.
37. $362.40 for 1 yr. 8 mo. 20 da. at 4>^%.
38. $271.35 for 4 mo. 24 da. at 7%.
39. $361.75 for 44 da. at 7%.
40. $285.60 for 2 yr. 8 mo. 24 da. at 7i%.
41. $397.80 for 7 mo. 12 da. at 5%.
42. $184.25 for 8 mo. 21 da. at 4^%.
43. $1495.60 for 1 yr. 2 mo. 13 da. at 6%.
44. $372.75 for 11 mo. 8 da. at 7%.
45. $175.43 for 2 yr. 7 mo. 14 da. at 7i%.
46. $295.60 for 9 mo. 24 da. at 6%.
47. $362.70 for 8 mo. 21 da. at 6%.
RATIONAL ARITHMETIC 75
48. $467.80 for 1 yr. 3 mo. 5 da. at 3%.
49. $284.60 for 9 mo. 13 da. at 8%.
50. $575.80 for 6 mo. 18 da. at 4^%.
References 335, 336.
55. Find the interest on :
1. $600 from Jmie 1, 1916 to Aug. 13, 1916 at 6%.
2. $360 from Oct. 3, 1914 to June 3, 1915 at 5%.
3. $180 from Dec. 6, 1915 to July 13, 1916 at 8%.
4. $840.75 from May 12, 1912 to Dec. 8, 1914 at 7%.
5. $454.54 from Jan. 16, 1916 to Oct. 28, 1916 at 5%.
6. $544.44 from April 5, 1916 to Jan. 1, 1917 at 1\%,
7. $850 from July 18, 1914 to Dec. 31, 1916 at 9%.
8. $809 from Sept. 13, 1912 to June 11, 1915 at 5%.
9. $256 from Nov. 28, 1913 to Mar. 12, 1914 at 1\%.
10. $660.80 from Aug. 17, 1914 to Jan. 3, 1916 at 7%.
11. $840 from April 1, 1914 to Jan. 3, 1916 at 7%.
12. $629 from Nov. 13, 1908 to Aug. 4, 1910 at \\%,
13. $548 from Jan. 30, 1912 to Sept. 28, 1914 at 7%.
14. $465.10 from Oct. 2, 1913 to Sept. 12, 1914 at 8%.
15. $654 from Feb. 12, 1914 to Dec. 21, 1916 at 6%.
16. $360 from June 16, 1909 to Sept. 30, 1914 at 4^%.
17. $126 from Aug. 28, 1907 to Feb. 16, 1912 at 4%.
18. $480 from April 6, 1910 to June 30, 1914 at 5%.
19. $1000 from Oct. 13, 1914 to Nov. 28, 1915 at 6%.
20. $975 from May 12, 1916 to Nov. 11, 1916 at 9%.
21. $649.24 from Jan. 1, 1914 to April 3, 1916 at 6%.
22. $100 from Oct. 2, 1915 to Feb. 12, 1916 at 5%.
76 RATIONAL ARITHMETIC
23. $654 from Dec. 12, 1915 to Aug. 7, 1916 at 5%.
24. $962 from March 8, 1915 to Aug. 7, 1916 at 4^%.
25. $269.05 from Dec. 3, 1909 to Sept. 8, 1915 at 4%.
26. $680 from April 21, 1914 to Nov. 2, 1914 at 6%.
27. $500 from Aug. 19, 1913 to May 28, 1915 at 8%.
28. $85 from Nov. 12, 1914 to April 3, 1915 at 4%.
29. $450 from Feb. 9, 1916 to Dec. 21, 1916 at 5%.
30. $240 from Jan. 8, 1913 to June 9, 1915 at 7%.
31. $400 from Sept. 12, 1909 to Oct. 9, 1914 at 7^%.
32. $560 from July 4, 1910 to Sept. 8, 1914 at 4i%.
33. $200 from March 2, 1913 to Sept. 27, 1913 at 9%.
34. $460 from Dec. 8, 1916 to Feb. 3, 1917 at 3%.
35. $296.50 from May 8, 1904 to June 11, 1913 at 8%.
36. $320.60 from Oct. 28, 1913 to July 7, 1915 at 4%.
37. $576 from Jan. 23, 1909 to Aug. 13, 1912 at 6%.
38. $320.60 from Dec. 6, 1914 to June 28, 1916 at 4>i%.
39. $720.14 from July 11, 1912 to Aug. 29, 1916 at 8%.
40. $365.40 from Sept. 14, 1911 to Feb. 18, 1915 at 7%.
41. $428.60 from May 11, 1916 to Nov. 8, 1916 at 12%.
42. $576.80fromJune20,1909toDec.31,1909atl0%.
43. $162.38 from Oct. 27, 1914 to Dec. 3, 1916 at 8%.
44. $316.20 from July 26, 1913 to Oct. 19, 1914 at 5%.
45. $483.90 from Jan. 8, 1915 to July 4, 1916 at 4i%.
46. $265.70 from Sept. 4, 1915 to May 30, 1916 at 9%.
47. $456.75 from Dec. 9, 1914 to June 30, 1916 at U%.
48. $195 from Nov. 19, 1913 to April 19, 1914 at 4%.
49. $362.80 from Sept. 12, 1914 to Aug. 8, 1915 at 6%.
50. $195.64 from Aug. 14, 1915 to May 3, 1916 at 3%.
RATIONAL ARITHMETIC 77
ACCURATE INTEREST
References S'2G, 327, 328 ; 337 to 343 inclusive.
While both methods are used in business, the one
explained in 340 is better because of the infrequency
with which accurate interest is used.
56. Find the accurate interest of :
1. $1436 for '295 days at 6% ; at 8%.
2. $484.50 for 193 days at 6% ; at 7i%.
3. $956.35 for 1 year 214 days at 6% ; at 4^%.
4. $632 for 462 days at 6% ; at 4^%.
5. $1284.50 from Aug. 8, 1916 to Jan. 12, 1917 at 8%.
6. $543.32 from Apr. 7, 1916 to Feb. 25, 1917 at 5%.
7. $246.50 from Jan. 5, 1915 to May 27, 1916 at 4i%.
8. $3432.40 from June 13, 1914 to Feb. 29, 1916 at
Si%.
9. £ 120 9s Sd for 214 days at 8%.
10. £253 ll5 10^ for 313 days at 5%.
11. £586 Us 4>d for 1 year 246 days at 4i%.
12. £732 Ids 9d for 2 years 97 days at 3%.
Note : Change English money to pounds ; see 232, and then
apply 343. Reduce resulting decimal to lower denomination, 231.
TO FIND TIME
References 346, 347, 348.
57. 1. In what time will $417.40 produce $7.43 at
3i% interest ?
2. How long will it take $325.80 on interest at 10%
to produce $30.86 ?
78 RATIONAL ARITHMETIC
3. $895.80 earned $16.50 at 7% interest. How long
was the money at interest ?
4. How long will it take $9500 to earn $1524.75 at
9%?
5. I loaned $592.25 to my brother at 5% interest.
He paid me $13.74 interest. How long did he have the
money ?
6. $182.40 drawing interest at 4% earned $9.48.
How long was it invested ?
7. $4150.30 was invested at 8% long enough to earn
$295.58. How long was it invested ?
8. $318.60 at 3% would require how long to produce
$7.38 interest.^
9. I loaned $7500 at 7%. When the loan was paid
I received $7702.71. For what time was the loan
made ?
10. I received a check for $533.85 to cancel a loan of
$519.75 effected at 4^%. How long had the loan been
standing ?
TO FIND RATE
References 349, 350.
58. 1. At what rate will $1836 earn $23,46 in 115
days ?
2. The interest on $852 for 18 days is $2.13. What
is the rate ?
3. In 93 days $75 increases at interest $1.55. What
is the rate ^
RATIONAL ARITHMETIC 79
4. In 144 days $375 amounts to $381.75. What is
the rate ?
5. At what rate would $982 be placed at interest
for 2 mo. 12 da. to earn $9.82 ?
6. A loaned $2160 for 5 yr. 9 mo. 1 da. It
amounted to $3091.95. What rate did he charge?
7. $588 on interest for 2 yr. 3 mo. and 18 da. earns
$67.62. What is the rate ?
8. At what rate must $3500 be placed at interest
for 4 mo. and 15 da. to amount to $3605 ?
9. $296 produced $3.70 in 45 days. At what rate
was it invested ?
10. $810 amounts to $829.44 in 6 months and 12
days. What was the rate ?
11. $750 on interest from March 1, 1916 to August 7,
1916 earns $26.50. What is the rate.^
12. From April 7, 1915 to October 2, 1915, $360
earns $16.02. What is the rate ?
TO FIND PRINCIPAL
References 351 to 356.
59. 1. What principal will be required to earn
$13.18 in 11 mo. 11 da. .^
2. W'hat principal will be required to earn $39.92
in 192 days at 3^% ?
3. How much money invested at 5^% for 86 days
will earn $24.62?
80 RATIONAL ARITHMETIC
4. The interest is $16, time 5 months and 18 days,
rate 8%. What is the principal ?
5. At 9% the interest for 3 mo. 27 da. is $150.07.
What is the principal ?
6. W^hat principal will in 6 mo. 13 da. at 6% amount
to $720.55 .?
7. What principal will in 8 mo. 25 da. at 5% earn
$15.73?
8. In 4 mo. 9 da. at 4% what principal will amount
to $591.61.^
9. At 6% what principal will in 1 yr. 9 mo. 28 da.
yield $74.99?
10. Money invested for 1 yr. 7 mo. 14 da. at 4^%
amounts to $531.30. What was the principal?
11. A certain sum of money on interest at 3% from
May 19, 1910 to July 15, 1915 earns $392.35. What is
the investment ?
12. W^hat sum was loaned on April 5, 1910 at 7% if
it were paid by check for $942.66 on Jan. 9, 1916?
13. What principal on interest from October 25,
1905 to May 21, 1912 at 12% would amount to
$566.44 ?
14. What sum on interest from July 15, 1915 to
October 19, 1915 at 7% will earn $14.90?
15. What sum on interest from May 13, 1910 to
August 25, 1919 would amount to $11,446.47?
RATIONAL ARITHMETIC 81
GENERAL PROBLEMS IN INTEREST
Before attempting to solve the following problems, the student
should be thoroughly familiar with the entire subject of interest as
presented in paragraphs 317 to 356 inclusive. The following prob-
lems comprise a series of tests on the subject of interest and are not
graded according to difficulty, but are arranged as such problems
might present themselves in business. Solve each in the simplest
possible way.
60. 1. A note for $852 dated May 2, 1915 with
interest at 5% was paid on Feb. 14, 1916 by certified
check. What was the amount of the check, time
being computed in exact days ?
2. I have just received a legacy of $5000. I have
an obligation of $395 which will be due in 1 yr. 6 mo.
and 14 da. from to-day. I have decided to set aside
enough of my legacy at 4% interest to pay the obliga-
tion when it is due. How much will I have to set aside ?
3. Brown borrowed of me $400 at 7%, Jones $647 at
5%, and Smith $398 at 4^%. Brown's loan ran 1 year
8 months and 11 days, Jones' ran 6 months and 19 days,
and Smith's ran 297 days. What were my total re-
ceipts for interest ?
4. After a loss by a fire the insurance company has
agreed to pay me $4340 in full settlement of the claim.
They will pay this amount in full at the end of 60 days
or will make a cash settlement, deducting 2%. Which
proposition should I accept and what will I gain by so
doing, money being worth 6% ?
5. On May 26, 1915 I gave my note for $1000 for
6 months at 5%. November 26, 1915 I paid the note
82 RATIONAL ARITHMETIC
and accumulated interest, reckoned on the basis of
exact days. How much did I pay?
6. A bill of $780.14 due on March 3, 1909, was not
paid until October 15, 1909, when it w^as settled with
interest at 6%. What was the amount paid, computing
time in exact days ?
7. A house costing $7500 rents for $60 a month.
What rate of interest does the investment pay if the
annual expenses, including repairs, taxes, etc., amount
to $250 ?
8. What will be the difference in the amount of
interest involved on a claim for $85, running from
May 12, 1909 to October 19, 1909 at 8%, between
the amount due by computing the time in exact da^^s
and computing the time in months and da^^s ?
9. On May 1, 1915, I bought a bill of hides at
$15,300, on 60 days' credit, 2% off for cash, and bor-
rowed the money at 6% on the hides as security to
accept the cash price, giving my note for 60 days. At
the expiration of the note I had tanned the hides at
an expense of $1500 and sold them at an advance of
25% on the price paid for them. After paying my
note, what was my net profit ?
10. I bought a bill of $1450 subject to a discount of
20%, 10%, and 5%, with an additional discount of ^Z%
for cash, and borrowed the money to pay for it at 5%.
After 45 days I sold the goods at the same list price,
subject to a discount of 25%, 2% extra for cash, re-
ceiving cash settlement, and paid my loan. What was
my profit ?
PARTIAL PAYMENTS
Study carefully 357 to 370 inclusive, before attempting to work
on partial payments.
References 371, 372, 373.
Use the United States Rule.
61. 1. What is the balance due on July 1, 1916 on
a note for $1500 dated February 1, 1913, upon which
the following payments were made : July 24, 1913,
$250; Aug. 7, 1913, $100; March 9, 1915, $50; Jan.
1, 1916, $300; interest at the rate of 5%?
2. What is the balance due on December 31, 1917
on a note for $1400 dated May 2, 1915, bearing in-
terest at 6% and having the following indorsements :
July 1, 1915, $200; September 25, 1915, $90; Febru-
ary 28, 1916, $175; May 19, 1917, $475?
3. On April 21, 1913, I gave my note for $3550,
payable in three years, interest at 4 5%. I paid as
follows: Oct. 15, 1913, $125; March 3, 1914, $125;
July 20, 1914, $875. How much will be required to
settle the note at maturitv ?
4. W. A. Jones gave a note on June 4, 1914 to Frank
Brown for $1285.50 with interest at 7%. He made
payments as follows: Dec. 15, 1914, $340; Feb. 17,
1915, $330; March 11, 1915, $400. What was due on
Mav 15, 1915 ?
83
84 RATIONAL ARITHMETIC
5. Find the value of a note for $2500, given Oct. 10,
1915, with interest at 4^%, and on which the following
payments have been indorsed : Jan. 5, 1916, $815 ;
June 18, 1916, $350; Oct. 10, 1916, $250; Jan. 17,
1917, $150. Settlement was made Sept. 6, 1917.
6. On a note of $3080, dated Oct. 1, 1915, the follow-
ing payments have been made, interest being at 6% :
Dec. 31, 1915, $300; Feb. 29, 1916, $50; June 5, 1916,
$500 ; Oct. 2, 1916, $700. What will be due on Dec. 31,
1916?
7. Find the amount due Feb. 24, 1917 on a note for
$3000, dated March 12, 1915, with interest at 7%, upon
which the following payments were made : Aug. 18,
1915, $235; April 9, 1916, $80; July 3, 1916, $400;
Dec. 5, 1916, $175.
8. What is the balance due on Jan. 1, 1917 on a note
for $500, dated July 15, 1916, bearing interest at 5%
and having the following indorsements : Aug. 20, 1916,
$27.50; Oct. 8, 1916, $125; Nov. 12, 1916, $110;
Dec. 11, 1916, $65?
References 374-375.
Use the Merchants' Rule.
62. 1. A note of $850 was dated May 25, 1914,
interest at 6%. It was indorsed Aug. 13, 1914, $50 ;
Nov. 7, 1914, $324.95. What was due March 25, 1915 ?
2. A note for $1250, dated Jan. 25, 1916, interest at
8%, bears the following indorsements : March 10, 1916,
$462.50; Aug. 4, 1916, $100; May 22, 1917, $556,
What was due on Jan. 1, 1918 to settle the note in full ?
RATIONAL ARITHMETIC 85
3. On a note for $550, dated Feb. 5, 1913, interest
at 6%, the following payments were made : Oct. 17,
1913, $66.10; March 5, 1914, $140. What was due
Nov. 11, 1914.^
4. A note for $2000 was dated Dec. 12, 1915, interest
at 7%. It was indorsed as follows : June 19, 1916,
$200; Dec. 6, 1916, $338; Aug. 21, 1917, $276.50;
Sept. 12, 1917, $60. What was due Oct. 15, 1917.^
5. I gave my note for $1080 with interest at 5% on
Jan. 25, 1914. I made the following payments :
Mar. 1, 1914, $364.40 ; May 13, 1914, $341.50 ; Sept. 1,
1914, $205. What was due on settlement, Jan. 25,
1915?
6. A note for $1500 was dated May 11, 1913, bearing
interest at 7.2%. The following payments were made :
Feb. 14, 1914, $150; Sept. 23, 1914, $300; July 8,
1915, $100; May 29, 1916, $200. What was due Sep-
tember 4, 1916?
7. On a note for $1120, dated August 7, 1914,
interest at 7%, payments were made as follows : Sept.
13, 1914, $80; Nov. 7, 1914, $200; Sept. 15, 1915,
$450. What was due Aug. 7, 1916?
8. The following pa^mients were made on a note
for $580, dated Oct. 17, 1915, bearing interest at 5% :
Aug. 5, 1916, $52.50; April 17, 1917, $49.30; Aug. 5,
1917, $250. What was due Sept. 9, 1917?
BANK DISCOUNT
Study 376 to 381 inclusive.
Reference 38*2.
63. Find the bank discount and net proceeds
Face
1. $^40
2. $300
3. $1000
4. $400
5. $250
6. $350
7. $500
8. $850
9. $600
10. $375
11. $460
12. $2500
13. $36500
14. $845
15. $280
16. $430
17. $375
18. $3000
19. $575
20. $490
21. $450
22. $340
23. $500
24. $1500
25. $475
Date Time
Jan. 3, 1916 60 da.
Sept. 8, 1916 2 mo.
June 1, 1915 90 da.
May 1, 1916 3 mo.
June 1, 1916 3 mo.
Dec. 30, 1914 4 mo.
Apr. 3, 1915 6 mo.
June 1, 1916 6 mo.
Mar. 3, 1915 90 da.
Feb. 3, 1916 60 da.
May 9, 1914 3 mo.
July 1, 1915 6 mo.
Mar. 3, 1916 4 mo.
Jan. 3, 1916 90 da.
Sept. 8, 1914 30 da.
Nov. 9, 1915 3 mo.
Oct. 12, 1916 2 mo.
Jan. 10, 1916 4 mo.
Dec. 4, 1915 3 mo.
Aug. 8, 1915 60 da.
Mar. 3, 1916 90 da.
May 2, 1915 6 mo.
July 5, 1916 60 da.
Dec. 3, 1914 4 mo.
Jan. 4, 1916 3 mo.
86
Date of Disc.
Jan. 4, 1916
Oct. 1, 1916
June 7, 1915
June 1, 1916
Aug. 3, 1916
Jan. 2, 1915
May 15, 1915
Oct. 14, 1916
Apr. 4, 1915
Feb. 20, 1916
May 12, 1914
Aug. 30, 1915
June 30, 1916
Feb. 8, 1916
Sept. 10, 1914
Nov. 20, 1915
Nov. 13, 1916
Jan. 12, 1916
Jan. 29, 1916
Sept. 18, 1915
Apr. 5, 1916
Aug. 3, 1915
July 10, 1916
Feb. 9, 1915
Jan. 20, 1916
Rate op
Disc.
6%.
7%.
5%.
6%.
0-
^0-
4%.
n%.
41%.
5%.
6%.
'0-
710/
'2/0-
0-
7%.
7%.
/o-
7i%.
4%.
5%.
0-
RATIONAL ARITHMETIC
87
Reference 383.
64. Find the bank discount and proceeds of the
following interest-bearing notes :
Face
1. $800
2. $400
3. $2240
4. $480
5. $1530
6. $285
7. $390
8. $2500
9. $460
10. $1400
11. $2150
12. $580
13. $490
14. $3400
15. $1560
16. $780
17. $?40
18. $1375
19. $650
20. $425
21. $730
22. $260
23. $475
24. $390
25. $1575
Date Time
Sept. 1, 1915 3 mo.
June 15, 1916 2 mo.
Jan. 10, 1916 90 da.
Jmie 1, 1916 6 mo.
Mav4, 1916 60 da.
Sept. 3, 1916 4 mo.
Feb. 6, 1916 30 da.
Mar. 10, 1916 3 mo.
June 12, 1916 4 mo.
Aug. 3, 1916 90 da.
July 5, 1916 60 da.
Mavl2, 1916 6 mo.
June 6, 1916 30 da.
Dec. 2, 1916 3 mo.
Mar. 3, 1916 5 mo.
Aug. 1, 1915 4 mo.
July 11, 1916 90 da.
Mar. 18, 1916 30 da.
June 19. 1916 5 mo.
Jan. 31, 1916 1 mo.
Oct. 6, 1916 2 mo.
Jan. 14, 1916 5 mo.
May 10, 1916 90 da.
Apr. 5, 1916 30 da.
July 1, 1916 1 mo.
'0
5%
6%
5%
Date Date of Disc.
6% Sept. 11, 1915
6% June 30, 1916
7% Feb. 8, 1916
5% Aug. 3, 1916
May 4, 1916
Sept. 20, 1916
Feb. 6, 1916
Mar. 15, 1916
Aug. 2, 1916
Sept. 1, 1916
Julv 7, 1916
May 12, 1916
Junes, 1916
Jan. 3, 1917
Mar. 15, 1916
Aug. 11, 1915
Aug. 1, 1916
Mar. 24, 1916
Aug. 1, 1916
Feb. 4, 1916
Oct. 6, 1916
Mar. 3, 1916
May 11, 1916
/o Apr. 12, 1916
7i% July 5, 1916
'0
6%
5%
4%
5%
6%
4%
o%
7%
^0
10%
Rate of
Disc.
'0-
5%.
0-
0-
4%.
5%.
'0-
7%.
^0-
6%.
5%.
6%.
7%.
5%.
7%.
7%.
6%.
5%.
6%.
References 384-385.
65. For what sum must I write my note in order to
yield the following proceeds if discounted on the date
of the note ?
88 RATIONAL ARITHMETIC
Proceeds
Time
Rate
1.
$385
90 da.
6%
2.
$450
3 mo.
5%
3.
$1285
60 da.
8%
4.
$370
4 mo.
7%
5.
$260
90 da.
8%
6.
$580
2 mo.
4i%
7.
$290
6 mo.
8%
8.
$365
4 mo.
7%
9.
$290
90 da.
6%
10.
$460
30 da.
8%
11.
$1340
2 mo.
7%
12.
$360
3 mo.
6%
13.
$1500
1 mo.
4%
14.
$2560
5 mo.
7i%
15.
$775
60 da.
6%
16.
$550
30 da.
7%
17.
$1250
2 mo.
8%
18.
$875
4 mo.
8%
19.
$1360
90 da.
6%
20.
$728
60 da.
5%
21.
$450
5 mo.
7%
22.
$1385
30 da.
8%
23.
$3760
60 da.
6%
24.
$1485
3 mo.
5%
25.
$960
90 da.
8%
RATIONAL ARITHMETIC 89
COMPOUND INTEREST
Study 386 to 388 inclusive.
Reference 388.
66. Find the compound interest '.
Principal
Rate
Time
Compounded
1.
$7800
6%
2 yr.
Annually
2.
$4600
5%
2yr.
Semi-annually
3.
$8400
8%
Syr.
Annually
4.
$9000
4%
Syr.
Quarterly
5.
$3500
6%
4 yr. 5 mo.
Semi-annually
6.
$4650
6%
1 yr. 8 mo.
Quarterly
7.
$3865
4%
1 yr. 2 mo. 15 da.
Quarterly
8. A note for $495.60, dated June 10, 1912, and
drawing interest at 6% per annum, compounded semi-
annually, was paid March 22, 1916. What was the
amount due, if no payments of either interest or prin-
cipal had been made ?
9. What amount will, on June 30, 1917, discharge
a note of $3560, dated Dec. 1, 1914, and drawing in-
terest at 8% per annum, compounded quarterly, no
previous payments having been made ?
10. What is the amount due April 1, 1916, upon a
note for $480.50, dated May 10, 1911, and drawing
interest at 8% per annum, compounded semi-annually,
no previous payments having been made ?
11. A young man deposited $200 in a savings bank
which paid 4% per annum, compounded quarterly.
If nothing was withdrawn, what amount was to his
credit at the end of the third year ?
90
RATIONAL ARITHMETIC
12. For the benefit of his son who is 12 years old,
Mr. A deposited in a savings bank $1000 at 4%, in-
terest compounded semi-annually. How much should
the son receive when he becomes 21 years old?
PERIODIC INTEREST
Study 389 to 390 inclusive.
67. Find the periodic interest :
1.
2.
3.
4.
5.
6.
7.
Principal
$5500
$450
$3000
$2850
$4650
$956
$380
Rate per
Annum
6%
'0
7%
4%
Time
4 yr.
3yr.
4 yr.
5 yr. 3 mo.
1 yr.
2 yr. 10 mo.
Interest Due
Annually
Annually
Semi-annually
Annually
Quarterly
Quarterly
Semi-annually
4 yr. 1 mo.
8. What amount will be due Feb. 1, 1922, on a note
of $3000, dated Jan. 1, 1920, and drawing interest at
6% per annum, payable semi-annually, if. the first four
interest payments are paid when due, and no subse-
quent payments made ?
9. What amount was due July 15, 1917, on a note
of $4600, dated March 13, 1913, drawing interest at 5%
per annum, payable semi-annually, no previous pay-
ments having been made ?
10. No interest having been previously paid, what
was the amount of a note of $1400 at 6%, interest pay-
able quarterly, dated Jan. 1, 1914, and paid Feb. 1,
1916?
RATIONAL ARITHMETIC 91
11. What sum was due Jan. 28, 1917, on a note of
$4000, dated May 18, 1913, and drawing interest at 5%
per annum, payable semi-annually ; no payments hav-
ing been made previous to that time ?
12. A merchant bought a store building for $9000,
giving his note without interest, payable 2 years from
date, and 8 separate non-interest-bearing notes for the
quarterly interest at 6% per annum. If nothing was
paid until the maturity of the note, what was the
amount then due ?
13. I purchased a $1250 mortgage on which interest
at 6% was due semi-annually on Jan. 15 and July 15.
Owing to the fact that no interest had been paid since
Jan. 15, 1917, I secured the mortgage at less than its
face value. On Oct. 27, 1919, I made arrangements
with the mortgagor whereby he paid the interest in
full and $500 on the face of the mortgage. How much
did I receive in all ?
14. What interest is due Jan. 7, 1920, on $875 from
Nov. 13, 1917, at 6%, interest due quarterly and none
having been paid ?
15. What is the total interest on $386.40 from Dec.
31, 1914, to Sept. 1, 1919, interest due annually and
none having been paid ? Rate 4^%.
16. What amount was due Jan. 8, 1920, on a note of
$2340 dated Sept. 1, 1917, drawing interest at 6%, in-
terest payable semi-annually, if the first two payments
were made when due and no subsequent payments
made ?
AVERAGE ACCOUNTS
Study 391 to 396 inclusive.
Reference 397.
68. Average the following :
1. Dr. Harold Chute
1916
Oct. 12
Dec. 20
1917
Jan. 5
Mar. 2
$ 67.85
71.15
143.50
116.20
Cr.
1917
Jan. 10
$316.20
Feb. 19
415.23
Mar. 24
99.
May 10
271.
2. Dr.
Fred Ellis
Cr.
1917
Apr. 18
$367.40
May 6
572.
May 23
923.
June 2
134.50
3. Dr,
Benjamin Jones
Cr.
92
4. Dr.
RATIONAL ARITHMETIC
Howard Colson
93
Cr,
1916
Dec. 1
$540.
Dec. 15
236.10
1917
Jan. 2
200.
Jan. 31
150.
5. Dr,
James Mullaney
Cr.
1917
Feb. 5
$1050.10
Mar. 10
826.
May 1
924.
May 31
186.
Reference 398
69. Average the following :
1. Find cash balance on April 18, 1915.
Dr.
Paul Duncanson
Cr,
1915
Jan. 27
30 da.
$420.
Feb. 17
10 da.
300.
Mar. 1
20 da.
540.
Apr. 12
30 da.
600.
94
RATIONAL ARITHMETIC
2. Find cash balance on Jan. 1, 1917.
Dr.
Arthur Bennett
Cr.
1916
Oct.
17
15 da.
$432.
Nov.
20
2 mo.
864.
Nov.
30
30 da.
286.
Dec.
19
10 da.
627.
3. Find cash balance on Dec. 31, 1916.
Dr
C. D. Adams
4. Find cash balance on Sept. 7, 1916.
Cr,
1916
Aug. 9
30 da.
$234.
Sept. 15
60 da.
562.
Nov. 29
10 da.
52.96
Dec. 21
15 da.
715.
Dr.
George Duncan
Cr,
1916
May 6
30 da.
$128.
June 30
20 da.
126.
July 19
10 da.
213.20
Sept. 3
30 da.
185.
RATIONAL ARITHMETIC
95
5. Find cash balance on May 1, 1915.
Dr. Paul Jones
70.
1. Dr.
References 399-400.
A. C. Davis
When is the above due by average ?
What was the cash balance Apr. 15, 1917.^
CV.
1915
•
Jan. 5
10 da.
$400.
Jan. 31
30 da.
90.60
Mar. 8
10 da.
150.
Apr. 25
2 mo.
86.12
Cr.
1917
1917
Jan.
1
2 mo.
$600.
Feb. 1
Cash
$200.
Feb.
2
30 da.
240.
Mar. 18
Cash
150.
Apr.
6
10 da.
360.
Apr. 3
Cash
75.
2.
Dr.
J. F. Howard
Cr,
1916
1916
May
18
60 da.
$209.70
June 1
Cash
$100.
June
3
30 da.
180.
June 30
Cash
50.
July
10
15 da.
750.
July 19
Cash
300.
Aug.
1
10 da.
280.50
When is the above due by average ?
What was the cash balance Aug. 21, 1916?
96
RATIONAL ARITHMETIC
3. Dr,
George Stevens
When is the above due by average ?
What was the cash balance July 1, 1915.'^
4. Dr.
Fred Ellis
When is the above due by average .^
What was the cash balance May 5, 1917.'^
Cr.
1915
1915
Jan.
20
2 mo.
$219.50
Feb.
25
Cash
$ 50.
Feb.
25
30 da.
218.75
Mar.
31
Cash
75.
Mar.
28
10 da.
413.
Apr.
30
Cash
200.
June
30
10 da.
216.
Cr.
1917
1917
Jan.
1
10 da.
$600.
Feb. 28
Cash
$400.
Feb.
2
10 da.
200.
Mar. 31
Cash
100.
Mar.
3
10 da.
350.
Apr. 30
Cash
150.
5. Dr.
William Walker
Cr.
1915
1915
Jan.
31
2 mo.
$540.
Feb. 15
Cash
$225.
Feb.
15
60 da.
450.
Mar. 1
Cash
345.
Mar.
30
10 da.
306.50
Mar. 10
Cash
295.
When is the above due by average ?
What is the cash balance Julv 3, 1915 ?
RATIONAL ARITHMETIC
97
6. D
r.
Benjamin Brown
Cr.
1916
1916
Apr. 2
10 da.
$150.
June 25
Cash
$300.
Mav 1
15 da.
540.
Julv 31
Cash
360.
June 3
10 da.
450.
Aug. 10
Cash
250.
July 2
10 da.
323.
When is the above due by average ?
What is the cash balance Aug. 29, 1916?
7. Dr.
Charles Smith
Cr.
1916
1916
Sept. 1
2 mo.
$315.60
Oct. 5
Cash
$200.
Oct. 25
60 da.
419.10
Nov. 1
Cash
150.
Nov. 16
10 da.
216.05
Dec. 2
Cash
375.
When is the above due by average '^
What was the cash balance Mar. 5, 1917 ?
8. D
r.
Robert Brown
Cr.
1915
1915
Jan. 2
1 mo.
$1800.
Feb. 18
Cash
$300.
30
10 da.
600.
Feb. 27
Cash
300.
Mar. 5
Cash
300.
When is the above due by average ?
What was the cash balance Mar. 10, 1915 ?
98
RATIONAL ARITHMETIC
9. Dr.
E. BOWDOIN
Cr.
1916
1916
Oct.
1
30 da.
$350.
Oct. 21
Cash
$300.
Nov.
8
10 da.
340.
Nov. 24
Cash
300.
Dec.
9
15 da.
210.
1917
Jan.
20
10 da.
116.
When is the above due by average ?
What was the cash balance Feb. 1, 1917?
10.
Dr.
Sidney
Berry
Cr.
1915
1915
June
1
60 da.
$410.
Aug. 1
Cash
$300.
July
5
30 da.
135.
Aug. 31
Cash
200.
Aug.
1
10 da.
216.39
Sept. 4
Cash
100.
Aug.
31
15 da.
162.54
When is the above due by average ?
What was the cash balance Oct. 6, 1915 ?
TAXES
The general principles of percentage are used in figuring taxes.
Study 401-404 inclusive.
References 242-*263 inclusive.
71. 1. What is the tax on property assessed for
$17,400, the rate of taxation being |% ?
2. What is the tax on property assessed for $8500,
rate of taxation being 16f mills on the dollar?
3. What is the tax on property assessed for $23,500,
the rate of taxation being $19.20 on the thousand .^^
4. What is the tax on property assessed for $7588,
the rate of taxation being $1.20 on the hundred.^
5. I own real estate worth $19,500 upon which I pay
a tax at the rate of $21.40 a thousand. I also pay an
income tax of 6% on a net taxable income of $1400 and^
a poll tax of $2. What is nay entire tax.^
6. My real estate is assessed at $6500, my personal
property at $1570 ; my net taxable income is $2400.
Tax on the tangible property is levied by the city at
the rate of $19.40 a thousand ; an income tax is levied
by the state at the rate of 3% ; my poll tax is $2. What
is my entire tax ?
7. The assessed value of real estate in a town is
$1,869,000; personal property is $2,450,000. It is
99
100 RATIONAL ARITHMETIC
necessary to raise by taxation $412,560. What would
be the rate a thousand if there are 1742 polls at $2 each ?
8. In a town whose valuation is $25,000,000, there
is an increase in the budget to cover additional expenses
of the public schools amounting to $40,000. How many
cents a thousand is the tax increased thereby? How
much will the improvement cost a citizen who is worth
$30,000 ?
CUSTOMS AND DUTIES
Ad valorem duties are estimated according to the value of the
goods in conformity to the principles of percentage. Study 405-422
inclusive.
References 242-263 inclusive.
Values of units of foreign currency expressed in United States
money will be found in 468.
72. 1. What is the ad valorem duty upon an im-
portation valued at £430, Ss, 9d, allowing 10% for
breakage, duty being at 25% ?
2. Find the ad valorem duty on an invoice of
15,834 marks at 23%,.
3. Find the ad valorem duty on an invoice of
3446.18 francs, duty being 4^3%.
4. Find the ad valorem duty on a bill of 1475
pesos if the duty is 24%.
6. What is the specific duty on 13 tons of tan bark
on which there is a duty of 3 cents a hundred pounds ?
6. What is the specific duty on an invoice amount-
ing to $760, allowing 10% for breakage, duty being
at 35% ?
RATIONAL ARITHMETIC 101
7. Find the duty at 10 cents a square yard and
40% ad valorem on a rug 12'X18', imported from Eng-
land and invoiced at £14.
8. What is the total duty on 140 cases of plate
glass, each containing 25 plates, 20''x48'' at 8^ a
square foot ?
9. What is the duty on an invoice of 2300 yards of
27-inch goods, invoiced at 8^ 9^ a yard, subject to an
ad valorem duty of 40% and a specific duty of 6^ a
square yard ?
10. What is the duty at 60% on a bill amounting
to £736 9s Sd ?
11. W^hat is the duty at 30% ad valorem on U\o
bales of burlap, each bale containing 40 webs, each
web being 48 yd. long and 30 in. wide, invoiced at 30^
per square yard ?
12. What is the duty at 20^ per square yard and
35% ad valorem on 1750 yards of cloth invoiced at
7 francs per yard ?
13. A merchant imported a lot of steel knives from
England as follows: 75 doz. at 12^ Qd; 50 doz. at 18.?
6d\ 30 doz. at £l 5s Qd; 20 doz. at £l 8^ 6d; 12 doz.
at £2 9s 6d; 10 doz. at £2 10^ 6d. The charges in
England amovmt to £7 12^* Qd. The consul's fee was
125 6d. Marine insurance was 20 (^ per hundred on the
value of the invoice. The cartage amounted to $2.50.
The duty was 30% ad valorem and 30 p per dozen.
Find the total cost of the invoice.
INSURANCE
Study 423-434 inclusive.
References M^-263 inclusive.
73. 1. My house is insured for $4500 for a period
of five years at 21%. What is the premium ?
2. A merchant insured his stock of goods for $5600
at the rate of li% per annum. What annual premium
does he pay ?
3. A factory is insured for $1 '25, 000 in four com-
panies. A carries i of the insurance, B carries i,
C carries i, and D i. A fire occurs causing a damage
of $50,000. For how much will each company be
responsible ? •
4. A stock of goods is insured in four companies as
follows : $1500 in A, $2400 in B, $3200 in C, and $2500
in D. The goods are damaged to the extent of $8000.
How much should each company pay.^
5. A building worth $85,000 was insured for $68,000,
and afterwards damaged by fire to the extent of $4500.
The policy contains the average clause. What amount
of insurance can be collected from the company ?
6. A vessel worth $50,000 is insured for $20,000 in
company A and $18,000 in company B. The vessel
is damaged to the extent of $20,000. What amount
is to be paid by each company ?
102
RATIONAL ARITHMETIC 103
7. I insured my building worth $80,000 for 80% of
its value at f% premium with the iEtna Insurance
Company. The iEtna Insurance Company later re-
insured $20,000 in the Niagara Insurance Company
and $18,000 in the Massachusetts Fire and Marine
Insurance Company. The property is damaged to
the extent of $30,000. What was the net loss to each
of the companies ?
8. I have a policy, containing the average clause,
for $7500 on merchandise in stock worth $9000,
upon which I have paid a premium of |%. A fire
occurs by which the goods are damaged to the ex-
tent of $4000. What was my total loss and the net
loss to the company ?
LIFE INSURANCE
Study 435-441 inclusive.
Reference 442,
74. 1. What would be the annual premium on a
policy for $2500, premiums payable annually during
life, at the age of 21 years ? 26 years ? 32 years ?
38 vears ?
2. What would be the annual premium on a fifteen -
year endowment policy for $5000, at the age of 23
years ? 27 years ? 32 years ? 37 years ?
3. What would be the annual premium on a policy
for $4000, premiums due annually for a period of ten
years, policy payable at death only, at the age of
20 vears ? 25 vears ? 35 vears ?
104 RATIONAL ARITHMETIC
4. What would be the annual premium on a twenty-
year endowment policy for $6500, age of insured at
nearest birthday 23 years ? 29 years ? 37 years ?
5. What would be the annual premium on an or-
dinary life policy for $3500, premiums to be paid
annually for twenty years, policy to mature at death,
age of insured at nearest birthday 28 years ? 23 years ?
38 years ?
6. A man insured his life at the age of 23 years on a
twenty-year endowment plan, payments to be paid
annually, amount of policy $5000. He died at the age
of 33 years. How much less would he have paid in
premiums if he had been insured by the ordinary life
plan?
7. A man at the age of 35 took out a fifteen -year
endowment policy for $2000. What annual premium
must he pay ? He lives 20 years and receives the face
of the policy. How much less will this amount to
than it would have if he had invested the premium
at 4% compound interest "^
8. A man at the age of 25 took out a $3000 twenty
payment life policy. He died after paying ten pre-
miums. What was the annual premium ? How much
more did his family receive than the premiums
amounted to, making no allowance for interest ?
9. Three men, aged 24, take a policy for $1000 each.
One takes an ordinary life policy, one a twenty-year
life policy, and one a twenty -year endowment policy.
At the end of five years how much had each paid in
premiums ?
EXCHANGE
DOMESTIC EXCHANGE
TO FIND THE VALUE OF A SIGHT DRAFT
Study 443-449 inclusive.
References 255-258 inclusive.
75. Find the value of the following drafts :
1. $2300 bought at i% discount.
2. $1400 bought at lf% premium.
3. $1740 sold at 1% premium.
4. $3000 bought at li% discount.
5. $2450 sold at li% premium.
6. $4500 bought at $1.50 premium.
7. $1240 sold at $1.25 discount.
8. $1450 bought at $.50 premium.
9. $4300 sold at |% discount.
10. $9000 sold at i% discount.
TO FIND THE VALUE OF A TIME DRAFT
76. To find the cost or selling price of a time draft :
Find the net proceeds of the draft according to the
principles of bank discount (381-382). From this
deduct the exchange discount, or to it add the exchange
premium, found as in 75.
105
106 RATIONAL ARITHMETIC
77. What is the cost of a
1. 60-day draft for $6000, |% premium, interest
at 6% ?
2. 30-day draft for $2200, i% premium, interest
at 7% ?
3. 15-day draft for $2500, f% discount, interest
at 7% ?
4. 30-day draft for $750, |% premium, interest
at 5% ?
5. 30-day draft for $5000 at |% discount, interest
at 6% ?
6. 90-day draft for $2300 at ^% premium, interest
at 4i% ?
7. 60-day draft for $2500 at |% discount, interest
at 4% ?
8. 30-day draft for $2350 at f% premium, interest
at 4i% ?
9. 60-day draft for $1240 at $1.25 premium,
interest at 5% .^
10. 30-day draft for $2350 at $1.50 discount, interest
at 6% ?
TO FIND THE FACE OF A DRAFT
78. Find the value" of a draft of $1 as explained in
75 and 76, and divide the given value by this.
79. What is the face of a sight draft which can be
bought for
1. $1207.50 if exchange is at f% premium ?
2. $1091.75 if exchange is at f% discount .^^
RATIONAL ARITHMETIC 107
3. $2453.28 if exchange is at i% premium?
4. $1636.02 if exchange is at f% discount?
5. $4234.62 if exchange is at i% discount?
80. What is the face vakie of a 30-day draft which
can be bought for
1. $1183.50 at i% discount, interest 6% ?
2. $1453.84 at i% premium, interest 5% ?
3. $2493.62 at $1.20 premium, interest 4i% ?
4. $3977.33 at $1.50 discount, interest 5%?
5. $2843.55 at f% premium, interest 6% ?
FOREIGN EXCHANGE
Study 450-452 inclusive.
Reference 468.
81. Find the exchange value of a bill for '
1. £540 at 4.83i.
2. £1476 at 4.85f.
3. £250 9s 8d at 4.85^.
4. £783 13s lid at 4.841.
5. 15,642 francs at 5.18|.
6. 8575.75 francs at 5.19.
7. 8462.73 francs at 5.20^.
8. 2648.55 francs at 5.19f.
9. 1284 marks at 94^.
10. 2556 marks at 95i.
11. 6742 marks at 94f.
12. 1287.5 marks at 94^.
108 RATIONAL ARITHMETIC
13. 789.7 guilders at 40i.
14. 2345 guilders at 40^.
15. 1286 guilders at 401.
16. 1286.5 guilders at 39J.
82. What is the face of an English bill of exchange
that cost
1. $2213.88 at 4.85i.^ 3. $585.37 at 4.84f,P
2. $6060.95 at 4.84^.^ 4. $1209.38 at 4.83f.?
83. What is the face of a French bill of exchange
that cost
5. $819.29 at 5\19f.? 7. $225.44 at 5.19 ?
6. $2316.04 at 5. 18i.? 8. $88.42 at 5.201.^
84. What is the face of a German bill of exchange
that cost
9. $348.60 at 95f? 11. $393.46 at 95i?
10. $2945.31 at 94i? 12. $13795.19 at 94|.^
85. What is the face of a Dutch bill of exchange
that cost
13. $226.10 at 40f? 15. $235.52 at 40i?
14. $458.97 at 39i ? 16. $6415.60 at 40i?
STOCKS AND BONDS
The general principles of percentage are involved in solving the
following problems (242-263).
Studv 453-464 inclusive.
86. 1. A railroad with a capital stock of $2,500,000
declared a dividend at the rate of 5%. What was the
total amount of the dividend.^ How much did A,
the owner of 350 shares, receive ?
2. What will be the total dividend at 5%, declared
by a $3,000,000 corporation ?
3. A manufacturing corporation with a capital of
$50,000 levies an assessment of 8% upon its stock-
holders. What is the total assessment, and what will
B be called upon to pay, if he holds 230 shares of $100
each ?
4. What dividend would I receive on 163 shares
of $100 stock at the rate of 5% ?
5. A corporation with a capital stock of $475,500,
divided $38,040 among its stockholders. What was
the rate of this dividend ?
6. A corporation of which I am a stockholder
declares a dividend of 4^%. My dividend check is
$652.50. How many shares, par value of $100, do I
own?
109
110 RATIONAL ARITHMETIC
7. A mining corporation of which I am a stock-
holder declares a dividend of 10%. I receive $125 as
my dividend. How many shares of $10 par value do
I own ?
8. A corporation with $1,500,000 capital had a
gross income of $975,000. Its total expenses were
$785,000. Its directors set $100,000 aside as a reserve
fund ; the rest was divided among the stockholders.
What per cent dividend was declared ?
9. What is the market value of 130 shares, par value
$100, of Q., O. & K. C. R. R. quoted at 113^.^
10. What is the market value of 640 shares, par
value of $10 each, of New England Manufacturing
Company, quoted at 85^ ?
11. How much must I pay for 75 shares ($100 par
value) B. & M. R. R. at 49i, brokerage i%?
12. What is the cost of 130 shares ($100 par value)
Bell Telephone at 418i, brokerage i% ?
13. Find the total cost of $1000 L. & E. R. R. 2d 4's
at 1021; $4000 C. & N. W. 5's at 102f ; 40 shares
($100 par value) of A. T. & S. F. R. R. at 43f ; 75
shares ($100 par value) B. & M. R. R. at 34f ; brokerage
on all i% ?
14. What is the net cost of 150 shares ($100 par
value) of B. & A. R. R. at 134f ; 75 shares ($100 par
value) M. C. R. R. at 104f ; $5000 S. E. L. Co. 6's at
95i ; brokerage on all i% ?
15. What is the proceeds of 450 shares ($50 par
value) sold at 102f , brokerage i% ?
RATIONAL ARITHMETIC 111
16. How much must I invest in U. S. 4's of 1932
to secure a quarterly income of $450, bonds selling at
108i, brokerage i% ?
17. I invested $3376.25 through my broker at i%
commission, in U. S. 4% bonds at 115f. What will be
my annual income ?
18. How much must I invest in U. S. 4's of 1935 to
secure a quarterly income of $600, bonds selling at
108f , brokerage i% ?
19. Sold 75 shares ($100 par value) railroad stock
through a broker and received $7388 net proceeds.
At what quotation did the broker sell the stock?
20. At what price may 6% stock be bought to re-
ceive 5% on the investment, brokerage i%?
21. What price can I afford to pay for 7% bonds in
order to realize 8% income on the investment, broker-
age i% ?
22. What price will I pay for 5% bonds bought
through a broker so as to bring in a net income of 4%
on the investment ?
23. At what quotation could 8% preferred stock be
bought through a broker to realize 5% income on the
investment ?
24. What price can I afford to pay for 7% bonds
bought through a broker so as to receive a net income of
6% on the investment .^^
25. What per cent income on the investment will
be realized if 4% stock is bought at 79|, brokerage i% ?
26. What per cent is realized on the investment if
6% stock is bought at 74 1, brokerage i% ?
112 RATIONAL ARITHMETIC
27. 5% bonds bought at 124 J would bring what per
cent on the investment, brokerage i% ?
28. Stock bought at 79|, brokerage i%, yields 4%
on the investment. What is the rate of dividend .^
29. Which is the better investment and how much :
stock paying 6% dividend, bought at 74|, or stock
paying 9% dividend, bought at 119J, brokerage i%?
30. I bought, through a broker, 52 shares of stock
at 84. I paid an assessment of 5% and then sold them
at 99f . How much did I gain, brokerage i% ?
31. I have $8000 to invest. I am offered bank
stock at 375 yielding 3^% each three months, or stock
in a shoe manufacturing company at 150 paying 4%
semi-annually. I have made up my mind to invest
in the stock which will give me the greater dividend.
Which shall I buy and what will be the total dividend
each year ?
32. By investing $18,750 in 150 shares of stock I am
able to realize 4% on the investment. What rate
of dividend does the stock pay ?
33. What can I afford to pay for 8% stock to realize
5% on the investment ?
34. I have been offered a block of 4v Libertv Bonds
at 98. What per cent would they yield on the invest-
ment ?
PART TWO
RATIONAL ARITHMETIC
PART TWO
87. Arithmetic is the measure of values or quantities
expressed in figures.
All arithmetic consists of increasing or decreasing
values or quantities.
88. Addition is the simple or basic operation of
increasing values or quantities.
The sign of addition is +, read plus.
89. Subtraction is the simple or basic operation of
decreasing values or quantities.
The sign of subtraction is — , read minus.
90. Multiplication is a short method of addition by
which quantities or values are increased at a fixed
ratio — by a given number.
The sign of multiplication is X, read times.
91. Division is a short method of subtraction by
which a certain quantity or value is reduced at a fixed
ratio — by a given number.
The sign of division is -^ , read divided by.
Inasmuch as these operations are quite different in
their applications, they are treated separately, and are
known as the four fundamental operations of arithmetic.
1
2 RATIONAL ARITHMETIC
NOTATION
92. For a thorough understanding of arithmetic it
is necessary to be famihar with the system of notation
used in expressing values and quantities in figures.
Ten characters (figures) are used, nine of which
have a positive or integral value. These are repre-
sented by the figures 12345678 9. The tenth
figure, (read cipher^ zero, or naught), represents nothing
and has no integral value. In other words, the figure 3
stands for three individual units ; 5, for five individual
units ; 7, for seven ; 9, for nine ; while the cipher is
used to visualize nothing.
These nine integral units, with their accompanying
cipher, are given a distinct value according to their
position in relation to a fixed line represented by the
decimal point. Thus, one in the units column — the
first column to the left of the line — means one whole
unit.
Move this 1 to the next column, one place to the
left ; fill in the space from which it has been taken
with a cipher to show that nothing is there; it then
represents the value of " ten " and is so read.
Move it one more column to the left and it repre-
sents ten times ten, or one hundred, and so on.
Writing the 1 in the second, or tens column, and the
figure 3 in the first, or units column, we show 10 units
(in the tens column) and 3 units (in the units column),
that is, 13, read thirteen.
93. It will readily be seen that the removal of a figure
one column to the left multiplies its value by ten. It will
RATIONAL ARITHMETIC
also be apparent that bringing it back one place to the
right divides its value by ten.
This is the governing principle of notation and may be appHed
on either side of the decimal hne. The figure 1 starting to the left
of the line, in the units column, and moving one place to the right
becomes yq^ of 1 ; moved another place to the right it becomes -^o oi
moved another place to the right it becomes j^
1
10'
w
hich
IS
1
100 '
of
or
and so on without limit.
10 0' "^ 10 00'
If a student has difficulty in learning the value of a figure to the
right of a decimal point (which is merely the line of division between
whole numbers and their fractional parts represented by tenths), it
is suggested that he take an ordmary sheet of writing paper, turn it
sidewise, write the names of the various places in the columns thus
formed, draw a heavy line to represent the decimal line, write the
decimal notation to the right of this line, and then place figures in
such columns as may appeal to him, calling them by the names of
the values written in the columns.
m
a
.2
'C
o
-a
•a
6
on
a
.2
«
3
en
C
4
C
i
-a
3
8
tn
i
a;
H
9
5
tn
a
CD
o
H
T3
OJ
f-
'^
c
3
6
d
!D
O
d
a
3
«3
T3
d
03
3
O
H
1
2
■c
d
3
K
1
1
4
d
1
1
1
8
c
<
'd
1
3
3
3
D
d
H
1
c
r
i
CO,
.d
-o
1
1
Ire
nil
lur
tn
d
03
tc
3
2
H
1
dt
ioi
idr
73
-d
-tj
-a
a
03
CO
3
O
-d
d
«
H
hir
a,
ed
m
d
o3
CO
3
O
-c
o
d
3
ty-
six
eij
73
d
fo
h
73
d
d
<u
H
iirl
un(
y-t
n
-d
-u
d
3
t^
d
3
hH
M-l
Dill
ire
hr
1
Read one
Read ten
Read thirteen
Read one hundred
Read one thousand, one
hundred thirteen
Read one-tenth
Read one one-hundredth
Read one one-thousandth
Read one trillion, six hun-
ion, eight hundred ninety-five
d thirty-two thousand, four
ee.
4 RATIONAL ARITHMETIC
Exercises of this kind are very valuable for students whose minds
are so constituted that they have difficulty in reading decimals.
Difficulty in reading decimals should not be ascribed to arithmetical
weakness, but rather to inability to use the imagination in repre-
senting figure pictures.
94. Figures to the left of the decimal line represent
whole numbers and are called Integers.
95. Figures to the right of the decimal line represent
parts (tenths, hundredths, etc.), and are called Deci-
mals , '
COMMON PROCESSES
ADDITION — INTEGERS
96. Addition is the process of combining several
numbers into one quantity that shall equal the value
of all.
(a) Only numbers representing like values or like quantities, or
parts of like values or like quantities, can be added, thus, 3 cows
can be added to 2 cows and the result will be 7 cows. .5 horses
added to 5 cows would produce 10 things, but they would be
neither horses nor cows — just 10 animals.
(b) Figures not used to measure value or quantity can be added
as a mere matter of counting. Furthermore, unlike things may be
added, if first reduced to common terms. The term "animal" is
common to both horses and cows.
97. The name Addend is applied to any of the in-
dividual quantities or values that are added.
98. The name Sum is applied to the total value of
the addends. It is the result of addition.
ILLUSTRATED SOLUTION
99. Problem : $324.23 + $89.96 + $742.05 + $23 +
$1.95 + $796.45= ?
RATIONAL ARITHMETIC
$324
23
89
96
742
05
23
1
95
796
45
$1977
64
the next, or
The sum of
total of the
both figures
Arrange the addends in a column, placing decimal
point under decimal point so as to form the decimal
line. Add either up or down. Beginning with
cents (first right-hand column) reading up and
combining values as we go, we have 5, 10, 15, 21,
24 cents. This is 2 tens-cents and 4 units-cents.
Write the 4 units-cents under the first column.
Carry 2 tens-cents to the next column. Adding
as before the result is 26. Write 6, carry 2. Adding
unit-dollars column, we have 27. Write the 7, carry 2.
the next column is 27. Write the 7, carry 2. The
next column is 19. As this is the last column write
100. To Check the Work: Add each column sepa-
rately, beginning with the first right-hand column.
Write the results as partial sums, one under the
other, each one place to the left of its predeces-
sor, thus :
First column adds
24
Second column adds
2
4
Third column adds
25
Fourth column adds
25
Fifth column adds
17
1977.64
or we may begin on the left and work in the opposite
direction, adding, either up or down, thus :
First column adds
17
Second column adds
25
Third column adds
25
Fourth column adds
2
4
Fifth column adds
24
1977.64
RATIONAL ARITHMETIC 7
101. In billing it is sometimes desirable to add
quantities written in a horizontal line, thus :
24 + 112+36+43 + '246 + 95 = 556
The secret of accurately adding in this way is to
add from left to right — because we read from left to
right and the eye naturally " picks up " the proper
quantities in traveling this way, and does not become
confused, as is often the case if we try to add from
right to left.
Thus, in the above, beginning with units in the first left-hand
quantity, we add with the following results, 4, 6, 12, 15, 21, 2G :
write 6 as units, carry 2. In the tens from left to right we add 2,
4, 5, 8, 12, 16, 25 : write the 5 as tens, carry 2. In the hundreds
from left to right we add 2, 3, 5 : write the 5 as hundreds, giving
55Q as the total.
Note. For practice problems in addition see pars. 1 to 8 inclusive.
SUBTRACTION — INTEGERS
102. Subtraction is the process of decreasing one
value, or quantity, by taking from it a smaller value,
or quantity.
103. The Minuend is the larger number ; the one
from which another number is taken.
104. The Subtrahend is the number that is de-
ducted from the minuend ; it is the smaller number
which is taken out of the larger.
105. The Difference, or Remainder, is the number
that is left after the subtrahend has been taken from
the minuend.
8 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
106. Prohlem: $2294.18-$346.23= ?
1 8 3 Write the subtrahend under the minuend so that
$2294 18 the decimal points will form the decimal line. Sub-
346 23 tract: 8 — 3 = 5. Write 5 in the proper column.
1^1947 95 Since 2 cannot be taken from 1 we must bring over 1
from the next column to the left. We know that one
in the third column is 10 of the second column (par. 93) ; hence we
now have 11 in this column. 11—2 = 9. Write 9. In the third
column we now have 3 — 6 which cannot be performed. Then take
1 from 9 in the next column, which would give us 13 — 6 = 7. Write
the 7. In the fourth column we now have 8—4=4. Write
the 4. In the fifth column we now have 2 — 3 which cannot be per-
formed. Take one from the next column which gives us 12 — 3 = 9.
Write the 9. In the last column we have 1—0 = 1. Write the 1.
107. To Check the Work: Add the subtrahend and
the remainder.
The result should be the minuend, thus :
$1947.95 + $346.23 = $2294.18
108. Another Method of Subtraction. From the
work used in checking, another method for finding
the difference between two quantities is derived. By
it we simply write the figiires that must be added to
the subtrahend to make it equal the minuend, thus :
109. Pro6/(?m; $2294.18— $346.23= ? iVns. $1947.95
Start with the subtrahend, |346.23. To the first right-hand
figure, 3, add 5 to make 8, the right-hand figure of the minuend.
Write 5 in the difference. To the second figure, 2, add 9, which
makes 11; write 9 in the difference.. : Change the third figure, 6,
to 7 by addmg the 1 carried from 11. To the newjthirfl figure, 7^
RATIONAL ARITHMETIC
9
add 7 to make 14. Write 7 in the difference. Change the fourth
figure, 4, to 5, by adding the 1 carried from 14. To the new fourth
figure, o, add 4 to make 9. Write 4 in the difference. To the last
figure, 3, add 19 to make 22. Write 19, making 1947.95 in all.
This process is especially valuable in balancing
accounts, thus :
Dr. cash Cr.
519
25
125
40
136
123
32
798.77
143
52
295.22
46
50
Balance
503
55
798
77
798
77
In the above, add each side of the account separately, setting
results in small figures, on proper side. Then add to the smaller
sum the figures required to make it equal the larger, inserting these
figures, as Balance. Then add both sides as a check.
Note. For practice problems in subtraction see par. 9.
MULTIPLICATION — INTEGERS
110. Multiplication is the process of increasing a
given value or quantity a given number of times.
111. Multiplicand is the name applied to the value
or quantity that is increased.
112. Multiplier is the name applied to the number
representing the number of times the multiplicand is
increased.
113. Product is the name applied to the final result
of increasing the multiplicand the number of times
indicated by the multiplier.
10 RATIONAL ARITHMETIC
114. Factor is a name applied to either the multi-
plicand or multiplier, because both are factors (or
makers) of the product.
ILLUSTRATED SOLUTIONS
115. Problem : What is the value of 6 times $4682 .'^
Begin with the units' column, 6X2 = 12. 12 is 2
tJ54Do/i units and 1 ten. Write the 2 units in the units' cohimn,
6 carry 1 ten. 6X8 = 48+1 (carried) =49 tens, which
$28092 is 9 tens and 4 hundreds. Write the 9 in the tens'
column, carry 4. 6X6 = 36+4 = 40. Write 0, carry 4.
6X4 = 24+4 = 28. As this is the last column, write 28.
116. To Check the Work: Multiply each figure
separately beginning with the left column. Write the
products under one another for addition, but remov-
ing each one place to the right, thus :
6X4 =
24
6X6 =
36
6X8 =
48
6X2 =
12
28092
117. Problem: 239X$8461=.?
$8461 First multiply 8461 by 9 units as explained above.
239 The result is 76149. Write this as units. Then mul-
rv/^l j^q tiply by 3 tens, which is 25383 tens. Then multiply
by 2 hundreds, which is 16922 hundreds. These are
called partial products. Write the partial products
each one place to the left of the previous one, and
25383
16922
$2022179 add. The result is the total product.
RATIONAL ARITHMETIC 11
118. To Check the Work: Let the multiplier and
multiplicand change places and proceed as before,
thus :
239
8461
239
1434
956
1912
2022179
DIVISION — INTEGERS
119. Division is the process of decreasing a given
value or quantity a given number of times.
120. Dividend is the name applied to the value or
quantity that is decreased or divided.
121. Divisor is the name applied to the number by
which the dividend is decreased.
122. Quotient is the number of times the divisor is
contained in the dividend ; the result of division.
123. Division is the reverse of multiplication. Hav-
ing the product and either of the factors given, the
other factor may be found by dividing the product
by the given factor. Division may, therefore, be
used as a method of checking multiplication, and vice
versa.
12 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
124. Problem: 1736^7=?
248
7 is contained in 17 twice, with 3 remaining. Write
the figure 2 over (or under) the 7 as the first figure of the
7)1736 quotient. The 3 hundreds left over, combined with the
Or, next figure, 3, makes 33 tens. 7 is contained in 33 four
7H736 times, and 5 are left. Write the quotient figure 4, plac-
^^7^ ing it over (or under) the 3 of the dividend. The
5 tens left from this operation, combined with the
6 units, makes 56 units. 7 is contained 8 times in 56. Place
the 8 over (or under) the 6 of the dividend and the final quotient is
complete, 248.
125. To Check the Work: Multiply the quotient
by the divisor. The result should be the dividend.
Thus, 248X7 = 1736.
126. Problem : 248963 ^ 139 = ?
Write the problem in proper form for di-
17Q1JJL —
^^^^139 vision, thus, 139)248963.
139)248963 139 is contained in 248 once. Write 1 over
139 the 8 to show this. 139X1 = 139. 139 sub-
1099 tracted from 248 leaves 109. To this annex
the next figure in the dividend, 9. 139 is
contained in 1099 seven times. Write 7 over
the 9. 7 X 139 = 973. 1099 - 973 = 126.
973
1266
^^^^ Bring down 6 and proceed as before, so con-
153 tinning until all the figures of the dividend
139 have been used. After the last figure is used,
I j^ the remainder, if any, should be written above
a line with the divisor below, thus, j^^-
RATIONAL ARITHMETIC 13
127. To Check the Work : Multiply the whole num-
ber of the quotient, 1791, by the divisor, 139, and add
the remainder, 14, thus :
1791
139
16119
5373
1791
14
248963
DECIMALS
128. All figures to the right of the decimal line repre-
sent parts of a unit. Each removal of the figure one place
from the line, to the right, divides its value by ten.
From the Latin decern, meaning ten, we derivq the name
decimal, which is applied to all values or quantities repre-
sented by figures written to the right of the decimal line.
From the fact that figures representing decimal values are written
in exactly the same way as to represent integral vakies, except that
they appear to the right of the decimal line, the actual processes of
addition, subtraction, multiplication, and division are the same as
for integers.
129. The only matter requiring special attention is
the decimal line or decimal point.
ADDITION AND SUBTRACTION — DECIMALS
130. Decimals are added and subtracted in exactly
the same manner as integers. The essential thing is
to place the decimal points under one another so as to
form the decimal line.
14 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
131. Problem: 248.15 + .43642 + 12.05 + 124.3096-f
85.03752 + 100.075 + 2.12465= ?
248
15
43642
12
05
124
3096
85
03752
100
075
2
12465
572
18319
After arranging the figures so as to form the
decimal Une, add as explained in par. 99.
132. Problem: 246.75-4.88625=?
946
4
241
75000
OQ^Q^ After arranging the figures so as to form the
decimal line, subtract as explained in par. 106.
86375
MULTIPLICATION — DECIMALS
133. The actual work of multiplying decimals is
exactly the same as for multiplying integers. It is
necessary, however, to be able to locate the decimal
line, or decimal point, in the product with unfailing
accuracy.
One tenth (.1) multiplied by five tenths (.5) equals five hun-
dredths (.05). Then one decimal place multiplied by one decimal
place produces tivo decimal places in the product.
Five hundredths (.05) multiplied by three tenths (.3) equals
fifteen thousandths (.015). That is, two decimal places multiplied
by one decimal place produces three decimal places in the product.
From this we see that the product contains as many decimal
places as the multiplicand and multiplier together contain.
RATIONAL ARITHMETIC 15
134. To Locate the Decimal Point in the Product :
Count the number of decimal figures in both factors
together. Place the decimal line, or point, that num-
ber of places from the right-hand end of the product.
ILLUSTRATED SOLUTION
135. Problem : Multiply .875 by .63.
.875
Multiply as explained in par. 117, regardless of the
point.
•^^ Since there are three decimals in one factor and two
'^Q^5 decimals in the other, counted together there will be five
5250 decimals in the product. Count five figures beginning
551*^5 with the right figure of the product. Place the decimal
line, or point, to the left of the fifth figure, thus, .55125
Note. For practice problems in the multiplication of decimals see
par. 10.
DIVISION — DECIMALS
136. The actual work of division of decimals is the
same as for division of integers, but it is necessary to
be able to place the decimal point in the quotient with
absolute accuracy.
Since the process of division is the direct opposite of that of
multiplication ; since the dividend of division is the product of
multiplication ; since the quotient of division is one of the factors
of multiplication ; and since the divisor is the other factor ; if we
take the number of decimals in the divisor (one factor) from the
number of decimals in the dividend (the product) it will give us the
number of decimals in the quotient (the other factor).
With this knowledge it is easy to understand the following rule,
which should be memorized and carefullv followed under all circum-
stances.
16 RATIONAL ARITHMETIC
137. To Locate the Decimal Point in the Quotient :
Imagine the divisor placed upon the dividend so that
the decimal line of the divisor covers the decimal line
of the dividend. Count the number of decimal places
" covered " in the dividend. Place the new decimal
line at this point, extending it up into the quotient.
Proceed as in ordinary division.
138. To he absolutely sure that the decimal line, or
point, is in the right place, it should be placed in the
quotient before any figures are written there.
ILLUSTRATED SOLUTIONS
139. Problem: 12.435^1.5=.^
Apply the above rule (137) : Imagine 1.5 placed
upon 12.435 in the expression 1.5)12.435 so that
decimal point covers decimal point. One figure,
4, in the dividend is covered.
Place the decimal line between 4 and 3 ex-
8
29
.5)12.4
35
12
43
30
135
1
35
tending it into quotient, thus 1.5)12.4
proceed as for integers (par. 126).
Ans. 8.29.
35 Now
140. Problem: 12.875 -M4 = .^
|9|9_9_ Apply the same rule: Imagine 14. placed
1 4
14 ')12!875 ^^ 12.875 in the expression 14.) 12.875 so
12 6 that decimal point covers decimal point. No
decimal figures in the dividend are covered.
Therefore, the decimal line remains unchanged.
27
14
^"^^ thus, 14.) 12 875 Now proceed as for integers
M? and the result will be .91 g^-^^.
^ Ans. .919i^.
RATIONAL ARITHMETIC 17
141. Problem: 13.5 --.875=?
Apply the same rule : Imagine .875
placed on 13.5 in the expression .875)13.5 so
that decimal point covers decimal point.
Three decimal places in the dividend are
covered. Two of these places must be
visualized with O's. Therefore, the decimal
line falls between the third and fourth places,
15 428+
875)13.500 000
8 75
4 750
4 375
375
350
25 00
17 50
7 500
7 000
thus, .875)13.500
Now proceed as for
integers and the result will be 15.428+.
500 Ans. 15.428+.
In this use + means " and more."
Note. For practice problems in the division of decimals see par. 11.
FACTORING
142. Every number can be divided by 1 and by
itself without leaving a remainder.
143. Numbers that can be divided exactly only by
themselves and by 1 are Prime Numbers. 1, 2, 3, 5,
7, 11, 13, 17, 19, etc., are prime numbers.
144. Numbers that can be divided exactly by num-
bers other than themselves and 1 are Composite
Numbers.
145. Composite numbers are made up of two or
more prime numbers multiplied together. These are
called Prime Factors. It is sometimes necessary to
find what prime factors go to make up a number.
18 RATIONAL ARITHMETIC
146. Problem : What are the prime factors of 420 ?
2)420
ILLUSTRATED SOLUTION
The smallest prime number contained in 4'20
is 2. It is contained 210 times. The smallest
^J^IO prime number contained in 210 is 2. It is con-
3)105 tained 105 times. The smallest prime number
5) 35 contained in 105 is 3. It is contained 35 times.
ry The smallest prime number contained in 35 is
5. It is contained 7 times. 7 is itself prime.
Therefore, the prime factors of 420 are 2, 2, 3, 5, 7.
Ans. 2, 2, 3, 5, 7.
147. To Check the Work : Multiply the prime factors.
The result will be the original quantity, thus, 2X2X3X
5X7 = 420.
LEAST COMMON MULTIPLE
148. A Common Multiple of several numbers is a
quantity that contains all of them.
(a) Thus : 48 is a common multiple of 6, 8, 4, 3, and 2, because
it contains all of them.
(b) The product of the several numbers is always a common
multiple of them. That is, 8X5X4 (160) is a common multiple of
8, 5, and 4.
149. The Least Common Multiple of several numbers
is the least quantity that contains all of them.
{a) Thus : 24 is the least common multiple of 6, 8, 4, 3, and 2,
because it is the least number that contains all of them.
(b) The product of the prime factors of several numbers is the
least common multiple of those numbers.
2)24
15
36
9
2)12
15
18
9
3) 6
15
9
9
3) 2
5
3
3
RATIONAL ARITHMETIC 19
150. To Find the Least Common Multiple of Several
Numbers : Find the prime factors of the numbers.
Multiply these factors together. The result will be
the least common multiple of the original numbers.
ILLUSTRATED SOLUTION
151. Problem : Find the least common multiple of
24, 15, 36, and 9.
Write the numbers in line. 2 is
the smallest prime factor contained
in anv two or more. Divide bv 2
where possible, bringing down num-
bers of which 2 is not a factor.
^^11 The result is 12, 15, 18, 9. 2 is a
2X2X3X3X2X5 = 360. factor of 12 and 18. Proceed as
before. The result is 6, 15,. 9, 9.
3 is a prime factor of 15 and 9. Proceed as before. The result of
this is 2, 5, 3, 3. 3 is a factor of 3 and 3. Proceed as before.
The result is 2, 5, 1, 1, each of which is a prime number. The
prime factors then are 2, 2, 3, 3, 2, 5. Multiply these and we ob-
tain 360, which is the least common multiple.
152. To Check the Work : Divide the least common
multiple by each of the original numbers, thus :
360-=-24 = 15 360^15 = 24
360-r-36 = 10 360^ 9 = 40
GREATEST COMMON DIVISOR
153. A Common Divisor of two or more numbers is
any number that will exactly divide each of them.
20 RATIONAL ARITHMETIC
154. The Greatest Common Divisor is the greatest
number that will exactly divide each of them.
7 is a common divdsor of 21, 35, and 14. It will divide each of
them exactly. Some numbers have no common divisor. No num-
ber will divide both 9 and 8 exactly. Such numbers are said to be
prime to each other.
155. To Find the Greatest Common Divisor : Divide
the greater number by the lesser. Then divide the
first divisor by the remainder, and so continue until
the division is exact, or there is a remainder of one.
If exact, the last divisor is the greatest connnon divisor.
If the remainder is one, there is no common divisor.
The numbers are prime to each other.
ILLUSTRATED SOLUTIONS
156. Problem : What is the greatest common divisor
of 351 and 459?
1
351)459
351
3
108)351
324
4
27)108
108
Divide 459 by 351 (par. 126). The re-
mainder is 108. Divide 351 by 108 and the
remainder is 27. Divide 108 by 27. There
is no remainder. Therefore, 27 is the
G. C. D. of 459 and 351.
G. CD. = 27.
157. To Check the Work :
459-^27 = 17 351^27 = 13
RATIONAL ARITHMETIC 21
158. Problem : Find the G. C. D. of 209, 247, and
456.
1
209)247
209 5
38)209
190 2
19)38
38
24
Find the G. C. D. of 209 and 247, as in
the previous problem. Result, 19.
19)456
^^ Find the G. C. D. of 19 and 456, which is
76 19. Therefore, 19 is the G. C. D. of all.
76
G. C. D. = 19.
159. Problem : Find the G. C. D. of 943 and 35.
26
35)943
70
243
210
1
33)35
33 16
2)33
2
13
12
1
No G.
CD.
Proceeding as above, we derive a final re-
mainder of 1. Therefore, the numbers are
prime to each other.
FRACTIONS
160. A Fraction is a part of an integer.
(a) Any quantity or value may be divided into any number of
equal parts. One or more of these equal parts constitutes a fraction.
(b) If a quantity be divided into four equal parts, each of them
will be known as one-fourth {^) and three of them would be three-
fourths (f ). If the whole were divided into a thousand equal parts,
each part would be called one one-thousandth (toVo") ^^^ ^^^.V num-
ber of these parts could be made to form the fractional value, as
748 thousandths (1^*7^).
161. A fraction is composed of two terms. One
term shows the number of parts into which the original
integer was divided. The other term shows the num-
ber of these equal parts taken to make the fraction.
162. Denominator is the name applied to the term
showing into hoiv many parts the integer was divided.
163. Numerator is the name applied to the term
showing the number of these equal parts used in the
fraction.
164. There are two methods of fractional notation,
common fractions and decimal fractions.
165. A Common Fraction is expressed by writing
the numerator above the denominator with a line
between, thus, f .
22
RATIONAL ARITHMETIC 23
166. A Decimal Fraction is any fraction whose
denominator is 10 or any multiple of 10.
It is not necessary to write the denominators of decimal fractions.
All that is required is to write the numerator and place it in the
proper decimal column to show what denominator is intended, thus^
six-tenths (y^) may be written .G. Six-thousandths may be ex-
pressed by placing the figure six in the proper decimal column, thus,
.006, and so on.
167. Every fractional value may be expressed either
as a common fraction or as a decimal fraction.
In some cases it is easier to use the common fraction form ;
while in others it is easier to use the decimal form expressing the
same value.
168. A Proper Fraction is a common fraction whose
numerator is smaller than its denominator. Its value
is less than one.
169. An Improper Fraction is a common fraction
whose numerator is larger than its denominator. Its
value is more than one.
170. A Mixed Number is an integral quantity and
a fraction taken together, as, 1^.
171. In its decimal form this quantity would be
called, a Mixed Decimal and would be written 1.25.
172. It is not always possible to reduce a fractional
value to an exact decimal. In such cases it is neces-
sary to retain a common fraction as part of a decimal,
thus, .16f, read sixteen and two-thirds hundredths.
173. A Complex Decimal is a decimal whose right-
hand place is a common fraction.
24 RATIONAL ARITHMETIC^
J
174. A Complex Fraction is a common fraction, the
right-hand place of whose numerator is itself a fraction.
thus,
161
100
175. Every common fraction represents an unper-
formed division.
f is the result of performing the operation 8^9. It may be read
" eight divided by 9 " or it may be read " eight-ninths." |- and also
2-i-3 may each be read "2 divided by 3."
176. Reduction of Fractions is the process of chang-
ing the form of a fraction without changing its value.
177. Reduction of fractions comprises :
Changing to lower terms.
Changing to higher terms.
Changing improper fractions to mixed numbers.
Changing mixed numbers to improper fractions.
Changing decimals to common fractions.
Changing common fractions to decimals.
CHANGE TO LOWER TERMS
178. Common fractions are in their lowest terms
when the numerator and denominator are prime to
each other ; that is, when they have no common
factors, no common divisor.
179. Decimal fractions, when written in the form
of common fractions, may be reduced to lower terms
when their numerator and denominator are not prime
to each other.
RATIONAL ARITHMETIC
15
The decimal fraction, four-tenths, if written in its common frac-
tion form ^, shows at once that its real or lowest value is f because
2 is a common factor of both 4 and 10. In other words, if an integer
is divided into ten parts and four of these parts are taken, the value
of the fraction would be exactly the same as if the integer had been
divided into five parts and two of these parts taken.
180. To Change to Loiver Terms : Strike out all
factors common to both the numerator and the de-
nominator; or, divide both the numerator and the
denominator by their G. C. D.
ILLUSTRATED SOLUTIONS
181. Problem : Reduce 414 to lowest terms.
4
^^
5
Ans.
Or,
5'
Seven is a factor of both 420 and o'l5. Dividing
the numerator, 420, by 7 gives a new numerator of
60. Dividing the denominator, 52o, by 7 gives a
new denominator of 75. Five is a common factor
of both 60 and 75. Dividing 60 by 5 gives a new
numerator of 12. Dividing 75 by 5 gives a new
denominator of 15. Three is a common factor of
12 and 15. Divide 12 by 3 and we have a new
numerator of 4. Dividing 15 by 3 gives a new
denominator of 5. No factor is common to 4 and 5.
Therefore, 4 is the lowest equivalent of ^^
420)525
420
105)525
525
4
105)420^4
105)525 5
Find the G. C. D. of 420 and 525 (pars. 155
and 156). The G. C. D. is 105. 420^-105 = 4.
Therefore, the new numerator is 4. 525 -M05 = 5.
Therefore, the new denominator is 5, making the
fraction ^.
26 RATIONAL ARITHMETIC
1-
182. Problem : Chanaje -^ to lowest terms.
^ 9
1^
n_
14
8
40
7
M_
7
4rar
20
20
v4n5.
7
2 n
This is a complex fraction. The first step is to
J^ =z ^ change the numerator and the denominator to the
9 27 same kind of parts, thirds. One unit = f ; then
Ans. 2T' lf~f ' ^ units = -^3^. Then the numerator equals
5 thirds and the denominator equals 27 thirds ;
■5 and 27 being prime to each other, -^^ is in its lowest terms.
2-
183. Problem : Chanaje — to lowest terms.
^ 8
Proceeding as before, we find that the complex
fraction is equal to ^. Both terms may be divided
by 2, producing -^q' ; 7 and 20 being prime to each
other, -^Q is in its lowest terms.
Note. For practice problems in reducing fractions to lowest terms
see par. 12.
CHANGING TO HIGHER TERMS
184. In handling common fractions in business
problems, it is sometimes necessary to change a com-
mon fraction from its lowest terms to an equivalent
common fraction of some higher denomination.
To do this it is necessary to introduce such factors
into both the denominator and numerator as will
change the value of the denominator from the given
figure to that required.
185. To Change a Common Fraction to a Required
Denominator: Divide the required denominator by
RATIONAL ARITHMETIC 27
the given denominator. Multiply both the numerator
and the denominator of the given fraction by the
quotient thus obtained.
ILLUSTRATED SOLUTION
186. Problem : Change f to 525ths.
105
5^5^5 Five is contained 105 times in 5'i5; therefore,
5 must be multiplied hv 105 to produce 515.
4X105 = 420 Multiplying 4 by 105 gives 420 for the new
^VlO^ — ^9^ numerator. Multiplying 5 by 105 gives 525,
. ^20 ^^^ required denominator.
./j-ito. .525*
Note. For practice problems in changing fractions to higher terms see
par. 13.
CHANGING AN IMPROPER FRACTION TO A MIXED
NUMBER
187. The denominator shows into how many parts
the original integer has been divided. If the number
of parts taken is greater than the number into which
the integer was divided, as is the case with every im-
proper fraction, then the value of every improper
fraction must be greater than the value of the original
integer. It will be as many times the original integer
as the denominator is contained in the numerator.
If the denominator is not contained in the numerator
an even number of times, the remaining number of
parts would constitute the number of fractional units
that are left.
28 RATIONAL ARITHMETIC
188. To Change an Improper Fraction to a Mixed
Number : Divide the numerator by the denominator.
ILLUSTRATED SOLUTION
189. Problem : Change -^- to a mixed number.
3-1
9)32
27
5
Ans. 34
The denominator, 9, is contained 3f times in
the numerator, 32. Therefore, %^=3^.
9'
Note. For practice problems in changing improper fractions to mixed
numbers see par. 14.
CHANGING A MIXED NUMBER TO AN IMPROPER
FRACTION
190. The denominator of a fraction shows the num-
ber of parts into which the unit has been divided.
Every unit of a mixed number then will contain as
many of these parts as are expressed by the denomina-
tor. The fractional value of the whole then may be
found by multiplying the integral number by the
denominator of the fraction and adding the numerator
of the fraction to this result.
191. To Change a Mixed Number to an Improper
Fraction: Multiply the integer by the denominator
of the fraction. To the result add the numerator of
the fraction. The total should be written as the
numerator of the new fraction. The denominator
remains unchanged.
RATIONAL ARITHMETIC 29
ILLUSTRATED SOLUTION
192. Problem : Change 3| to an improper fraction.
^y\c> = ^^ The value of three units is twenty-four eighths.
24-(-5 = 29 Twenty-four eighths pUis five eighths equals
j4_flS, ^, twenty-nine eighths. Therefore, 3f equals -^^.
Note. For practice problems in changing mixed numbers to improper
fractions see par. 15.
CHANGING A DECIMAL FRACTION TO A COMMON
FRACTION
193. To Change a Decimal Fraction to a Common
Fraction: Write the decimal in its common fraction
form and then reduce to lowest terms.
ILLUSTRATED SOLUTIONS
194. Problem: Change .125 to a common fraction.
1 Write •I'^o in its fraction form. It is apparent
■^^<^ _ t that 1*25 is a common divisor of both the numerator
iOOO" 8 and the denominator. 125 is contained once in
8 the numerator. U25 is contained eight times in the
A.ns. ^. denominator.
195. Problem: Change .16f to a common fraction.
Writing .16§ as a common fraction produces a
complex fraction. Reduce this complex fraction to
its lowest terms (par. IS'i). The numerator con-
tains fifty thirds, the denominator contains three
hundred thirds, making -^^-q or ^.
Note. For practice problems in changing decimal fractions to common
fractions see par. 16.
100 =
= 300
1
m _
_1
%m 6
6
Ans. -5-
30 RATIONAL ARITHMETIC
CHANGING A COMMON FRACTION TO A DECIMAL
FRACTION
196. Every common fraction is the statement of
an unperformed division (par. 175). The result of
performing this division is a decimal.
(a) The principle involved in changing a common fraction to a
decimal is practically the same as that for changing an improper
fraction to a mixed number.
(6) In changing from an improper fraction to a mixed number all
the work is on the integral, or left, side of the decimal line ; while in
changing from a common fraction to a decimal fraction the work is
all to the right of the decimal line.
(c) In changing from mixed numbers to mixed decimals, the
work is on both sides of the line.
197. To Change a Common Fraction to a Decimal
Fraction : Divide the numerator by the denominator.
Decimal values are seldom carried beyond the sixth decimal place.
Any fraction remaining at this point is usually disregarded, although
when absolute accuracy is desired the decimal should be carried out
until exact, or the fraction should be retained, making a complex
decimal.
ILLUSTRATED SOLUTIONS
198. Problem : Change i to a decimal fraction.
1 125 Divide the numerator by the denominator,
8)1 1 000 fi^^* placing the decimal point in the quotient,
Ans 125 ^^ shown in pars. 139, 140, 141. Three decimal
places will be used, making the result .l!25.
RATIONAL ARITHMETIC 31
199. Problem : What is the decimal value of {i ?
lftRR2 Solve this problem in the same way as the
previous one. After two decimal places have
1 fir \ 1 o f\f\f\
ID) lo uuu been used, we find that the remainder will
i-^ ^ continue to repeat itself. This shows that
1 00 the division will never be exact. No matter
90 how far carried, the decimal figure will be 6.
Tq We may, therefore, stop at any decimal place,
J ftf»fi2 retain the fraction ^, and make the result a
^* complex decimal, .866|^, .8666|^, or .86§, etc.
200. Problem : Change 2^ to its decimal form.
P Reducing the common fraction ^ as shown above,
5)4 10 the result is .8. The integer 2 remains the same
Ans. 2.8. ^^^ t^^^ result is, therefore, 2.8.
Or,
^5 — 5
2|8 Changing 2f to fifths, we have ^, which, when
5)1410 reduced according to rule, gives 2.8.
Ans. 2.8.
Note. For practice problems in changing common fractions to decimal
fractions see par. 17.
ADDITION OF FRACTIONS
201. Only numbers representing like values or like
quantities or like parts of such values and quantities
can be added (par. 96, a, b).
(a) In adding decimals all that is necessary is to carefully arrange
the numbers so that the decimal points form a decimal line. In
32 RATIONAL ARITHMETIC
this way tenths come over tenths, hundredths over hundredths, and
so on, thus making it possible to add Uke parts.
(6) In adding common fractions it is necessary to change all frac-
tions to equivalent fractions having the same denominator.
202. To Add Common Fractions : Find the L. C. M.
of all the denominators (par. 150). Use this L. C. M.
as a common denominator. Change each given frac-
tion to a fraction having this denominator (par. 185).
Add the numerators of the new fractions. The result
is the numerator of the sum. The L. C. M. of the
given denominators is the denominator of the sum.
Reduce the sum-fraction to its lowest terms.
ILLUSTRATED SOLUTIONS
203. Problem: f+f+TV+l= ?
2 )3-8-12-6
2 )3-4- 6-3
3)3_2— 3 — 3 We find the L. C. M. of the given de-
1 _Q \ —\ nominators, 3, 8, 1 "2, 6, to be 24 (pars. 150,
/-. /-» r. ./-» r»j 5k 151). Change each of the given fractions
^X^X^X^ ^^ to twenty-fourths, f = if , f = M' T2 =M
--16 andf = |f.
3 As the denominators are all the same
5 ^ and we are to add only the numerators,
8 it will save time and confusion if we simply
7 write the numerators 16, 15, 14, 20.
]^2 Added, we have fl^, which reduced to a
K mixed number (pars. 188, 189) equals 2^x«
65
24
* This L. C. M. can be determined mentally.
a,=^ii Ans. m
24
RATIONAL ARITHMETIC 33
204. Problem: 12t+9i+23f+19f = ?
2)5-2-8-6
2 )5-1-4-3
5 — 1 — 2 — 3 Arrange the mixed numbers in a
2X2X5X2X3 = 120 column. Add the fractions as ex-
-JQ4. qrj plained in the previous solution. The
^^ result will be 3 0j. = £>_6_i_
qj_ r^r\ lesLUL will ue J 20 ^12 0'
^ The Y2V ^i^^ ^^ t^^ fraction of the
238 — 45 gj^j^l gyj^^
19f — 100 Add the 2 with the given integers.
2 tI i = 2 1^ The total sum is Q5^\.
^^1 2
./xTlS . \)D -^ 2 •
205. Problem: 12.8 + 19i+14.875+5.8i= ?
12.8 Mixed numbers, mixed decimals and com-
jg ^ plex decimals can be added, but it is neces-
sary first to change the mixed numbers to
mixed decimals ; that is, 19| must be changed
to 19.5. The complex decimals must be re-
14.875
5.833^
oJ.UUo3^ duced to the same order as the longest
Ans. 53.008^. decimal; that is, 5.8^ must be changed to
5.833^. x\fter these reductions have been
made, add as explained in the previous solutions.
Note. For practice problems in addition of fractions see par. 18.
SUBTRACTION OF FRACTIONS
206. Subtraction of fractional values is performed
in the same way as subtraction of integral quantities.
As in addition, it is necessary to change the given fractions to
fractions having a common denominator.
34 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTION
207. Problem: 412i-204f = ?
The given denominators are 9 and 2. The
^ ^' common denominator is 18. ^ = ^8^5 l^~T§-
41/2^ p Ten cannot be taken from 9. Take one from
204|- 10 the units cohimn of the integers. l=Tf-
rtf\iy iT _9 -|-J_8___2 7 27_X0_17 ryyi rlifff^rpnp*^
207 T¥ 18^ I r8~T8- T¥ T8^ — T¥- ^'^^ omercnce
between the integers is 207. The total dif-
Ans. 207ii.
Note. For practice problems in subtraction of fractions see par. 19.
ference is 207^1^.
MULTIPLICATION OF FRACTIONS
208. When an integer is multiplied by an integer,
the product shows a value greater than either.
209. When an integer is multiplied by a fraction,
the product shows a value less than the integer.
(a) 5X5 = ^^5. Both factors are integral numbers and the pro-
duct, 25, shows an increase in valuCo
(6) ^X2 = f. f is a fraction and the product shows a value less
than 2.
(c) The expression ^X2 may be read ^ of 2. |^X2, and ^ of 2
represent the same arithmetical operation.
(d) The sign "X" and the word "of" are interchangeable in
fractional computation.
210. To Multiply a Fraction by a Fraction : Multiply
the numerators. The result will be the numerator of
the product. Multiply the denominators. The result
will be the denominator of the product. Reduce the
new fraction to its lowest terms.
The work may be simplified by first casting out factors that may
be common to one of the numerators and one of the denominators.
RATIONAL ARITHMETIC 35
ILLUSTRATED SOLUTIONS
211. Prohlem : Multiply f by |.
Multiply the numerators, 2X4 = 8. This is
%yi^ = ^j the numerator of the product. Multiply the de-
Ans. #y. nominators, 3X9 = 27. This is the denominator
of the product.
212. Problem: f of|ofT'o=?
- of - of — = — Multiply the numerators, 5X2X7 = 70.
o o lu /*-*u This is the numerator of the new fraction.
70 ^^ 8X3X10 = 240. This is the denominator
24^ '24 of the new fraction. 2^*0 reduced to the
J 'no JL- lowest terms equals ^V-
Or,
1 1 The solution may be simplified by cast-
^ f ^ £ 7 _ 7 ing out factors as follows : Five is con-
8 3 1(^ 24 tained in the numerator of the fraction |^
of and in the denominator of the fraction yo .
\ Casting out makes the first fraction ^ and
J >7 9 -J- the last ^. Two is a common factor of the
' numerator of f and the denominator of ^
and may be cast out of each. Then
multiply 1X1X7 = 7 for the numerator of the product. 8 X3 = 24
for the denominator of the product.
213. To Multiply an Integer by a Fraction : Multiply
the integer by the numerator and divide the product
thus obtained by the denominator of the fraction.
36 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTION
214. Problem : Multiply 324 by -.
5
324
4
5 Applying the above rule, 4 X324 = 1296.
5 )1296 * 1296-^5 = 259i.
259i
Ans. 259i.
215. To Multiply Mixed Numbers : Multiply the
fractions ; multiply the integers ; combine the results.
ILLUSTRATED SOLUTIONS
216. Problem : Multiply 246 by 23^.
246 1964
First multiply the integer by the fraction.
23- 5)984 "x.vfe^x ^j ...V. x.«.^..xwxx.
' The product of 246 by the numerator, 4, is
1^"5 984, divided by 5 gives 196f as the partial
738 product by \.
492 Multiply 246 by 3 and by 2 as explained in
5854^ P^^* ^^^' ^^^ the results. The answer is
Ans. 5854t. ^^''^^•
217. Problem : Multiply 68f by 24. .
Multiply as in the previous problem. Multi-
plying the integer 24 by the numerator, 2, of
the fraction f , gives 48. Dividing this by 3
gives 16. This is the partial product by f.
Multiply 68 by 24 (par. 117). Add, and the
1648 result is 1648.
Ans, 1648.
68f
16
24
3)48
16
272
136
RATIONAL ARITH^ylETIC 37
218. Problem : Multiply 243| by 42
2431
2k/ i — 8
3 A 5 "~ 15
42t
5)972
8
1 5
8
1941
1941
6
3)84
28
28
486
972
10428
1 4
1 5
Multiply the fraction of the
multiplicand by the fraction of
the multiplier. The product of
the numerators, 4 and 2, equals 8.
The product of the denominators,
5 and 3, equals 15. The first
partial product is j-^.
Multiply the integral part of
the multiplicand by the fraction
of the multiplier. Four times
243 equals 972 (par. 214). 972
Ans. 10428|i. divided by 5 equals 194f, which
is the second partial product.
Multiply the fractional part of the multiplicand by the integral
part of the multiplier (par. 214). 42X2 equals 84. 84 divided
by 3 equals 28, which is the third partial product.
Multiply the integer of the multiplicand by the integer of the
multiplier (par. 117). Add the partial products and the total is
10428if.
Note. For practice problems in multiplication of fractions see par. 20.
DIVISION OF FRACTIONS
219. Division of fractions, like division of integers,
is the direct opposite of multiplication,
220. When an integer is divided by an integer, the
result shows a decrease. When an integer is divided
by a fraction the result shows an increase.
Ten divided by 5 equals 2. That is, one integer
divided by another gives a decrease. Ten divided
by one-half equals 20, for in 10 units there are 20
halves.
38 RATIONAL ARITHMETIC
221. To Divide Fractions by Fractions : Multiply the
numerator of the dividend by the denominator of the
divisor. The result is the numerator of the quotient.
Multiply the denominator of the dividend by the
numerator of the divisor. The result is the denomina-
tor of the quotient. Reduce the quotient to its lowest
terms.
Or : Invert the divisor and proceed as in multi-
plication.
ILLUSTRATED SOLUTIONS
222. Problem : Divide t by f.
-|-i-f = ij=li The numerator of the dividend is 4 and the
J -ii denominator of the divisor is 3. 4X3 = 12,
^' which is the numerator of the quotient. The
denominator of the dividend is 5 and the numerator of the divisor
is 2. 2X5 = 10. The denominator of the quotient is 10. f| is
an improper fraction, reduced to a mixed number equals l^^.
Or,
t"^f = ? In the problem f ^f, the dividend is f and
tX f = i^ = li the divisor f . Inverting f gives f . 4X3 = 12.
223. To Divide Mixed Numbers : Change both the
divisor and the dividend to improper fractions having
a common denominator. Divide the numerator of the
dividend by the numerator of the divisor. The result
will be the quotient required.
RATIONAL ARITHMETIC
39
ILLUSTRATED SOLUTIONS
224. Problem : Divide $124.50 by 23f.
5 26
71)373 50
355
18 5
14 2
4 30
4 26
In tliis problem, both the dividend and
the divisor must be changed to thirds.
$124.50 should be changed to thirds by
multiplying by three, which gives 373.50
thirds.
23f = ^ (pars. 191, 192). Divide 373.50
by 71 (par. 140). The result is $5.26y*Y-
A
71
Ans. $5.26-^.
Problem : Divide 248^ by 34.
n'2 7
85
170)1244
1190
54 27
Changing the dividend, 2484, to fifths equals
1244 fifths. Changing the divisor, 34, to fifths
equals 170 fifths. Divide the numerator of the
dividend, 1244, by the numerator of the divisor,
170. The result is 7||.
170 85
225. Problem : Divide 433^ by 18f .
23
112)2601
224
361
336
25
Change the fraction in the dividend and
the fraction in the divisor to fractions having
a common denominator. 433^ = 433|^ and
18f =18|. Then 433i-M8f is the same as
433f divided by 18|^. Changing the divi-
dend to sixths, we have 2601 sixths. Chang-
ing the divisor to sixths, we have 112 sixths.
25
Ans. 23i^. 112 Dividing 2601 by 112, gives 23^
Note. For practice problems in division of fractions see par. 21.
For practice problems involving the use of fractions and decimals see
par. 22.
DENOMINATE NUMBERS
226. A Denominate Number is a number expressed
in units of weight, measure, or value.
227. A Simple Denominate Number is a quantity
expressed in a single denomination.
4 pounds, 5 bushels, 2 quarts are simple denominate numbers.
228. A Compound Denominate Number (usually
called a compound number) is a quantity expressed
in two or more different denominations.
(a) 1 year, 9 months, and 9 days ; 3 pounds, 9 shillings, and 4
pence are compound numbers.
(6) Tables of weight, measure, distance, values, etc., will be found
on pages 130 to 146.
(c) In the follow^ing illustrated solutions, English money is used
throughout for the purpose of uniformity. The solutions are equally
applicable to any or all the tables.
REDUCTION OF
DENOMINATE NUMBERS
ILLUSTRATED SOLUTION
229. To Reduce a Compound Denominate Number
io a Simple Denominate Number of Equivalent Value:
40
RATIONAL ARITHMETIC
41
Problem : Reduce £5 Ss 9d to pence.
5
100
8
108
1296
9
1305
= £
= s
= d
J.
ins. ISOBd.
Since there are 205 in £l, in £5 there are 5
times 20 or IOO5. £0 Ss equals lOSs. Since
there are 12(Z in Is, in IO85 there are 12 times
108, which is 1296J. If there are UQQd in
£5 8s, in £5 8s 9d there are lS05d.
Note. For practice problems in reducing compound denominate num-
bers to simple denominate numbers of equivalent value see par. 23.
ILLUSTRATED SOLUTION
230. To Change to Higher Denomination:
Problem : Reduce 1305 pence to higher denomina-
tion.
Since there are 12cZ in \s, in ISOofZ there
are as many shillings as 12 is contained in
1305<Z which is 108^ and 9d left. Since
there are 20^ in £1, in 108* there are as
many pounds as 20 is contained in IO85
which is £5 and 85 left. Therefore, 130oc^
equals £0 Ss 9d.
n )lS05d
20) 108^ 9d
£5 Ss
Ans. £5 Ss 9d.
Note. For practice problems in changing simple denominate numbers
to higher denominations see par. 24.
42 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTION
231. To Change to Lower Denomination:
Problem : Reduce £.575 to lower denomination.
.575£
20
11
Since there are 205 in £l, in £.575 there are
ojOyis 20 times £.575, which is 11.5^. Since there
12 are 12c/ in Is, in .5s there are 12 times .5 which
Q\Od ^ is 6d.
Ans. lis 6d.
Note. For practice problems in changing denominate numbers to lower
denominations see par. 25.
ILLUSTRATED SOLUTION
232. To Change a Compound Denominate Number
to a Simple Denominate Number:
Problem : Reduce ll6* 6c? to a decimal of a pound.
\5s
12)6|0c^ Since there are 12c? in 1^, in Qd there are as
many shillings as 12 is contained times in 6
£|575 which is .OS. \\s 6d equals 11.5*. Since there
20) 1 1 15005 are 205 in £l, in 11.5* there are as many pounds
as 20 is contained in 11.55 which is £.575.
Ans. £.575
Note. For practice problems in changing compound denominate num-
bers to simple denominate numbers see par. 26.
RATIONxVL ARITHMETIC 43
ADDITION OF COMPOUND NUMBERS
ILLUSTRATED SOLUTION
233. Problem : £125 13^ 9rf+£23 86' 7d-{-£lS 5d= ?
Arrange the compound numbers so
that similar denominations fall in the
same column. Adding the column
representing the lowest denomination,
we have 21c?, ^Id equals Is and 9^.
Write the 9d under the proper column.
Carry Is to the column of shillings.
Adding we obtain 22.? which equals £l
and 25, Write 25, carry £l to the pound column. Adding, we
have £162.
£
s
d
125
13
9
23
8
7
13
5
162
2
9
Ans. £162 2^ 9d.
SUBTRACTION OF COMPOUND NUMBERS
ILLUSTRATED SOLUTION
234. Problem : Subtract £15 125 Sd from £48 7^ 6d,
Beginning with the column represent-
ing the lowest denomination, we sub-
tract Sd from 6d, which cannot be
performed. Take Is from 7s. One
shilling equals 12c?, making 18(/ in all.
18 minus 8 leaves lOd. 12^ from 65 can-
not be performed. Take £l from £48.
£1 equals 205, plus 6 equals 265, 26.s minus 12^ equals 145, £47
minus £15 equals £32, The remainder then is £32 145 lOd.
Note. For practice problems in subtraction of compound numbers see
pars. 52, 53.
7
6
£4^
P
6^
15
12
8
^32
14
10
Ans. £32 14^ 10^.
44 RATIONAL ARITHMETIC
MULTIPLICATION OF COMPOUND NUMBERS
ILLUSTRATED SOLUTION
235. Problem : Multiply £26 16^ 9d by 23.
£26 IQs 9d Beginning with the lowest denomina-
23 tion, multiply 9d by 23. This is 207c?.
£xQQ S689 ^01 d ^^ times I65 equals 3685. 23 times
£26 equals £598.
£617 5s Sd Reduce 207(/ to shillings (par. 230)
Ans £617 55 M. ™^l^"^g ^'^^ ^^- ^^rite M as part of
the final product.
175+3685 = 3855. Reduce 3855 to pounds (par. 230). Result.
19c? 55. Write 5s as part of the final product.
Add £19 to £598. Result, £617. Write £617 as the last of
the final product.
DIVISION OF COMPOUND NUMBERS
ILLUSTRATED SOLUTION
236. Problem :
Divide i
£26
I65
23)617
23)3855
46
23
157
155
138
138
£19
lis
20
12
380^
204^
5
3
9^
23)207
207
Ans, £26 I65 9d.
3855 207^
Commencing with the highest denomination, divide £617 by
23. This gives £26 with an undivided remainder of £19. Write
the £26 as part of the final quotient.
RATIONAL ARITHMETIC 45
Reduce £19 to shillings by multiplying by 20 (par. 229). This
equals 3805. 3805 plus 5s equals 3855. Divide 3855 by 23, which
gives 165, with an undivided remainder of 175. Write the I65 as
part of the final quotient.
Reduce the 175 to pence (par. 229).
This equals 204c?, plus 3d is 207(Z. 207d divided by 23 equals
9c?. Write 9cZ as part of the final quotient, making the complete
quotient £26 I65 9d.
COMPUTING TIME
237. In business it is often necessary to compute the
interval of time between two given dates.
Two methods are followed : Compound Subtraction
and Exact Days.
238. To Find the Time by Compound Subtraction:
ILLUSTRATED SOLUTIONS
Problem : Find the time between October 15, 1908,
and July 1, 1916.
1916 7 1 Use the later date as the
1908 10 15 minuend and the earlier date as
rv Q T~^ the subtrahend. The minuend
will be the 1916th year, 7th
Ans. 7 yr. 8 mo. 16 da. month, and 1st day. The sub-
trahend will be the 1908th year,
10th month, and 15th day. Write the subtrahend beneath the
minuend and proceed as for compound subtraction (par. 234).
Note. For practice problems in finding the time by compound sub-
traction see par. 52.
46 RATIONAL ARITHMETIC
239. To Find the Time in Exact Days:
Problem : How many days between January 8, 1916,
and May 19, 1916?
Jan. 23
17 K ciCk J V Subtract the date (8) from the
JjeD. y4/\) Lieap Year p n i <• i • ,^ n .
lull number oi days m the nrst
Mar. 31 month (31 days), leaving 23 days in
Apr. 30 January. February has 29 days
May 19 (Leap Year), March 31, April 30,
132 days ^^^ May 19. Write these in a
column and add.
Ans. 132.
Note. For practice problems in finding the time in exact days see par.
53.
ALIQUOT PARTS
240. The term Aliquot Part is applied to any number
that is contained an even number of times in a given
value or quantity.
The aliquot parts of 100 may be used to great
advantage in many of the arithmetical operations
involved in business transactions. Therefore, the
term aliquot parts usually means the aliquot parts
of 100.
241. The following table shows the aliquot parts of
one hundred that are of practical use. They are ex-
pressed as common fractions, as decimals, and as cents,
and should be memorized and used whenever prac-
ticable.
RATIONAL ARITHMETIC
47
Fraction
X
2
3
3
4
4
1
5
2.
5
^
5
4.
5
i
6
5.
6
JL
7
7
7
7
5.
7
6.
7
8
3.
8
5.
8
7.
8
9
2.
9
Decimal
.0
.O3
.U3
.875
.Hi
Cents
50
33^
66f
25
25
75
75
2
20
4
40
6
60
8
80
16f
16f
m
m
14f
14f
28-f
284-
42f
42f
574^
571
71|-
71f
85f
85f
125
12i
375
m
625
621
87J
Hi
Fraction
A
9
5.
9
7_
9
8.
9
1
10
3
10
7
10
9
10
1 1
2
1 1
3
1 1
4
1 1
5
1 1
6
1 1
7
1 1
8
1 1
9
1 1
10
1 1
1 2
5
12
7
1 2
1 1
12
16
3
16
5
1 6
7
16
Decimal
.441
.509^
.77i
.889^
.1
.3
.7
.9
•09iV
.18^
.27^
•36A
•54A
•63t^
•72^
.81A
.0^
.41f
.58^
.91f
.06i
.18f
.31^
.43f
Cents
44A
m
88f
10
30
70
90
9iV
18T?r
27T3r
36A
45A
54A
63Jr
72tL
81A
41f
58^
91t
6i
18f
311
43f
48
Rx\TIONAL ARITHMETIC
Fraction
9
16
1 1
16
1 3
16
1 5
16
Decimal
.56i
.68f
.81^
Cents
5Cii
68f
81i
93f
Fraction
1
20
3
20
7
20
9
20
a
13
20
17
20
19
20
Decimal
.05
.15
.35
.45
.55
.65
.85
.95
Cents
5
15
35
45
55
65
85
95
ILLUSTRATED SOLUTION
Application of Aliquot Parts
Problem: Find the cost of 640 lb. of tea at 37^-
S7U = ioi$l
I of 640 = 240
640 lb. at $1 a pound would cost $640. At f of a dollar a
pound, 640 lb. would cost f of $640, which is $240.
To avoid fractions, first multiply by the numerator and then
divide bv the denominator.
Note. For practice problems in aliquot parts see par. 27 to 31 inclusive.
PERCENTAGE
242. The words per cent and the name percentage
are derived from two Latin words, per centum, meaning
hy the hundred.
243. Percentage is a system of measurement in
which 100 is used as the standard of comparison.
Every unit contains 100 one-hundredths or 100 per
cent. The words per cent are used instead of the
denominator, one-hundredths. Thus : Instead of say-
ing five one-hundredths, we say 5 per cent. Nine per
cent would be xo^ oi' -09. It could be used in either
its fractional or decimal form.
244. The per cent sign (%) is generally used instead
of the words " per cent " and instead of the decimal
point, just as the cent sign (^) is used instead of the
word '* cents " and instead of the decimal point in ex-
pressing United States money, which is really another
use of the percentage system.
(a) Every per cent may be expressed in four different ways ; as
a common fraction with 100 for a denominator ; as an equivalent
common fraction in its lowest terms ; as a decimal ; and by the use
of the sign.
(6) The sign and the decimal should never be used together except
to designate a part of one per cent.
49
50 RATIONAL ARITHMETIC
(c) 25 /o — .25 — xbo
16f% = .16
4
2_ _ 3 _ J,
3 100 6
1
qql(7/ _ qql— 3 _ l
•^^^ 3 /C — -^^ 3 ~ 1 ~ 3
OQ 07 — OQ —23
-^^ /0~'"'*^ ~ 100
8
•8 % = To % = -008 = YJ5" =10
245. In solving problems in percentage that form
should he used ivhich makes the solution easiest.
246. Several special terms are used in the subject
of percentage. These are Base, Rate, Percentage,
Amount, and Difference.
247. The Base is the value, or quantity, represented
by 100%. It is the basis of comparison.
248. The Rate is the number of one-hundredths used
in the comparison.
249. The Percentage is the value of the rate. It is
the part of the base equal to the number of hundredths
represented by the rate. It is the product of the base
by the rate.
250. The Amount is the base plus the percentage.
251. The Difference is the base minus the percentage.
252. In the subject of aliquot parts (pars. 240, 241)
a table showing the aliquot parts of one hundred is
given. The aliquot parts of 100% are the same.
253. In the following table the fractional values of
the easier rates are given. It will be well to use the
common fraction form for any of these rates.
RATIONAL ARITHMETIC
51
-^2 /O — 4
10 % = tV
334% = i
62i%
5 % — 20
121-% = i
374 % = 1
661%
6i% = iV
161% = i
40 % = i
75 %
6|% = tV
20 % = i
50 % = ^
80 %
83%— 12
25 % = i
60'% = f
87i%
5^
8
2^
3
3.
4
:4
5
7^
8
254. All operations in percentage are solved in
accordance with the general principles of multiplica-
tion and division. They may be expressed as follows :
(a) Base X Rate = Percentage.
Percentage -^ Rate = Base.
Percentage 4- Base = Rate.
(6) Because the smallest coin used in the United States is one
cent, final answers should be expressed in the nearest cent ; that is,
5 mills (.005) or more would be called another cent, less than 5 mills
would be disregarded.
255. To Find the Percentage:
In solving problems in this case always use the easiest
method.
ILLUSTRATED SOLUTIONS
Problem : Find 37i% of $342.
$342
3.42
37i
171
23 94
102 6
$128.25
Ans. $128.25.
$342 is the standard quantity or 100%.
One per cent would be -j^q of this which is
SA'l and 37^% would then be found by
multiplying 3.42 by 37^ (par. 135). The
result is $128.25.
52 RATIONAL ARITHMETIC
Or,
$342
cyrvx Thirty-seven and one-half per cent of
171
$342 is the same as .37i of 342, which
means .37^X342. Performed as explained
^^ ^^ in par. 135, the result is $128.25.
102 6
$128.25 Ans. $128.25.
Or,
'*J I 9. ft
$342
3
8)1026
Thirty-seven and one-half per cent
equals |. | of $342 (par. 214) is $128.25.
128.25 Ans. $128.25.
All of these methods will be found to apply to any problem in
which it is necessary to find the percentage. In some cases one
method will be easier than another. Always use the easiest method.
256. Problem : A man who is worth $8465 has 65%
of his property in real estate, 15% of it in a mortgage,
and the remainder in cash. How much cash has he.^^
100% = $8465 For purpose of comparison in the
Qo% 4- 15% = S0% above problem it will be seen that the
100^ —809" =209^ man's entire property is 100%. Sixty-
oncy — 1 ^^^ P^^ ^^^^ ^^^ ^^% (80%) are known
^^ ^ to be invested in real estate and mort-
i of $8465 = $1693 gage. The remainder of the 100%, or
20%, is in cash. Our problem then is
Ans, $1693. to find 20% of the property, $8465, by
any of the above explained methods.
Since 20% is one-fifth, the easiest way is to find one-fifth of $8465,
which is $1693.
RATIONAL ARITHMETIC
53
257. Problem: I invested $7460 in business.
During the first year I gained 25%. How much
did I have invested in the business at the end of
that time?
4)
7460
1865
9325
Ans. $9325.
Twenty-five per cent is one-fourth. One-
fourth of $7469 is $1865. If I invested $7460
and gained $1865, I must have now $9325.
Or,
$7460
1.25
37300
14920
7460
$9325.00
Ans. $9325.
The original investment was 100% of itself.
If it is increased 25%, then the new capital
would be 125% (1.25) of the original invest-
ment. Find 125% of $7460 by multiplying by
1.25. Result, $9325.
It is sometimes easier to combine the various methods illustrated
in the above solutions.
258. Problem : Find 27% of $6240.
4) 6240
1560
124.80
$1684.80
Ans. $1684.80.
Twenty-seven per cent is made up of
two easy rates, 25% and 2%. Twentv-
0-
five per cent is \ (1560). Two per cent is
twice 1%. One per cent is $62.40 and 2%
is $124.80 ; adding this amount to $1560,
we have $1684.80.
Note. For practice problems in finding the pei'centage see par. 33.
54 RATIONAL ARITHMETIC
259. To Find the Base:
In solving problems, always use the easiest method.
ILLUSTRATED SOLUTIONS
Problem : $1172.34 is 27% of what?
$43142
27) 1172|34
108
92 Since $1172.34 is 27%, 1% would be Jy o^
g-^ ) $1172.34. Divide $1172.34 by 27, which is
^-ya $43.42. This would be 1% of the original
value. One hundred per cent, or the whole
1—— of the original value, would be 100 times 1%,
^4 which can be found by moving the figures in
54 $43.42 two places to the left. Therefore,
100% would be $4342.
$43.42 = 1%)
$4342 = 100%
Ans, $4342.
Or,
$4342 I
.27) 1172.34|
■*^"" $1172.34 is .27 of some number and was
92 found by multiplying the original number
81 by .27. In other words, $1172.34 is the
]^|3 product of one factor by .27. If we divide
-jQo $1172.34 (the product) by .27 (one factor),
— —7 the result will be the other factor. Perform
^^ this division. The result is $4342.
54
Ans. $4342.
RATIONAL ARITHMETIC 55
260. Problem : $94.65 is 62i % of what?
5) 94 65 = f Sixty-two and one-half per cent is f of
1 qIqq _ j^ the required quantity. Since 94.65 is f , we
will get ^ of the quantity by dividing 94.65
- by 5, which is 18.93. If 18.93 is ^, the
tpl51.44 = 8^ whole quantity, or f, would be 8 times 18.93,
which is 151.44, the value of the original
Ans. $151.44. quantity or 100%.
The problem given above could be solved according to either of
the preceding methods.
261. Problem: $235.64 is 8% more than what?
$218|185
1.08)235. 64|000
216
196
108
ooj^ The original quantity was 100%. $235.64
equals this and 8% more. Therefore,
$235.64 is 108% of what? Solving this by
any of the above methods, we find the
864
200
108 original number to be $218,185, which, ex-
920 pressed in the nearest equivalent cent, is
864 $218.19.
"560
540
20
Ans. $218.19.
Note. For practice problems in finding the base see par. 34.
56 RATIONAL ARITHMETIC
262. To Find the Rate:
In solving problems, always use the easiest method.
ILLUSTRATED SOLUTIONS
Problem : $128.25 is what per cent of $342?
In the above problem we simply want to
^|5 = I know what part $128.25 is of $342.
'^ 3^ 371. $128.2o is iffff of $342. Reduced to
lowest terms |ffff equals f . Then $128.25
Ans, 37i%. is f of $342. f equals 37i%. Therefore,
> $128.25 is 37i% of $342.
Or, Inasmuch as $128.25 is a certain
or.2. per cent of $342, it ($128.25) is the
QzLQ MQQ QK product obtained by multiplying one
lOQ r ^^^*^^ *^^^^^ desired rate) by another
I5?_^ factor ($342). Therefore, if we di-
25 65 vide $128.25 (the product) by 342
23 94 (one factor) , we shall obtain the other
171 1 factor (par. 123). Perform this
349 ~ Q operation as described in par. 140.
The result is .37^, which equals
Arts. .37ior37i%.
263. Problem : $9072 is what per cent of $7560 ?
1|20 $9072 is the product obtained by
7560)9072|00 multiplying one factor (7560) by
another factor (the desired rate).
If we divide 9072 (the product) by
7560 (one factor), we shall obtain the
other factor (par. 123). The result,
7560
1512
1512
Arts. 1.20 or 120%. 1.20, equals 120%.
Note. For practice problems in finding the rate see par. 35.
For general problems in percentage see par. 36.
PROFITS AND LOSSES
264. In measuring and comparing profits and losses,
it has been found best to do so by means of percentage.
265. For this purpose it is necessary to understand the
exact meaning of the special terms used in connection
with this subject.
266. Cost is the value of the investment.
267. Prime Cost of an article is the amount actually
paid for it. The prime cost is sometimes called the
net cost and also the^r^^ cost.
268. The Gross Cost of an article is the total amount
invested in it, and includes the prime cost and the
incidental expenses, such as freight, cartage, insurance,
etc. The simple term " cost " usually means the
gross cost.
269. The Gross Selling Price is the total amount
received for the goods sold.
270. The Net Selling Price is the amount of the
gross selling price left after incidental expenses of the
sale have been deducted, such as freight, commission,
insurance, etc.
57
58 RATIONAL ARITHMETIC
271. Profit is the difference between the net seUing
price and the gross cost, ivhen the selling price exceeds
the cost.
272. Loss is the difference between the net selHng
price and the gross cost, when the cost exceeds the selling
price.
273. In the subject of profit and loss :
Base = Gross Cost
Rate = Rate
Percentage = Profit or Loss
Cost =100%
In solving problems in this subject, always use the easiest method.
274. To Find Profit, Loss, or Selling Price:
ILLUSTRATED SOLUTIONS
Problem : A farm costing $4200 increased in value
^i% when it was sold. Find the profit and the selling
price.
$4200
Qgj, The cost is the standard of
value, 100%. Then $4200 is
100%. Eight and one-third
per cent may be found, as ex-
$350.00 Profit plained in par. 255, to be $350.
4200.00 If the cost is $4200 and the profit
$4550.00 Selling Price is $350, the selhng price must be
the sum of the two, $4550.
1400
33600
Ans.
\ $350.
$4550.
RATIONAL ARITHMETIC
59
Or,
12 )4200
350
4550
Arts.
Profit
Selling Price
f $350.
[$4550.
Since 8^% is -^^, the profit is yV of
the cost. One-twelfth of $4200 is
$350. Added to the cost, the
amount is $4550.
275. Problem: 1200 bushels of wheat were pur-
chased at 75 cents per bushel and later sold at a loss
of 17%. What was lost, and for what was the wheat
sold?
1200 bu. @ $.75 = $900
$900
.17
$153.00 Loss
$900
153
$747 Selling Price
$153.
1200 bushels of wheat at $.75 cost
$900 (found by the use of aliquot
parts, par. 241). 17% of $900
(par. 255) equals $153.
Since the goods cost $900 and are
sold for $153 less than cost, they will
be sold for $900 minus $153, which
is $747.
Ans. \
[ $747.
Note. For practice problems in finding the profit or loss and selling
price see par. 37.
276. To Find Cost:
ILLUSTRATED SOLUTIONS
Problem : My profit on a certain transaction,
figured at 12^%, would be $720. What was the cost
of the transaction ?
60
RATIONAL ARITHMETIC
5 760
.125)720.000
6^5
95
87 5
7 50
7 50
In this problem, the rate is
12|% and the percentage $720,
The base equals the cost, which
may be found as explained in
par. 259 or 260.
Ans. $5760.
— i
Or
12i
$720
8
$5760 Cost
Ans. $5760.
Twelve and one-half per cent equals ^.
Since ^ is gained and the gain is $720, then
$720 is i of the cost. Therefore, the cost
would be 8 times $720.
277. Problem : I bought goods and afterwards sold
them at a loss of 23%, receiving $412.80 for them.
What did they cost?
100%
23%
77%
5 361103
.77)412.80 000
385
27 8
23 1
4 70
4 62
80
77
300
231
The base, or cost, is 100%, from which
Vq is lost, leaving 77%, the measure of
the value for which the goods were sold.
Then $412.80 equals 77% of the cost, which
may be found as explained in pars. 259,
260, 261.
69 Ans. $536.10.
RATIONAL ARITHMETIC
61
278. Problem : Goods are sold for $126, which
shows a loss of lli%. What did they cost?
100
IH
88f
126 = .88f
$141|75
8.00)1134|OO
Or,
8 )126.00
15.75
9
Arts. $141.75.
One hundred per cent is the
standard of measure, or cost ;
tl^% has been lost. The
goods then sold for 88 1% of
the cost. The cost may be
found as explained in par. 261.
Eleven and one-ninth per cent is equal to
If ^ is lost, the goods
Therefore, $126 is f of the cost.
If $126 is f of the cost, ^ of the cost would be
found by dividing $126 by 8, which is $15.57.
If $15.75 is ^ of the cost, the whole cost
the common fraction ^
are sold for |^.
141.75
Ans. $141.75. would be 9 times $15.75, which is $141.75.
Note. For practice problems in finding the cost see par. 38.
279. To Find Rate of Profit or Loss:
ILLUSTRATED SOLUTIONS
Problem : Goods costing $723.45 are sold so as to
gain $241.15. What is the gain per cent?
1
J7^37*5-
3
= - = 33^
3/0
M^tr^ 1
3
Ans. 33i%.
The cost, or base, is $723.45. The
gain, or percentage, is $241.15. Find
the rate as explained in par. 262.
Or,
I33i
723.45)241. 15|00
217 03 5
24 11 50
21 70 35
2 41 15 Ans, 33i%.
Use the fractional method
when its reduction may be
determined at a glance ; use
the decimal method when
this is not the case.
m RATIONAL ARITHMETIC
280. Problem : Goods that cost $414 are sold for
$492.66. What per cent is gained ?
$492.66
414.
$ 78.66
If the goods cost $414 and sell for $492.66,
1^" the difference, $78.66, must be gain or profit.
414)78|66 The problem then is: $78.66 is what per cent
41 4 of $414, the cost.'* This can be ascertained as
37 26 explained in par. 262.
37 26
Ans. 19%.
Note. For practice problems in finding the per cent of gain or loss see
par. 39.
For general problems in profit and loss see par. 40.
DISCOUNT
281. A Discount is an amount deducted from a sum
owed by one person to another. In measuring dis-
counts the principles of percentage are used. There
are two kinds of discount, trade discount and time
discount.
282. Trade Discount is the discount allowed by a
manufacturer or jobber to a retail dealer.
283. Time Discount is a discount allowed as a con-
sideration for paying an amount during a certain time.
TRADE DISCOUNT
284. Manufacturers, wholesalers, and others doing
business of a similar nature, and handling goods the
value of which is likely to fluctuate from time to time,
have a fixed list price for their goods. These list prices
remain fixed, and fluctuations in market rates are met
by allowing different discounts from time to time ;
that is, if the value drops, the discount is made larger,
and if the price of the goods rises, the discount is made
smaller.
285. The List Price of goods is the price at which
they are listed in the catalogue and from which dis-
counts are allowed.
Goods are always billed at the list price and then the discount
is deducted from the total.
63
64 RATIONAL ARITHMETIC
286. The Gross Amount of the bill is the total
amount before any discounts have been deducted.
287. The Net Amount is the amount to be paid
after all discounts have been deducted.
288. The Discount is the sum deducted from the
gross amount.
289. A Discount Series is several discounts deducted
one after another ; as, 25%, 10%, and 5%.
The first discount is deducted from the gross amount ; the second
from the remainder, and so on ; the final remainder being tlie net
amount.
290. The principles of percentage are used in per-
forming all operations in discount.
Base = Gross Amount
Rate = Rate
Percentage = Discount
Difference = Net Amount
Gross Amount = 100%
291. To Find the Discount or Net Amount:
Always use the easiest possible method.
ILLUSTRATED SOLUTIONS
Problem : Goods listed at $420.34 are sold at a
discount of 28%. What is the discount and what is
the net amount of the bill ?
RATIONAL ARITHMETIC 65
$420.34
.28
3362 72
8406 8
$117.69 52 Discount
$420.34
117.70
$302.64 Net Amount
[$117.70.
^''^- ($302.64. .
The gross amount of the bill,
$420.34, is 100%. Find 28% of this
bv the easiest method. This will
give $117.70. The difference will
equal the net amount, $302.64.
292. Problem : A certain line of goods is sold at a
discount of 25%, 20%, 10%, and 5%. What would be
the net amount of a bill of $1214.43, purchased under
these terms ? What would the discount amount to ?
$1214.43 rpj^jg problem involves a dis-
303.60/.^ 25% count series. The first discount is
910.82^^ deducted from the gross amount.
182.16^ 20% Twenty-five per cent of $1214.43
ifoQ gg is $303.61 ; deducted from the
79 87 100/ gross amount leaves $910.82.
^ J — Twenty per cent of this, found in
^^^•^^ the easiest way, is $182.16; de-
32.78 5% ducted, leaves $728.66. Proceed-
$623.01 Net Amount ing in the same way, deducting
$1214.43 Gross Amount ^^^^ ^^^ ^^^^ ^^^' ^^^ "^^ ^^^^^^
nr^c ^-. TVT . A . of the bill is $623.01. If the net
-^?M1 Net Amount ^^^^^^^ ^^ ^^^ ^.^^ .^ ^^,3^^ ^^^
$591.42 Discount the gross amount is $1214.43, the
[ $623 01. difference must be the discount,
^"*- i $591.42. «^91-*2-
Note. For practice problems in finding the net amount see par. 41.
66 RATIONAL ARITHMETIC
293. To Find a Single Rate of Discount Equal to a
Discount Series:
ILLUSTRATED SOLUTION
Problem : What single discount is equal to a dis-
count series of 25%, 20%, 10%, and 5% ?
4)100% One hundred per cent represents the gross
25 amount of the bill. The first discount is 25%
or \ of this. One-fourth of 100% equals 25%.
Deduct the first discount (25%) from 100%.
This leaves 75% from which to deduct the
second discount. The second discount is 20%
or one-fifth. One-fifth of 75% equals 15%.
Deduct this from 75%. This leaves 60%.
The next discount in the series is 10% or one-
tenth. One-tenth of 60% equals 6%. 60%
minus 6% equals 54%. The last discount
must then be deducted from 54%. The
last discount is 5% or one- twentieth. One-
48.7% twentieth of 54% equals 2.7%. 54% minus
4 dft 70/ ^•'^% equals 51.3%. This is the net amount
* ^^* to be paid. Then the discount is the differ-
ence between the gross amount (100%) and the net amount (51.3%).
This is 48.7%.
Note. For practice problems in finding a single rate of discount equal
to a discount series see par. 42.
294. To Find the Gross Amount or the List Price:
ILLUSTRATED SOLUTIONS
Problem : On a bill of goods sold at a discount of
33 i% the discount equals $55.40. What is the gross
amount of the bill ?
5)
75
15
10)
60
6
^0)
54
2.7
51.3
100
51.3
RATIONAL ARITHMETIC
67
33i%of ? = $55.40
$55.40 33^%
3
$166.20 100%
Ans. $166.20.
The discount, 33^%, is $55.40. If
$55.40 is 33^%, 100% found as ex-
plained in par. 259, is $166.20.
295. Problem : A check for $953.80 was given in
full payment for a bill of goods bought at 24% dis-
count. What would the gross amount of the bill be ?
$12 55
.76)953.80
76
193
152
418
38
3 80
3 80
The gross amount of the bill is 100% and
the discount 24%. The net amount of the
bill must be 76%. Therefore, $952.80 equals
76% of the gross amount. Find the gross
amount as explained in par. 259.
Ans. $1255.
296. Problem : The net amount of a bill of goods
sold at a discount of 40%, 30%, and 20% was $114.24.
Find the gross amount of the bill.
100%
40
60
42-
8.4
33.6%
Goods sold at a discount of 40%, 30%, and
20% are sold at a net price which equals 33,6%
of the original bill (par. 293). If $114.24
equals 33.6%, 100%, the gross amount, may be
found, as explained in par. 259, to be $340.
68
RATIONAL ARITHMETIC
$340
.336)114.240
100 8
13 44
13 44
Ans. $340.
Note. For practice problems in finding the gross amount or the list
price see par. 43.
297. To Find the Rate of Discount:
ILLUSTRATED SOLUTIONS
Problem : The gross amount of a bill of goods is
$346.40. The net amount is $259.80. What is the
rate of discount ?
$346.40
259.80
$86.60
The difference between the gross amount
25 and the net amount equals the discount,
346 40)86.60 00 which is $86.60. The problem then is :
6Q^8 $86.60 is what per cent of $346.40.? This is
iiWoo ^'''''''^ ^"^ ^^ ^^^"^ ^^^'' ^^^^*
17 32 00
or
1
B600r _l
^04^ 4
4
Ans. 25%.
or 25%.
Note. For practice problems in finding the rate of discount see par. 44.
RATIONAL ARITHMETIC 69
298. To Find What Price to Mark Goods in Order to
Allow a Certain Discount and Still Make a Certain Profit:
ILLUSTRATED SOLUTION
Problem : What price must we mark goods costing
$*214 in order that we may allow a discount series of
20%, 10%, and 5% and still make a profit of 25% ?
$214
53.50
$267.50
100%
20
80
g By the principles of profit and loss, we see
^ that to make a profit of 25%, goods costing
o ^ $214 must be sold for $267.50 (par. 274).
— - — ^ Then we must sell the goods for a net amount
68.4% Qf $267.50. If a discount series of 2q^c,
$391 108 10%, and 5% is to be allowed from the gross
684^267 500 loo ^^ount of the bill, the net amount of the bill
oQx Q will be 68.4% of the gross amount (par. 293).
Therefore, $267.50 equals 68.4% of the gross
amount, or asking price, which will be found,
as explained in par. 259, to be $391.08.
62 30
61 56
740
684
56 00
54 72
128
Ans. $391.08.
Note. For practice problems in finding what price to mark goods in
order to allow a certain discount and still make a certain profit see par. 45.
For general problems in trade discount see par. 46.
COMMISSION AND BROKERAGE
299. A Commission Merchant is a person or firm who
buys or sells merchandise for another person or firm.
A commission merchant actually handles the goods and buys
and sells them as if for himself, but in reality for another person or
firm.
300. A Broker is a person or firm who arranges
transactions between other persons.
A broker does not handle the merchandise himself, but simply
brings the buyer and seller together in the interest of one or the
other.
301. The Principal is the person or firm for whom
the business is transacted.
302. The Commission is the compensation allowed
the commission merchant or the broker.
303. The Gross Proceeds of a sale or collection is
the entire amount received from the purchaser or
debtor by the commission merchant.
304. The Charges are the incidental expenses of the
sale or purchase.
305. The Net Proceeds is the amount remaining
after the charges have been deducted from the gross
proceeds. It is the amount to be returned by the
commission merchant to his principal.
70
RATIONAL ARITHMETIC 71
306. Prime Cost is the first cost of goods purchased
by the commission merchant in the interest of his
principal.
307. The Gross Cost is the prime cost plus the
charges incidental to the purchase.
308. Account Sales is an itemized statement of
sales of merchandise by a commission merchant. It
shows the amount for which the goods were sold, the
charges, and the net proceeds of the sale.
309. Account Purchase is an itemized statement
covering the merchandise purchased by a commission
merchant and shows the prime cost plus the charges.
310. In solving problems in commission, the general
principles of percentage are used.
Base = Gross Sales or Prime Cost
Rate = Rate of Commission
Percentage = Commission
Net Proceeds = Difference
Gross Cost = Amount
Gross Sales or Prime Cost = 100%
311. To Find the Commission and Net Proceeds or
Gross Amount of Purchase :
ILLUSTRATED SOLUTIONS
Problem : A commission merchant sold goods for
$2346.40. His commission was 2^%. Charges for
insurance, freight, etc., amounted to $98.40. Find
the commission and net proceeds. •
72
RATIONAL ARITHMETIC
$2346.40
58.66
98.40
157.06
$2189.34
Ans.
2i% Com-
mission
Charges
Net Proceeds
$58.66.
$2189.34.
Gross sales, 100%, is $2346.40.
Two and one-half per cent of this,
the commission, is $58.66. Since
the commission merchant spent
$98.40 for expenses and kept $58.66
for commission, the sum of these,
$157.06, must be deducted from
$2346.40, leaving a net proceeds
of $2189.34.
312. Problem : A commission merchant bought
500 barrels of apples at $2.75 on a commission of 5%
and paid $15 for cartage and $52.50 for cooperage.
For what sum must his principal write a check to cover
the entire transaction ?
500 bbl. @ $2.75 = $1375
$1375
68.75
15.
52.50
5%
$1511.25
Ans. $1511.25.
Five hundred barrels of apples
at $2.75 cost $1375. The com-
mission, 5%, equals $68.75. The
principal will, therefore, have to
send the commission merchant
$1375 to pay for the goods, $68.75
for his commission, $15 to pay for
cartage, and $52.50 for cooperage,
or $1511.25 in all.
Note. For practice problems in finding the commission and net pro-
ceeds or gross amount of purchase, see pars. 47 and 48.
313. To Find the Gross Sales or Net Purchase Price:
ILLUSTRATED SOLUTIONS
Problem : A commission merchant working on 2^%
commission received $134.50 for selling a consignment
of flour. What did- the flour sell for.^^
RATIONAL ARITHMETIC 73
$134.50 = 2i%
$5 380
.025)134.500
125 $134.50 is ^%. Find 100%
as explained in par. 259.
Ans. $5380.
Or,
QJ-Or — JL
^2/0 — 4
134.50 = A
40
95
75
2 00
2 00
$5380.00 Ans. $5380.
314. Problem : The net proceeds is $568.40. The
charges for freight, insurance, etc., are $27.40. The
commission is 3%. For what were the goods sold ?
$568.40
27.40
$595.80
6 14 22
.97)595.80 00
582
13 8
97
4 10
3 88
22
19 4
2 60
194
The net proceeds is $568.40. Charges for
freight, insurance, etc., are $27.40; added to
$568.40 equals $595.80. This is the amount
remaining from the sales after the commissioin
alone has been deducted. If the commission
is 3%, then $595.80 is 97%. Find the total
amount of the sales as explained in par. 259,
which is $614.22.
Ans. $614.22.
74
RATIONAL ARITHMETIC
315. Problem : I sent a commission merchant $927
to invest in apples. How many barrels at $3.75 can
be purchased after deducting a commission of 3%?
9 00
1.03)927.00
927
2 40
3.75)900.00
750
150
150
Ans. 240.
$927 includes the amount of purchase and
the commission of 3%. Therefore, $927 is
103%. 100% may be found, as explained in
par. 261, to be $900.
If one barrel of apples cost $3.75, for $900 we
can buy as many barrels as $3.75 is contained in
$900, which is 240.
Or,
3% of $3.75 = .1125, commission on 1 bbl.
$3.75 Cost of 1 bbl.
.1125 Commission on 1 bbl.
$3.8625 Gross cost of 1 bbl.
240
3.8625)927.0000
772 50
154 500
154 500
Ans. 240.
Since the market price of 1 bbl. is $3.75,
the commission on one barrel would be 3%
of $3.75 which is .1125, and the gross cost
of one barrel would be $3.8625. For $927
the commission merchant could buy as many
barrels as $3.8625 is contained in $927, which
is 240.
Note. For practice problems in finding the gross sales or net purchase
price see par. 49.
RATIONAL ARITHMETIC 75
316. To Find the Rate of Commission:
ILLUSTRATED SOLUTION
Problem : A commission merchant charges $80.16
for selling a bill of goods for $1336. What is his rate
of commission ?
|06
lSS6')80ll6 Since the commission is figured as a certain
en 1 « per cent of the amount of sales, this problem
— really is: $80.16 is what per cent of $1336,
which, solved as explained in par. 262, is .06 or 6%.
Ans. 6%.
Note. For practice problems in finding the rate of commission see
par. 50.
For general problems in commission see par. 51.
INTEREST
317. Interest is the amount paid for the use of
money.
(a) When the use of real estate is allowed to someone other than
the owner, the compensation is called rent ; the compensation for
the use of personal property is called hire; the compensation for
the use of manual labor is called wages; the compensation for the
use of mental labor and time is called salary.
(b) The amount of interest to be paid depends upon the time
that the money is used, the sum that is used, and the way it is
used (risk involved).
(c) The first two are self-fixing. The third is subject to agree-
ment between the parties. If there is danger of the original sum
being lost, a larger rate should be paid for its use. Also when
money is plentiful and easy to hire, the rate should be lower than
if it were scarce and hard to hire.
(d) It has been found that the best way of figuring interest is on
a percentage basis. Therefore, the principles of percentage with
another element, time, govern the subject of interest.
318. Principal is the sum for the use of which interest
is charged.
319. Rate is the per cent of the principal charged
for the use of the principal for one year.
320. The Legal Rate is the rate fixed by law to be
understood when no rate is mentioned by the parties.
It differs in different states. The legal rate in a
majority of the states is 6%.
76
RATIONAL ARITHMETIC 77
Charging more than a reasonable compensation is called usury.
Some of the states name a definite maximum rate ; in such states
to charge more is usury. Where no maximum rate is fixed by law,
it is a question for the courts to decide whether or not usury is
charged in a given case.
321. Time is the period for which the principal is used.
322. The Amount is the sum of the principal and
interest.
323. The interest on $1 for one year at 6% would
be 6% of $1, which is $.06. The interest on $1240
for one year at 6% would be 6% of $1240, which is
$74.40. The interest for one year, then, is always
equal to the percentage of the principal represented by
the rate.
324. The general principle on which all interest is
based is :
Principal X Rate X Time = Interest
In figuring the interest for part of a year, two dif-
ferent methods arise : Accurate Method and Ordinary
Method.
ACCURATE INTEREST
325. Accurate Interest gives a year, or any part of
a year, its exact value.
(a) In computing the accurate interest for any part of a year the
time is counted in days, and each day given its actual value, 3^^ of
a year.
(6) This method is used only in figuring interest on United States
bonds, on foreign moneys, and by special agreement.
326. Accurate Interest is found by applying the
general principle of interest with absolute accuracy.
78 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
327. Problem : Find the accurate interest of $1140
for 93 days at 6%.
(h-, 1 j^Q Six per cent of $1140 equals $68.40. The interest on
$1140 for one year is $68.40. The mterest for 93 days
^ — equals -^^^ of $68.40, which, found as explained in par.
$68.40 214, is $17.43.
Find ^^5 of $68.40.
$68.40
$17 427
93
365)6361 200
205 20
365
6156
2711
$6361.20
2555
156 2
146
10 20
7 30
2 900
2 555
345
Ans. $17.43.
328. Problem : Find the accurate interest of $1140
for 458 days.
458 days = 1 year 93 days.
* This problem differs from the preceding
^ ' •'*'^ one only in the matter of time. The interest
$85.83 for one year is $68.40. For 93 days the
Ans. $85.83. interest, found as in the previous problem, is
$17.43. The sum of the two will equal the
interest for 1 year and 93 days.
RATIONAL ARITHMETIC 79
ORDINARY INTEREST
329. Ordinary Interest Method is that used by busi-
ness men under ordinary circumstances.
It differs from exact interest simply in that one day is roughly
considered one three-hundred-sixtieth of a year.
This is arrived at in this way : one-twelfth of a year is called one
month, and one-thirtieth of a month is called a day. So that one
day is one-thirtieth of one-twelfth, or one three-hundred-sixtieth of
a year.
330. To figure the ordinary interest for parts of a
year, business men sometimes count the time in months
and days, considering each month to have thirty days ;
sometimes the time is figured in exact days.
The latter plan is always followed by banks. Some authorities
divide ordinary interest into two classes. Common and Bankers' .
331. The best method of calculating ordinary in-
terest is by what is known as the Sixty-day Method.
By this method the interest is always found at 6%
first and then changed to the rate desired.
EXPLANATION
At 6%, the interest on any sum for one year would
be six one-hundredths of the principal. One one-
hundredth of the principal, then, would be the interest
at 6% for one-sixth of a year. One-sixth of a year is
2 months, or 60. days. Then the interest on any
principal for 60 days, or 2 months, at 6%, would be
one one-hundredth of itself. Thus the interest at 6%
on $2420 for 60 days is $24.20. For 12 days it would
be one-fifth of the interest for 60 days, or $4.84, and
80 RATIONAL ARITHMETIC
so on. The interest at 1% would be one-sixth of the
interest at 6%, or $4.0333 for 60 days. The interest
at 3i% would be 3i times $4.0333, which is $14.11 for
60 days.
SIXTY-DAY METHOD — ORDINARY INTEREST
RULE
332, Write the principal. Set off the interest for
2 months, or 60 days, at 6%. Using the interest for
60 days as a- basis, the interest at 6% for any other
period may be easily ascertained. When the interest
for the desired period has been found at 6%, change to
the rate desired.
Interest in partial results should always be carried to the fourth
decimal place.
ILLUSTRATED SOLUTIONS
333. Problem : Find the interest on $1246.40 for
1 year 9 months 16 days at 6% ; at 4%.
$1246.40 1 yr. 9 mo. 16 da. at 6% ; at 4%
12.4640 Int. for 60 da. at 6%
74.7840 Int. for 1 yr. at 6%
49.8560 Int. for 8 mo. at 6%
6.2320 Int. for 1 mo. at 6%
2.0773 Int. for 10 da. at 6%
1.2464 Int. for 6 da. at 6%
6 )134.1957 Int. for 1 yr. 9 mo. 16 da. at 6%
22.3659 Int. for 1 yr. 9 mo. 16 da. at 1%
4
$89.4636 Int. for 1 yr. 9 mo. 16 da. at 4%
RATIONAL ARITHMETIC 81
Set off the interest for 2 months or 60 days at 6%, $12.4040.
We now find the interest for 1 year, 8 months and 1 month, 10 days
and 6 days. One year is 6 times 2 months. Eight months is 4
times 2 months. One month is one-half of 60 days. Ten days is
one-sixth of 60 days. Six days is one-tenth of 60 days.
12.4640X 6 = $74.7840, ^Yhich is interest for 1 yr. at 6%
12.4640 X 4 = $49.8o60, which is Interest for 8 mo. at 6%
12.4640^ 2 = $6.2320, which is interest for 1 mo. at 6%
12.4640^ 6 = $2.0773, which is interest for 10 da. at 6%
12.4640 ^10 = $1.2464, which is interest for 6 da. at 6%
Adding these partial results, we have a total of $134.1957, which
is the interest for 1 year 9 months 16 days at 6%. Dividing this
by 6 gives us the interest at 1%, which is $22.3659. Multiplying
this by 4 gives $89.4636, which is the interest for 1 year 9 mouths
16 days at 4%.
Or,
$134 1957 at 69^ Divide the interest at 6% by 3, which
^ gives $44.7319, which is the interest at 2%.
Subtract this from the interest at 6%. The
44.7319 at 2%
$89.4638 at 4% difference is $89.4638, the interest at 4%.
334. Problem : What is the interest on $428.75 for
214 days at 6% ?
$428.75
214 da.. at 6%
4.2875
Int. for 60 da. at 6%
12.8625
Int. for 180 da. at 6%
2.1437
Int. for 30 da. at 6%
.2143
Int. for 3 da. at 6%
.0714
Int. for 1 da. at 6%
$15.2919
Int. for 214 da. at 6%
ns. $15.29.
82 RATIONAL ARITHMETIC
In this problem, the most convenient periods of time would be
180-30-3-1 days. Set off the interest for 60 days, which is $4.2875.
One hundred eighty days is 3 times 60. Thirty days is one-half
of 60. Three days is one-tenth of 30. One day is one-third of
3 days. Adding, gives us $15.2919, the interest for 214 days
at 6%.
Note. For practice problems in finding the interest see par. 54.
335. Problem : Find the Interest on $1234.28 from
January 3, 1915, to August 1, 1916, at 6%.
1916
8
1
$1234.28
1 yr. 6 mo. 28 da. at 6%
1915
1
3
12.3428
Int. for 60 da. at 6%
1
6
28
74.0568
37.0284
4.9368
.8228
Int. for 1 yr. at 6%
Int. for 6 mo. at 6%
Int. for 24 da. at 6%
Int. for 4 da. at 6%
$116.8448
Int. for 1 yr. 6 mo. 28 da.
at 6%
Ans. $116.8448.
In this problem it is necessary to find the time between Janu-
ary 3, 1915, and August 1, 1916. This will be found (par. 238) to
be 1 year 6 months and 28 daj's.
One year is 6 times 2 months, 6 months is 3 times 2 months,
24 days is 4 times 6 days (6 days being one- tenth of 60 days).
4 days is one-sixth of 24. The total is $116.8448, which is the
interest at 6% for 1 year 6 months and 28 days.
336. Problem : Find the interest on $514.95 from
January 18, 1915, to June 7, 1915, at 4i%. Time
computed in exact days.
RATIONAL ARITHMETIC 83
Jan. 13 $514.95 140 da. at 4>i%
Int. for 60 da. at
Feb. 28
Mar. 31
Apr. 30
5.1495
10.2990
1.7165
Mav 31
June 7
4)12.0155
3.0038
140 days
Ans. $9.0117.
$9.0117
Int. for 120 da. at
Int. for 20 da. at
Int. for 140 da. at 6%
Int. for 140 da. at li%
Int. for 140 da. at 4^%
Find the exact number of days between January 18, 1915, and
June 7, 1915 (par. 239), which is 140 days. This consists of 120
days and 20 days. One hundred twenty days is twice 60 days.
Twentv davs is one-third of 60 davs, making the total interest
$12.0155 for 140 days at 6%. The difference between 6% and
4^% is 1^%. One and one-half per cent is one-fourth of 6%.
Divide $12.0155, the interest at 6%, by 4, which gives us $3.0038,
the interest at 1^%. Subtract this from the interest at 6%, which
leaves $9.0117, the interest at 4^%.
Note. For practice problems in finding the interest when the time has
to be found either in exact days or by compound subtraction see par. 55.
SIXTY-DAY METHOD — ACCURATE INTEREST
337. The difference between ordinary interest and
accurate interest for any part of a year is one seventy-
third of the ordinary interest.
338. The interest for one year or any number of
years is the percentage yalue of the rate and is the same
for ordinary and accurate because it is the standard for
both methods.
339. Ordinary interest for any part of a year is
greater than accurate interest for the same time. •
84
RATIONAL ARITHMETIC
340. To change the ordinary interest for any part of
a year to the accurate interest for the same time,
divide the ordinary interest by 73. Subtract this from
the ordinary interest.
341. Problem : Find the accurate interest on $246
for 73 days at 6%.
$240.
2.40
.48
.04
73)2.92
.04
73 da. at yjyo
Int. for 60 da. at
Int. for 12 da. at
Int. for 1 da. at
Ordinary interest
c
2.88 Accurate interest
Ans. $2.88.
First find the ordinary
interest, as explained in
par. 334, on $240 for 73
days at 6%, which is $'2.92.
One seventy-third of
$2.92 is $.04. The ordi-
nary interest, therefore,
is $.04 larger than the ac-
curate interest, which is
found by subtracting $.04
from $2.92.
342. Prohlem : Find the accurate interest for 41^
days at 4i% on $425.
419 days = l year and 54 days
$425.
4>.^5
2.125
1.700
73)3.825
.0523
3.7727
25.50
4)29.2727
7.3181
$21.9546
Ans. $21.95.
1 yr. 54 da. at 4^'
Int. for 60 da. at
Int. for 30 da. at
Int. for 24 da. at uyo
Ordinary interest for 54 da.
Accurate interest for 54 da. at
Accurate interest for 1 yr. at
o
RATIONAL ARITHMETIC 85
This problem differs from tlie previous one in that the time is
more than a year. Three hundred sixty-five days should be sub-
tracted from 419, leaving 54 days more than a 3''ear, so that the full
time is 1 year and 54 days.
Find the interest by the ordinary interest method for 54 days (par.
334). This is $3,825 at 6%. Change this to accurate interest as ex-
plained above. The accurate interest for 54 days at 6% is $3.7727.
Six per cent of $425 gives the interest for one year (both accurate
and ordinary). Six per cent of $425 is $25.50. Add this to the
accurate interest for 54 days and we have the accurate interest
for 1 vear and 54 davs, which is $29.2727.
Dividing this by 4 gives the interest at 1^%, which is $7.3181.
Deduct this from the accurate interest at 6% and we have $21.9546,
the accurate interest for 1 year and 54 days at 4^%.
343. Prohlem : Find the accurate interest on £214
\\s 9d for 115 days at 5%.
12)9
11
20)11
d
75s
75s
5875£
£214.5875 115 da. at 5%
2.1458 Int. for 60 da. at
2.1458 Int. for 60 da. at
1.0729 Int. for 30 da. at 6%
.7152 Int. for 20 da. at 6%
.1788 Int. for 5 da. at 6%
73) £4.1127 Ordinary interest at 6%
.0563
4.0564 Accurate interest at 6%
.6760 Accurate interest at 1%
£3.3804 Accurate interest at 5%
86 RATIONAL ARITHMETIC
Reduce £'^214 lis 9d to pounds (par.
£3804 .608 ^32), which is £214.5875. Find the
QQ TQ ordinary interest (par. 334). Change to
^ „„„^ ^ ^^^ 7 accurate interest by subtracting one
7.60805 7.29Da + +i • i ^ •. u / onx a.,.
seventy-third oi itseli (par. 341). Ihis
Ans. i/O tS la. equals £4.0564, which is the accurate
interest at 6%. Subtract one-sixth (par.
333). This equals accurate interest at 5%.
Note. For practice problems in accurate interest see par. 56.
344. The best combinations for finding the interest
by the sixty-day method for from one to thirty days :
1 day = eV of 60, or ^ of 6
2 days = 3V of 60, or i of 6
3 days = 2V of 60, or i of 6
4 days = 3 days + 1 day
5 days = iV of 60 days
6 days = ro of 60 days
7 days = 6 days + 1 day
8 days = 6 days + 2 days
9 days = 6 days + 3 days
10 days = i of 60 days
11 days = 10 days + 1 day
12 days = i of 60 days
13 days = 10 days + 3 days
14 days = 12 days + 2 days
15 days = i of 60 days or i of 30
9
RATIONAL ARITHMETIC 87
16 days = 10 days + 6 days
17 days = 15 days + 2 days
18 days = 12 days + 6 days or 3X6
19 days = 15 days + 3 days + 1 day
20 days = |^ of 60 days
21 days = 20 days + 1 day
22 days = 20 + 2
23 days = 20+3
24 days = 20 + 4 or 4X6
25 days = 20 + 5
26 days = 20 + 6
27 days = 24 + 3 (i of 24)
28 days = 24 + 4 (i of 24)
29 davs = 24+5
30 days = i of 60
345. To Find the Interest at Common Rates Other Than
0'
\% = A of 6% 2% = i of 6% 5% = 6% - 1%
i% = \ of i% 2i% = 2% + ¥7o 7% = 6% + 1%
1% = 4+1% 3% = i of 6% 8% = 6%+2%
1% = i of 6% 4% = 6% - 2% 9% = 6% + 3%
H% = 1 of 6% 4i% = 6% - U% 10% = lOX 1%
88
RATIONAL ARITHMETIC
346. To Find the Time : Divide the given interest
by the interest on the principal for one year, at the
given rate.
(a) The divisor should be carried to the fourth decimal place if
necessary.
(6) The quotient should be carried to the fourth decimal place.
(c) In the final result, more than half should be considered
another day.
ILLUSTRATED SOLUTIONS
347. Problem : In what time wih $532.56 produce
$48, interest at 5% ?
$532.56
.05
$26.628C
1
1
8026
26.628)48.000
0000
26 628
21 372
21 302 4
69 600
53 256
16 3440
15
9768
3672
1.8026 yr.
12
9.6312 mo.
30
18.9360 da.
Five per cent of $532.56 is
the interest on $532.56 for one
year. It is $26.6280.
If $532.56 earns $26.6280 in
one year, it will take as manv
years to earn $48 as $26.6280
is contained in $48, which is
1.8026 years.
Reducing 1.8026 j^ears to
years, months, and days (par.
231), we have 1 year 9 months
and 19 days.
Ans. 1 yr. 9 mo. 19 da.
KATIONAL ARITHMETIC 89
348. Problem : In what time will $560 amount to
$625.71 at 6% interest?
$625.71 $560
560. .06
$
65.71 $33.60
1 9556
33.60)65.71 10000
33 60
32 11
30 24
187 00
1 68 00
19 000
16 800
2 2000
2 0160
First find the interest by
deducting the principal, $560,
from the amount, $625.71. This
shows that the interest is $65.71.
The interest at 6% on $560
for one year is $33.60.
Proceeding as in the previous
illustrated problem, we find the
time required to be 1.9556
years. Reduced to years,
months, and days, this equals
1840 1 year 11 months and 14 days
1.9556 yr. (par. 231).
I 12
11.4672 mo.
I 30
14.0160 da.
Ans. 1 yr. 11 mo. 14 da.
Note. For practice problems in finding the time see par. 57.
349. To Find the Rate: Divide the interest by the
interest on the principal for the given time at 1%.
The divisor should be carried to the fourth decimal place if
necessarv.
^0
RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
350. Problem : At what rate will $475 earn $8.84 in
134 days ?
$475.
4.75
9.50
.95
^ .1583
6)10.6083
$1.7680
134 da.
Int. for 60 da. at u/o
Int. for 120 da. at 6%
Int. for 12 da. at 6%
Int. for 2 da. at 6%
Int. for 134 da. at 6%
Int. for 134 da. at 1%
1.768)8.840
8.840
j^lXS , O /q.
The interest on $475 for 134 days at 1%
is $1.7680. The interest at 1% is contained in
the given interest 5 times ; therefore the inter-
est must be reckoned at 5% in order to produce
$8.84 in 134 days.
Note. For practice problems in finding the rate see par. 58.
351. To Find the Princijpal: Divide the Given
Interest by the hiterest on one dollar for the given
time at the given rate.
352. Divide the Given Amount bv the Amount of
one dollar for the given time at the given rate.
The divisor must be absolutely correct. The interest on $1
should be carried to three places and all fractions must he retained.
RATIONAL ARITHMETIC 91
ILLUSTRATED SOLUTIONS
353. Problem : What principal will be required to
earn $152.50 in 1 year 7 months 20 days at 8% ?
1.
.01 Int. for 2 mo. at 6%
.06 Int. for 1 yr. at 6%
.03 Int. for 6 mo. at 6%
.005 Int. for 1 mo. at 6%
.003i Int. for 20 da. at 6%
3).098i Int. for 1 yr. 7 mo. 20 da. at K^yo
.032J - Int. for 1 yr. 7 mo. 20 da. at 2%
$.131i Int. for 1 yr. 7 mo. 20 da. at 8%
.13H)152.50
$11 63|135
1.180)1372.5O|OOO
118
74 5
70 8
IQQ First find the interest on $1 for 1
1 1 o year 7 months and 20 days at 8% by
the sixty-day method, retaining all
fractions (par. 333). This is $.131^.
If $1 produces $.131^ in 1 year
^'^^ 7 months and 20 days at 8%, it will
•^ ^^ take as many dollars to produce $152.50
16 as $.131i is contained in $152.50.
11 8 Performing this division (par. 224), we
4 20 find that $1163.14 is the principal
3 54 required.
660
590
70
Arts. $1163.14.
92
RATIONAL ARITHMETIC
354. Problem : What principal will amount to $1250
in 287 days at 6% ?
1.
.01
.04
.005
.002i
.000^
1.047
Int. for 60 da. at \t-/o
Int. for 240 da. at 6%
Int. for 30 da. at 6%
Int. for 15 da. at 6%
Int. for 2 da. at 6%
Amount on $1 for 287 da. at
1.0471)1250.
$1 192 937
6.287)7500.000 000
6287
1213
628 7
584 30
565 83
18 470
12 574
5 896
5 658 3
237 70
188 61
49 090
44 009
First find the interest on $1 for
287 days at 6%. This is $.047f.
Add this to $1 and we find that $1
will amount to $1.047f.
It will take as many dollars to
amount to $1250 as $1.047f is con-
tained in $1250, which is $1192.94.
5 081
Ans. $1192.94.
RATIONAL ARITHMETIC 93
355. The work of finding the interest on $1 at 6%
may be simpHfied by using the following rule :
Multiply the number of years by 6. Call the result
cents.
Divide the number of months by 2. Call the result
cents.
Divide the number of days by 6. Call the result
mills, or tenths of a cent.
356. Applying this rule in the above illustrated
solutions we would have :
In the first : Find the interest on $1 for 1 year 7
months 20 days at 6%, thus :
6X1 = 6 written .06
7 -f- Q = 3i written .035
20 -^ 6 - 3i written .0331
.128^ = Int. on $1 at 6%
In the second : To find the interest on $1 for 287 days
at 6%, thus :
287-^6 = 471 written .0471 = Int. on $1 at 6%
Note. For practice problems in finding the principal see par. 59.
For general problems in interest see par. 60.
COMMERCIAL PAPERS
357. Commercial Papers comprise notes, drafts, and
checks.
358. A Note is a written promise of one party to
pay a second party a certain sum at a certain time.
There are two parties to a note.
359. The Maker is the party who promises to pay.
360. The Payee is the party to whom, or to whose
order, payment is to be made.
361. A Draft is a written order of one party telling
a second party to pay a third party a certain sum at a
certain time.
There are three parties to a draft.
362. The Drawer is the party requesting payment.
363. The Drawee is the party to whom this request
is addressed : the party told to pay.
364. The Payee is the party to whom, or to whose
order, payment is to be made.
365. A Sight Draft is a draft which by its terms is
to be paid by the drawee immediately upon its presen-
tation to him.
(a) In some states, three days, called days of grace, are allowed
on sight drafts.
94
RATIONAL ARITHMETIC 95
366. A Time Draft is a draft payable at a certain
time after presentation to the drawee, or after date.
367. The Date of Maturity of either a note or a
draft is the date upon which the payment is due.
(a) To fix the maturity of a time draft due "after sight," it is
necessary to present the draft to the drawee to see if he is wiUing
to pay it. If he is wiUing to pay it, he so indicates by writing the
word "accepted," together with the date, over his signature, across
the face.
(b) The maturity of a note is ascertained by reckoning the
specified time from date of the note.
(c) The maturity of a draft drawn "after sight" is ascertained
by reckoning the specified time from the date of acceptance.
(d) The maturity of a draft drawn "afterdate" is ascertained
by reckoning the specified time from the date of the draft.
368. The Face of a note or draft is the amount of
money mentioned in it.
369. The Amount due at maturity is the sum that
is to be paid. This may be the face, or it may be the
face plus interest.
370. A Check is an order by one who has funds on
deposit in a bank, telling the bank to pay a certain
sum from this deposit to a certain party. Checks are
treated as cash.
(a) The drawer of a check is the depositor.
(6) The drawee of a check is the bank where the funds are on
deposit.
(c) The payee of a check is the party in whose favor the check is
made : The party to whom funds from the deposit are to be paid
by the bank.
PARTIAL PAYMENTS
371. It is sometimes necessary to make a part pay-
ment on a promissory note or other obligation. It is
usually customary in such cases to cancel the original
note and issue another for the reduced amount. When
it is not feasible to do this, the amount of the part
payment, together with the date, is indorsed on the
back of the original instrument.
Various rules are in use for finding the balance due
on obligations upon which part payments have been
indorsed.
The more important of these are the United States
Rule and the Merchants' Rule. The United States
Rule has been sanctioned by the Supreme Court of
the United States. The Merchants' Rule is used by
most bankers and business men because of its sim-
plicity.
THE UNITED STATES RULE FOR PARTIAL PAYMENTS
372. To Find the Balance Due at a Given Time:
Find the interest on the principal from the date of
the instrument to the date of the first payment. If
this interest is less than the first payment, add the
interest to the original principal and subtract the
payment from this amount. Treat the remainder as
96
RATIONAL ARITHMETIC 97
a new principal and proceed as before, so continuing
until the date of settlement is reached.
If at any time the interest is greater than the pay-
ment, the interest should be disregarded and the
interest found to such date as the sum of the payments
exceeds the interest.
ILLUSTRATED SOLUTION
373. Problem : What is the balance due July 1, 1916,
on a note of $1200, dated January 1, 1914, upon which
the following payments have been made : June 24,
1914, $250 ; August 16, 1914, $100 ; July 8, 1915, $40 ;
January 1, 1916, $300.^
$1200 Face Jan. 1, 1914
34.60 Int. to June 24, 1914, 5 mo. 23 da.
$1234.60
250. 1st payment
984.60 New principal June 24, 1914
8.53 Int. to Aug. 16, 1914, 1 mo. 22 da.
993.13
100. 2d payment
893.13 New principal Aug. 16, 1914
73.68 Int. to Jan. 1, 1916, 1 yr. 4 mo. 15 da.
(Third payment not equal to interest)
966.81
340. 3d and 4th payments
626.81 New principal Jan. 1, 1916
18.8 Int. to July 1, 1916, 6 mo.
$645.61
Ans, $645.61.
98 RATIONAL ARITHMETIC
$1200 draws interest from January 1, 1914, to June 24, 1914.
We find this time (par. 238) to be 5 months and 23 days. The
interest on $1200 for 5 months and 23 days is $34.60 (par. 335),
making the amount due on June 24, 1914, $1234.60, upon which
$250 is paid, leaving the balance of $984.60. This is on interest
from June 24, 1914, to August 16, 1914, a period of 1 month and
22 days. The interest on $984.60 for 1 month and 22 days is
$8.53, making $993.13 due August 16, 1914. On this $100 is paid,
leaving a balance of $893.13 to draw interest from August 16, 1914.
The next payment is made on July 8, 1915. The interest on
$893.13 from August 16, 1914, to July 8, 1915, is greater than
the payment ; therefore we disregard the interest at this time and
find the interest to the date of the next payment, January 1, 1916.
The time from August 16, 1914, to January 1, 1916, is 1 year 4
months and 15 days. The interest on $893.13 for 1 year 4 months
and 15 days is $73.68, making $966.81 due. On this $40 was paid
on July 6, 1915, and $300 on January 1, 1916, which is $340 in all,
leaving $626.81 to draw interest from January 1, 1916, to the date
of settlement, July 1, 1916, 6 months. The interest on $626.81 for
6 months is $18.80, making $645.61 due on July 1, 1916.
Note. For practice problems in partial payments (United States Rule)
see par. 61.
MERCHANTS' RULE
374. To Find the Balance Due at a Given Time:
Find the interest on the face of the obligation from the
date at which it begins to draw interest to the date of
final settlement. Add this interest to the face of the
debt. Find the interest on each payment from the
date of the payment to the date of final settlement.
Add the payments and the interest on the payments.
Subtract this sum from the amount of the principal
and interest. The difference will be the balance
due.
RATIONAL ARITHMETIC 99
ILLUSTRATED SOLUTION
375. Problem : What is the balance due Julv 1,
1916, on a note of $1200, dated January 1, 1914, upon
which the following payments have been made : June
24, 1914, $250; August 16, 1914, $100; July 8, 1915,
$40; January 1, 1916, $300?
$1200 Face Jan. 1, 1914
180 Int. to July 1, 1916, 2 yr. 6 mo.
$1380
$250. Paid June 24, 1914
30.29 Int. to July 1, 1916, 2 yr. 7 da.
100. Paid Aug. 16, 1914
11.25 Int. to July 1, 1916, 1 yr. 10 mo. 15 da.
40. Paid Julv 8, 1915
2.35 Int. to July 1, 1916, 11 mo. 23 da.
300. Paid Jan. 1, 1916
9. Int. to Julv 1, 1916, 6 mo.
$742.89
$1380
742.89
$637.11 Balance due July 1, 1919.
A $1200 note given January 1, 1914, would earn $180 interest
to July 1, 1916, making its value at maturitj^ $1380. $250 paid
June 24, 1914, would earn $30.29, interest to July 1, 1916, a period
of 2 years 7 days. $100 paid August 16, 1914, would earn $11.25
to July 1, 1916. $40 paid on July 8, 1915, would accrue $2.35 in-
terest to July 16, 1916. $300 would accrue $9 interest to July 1,
1916. The payments and accrued interest amount to $742.89.
The value of the note at maturity, $1380, minus the payments
and accrued interest $742.89, leaves $637.11 due on July 1, 1916.
Note. For practice problems in partial payments (Merchants' Rule)
see par. 62.
BANK DISCOUNT
376. Bank Discount is the charge made by a bank
for cashing an obHgation before it is legally due. It is
the interest on the amount due at maturity for the
unexpired time.
377. The Maturity of a debt is the date upon which
it becomes legally due.
{a) A few states allow three days in addition to the time men-
tioned in a note or draft. Tliese are called days of grace.
{b) Most states allow days of grace on sight drafts only.
378. The Term of Discount is the number of days
between the date of discount and the date of maturity.
379. The Bank Discount is the interest on the
amount due at maturity for the term of discount.
380. The Proceeds is the difference between the
amount due at maturity and the bank discount. It
is the cash value of the debt on the date of discount.
381. To Find the Proceeds : Find the date of
maturity. Ascertain the amount due at maturity.
Find the time in exact days from the date of discount
to the date of maturity. Compute the interest on the
amount due at maturity for this time. The result will
be the bank discount. Deduct the bank discount
from the amount due at maturity. The result will be
the proceeds.
100
RATIONAL ARITHMETIC ' ^ ^ X5l - ,' :>
ILLUSTRATED SOLUTIONS , >'^ '■> '-> >'\}l^} i\5 > .V
382. Problem : Find the bank discount and the
proceeds of a note for 60 days for $5000, dated March 3,
1916, discounted April 1, 1916, at 5%. |
March 3, 1916 + 60 days = May 2, 1916 = date of i
maturity. i
April 1, 1916, to May 2, 1916 = 31 days = term of
discount '
A note given March 3,
$5000. 1916, for 60 days would
50. fall due on May 2, 1916,
25 30 days which would be the date
~ '8333 Idav f '"T'f, ^,t/™'
- — — " from April 1, 1916, to
6)25.8333 6^ jy^^^ 2^ l9jg^ j^ 3j ^^^.^
4.30oo 1% This is the term of dis-
21.5278 5% Bank Discount count. The interest on
$5000 for 31 days at 5%
$5000. is $21.53. This is the
21 53 bank discount. The face
$4978.47 Proceeds °f the note was $5000,
the bank discount $21.53 ;
Ans. $4978.47. the net proceeds would be i
the difference, $4978.47.
Note. For practice problems in finding the bank discount and proceeds
of non-interest-bearing notes see par. 63.
I
383. Problem : Find the bank discount and proceeds
of a sixty-day note for $5000, dated January 1, 1915, .
bearing interest at 6%, discounted February 6, 1915,
at- o /q.
January 1, 1915 + 60 days = March 2, 1915 j
February 6, 1915, to March 2, 1915 = 24 days
$5000 Face
50 Interest
$5050 Due
at Maturity
50.50
60 days
6)20.20
24 days at 6%
3.3666
$16.8334
Bank Discoun
$5050
16.83
■>
102 RATIONAL ARITHMETIC
A note, dated January 1,
1915, to run 60 days would
fall due on March 2, 1915.
If it were discounted on
February 6, 1915, it would
then have 24 days to run.
The term of discount, there-
fore, is 24 days. xA.s this
note is given with interest, its
face value, plus 60 days' inter-
est on $5000, is $5050. As
the amount due at maturity
^5033 17 Proceeds ^^ $5050, the bank discount
would be figured on this
Ans. $5033.17. amount for 24 days at 5%,
which is $16.83. The
amount due at maturity being $5050 and the bank discount
$16.83, the net proceeds would be the difference, or $5033.17.
Note. For practice problems in ifinding the bank discount and proceeds
of interest-bearing notes see par. 64.
384. In making a loan at a bank when a definite
amount is desired, the note must be made for a sum
that, when discounted, will leave as the net proceeds
the amount of loan desired.
To Find the Sum for Which a Note Must Be Drawn
So That, if Discounted at Date, the Proceeds Will Be a
Given Sum : Find the proceeds of a note for $1 for
the given time at the given rate. Divide the given
proceeds by this. The quotient will be the face
required.
The divisor must be absolutely correct. Carry the discount on
$1 to the third place and retain all fractions.
RATIONAL ARITHMETIC 103
ILLUSTRATED SOLUTION
385. Problem : For what sum must a ninety-day
note be drawn so that if discounted on its date at 4^%
the proceeds may be $1875 ?
$1.00
.01 60 days
.005 30 days
4).015 6%
.0031 li%
.01 li Bank Discount on $1
.9881 Proceeds on $1
$1875 --.9881
1 896 333
3.955)7500.000 000
3955
3545
3164
38100
355 95
25 050
23 730
1 320
1 186 5
133 50
118 65
14 850
11865
The bank discount on a note of $1 for
90 days at ^% would be $.01 1^. The
net proceeds of $1 would be $.988f.
If $1 yields proceeds of $.988f, it
would take as many dollars to yield
$1875 as $.988f is contained in $1875,
which is 1896.33, the face of the note
required.
2 985 Ans, $1896.33.
Note. For practice problems in finding the sum for which a note must
be drawn so that if discounted at date the proceeds will be a given sum, see
par. 65.
COMPOUND INTEREST
386. Compound Interest is interest on the principal
and interest combined, as fast as the interest falls
due.
(a) Compound interest can only be charged by special agreement,
and then care must be exercised that the laws of usury are not
violated.
{b) Interest is usually compounded annually, semi-annually, or
quarterly.
(c) Compound interest is little used except in savings banks.
387. To Find Compound Interest : Find the amount
of the principal and interest at the end of the first
interest period. Use this amount as a new principal
for the next period, and so on. Deduct the original
principal from the final amount. The difference will
be the compound interest.
ILLUSTRATED SOLUTION
388. Problem : Find the compound interest on
$1400 from March 8, 1912, to June 15, 1916, at 5%,
interest compounded annually.
1916 6 15
1912 3 8
4 3 7 = Four full periods and 3 mo. 7 da. extra
104
RATIONAL ARITHMETIC
105
$1400
70
$1470
73.50
$1543.50
77.18
$1620.68
81.03
$1701.71
17.0171
17.0171
8.5085
1.7017
.2836
6)27.5109
4.5851
22.9258
$1724.6358
1400
Original Principal
Int. at 5% for 1st period
New Principal
Int. at 5% for 2d period
New Principal
Int. at 5% for 3d period
New Principal
Int. at 5% for 4th period
New Principal
Int. for 2 mo. at
Int. for 2 mo. at
Int. for 1 mo. at
Int. for 6 da. at
Int. for 1 da. at u/o
Int. for 3 mo. 7 da. at u/o
Int. for 3 mo. 7 da. at 1%
Int. for 3 mo. 7 da. at 5%
Final Amount
$324.6358 Compound Int. Ans. $324.64.
Note. For practice problems in compound interest see par. 66.
PERIODIC OR ANNUAL INTEREST
389. Periodic Interest, often called annual interest,
is interest on the principal and interest on each overdue
payment of interest.
(a) It is the result of a business custom in certain lines, and is
merely an application of the general principles of simple interest.
(6) It is not entitled to be considered as a separate arithmetical
subdivision, and is only so treated in this book because many authors
have seen fit to introduce it as a separate kind of interest.
(c) It has no legal status.
106 RATIONAL ARITHMETIC
ILLUSTRATED SOLUTIONS
390. Problem : $1400 was loaned on January 8,
1915, for 2 years at 6%, interest payable semi-annually ;
each installment of interest to draw interest at 6%
from its due date until paid. What sum would be
required to cancel the debt and all interest on January 8,
1917, nothing having been paid previously ?
$1400.
14. Int. for 2 mo. at 6%
$42. Int. for 6 mo. at 6%
4.
$168. Int. for 2 yr. at 6%
1 yr. 6 mo. — 1st installment overdue
1 yr. — 2d
6 mo. — 3d
2 yr. 12 mo. = 3 yr.
$42.
'0
.42 Int. on interest for 2 mo. at 6%
2.52 Int. on interest for 1 yr. at
5.04 Int. on interest for 2 yr. at
7.56 Int. on interest for 3 yr. at
168. Int. on the Principal
1400. Principal
$1575.56 Amount due Jan. 8, 1917
Ans. $1575.56.
The interest on $1400 for 6 months at 6% is $42. Then $42
should be paid every 6 months. There would be four such pay-
ments due in 2 years. 4 X42 = $168, which must be paid as interest
RATIONAL ARITHMETIC 107
on the principal. The first installment of this, amounting to $42,
was due on July 8, 1915. Tliis is overdue 1 year 6 months. The
next installment of $42 is overdue 1 year ; the next, 6 months ;
and the last is just due. Besides the interest on the principal,
then, there is interest on the interest due. This last is the interest
on $42 for 1 year 6 months, and for 1 year, and for 6 months. In
other words, the interest on the interest equals the interest on $42
for 3 years, w^iich is $7.56. Adding interest on interest, interest
on principal, and principal gives $1575.56, the amount due at
maturity.
Note. For practice problems in periodic interest see par. 67.
AVERAGING ACCOUNTS
391. Averaging of Accounts is the process of ascer-
taining the date on which an account may be paid
without loss of interest to either the debtor or the
creditor. ^
Averaging of accounts has been abandoned by most lines of busi-
ness. It is now the general custom to settle each item, separately,
at maturity. The subject, however, is not entirely obsolete.
392. Cash Balance is the amount of cash required
to settle an account without loss of interest to either
party on any date other than the average due date.
GENERAL PRINCIPLES OF AVERAGE
393. If an account is paid before it is due, the payer
loses the interest on the sum paid, and the receiver
gains it.
394. If an account is paid after it is due, the payer
gains the interest on the money paid and the receiver
loses it.
395. The average due date of several items, due at
different dates, is the date when the payer's losses of
interest and gains of interest would be equal, or within,
one half day's interest of being equal.
108
RATIONAL ARITHMETIC
109
396. To Average an Account: Assume a date of
settlement. Find the net gain or loss of interest by
paying on that date. Find how long it will take the
amount to be paid to earn this interest. Count that
time forward or backward from the assumed date
according to whether the payer would lose or gain.
(a) The assumed date Is called the focal date.
{h) The focal date may be any date.
(c) The easiest focal date to use is the zero date of the earliest
month in which any one of the items is due ; thus, if items are due
June 8, July 15, and August 9, the easiest focal date to use would
be June 0, which is, in reality. May 31.
ILLUSTRATED SOLUTIONS
397. Problem : Average the follow^ing :
Charles S. Chase
1916
Jan. 5
$434.
27
123.50
Feb. 8
215.
Apr. 9
310.65
Focal date Jan. 0.
Jan.
5
434.
5 da.
.3616
27
123.50
27 da
.5557
Feb.
8
215.
39 da.
1.3975
Apr.
9
310.65
99 da.
5.1256
1083.15
7.4404
110 RATIONAL ARITHMETIC
41
6 )1.0831 Int. for 6 da. .1805)7.4404
.1805 Int. for 1 da. 7 220
2204
1805
399
Ans. Jan. 0+41= Feb. 10.
Assuming Januarj^ as the focal date, on the first item, $434,
Chase would lose 5 days' interest, because he would pay it 5 days
before it was due. The interest on $434 for 5 days is $.3616. By
paying $123.50 on January 0, the interest for 27 days would be
lost, which would be $.5557. By paying $215, due February 8, on
January 0, the interest for 39 days would be lost, which would equal
$1.3975. If $310.65 is paid 99 days before it is due, the interest
lost would be $5.1256. Then by paying $1083.15 on January 0,
$7.4404 interest would be lost. The interest on $1083.15 for one
day is $.1805. It will take as many days to earn $7.4404 as
$.1805 is contained in $7.4401, which is 41 days. Then Chase
should pay the money 41 days later than January 0, which is
February 10.
Note. For practice problems in averaging accounts see par. 68.
398. Problem : Average the following :
Find cash balance on Jan. 1, 1916.
1915
Sept. 4
$625
for 2 mo.
Credit
Oct. 23
350
for 30 da.
Credit
Nov. 18
215
for 10 da.
Credit
Dec. 8
643
for 60 da.
Credit
RATIONAL ARITHMETIC 111
Sept.
4
2 mo.
Nov.
4
$ 625
4 da.
$ .4166
Oct.
23
30 da.
Nov.
22
350
22 da.
1.2832
Nov.
18
10 da.
Nov.
28
215
28 da.
1.0032
Dec.
8
60 da.
Feb.
6
643
98 da.
10.5023
Focal date Nov. 0.
$1833 $13.2053
43
)1.833
Int. for 6 da.
.3055)13.2053
.3055
Int. for 1 da.
12 220
9853
9165
688
Nov. + 43 da. = Dec. 13.
Dec. 18 1833. 19 days at 6%
Jan. J. 18.33 Int. for 60 da. at 6%
19 days 4.5825 Int. for 15 da. at 6%
.9165 Int. for 3 da. at 6%
.3055 Int. for 1 da. at 6%
5.8045 Int. for 19 da. at 6%
1833
$1838.80 Cash Bal. Jan. 1, 1916
. f Dec. 13, 1915.
^^^' I $1838.80.
Goods billed on September 4 for 2 months' credit would be due
on November 4. A bill bought October 23 on 30 days' credit
would be due November 22. A bill bought November 18 on 10
days' credit would be due November 28. A bill bought December 8
on 60 days' credit would be due on February 6, Assuming Novem-
ber as the focal date and averaging as in par. 397, we find the
average due date to be December 13, 1915.
If the debtor should pay $1833 on December 13, but did not do
112 RATIONAL ARITHMETIC
so until January 1, he would owe $1833 plus 19 days' interest. The
interest on $1833 for 19 days is $5.8045, making the amount due
January 1, $1838.80.
Note. For practice problems in averaging accounts and finding the
cash balance see par. 69.
399. Problem :
Dr. C. E. Batchelor Cr.
1915
1915
May 8
30
days
$525.30
May
15
Cash
$300.
June 7
60
days
415.40
June
5
Cash
425.50
When is the above due by average ?
What was the cash balance December 15, 1915 .^^
May 8+30 da. = June 7 $525.30 38 $3.3269
June 7 + 60 da. = Aug. 6 415.40 98 6.7848
940.70
10.1117
725.50
3.303
$215.20
$6.8087
May 15 $300
15 $.75
June 5 425.50
36 2.553
$725.50 $3,303
Focal date May 0.
190
6 ).2152 Int. for 6 da. at 6% .0358)6.8087
.0358 Int. for 1 da. at 6% 3 58
3 228
3 222
67
RATIONAL ARITHMETIC 113
May + 190 days = Nov. 6, 1915.
Nov. 6 to Dec. 15 is 39 days.
$^15.20
2.15 Int. for 60 da. at 6%
1.076 Int. for 30 da. at 6%
.2152 Int. for 6 da. at 6%
.1076 Int. for 3 da. at 6%
1.3988 Int. for 39 da. at 6%
215.20
$216.60 Cash Bal. Dec. 15, 1915
Ans.
Nov. 6, 1915.
$216.60.
In the above account C. E. Batchelor owes the items on the
debit side of the account and we owe him, theoretically, the items
on the credit side. The debit side being the larger, he owes us the
balance of the account. First find the due dates of all the items.
On the debit side $5'25. 30 is due June 7. $415.40 is due August 6.
On the credit side $300 is due May 15. $4*25.50 was due June 5.
Assuming the zero date of the earliest month as the focal date, this
w^ould be May 0. Batchelor, by paying $525.30 on May 0, would
lose 38 days' interest, which is $3.3269, and by paying $415,40 on
May he would lose 98 daj's' interest, which is $6.7848. If the
account were settled on May 0, Batclielor would gain the interest
on $300 for 15 days, which is $.75 and he would gain the interest
on $425.50 for 36 days, which is $2,553. This would amount to
$3,303. The balance of the account is $215.20. If Batchelor paid
this on May 0, he would lose $10.1117 and gain $3,303, or he would
make a net loss of $6.8087. The interest on $215.20 for one day
is $.0358. To make up $6.8087, it would take as many days as
$.0358 is contained in $6.8087, which is 190 times, or 190 days.
Since by settling the account on May the interest for 190 days is
lost, it should be settled 190 days later, which would be Novem-
ber 6.
114
RATIONAL ARITHMETIC
Whenever the balance of interest and balance of
account fall on the same side, the payer will lose the
interest and the time should then be counted forward.
400. Problem
Dr.
F. H. Bray
Cr.
1916
1916
June
6
2
mo.
$2200
Aug.
12
Cash
$ 108
Aug.
12
12
da.
1400
Sept.
o
Cash
2892
Find the cash balance for Sept. 12, 1916, at ytyc.
Focal date August 0.
Aug. 6
24
$2200
1400
$3600
3000
$ 600
6
24
2.20
5.60
7.80
Aug. 12 $ 108 12 .216
Sept. 5 2892 36 17.352
$3000
17.568
7.80
9.768
6 ). 600 Int. for 6 da. at u/o
.10 Int. for 1 da. at 6%
97
.10)9.76
90
76
70
6
8
68
RATIONAL ARITHMETIC 115
97.6 days = 98 days.
98 days counted backward from Aug. 0= April 24,
Average Date.
April 24 to Sept. 12 = 141 days.
Balance of account
600
$600.
6.
14.10
12. --
= 120
$614.10
2. =
= 20
.10 =
= 1
14.10 =
= Int. for 141 da
•
Ans. $614.10.
The above problem is similar to 399, except that the balance
of account and the balance of interest fall on opposite sides. This
amount shows that F. H. Bray owes the balance of $600. By pay-
ing this balance on August he would lose the interest on $2''200
for 6 days and the interest on $1400 for 24 days, or $7.80 on both
items, because he would be paying before due. However, his loss
of interest on the items on the debit side of the account is more
than offset by his gain of interest on the items on the credit side
(the interest on $108 for 12 days and the interest on $2892 for 26
days) which is $17,568. The difference between $17,568 and $7.80
is $9,768, which is the amount of interest that Bray would gain by
paying the balance on August 0. The balance, $600, requires 98
days to earn $9,768. Then Bray, in order to neither gain nor
lose, should settle the account 98 days before iVugust 0, which is
April 24.
When the balance of account and the balance of
interest fall on opposite sides, count the time backward
from the focal date.
Note. For practice problems in finding the amount due by average and
the cash balance of two-sided accounts see par. 70.
TAXES
401. A Tax is a sum of money levied upon a citizen
or his property to meet the expenses of maintaining
the Government.
(a) Taxes are levied to pay the expenses of the city, county,
state, and United States.
(h) The first three are levied directly upon the person, property,
and, in some cases, the income of the individual.
(c) In case of the United States, the tax is levied through duties
and customs and by a tax on incomes.
402. A Poll Tax is a tax levied on the person, and
in most states is assessed upon all male citizens of
20 years of age or more.
403. A Property Tax is a tax assessed on property,
either real or personal, and is levied upon all persons
owning taxable property, irrespective of age or sex.
404. An Income Tax is a tax upon incomes, and is
levied alike on all citizens receiving certain incomes,
regardless of age or sex.
Income taxes are levied by the United States Government and by
some states, in accordance with laws passed by Congress or State
Legislatures.
Note. For practice problems in taxes see par. 71.
116
RATIONAL ARITHMETIC 117
DUTIES AND CUSTOMS
405. Duties and Customs are taxes assessed by the
United States Government on imported merchandise.
406. An Ad Valorem Duty is a certain percentage
of the net cost (value) of the importation.
407. A Specific Duty is a specified sum levied on
each article, or on each unit of measure, regardless of
the value.
(a) Ad valorem duties are not computed on fractions of a dollar.
Cents are disregarded for less than 50 and are considered another
dollar for more than 50.
(b) Some articles are subject to both ad valorem and specific
duties.
(c) Specific duties are not computed on fractions of a unit. The
long ton, or 2240 pounds, is used in computing specific duties.
408. A Tariff is a schedule showing the different
rates of duties imposed by Congress on different articles.
409. A Free List is a schedule of articles upon which
no duties are to be levied.
410. A Customhouse is a branch office of the
Treasury Department of the United States Govern-
ment.
Customhouses are established at various ports ; each custom-
house has jurisdiction over certain territory.
411. A Port of Entry is a port where a customhouse
is established.
412. All ports, whether of entry or otherwise, are
called ports of delivery.
118 RATIONAL ARITHMETIC
413. The Customhouse Business is distributed
among three departments.
414. The Collector's Office takes charge of entries
and papers, issues permits, and collects the duties.
415. The Surveyor's Office takes charge of the
vessels and cargoes, receives the permits, ascertains
the quantities, and delivers the merchandise to the
importer.
416. The Appraiser's Office examines the mer-
chandise and determines the value and rate of duty
on the goods.
417. Internal Revenue is a revenue raised by the
Government by placing duties on such articles of
luxury as may be determined by Congress. These
duties are collected by the Treasury Department and
vary from time to time according to the needs of the
country.
418. A Manifest is a memorandum, signed by the
master of the vessel, showing the name of the vessel,
its cargo, and the names and addresses of the con-
signors and consignees.
419. An Invoice is a detailed statement showing the
items and value of the goods imported and is made
out in the weights and measures of the country of
export.
The values of foreign moneys are periodically proclaimed
by the Secretary of the Treasury, and these values must be taken
in estimating duties. See par. 468.
RATIONAL ARITHMETIC 119
420. A Bonded Warehouse is a warehouse provided
for the storage of goods upon which duties have not
yet been paid.
(a) Any importer may deposit goods in the warehouse by
giving bond for the payment of duties on the goods thus stored.
(6) On goods remaining in bond more than a year, 10% addi-
tional duty is charged.
(c) Goods left in a government warehouse for three years are
forfeited to the Government and sold at auction.
(d) Goods may be withdrawn from a warehouse for export
without payment of duty.
421. An Excise Duty is a tax levied upon goods pro-
duced and consumed in the United States.
In this class come taxes upon tobacco and such articles of
luxury as Congress may from time to time prescribe.
422. If goods on which either excise or import
duties have been paid are exported, the amount of duty
is refunded. This is called a drawback.
Note. For practice problems in duties and customs see par. 72.
INSURA.NCE
•
423. Insurance is a contract bj^ which one party
(the insurer) agrees to reimburse another party (the
insured) in case of damage or loss to the latter's prop-
erty or person.
424. The insurance business is usually conducted
by corporations called Insurance Companies, which
limit their operations to certain classes of risks. Some
companies handle fire insurance, others marine in-
surance, others accident insurance, others life insurance,
etc.
425. There are two kinds of insurance companies,
stock companies and mutual companies.
426. A Stock Company is one whose capital is owned
by stockholders who share the profits and who are
liable for the losses.
427. A Mutual Insurance Company is one in which
there are no stockholders, but in which the parties
insured share the profits and losses.
428. A Policy is the contract of insurance.
429. A Premium is the sum paid for the insurance.
It is the consideration.
120
RATIONAL ARITHMETIC 121
(a) Sometimes the rate is expressed as a certain percentage
of the value insured and sometimes at so much per hundred dollars
of insurance.
(b) The rate of premium depends upon the amount and the
nature of the risk and the length of time for which the risk is taken.
430. Fire Insurance is insurance against loss or
damage by fire, or from the means employed for ex-
tinguishing it, or to prevent its spread.
431. Owners of property may insure in one or more
companies. When the risk is placed in several com-
panies, care should be taken to have the policies
uniform in every particular. Each company will then
pay such part of the total loss as its risk is of the total
risk,
432. The Average Clause, contained in manj^ policies,
is to the effect that the liability of the company in case
of a partial loss shall be such part of the loss as the insured
value is of the actual value of the property.
Thus, a building worth $20,000 is insured for $18,000, which
is nine-tenths or 90% of the value. In case of loss the company
would pay, under the average clause, 90% of the loss.
433. Short Rate is the rate for less than a year.
434. Marine Insurance is insurance against loss or
damage to a vessel or her cargo by storm or other
dangers of the sea.
Marine policies always contain the average clause.
Note. For practice problems in insurance see par. 73.
122 RATIONAL ARITHMETIC
LIFE INSURANCE
435. Under this head are considered life insurance^
accident insurance, and health insurance.
436. Life Insurance is indemnity for loss of life.
437. Accident Insurance is indemnity for loss or
disability caused by accident.
438. Health Insurance is indemnity for loss oc-
casioned by sickness.
439. There are two kinds of life insurance policies,
life policies and endowment policies.
(a) Under the ordinary life policy, premiums are paid annually
during the life of the insured.
{b) There is another kind of life policy known as the Limited
Payment Life Policy. Under this policy the premiums are paid
during a certain number of years only.
440. A Life Policy is a contract on the part of the
insurance company to pay the beneficiary a designated
sum upon the death of the insured.
441. An Endowment Policy is a contract on the part
of the insurance company to pay the beneficiary at
the death of the insured, or after the lapse of an
agreed period of time, if the insured is then alive.
442. The following table shows the rates charged by
an insurance company :
RATIONAL ARITHMETIC
ns
LIFE
PREMIUMS
ENDOWMENT PREMIUMS
Insurance of $1000, payable at death only
Insurance of $1000, payable as
specified or on prior decease
Annual Premiums
DURING
Annual Payments
Age
Age
Life
10 Years
20 Years
In 15 Years
In 20 Years
20
$18.95
$43.85
$27.65
20
$68.10
$49.45
21
19.35
44.55
28.10
21
68.20
49.55
22
19.75
45.25
28.55
22
68.25
49.65
23
20.20
46.00
29.00
23
68.35
49.75
24
20.65
46.75
29.55
24
68.45
49.85
25
21.15
47.55
30.05
25
68.55
50.00
26
21.65
48.40
30.60
26
68.70
50.10
27
22.20
49.25
31.15
27
68.80
50.25
28
22.75
50.15
31.75
28
68.95
50.40
29
23.35
51.10
32.35
29
69.10
50.55
30
23.95
52.05
33.00
30
69.25
50.75
31
24.60
53.05
33.65
31
69.40
50.95
32
25.30
54.10
34.35
32
69.55
51.15
33
26.05
55.20
35.05
33
69.75
51.35
34
26.80
56.30
35.80
34
69.95
51.60
35
27.65
57.45
36.60
35
70.20
51.90
36
28.50
58.65
37.45
36
70.40
52.15
37
29.40
59.95
38.30
37
70.70
52.50
38
30.35
61.25
39.20
38
71.00
52.85
39
31.40
62.60
40.15
39
71.30
53.25
40
32.50
64.00
41.20
40
71.65
53.70
By this table the premium on an ordinary life policy of $ 1000 at
the age of 20 would be $18.95, for a $5000 policy the premium
would be 5 X $18.95, or $94.75.
At the same age, a $1000 ten-payment life policy would cost
$43.85 per year, and a $5000 policy 5 X $43.85, or $2 19. 25 per year.
A $1000 fifteen-year endowment policy would cost $68.10 a year,
and a $5000 policy would cost $340.50.
Note. For practice problems in life insurance see par. 74.
EXCHANGE
443. Exchange is a system of paying debts in distant
places by means of drafts.
444. A Draft, or Bill of Exchange, is an order of
one party directing a second party to pay a third
party a certain sum of money at a certain time.
445. There are two kinds of bills of exchange, do-
mestic and foreign.
446. A Domestic Bill of Exchange, sometimes called
an inland hill of exchange, is one drawn and payable
in the same state or country.
447. Foreign Bills of Exchange are those drawn in
one state or country and payable in another state or
country.
448. The Face or Par Value of a bill of exchange is
the sum of money for which it is written.
Bills of exchange are always written in the coinage of the country
in which they are to be paid.
DOMESTIC EXCHANGE
449. Domestic Exchange quoted at a premium is
worth the given percentage of the face more than the
face.
124
RATIONAL ARITHMETIC 125
Domestic Exchange quoted at a discount is worth
the quoted percentage of the face less than the face.
(a) Thus, ^% premium means that $1 would cost $1,005. Ex-
change quoted at |% discount would mean that $1 would cost
$.995.
(b) When Boston owes New York the same that New York
owes Boston, exchange will be at par in botli places. AMien Boston
owes New York more than New York owes Boston, exchange on
New York will be at a premium in Boston, since there will be more
buyers of New York exchange than sellers ; and when New York
owes Boston more money than Boston owes New York, exchange on
New York will sell at a discount in Boston and exchange on Boston
will sell at a premium in New York.
The general principles of percentage (pars. 242-254)
are used in solving problems in exchange.
Note. For practice problems in domestic exchange see pars. 75 to 80
inclusive.
FOREIGN EXCHANGE
450. Foreign Exchange is exchange drawn in one
country and payable in another country.
Foreign bills are always made in the coinage of the country
where they are to be paid.
451. The Intrinsic Par of Exchange is the actual
value of the money of one country expressed in the
monev of another.
Intrinsic value of foreign money expressed in the money of the
United States will be found in the table, par. 468.
452. Commercial Rate of Exchange is the market
value of the money of one country expressed in the
money of another.
126 RATIONAL ARITHMETIC
(a) This value changes from time to time, according to the
demand that may exist and according to the different conditions
of commerce that may arise.
(&) The rate of exchange on Great Britain is expressed by giv-
ing the market value of a pound in United States money.
(c) On France, Belgium, and Switzerland the rate of exchange
is expressed by giving the number of francs that may be secured
for $1.
(d) On Germany the rate of exchange is expressed by giving
the market value of 4 reichsmarks in United States money.
(e) On Holland the rate of exchange is expressed by giving the
value of 1 guilder in United States money.
(/) Gold is exported at a profit when the cost of foreign ex-
change is enough greater than the intrinsic value of the bill to pay
the cost of safe shipment and yet leave a margin ; and gold is im-
ported at a profit when the cost of exchange is enough less than the
intrinsic value of the bill to pay the same expenses and leave a
margin. Thus under normal conditions, the commercial rate is not
allowed to vary from the intrinsic par by more than enough to pay
the expense of shipping gold.
Note. For practice problems in foreign exchange see pars. 81 to 85 in-
clusive.
STOCKS AND BONDS
453. A Corporation is an association of persons
authorized by law to act as one person.
454. The Capital Stock of a corporation is the value
of its investment. This is divided into equal parts
called shares.
455. The Par Value of a share of stock is the value
placed upon each share at the time of the original
division of its capital stock.
The usual par value of one share of stock is $100. Stock is
frequently issued in other sized shares, however, usually $50, $25,
$10, $5, or $1.
456. A Stock Certificate is a document issued by
the company to the shareholder specifying the number
and par value of the shares to which he is entitled.
457. The Market Value of a share is the sum for
which it will sell in the open market.
Sometimes stock is worth more than par and sometimes less.
This depends upon the condition of the business.
458. A Dividend is that part of the net earnings of a
corporation that is divided among its stockholders.
127
128 RATIONAL ARITHMETIC
459. An Assessment is a sum levied upon the stock-
holders to make up losses.
There are two kinds of stock, common and preferred.
460. Common Stock participates in the net earnings
of the company, after all other expenses have been
met, in such proportion as the directors of the cor-
poration may determine.
461. Preferred Stock participates in the net earnings
of the corporation at a fixed rate before any dividend
may be declared on the common stock.
462. A Bond is an obligation of a corporation to
pay money on a long term of credit.
(a) Bonds are usually secured by deeds of trust and mortgages.
They are generally issued as securities for loans. They are similar
to promissory notes, but are more formal and are also made under
seal.
(6) Bonds are usually issued in $500, $1000, or multiples thereof.
(c) Quotations on bonds are given on $100 par value.
{d) Bonds are issued in two classes, registered and coupon.
463. Registered Bonds are those payable to the
order of the owner and can be transferred only by
acknowledged assignment.
Interest on registered bonds is paid by check from the corpo-
ration made to the holder of record.
464. A Coupon Bond is one made payable to the
bearer, and has interest certificates attached. These
certificates, called coupons, are to be cut off as they
RATIONAL ARITHMETIC 129
become due and presented at the designated place for
payment.
(a) Bonds are named from the nature of the security ; the
name of the corporation issuing tliem ; the date on which they are
payable ; the rate of interest they bear ; or the purpose for which
they are issued.
(h) Both stocks and bonds are quoted at some per cent of par
value.
(c) The regular commission allowed to brokers for buying or
selling either stocks or bonds is |% of the par value. There is a
minimum charge for small transactions, however.
(d) Dividends and assessments are always figured on the par
value of the stock.
Note. For practice problems in stocks and bonds see par. 86.
TABLES
UNITED STATES MONEYS
465. United States Money consists of gold coins,
silver coins, United States Treasury notes and certifi-
cates, and national bank notes.
The unit of measure is the gold dollar of 25.8 grains.
10 mills =lcent(^) $ .01
10 cents =1 dime (d.) $ .10
10 dimes =1 dollar ($) $ 1.
10 dollars = 1 eagle (e.) $10.
20 dollars = 1 double eagle (d. e.) $20.
ENGLISH MONEYS
466. English or Sterling Money is the legal currency
of Great Britain.
The unit of measure is the pound, worth $4.8665 in
United States money.
4 farthings = 1 penny (d)
12 pence = 1 shilling (s)
20 shillings = 1 pound (£)
Note. 21 shillings = 1 guinea (used in the retail trade).
130
RATIONAL ARITHMETIC 131
FOREIGN MONEYS
467. Once each year the Director of the United
States Mint is required to compare the values of foreign
coins with the United States Gold Dollar and certify
the result of his comparison to the Secretary of the
Treasury, who then proclaims the value of foreign
money thus found to be the value to be used in esti-
mating the worth of all foreign merchandise imported.
Values thus found are called intrinsic or real values
and should be distinguished from commercial or ex-
change values.
468. The table of values on pages 132 and 133 was
proclaimed Oct. 1, 1918.
WEIGHT
Troy Weight
469. Troy Weight is used in weighing precious
metals.
TABLE
24 grains (gr.) = 1 pennyweight (dwt.)
20 penny weights = 1 ounce (oz.)
12 ounces = 1 pound (lb.)
Diamond Weight
470. Diamond Weight is used in weighing precious
stones.
The unit is 3^ Troy grains and is called a carat.
This carat is not the same as that used in estimating the rela-
tive purity of gold in coins and jewelry. Pure gold is 24 carats
fine ; 18 carats fine means ^f pure gold and ^ alloy.
132
RATIONAL ARITHMETIC
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134 RATIONAL ARITHMETIC
Apothecaries' Weight
471. Apothecaries' Weight is used by physicians and
apothecaries in writing and preparing prescriptions for
dry medicines.
TABLE
20 grains (gr.) = 1 scruple (sc. or 3)
3 scruples = 1 dram (dr. or 3)
8 drams = 1 ounce (oz. or §)
12 ounces =1 pound (lb.)
Avoirdupois Weight
472. Avoir d 'pois Weight is used in commerce in all
cases excepting those requiring Troy or Apothecaries'
weight.
TABLE
16 ounces =1 pound (lb.)
25 pounds = 1 quarter (qr.)
4 quarters = 1 hundredweight (cwt.)
20 hundredweights = 1 ton (T.)
2240 pounds = 1 long ton
473. Comparison of Troy and Avoirdupois Weights.
1 pound Troy = 5760 grains
1 pound Avoirdupois = 7000 grains
1 ounce Troy = 427 grains
1 ounce Avoirdupois = 480 grains
474. The following table shows the weight of a
bushel used commercially in measuring grain and
other farm products :
RATIONAL ARITHMETIC
135
Barley
48 1b.
Oats
321b
Beans
60 "
Onions
57 "
Buckwheat
48 "
Peas
60 "
Clover Seed
60 "
Potatoes
60 "
Corn, shelled
56 "
'I'imothy Seed
45 "
Corn, in the ear
70 "
Rye
56 "
Corn Meal
50 "
Rye Meal
50 "
Flaxseed
56 "
Wheat
60 "
Hemp Seed
44 "
Wheat Bran
20 "
Malt
34 "
Liquid Measure
475. Liquid Measure is used in measuring liquids.
TABLE
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
476. Standard liquid gallon contains 231 cubic
inches.
There are various kinds of casks for containing
liquids. In commerce each is gauged and its capacity
marked upon it. The various kinds of casks are :
1 lerce
about
42 gal.
Puncheon
84 "
Pipe
126 "
Butt
126 "
Tun
252 "
Hogshead (hhd.)
63 "
136 RATIONAL ARITHMETIC
Apothecaries' Liquid Measure
477. Apothecaries'' Liquid Measure is used in pre-
scribing and compounding liquid medicines.
TABLE
60 minims ("l) =1 fluid drachm (f3)
8 fluid drachms = 1 fluid ounce (f i)
16 fluid ounces = 1 pint (O)
8 pints = 1 gallon (Cong.)
The gallon of this measure is the same as the gallon
of the liquid measure.
Dry Measure
478. Dry Measure is used in measuring grain, fruits,
vegetables, etc., which are not sold by weight.
TABLE
2 pints (pt.) = 1 quart (qt.)
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
Long Measure
479. Long Measure is used in measuring lengths, or
distances.
TABLE
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
5i yards (16i ft.) = 1 rod (rd.)
40 rods = 1 furlong (fur.)
8 furlongs (320 rods) = 1 mile (mi.)
RATIONAL ARITHMETIC 137
Surveyors' Long Measure
480. Surveyors' Long Measure is used by surveyors
in measuring distances.
TABLE
7.92 inches = 1 link (1.)
100 links = 1 chain (ch.)
80 chains = 1 mile (mi.)
Square Measure
481. Square Measure is used in measuring extent of
surfaces.
TABLE
144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
30i square yards = 1 square rod (sq. rd.)
40 square rods = 1 rood (R.)
4 roods = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
Surveyors' Square Measure
482. Surveyors' Square Measure is used by surveyors
in finding the area of land.
TABLE
625 square links (sq. 1.) = 1 square rod or pole (sq. rd.
or p.)
16 poles =1 square chain (sq. ch.)
10 square chains = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
138 RATIONAL ARITHMETIC
Cubic Measure
483. Cubic Measure is used in measuring the con-
tents of anything which has length, breadth, and
thickness.
TABLE
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
Wood Measure
484. Wood Measure is used in measuring wood.
TABLE
16 cubic feet = 1 cord foot (cd. ft.)
8 cord feet (128 cu. ft.) = 1 cord (cd.)
A cord of wood is a pile 8 feet long, 4 feet wide, and
4 feet high, or its equivalent.
485.
TIME
60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day (da.)
7 days = 1 week (wk.)
30 days = 1 month (mo.)
52 weeks = 1 year (yr.)
12 months = 1 year (yr.)
365 days = 1 common year
366 days = 1 leap year
100 years = 1 century
RATIONAL ARITHMETIC 139
486. The day is the time during which the earth
makes one revolution on its own axis.
487. The Solar Year is the time the earth requires
to make one complete revolution around the sun. It
actually takes the earth 365^ days to make this revolu-
tion. Therefore, every fourth year is given 366 days.
This extra day is added to the month of February, the
shortest month, and the year is called Leap Year.
These figures are not absolutely accurate but are practically
so.
488. Any year whose number can be divided by 4
is a leap year, except that a century year must be
divisible by 400.
(a) The year 1916 could be divided by 4 and was a leap year,
while 1915 could not be divided bv 4 and was an ordinarv year.
(b) The year 1900 was divisible by 4 but not by 400 and was
not, therefore, a leap year. The year 2000 being divisible by 4
and 400 will be a leap year.
489. Months of the Year, and Days in Each :
1.
Januarv
31
7.
July
31
2.
February
28 or 29
8.
August
31
3.
March
31
9.
September
30
4.
April
30
10.
October
31
5.
Mav
31
11.
November
30
6.
June
30
12.
December
31
140 RATIONAL ARITHMETIC
MISCELLANEOUS
490. Some articles are sold by quantity according
to the following table :
TABLE
12 units = 1 dozen (doz.)
12 dozen = 1 gross (gr.)
12 gross = 1 great gross (g. gr.)
20 units = 1 score
Paper Measure
491. Paper is measured according to the following
table :
TABLE
24 sheets = 1 quire (qu.)
20 quires = 1 ream (rm.)
2 reams = 1 bundle (bdl.)
5 bundles = 1 bale (bl.)
THE METRIC SYSTEM
492. The Metric System is a decimal system of
weights and measures, similar to the decimal system
used in measuring United States money. It was
originated in France early in the nineteenth century,
and has been adopted by nearly all the commercial
nations except United States and England.
The Metric System was made legal in the United
States in 1866, but is not generally used except in
scientific work.
RATIONAL ARITHMETIC 141
493. The Meter. The basic unit is the meter.
The other units, those of weight and of capacity, are
based on it.
494. The length of the meter was originally deter-
mined by taking one ten-millionth of the distance
from the equator to the pole. This length is 39.37
inches.
495. The primary units are :
For length — meter
For capacity — liter
For weight — gram
496. The desired integral multiples of these are
formed by using the following Greek prefixes :
Deca =10 (decameter = 10 meters)
Hecto = 100 (hectometer = 100 meters)
Kilo = 1000 (kilometer = 1000 meters)
Myria= 10,000 (myriameter= 10,000 meters)
497. To designate decimals of a meter, the following
Latin prefixes are used :
Deci = To (decimeter = ro meter)
Centi = -TWO (centimeter = two meter)
Milli = ToVo (millimeter = x^oo meter)
The most commonly used denominations in the
following tables are indicated by heavy-faced type.
142 RATIONAL ARITHMETIC
Linear Measure
498. The unit of Linear Measure is the meter.
TABLE
10 millimeters (mm.) = 1 centimeter (cm.)
10 centimeters = 1 decimeter (dm.)
10 decimeters = 1 meter (m.)
10 meters = 1 decameter (dm.)
10 decameters = 1 hectometer (hm.)
10 hectometers = 1 kilometer (km.)
10 kilometers = 1 myriameter (mm.)
Square Measure
499. The unit of Square Measure is the square meter.
TABLE
100 square milHmeters = 1 square centimeter (cmq.)
100 square centimeters = 1 square decimeter (dmq.)
100 square decimeters = 1 square meter (mq.)
100 square meters = 1 square decameter (dcmq.)
100 square decameters = 1 square hectometer (sq. hm.)
100 square hectometers = 1 square kilometer (sq. km.)
Land Measure
500. The unit of Land Measure is the are.
TABLE
100 centiares (ca.) = 1 are (a.) = 100 sq. m.
100 ares = 1 hectare (ha.) = 10,000 mq.
RATIONAL ARITHMETIC 143
Cubic Measure
501. The unit of volume is the cubic meter.
TABLE
100 cubic millimeters (cmm.) = 1 cubic centimeter (cmc.)
100 cubic centimeters = 1 cubic decimeter (dmc.)
100 cubic decimeters = 1 cubic meter (mc.)
Wood Measure
502. The unit of wood measure is the stere.
TABLE
10 decisteres (dst.) = 1 stere (st.) = 1 cu. m.
10 steres = 1 decastere (dast.) = 10 cu. m.
Measure of Capacity
503. The unit of capacity for either solids or liquids
is the liter, which is equal in volume to 1 cu. dm.
TABLE
10 milliliters (ml.) = 1 centiliter (cl.)
10 centiliters = 1 deciliter (dl.)
10 deciliters = 1 liter (1.)
10 liters = 1 decaliter (dl.)
10 decaliters = 1 hectoliter (hi.)
10 hectoliters = 1 kiloKter (kl.)
Measure of Weight
504. The unit of weight is the gram, which is the
weight of 1 dmc. of distilled water in a vacuum, at its
greatest density. It weighs 15.4324 gr.
144
RATIONAL ARITHMETIC
TABLE
10 milligrams (mg.) =
10 centigrams =
10 decigrams =
10 grams =
10 decagrams =
10 hectograms =
10 kilograms =
10 myriagrams =
10 quintals =
centigram (eg.)
decigram (dg.)
gram (g.)
decagram (dg.)
hectogram (hg.^
kilogram (kg.)
myriagram (mg.)
quintal (q.)
tonneau (t.)
505.
TABLES OF EQUIVALENTS
Convenient Equivalent Values
1 cu. cm. of water = 1 ml. of water, and weighs 1 gram
= 15.432 gr.
1 cu. dm. of water = 1 1. of water, and weighs 1 kg.
= 2.2046 lb.
1 cu. m. of water =1 kl. of water, and weighs 1 tonneau
= 2204.6 lb.
506.
Measures of Weight
1 grain, Troy = .0648 of a gram
1 ounce, Troy =31.104 grams
1 ounce, Avoir. =28.35 grams
1 lb. Troy = .3732 of a kilogram
1 lb. Avoir. = .4536 of a kilogram
1 ton (short) = .9072 of a tonneau or ton
RATIONAL ARITHMETIC
145
1 gram
1 gram
1 gram
1 kilogram
1 kilogram
1 tonneau
= 15.432 grains, Troy
= .03215 of an oz. Troy
= .03527 of an oz. Avoir.
= 2.679 lb. Troy
= 2.2046 lb. Avoir.
= 1.1023 tons (short)
507.
Measures of Capacity
1 dry quart =1.101 liters
1 liquid quart
1 liquid gallon
1 peck
1 bushel
= .9463 of a liter
= .3785 of a decaliter
= .881 of a decaliter
= .3524 of a hectoliter
1 liter
1 liter
1 decaliter
1 decaliter
1 hectoliter
= .908 of a dry quart
= 1.0567 liquid quarts
= 2.6417 liquid gal.
= 1.135 pecks
= 2.8377 bushels
508.
1 inch
1 foot
1 yard
1 rod
1 mile
Linear Measure
= 2.54 centimeters
= .3048 of a meter
= .9144 of a meter
= 5.029 meters
= 1.6093 kilometers
146 RATIONAL ARITHMETIC
1 centimeter = .3937 of an inch
1 decimeter = .328 of a foot
1 meter =1.0936 yards
1 dekameter =1.9884 rods
1 kilometer = .62137 of a mile
509.
Surface Measure
sq. inch =6.452 sq. centimeters
sq. foot = .0929 of a sq. meter
sq. yard = .8361 of a sq. meter
sq. rod =25.293 sq. meters
acre =40.47 ares
sq. mile =259 hectares
sq. centimeter = .155 of a sq. inch
sq. decimeter =.1076 of a sq. foot
sq. meter =1.196 sq. yards
are =3.954 sq. rods
hectare =2.471 acres
sq. kilometer = .3861 of a sq. mile
510.
Cubic Measure
1 cu. inch = 16.387 cu. centimeters
1 cu. foot =28.317 cu. decimeters
1 cu. yard = .7646 of a cu. meter
1 cord =3.624 steres
1 cu. centimeter = .061 of a cu. inch
1 cu. decimeter = .0353 of a cu. foot
1 cu. meter =1.308 cu. vards
1 stere = 275) of a cord
INDEX
(Figures refer to paragraph numbers.)
Accident insurance, 437.
Account purchase, definition, 309.
Account purchases, problems in
making, 48.
Account sales, definition, 308 ; prob-
lems in making, 48.
Accurate interest, definition, 325 ;
illustrated solutions, 327, 328, 341,
343; problems in finding, 56.
Addend, definition, 97.
Addition, compound numbers, 233;
decimals, 120, 131 ; definition,
88; fractions, 18, 201, 205; in-
tegers, 1, 92, 99; proof of, 100.
Ad valorem duty, 406.
Aliquot parts, definitions, 240; illus-
trated solution, 241 ; problems,
27, 30; table of, 241.
Amount, 250, 322, 369.
Amount of purchase, to find, illus-
trated solution, 311.
Annual interest, 389; illustrated
solution, 390.
Apothecaries' liquid measure, 477.
Apothecaries' weight, 471.
Appraiser, 410.
Arithmetic, definition, 87.
Asking price, problems in finding, 45.
Assessment, 459.
Average clause, 432.
Averaging accounts, definition, 391 ;
general principles of. 393, 395 ;
illustrated solutions, 397, 400;
problems, 68; rule, 396.
Avoirdupois weight, 472.
Balancing accounts, 109 ; problems,
9.
Bank discount, definition, 376, 379;
illustrated solution, 382, 385 ; to
find face, 64 ; to find proceeds, 63.
Base definition, 247; problems in
finding, 34; to find, illustrated
solution, 259, 261.
Bill of exchange, 444.
Billing, exercises in, 31 ; problems,
46.
Bond, 462 ; coupon, 464 ; registered,
463.
Bonded warehouse, 420.
Broker, definition, 300.
Brokerage, see Commission.
Butt, 476.
Capital stock, 454.
Cash balance, 392 ; problems in find-
ing, 68.
Casks, 476.
Charges, definition, 304.
Check, definition, 370.
Collector of customs, 414.
Commercial paper, definition, 357.
Commercial rate of exchange, 452.
Commission, definitions, 302, 310;
to find, illustrated solution, 310,
311.
Commission and brokerage, illus-
trated solutions, 311, 316.
Commission merchant, definition,
299.
Commission problems, to find com-
147
148
INDEX
mission, 47 ; to find gross cost, 47 ;
to find gross proceeds, 49; to find
net proceeds, 47 ; to find rate, 50 ;
general problems, 51.
Common divisor, definition, 153.
Common fraction, definition, 165 ;
changing to, 193.
Common multiple, definition, 148.
Common stock, 460.
Complex decimal, definition, 173.
Complex fractions, definition, 174.
Composite numbers, definition, 144.
Compound interest, definition, 386;
illustrated solution, 388 ; problems
in finding, 66 ; rule, 387.
Compound subtraction, problems in,
52.
Corporation, 453.
Cost, definition, 266, 268; problems
in finding, 38 ; to find, illustrated
solution, 276, 278.
Coupon bond, 464.
Cubic measure, 483.
Customhouse business, 410, 416.
Customs, 405.
Customs and duties, problems, 72.
Date of maturity, definition, 367.
Decimal fraction, definition, 166.
Decimal fractions, changing to, 196.
Decimals, 10; division problems, 11 ;
multiplication problems, 10; com-
plex, 173; mixed, 171.
Denominate numbers, addition, 233;
change to higher denomination,
230; change to lower denomina-
tion, 231 ; changing to simple,
232 ; definitions, 226, 228 ; divi-
sion, 236 ; multiplication, 235 ;
reduction, illustrated solution, 229 ;
reduction problems, 23, 24, 25, 26 ;
subtraction, 234.
Denominator, definition, 192.
Diamond weight, 470.
Difference, definition, 251, 105.
Discount, 288; definitions, 281, 283;
series, 289; to find, illustrated
solution, 291, 292.
Dividend, 458.
Dividend, definition, 120.
Division, compound numbers, 236;
decimals, 11, 136; decimals, illus-
trated solutions, 139, 141 ; deci-
mals, rule, 137 ; definition, 91 ;
fractions, 219 ; fractions, problems,
21; integers, definition, 119; in-
tegers, illustrated solutions, 124,
126; integers, proof, 125, 127.
Divisor, definition, 121.
Domestic bill of exchange, 446.
Domestic exchange, 449.
Draft, 444; definition, 361.
Drawee, definition, 363.
Drawer, definition, 362.
Dry measure, 478.
Duties, 405.
Endowment policy, 441.
English money, 466.
Exact days, problems in finding, 53.
Exchange, definition, 443 ; domestic,
to find face value of a draft, 78, 79
to find value of sight draft, 75
to find value of time draft, 76
foreign, 450; to find value of
draft, 81.
Excise duty, 421.
Face, 448.
Factor, definition, 114.
Factoring, 142.
Fire insurance, 430 ; problems, 73.
Foreign bill of exchange, 447.
Foreign exchange, 450.
Fraction, definition, 160.
Fractions, addition, 201 ; addition,
illustrated solution, 203, 205;
addition, problems, 18; addition.
INDEX
149
rule, 202; changing to common,
illustrated solution, 195; chang-
ing to common, rule, 190; illus-
trated solution, 198, 200 ; changing
to a decimal, rule, 197; division
problems, 21; division, rule, 221,
223; changing to higher terms,
illustrated solutions, 186 ; chang-
ing to higher terms, rule, 185 ;
changing to improper fractions,
rule, 191 ; change to lower terms,
illustrated solutions, 181, 183;
change to lower terms, rule, 180;
changing to mixed numbers, illus-
trated solution, 189; changing
to mixed numbers, rule, 188 ; com-
plex, 174 ; general problems, 22 ;
multiplication, illustrated solution,
211, 212, 214, 216, 218; multipli-
cation problems, 20; multiplica-
tion, rules, 210, 213, 215; prob-
lems in reduction, 12, 22; sub-
traction, 206 ; subtraction, illus-
trated solution, 207.
Free list, 409.
Gain, see Profit.
Gram, 495, 504.
Greatest common divisor, definition,
154 ; illustrated solution, 156, 158,
159.
Gross amount, definition, 286 ; prob-
lems in finding, 43; to find, illus-
trated solution, 294, 296.
Gross cost, definition, 268, 307.
Gross proceeds, definition, 303;
problems in finding, 49.
Gross sales, to find, illustrated solu-
tion, 312, 314.
Gross selling price, definition, 269.
Health insurance, 438.
Higher terms, changing to, 184.
Hogshead, 476.
Improper fraction, 190; definition,
169.
Income tax, 404.
Insurance, definition, 423 ; problems,
73; table, 442.
Interest, combinations of time, 344 ;
definition, 319; problems, com-
pound interest, 66; general prob-
lems, 60 ; periodic interest, 67 ;
to find accurate interest, 5Q; to
find interest, 54, 55, 56; to find
principal, 59; to find rate, 58;
to find time, 57 ; rates other than
six, 345; to find interest on $1,
rule, 355 ; illustrated solution, 356 ;
to find principal, illustrated solu-
tions, 352, 354 ; to find principal,
rules, 351, 352; to find rate, illus-
trated solutions, 350 ; to find rate,
rule, 349; to find the time, illus-
trated solutions, 347, 348 ; to find
the time, rule, 346.
Internal revenue, 417.
Intrinsic par, 451.
Invoice, 419.
Least common multiple, definition,
149; illustrated solution, 151.
Legal rate, definition, 320.
Life insurance, 436; problems, 74.
Life policy, 440.
Liquid measure, 475.
List price, definition, 285 ; to find,
illustrated solution, 294, 296.
Liter, 495, 503.
Long measure, 479.
Loss, definition, 272; to find, 274.
Lower terms, changing to, 180.
Maker, definition, 359.
Manifest, 418.
Marine insurance, 434.
Market value, 457.
150
INDEX
Marking price, to find, illustrated
solution, 298.
Maturity, date of, 367; definition,
377.
Merchants' rule, partial payment,
374; problems, 62.
Meter, 493.
Metric equivalents, 505, 509.
Metric system, 492, 510.
Minuend, definition, 103.
Miscellaneous measures, 490.
Mixed decimal, definition, 171.
Mixed numbers, changing to, 188,
189; definition, 170.
Money, English, 466; foreign, 467;
table of, 468 ; United States, 465.
Multiplicand, definition. 111.
Multiplication, compound numbers,
235 ; problems in decimals, 10 ;
problems in fractions, 20; deci-
mals, 133; decimals, illustrated
solution, 135; decimals, rule, 134;
definition, 90 ; fractions, 208 ;
integers, definition, 110; integers,
illustrated solutions, 115, 117;
proof, 116, 118.
Multiplier, definition, 112.
Mutual Insurance company, 427.
Net amount, definition, 287 ; to find,
illustrated solution, 291, 292.
Net proceeds, definition, 305 ; prob-
lems in finding, 47, 48; to find,
illustrated solution, 313.
Net selling price, definition, 270.
Notation, 92.
Note, definition, 358.
Numerator, definition, 162.
Ordinary interest, definition, 329;
rule, 332 ; sixty-day method, 331 ;
sixty-day rule, illustrated solu-
tions, 3.S2, 336.
Paper measure, 491.
Par value, 448, 455.
Partial payments, merchants' rule,
374; problems in merchants'
rule, 62, 375; problems in United
States rule, 61, 373; problems,
371, 374; liuited States rule,
372.
Payee, 364.
Pavee, definition, 360.
Percentage, definitions, 243, 254;
general problems, 36; to find, 33,
255, 258; to find base, 34, 259,
261 ; to find rate, 35, 262, 263.
Periodic interest, 389; illustrated
solution, 390.
Pipe, 476.
Policy, 428.
Poll tax, 402.
Port of delivery, 412.
Port of entry, 411.
Preferred stock, 461.
Premium, 429.
Prime cost, definition, 267, 306.
Prime factors, definition, 145 ; illus-
trated solution, 146.
Prime numbers, definition, 143.
Principal, definition, 301, 318; prob-
lems in finding, 59.
Proceeds, 380; definition, 303, 305;
problems in finding, 63, 64 ; to
find, 381.
Product, definition, 113.
Profit, definition, 271; to find,
illustrated solution, 274.
Profit and loss, general problems,
37, 38, 39, 40; illustrated solu-
tions, 274, 280; problems, 42;
to find cost, 38; to find profit or
loss, 37; to find rate of profit or
loss, 39, 279, 280.
Profits and losses, definitions, 264,
272.
Proper fraction, definition, 168.
INDEX
151
Property tax, 403.
Puncheon, 476.
Quotient, definition, 122.
Rate, definitions, 248, 319, 320;
problems in finding, 35, 39, 44, 50,
58; to find, illustrated solutions,
262, 263, 297.
Rate of discount, to find single rate
equal series, illustrated solution,
291.
Reduction, denominate numbers,
problems, 23, 24, 25, 26 ; fractions,
12, 13, 14, 15, 16, 17.
Reduction of fractions, definition,
176.
Registered bond, 463.
Remainder, definition, 105.
Selling price, definition, 269, 270;
to find, illustrated solution, 274,
275.
Short rate, 433.
Sight draft, 365.
Sixty-day method, ordmars'' interest,
illustrated solution, 333, 336;
ordinary interest, rule, 332.
Specific duty, 407.
Square measure, 481.
Stock, 454; certificate, 456; com-
mon, 460; company, 426; pre-
ferred, 461.
Stocks and bonds, 453, 464 ; general
problems, 86.
Subtraction, compound numbers,
234; decimals, 120; definition,
89; fractions, 207; fractions,
problems, 19; illustrated solu-
tion, 132; integers, 9, 102; illus-
trated solution, 106, 109; proof,
107.
Subtrahend, definition, 104.
Sum, definition, 98.
Surveyor, customs, 415.
Surveyor's long measure, 480 ; square
measure, 482.
Tariff, 408.
Taxes, 401 ; problems, 71.
Term of discount, 378.
Tierce, 476.
Time, 485 ; definition, 321 ; method?
of computing, 238, 239; problems
in finding, 52, 53.
Time discount, definition, 283.
Time draft, 365.
Trade discount, definition, 281, 284^
289; general problems, 46; illus-
trated solutions, 291, 298; tc
find asking price, 45; to find
gross amount, 43; to find net
amount, 50 ; to find rate, 44 ;
to find single discount equal to a
series, 45.
Troy weight, 469.
Tun, 476.
United States money, 465.
United States rule, partial payment,
illustrated solution, 373 ; problems
in partial payments, 61 ; partial
payments, rule, 372.
Warehouse, bonded, 420.
Weight, apothecaries', 471 ; avoir-
dupois, 472 ; comparison, 473 :
diamond, 470; troy, 469.
Weights, miscellaneous, 490.
Wood measure, 484.
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UNIVERSITY OF CALIFORNIA LIBRARY