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tion the divisors will be smaller. Aliquots above 16 which are not multiples of 10 are inconvenient.

Decimals should always be used as their use avoids all disagreeable fractions of a penny in the addition, and makes multiples of 10 convenient aliquots.

Carry the lines to 4 places, making allowance for the 5th—the addition will then be correct to 3 places.

Prices may be thus aliquotised without regard to the quantities. Hence if any prices are of frequent occurrence their aliquot-formulae may be made and kept for use.

Example 1. £3.78. 11}d.= £3—426 (10).
Example 2. 48.7}d.= £0-584.
Example 3. £1. 68.52d.= £1—453422.
Example 4. 211d.= £0-(120) 442.
Examples of Prices of frequent occurrence.
(1) 3 d.= £0-(80)64.

(2) 575d. = £0—(60) 4422. (3) 18. 316d.= £0—(20)4 (12) 4. (4) £3. 178. 103d.= £3—222(10)2. Example. 824 things at 18. 318d.

824 41.2 10:3

.8583

2146
£52.5729= £52. 118.51d.

EXAMPLES.

1. Construct Aliquot formulae for these prices. (1) £1. 188. 5 d.

(2) 19s. 4 d. £3. 16s, 5 d.

(4) £17. 158. 4d. 178. 61d.

(6) 278. 9 d.
188. 4 d.

56s. 7d.
(9)
398. 8 d.

(10) 658. 4£d.
2. Construct Aliquot formulae for these prices.
(1) 3,740.

(2) 41 d. (3) 6 d.
(4) 18. 3 d. (5) 27d. (6) 91d.

DIRECT ALIQUOT'S.

41

3. Calculate

(1) Prices of 8315 things at 58. 7]d., 88. 101d., £1.78. 10d. (2) Prices of 9640 things at 3 d., 43.d., 9:9d., £2. 88. 4.d. (3) Prices of 856 things at 7:39.d., 97.d., £2.78.6}d.

Prices of 932 things at 58. 64d., 713d., 9., d., 11 d.

(5) Prices of 1000 things at 8s. 11?d., 18.3. d., 10. d., 111d.

(6) Prices of 1250 things at 108. 1{d., 18. 5,8 d., 28. 1 d., 38. 218 d.

(7) Prices of 3000 things at 47d., 1.:,d., 31.d., 1}}d., 18. 2 d.

(8) Prices of 784 things at 38. 8d., 3s. 2 d., 10 d., 18. 13d.

(9) Prices of 940 things at 5.1.d., 7;8d., 573d., 18. 3ļd., 58. 7 d.

(10) Prices of 2000 things at 73d., 138d., 913d., 88. 4 d., 98. 6d.

This Method of Direct Aliquotation may be applied

Weights and Measures with Decimalisation of the Price.

Example. 17 cwts. 3 qrs. 16 lbs. at 17s. 71d. per cwt.
Quantity=1741 cwts. £ •88125
=17--2227

17
6.16876
8.81250
44063
.22032
•11
01574

£15•76810

15s. 44d. It is in general advantageous to Decimalise the Price.

EXAMPLES. 1. 37 cwts. 2 qrs. 19 lbs. at 19s. 5d. per cwt. 2. 3 tons 19 cwts. 8 lbs. at £3. 88. 7d. per ton. 3. 841 yds. 1 ft. 10 inches at 228. 8d. per yd. 4. 95 acres 2 roods 37 poles at £5. 12s. 6d. per acre. 5. Construct the Aliquots for 1 cwt. from 40 to 50 lbs.

2o. Inverse Aliquotation,

Aliquotation of the price is advantageous when the number of articles constantly varies, the prices being nearly the same but when the prices vary largely and the number of articles is well known Inverse Aliquotation is to be preferred.

The Rule may be stated in two parts.

(1) Reckon pence, farthings, eighths, or sixteenths as shillings and divide the given number of articles by 12, 48, 96, 192 respectively—aliquotising the remainder.

(2) Reckon shillings, pence, farthings, eighths, or sixteenths as pounds and divide the given number of articles by 20, 240, 960, 1920, 3840 respectively-aliquotising the remainder. Example 1. Price of 840 articles at 18.58d.=174d. $18=31.

£ 8. d. 17 15 0

3

53 5 0

8 17 6 £62 2 6

Example 2. Price of 713 articles at 5*.d.= id.
731=31;}=3—2348. £4.15

8.30
2.075
.6916
1729

·0216
£15.4111

88. 21d.

Example 3. Price of 1356 articles at 28.57.d.=1174 farthings.
He=1389=1-3582. £ 8. d.

117 15 0
39 5 0
7 17 0
19 77

9 92 £166 6 51

IN VERSE ALIQUOTS.

43

The most useful bases to take are 192, 240, 960.

By using Decimals we avoid difficult fractions of a penny.

If the remainder is small (under 12) it is shorter to obtain the price direct. This is always done in the case of dozens and score by commercial men.

If the remainder is large we may adopt the Principle of Interchanges (that is reckon the price something else for the remainder) in preference to Aliquotation.

The Aliquot-formulae for any quantities may be calculated for the various bases without regard to the Prices. Hence if certain quantities are of frequent occurrence suitable Inverse-Aliquotations may be made and kept for use. e.g. 112 to base 240=0—351; hence if price per lb. is 27d.

£ 8. d. price per cwt.= 2 15 0

18 4
3 8

£1 5 8

EXAMPLES
1. Price of 854 things at 18. 9 d., 28. 7.d., 3s. 1d.
2. Price of 1260 things at 58. 7d., 93. 6{d., 18.7d.
3. Price of 3000 things at 4s. 11d., 58. 9£d., 38. 2}d.
4. Price of 972 things at 38. 114d., 28. 94d., 43. 3 d.
5. Price of 1008 things at 78. 94d., 51/d., 33. 14d.
6. Price of 1892 things at 3s. 5 d., 48. 298d., 28. 114d.
7. Price of 736 things at 28. 318d., 78. 33d., 43. 5iod.
8. Price of 4050 things at 3s. 3yd., 38. 418d., 1s. 11%d.
9. Price of 1500 things at 28. 9£d., 58. 1}d., 18. 54d.
10. Price of 1648 things at 3s. 11d., 38. 25d., 18. 10fd.
11. Price of 1250 things at 1s. 7 d., 28. 3 d., 38. 1{d.
12. Price of 974 things at 28. 3 d., 48. 7 d., 18. 9}d.
13. Price of 2546 things at 3s. 5 d., 28. 4£d., 18. 5}3d.
14. Price of 3120 things at 18.73d., 18. 9{d., 18. 11 d.
15. Price of 760 things at 2s. 11}d., 38. 15d., 43. 2£d.

5. Interchanges.

In preference to Aliquotation the Method of Interchanges may be employed, i.e. we may split the number into convenient parts and consider the price as changed from one denomination to another to suit the parts chosen.

An example will show the principles involved. Price of 844 arts. at 2s. 3d. 28. 3d.=274d. = 109 f. 844=720+124=720+120+4=3 x 240 + 10 x 12+4. :: price=3 x £27.58. Od. (d. as £)+10 x £1. 7s. 3d. (d. as s.) +98.1d.

£ 8. d. 81 15 0 13 12 6

9 1 £95 16 7

The Divisions practically useful are these :

(1) Into 960ths—reckon f. as £.
(2) Into 240ths—reckon d. as £.
(3) Into 48ths—reckon f. as s.
(4) Into 20ths-reckon s. as £.

(5) Into 12ths—reckon d. as s. However unless the price is an easy fraction of a shilling (4) is not to be employed.

For very large quantities we may divide into 1920ths, 3840ths, reckoning eighths as £ and sixteenths

as £.

The Best Rule to observe is :-
Write price as shillings, pence, farthings.

Divide number of articles by 960, remainder by 240, remainder by 48, remainder by 12.

Multiply appropriately.

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