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INVERSION TABLES.

35

6. From the Tables in 4 calculate the price

(1) per perch-given price per acre £4. 10s.Od., £13. 158.6d., £9. 78. 6d.

(2) per lb.---given price per ton £3. 178. 6d., £1. 98. 4d., £2. 8s. 10d.

(3) per gallon--given price per Imp. qr. 338.4d., 358, 6d., 41s. 10d. 7. Construct Tables for the following prices

(1) 316d. (2) 4 d. (3) 15.7. (4) 27d. (5) 511d. (6) 18. 23d. (7) 28.3%

.. (8) 18. 9d. (9) 71 d. (10) 211.

And calculate the prices of 320, 94, 87, 201, 416, 512, 1000, 745, 621, 83, 192 articles at each price. 8. Construct Tables for these Exchanges and Silver Prices

(1) 18.3}d. (2) 18.11%. (3) 28.4. (4) 38.75d. (5) 3918d. (6) 411d. (7) 403d. (8) 58. Opod. (9) 48. 101 d. (10) 2235d.

And find value of 841, 325, 612, 7518, 8619, 25000, 3000 foreign coinage-units quoted at above prices.

9. Construct Tables for these prices and find value of quantities given.

(1) 183. 9d., 278. 6fd., 338. 4d., 578. 8d. per doz.--27, 45, 300, 80, 76 dozens.

(2) £3. 158. 7d., £5. 8s. 9d., £20. 14s. 6d. per gross-30, 52,

65 gross.

3. The Method of Inversion,

A most convenient way of finding the price of common quantities consists in constructing a small table for each in the following manner.

Consider the quantity in succession shillings, pence, farthings, eighths and sixteenths of a penny and find the corresponding sums of money.

Keep this little table for use.

The following quantities are of frequent occurrence and their Tables are given as examples, but any other quantities may be dealt with similarly as the requirements of business render advisable. All such should be carried in the pocket ready for use.

8.

e.

(1) The cental (100 lbs.).

£ 8. d. Example 1. Price of 1 lb. = 1s. 14d.
5 0 0

£ 8. d.
.
8 4

.. Price per cental= 5 0 0
f.
2 1

8 4
1 01

6 3
st.
63

£5 14 7
Example 2. Price of 1 lb.=7Pd.

£ 8. d. :: Price per cental= 2 18 4

2 77

£3 0 111
(2) The cwt. (112 lbs.).

£ 8. d. Example. Price of 1 lb. 18. 3 d.
5 12 0

d.
d.
9 4

:. Price per cwt. 5 12 0
f.
2 4

1 8 0
1 2

5 10
st.
7

£7 5 10
(3) The gross (144).

d. Example. Price of 1 art. 8 d.
7 4 0

d.
d
12 0

:. Price per gross=4 16 0
3 0

6 9
1 6
st.
9

£5 2 9

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(4) The Lisbon Pipe (115 gallons).

£ 8. d. Example. Price per gallon 18.9fd.
5 15 0

£ 8. d.
d.
9 7

:. Price per Pipe = 5 15 0
f.
2 49

4 6 3
1 23
st
736

£10 6 0?

e.

4 91

The Principle of Inversion on which the above Tables are based is one of great value in all price calculations.

It may be stated thus :—Consider the number of articles whose price is wanted as shillings, pence, farthings, eighths, or sixteenths (one only) and the

SINGLE INVERSIONS.

37

price in the denomination taken as the number of articles—reduce the former to £. s. d. f. and multiply by the latter.

Which denomination is taken depends on the price. Examples. 1. 37 arts. at 61d.=64 arts. at 3s. 1d.

=188. 6d. +18.64d. = 208.04d. 2. 23 cwts. at 58. 6d. per cwt.=5} cwts. at £1.38.0d.

= £5.158. Od. + 11s. 6d.= £6. 6s. 6d. 3. 112 lbs. at 31 d. per lb. =51 lbs. at 7d.

= 355d.= £1.98. 7d.

EXAMPLES (to be done by making the necessary

Inversion Tables).

73d. per

1. Find price of a cental at 18. 2{d., 3s. 1fd., 9}d., 28. 4}d.,

lb. 2. Find price of a cwt. at 3s. 4d., 11{d., 18.788d., 58. Ojd., 10d. per lb.

3. Find price of 50 dozen at 3 d., 2 d., 18. 1}d., 104d., 11}d. each.

4. Find price of 812 dozen at 28. ld., 1s. 6fd., 1s. 3}d., 10d., 53. ld. each.

5. Find price of a Lisbon Pipe at 28. 7d., 3s. 1}d., 58. 4£d., 98. 7d., 6s. 3fd. per gallon.

6. Find price of 1000 yds. at 38. 17d., 18. 111d., 98. 101d., 18. 5 d., 28. 744. per yd.

7. Find price of 950 things at ls. 13d., 847 things at 28. 5{d., 912 things at 8/d.

8. Find price of 1000 things at 28. 31d., 642 things at 18. 41 d., 900 things at 38. 54d.

9. Find price of 512 gallons at 138.71d., 108. 9d., 88.43d., 58. 6 d., 9s. 8d. per gallon.

10. Find price of 752 grains at 38. 14d., 88. 6£d., 43. 11d., 158.3d., 18. 101d. per grain.

11. Find price of a firkin of butter at 18. 2d., 18. 3 d., 1s. 5d., 11 d., 18. 6 d. per lb.

12. Find price of a puncheon of prunes at 24d., 3}d., 74d., 6 d., 41d. per lb.

13. Find price of a barrel of anchovies at 5 d., 18. 1 d., 10}d., 1s. 77d., 12fd. per lb.

4. Aliquotation.
An aliquot part is a fraction whose numerator is 1.

The principle has been long used in Practice of dividing the Price into successive portions which are aliquot parts of one of the preceding portions and thus the Price is calculated by a series of short divisions.

The extension of this principle so as to get a series of fractions each an aliquot part of the preceding one as an equivalent for a given fraction produces great simplifications in various calculations.

It is of special use in its application to Prices and Percentages.

We give here the Method of Aliquotation and the Notation adopted in this book.

4+2+1 1. To aliquotise

=5+3(1)+1{2(3)}

=0—222 (Notation).

8+4+1 2. To aliquotise 11t=1+ =1+1+1(1) ++{1()}

16

=1-224 (Notation).

32 + 16+1 3. To aliquotise 314=3+

=3+1+1 (1)+1{}(x)}

=3-22 (16) (Notation).

10+5 4. To aliquotise 15 p.c.=1 = = 16+1(1)=0-(10)2.

100 The Method employed, then, is to split up the numerator in such a way that each part is an aliquot of the preceding.

The Notation adopted is the following :-the denominators of the successive aliquots are written down in order without any marks of division-numbers above 9 being enclosed in brackets and multiples being placed before the dash.

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ALIQUOTATION.

39

Occasionally the aliquotation is shorter if the aliquots are not all successive but two or more aliquots of a previous one. This is denoted by a plus between them with a bar above.

5+4 Example. Do = +1=0-4+ 5.

20

It
may

also be advisable to attach successive further aliquots to one of two or more under a bar—this is expressed thus.

480+ 320+64 +8+4 Example. 87=

= 0-2 +3582. 960

When 1 occurs among the aliquots it denotes that the previous line of division is to be repeated-a device of frequent occurrence in practice.

There are often two or three ways of aliquotising the same fraction.

EXAMPLES

1. Aliquotise the fractions 1 1, 3,31, 13334, 113, .

2. Aliquotise 89, 77, 292, 512, 786 to the bases 144, 240, 384, 960, 192.

3. Aliquotise 13, 23, 32, 40 p.c.
4. Aliquotise 74, 81, 59, 61 per mille.

1°. Direct Aliquotation (Practice).

The ordinary Method of Practice is to aliquotise parts of the price in succession till it is exhausted. This is as a rule the most convenient and shortest method. Sometimes however difficulties arise as to the proper parts to take at the end. These may be avoided by the repetition of lines and other devices.

There may be more aliquots in one way of aliquotising a price than in another but as a compensa

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