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3. Find prices of 2, 5, 7, 11, 12, 50, score at 11d., 77d., 18. 24d., 38. 44d., 58. 6d. each.

4. Find prices of 5, 7, 10, 24, 100, gross at 7{d., 6£d., 18.51d., 28. 1 d., 3, d. each.

5. Find prices of 720, 320, 600, 360, 160, articles at 1s. 1}d., 48. 24d., 1s. 7 d., 23. 04d., 58. 13d.

6. Find prices of 10, 100, 1000, 50, 80, things at 1s. 4 d., 28. 31d., 3s. 15d., 41d., 58.9}d.

7. Find prices of 837, 965, 65, 74, 68, 312, 198, things at 28. 2}d., 11 d., 3s. 4}d., 58. 136d., 38. 11įd., 18. 11}d., 1s. 101d.

8. Find price of a stone, cwt., ton, pack and sack at 1s. 5d., 9 d., 1s. 5{d., 28. 914d., 3{d. per lb.

9. The price of 12, 20, 80, 64, 1920, things is £3. 10s. Od., £4.58. 11 d., £18. 98. 6d., £75. Os. 1d., £214.78. 9d. respectively; find the price of 1.

10. Construct a Table of Prices for a dozen, score, and gross at these prices-3}d., 5{d., 6}d., 10 d., 11{d., 1s. 11d., 18. ltd., 18. 11 d., 28. 13d., 48. 11d.

2. The Method of “Nine Multiples” and “Moving the Points."

The construction of small Tables containing 1, 2, 3, 4, 5, 6, 7, 8, 9 times any unit, quantity: price or decimal expression enables us to perform the conversion of quantities from one unit to another, and the calculation of prices and exchanges without further labour than addition by simply moving the decimal points—to any required accuracy.

1°. Conversion of Units. English.
(1) Grains to Troy ounces for Bullion Operations.
1= .002083

Example. 325 oz. 10 dwts. 15 grs.
2 •00410

= 325 oz. 255 grs.
3 .00625
4 .0083

255 gr8=.416666
5 •010416

•104166 6 .0125

·010416 7 014583

-53125 8 .016

Ans. 325.53125. 9 01875

MOVING THE POINTS.

31

(2) lbs. to cwts. for Commerce generally.

1=.0089285714 Example. 79 cwts. 2 qrs. 17 lbs.
2=.0178571428

=79 cwts. 73 lbs.
3 •0267857142
4 •0357142857

79.625
5 •0446428571

•0267...
6 :0535714285

79.6517...
7 •0625
8 •0714285714
9 .0803571428

(3) sq. inches to sq. feet for Engineers etc.
1=:00694

Example. 27 sq. ft. 97 sq. miles.
2 •01388

= 27.625
3 .02083

·04861
4 •02771
5 .03472

27-67361
6 •04166
7 04861
8 •05555
9.0625

(4) cubic ins. to cub. ft. for Engineers etc.

1=.00057865740 Example 91 cub. ft. 347 cub. inches. 2 .00115731481

=91.173597 3 .00173597222

·023146
4 .00231462962

.004051
5 .00289328703
6 .00347194444

91.200794
7 .00405060184
8 •00462925925
9 .00520791666

For English-Metric conversions see the Chapter on the Metric System.

Multiples like (2) may also be used to calculate the price per lower unit, given the price of the higher. Example. Suppose price per cwt. is 188. 9d.—to find price per lb.

188. 9d.=225d. Now use (2).
:: Price per lb. = 1.79

•18
-04
2.11=21d.

2°. Calculation of Prices by Multiples of the Quantities (Inverse Method).

The use of Multiples of well-known quantities is not very advantageous with our coinage because it is not a decimal one. By constructing the multiples of these quantities decimalised to certain bases, however, and employing denomination-changes, full advantage may be taken of the principle of moving the points.

The best bases to use are 960 (f. as £) and 3840 (16ths as £). We give both sets of multiples, the actual and the decimalised, and the examples are worked in

both ways.

With the actual multiples it is easiest to use pence -afterwards changing to £, e., d.

When we come to metric quantities at foreign prices we shall see the value of the actual multiples. They would be of as great importance with us if we used mils as the division of the £.

(1) cwts. to lbs.-base 3840. 1= 1121:02916

Example 1. Price per lb. is 3 d. = 13d. 2= 224 .0583

:: Price per cwt.= 336 1.458 3 336:0875

35 1

•088 4= 448 -116 5= 560 -14583

371d. = £1.546 (10s. 11d.). 6= 672 .175 Example 2. Price per lb. is ls. 13d.=41d. 7 = 784 -20416

:. Price per cwt. =1120 5.833 8= 896 •23

336 •292 9=1008 •2625

70 •233

1526d. = £6.358 (78. 2d.). (2) sacks to lbs. (flour)-base 960. 1= 280 2916 Example 1. Price per lb. is 25d.=101f. 2= 560 583

:: Price per sack=560 2.917 3= 840 875

175 •146 4=1120 1.16 5=1400 1.4583

735d. = £3.063 (18. 3d.). 6=1680 1.75 Example 2. Price per lb. is 97} mils. 7=1960 2:0416

:. Price per sack=25200 8=2240 2-3

1960 9=2520 ! 2.625

140 £27.300

PRICES BY MULTIPLES.

33

These and such similar multiples as may be required should be carried in the pocket-book ready for use.

The bases 240 (d. as £) and 1920 (8ths as £) may also be used, but 960 is the most convenient for all kinds of prices.

3. Calculation of Prices by Multiples of the Price (Direct Method).

The Price is first converted into a decimal of a £ and then the nine multiples are taken.

With commonly-occurring prices the method is very advantageous as its speed is always great and in many cases the working is instantaneous.

It is obvious also that the method can equally well apply to dozens, scores, cwts., gross, or any special quantities in which goods are quoted.

(1) Price = 3 d.
1= £.01510416

Examples. 1. 5 doz. = £.906 =188. 14d.
2 •0302083
3 •0453125

2. 73 arts. = 1.057
4 ·060416

.045 5 07552083

1.102 - £1. 28.01d. 6 090625

3. 150 things=} (300) 7 •10572916

= £2•265= £2. 58. 3 d. •12083 9 •1359375

4. 27 thing8=(3 x 9)

= £.408=8s. 2d. (2) Bar Silver per oz. = 4076d. 1=£ •168489583 Examples. 1. 4000 oz. =

= £673.958 2 •33697916

= £673. 198. 2d. 3 •50546875 4 -6739583

2. 3125 oz. = 505.469 5 •842447916

16.849
6

3.370
1.0109375
7 1179427083

.842
1.347916

£1200.480 9 1.51640625

=£1200. 98. 74d.

8

8

With practice all two-figure quantities can be written down at sight after mental addition.

The method can be applied to Sterling Exchanges with great effect.

(3) Rupee = ls. 53 d.

1= £.073046875 Example. 8560 rupees=584.375
2 •146093750

36.523
3 .219140625

4.383 4 •292187500

£625•281
5 •365234375

58. 74d.
6 •438281250
7 -511328125
8 •584375000

9 -657421875 Such exchanges fluctuate as a rule within very small limits and therefore the Tables required are very few. Hence those whose dealings with the East are frequent can readily make tables of the kind as they are required-keeping them for future use.

EXAMPLES (to be done by Multiples). 1. Decimalise in ounces (1) 612 oz. 19 dwts. 23 grs.

(2) 481 oz. 13 dwts. 20 grs.

(3) 1362 oz. 9 dwts. 16 grs. 2. Decimalise in cwts. (1) 2 tons 13 cwts. 3 qrs. 17 lbs.

(2) 83 cwts. 1 qr. 19 lbs.

(3) 980 cwts. 2 qrs. 16 lbs. 3. Decimalise (1) in

sq.
ft. 823

sq.

ft. 119 sq. in. (2) in sq. yds. 28 sq. yds. 7 sq. ft. 56 sq. in. (3) in cub. ft. 141 c. ft. 953 c. inches. (4) in sq. ft. 71 sq. ft. 91 sq. in.

(5) in cub. ft. 84 c. ft. 1152 c. in. 4. Construct the Tables for Converting

(1) perches to acres. Ex. 19 ac. 3 ro. 29 poles.
(2) sq. in. to sq. yds. 27 sq. yds. 7 sq. ft. 110 sq. in.
(3) lbs. to tons.

15 tons 13 cwts. 59 lbs.
(4) gallons to quarters (Imp.). 18 qrs. 7 bushels 5 galls.
(5) links to yards.

81.96 chains.
5. Construct the Tables for Reducing
(1) tons to lbs.

Ex. 31d., 21 d., 1s. 3 d.
(2) acres to sq. yds.

15. 6}d., 28. 11d., 9/d.
(3) puncheons of prunes to lbs. 27d., 3 d., 53d.
(4) barrels of soap to lbs. 1 d., 75d., 4d.

(5) Lisbon pipes to gallons. 8jd., 13. 11d., 28. 9 d. Find the prices per higher unit at the given prices per lower unit in each case.

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