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use the Absolute Rule. The difference in the labour involved is not great.

In the Examples following each form is shown.

It should be noticed that form (3) is only to the same accuracy as form (2). To get results to the same accuracy as form (1) the multiplication should begin one place further to the right than the given form of the rule directs.

The importance of this method of Approximation cannot be over estimated.

Example 1. 373.8615 x 27.195 to 3 places. 373.86150

373.8615 59172

59172 74772300

7477230 26170305

2617031 373862

37386 336475

33647 18693

1869 10167 1633

10167.163

373.8615

27.195
7477.230
2617.031

37.386
33.647

1.869
10167.163

Example 2. £37569. 198. 77d. by 87 correct to pence. 37569.9802083

37569.98020

37569.9802083 78

78

87.000 30055984166

3005598416

3005598-416 2629898614

262989861

262989.861 £3268588.2780

£3268588.277 £3268588.277 5s, 6 d.

58. 64d.

58. 61d.

Example 3. Value of 1751.96875 tons at £13. 78. 8 d. per ton.
1751.96875
1751.96875

1751.96875
57348331
57348331

13.384375
175196875
17519688

17519.688
52559063
5255906

5255.906
5255906
525590

525-590
1401574
140157

140.157
70078
7008

7.008
5256
525

.52
1226
123

•123
88

9

·009
£23449.0066
£23449.006

£23449.006

DE MORGAN'S RULES.

21

EXAMPLES.

1. Multiply 876.314 by 38:72 correct to 3 places. 2. Multiply 9615•83416 by .001375 correct to 5 places. 3. Multiply 3751.407 •6158 correct to 4 places. 4. Multiply 765103 by 8.91035 correct to 3 places. 5. Multiply .008135 by 71.324 correct to 5 places. 6. Multiply 3125.006 by 1.834 correct to 3 places. 7. Multiply .0001417 by 831 correct to 6 places. 8. Multiply 97•125 by :0000456 correct to 7 places. 9. Multiply ·00005678 by .000375 correct to 9 places. 10. Multiply 61.32174 by 89371.25 correct to 2 places. 11. Find price of 876 things at £3. 78. 10£d. each. 12. Find price of 3725} things at £7. 138. 4žd. each. 13. Find price of 8000 gross at £51. 178. 9 d. per gross. 14. Find price of 715 dozen at £3. 88. 9£d. per dozen. 15. Find price of 45225 ozs. of gold at £3. 178. 101d. per oz. 16. Find price of 9124 tons at £84. 198. 1 d. per ton.

17. Find price of 3225 pipes of wine at £46. 13s. 9d. the pipe.

18. Find price of 727 hides of land at £115. 188. 11 d. per hide.

19. Find price of 1959 barrels of herrings at £2. 158. 7}d.

per barrel.

20. Find price of 5625 yds. of silk at £1. 178. 8£d. per yd. 21. Find price of 473 tons 19 cwts. 57 lbs. at £116. 8s. 4£d.

per ton.

22. Find price of 3125 qrs. 7 bushels 1 peck at £2. 118. 5{d. per qr.

23. Find price of 825000 things at £731. 198. 41d. per thousand.

24. Find price of 383000 dozens at £905. 128. 31d. per thousand dozen.

25. Find price of 751 rods of brickwork at £31. 108. 5}d.

per rod.

26. Find price of 825 sq. chains 1375 sq. links at £205.14s. 9fd. per sq. chain.

27. Find price of 1372 acres 3 roods 29 poles at £93. 78. 11d. per acre.

28. Find price of 247 miles 1312 yards of railway at £8351. 178. 6d.

per

mile. 29. Find price of 7032 cubic yds. of earth at £5. 168. 74d. per cubic yd.

30. Find price of 8743 cwts. at £121. 98. 8{d. per cwt.

4°. Division. De Morgan's Method.

To find a quotient correct to a given number of decimal places.

By inspection decide on the number of digits in the integral part of the quotient.

This added to the no. of dec. places required gives the no. of digits in the quotient.

Proceed in the ordinary way until the no. of quotient-digits to be found is one less than the no. of divisor-digits.

Then strike off a digit from the end of the divisor for each new figure in the quotient (making allowance for the figure struck off in the usual way) instead of bringing down the remaining figures of the dividend.

If there are more divisor-digits than there are to be quotient-digits, retain one more and strike out the rest.

Example 1. Divide 373.81956 by 87.243 correct to 3 places.
No. of places in

8,7,2,4,3 ) 373819:56 ( 4.284 quotient=1+3=4.

2484 No. places

739 divisor=5.

41

DIVISION-APPROXIMATION.

23

Example 2. Divide 2:7183615 by 4:1451297 correct to 8 places.

No. of places in 4,1,4,5,1,2,9,7 ) 271836150 ( •65579640 quotient=0+8=8.

23128368 No. of places in

2402719 divisor=8.

330154
39996
2690
203

39 Example 3. Divide £731. 10s. 74d. by 85.643 correct to pence.

8,5,6,413 | 731:530 | £8.541
4639

10s. 10d.
357
15

6 5. Modification of De Morgan's Method for Division,

Decide by inspection the no. of digits in the integral part of the quotient or the no. of cyphers following the dec. point.

Make the no. of figures in the divisor equal to the no. of required places in the quotient + the no. of integral digits in the quotient or – the no. of cyphers in the quotient.

Then proceed by the method at once.
Example 1. Divide 373.8651 by 8514:37 to 4 places.
No. of cyphers=1. :. Take (4 – 1)=3 figures in divisor.

8,5,1/437 ) 373.8651 ( 0439

332
77

Example 2. Divide 7431.26 by 351 to 3 places.
No. of integers = 2. :. Take (3+2)=5 figures in divisor.

3,5,1,0,0 ) 743126 ( 21:171

4112 602 251

5 1

[ocr errors]

This is a most valuable method in Division of Money.
Example 3. Divide £73965. 198. 87d. by 8901.
No. of integers =1. :: Take 4 figures in the divisor.

8,9,0,1) 73965 ( £8:309
2757

68. 2d.
87
7

Example 4. Divide £83. 9s. 77d. by 203.
No. of ciphers=0. :: Take 3 figures in the divisor.

2,0,3 ) 83.4 ( £•411

22

88. 24d.

.

In the case of Interest (which involves Division by 36500] this method may be used but the Method of Prediction following supersedes it.

6o. Method of Prediction,

To determine the number of places which must be correct in a Multiplication so as to ensure an ensuing Division being correct to a given number of places.

Decide by inspection the number of integers resulting from the Multiplication.

Then from this and the given divisor decide the number of integers which may arise in the final answer.

Add to this number, the number of places required correct.

Subtract from this the number of integers in the multiplication-result and you will thus get the no. of places which must be correct in the multiplication.

Arrange accordingly.
Example. Int. on £311. 8s. 6 d. for 174 days at 3} per cent.
Equivalent to £.311426 x 1218-73.
No. of Integers in Multiplication=3 . no. of Integers which may
No. of Integers in Divisor

occur in Quotient is 2. No. of Places required correct is 3. 2+3=5.

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