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 2. £37569. 19s. 74d. by 87 correct to pence.

Example 3. Value of 1751.96875 tons at £13. 7s. 81d. per ton.

DE MORGAN'S RULES.

21

176

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 by 6158 correct to 4 places. 4. Multiply 76510.3 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. 7s. 10‡d. each. 12. Find price of 3725 things at £7. 13s. 4 d. each. 13. Find price of 8000 gross at £51. 178. 9ąd. per gross. 14. Find price of 715 dozen at £3. 8s. 94d. per dozen. 15. Find price of 45225 ozs. of gold at £3. 178. 10 d. per oz. 16. Find price of 9124 tons at £84. 19s. 14d. 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. 18s. 111d. per hide.

19. Find price of 1959 barrels of herrings at £2. 15s. 71⁄2d. per barrel.

20. Find price of 5625 yds. of silk at £1. 17s. 8d. per yd. 21. Find price of 473 tons 19 cwts. 57 lbs. at £116. 8s. 44d. per ton.

22. Find price of 3125 qrs. 7 bushels 1 peck at £2. 11s. 5ąd. per qr.

23. Find price of 825000 things at £731. 19s. 4 d. 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. 10s. 5d. per rod.

26. Find price of 825 sq. chains 1375 sq. links at £205.14s. 94d. per sq. chain.

27. Find price of 1372 acres 3 roods 29 poles at £93. 7s. 11d. per acre.

28. Find price of 247 miles 1312 yards of railway at £8351. 17s. 6d. per mile.

29. Find price of 7032 cubic yds. of earth at £5. 16s. 73d. per cubic yd.

30. Find price of 8743 cwts. at £121. 9s. 8d. 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. 8,7,2,4,3) 373819.56 ( 4.284

No. of places in

quotient=1+3=4. No. of places in

divisor = 5.

2484

739

41

6

DIVISION-APPROXIMATION.

Example 2. Divide 2.7183615 by 4.1451297 correct to 8 places. 4,1,4,5,1,2,9,7) 271836150 (·65579640

No. of places in

quotient 0+8=8.

No. of places in

23128368

divisor = 8.

2402719

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

23

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

This is a most valuable method in Division of Money.

Example 3. Divide £73965. 19s. 81d. by 8901.

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

6°. 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. 6d. for 174 days at 3 per cent.
Equivalent to £311426 × 1218÷73.

No. of Integers in Multiplication=3 .. no. of Integers which may
No. of Integers in Divisor

=2

No. of Places required correct is 3.

occur in Quotient is 2. 2+3=5.