CONTRACTIONS IN DIVISION.

To divide by 25.-Multiply the dividend by 4, and point off two figures to the right-hand of the product as so many hundredths, or take of the two right-hand figures of the product as so many twenty-fives.

To divide by 125.-Multiply the dividend by 8, and point off 3 figures to the right of the product as so many thousands, or take of those 3 figures as so many 125ths.

11. Divide 7643275 by 125.

12. Divide 4976372 by 125. 13. Divide 9874625 by 125. 14. Divide 9846375 by 125. 15. Divide 173964875 by 125.

Ans. 61146 Ans. 39810122. Ans. 78997.

Ans. 78771. Ans. 1391719.

NOTE TO THE LEARNER.-Your attention will now be directed to

an important consideration. We propose to teach you to perform long division with less than half the number of figures required by the ordinary process.

Observe, that the first example requires 25 figures in the work, while the short method requires but 8. The second example requires 37 figures in the work by the common method, whilst the short method requires but 13.

The only difference between these two methods, is, that in the one case, the product of each quotient figure is set down, and then subtracted, whereas in the other, we subtract each figure of the product as it is formed. A little practice will enable any one to perform division by the short method, as

readily as in the ordinary way, and consequently in less than half the time, as he will need to make less than half the number of figures in the work.

TO DIVIDE BY THE SHORT METHOD.

Subtract each product figure, as it is formed, (that is the right hand figure of the product,) and when it is greater than the figure from which you subtract, carry one more to the next product figure than you would otherwise carry.

18. Divide 15341 by 29.

Here we say, 5 times 9 are 45—

5 from 13 leaves 8, then 5 times 2 are 10, and 5 to carry make 15, being 1 more than the multiplication requires, because the 5 in the product

Operation.
29)15341(529
8

26

was greater than the 3, from which you subtract, &c. 19. Divide 6283459 by 29.

do.

Ans. 74-2 figs. in the work.
Ans. 432-3 figs. in
Ans. 948-4 figs. in

do.

23. Divide 2738 by 37.
24. Divide 15552 by 36.
25. Divide 74892 by 79.
26. Divide 112896 by 84.
27. Divide 245551 by 89.
28. Divide 668039 by 97.
29. Divide 2248704 by 192.
30. Divide 2755116 by 174.
31. Divide 2686211248 by 296.
32. Divide 175152 by 246.
33. Divide 43859556 by 234.
34. Divide 11133936 by 396.
35. Divide 15351975 by 465.
36. Divide 121932631112635269 by 123456789.

Ans. 1344-6 figs. in do.
Ans. 2759-6 figs. in do.
Ans. 6887-6 figs. in do.
Ans. 11712-9 figs. do.

Ans. 15834-10 figs.
Ans. 9075038-10 figs.
Ans. 712—4 figs.

Ans. 187434-13 figs.
Ans. 28116-10 figs.
Ans. 33015-7 figs.

Ans. 987654321-57 figs. in the work.

37. Divide 3973580210754 by 32186.

Ans. 123456789-39 figs. in the work.

PROPERTIES OF NUMBERS.

The following properties of figures and numbers will render important assistance, by way of abridging the work in calculations, and the learner will be abundantly repaid for all the labour which it may be necessary to bestow upon them, in order to profit by their friendly services.

When one number will divide another without a remainder, it is said to measure that other number.

An odd number is one whose right-hand or unit figure is either 1, 3, 5, 7, or 9.

An even number is one whose right-hand or unit figure is either 2, 4, 6, 8, or 0.

1-Has this peculiar property, that as a multiplier or divisor it has no power.

2-Will measure all even numbers.

3-Will measure any number, if the sum of the figures added horizontally be a multiple of 3, as 42612, or 741, the

figures of the first of these numbers added horizontally make 15, and of the others, 12 each a multiple of 3.

4-Will measure any number, if it will measure the two right-hand figures, as 371432, here the two right-hand figures being 32, a multiple of 4, the number 371432 can be measured by 4.

5-Will measure any number of which the right-hand figure is 5 or 0.

6-Will measure any even number which can be measured by 3.

7-Will measure any number of 2 or 3 figures, if the right-hand figure beor of the left-hand figure or figures, as 84, 126, 168, 91, 364, 455, &c.

It will measure a number of 3 or 4 figures, if the two righthand figures be either 5 times or of the left; when they are , 43 will also measure the number; again if two ciphers be inclosed by equal figures, 7, 11, and 13 will also measure a number of 5 figures, if a cipher be inclosed by equal numbers. Thus, 7, 11, and 13 will measure 8008, 3003, 43043, 79079, &c.

Again, 7 will measure a number of 5 figures, if the two right-hand figures be of the three left-hand ones, 43 will also measure in this case, as 16254, 28595, 29498, &c.

In a number of 5 or 6 figures, 7 will measure, if the difference between the three right-hand figures and the two or three left-hand ones be 0, 7, or a multiple of 7, as 65, 121, 79,863, 38,633, 458,458, 384,447, 594,685, 829,955, &c.

8-Will measure any number, if it will measure the two right-hand figures when the hundreds figure is even, or the three right-hand figures when the hundreds figure is odd; thus, 8 will measure 3197824, 273672, 5397656, because it will measure the two right-hand figures, the hundreds figure being even. It will measure 4573536, 3592784, &c., because it will measure the three right-hand figures, the hundreds figure being odd.

9-Will measure any number, if the sum of the figures added horizontally can be measured by 9, as 4575, the sum of whose figures is 18, 46782 the sum of whose figures is 27, 14684787, the sum of the figures being 45, 4375683, &c.

11-Will measure a number of 3 figures, if the middle