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4. Find the equated time for paying the balance of the following account:

Dr.

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1892.

1892.

66

66

May 4 To Mdse., 20 da. $800 June
June
July 10

6

500 July

30 da. 600 Aug.

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John L. Shroy.

5. Find the equated time for paying the balance of the following account:

Dr.

30 da.

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1891.

$600 Jan. 18
500 Mar. 16
700 May 1

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Geo. W. Bickel.

1893.

1893.

66 1

66

66

66

Mar. 4 To Mdse., 30 da. $600 Apr. 5 By Draft, 30 da.
Apr.
800 May 14
15 da.
June 12
500 July 1
Aug. 18
400 Aug. 30

By Cash,
"Draft, 15 da.
"Cash,

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20 da.

6. Find the equated time for paying the balance of the following account:

Dr.

J. L. Lindsay.

Cr.

Cash,
"Draft, 20 da.

$400

600

500

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Cr.

$700

500

400

Cr.

$1000

300

500

300

NOTE 1.-Any other date than the earliest might be selected as the focal date. If we reckon from the last date at which any item becomes due, as a focal date, and the balances are on the same side of the account, the equated time is before the focal date; if they are on opposite sides, the equated time is after the focal date.

NOTE 2.—An account may be averaged by finding the interest at any per cent. on each item, and dividing the balance of interest by the interest on the balance of the account for one day; the quotient will be the number of days to be added to or subtracted from the focal date.

SECTION X.

INVOLUTION AND EVOLUTION.

INVOLUTION.

212. Involution is the process of raising a number to any required power.

213. The Power of a number is the product obtained by using the number several times as a factor.

The Power is indicated by a small figure, called an Exponent, placed at the right of and a little above the number.

Thus,

the first power of 4.

16, the second power of 4, or the square of 4. 4 X 4 X 4 = 4= 64, the third power of 4, or the cube of 4.

48

256, the fourth power of 4, etc.

4 = 4', 4X4 42

=

=

4 X 4 X 4 X 4 = 4' =

ORAL EXERCISES.

1. Square 2, 3, 4, 5, 6, 7, 8, 9, 10.
2. Cube 2, 3, 4, 5, 6, 7, 8, 9, 10.
3. Square 20, 25, 30, 40, 50, 60, 70.
4. Cube 20, 30, 40, 50, 60, 70, 80.

5. Find the value of 42, 43, 4, 52, 53, 8".
6. Square .2, .3, .4, .6, .8, .9, .03, .05.
7. Cube .2, .3, .4, .5, .6, .7, .8, .9.

8. Find the value of (3)3.

We have (3)3

27.

Hence to raise a fraction to any power, raise both numerator and denominator to the required power.

9. Find the value of 10. Find the value of 11. Find the value of 52 × 53.

We have 52 × 53

What power is

12. 82 × 83?

13. 73 X 72? 14. 94 X 95 ? 15. 35 × 36?

What power is—

25. (22)2?

26. (33)3?

27. (42)3?

28. (53)2?

=

56

59

=

=

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(3)2, (§)3, (1)2, (3)3, (3)2.
(†)3, (19)3, (§)3.

(.4),

()3,

Hence the product of any two powers of a number equals a power of the number denoted by the sum

of the exponents.

5 x 5 times 5 x 5 x 5

=3

16. (3)*X()?

17. 42 × 43 × 45?
18. 22 X 23 × 24?
19. 53 X 53 X 53?

=

24. What power is (42)3?

We have (42)3 Hence a power of a number raised to any power equals a power of the number denoted by the product

of the exponents.

29. (52)5?

30. (73)*?

31. (63)?

32. (83)+?

==

42 x 42 x 42 46.

=

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20. 610 612? 21. 12 X 123 ? 22. 18 X 18"? 23. 203 X 208?

NOTE.-By means of this principle we may abbreviate the process of involution. Th

We may raise 5 to the 6th power by squaring its cube, or to the 9th power by cubing its cube, etc.

(53) 2 (125)2 = 15625.
(53)3 = (125)3 — 1953125.

33. (123)*?

34. (15*)*?

35. (202)*?

36. (306)3?

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44. 84.

45. 92.

46. 123.

20

30. 3622.

31. 6.052.

32. 42.12.

33. 2.372.

34. 10012.

35. 10.012.

36. 1.0012.

Draw a figure and square the following numbers:

38. 32.

41. 53.

39. 45.

42. 63.

40. 48.

43. 72.

52

That is, the square of any number composed of two figures is equal to the square of the tens, plus twice the product of the tens into the units, plus the square of the units.

20 X 5

H 5

47. 245.

48. 357.

49. 582.

B

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